Applied Science, Faculty of
Mechanical Engineering, Department of
DSpace
UBCV
Cherom Kheirabadi, Ali
Doctor of Philosophy - PhD
University of British Columbia
Due to a phenomenon termed the wake effect, wind turbines that are placed in close proximity within wind farms interact aerodynamically. In short, each turbine generates a wake within which wind speeds are reduced, and these wakes overlap with the rotors of machines located downstream. This interaction diminishes power production in wind farms by up to 60%. Using a process referred to as wind farm control, individual wind turbines may be operated in a manner that increases power production from the collective. This thesis investigates the potential of a wind farm control strategy named yaw and induction-based turbine repositioning (YITuR) that is specifically compatible with floating offshore wind farms. Since floating platforms are anchored to the seabed using slack mooring line cables, each turbine may be repositioned in real-time using the aerodynamic forces exerted on its rotor. By relocating floating platforms accordingly, the overlap area between the wakes generated by upstream turbines and the rotors of downstream machines may be reduced; leading to an increase in wind farm efficiency. The potential of YITuR is assessed through several steps. First, a steady-state model of floating offshore wind farms is constructed and stationary optimization studies are carried out to determine the potential of YITuR under idealized steady wind conditions. Major findings from this study are that wind farm efficiency may increase by more than 40% using YITuR over traditional wind farm operation; however, these benefits are strongly influenced by mooring system designs. Second, a dynamic floating wind farm model is developed to evaluate the performance of real-time control systems. Third, due to the non-convexity of the YITuR control problem, novel distributed economic model predictive control (DEMPC) theory is developed to guarantee power maximization. Existing DEMPC algorithms do not offer such a guarantee in the presence of non-convex objective functions. Finally, the DEMPC algorithm is evaluated using the dynamic simulation tool. Neural networks are used to estimate the dynamics of floating platforms in order to expedite decision-making in DEMPC. Simulation results indicate gains of 20% in energy production when YITuR replaces traditional wind farm operation.
Theoretical Foundations for Optimal Control of FloatingOffshore Wind FarmsbyAli Cherom KheirabadiB.Eng., Dalhousie University, 2014M.A.Sc., Dalhousie University, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)December 2020© Ali Cherom Kheirabadi, 2020The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Theoretical Foundations for Optimal Control of Floating Offshore WindFarmssubmitted by Ali Cherom Kheirabadi in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Mechanical Engineering.Examining Committee:Ryozo Nagamune, Mechanical EngineeringSupervisorRajeev Jaiman, Mechanical EngineeringSupervisory Committee MemberGuy Dumont, Electrical EngineeringUniversity ExaminerMartin Ordonez, Electrical EngineeringUniversity ExaminerJan-Willem van Wingerden, Mechanical, Maritime, and Materials Engineering,Delft University of Technology, The NetherlandsExternal ExaminerAdditional Supervisory Committee Members:Anasavarapu Srikantha Phani, Mechanical EngineeringSupervisory Committee MemberBhushan Gopaluni, Chemical EngineeringSupervisory Committee MemberiiAbstractDue to a phenomenon termed the wake effect, wind turbines that are placed inclose proximity within wind farms interact aerodynamically. In short, each turbinegenerates a wake within which wind speeds are reduced, and these wakes overlapwith the rotors of machines located downstream. This interaction diminishes powerproduction in wind farms by up to 60 %. Using a process referred to as wind farmcontrol, individual wind turbines may be operated in a manner that increases powerproduction from the collective.This thesis investigates the potential of a wind farm control strategy namedyaw and induction-based turbine repositioning (YITuR) that is specifically com-patible with floating offshore wind farms. Since floating platforms are anchoredto the seabed using slack mooring line cables, each turbine may be repositioned inreal-time using the aerodynamic forces exerted on its rotor. By relocating floatingplatforms accordingly, the overlap area between the wakes generated by upstreamturbines and the rotors of downstream machines may be reduced; leading to anincrease in wind farm efficiency.The potential of YITuR is assessed through several steps. First, a steady-statemodel of floating offshore wind farms is constructed and stationary optimizationstudies are carried out to determine the potential of YITuR under idealized steadywind conditions. Major findings from this study are that wind farm efficiency mayincrease by more than 40 % using YITuR over traditional wind farm operation;however, these benefits are strongly influenced by mooring system designs. Sec-ond, a dynamic floating wind farm model is developed to evaluate the performanceof real-time control systems. Third, due to the non-convexity of the YITuR controlproblem, novel distributed economic model predictive control (DEMPC) theory isiiideveloped to guarantee power maximization. Existing DEMPC algorithms do notoffer such a guarantee in the presence of non-convex objective functions. Finally,the DEMPC algorithm is evaluated using the dynamic simulation tool. Neural net-works are used to estimate the dynamics of floating platforms in order to expeditedecision-making in DEMPC. Simulation results indicate gains of 20 % in energyproduction when YITuR replaces traditional wind farm operation.ivLay SummaryWind turbines that are clustered in close proximity within wind farms interact aero-dynamically. This interaction decreases the efficiency of wind farms and results in20 to 30 % loss of annual revenue. One approach to recuperating these losses iswind farm control. This strategy involves operating individual turbines in a man-ner that mitigates the negative effects of aerodynamic interaction; thus increasingwind farm efficiency.This thesis investigates the potential of one particular wind farm control tech-nique that is compatible with floating offshore wind farms and is referred to asturbine repositioning. In this method, each floating wind turbine uses the aerody-namic force of the wind to reposition itself along the ocean surface with the goal ofreducing the scale of aerodynamic interaction and increasing wind farm efficiency.This research therefore delivers economic and environmental benefits by raisingthe competitiveness of floating offshore wind farms relative to fossil fuel-basedenergy generation methods.vPrefaceThe general topic of research investigated in this thesis was recommended by theauthor’s supervisor, Ryozo Nagamune. Beyond this task, all efforts related to thedesign and implementation of the conducted research are the sole work of the au-thor, Ali Cherom Kheirabadi. Further, the preparation of this thesis is also the solework of the author with guidance and review offered by the research supervisor.The main body chapters of this thesis comprise a collection of works that havebeen or are intended for publication in peer-reviewed journals. The following listcontains information on each article including the corresponding chapter:• The content from Chapter 2 has been published and is available in:– A. C. Kheirabadi, and R. Nagamune. A quantitative review of windfarm control with the objective of wind farm power maximization.Journal of Wind Engineering and Industrial Aerodynamics, 192(May),45–73, 2019.• The content from Chapter 3 has been published and is available in:– A. C. Kheirabadi, and R. Nagamune. Real-time relocation of floatingoffshore wind turbine platforms for wind farm efficiency maximiza-tion: An assessment of feasibility and steady-state potential. OceanEngineering, 208(May), 2020.• The content from Chapter 4 has been submitted for review with the followingtitle and authorship information:vi– A. C. Kheirabadi, and R. Nagamune. A low-fidelity dynamic windfarm model for simulating time-varying wind conditions and floatingplatform motion.• The content from Chapter 5 has been submitted for review with the followingtitle and authorship information:– A. C. Kheirabadi, and R. Nagamune. Distributed economic model pre-dictive control – Addressing non-convexity using social hierarchies.• The content from Chapter 6 will be submitted for review with the followingtitle and authorship information:– A. C. Kheirabadi, and R. Nagamune. Social hierarchy-based distributedeconomic model predictive control of floating offshore wind farms.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The wake effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Yaw and induction-based turbine repositioning (YITuR) . . . . . . 21.3 Research question and thesis outline . . . . . . . . . . . . . . . . 32 A Quantitative Review of Wind Farm Control with the Objective ofWind Farm Power Maximization . . . . . . . . . . . . . . . . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.1 Status of wind energy . . . . . . . . . . . . . . . . . . . . 82.1.2 Increasing wind farm efficiency . . . . . . . . . . . . . . 10viii2.1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Wind farm control . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 Control concepts . . . . . . . . . . . . . . . . . . . . . . 142.2.2 Controller properties . . . . . . . . . . . . . . . . . . . . 182.3 Wind farm modeling . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Low-fidelity wake models . . . . . . . . . . . . . . . . . 252.3.2 High-fidelity wake models . . . . . . . . . . . . . . . . . 292.3.3 Medium-fidelity wake models . . . . . . . . . . . . . . . 302.4 Review of wind farm control literature . . . . . . . . . . . . . . . 322.4.1 Steady-state potential of power de-rating . . . . . . . . . 322.4.2 Controllers developed for power de-rating . . . . . . . . . 372.4.3 Steady-state potential of yaw-based wake redirection . . . 442.4.4 Controllers developed for yaw-based wake redirection . . 482.4.5 Steady-state potential of turbine repositioning . . . . . . . 522.5 Quantitative overview of published results . . . . . . . . . . . . . 542.5.1 Effects of evaluation techniques on predicted controller po-tential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.5.2 Effects of simulated wind conditions on predicted controllerpotential . . . . . . . . . . . . . . . . . . . . . . . . . . 582.5.3 Effects of controller properties on performance . . . . . . 602.6 Conclusions and recommendations for future research . . . . . . . 633 Real-time Relocation of Floating Offshore Wind Turbine Platformsfor Wind Farm Efficiency Maximization: An Assessment of Feasi-bility and Steady-state Potential . . . . . . . . . . . . . . . . . . . . 673.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.2 Mathematical model - FOWFSim . . . . . . . . . . . . . . . . . . 713.2.1 Wind farm description . . . . . . . . . . . . . . . . . . . 723.2.2 Wind farm power output and efficiency . . . . . . . . . . 743.2.3 Wind farm aerodynamics - FLORIS . . . . . . . . . . . . 753.2.4 Floating turbine displacement model . . . . . . . . . . . . 763.2.5 Mooring line tensions . . . . . . . . . . . . . . . . . . . 793.3 Optimization problem . . . . . . . . . . . . . . . . . . . . . . . . 84ix3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 853.4.1 Effects of anchor location and cable length . . . . . . . . 863.4.2 Effects of mooring system orientation . . . . . . . . . . . 913.4.3 Effects of turbine spacing and wind farm size . . . . . . . 963.4.4 Wind farm efficiency studies . . . . . . . . . . . . . . . . 983.5 Conclusions and recommendations for future research . . . . . . . 1004 A Low-fidelity Dynamic Wind Farm Model for Simulating Time-varying Wind Conditions and Floating Platform Motion . . . . . . . 1034.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.2 Mathematical model - FOWFSim-Dyn . . . . . . . . . . . . . . . 1074.2.1 Wind farm description . . . . . . . . . . . . . . . . . . . 1074.2.2 States and inputs . . . . . . . . . . . . . . . . . . . . . . 1094.2.3 Solver block diagram . . . . . . . . . . . . . . . . . . . . 1104.2.4 Wind farm power output . . . . . . . . . . . . . . . . . . 1114.2.5 Floating wind turbine motion . . . . . . . . . . . . . . . 1134.2.6 Single wake model . . . . . . . . . . . . . . . . . . . . . 1164.2.7 Wake interaction model . . . . . . . . . . . . . . . . . . 1204.2.8 Model limitations . . . . . . . . . . . . . . . . . . . . . . 1224.3 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.3.1 Mesh sensitivity analysis . . . . . . . . . . . . . . . . . . 1244.3.2 Validation of platform dynamics . . . . . . . . . . . . . . 1254.3.3 Steady-state validation of wake transport . . . . . . . . . 1294.3.4 Dynamic assessment of wake transport . . . . . . . . . . 1314.4 Conclusions and recommendations for future research . . . . . . . 1355 Distributed Economic Model Predictive Control – Addressing Non-convexity Using Social Hierarchies . . . . . . . . . . . . . . . . . . . 1385.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . 1395.1.2 Distributed economic model predictive control . . . . . . 1395.1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . 1425.1.4 Chapter organization . . . . . . . . . . . . . . . . . . . . 142x5.2 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . 1435.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 1435.2.2 Dynamic model . . . . . . . . . . . . . . . . . . . . . . . 1435.2.3 Control objective . . . . . . . . . . . . . . . . . . . . . . 1455.2.4 Conflict under non-convexity . . . . . . . . . . . . . . . . 1475.3 Social hierarchy-based DEMPC algorithm . . . . . . . . . . . . . 1505.3.1 Social hierarchy framework . . . . . . . . . . . . . . . . 1505.3.2 DEMPC coordination algorithm . . . . . . . . . . . . . . 1515.3.3 Closed-loop properties . . . . . . . . . . . . . . . . . . . 1575.3.4 Determining social hierarchy properties . . . . . . . . . . 1635.4 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . 1645.4.1 Problem description . . . . . . . . . . . . . . . . . . . . 1645.4.2 Social hierarchy-based DEMPC properties . . . . . . . . 1655.4.3 Simulation results . . . . . . . . . . . . . . . . . . . . . 1665.5 Conclusions and recommendations for future research . . . . . . . 1716 Social Hierarchy-based Distributed Economic Model Predictive Con-trol of Floating Offshore Wind Farms . . . . . . . . . . . . . . . . . 1746.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1756.1.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . 1756.1.2 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 1766.1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . 1786.1.4 Chapter organization . . . . . . . . . . . . . . . . . . . . 1786.2 Simulation tool - FOWFSim-Dyn . . . . . . . . . . . . . . . . . . 1796.2.1 Floating turbine dynamics module . . . . . . . . . . . . . 1796.2.2 Aerodynamics module . . . . . . . . . . . . . . . . . . . 1816.3 Controller design . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.3.1 Controller objective functions . . . . . . . . . . . . . . . 1816.3.2 Feed-forward artificial neural network . . . . . . . . . . . 1846.3.3 Social hierarchy-based DEMPC algorithm . . . . . . . . . 1886.3.4 Effects of dynamic coupling . . . . . . . . . . . . . . . . 1896.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 1916.4.1 Wind velocity variation . . . . . . . . . . . . . . . . . . . 191xi6.4.2 Social hierarchy-based DEMPC properties . . . . . . . . 1926.4.3 Effects of wind velocity variations . . . . . . . . . . . . . 1936.4.4 Performance of different wind farm sizes . . . . . . . . . 1976.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2007 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2037.1 Steady-state assessment of YITuR . . . . . . . . . . . . . . . . . 2047.2 Dynamic modeling of floating offshore wind farms . . . . . . . . 2047.3 Real-time YITuR controller implementation . . . . . . . . . . . . 2047.4 Final comments on YITuR . . . . . . . . . . . . . . . . . . . . . 206Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208A Wind Farm Control Review Tables . . . . . . . . . . . . . . . . . . . 230xiiList of TablesTable 3.1 Properties of the floating wind turbine detailed by Jonkman etal. [96] and Robertson et al. [152]. . . . . . . . . . . . . . . . 86Table 4.1 Computation times and maximum normalized velocity deficitscorresponding to different simulated mesh element sizes fromFigure 4.7. The final column lists the convergence of the maxi-mum normalized velocity deficit. In other words, it contains therelative difference in the maximum normalized velocity deficitthat would be obtained if each element size was halved. Forinstance, if the element size were to be reduced from 8D to 4D,the predicted velocity deficit would change by 19.46%. Thecomputation times correspond to 5sec long simulations. . . . . 126Table 4.2 List of floating wind farm properties used during dynamic sim-ulations. All wind turbines are based on the NREL 5MW base-line design presented by Jonkman et al. [96], and all floatingplatforms and mooring subsystems are modeled after the designdescribed by Robertson et al. [152]. . . . . . . . . . . . . . . . 127Table 6.1 List of RMSE values between neural network predictions andFOWFSim-Dyn predictions. . . . . . . . . . . . . . . . . . . . 188Table 6.2 List of properties relevant to the social hierarchy-based DEMPCalgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193xiiiTable A.1 Summary of wake models and simulation tools used within windfarm control studies that have been reviewed in the current chap-ter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231Table A.2 Summary of studies that have assessed the steady-state potentialof power de-rating. . . . . . . . . . . . . . . . . . . . . . . . . 233Table A.3 Summary of studies that have developed wind farm controllersbased on the concept of power de-rating. . . . . . . . . . . . . 237Table A.4 Summary of studies that have assessed the steady-state potentialof yaw-based wake redirection. . . . . . . . . . . . . . . . . . 241Table A.5 Summary of studies that have developed wind farm controllersbased on the concept of yaw-based wake redirection. . . . . . . 244Table A.6 Summary of studies that have assessed the steady-state potentialof turbine repositioning. . . . . . . . . . . . . . . . . . . . . . 247xivList of FiguresFigure 1.1 Schematic demonstrating the phenomenon of aerodynamic cou-pling between wind turbines that are aligned with the wind. . . 2Figure 1.2 Schematic demonstrating the use of yaw and induction basedturbine repositioning (YITuR) to minimize wake overlap infloating offshore wind farms. . . . . . . . . . . . . . . . . . . 3Figure 1.3 Layout of the thesis and the role of each chapter. . . . . . . . 4Figure 2.1 Trends in installed global wind power capacity and global in-vestment in the wind energy sector since the start of the cen-tury. Data have been obtained from the International Renew-able Energy Agency (IRENA) [87], [90]. . . . . . . . . . . . 9Figure 2.2 Trends in levelized energy and investments costs for onshoreand offshore wind power since 2010. Data have been obtainedfrom IRENA [88]. . . . . . . . . . . . . . . . . . . . . . . . 11Figure 2.3 The phenomenon of aerodynamic coupling between two windturbines aligned with the free stream wind. . . . . . . . . . . 12Figure 2.4 Overview of three notable wind farm control techniques pro-posed for mitigating the wake effect. The schematic shows atop-view of two wind turbines aligned with the incoming freestream wind. Green elements highlight changes in turbine op-eration and wake conditions associated with each control con-cept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15xvFigure 2.5 The closed-loop feedback structure of a general control systemfor wind farm power maximization. Listed measurements andturbine set-points do not encapsulate all possible signals; theyare examples of commonly used data for wind farm control. . 19Figure 2.6 The closed-loop feedback structure of a model-based wind farmcontroller with wake model parameter and state estimation,and wind field estimation functions. . . . . . . . . . . . . . . 21Figure 2.7 Comparison between the closed-loop feedback structures of(a) centralized and (b) distributed wind farm controllers. . . . 24Figure 2.8 The control volume analysis used for computing momentumconservation in parametric wake models that are based on in-tegral relations from fluid mechanics. . . . . . . . . . . . . . 26Figure 2.9 A demonstration of steady-state solutions being transported inthe downstream direction as a simplified means of approximat-ing dynamic wake transport. . . . . . . . . . . . . . . . . . . 28Figure 2.10 Schematic demonstrating the concept of yaw and induction-based turbine repositioning (YITuR). . . . . . . . . . . . . . 54Figure 2.11 Bivariate histogram demonstrating the quantity of studies as-sessing power de-rating that have reported different wind effi-ciency gains using various evaluation models and techniques. . 55Figure 2.12 Bivariate histogram demonstrating the quantity of studies as-sessing yaw-based wake redirection that have reported differ-ent wind efficiency gains using various evaluation models andtechniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Figure 2.13 Bivariate histogram demonstrating the quantity of studies as-sessing turbine repositioning that have reported different windefficiency gains using various evaluation models and techniques. 58Figure 3.1 Schematic demonstrating the concept of yaw and induction-based turbine repositioning (YITuR). The faded diagram cor-responds to a floating offshore wind turbine subject to greedyoperation (i.e. agreedy = 1/3 and γgreedy = 0deg). . . . . . . . 71xviFigure 3.2 Schematic demonstrating the representation of a floating off-shore wind farm as a system of N particles distributed alongthe two-dimensional ocean surface. The loads and local windvelocity Ui that are incident on a single turbine i are also shown. 73Figure 3.3 Block diagram showing the computation modules of FOWF-Sim along with information transfer routes. . . . . . . . . . . 77Figure 3.4 Schematic demonstrating the position vectors that are used todescribe the mooring system for turbine i. . . . . . . . . . . . 78Figure 3.5 Schematic demonstrating the three zones in which mooringline fairleads may be located. The zone that is active deter-mines the cable profile and tension. . . . . . . . . . . . . . . 80Figure 3.6 Schematic demonstrating the forces acting on different cableportions when the fairlead of the mooring line is located within(a) zone 2 and (b) zone 3. . . . . . . . . . . . . . . . . . . . . 82Figure 3.7 Schematic demonstrating the baseline orientation of a floatingplatform and mooring system relative to the incoming winddirection. The orientation of a mooring system that is rotatedby a value θmoor is also presented in the faded diagram. . . . . 87Figure 3.8 Relative gain in wind farm efficiency using YITuR operationwith different mooring line length factors αcable and anchordistances from turbine neutral positions LA→G,neutral. Simula-tion properties: 1× 2 wind farm size, free stream wind speedU∞= 8m/s, inter-turbine spacing of 7D, baseline mooring sys-tem orientation θmoor = 0deg. . . . . . . . . . . . . . . . . . 89Figure 3.9 Floating wind turbine positions corresponding to optimizationresults presented in Figure 3.8. Simulation properties: 1× 2wind farm size, free stream wind speed U∞ = 8m/s, inter-turbine spacing of 7D, baseline mooring system orientationθmoor = 0deg, distance from anchors to turbine neutral posi-tions LA→G,neutral = 900m. . . . . . . . . . . . . . . . . . . . 90Figure 3.10 Horizontal and vertical components of cable tension correspond-ing to a mooring line of length Lc = 925m with distance fromanchor to platform neutral position LA→G,neutral = 837.6m. . . 91xviiFigure 3.11 Schematic demonstrating the effect of increased anchor dis-tance from floating platform neutral position. . . . . . . . . . 92Figure 3.12 Relative gain in wind farm efficiency using YITuR operationwith different cable length factors αcable and mooring systemorientations θmoor. Simulation properties: 1× 2 wind farmsize, free stream wind speed U∞ = 8m/s, inter-turbine spac-ing of 7D, distance from anchors to platform neutral positionsLA→G,neutral = 837.6m. . . . . . . . . . . . . . . . . . . . . . 93Figure 3.13 Schematic demonstrating the effects of mooring system ori-entation on the crosswind mobility of floating offshore windturbines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Figure 3.14 Floating wind turbine positions corresponding to optimizationresults presented Figure 3.12. Simulation properties: 1× 2wind farm size, free stream wind speed U∞ = 8m/s, inter-turbine spacing of 7D, distance from anchors to platform neu-tral positions LA→G,neutral = 837.6m, cable length factor αcable =0.8 (i.e. cable length Lc = 950m). . . . . . . . . . . . . . . . 95Figure 3.15 Schematic demonstrating the origin of bias in the steady-statedisplacements of a floating offshore wind turbines when moor-ing system orientations deviate from the baseline shown in Fig-ure 3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 3.16 Efficiency ηfarm of a wind-aligned row of N turbines obtainedfrom greedy and YITuR operation for different inter-turbinespacings and wind turbine quantities N. Simulation proper-ties: free stream wind speed U∞ = 8m/s, mooring line cablelength Lc = 925m, distance from anchors to platform neutralpositions LA→G,neutral = 837.6m, baseline mooring system ori-entation θmoor = 0deg. . . . . . . . . . . . . . . . . . . . . . 98Figure 3.17 Floating wind turbine positions corresponding to optimizationresults presented Figure 3.16. Simulation properties: free streamwind speed U∞= 8m/s, mooring line cable length Lc = 925m,distance from anchors to platform neutral positions LA→G,neutral =837.6m, baseline mooring system orientation θmoor = 0deg. . 99xviiiFigure 3.18 Efficiency ηfarm of wind farms with grid layouts and differ-ent sizes under greedy and YITuR operation subject to dif-ferent wind directions. The radial axes represent wind farmefficiency ηfarm and the circumferential axes denote the winddirection with 0deg indicating wind flow to the right. Simu-lation properties: free stream wind speed U∞ = 8m/s, inter-turbine spacing of 7D in x and y directions, mooring line cablelength Lc = 925m, distance from anchors to turbine neutralpositions LA→G,neutral = 837.6m, baseline mooring system ori-entation θmoor = 0deg. . . . . . . . . . . . . . . . . . . . . . 101Figure 4.1 Schematic of a general floating offshore wind farm with semi-submersible platforms used as a basis for FOWFSim-Dyn’smathematical model. . . . . . . . . . . . . . . . . . . . . . . 108Figure 4.2 Block diagram showing the computation modules of FOWFSim-Dyn along with information transfer routes. . . . . . . . . . . 111Figure 4.3 Schematic of floating platform velocity vector vi(t), incidentwind velocity vector Vi(t), and the relative incident velocityvector Vrel,i(t) at the location of turbine i. . . . . . . . . . . . 112Figure 4.4 Schematic of aerodynamic thrust force Fa,i(t), hydrodynamicdrag force Fh,i(t), and mooring line forces Fm,i,k(t) acting onwind turbine i with a semi-submersible floating platform. . . . 114Figure 4.5 Schematic of position vectors that are relevant for calculatingthe force in mooring line k of turbine i. . . . . . . . . . . . . 116Figure 4.6 Schematic of characteristics necessary for modeling the wakegenerated by turbine i. The wake centerline position yw,i(xˆi, t),average wake velocity vw,i(xˆi, t), and wake diameter Dw,i(xˆi, t)are defined within the reference frame that is fixed to turbine i. 116Figure 4.7 Effects of various finite difference mesh element sizes on a) thesteady-state wake velocity profile at a downstream distance of7D, and b) the steady-state wake centerline evolution. Simula-tion parameters: D = 15cm, U∞ = 8m/s, a = 1/3, γ = 20deg,kx = 0.08, σ = 0.025xˆ+0.396m. . . . . . . . . . . . . . . . 125xixFigure 4.8 Comparison between FOWFSim-Dyn and FAST predictionsof platform motion for a single floating wind turbine basedon the baseline designs described in ??. The curve identifiedby FOWFSim-Dyn* utilizes the dynamics lumped-parametermooring line model proposed by Hall et al. [79] in place ofthe static catenary model described in Chapter 3. Simulationparameters: U∞ = 10m/s, a = 1/3, γ = 15deg, L = 850m. . . 128Figure 4.9 Comparison between FOWFSim-Dyn predictions and exper-imental results reported by Bastankhah and Porte´ Agel [18].Each figure shows steady-state wake centerlines and normal-ized velocity profiles corresponding to yaw angles of a) γ =0deg, b) γ = 10deg, and c) γ = 20deg. Normalized veloc-ity profiles range from zero to one using the same scaling asthe xˆ/D axis, but have been shifted to the downstream locationwhere they are measured. Simulation parameters: D = 15cm,U∞ = 8m/s, a = 1/3, kx = 0.08, σ = 0.025xˆ+0.396m. . . . 130Figure 4.10 Schematic of the 1×3 wind farm with inter-turbine spacings of7D used for dynamic simulations. All wind turbines are basedon the NREL 5MW baseline design presented by Jonkman etal. [96], and all floating platforms and mooring subsystems aremodeled after the design described by Robertson et al. [152]. . 131Figure 4.11 Velocity contours at various time-steps of simulation scenario 1(i.e. fixed wind condition and turbine operating parameters,while platform motion is permitted). The white + symbolsrepresent the neutral positions of the floating platforms. Allfloating platforms are held fixed at their respective neutral po-sitions for the first 1000sec of simulation. Simulation param-eters: U∞(t) = 8m/s, V∞(t) = 0m/s, a1(t) = a2(t) = a3(t) =1/3, γ1(t) = γ3(t) = −20deg and γ2(t) = +20deg, kx = 0.08,σ = 0.025xˆ+0.396m. . . . . . . . . . . . . . . . . . . . . . 133xxFigure 4.12 Velocity contours at various time-steps of simulation scenario 2(i.e. fixed wind condition and sinusoidally varying yaw angles,while platform motion is permitted). The white + symbolsrepresent the neutral positions of the floating platforms. Allfloating platforms are held fixed at their respective neutral po-sitions for the first 1000sec of simulation. Simulation param-eters: U∞(t) = 8m/s, V∞(t) = 0m/s, a1(t) = a2(t) = a3(t) =1/3, γ1(t) and γ3(t) defined in Eq. (4.50) and γ2(t) defined inEq. (4.51), kx = 0.08, σ = 0.025xˆ+0.396m. . . . . . . . . . 134Figure 4.13 Velocity contours at various time-steps of simulation scenario 3(i.e. fixed turbine operating conditions and fluctuating windspeed in the yˆ direction, while platform motion is prohibited).The white + symbols represent the neutral positions of thefloating platforms. The white arrows denote the free streamwind direction. All floating platforms are held fixed through-out the simulation. Simulation parameters: U∞(t) = 8m/s,V∞(t) defined in Eq. (4.52), a1(t)= a2(t)= a3(t)= 1/3, γ1(t)=γ2(t) = γ3(t) = 0deg, kx = 0.08, σ = 0.025xˆ+0.396m. . . . . 136Figure 5.1 A visual explanation of conflict generated as a result of non-convexity in distributed parallel optimization. . . . . . . . . . 148Figure 5.2 Schematic demonstrating the organization structure of the so-cial hierarchy framework. . . . . . . . . . . . . . . . . . . . 151Figure 5.3 Schematic of the plate overlap problem used in the numericalexample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Figure 5.4 Interaction graph of the plate overlap problem with vertex col-oring used to identify a universal social hierarchy. . . . . . . . 165Figure 5.5 Evolution of the cumulative number of social hierarchy changesfor five simulations of ten plates. Each color corresponds to adifferent simulation. . . . . . . . . . . . . . . . . . . . . . . 167Figure 5.6 Evolution of the global cost function for five simulations of tenplates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168xxiFigure 5.7 Evolution of the mean system-wide stationary target positionfor five different simulations involving ten plates. . . . . . . . 169Figure 5.8 Plate locations at the final sampling time in (from top to bot-tom) simulations 1 to 5. . . . . . . . . . . . . . . . . . . . . . 170Figure 5.9 Evolution of the global cost function V using a parallel DEMPCalgorithm for five simulations involving different quantities ofplates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171Figure 5.10 Evolution of the global cost function V using the proposed so-cial hierarchy-based DEMPC algorithm for five simulations in-volving different quantities of plates. . . . . . . . . . . . . . . 172Figure 6.1 Schematic describing the use of aerodynamic thrust forces topassively relocate floating wind turbines in real-time. . . . . . 176Figure 6.2 Block diagram showing the computation modules of FOWFSim-Dyn along with information transfer routes. . . . . . . . . . . 179Figure 6.3 Schematic of the general floating offshore wind farm used inFOWFSim-Dyn. . . . . . . . . . . . . . . . . . . . . . . . . 180Figure 6.4 Schematic of the local neural network structure used to identifythe dynamic model of each floating wind turbine. . . . . . . . 185Figure 6.5 Schematic of a two-turbine floating offshore wind farm sim-ulated in the current work. Wind turbine and floating plat-form properties are based on the National Renewable EnergyLaboratory’s (NREL’s) baseline designs presented by Jonkmanet al. [96] and Robertson et al. [152]. The sole modificationis that mooring line lengths have been increased from 835 to950m to permit greater platform displacement. . . . . . . . . 186Figure 6.6 Sample validation data for the neural network of the upstreammachine (i.e. turbine 1) in a two-turbine floating wind farm. . 187Figure 6.7 Sample validation data for the neural network of the upstreammachine (i.e. turbine 2) in a two-turbine floating wind farm. . 187Figure 6.8 Schematic showing the impacts of disturbance/model uncer-tainty on terminal equality constraint satisfaction in EMPC . . 190xxiiFigure 6.9 Schematic demonstrating the method of constraint tighteningfor process constraint satisfaction in EMPC. . . . . . . . . . . 191Figure 6.10 Sample randomly generated free stream wind speed evolutionsin the xˆ and yˆ directions corresponding to a perturbation param-eter of σ∞ = 5 %. . . . . . . . . . . . . . . . . . . . . . . . . 192Figure 6.11 Evolution of yˆ positions in a two-turbine floating wind farmthat is optimally controlled by the DEMPC from Section 6.3.The four plots correspond to different wind velocity perturba-tions of a) σ∞= 5 %, b) σ∞= 10 %, c) σ∞= 15 %, and d) σ∞=20 %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194Figure 6.12 Evolution of power production in a two-turbine floating windfarm that is optimally controlled by the DEMPC from Sec-tion 6.3. For comparison, power production under greedy op-eration is also demonstrated. The four plots correspond to dif-ferent wind velocity perturbations of a) σ∞ = 5 %, b) σ∞ =10 %, c) σ∞ = 15 %, and d) σ∞ = 20 %. . . . . . . . . . . . . 196Figure 6.13 Evolution of xˆ positions in a two-turbine floating wind farmthat is optimally controlled by the DEMPC from Section 6.3.The four plots correspond to different wind velocity perturba-tions of a) σ∞= 5 %, b) σ∞= 10 %, c) σ∞= 15 %, and d) σ∞=20 %. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197Figure 6.14 Evolution of power production in floating wind farms of dif-ferent sizes that are optimally controlled by the DEMPC fromSection 6.3. For comparison, power production under greedyoperation is also demonstrated. The wind velocity perturbationis fixed at σ∞ = 5 %. The four plots correspond to differentwind farm configurations of a) 1×2, b) 1×3, c) 1×4, and d)1×5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198Figure 6.15 Evolution of yˆ positions in floating wind farms of differentsizes that are optimally controlled by the DEMPC from Sec-tion 6.3. The wind velocity perturbation is fixed at σ∞ = 5 %.The four plots correspond to different wind farm configura-tions of a) 1×2, b) 1×3, c) 1×4, and d) 1×5. . . . . . . . . 199xxiiiFigure 6.16 Evolution of xˆ positions in floating wind farms of differentsizes that are optimally controlled by the DEMPC from Sec-tion 6.3. The wind velocity perturbation is fixed at σ∞ = 5 %.The four plots correspond to different wind farm configura-tions of a) 1×2, b) 1×3, c) 1×4, and d) 1×5. . . . . . . . . 200xxivNomenclatureChapter 1AcronymsCFD Computational fluid dynamicsDEMPC Distributed economic model predictive controlFOWFSim Floating Offshore Wind Farm SimulatorFOWFSim-Dyn Dynamic Floating Offshore Wind Farm SimulatorYITuR Yaw and induction-based turbine repositioningChapter 2a Axial induction factorCP Power coefficientCT Thrust coefficientP Turbine power output, WTgen Generator torque, N·mx Downwind displacement of a wind turbine, my Crosswind displacement of a wind turbine, mGreek lettersβ Blade pitch angle, radγ Nacelle yaw angle, radλ Tip-speed-ratioAcronymsxxvAEP Annual energy productionCFD Computational fluid dynamicsDEL Damage equivalent loadingECN Energy Research Center of the NetherlandsFAST Fatigue, Aerodynamics, Structures, and TurbulenceFLORIDyn Flow Redirection and Induction DynamicsFLORIS Flow Redirection in Steady-stateFOWFSim Floating Offshore Wind Farm SimulatorGHG Green house gasIRENA International Renewable Energy AgencyLCOE Levelized cost of energyLES Large eddy simulationNREL National Renewable Energy LaboratoryRANS Reynolds-averaged Navier-StokesRMS Root-mean-squareRMSE Root-mean-square-errorSOWFA Simulator for Wind Farm ApplicationsUTD-WF University of Texas at Dallas-Wind FarmWFSim Wind Farm SimulatorYITuR Yaw and induction-based turbine repositioningChapter 3a Vector containing the axial induction factors of all turbinesA Rotor swept area of turbine rotors, m2Ac Mooring line cable cross-sectional area, m2ai Axial induction factor of turbine iCP,i Power coefficient of turbine iCP,max Power coefficient of turbine i under greedy operationCthrust,i Thrust coefficient of turbine iD Diameter of turbine rotors, mDc Mooring line cable diameter, mE Mooring line elastic modulus, N/m2F Set containing the indices of all turbines in the wind farmxxvif1(·) Newton-Raphson function for computing tensions of a mooring linethat is partially resting along the seabed, Nf2(·) Newton-Raphson function for computing tensions of a mooring linethat is fully lifted above the seabed, Nfmoor(·) Mooring line tension lookup function, NFmoor,i Total mooring line force vector acting on the platform of turbine i, NFmoor,i, j Mooring line force vector exerted by cable j on the platform of tur-bine i, NFs Static friction force acting on the portion of a mooring line cable thatis resting along the seabed, NFthrust,i Aerodynamic thrust force vector acting on the rotor of turbine i, NFw Total weight of mooring line cable, NH Horizontal tension along mooring line cable, NHA Horizontal tension at mooring line anchor, NHF Horizontal tension at mooring line fairlead, NHF,2→3 Horizontal tension that causes mooring line cable to fully lift off of theseabed, NHF,i, j Magnitude of the horizontal tension along mooring line j of turbine i,NL Stretched mooring line cable length, mLA→F,i, j Distance from anchor to fairlead for mooring line j of turbine i, mLA→G,neutral Distance from anchors to platform centers in their neutral positions, mLc Unstretched mooring line cable length, mLc,min Shortest permissible mooring line cable length, mLc,max Longest permissible mooring line cable length, mLG→F Distance from platform center to fairlead locations, mLs Length of cable resting along the seabed, mM Set containing the indices of mooring line cables for each platformN Number of turbines in the wind farmPfarm Wind farm power output, WPfarm,∞ Wind farm power output in the absence of the wake effect, WPi Power output of turbine i, WpP Tuning parameter for estimating turbine power coefficientsxxviiR Radius of turbine rotors, mrA→F,i, j Position vector from anchor to fairlead for mooring line j of turbine i,mrA,i, j Position vector identifying location of the anchor of mooring line j ofturbine i, mrG→F,i, j Position vector from platform center to fairlead for mooring line j ofturbine i, mri Position vector indicating the location of turbine i, ms Arbitrary arc length along a mooring line cable, mT Tension along mooring line cable, NU∞ Free stream wind speed, m/sUi Effective incident wind speed on the rotor of turbine i, m/sUi Set containing the indices of all machines located upstream of turbine iV Vertical tension along a mooring line cable, NVA Vertical tension at a mooring line anchor, NVF Vertical tension at a mooring line fairlead, Nw Weight per-unit-length of mooring line cables in ocean water, N/mx Axis aligned with the free stream wind directionx Vector containing positions of all turbines along the x axis, mxF Distance from an anchor to a fairlead in a catenary analysis, mxF,1→2 Distance from an anchor to a fairlead that first leads to a catenary pro-file, mxF,2→3 Distance from an anchor to a fairlead that fully lifts a cable off of theseabed, mxF,neutral Distance from anchors to fairleads while platforms are in their neutralpositions, mxi Position of turbine i along the x axis, mxs Distance along a mooring cable portion resting on the seabed wherethe static friction force balances with the horizontal cable tension, my Axis perpendicular to the free stream wind directionyi Position of turbine i along the y axis, my Vector containing positions of all turbines along the y axis, mz Axis pointing upward from the ocean surfacexxviiizF Vertical location of mooring line fairleads above the seabed, mGreek lettersαcable Modification factor used in mooring line cable length studiesγ Vector containing the yaw angles of all turbines in the wind farm, degγi Yaw angle of turbine i, rad∆ηfarm,rel Relative gain in wind farm efficiency, %ηfarm Wind farm efficiency, %ηP Turbine generator efficiencyθmoor Rotation of the mooring system orientation relative to the baseline de-sign, radµs Coefficient of static friction between mooring line cables and theseabedρ Density of air, kg/m3AcronymsCFD Computational fluid dynamicsFOWFSim Floating Offshore Wind Farm SimulatorFLORIS Flow Redirection in Steady-stateLES Large eddy simulationSOWFA Simulator for Wind Farm ApplicationsYITuR Yaw and induction-based turbine repositioningChapter 4A Generalized symbol to signify area-based integration, m2Aa,i, j Added mass reference area of submerged component j of turbine i, m2Ad,i, j Drag reference area of submerged component j of turbine i, m2ai(t) Axial induction factor of turbine iAi Swept area of the rotor of turbine i, m2Ca,i, j Added mass coefficient of submerged component j of turbine iCd,i, j Drag coefficient of submerged component j of turbine iCp,i(t) Power coefficient of turbine ixxixCt,i(t) Thrust coefficient of turbine iDi Diameter of the rotor of turbine i, mDi Set containing the indices of all submerged components of turbine iDw,i(xˆi, t) Diameter of the wake produced by turbine i over space and time, mDw,i,p(t) Wake diameter at point p of the wake produced by turbine i, mDw,q→i(t) Diameter of the wake produced by turbine q at the location of the rotorof turbine i, mf (·) Nonlinear continuous-time state-space model of the wind farmF Set containing the indices of all turbines in the wind farmFa,i(t) Aerodynamic force vector acting on the rotor of turbine i, NFh,i(t) Hydrodynamic force vector acting on the platform of turbine i, NFi(t) Total force vector acting on turbine i, NFm,i(t) Total mooring line force vector acting on the platform of turbine i, NFm,i,k(t) Force vector of mooring line k of turbine i, NHF,i,k(t) Horizontal tension along mooring line k of turbine i, Nkt Temporal wake expansion constant, m/skx Spatial wake expansion constant, m/mLi(xˆi, t) Linear momentum vector per unit length of the wake produced by tur-bine i over space and time, kg/sma,i Effective added hydrodynamic mass resulting from the acceleration ofthe platform of turbine i, kgmi Total mass of turbine i including the floating platform mass, kgMi Set containing the indices of the mooring lines attached to the platformof turbine iN Number of turbines in the wind farmn∞(t) Unit vector aligned with the free stream wind directionNh,i Total number of submerged components of turbine ini(t) Unit vector pointing outward from and normal to the rotor of turbine iNm,i Total number of mooring lines attached to the platform of turbine iNp,i Number of points used to spatially discretize the wake produced byturbine iPfarm(t) Power output of the wind farm, WPi(t) Power output of turbine i, Wxxxr Arbitrary radius along the radial axis used to define Gaussian wakevelocity profiles, mrA,i,k Position vector identifying the location of the anchor of mooring line kof turbine i, mrF/A,i,k(t) Position vector from anchor to fairlead for mooring line k of turbine i,mrF/G,i,k Position vector from platform center to fairlead for mooring line k ofturbine i, mri(t) Position vector identifying the location of turbine i, mU∞(t) Free stream wind speed along the xˆ axis, m/sU∞,ref Reference free stream wind speed used to convert between spatial andtemporal wake expansion constants, m/sufarm(t) Vector containing the inputs of all turbines in the wind farmui(t) Vector containing the inputs of turbine iUi(t) Component of Vi(t) aligned with the xˆ axis, m/sUi Set containing the indices of all machines that are located upstream ofturbine iuw,i(xˆi, t) Average speed of the wake produced by turbine i in the xˆ direction overspace and time, m/suw,i,p(t) Average wake speed at point p of the wake produced by turbine i inthe xˆ direction, m/sV∞(t) Free stream wind speed along the yˆ axis, m/sV∞(t) Free stream wind velocity vector, m/svi(t) Velocity vector of turbine i, m/sVi(t) Component of Vi(t) aligned with the yˆ axis, m/sVi(t) Effective wind velocity vector that is incident on the rotor of turbine imeasured in the global stationary frame, m/sVrel,i(t) Effective wind velocity vector that is incident on the rotor of turbine imeasured by an observer who is fixed to turbine i, m/svw,i(xˆi, t) Average speed of the wake produced by turbine i in the yˆ direction overspace and time, m/svw,i(xˆi, t) Average velocity vector of the wake produced by turbine i over spaceand time, m/sxxxivw,init,i(t) Average velocity vector of the wake produced by turbine i immediatelydownstream of its rotor, m/svw,i,p(t) Average wake speed at point p of the wake produced by turbine i inthe yˆ direction, m/sv˘w,q→i(r, t) Gaussian velocity distribution of the wake produced by turbine q at thelocation of the rotor of turbine i, m/svw,q→i(t) Effective velocity of the wake produced by turbine q that is incidenton the rotor of turbine i, m/svx,i(t) Velocity of turbine i along the xˆ axis, mvy,i(t) Velocity of turbine i along the yˆ axis, mw(t) Ocean current velocity vector, m/sxˆ Global axis aligned with the predominant free stream wind directionx(t) Vector containing the states of all turbines in the wind farmxfarm(t) Vector containing the states of all wakes and turbines in the wind farmxi(t) Position of turbine i along the xˆ axis, mxˆi Local axis aligned with the predominant free stream wind directionand fixed to turbine ixw(t) Vector containing the states of all wakes in the wind farmxw,i(t) Vector containing the states of the wake produced by turbine ixw,i,p(t) Vector containing the states of point p of the wake produced by tur-bine iyˆ Global axis perpendicular to the predominant free stream wind direc-tionyi(t) Position of turbine i along the yˆ axis, myˆi Local axis perpendicular to the predominant free stream wind directionand fixed to turbine iyw,i(xˆi, t) Centerline position of the wake produced by turbine i in the yˆ directionover space and time, myw,i,p(t) Centerline position at point p of the wake produced by turbine i in theyˆ direction, mGreek lettersxxxiiγi(t) Yaw angle of the nacelle of turbine i measured relative to the xˆ axis,radγrel,i(t) Yaw angle of the nacelle of turbine i relative to Vrel,i(t), radθi(t) Angle of the vector Vrel,i(t) measured relative to the xˆ axis, radξw,init,i(t) Skew angle of the wake produced by turbine i immediately past itsrotor used in momentum conservation calculations, radρa Density of air, kg/m3ρw Density of ocean water, kg/m3σ Standard deviation of Gaussian wake velocity profiles, mχi(t) Skew angle of the wake produced by turbine i immediately past itsrotor used in vortex theory calculations, radAcronymsFLORIDyn Flow Redirection and Induction DynamicsFLORIS Flow Redirection in Steady-stateFOWFSim Floating Offshore Wind Farm SimulatorFOWFSim-Dyn Dynamic Floating Offshore Wind Farm SimulatorLES Large eddy simulationNREL National Renewable Energy LaboratoryRMSE Root-mean-square-errorWFSim Wind Farm SimulatorChapter 5fi(·) Discrete-time state-space model of agent iH Number of time-steps along the prediction horizonI Set containing the indices of all agentsJ(·) Global stage cost functionJi(·) Neighborhood-cooperative stage cost function of agent ik Time-step number along the prediction horizon`i(·) Local stage cost function of agent imi Number of inputs of agent iN Number of agentsxxxiiiN−i Set containing the indices of all agents that influence `i(·)N−i|J Set containing the indices of all agents that influence Ji(·)N+i Set containing the indices of all agents whose local cost functions areinfluenced by the states and inputs of agent iN+i|J Set containing the indices of all agents whose neighborhood-cooperative cost functions are influenced by the states and inputs ofagent ini Number of states of agent iNq Total number of social hierarchy levelsNs Total number of possible social hierarchy levels that will resolve con-flictp Iteration number in Algorithm 1P Probability that all conflicts will be resolved within a single iterationq Current social hierarchy level being implemented by Algorithm 1qi Social hierarchy level of agent iU−i Set containing all permissible input vectors of neighboring agents inthe setN−iuk−i|J Vector containing the inputs of all neighboring agents in the setN−i|Jat time-step k along the prediction horizonuˆk−i|J Array containing the input vectors of all neighboring agents in the setN−i|J at time-step k along the prediction horizon that agent i assumeswhile computing optimal decisionsu˘k−i|J Array contain optimal input vectors of all neighboring agents in the setN−i|J at time-step k along the prediction horizonuˆs−i|J Array containing the steady input vectors of all neighboring agents inthe setN−i|J that agent i assumes while computing optimal decisionsu˘s−i|J Array containing optimal steady input vectors of all neighboringagents in the setN−i|Jui Current input vector of agent iui Input trajectory of agent i along the prediction horizonUi Set containing all permissible input vectors for agent iU i Set containing all permissible input vector trajectories for agent iu∗i Optimal input vector trajectory of agent i along the prediction horizonxxxivuki Candidate input vector of agent i at time-step k along the predictionhorizonu˘ki Optimal input vector of agent i at time-step k along the prediction hori-zonusi Candidate steady input vector of agent iu˘si Optimal steady input vector of agent iu˜ j The input vector trajectory along the prediction horizon that agent iassumes for agent juˆsj The steady input vector that agent i assumes for agent jV (·) Global stage cost function value summed over the prediction horizonVˆi Naive stage cost function value of agent i summed over the predictionhorizonV˘i Informed stage cost function value of agent i summed over the predic-tion horizonVˆ si Naive stage cost function value of agent i at steady-stateV˘ si Informed cost function value of agent i at steady-stateX−i Set containing all permissible state vectors of neighboring agents inthe setN−ixk−i|J Vector containing the states of all neighboring agents in the set N−i|Jat time-step k along the prediction horizonxˆk−i|J Array containing the state vectors of all neighboring agents in the setN−i|J at time-step k along the prediction horizon that agent i assumeswhile computing optimal decisionsx˘k−i|J Array contain optimal state vectors of all neighboring agents in the setN−i|J at time-step k along the prediction horizonxˆs−i|J Array containing the steady state vectors of all neighboring agents inthe setN−i|J that agent i assumes while computing optimal decisionsx˘s−i|J Array containing optimal steady state vectors of all neighboring agentsin the setN−i|Jxi Current state vector of agent iXi Set containing all permissible state vectors for agent iX i Set containing all permissible state vector trajectories for agent ix∗i Optimal state vector trajectory of agent i along the prediction horizonxxxvx+i State vector of agent i at the subsequent sampling timeX 0→si Set containing all initial state vectors that may be feasibly steered to astationary point for agent ixki Candidate state vector of agent i at time-step k along the predictionhorizonx˘ki Optimal state vector of agent i at time-step k along the prediction hori-zonxsi Candidate steady state vector of agent ix˘si Optimal steady state vector of agent iX si Set containing all feasible steady state vectorsx˜ j The state vector trajectory along the prediction horizon that agent iassumes for agent jxˆsj The steady state vector that agent i assumes for agent jZ 0→si Set containing all initial state vectors, input vector trajectories, andsteady state vectors that satisfy system dynamics and constraints foragent iZ si Set containing all feasible stationary points for agent iAcronymsDMPC Distributed model predictive controlDEMPC Distributed economic model predictive controlEMPC Economic model predictive controlMPC Model predictive controlChapter 6ai Axial induction factor of turbine iai,k Axial induction factor of turbine i at time-step k along the predictionhorizonai,s Axial induction factor of turbine i at the candidate stationary pointAOL,i→ j(·) Function that outputs the normalized overlap area between the rotorsof two adjacent turbinesxxxvifi(·) Discrete-time state-space model of turbine i that is identified usingfeed-forward neural networksH Number of time-steps along the prediction horizonk Time-step number along the prediction horizonNi Set containing the indices of all machines that are adjacent to turbine iQ Input penalty weighting matrixui Input vector trajectory of turbine i along the prediction horizonui,k Input vector of turbine i at time-step k along the prediction horizonui,s Input vector of turbine i at the candidate stationary pointV∞ Free stream wind velocity vector, m/sV∞ Mean free stream wind velocity vector, m/svx,i Velocity of turbine i along the xˆ axis, m/svx,i,k Velocity of turbine i along the xˆ axis at time-step k along the predictionhorizon, m/svy,i Velocity of turbine i along the yˆ axis, m/svy,i,k Velocity of turbine i along the yˆ axis at time-step k along the predictionhorizon, m/sxˆ Axis aligned with the predominant free stream wind directionxi Position of turbine i along the xˆ axis, mxi State vector trajectory of turbine i along the prediction horizonxi,k Position of turbine i along the xˆ axis at time-step k along the predictionhorizon, mxi,k State vector of turbine i at time-step k along the prediction horizonxi,s State vector of turbine i at the candidate stationary pointyˆ Axis perpendicular to the predominant free stream wind directionyi Position of turbine i along the yˆ axis, myi,k Position of turbine i along the yˆ axis at time-step k along the predictionhorizon, mGreek lettersγi Yaw angle of the nacelle of turbine i measured relative to the xˆ axis,radγi,k Yaw angle of turbine i at time-step k along the prediction horizon, radxxxviiγi,s Yaw angle of turbine i at the candidate stationary point, rad∆ui,k Change in the input vector of turbine i relative to greedy referencevaluesσ∞ Parameter used to fluctuate the free stream wind velocity vector rela-tive to its mean valueAcronymsDEMPC Distributed economic model predictive controlEMPC Economic model predictive controlFOWFSim-Dyn Dynamic Floating Offshore Wind Farm SimulatorMPC Model predictive controlNREL National Renewable Energy LaboratoryRMSE Root-mean-square-errorYITuR Yaw and induction-based turbine repositioningChapter 7AcronymsDEMPC Distributed economic model predictive controlFOWFSim Floating Offshore Wind Farm SimulatorFOWFSim-Dyn Dynamic Floating Offshore Wind Farm SimulatorLES Large eddy simulationYITuR Yaw and induction-based turbine repositioningxxxviiiAcknowledgmentsI am first grateful to my research supervisor Prof. Ryozo Nagamune for the oppor-tunities and guidance that he has offered to me over the past three years. Second, Iwould like to extend my appreciation to the supervisory committee members Prof.Anasavarapu Srikantha Phani, Prof. Rajeev Jaiman, and Prof. Bhushan Gopaluni,the university examiners Prof. Guy Dumont and Prof. Martin Ordonez, and the ex-ternal examiner Prof. Jan-Willem van Wingerden for offering their time, support,and useful feedback, which has led to substantive improvements to this thesis.I am also indebted to my close friends Moe Kabbara, Corey Yeadon, andPatrick MacInnis, and to my lab mates at the Control Engineering LaboratoryPouya, Sara, Eduardo, Alejandrina, Yue, Christopher, Marcin, Jacob, Nafise, Ji-hoon, and Naho for their support and encouragement.Finally, I am thankful for the financial support provided by the Natural Sciencesand Engineering Research Council of Canada (NSERC) and by the Province ofBritish Columbia.xxxixChapter 1IntroductionThis thesis presents various aspects of research aimed at maximizing power pro-duction in floating offshore wind farms. The current chapter introduces the readerto the main problem that causes inefficiency in wind farms, and then describes thesolution approach of interest for mitigating this problem. Subsequent chapters thendescribe the different steps taken to implement and assess the potential, feasibility,and challenges of the proposed solution. Each chapter is authored such that it maybe read independently of the rest of the text. The current chapter therefore alsoprovides an outline of the thesis while discussing the purpose and contributions ofeach chapter in the context of the overarching objective.1.1 The wake effectWind turbines are clustered into wind farms to maximize energy extraction fromregions with high-resource winds. One consequence of this siting approach isthat machines that are in close proximity interact aerodynamically and experiencediminished performance. Referring to Figure 1.1, viscous interaction along theblades of a wind turbine generates a region of airflow downstream of its rotor thatis characterized by reduced wind speeds; this region is referred to as a wake. Anywind turbines located downstream and aligned with the wake produced by an up-stream machine therefore produce less power than they would in the absence ofthe wake. This phenomenon is termed the wake effect, and may lead to reductions1Turbine 1 Turbine 2Incoming free stream wind speed profileViscous interaction at the blades of Turbine 1 generates wake regionWind speed is reduced at location of Turbine 2Wake generated by Turbine 1Reduced power output at Turbine 2Figure 1.1: Schematic demonstrating the phenomenon of aerodynamic cou-pling between wind turbines that are aligned with the wind.of up to 60 % in the power output of downstream turbines [141], while decreasingannual energy production by 20 to 30 % [30].1.2 Yaw and induction-based turbine repositioning(YITuR)One solution to mitigating the wake effect is wind farm control, which simplydescribes the process of operating individual wind turbines in a manner that raisespower production from the collective. The primary objective of this thesis is toinvestigate the potential of a specific wind farm control approach that is relevant tofloating offshore wind farms and is referred to as yaw and induction-based turbinerepositioning (YITuR).As shown in Figure 1.2, by varying the nacelle yaw angle of a wind turbine,the direction of the aerodynamic thrust force acting on its rotor may be alteredas desired. Further, by adjusting other parameters, such as the pitch angles of theblades and shaft rotation speed, the magnitude of the aerodynamic thrust force maybe manipulated as well. Floating offshore wind turbines that are anchored to theseabed using slack mooring lines possess some finite range of mobility along the2Wind directionThrust forceThrust forceAnchorsMooring line cablesThe wake produced bythe upstream turbineRepositioning the turbines reducesthe wake overlap at the location ofthe downstream machineFigure 1.2: Schematic demonstrating the use of yaw and induction based tur-bine repositioning (YITuR) to minimize wake overlap in floating off-shore wind farms.ocean surface. Variations of thrust forces thus permit relocation of floating plat-forms in real-time. The permissible range of this motion is determined by the equi-librium established between the aerodynamic thrust forces and restoring mooringline tensions acting on each platform. Given this repositioning mechanism, indi-vidual floating wind turbines may be relocated in real-time to minimize the overlapbetween the rotors of downstream machines and the wakes produced by upstreamcounterparts.1.3 Research question and thesis outlineThe research question addressed in this thesis may be stated as follows: Does theYITuR wind farm control technique offer sufficient economic incentive to be imple-mented in practice? And if so, what challenges must be overcome along the roadto industrial implementation?This question is answered through a series of steps. The efforts and resultsassociated with each step are outlined in subsequent chapters. An overview ofthe role of each chapter is provided in Figure 1.3. The chapters are listed in theorder that is most suitable for comprehension; however, each chapter is writtenwithout dependence on content or knowledge contained in other parts of the thesis.3Literature review ofwind farm controlSteady-state floatingwind farm modelingSteady-state optimizationstudy of YITuR+Dynamic floatingwind farm modelingDevelopment ofnovel control theoryReal-time implementionof YITuR Chapter 2Chapter 3Chapter 4Chapter 5 Chapter 6IntroductionChapter 1ConclusionChapter 7Figure 1.3: Layout of the thesis and the role of each chapter.The reader may therefore skip specific chapters that are not of interest withoutrelinquishing any information that is vital for their apprehension. The purpose andcontributions of each chapter in the context of the objective of the thesis are listedbelow:• Chapter 2: The second chapter provides a comprehensive review of windfarm control literature. The current status of the wind energy sector is firstdiscussed, followed by detailed descriptions of the wake effect and variouswind farm control techniques. Next, vital information from existing studies,such as modeling methodologies and reported wind farm efficiency gains, isaccumulated in review tables while the individual studies are summarized.Finally, the collected data are used to offer insight into the gaps in existingresearch practices. This work has been published in the Journal of WindEngineering and Industrial Aerodynamics [102].• Chapter 3: Prior to dedicating effort to the development of dynamic mod-els and controllers, it is necessary to establish, using a relatively simplifiedanalysis, whether turbine repositioning is a beneficial wind farm control ap-proach. The third chapter thus presents a steady-state assessment of the po-4tential benefits of YITuR. This step involves the development of the firststeady-state floating offshore wind farm model that couples platform dis-placement with wake aerodynamics. This simulator is name the FloatingOffshore Wind Farm Simulator (FOWFSim). Then, optimization studiesare carried out for different wind settings and mooring system design pa-rameters to assess the efficiency gains that result from YITuR. The analy-sis demonstrates that turbine repositioning may lead to significant gains inpower production under specific conditions, while also establishing its limi-tations. This work has been published in Ocean Engineering [103].• Chapter 4: In order to assess the performance of a YITuR controller in real-time, the fourth chapter extends FOWFSim to capture floating wind farmdynamics; hence the name FOWFSim-Dyn. To this end, the first paramet-ric dynamic floating offshore wind farm model is developed. Parametricwake models offer computational expediency by utilizing assumptions con-cerning flow behaviour which simplify the governing equations of fluid flow.The trade-off, however, is reduced accuracy in comparison to computationalfluid dynamics solvers. FOWFSim-Dyn predicts floating wind turbine mo-tion along the ocean surface coupled with wake transport. This simulator isalso the first parametric tool to capture the effects of time-varying wind speedand direction on wake behaviour. The model is validated under steady-stateconditions using published experimental results.• Chapter 5: The fifth chapter develops novel theory in the field of distributedeconomic model predictive control1 (DEMPC). The turbine repositioningoptimal control problem is non-convex since there exist multiple trajecto-ries that individual turbines may follow to reduce wake overlap. One majorgap among existing DEMPC algorithms is that, due to non-convexity, theycannot offer a guarantee that the optimal decisions made by the individualturbines will converge. In other words, they offer no guarantee that individ-ual floating turbines will make decisions that ultimately lead to a rise in totalpower production. As a result, the development of novel DEMPC theory ca-pable of providing such a guarantee is necessary for delivering a generalized1The reasons for selecting DEMPC for turbine repositioning are clarified in Chapter 6.5algorithm that implements YITuR.• Chapter 6: The sixth chapter pieces together the contributions of Chapter 4and Chapter 5 by using the novel DEMPC algorithm to implement YITuRin real-time in the simulated environment of FOWFSim-Dyn. The DEMPCalgorithm maximizes wind farm power production by solving dynamic op-timization problems at each time-step. In order to increase the speed ofdecision-making, feed-forward neural networks are used to estimate the dy-namic behaviour of floating wind turbine platforms during the optimizationprocess of DEMPC. These neural networks are tuned using data generated byFOWFSim-Dyn. Therefore, an additional contribution of this chapter is thefirst use of neural networks to predict floating turbine dynamics. Finally, theperformance of the DEMPC algorithm and the benefits of real-time YITuRare assessed for different wind farm sizes and time-varying wind conditions.• Chapter 7: The final chapter concludes the thesis by summarizing the majorfindings from each preceding research step. Additionally, recommendationsregarding future research steps are offered.6Chapter 2A Quantitative Review of WindFarm Control with the Objectiveof Wind Farm PowerMaximizationThis chapter presents a review of control strategies for maximizing power produc-tion within wind farms. Discussions focus on three notable concepts; power de-rating, yaw-based wake redirection, and turbine repositioning. Existing works thathave examined the potential of these concepts via optimization studies, numericalsimulation, experimentation, as well as those that have developed and evaluatedcontrol algorithms, are reviewed thoroughly and quantitatively. Criteria for this re-view process include the evaluation methods employed, simulated wind conditions,controller properties such as model dependency and communication architecture,and the resulting relative rise in wind farm efficiency. The data collected fromexisting literature is then utilized to draw conclusions regarding the influence ofeach of these criteria on the potential and performance of wind farm controllers.Appropriate recommendations for future modeling and controller design researchare then offered based on these conclusions.72.1 IntroductionThis introductory section summarizes the status of the wind energy industry andoutlines the phenomenon of aerodynamic interaction among wind turbines and itseffect on wind farm efficiency. Possible solutions for mitigating the wake effect arediscussed prior to tackling the main subject of this review, which is wind farm con-trol. A brief overview of existing literature surveys on this subject is also presentedas build-up to the objectives of the current chapter.2.1.1 Status of wind energyThe environmental and economic significance of wind power in the 21st centuryis not only palpable to members of scientific and green-tech communities, but isnow mainstream knowledge. What is more, the initial goals and ambitions thathave propelled the wind energy industry since the start of the century are graduallybeing realized. For instance, the primary incentive for investment and technologi-cal progress in the wind energy sector has been to reduce green house gas (GHG)emissions while continuing to meet global electricity demands [13]. From 2000 to2016, the global share of annual electricity production from wind power increasedfrom 0.20 to 3.84 %1 [87, 167]. Based on electricity production methods in theUnited States, the recent end of this range corresponds to 712 million metric tonsof avoided CO2 emissions [57], and accounts for 1.97 % of total global CO2 emis-sions from 2015 [143].In addition to mitigating environmental concerns, investment in wind energyoffers a multitude of social, political, and economic benefits; the diversification ofnational energy portfolios, which strengthens energy security and prevents globalconflicts over natural resources; the reduction of poverty through greater accessibil-ity to low-cost energy; and innovation and job creation [13]. On this final point, theglobal employment count in the wind energy sector rose from 0.75 to 1.16 millionbetween 2012 and 2016 [89]. The global wind energy sector therefore employedmore individuals than the traditional power generation industries (i.e. coal, oil, andgas) of the United States, which collectively employed 1.1 million individuals in1Data on global electricity production from wind were obtained from IRENA [87]. Data onoverall global electricity production were obtained from Enerdata [167].820406080100120140New annual global investment in thewind energy sector (billion USD)2000 2002 2004 2006 2008 2010 2012 2014 2016Year100200300400500600Cumulative installed global windpower capacity (GW)Onshore capacityOffshore capacityAnnual investmentFigure 2.1: Trends in installed global wind power capacity and global invest-ment in the wind energy sector since the start of the century. Datahave been obtained from the International Renewable Energy Agency(IRENA) [87], [90].2016 [173].These environmental and economic benefits, along with government mandatesand support, have generated the substantial levels of growth and investment in thewind energy sector that are shown in Figure 2.1. Cumulative installed global windpower capacity has increased from 16.9 to 513.5 GW from 2000 to 2017 [87].Meanwhile, new annual global investments in the wind energy sector have risenfrom 19.6 to 112.5 billion USD from 2004 to 2016 [90]. These trends are indica-tive of the considerable efforts being made to mature and proliferate wind powertechnology on a global scale.Figure 2.1 also reveals growing interest in locating wind farms offshore, forwhich the motivations are plentiful; environmental impact is limited since the con-struction of roads for transport is unnecessary; visual and auditory concerns aremitigated as turbines are located far from residential areas; obtaining land permitsis less problematic; and most major cities are located close to shore, which permitsshorter transmission lines [121]. Offshore wind is also stronger and less turbulent9due to the absence of obstructions and uneven terrain [8]. Consequently, offshorewind turbines accumulate less fatigue damage and deliver more electricity relativeto their onshore counterparts. On this final point, the global average capacity fac-tor (the ratio of energy produced to the maximum producible amount over a givenperiod) for offshore wind power was 30 % greater than that of onshore wind powerin 2017 [90].Despite these benefits, offshore wind power lacks competitiveness against on-shore wind and fossil fuel-based alternatives due to its infancy and harsh operat-ing conditions [30]. Investment trends and the levelized cost of energy (LCOE)for onshore and offshore wind power are compared in Figure 2.2. In 2017, theglobal average investment cost for offshore wind power per kW of installed ca-pacity was 4,239 USD/kW, while that for onshore wind power was 65.2 % lowerat 1,477 USD/kW [88]. Likewise, the global average LCOE for offshore windpower at this time was 0.14 USD/kWh; the LCOE for onshore wind power was57.1 % lower at 0.06 USD/kWh [88], and that of coal-fired power plants rangedfrom 0.0764 to 0.097 USD/kWh in 2015 [174]. Increased competitiveness is there-fore imperative for the proliferation of offshore wind power, and it is a prioritywithin the wind energy research community [24, 161].2.1.2 Increasing wind farm efficiencyOne approach to raising the competitiveness of a wind farm is to mitigate efficiencylosses resulting from aerodynamic coupling between individual wind turbines [24].The concept of aerodynamic coupling is depicted in Figure 2.3, which shows twowind turbines aligned with the free stream wind. Viscous interaction between theblades of turbine 1 and the incoming wind generates a downstream region of air-flow that is characterized by low velocity and increased turbulence intensity [115].This downstream region is referred to as a wake. Turbine 2, which is aligned withthe wake that is generated by turbine 1, therefore produces less power and accu-mulates greater fatigue damage over time [115]. The overall process describedin Figure 2.3 is commonly referred to as the wake effect, and may reduce powerproduction from individual downstream turbines by up to 60 %, while diminishingpower production from an entire wind farm by as much as 54 % [141]. Annu-101500200025003000350040004500500055006000Global weighted average investment cost per kW ofinstalled wind power capacity (2016 USD/kW)0.020.040.060.080.10.120.140.160.180.20.220.24Global weighted average levelized cost ofwind energy (2016 USD/kWh)2010 2011 2012 2013 2014 2015 2016 2017YearOnshore invest. costOffshore invest. costOffshore LCOEOnshore LCOEFigure 2.2: Trends in levelized energy and investments costs for onshoreand offshore wind power since 2010. Data have been obtained fromIRENA [88].ally, the wake effect may result in cumulative revenue losses ranging from 20 to30% [30].Thus far, three methods have either been proposed or employed to mitigatelosses resulting from the wake effect. The first approach involves siting wind tur-bines such that there exists appropriate spacing between two adjacent machinesin the predominant wind direction [30]. Large inter-turbine spacing permits suffi-cient distance over which the process of turbulent mixing re-energizes wakes andreduces velocity deficits [115]. Although this method has been widely employedwithin the wind energy industry, it conflicts with the fundamental rationale behindwind farms; that is, to cluster wind turbines in close proximity to minimize cap-ital and operational expenses [121]. Furthermore, the 20 to 30 % annual revenuelosses alluded to earlier are observed with current industry-standard inter-turbinespacings of seven to ten rotor diameters in the predominant wind direction [30].There is therefore growing interest in alternative methods that will further mitigatesuch losses.The second solution for increasing wind farm power production is layout opti-11Turbine 1 Turbine 2Uniform free stream wind speed profileViscous interaction at the blades of Turbine 1Reduced wind speed profile at Turbine 2Wake generated by Turbine 1Reduced power output at Turbine 2Figure 2.3: The phenomenon of aerodynamic coupling between two windturbines aligned with the free stream wind.mization. Engineering wake models are used to estimate wind farm power produc-tion as a function of historical wind velocity data spanning the entire year [161].Optimization algorithms are then used to determine ideal turbine siting such thatwind farm annual energy production is maximized [161]. Despite the large bodyof academic work that has been dedicated to layout optimization, the largest oper-ational offshore wind farms possess gridded layouts with appropriate inter-turbinespacing in the predominant wind direction2. The absence of optimized wind farmlayouts within the industry may be due to the novelty of this field of research3, thelow fidelity of engineering wake models [24], or a wind farm developer’s consid-eration of additional factors when designing wind farms, which may include elec-trical connectivity costs, maintenance procedures, and local terrain properties [30].The third strategy for alleviating revenue losses resulting from the wake effect,which is the focus of the current chapter, is wind farm control. In the broadestsense, a wind farm controller uses the available degrees of freedom of individualwind turbines to manipulate the wind field such that a wind farm level objectiveis achieved [24]. This wind farm level objective may be power maximization, or2The reader may investigate the London Array, Greater Gabbard, Bard 1, Anholt, Walney, Thorn-tonbank, Sheringham Shoal, Thanet, Centruica Lincs, and Horns Rev 2 wind farms.3Research into wind farm layout optimization was initiated in 1994 by Mosetti et al. [133].12power set-point tracking with load reduction [24]. Contrary to the aforementionedsolutions to the wake problem, wind farm control does not conflict with the designobjectives of wind farm developers as it requires no adjustment to turbine siting.Additionally, wind farm control may be implemented within existing operationalwind farms, whether they possess a gridded or optimized layout. The demonstratedpotential of such control systems has motivated a 2.7 million USD project, man-aged by the United Kingdom’s Carbon Trust, involving the implementation of windfarm control trials within operational wind farms [34].2.1.3 ObjectiveIn 2015, Knudsen et al. [107] reviewed wind farm control literature with objectivesof power maximization and power tracking. The various optimization and controlalgorithms that have been implemented in the existing literature were briefly dis-cussed, and results related to power gains for these studies were presented. Majorchallenges related to wind farm control were also highlighted. The 2017 tutorial byBoersma et al. [24] provided greater detail regarding the wake effect, wake model-ing, and wind farm control from a practical standpoint. Discussions covered a widearray of topics including actuation, sensors, control strategies, and observer designfor wind farm controllers, as well as wind turbine modeling and control, and wakemodel theory, applicability, and fidelity. These review articles provide adequateinformation for engineers and researchers to gain familiarity with the field of windfarm control. What is lacking in this field is a quantitative assessment and com-parison of published studies that identifies the benefits and drawbacks of specificcontrol techniques and evaluation methods.The objective of the current chapter is to provide such a quantitative reviewof wind farm control literature with focus on wind farm power maximization. Themethods and results presented in existing studies are examined and compared basedon a set of relevant criteria, and then used to draw conclusions regarding the effec-tiveness of control strategies in addition to the influence of evaluation techniquesand simulated wind conditions. Furthermore, the current chapter quantitativelyreviews recent publications involving field tests and high-fidelity simulations thathave not been covered in existing review articles. It is our intention to first pro-13vide engineers and researchers with a readable and focused introduction to windfarm control for power maximization, to then offer a recent and comprehensiveoverview of progress in this field, and to finally present definitive conclusions re-garding controller design and evaluation that will establish appropriate directionsfor future research.The remainder of this chapter is organized as follows; Section 2.2 providesan introduction to various wind farm control concepts and controller properties;Section 2.3 reviews commonly employed aerodynamic models used to simulatewind farms; Section 2.4 represents the major contribution of the current chapter,and provides a detailed review of wind farm control literature; Section 2.5 deliv-ers a quantitative overview that pieces together the major findings from reviewedworks, and draws conclusions regarding the current state of research in the field;and Section 2.6 concludes the chapter with a summary of major findings and rec-ommendations for future paths of research in wind farm control.2.2 Wind farm controlThis section introduces various wind farm control concepts that have been pro-posed and investigated within existing literature. Additionally, properties that havebeen used to distinguish between wind farm controllers are discussed.2.2.1 Control conceptsThree notable wind farm control concepts have thus far been proposed for mit-igating the effects of aerodynamic coupling within wind farms. These conceptsinclude power de-rating, yaw-based wake redirection, and turbine repositioning;all of which are depicted in Figure 2.4.Power de-ratingThe oldest and most prevalent approach within existing literature, which was firstproposed in 1988 by Steinbuch [169], is power de-rating (also commonly referredto as axial induction-based control). The axial induction factor is a measure of themomentum deficit experienced by air flowing past a wind turbine. It is directlyrelated to the power and thrust coefficients of a wind turbine rotor. Referring to14Turbine 1 Turbine 2De-rate Turbine 1 to reduce the thrust force exerted on the windTurbine 2Turbine 1Turbine 2Mean wake velocity is increased at Turbine 2Operate Turbine 1 with nacelle yaw offsetWake is deflected and Turbine 2 is exposed to higher speed windReposition turbines in the crosswind directionThrust forceWake overlap area is reduced and Turbine 2 is exposed to higher speed windTurbine 1a)b)c)Thrust forceYaw offsetFigure 2.4: Overview of three notable wind farm control techniques proposedfor mitigating the wake effect. The schematic shows a top-view of twowind turbines aligned with the incoming free stream wind. Green ele-ments highlight changes in turbine operation and wake conditions asso-ciated with each control concept.15Figure 2.4a, de-rating turbine 1 reduces its axial induction factor, which dimin-ishes power output and also decreases the magnitude of the overall thrust force thatis exerted on the wind by its rotor [95]. This process lessens the momentum deficitwithin the wake that is generated [95]. Turbine 2, which is located further down-stream and is aligned with this wake, is exposed to greater wind speeds as a result,and outputs electrical energy at a higher rate in relation to a baseline scenario inwhich turbine 1 had not been de-rated [95]. This concept is theoretically validsince, with the appropriate scale of power de-rating, the power loss associated withde-rating turbine 1 is outweighed by the power gain of turbine 2 [95].Yaw-based wake redirectionA wind turbine operating with nacelle yaw offset deflects the downstream path ofthe generated wake in the crosswind direction [93, 181]. This phenomenon re-sults from unsteady and asymmetric aerodynamic loads that are generated alongthe blades of a yawed rotor [29]. Due to misalignment of the rotor plane with theincoming wind, blades that are located in the top and bottom halves of the rotorplane experience different aerodynamic loads instantaneously [29]. As a result,load projections along the rotor plane are not in balance. This imbalance impartsforces that cause the wind to gain momentum in the crosswind direction [29]. Vor-tices generated in the wind field by these loads have also been shown to contributeto wake deflection [60].The phenomenon of wake deflection forms the basis of the concept of yaw-based wake redirection. Referring to Figure 2.4b, operating turbine 1 with nacelleyaw offset causes the generated wake to be redirected in the direction of yaw [29].The overlapping area between this deflected wake and the rotor of turbine 2 isreduced as a result, and a portion of the rotor of turbine 2 is exposed to higherspeed wind that is less disturbed by the wake effect [58]. At the optimal nacelleyaw offset, turbine 2 experiences a power output rise that exceeds the power losscorresponding to the operation of turbine 1 with yaw misalignment [58].16Turbine repositioningThe final notable concept that has recently begun to gain attention is reposition-ing of floating offshore wind turbines. Regardless of the mechanism employed,the concept of turbine repositioning involves altering the locations of individualwind turbines within a wind farm in real-time with the aim of minimizing wakeoverlap [24]. Referring to Figure 2.4c, shifting the locations of the two turbines inopposite crosswind directions decreases the wake overlap at turbine 2, and raisesits power output due to exposure to higher speed wind [58]. While the concepts ofpower de-rating and yaw-based wake redirection have received considerable atten-tion from the wind farm control community, publications investigating the potentialof turbine repositioning are limited. This trend stems from the narrow applicabilityof turbine repositioning to the floating offshore wind power sector [24], which isyet in an early stage of development [30]. Additionally, there is currently an ab-sence of wind farm simulation tools capable of incorporating non-stationary windturbines.Active technologies that have been proposed for controlling the position offloating offshore wind turbines include under-water thrusters, which are commonlyemployed by the offshore oil and gas industry [165], winch mechanisms that areattached to floating offshore platforms and anchored to the seabed [153], and po-tentially thermally-actuated sewing thread artificial muscles that utilize electricalcurrent to alter tension within mooring lines [114]. An alternative passive solu-tion is yaw and induction-based turbine repositioning (YITuR), which involvesusing nacelle yaw and blade pitch control to manipulate the magnitude and direc-tion of aerodynamic forces for the purpose of controlling floating platform posi-tions [80, 81]. Due to these changes in yaw angles and thrust force magnitudes, thewind velocity field will also be manipulated in the process. A wind farm controllerbased on YITuR must therefore perform dynamic optimization in consideration ofboth turbine motion and wind field changes.Other conceptsAdditional wind farm control concepts include wake redirection via individualblade pitch control and nacelle tilt. Using nacelle tilt to redirect wakes employs17the same underlying physics that permit yaw-based wake redirection, except wakesare redirected in the vertical direction rather than the crosswind direction [58]. Al-though redirecting wakes using nacelle tilt has shown greater capacity for windfarm power maximization relative to using nacelle yaw [58], the nacelle tilt de-gree of freedom is unavailable in modern wind turbine designs [62]. It is worthmentioning that if an optimal nacelle tilt angle is identified across a spectrum ofpotential wind speeds and atmospheric turbulence intensities, nacelle tilt actuationmay not be necessary for wind farm power maximization. Standard utility-scalewind turbines currently possess a fixed shaft tilt offset to increases clearance be-tween blades and the turbine tower [29]. This shaft tilt offset may be fixed at apredetermined optimal value for maximizing wind farm power production.Individual blade pitch control may be used to redirect wakes by generating anuneven distribution of drag and lift forces across the area spanned by a wind tur-bine rotor [62]. The difficulty in realizing such a controller is that individual bladepitch control is conventionally used by turbine-level controllers for providing in-dependent aerodynamic rotor braking and reducing asymmetrical blade loads [29].As a result, an effective wind farm controller based on individual blade pitch con-trol must simultaneously consider these turbine-level objectives in addition to wakeredirection for wind power maximization [62]. This requirement yields an inter-esting yet challenging problem which has consequently subsided research on thisspecific control concept.2.2.2 Controller propertiesProperties of importance when assessing and designing wind farm controllers in-clude their closed-loop structure, which governs requisite measurements and con-trol signals, their dependency upon analytical or estimated wake models, the al-gorithms they use for characterizing the wind field, and the scope of permittedcommunication among wind farm level control units. Understanding these proper-ties and their effects is necessary for justifying differences in the performances ofdifferent wind farm controllers.18Measurements- Turbine power outputs- Nacelle yaw angles- Local wind speeds- Local wind directionsTurbine set-points- Power outputs- Nacelle yaw anglesDisturbances- Free stream wind speed- Free stream wind direction- Atmospheric turbulence Wind farm- Wind turbine dynamics- Wind turbine control- Fluid-structure interaction- Wake propagation- Terrain effectsWind farm controller- Wind field estimation- State estimation- System identification- Dynamic optimizationFigure 2.5: The closed-loop feedback structure of a general control systemfor wind farm power maximization. Listed measurements and turbineset-points do not encapsulate all possible signals; they are examples ofcommonly used data for wind farm control.Closed-loop structureBased on the studies reviewed in this chapter, a general closed-loop structure forwind farm controllers has been formulated and is presented in Figure 2.5. Theplant to be controlled is the wind farm, whose dynamic behavior is a functionof several underlying physical subsystems; wind turbine dynamics and control,which govern the motions of individual wind turbine components in response toaerodynamic and hydrodynamic loads; fluid-structure interaction, which influencesthe magnitudes of these loads, as well as their local effects on the wind field; andwake transport, which affects temporal and spatial changes in the wind field thatresult from mass and momentum conservation within a turbulent flow field. Thisflow domain contains mixing co-flowing regions and is subject to terrain effects.The velocity and turbulence intensity of the free stream wind are regarded asdisturbances acting on the plant since these parameters are undisturbed by wind tur-bines and are therefore considered as uncontrollable inputs. In general, wind farmmeasurements include local wind velocities and magnitudes of control degrees offreedom associated with individual wind turbines. Among studies pertaining towind farm power maximization, these degrees of freedom are commonly turbine19power outputs (or axial induction factors, which are directly related to the powercoefficients of ideal actuator discs) and nacelle yaw angles, depending on the con-trol concepts being implemented. Few studies in this research area are concernedwith turbine-level control degrees of freedom such as blade pitch angles, tip-speedratios, or generator torque values. It is preferential for wind farm controllers tocompute power output set-points while individual wind turbine controllers deter-mine the optimal combinations of these turbine-level degrees of freedom that de-liver the desired power output at minimal mechanical loading.Regardless of specific wind turbine measurements, wind farm controllers solvedynamic optimization problems in order to identify optimal turbine set-points thatdynamically manipulate the wind field in such a way that the wake effect is mit-igated and power losses are minimized. These set-points, which commonly en-compass power outputs and nacelle yaw angles of individual wind turbines, arethen tracked by wind turbine-level controllers. In addition to dynamic optimiza-tion, wind farm control systems may encompass wind field estimation algorithmsfor conditioning wind field measurements [10], correcting wake model inaccura-cies [69], and estimating wake model parameters [72, 186]. These topics are brieflyreviewed in Section 2.2.2.Model dependencyIn solving a dynamic optimization problem, wind farm controllers may utilizemodel-based or model-free approaches. In the case of model-based control, forwhich the general feedback structure is shown in Figure 2.6, dynamic optimizationis performed on a wind farm model (which may be dynamic or steady) at each timestep in order to determine optimal turbine set-points. The wind farm model may beanalytically derived based on integral relations of fluid mechanics and assumptionsregarding wake expansion [98], or its parameters may be estimated using mea-sured data [72]. Machine learning methods such as neural networks may also beused for predicting wake behavior and wind turbine power production [132, 186].Furthermore, model predictions may be corrected using state estimators such asKalman filters [49]. Model-predictive control is commonly employed for model-based wind farm controller design due to its compatibility with constrained dy-20Dynamic optimizationMeasurementsTurbine set-pointsWind fieldestimationWind farmmodelSimulated turbine set-points Model parametersSystemidentificationStateestimationEstimated wind conditionsSimulated statesCorrectedstatesDisturbancesCorrectedstatesFigure 2.6: The closed-loop feedback structure of a model-based wind farmcontroller with wake model parameter and state estimation, and windfield estimation functions.namic optimization.Contrarily, model-free control involves dynamic optimization driven purely bymeasured data with no dependency on analytical or estimated wind farm mod-els [24]. Such algorithms iteratively or continuously probe systems and computefuture control actions based on observed changes in some objective function. Com-mon implementations within wind farm control literature include gradient-ascentschemes, game theory, and extremum-seeking control. The rising popularity ofsuch controllers within the wind farm control community is often attributed to thechallenge of accurately modeling aerodynamic interaction within wind farms usingcontrol-oriented computationally inexpensive wake models [122]. Additionally,using higher-fidelity computational modeling techniques such as large eddy sim-ulations (LES) is infeasible for online dynamic optimization due to the excessivecomputational resources required [77]. Model-free control should not be conflatedwith data-driven control. A data-driven controller, which may be model-free ormodel-based, computes control actions either directly from measured data or via a21model estimated using such data [155].Wind field estimation algorithmsWhether a wind farm controller uses a Kalman filter to correct wake model inac-curacies or system identification techniques to estimate wake model parameters,accurate measurements of wind speed and direction across wind farms are nec-essary [33]. The challenge presented here is that these measurements cannot beused directly and require conditioning. For instance, readings from wind vanesand anemometers mounted atop wind turbine nacelles are distorted by turbulenceand the effects of moving blades on the nearby wind field [82]. In the absence ofLiDAR technology and dedicated wind masts, appropriate algorithms are requiredfor estimating wind conditions at different wind farm locations [10]. These algo-rithms are represented by the wind field estimation block in Figure 2.6. Althoughnot the focus of the current chapter, wind field estimation is a broad and active areaof research with significant implications in the fields of both wind turbine and windfarm control.One recent effort implementing wind field estimation at the wind farm levelis presented by Annoni et al. [10], who estimated the wind direction at the lo-cation of any wind turbine within a wind farm using wind vane measurementsfrom the turbine itself and from those of its neighbors. Knowledge of local winddirection is crucial for identifying which pairs of wind turbines interact aerody-namically. Another example by Bottasso et al. [28] used blade bending momentsto estimate wind speeds within different regions along a wind turbine rotor. Thistechnique was then implemented experimentally in wind tunnel tests by Schreiberet al. [158] and Campagnolo et al. [33] to estimate the centerline location of a wakethat was overlapping with a downstream turbine rotor. Estimates of this centerlinewake location may be used to tune or correct parametric wake models intended forsimulating wake redirection resulting from yaw misalignment.Regardless of the means of obtaining wind speed and direction estimates, thisinformation may ultimately be used for tuning wake model parameters as sug-gested by Gebraad et al. [72], training neural networks as performed by Yan [186],or correcting wake model inaccuracies using state-estimators as presented by Ge-22braad et al. [69]. Data obtained from wind field estimation may also be used forjoint parameter-state identification. Doekemeijer et al. [49] designed state andparameter-state estimators for the medium-fidelity Wind Farm Simulator (WFSim)using various forms of Kalman filters. These estimators corrected wind speedpredictions across the flow domain while identifying WFSim parameters. Theyevaluated their algorithms by simulating a 3×3 wind farm using the high-fidelitySimulator for Wind Farm Applications (SOWFA); the results from these simula-tions were treated as true measurements. Upon implementing Kalman filters, theyreported convergence between wind speeds predicted by the two models within300 sec. Beyond this point, errors between calibrated WFSim and SOWFA predic-tions were predominantly less than 1 m/s across the flow domain through which thefree stream wind speed was 8 m/s.Communication architectureThe final property of interest is the communication architecture of a wind farm-level controller, which may be centralized or distributed. A comparison of thesearchitectures is provided in Figure 2.7. In a centralized control system, the windfarm controller receives measured wind velocity and control degree of freedommagnitudes from all wind turbines, and likewise transfers control set-points to allmachines [122]. In a distributed wind farm control system, there is no centralizedwind farm controller. Instead, the set-points of individual wind turbines are regu-lated by their respective distributed control units [70]. In the specific arrangementshown in Figure 2.7b, each turbine’s wind farm-level control unit is permitted tocommunicate with those of the turbine’s immediate neighbors. Alternatively, com-munication may be extended to an arbitrary collection of neighbors [122], or it maybe further narrowed to solely a single downstream or upstream neighbor [70].The motivation for developing distributed wind farm controllers is to reducethe size and complexity of the dynamic optimization problem being solved by anygiven control unit. Reducing the number of optimization variables enhances com-putational time-efficiency for each control unit; however a global optimal solutionis less likely to be reached in comparison to using centralized control systems [70].An additional benefit to developing distributed controllers is that they deliver a23Distributed control unitDistributed wind farm controller Centralized wind farm controller Measurements Turbine set-pointsa)b)Distributed control unitDistributed control unitDistributed control unitFigure 2.7: Comparison between the closed-loop feedback structures of (a)centralized and (b) distributed wind farm controllers.generalized solution that may be implemented in wind farms of any size withoutaltering computational efficiency and structure.2.3 Wind farm modelingEvaluating the performance of control systems on operational or experimental windfarms is an expensive and time-intensive task. Appropriate models for simulatingwind farm aerodynamics are therefore necessary at the early stages of controllerdevelopment [24]. Typically, such wake models are categorized along a spectrumranging from low to high fidelity based on their numerical accuracy in predictingwind farm and wind turbine power outputs [24]. The intention of this section isnot to review wake modeling literature in detail, but rather to provide the readerwith sufficient knowledge regarding the simplifying assumptions, computational24requirements, and accuracy of simulation tools that are commonly used to evalu-ate wind farm controllers. A summary of these wake models is provided in Ta-ble A.1. More comprehensive reviews of existing wind farm models are presentedby Boersma et al. [24], Go¨c¸men et al. [75], and Vermeer et al. [179].2.3.1 Low-fidelity wake modelsLow-fidelity wake models are based on integral relations of fluid mechanics, wherethe rates of change of fluid momentum and mass must be conserved across a spec-ified control volume [56]. Such models are often referred to as kinematic, en-gineering, or parametric wake models, and they are best suited for offline staticoptimization and online dynamic optimization due to their simplicity and econom-ical computational requirements [24]. The primary drawback of these models isthat they provide no detail regarding velocity and pressure gradients within controlvolumes [56]. The phenomenon of wake expansion, which is the physical result ofturbulent mixing between flow regions internal and external to wakes, is dependentupon velocity and pressure gradients within the fluid domain [115]. As a result,wake models based on integral relations rely upon parametric assumptions con-cerning the rate of wake expansion or the velocity distribution within wakes [24].The earliest and simplest implementation of integral relations for constructinga wake model was presented by Jensen [91] in 1983. The methodology and as-sumptions inherent to this model are shown in Figure 2.8. Assuming uniformlydistributed incoming flow that is subject to a uniform axial thrust force (actuatordisc theory [121] is invoked here), the wind speed immediately downstream of therotor is reduced by an amount that respects linear momentum conservation acrossthe rotor. Upon defining a cylindrical control volume whose length spans from therotor to some arbitrary downstream distance and whose diameter is equal to thewake diameter at this downstream location, the uniform wind speed at this loca-tion may also be found via linear momentum conservation. A major assumptionnecessary for this final step is a linearly expanding wake with a known expansionconstant. A second assumption is zero radial flow through the cylindrical boundaryof the control volume.The Jensen wake model has since been modified and extended for improved25Uniform free stream wind speed profileReduced wind speed profile at an arbitrary downstream distanceThrust forceControl volume boundaryReduced wind speed profile immediately past rotor resulting from thrust forceLinearly expanding wake boundaryFigure 2.8: The control volume analysis used for computing momentum con-servation in parametric wake models that are based on integral relationsfrom fluid mechanics.accuracy and the capability to capture additional aerodynamic phenomena. Katic´et al. [98] modeled the interaction of multiple wakes by assuming that the kineticenergy deficit at any point in the wind field is equal to the sum of kinetic energydeficits of all overlapping wakes. As a result, the velocity deficit at any point isequal to the root-sum-square of pertinent wake velocity deficits. This extendedmodel was named the Park wake model. Despite its simplicity and the absenceof ground effects and wake meandering, the Park wake model has been shown topredict experimental wind farm power production with relative errors ranging from0.1 to 15.9 % across a range of wind directions [98].Gebraad et al. [72] incorporated the effects of nacelle yaw on the evolutionof wakes. An analytical derivation by Jime´nez et al. [93] was used to calculatethe magnitude and direction of wake centerline deflection as a function of the na-celle yaw angle and axial induction factor. Additionally, while the Jensen wakemodel assumed uniform velocity distributions within wakes, Gebraad et al. [72]segmented wakes into three concentric zones to better predict velocity distribu-tions. Finally, the parameters of this wake model were identified using high-fidelitysimulation results obtained from SOWFA. This newly developed simulation toolwas named the Flow Redirection in Steady-state (FLORIS) model and was re-26cently released as standalone software by the National Renewable Energy Labo-ratory (NREL) for controller design and evaluation research. Recent releases ofthis software have incorporated Gaussian velocity profiles [17] and vorticity ef-fects [124]. FLORIS has been shown to match wind farm power predictions fromSOWFA with relative errors ranging from 0.1 to 5.3 % across various wind direc-tions and turbine yaw settings [72].Gebraad and van Wingerden [67] also introduced the Flow Redirection andInduction Dynamics (FLORIDyn) model, which extended FLORIS by includingdynamic wake transport effects. As the nacelle yaw angle or axial induction factorof an upstream turbine are altered, and the generated wake is altered accordingly, acertain period of time must pass before this change in wake behavior is transportedto a downstream machine. The FLORIDyn model considers this spatial and tempo-ral delay by transporting steady-state wake solutions in the downstream direction.The absence of partial differential equations results in a dynamic control-orientedsimulation tool for dynamic wind farm optimization.The process of wake transport used in FLORIDyn is visualized in Figure 2.9.Assuming that a wind turbine has been facing the free stream wind with zero na-celle yaw for a long period of time, the steady-state wake profile will exist as shownin Figure 2.9a. Then, assuming that the wind turbine instantaneously yaws counter-clockwise at time t, a deflected steady-state wake profile will result. In FLORIDyn,this deflected wake profile does not instantaneously take form downwind of the tur-bine. Instead, the downstream region is gradually updated with the deflected wakeprofile according to the distance traveled by the wake over an elapsed time-span.The wake profiles shown in Figure 2.9b and Figure 2.9c show the distance traveledby the deflected wake after elapsed time-spans of ∆t1 and ∆t2. Simulation toolsthat utilize transport delays in such a manner are categorized as quasi-dynamicwake models.Gebraad [67] validated FLORIDyn against dynamic SOWFA simulations con-ducted over a 600 sec time-span with turbine yaw settings that were adjusted mid-simulation. FLORIDyn results deviated from SOWFA turbine power predictionswith root-mean-square error (RMSE) values ranging from 7.9 to 11.7 % across dif-ferent wind turbines, whereas FLORIS results deviated from SOWFA’s with RMSEvalues ranging from 11.1 to 14.4 %. Approximating wake dynamics by transport-27a)b)c)Steady wake profile generated by yawed rotorSteady wake profile generated by rotor with zero yawAssume that the nacelle is instantaenously yawed at some time tDownstream distance over which the deflected steady wake profile has been transported at some time t + t2Wake transport velocityWake transport velocityDownstream distance over which the deflected steady wake profile has been transported at some time t + t1Figure 2.9: A demonstration of steady-state solutions being transported in thedownstream direction as a simplified means of approximating dynamicwake transport.28ing steady-state wake solutions in the wind direction therefore reduced absolutesimulation error by as much as 6.5 %4.Our recent work [101] extended FLORIS to model platform motion in float-ing offshore wind farms. The modified simulation tool was named the FloatingOffshore Wind Farm Simulator (FOWFSim). The wind farm was represented by asystem of particles that were subject to aerodynamic thrust and mooring line forces.Similar to FLORIS, aerodynamic thrust forces were computed using actuator disctheory. Mooring line tensions were determined by solving the differential equa-tions describing a static catenary that is either fully elevated or partially restingalong the seabed. Cable stretch and seabed contact friction were also consideredin this calculation. Finally, steady-state wake aerodynamics were captured usingFLORIS.2.3.2 High-fidelity wake modelsHigh-fidelity wake models rely upon differential relations of fluid mechanics. Inparticular, they solve various forms of the turbulent Navier-Stokes equations, whichare a set of partial differential equations that describe fluid momentum and massconservation based on assumptions regarding turbulence modeling [56]. Unlikeintegral relations, solutions to differential relations provide details regarding ve-locity and pressure gradients within fluid domains [56]. As a result, phenomenathat are dependent upon this information, such as turbulent mixing, wake expan-sion, and ground effects, are inherently captured by the partial differential equa-tions of motion; no parametric assumptions are necessary. The primary drawbackof higher-fidelity models is increased computational cost, since simulations mayrequire from 103 to beyond 106 states, which render these tools less practical orinfeasible for optimization and control [24].High-fidelity wind farm models generally employ LES, which solve tempo-rally and spatially-filtered forms of the three-dimensional Navier-Stokes equationsthat only capture eddies of relevant scale. One prominent high-fidelity simulationtool that is commonly used for wind farm controller evaluation is SOWFA, whichis publicly available via NREL [40]. SOWFA couples an LES-based flow field4Potential reduction in absolute simulation error was obtained from Figure 5a by calculating themaximum reduction in RMSE obtained for any single turbine’s power production results [67].29model with NREL’s Fatigue, Aerodynamics, Structures, and Turbulence (FAST)code, which models flexible multi-body wind turbine dynamics using the princi-pal of virtual power (or Kane’s method). The aerodynamic interaction betweenthe wind and turbine blades is modeled using actuator line potential flow theory,which captures additional flow phenomena such as blade root and tip vortices incomparison to LES-based tools that model turbines using actuator disc theory; al-beit at a higher computational cost with greater mesh density requirements. Wanget al. [182] compared SOWFA turbine power predictions to wind tunnel experi-mentation results and calculated relative discrepancies ranging from 0.2 to 14.0 %across various turbines with different yaw settings.Other three-dimensional LES-based simulators developed specifically for windfarm applications include SP-Wind [7] and University of Texas at Dallas-WindFarm (UTD-WF) [123]. SP-Wind reduces the computational cost of simulations bymodeling wind turbines using actuator disc theory, whereas UTD-WF offers bothactuator disc and actuator line modeling options. Martı´nez-Tossas et al. [123] com-pared LES-based wind farm modeling using actuator disc and actuator line modelsand showed that power predictions from the two methods relatively differ by lessthan 1.0 %, and that differences in velocity profiles are less pronounced in far-wakeregions. Additional general-purpose computational fluid dynamics (CFD) solversthat have been used in wind farm control literature include EllipSys3D [166] andSTAR-CCM+ [168].2.3.3 Medium-fidelity wake modelsMedium-fidelity models solve simplified forms of the turbulent Navier-Stokes equa-tions in order to reduce computational complexity at the cost of neglecting cer-tain flow physics. WFSim, developed by Boersma et al. [23, 25], solves a two-dimensional form of the unsteady turbulent Navier-Stokes equations along a hor-izontal plane located at the hub height of the wind turbines within a wind farm.Actuator disc theory is used to model thrust forces exerted on the wind by tur-bine rotors, and a mixing length turbulence model is used to compute momentumtransfer via Reynolds stresses. The assumption of two-dimensional flow does notcapture ground effects and wake rotation, which has been shown to influence wake30centerline deflection [60, 72]. Nevertheless, Boersma et al. [23, 25] validated WF-Sim against SOWFA simulation results and showed that the medium-fidelity wakemodel provides a reasonable approximation of wake centerline deflection, whichrenders the tool useful for wake-redirection control studies. It was additionallyshown that WFSim predicts spatially-averaged wake velocity profiles with up to an82.1 % variance accounted for (VAF) match to SOWFA results.The Ainslie wake model [4] solves the steady turbulent Navier-Stokes equa-tions in cylindrical coordinates with the primary assumption that flow within awake is axisymmetric with zero circumferential velocities. Additional propertiesof this model include the assumption of thin shear layers, which eliminate vis-cous terms from the momentum conservation equations, and the usage of an eddyviscosity turbulence model. Wake meandering is also accounted for by using a cor-rection factor. Similar to WFSim, the Ainslie model captures neither wind shearnor wake rotation, and is not capable of modeling wake redirection due to yaw mis-alignment. When validated against wind tunnel experimental results, this medium-fidelity wake model predicted wake centerline velocity deficits with relative errorsranging from 0.5 to 8.4 % across different instances in time.FarmFlow, which was developed by the Polytechnic University of Madrid [45],solves the parabolized form of the steady turbulent three-dimensional Navier-Stokesequations with turbulent transport coefficients computed using the k-ε turbulencemodel. A number of assumptions were made to render the flow problem parabolic,particularly the omission of streamwise diffusion5, which eliminated the influenceof downstream fluid conditions on upstream points in the fluid domain. This typeof flow problem with one-directional influence, similar to an elementary boundarylayer problem, is solvable using a computationally-inexpensive marching proce-dure [154]. The assumption of negligible streamwise diffusion is only valid inflows where boundary layer separation in the streamwise direction is insignifi-cant [154]. Upon validation against wind tunnel experimental results, FarmFlowpredicted maximum wake velocity deficits with relative errors ranging from 0.2 to26.9 % across different tip-speed ratios and downstream distances [45].5In the Navier-Stokes equations, streamwise diffusion refers to second-order velocity gradientsin the wind direction.312.4 Review of wind farm control literatureThis section provides a detailed review of studies that have contributed to windfarm control research. Each of the three notable wind farm control concepts dis-cussed in Section 2.2.1 are reviewed separately. Furthermore, referenced worksare grouped into two categories; those that assess the steady-state potential of windfarm control concepts by conducting static optimization studies, numerical sim-ulations, or experiments; and those that develop dynamic wind farm controllerscapable of responding to real-time changes in wind conditions. Two subsectionsare therefore dedicated to each control concept; the first section concerns its steady-state potential, while the second details implemented controllers and dynamic op-timization algorithms.Summaries of all referenced studies are presented in Table A.2–Table A.6,while only works with significant findings are reviewed in detail to maintain brevity.These tables list properties of significance that are used to draw quantitative con-clusions regarding controller performance and evaluation. These properties includeoptimization or control algorithms, input parameters, models used for optimiza-tion or control, models used for evaluation of optimal settings or controller perfor-mance, simulated wind conditions, and the resulting relative increase in wind farmpower efficiency. Each listing under the Relative efficiency gain heading is accom-panied by an endnote that highlights the origin of the result in the cited publication.The listings in all tables are sorted in order of increasing fidelity of the model ormethod used for evaluating controller performance or optimization results.2.4.1 Steady-state potential of power de-ratingStudies evaluating the steady-state potential of power de-rating are summarized inTable A.2. A large number of studies have used low and medium-fidelity wakemodels (particularly the Park wake model) to determine optimal wind turbine set-points for power de-rating under steady wind conditions. These works may bedistinguished based on the optimization scheme used and the turbine inputs thatare varied, which may consist of tip-speed ratios λ and/or collective blade pitchangles β , axial induction factors a, thrust coefficients CT , power coefficients CP,turbine power output targets P, or generator torque values Tgen. The general con-32sensus that may be drawn from literature based on parametric modeling is thatpower de-rating offers the potential to increase the instantaneous efficiency of awind farm by up to 25 %, with no risk of efficiency loss. As higher fidelity wakemodels are employed however, and as additional aerodynamic phenomena such asturbulence and time-varying wind velocities are considered, the range of poten-tial wind farm efficiency gains narrows or attains negative values (i.e. wind farmefficiency decreases relative to baseline control scenarios6).Annoni et al. [9] delineated the discrepancies observed between results ob-tained using steady parametric wake models and high-fidelity LES-based solvers.They used both FLORIS and SOWFA to simulate a two-turbine wind farm subjectto constant mean wind speed; SOWFA simulations considered a turbulence inten-sity of 6 %. A grid search optimization scheme applied to the steady Park modelwas used to calculate the optimal axial induction factor for the upstream turbine.This value was then implemented in FLORIS directly, while being converted intocollective blade pitch angle adjustments for SOWFA. Simulation results showedthat while the parametric FLORIS model predicted a wind farm efficiency gain of24.82 %, SOWFA results indicated an efficiency loss of 9.35 %; this discrepancywas attributed to two phenomena. First, when an upstream turbine is de-rated, theincreased kinetic power in the wind is predominantly concentrated along the wakeboundary, which lies beyond the area swept by wind turbine rotors located directlydownstream. As a result, these downstream wind turbines do not benefit whentheir upstream neighbors are de-rated. Parametric wake models fail to capture thisphenomenon due to their assumption of uniform velocity distribution within sim-ulated wake zones. Second, reducing the thrust force exerted on the wind by anupstream wind turbine decreases the turbulence intensity within its wake, which inturn diminishes the rates of wake expansion and momentum recovery. Parametricwake models fail to reproduce this observation as they rely upon wake expansionconstants that are independent of turbine thrust forces.Dilip and Porte´-Agel [48] further validated the second conclusion drawn byAnnoni et al. [9]. They used an LES-based solver coupled with a rotating actua-6The baseline control scenario that is universally used in wind farm control literature is referred toas greedy control. This control scenario involves each turbine maximizing its own power productionlocally without consideration of its effects on the wind field or its neighboring turbines.33tor disc model to simulate a two-turbine wind farm subject to constant mean windspeed and different turbulence intensities. The collective blade pitch angle of theupstream turbine was varied by increments of 1 deg in the direction of both featherand stall. It was shown that power de-rating by pitching to stall (i.e. increas-ing thrust) was preferential over pitching to feather (i.e. decreasing thrust), sincepitching to stall increased wake turbulence and resulted in faster wake expansionand momentum recovery further downstream. Despite this observation, conclu-sions regarding the potential of power de-rating as a wind farm control techniqueechoed those of Annoni et al. [9]. At the lower simulated turbulence intensity of4.3 %, a maximum wind farm efficiency gain of 2.8 % was observed, while thisvalue reduced to 0.5 % at the higher turbulence intensity of 6.3 %. The larger levelof atmospheric turbulence offset the benefits of pitching the blades of the upstreamturbine in the direction of stall. This outcome upheld the long-established under-standing that downstream wake behavior is primarily influenced by atmosphericturbulence, not turbulence generated by viscous interaction [115].Other studies employing CFD to assess power de-rating potential offer simi-lar conclusions regarding its efficacy. Santoni et al. [156] used UTD-WF and thesteady Park model to simulate a three-turbine wind farm subject to constant meanwind speed with an unspecified level of atmospheric turbulence. The steady Parkmodel was used to determine the optimal tip-speed ratios of the turbines for max-imizing the wind farm power output. The wind farm’s performance with theseoptimal set-points was then evaluated using both the steady Park model and UTD-WF. The former parametric model predicted a wind farm efficiency gain of 5.7 %,while a value of 0.2 % was obtained using UTD-WF. Nilsson et al. [141] usedEllipSys3D, an LES-based simulation tool that models wind turbine rotors usingactuator disc theory, to simulate power production from the Lillgrund wind farmsubject to constant mean wind speed and a turbulence intensity of 5.7 %. The windplant contained 48 wind turbines organized in a gridded layout. The front row tur-bines were de-rated by altering their collective blade pitch angles by increments of2 deg. It was shown that these adjustments resulted in wind farm efficiency lossesranging from 2.2 to 7.7 %. Across all simulated blade pitch angles, the poweroutput rise of the second-row wind turbines was superseded by the power loss as-sociated with pitching the blades of the front row turbines.34Kazda et al. [99] used MULTI3, an in-house CFD code based on the Reynolds-Averaged Navier-Stokes (RANS) equations and the k-ω turbulence model, to sim-ulate aerodynamic interaction among four wind turbines within the Mont Crosinwind farm. A steady mean wind speed was simulated with a turbulence intensity of13 %. The four wind turbines were situated in a single row, except the two trailingmachines were shifted in the crosswind direction by an unspecified amount. Thisstudy was then unique in the sense that not all wind turbine rotors were aligned withthe simulated wind direction. When assessing the effect of power de-rating withinthe upstream pair of turbines, altering the collective blade pitch angle and tip-speedratio of the leading machine raised the pair’s efficiency by 2.5 %. When consider-ing all four machines however, the wind farm efficiency increased by 9.7 %. Thislarge rise in efficiency may be attributed to the crosswind misalignment betweenthe trailing pair of turbines and the leading turbine, which offers credence to con-clusions drawn by Annoni et al. [9]. The trailing pair of turbines benefited fromthe increased kinetic power along the wake boundary of the leading turbine, sincethey were not directly aligned with the rotor of this leading machine.Similar to high-fidelity simulations, experimental wind tunnel tests using scaledwind turbines have shown less potential for power de-rating in comparison to low-fidelity parametric simulations. Corten and Schaak [43] examined a 3× 8 windfarm containing scaled wind turbines with rotor diameters of 25cm subject to con-stant mean wind speed. The effect of power de-rating was assessed by adjustingthe collective blade pitch angles of the leading two columns of turbines. A windfarm efficiency gain of 4.6 % was produced, with the most significant rise in poweroutput corresponding to the second column of three turbines. Although this ef-ficiency gain is sizable relative to predictions obtained from high-fidelity simula-tions reviewed in this chapter, it was acknowledged that the low Reynolds numberof the blades may have inflated power output measurements. McTavish et al. [127]showed experimentally that reduced Reynolds numbers in scaled wind tunnel testsdecrease the rate of wake expansion, which produces smaller wake diameters atthe locations of downstream turbines. Coupling this phenomenon with observa-tions reported by Annoni et al. [9], the added kinetic power attained from powerde-rating, which is concentrated along wake boundaries, will be clustered closerto the swept rotor areas of downstream machines. The relative efficacy of power35de-rating is therefore likely to be overestimated in scaled wind tunnel tests.Other experiments have yielded similar efficiency gains. Machielse et al. [118]performed experiments on the same wind tunnel setup with a single row of sevenscaled wind turbines. A maximum wind farm power gain of 3.46 % was obtainedby pitching the blades of the leading two turbines by 2.5 deg in the direction offeather. Adaramola and Krogstad [2] performed wind tunnel experiments on atwo-turbine wind farm containing scaled wind turbines with rotor diameters of90 cm subject to constant mean wind speed. Maximum wind farm efficiency gainsof 3.26 and 3.81 % were attained by increasing the tip-speed ratio and collectiveblade pitch angle of the upstream turbine, respectively. The Reynolds numberbased on blade chord length was an order of magnitude lower than typical valuespresent among utility-scale wind turbines.Campagnolo et al. [31] experimented with a wind farm containing three wind-aligned scaled turbines with rotor diameters of 1.1 m subject to constant mean windspeed with a turbulence intensity of 2 %. They reported a wind farm efficiencygain of 0.9 % when considering only the leading two turbines, and no improve-ment when including the third turbine in their calculations. The poor performanceof the third wind turbine was attributed to possible wall interference. Bartl and Sæ-tran [16] conducted wind tunnel experiments using two scaled wind turbines withrotor diameters of 90 cm subject to constant mean wind speed and turbulence in-tensity that decayed from 10 to 5 % across the flow domain. They observed no im-provement in wind farm efficiency when power de-rating was implemented usingeither tip-speed ratio or blade pitch adjustments. These findings were attributed tothe observation reported by Annoni et al. [9] regarding the concentration of addedkinetic power beyond the rotor swept area of the downstream turbine.Schepers and van der Pijl [157] and Boorsma [26] conducted field tests ontwo 2.5 MW wind turbines located in a research wind farm operated by the En-ergy Research Center of the Netherlands (ECN). The former study examined theeffects of noise reduced operation on the total power output of two machines. Thismethod of operation decreased the collective blade pitch angle and tip-speed ra-tio of the upstream turbine, which reduced its axial induction factor, albeit not byan amount that was optimal for wind farm efficiency gains. Results showed thatpower losses from the upstream turbine were recovered by the downstream ma-36chine, which validated power de-rating as a control concept, yet no demonstrablerise in wind farm power output was observed. The latter work varied the collectiveblade pitch angle of the upstream machine to simulate a reduction in axial induc-tion, yet also produced inconclusive results. The absence of efficiency gains wasattributed to several challenges in conducting field tests; varying wind direction,which rendered power de-rating ineffective as a wind farm control concept whentwo turbines were not nearly aligned with the wind; data collection from differentperiods in time, which introduced errors resulting from differing atmospheric con-ditions; and time-varying wind conditions that introduced transience in power andvelocity measurements, despite the partitioning of data into bins based on windspeed and direction.2.4.2 Controllers developed for power de-ratingA list of studies that have developed wind farm controllers based on the conceptof power de-rating is provided in Table A.3. Abbes and Allagui [1], Gionfra etal. [73], Heer et al. [83], Kim et al. [106], and Kazda et al. [100] developed model-based power de-rating controllers that solved static offline optimization problemsusing low and medium-fidelity wake models. These controllers were then evalu-ated using similar wake models. Static optimization algorithms were implementedat specific wind speeds and directions to compute optimal wind farm set-points,which were then transferred to and dynamically tracked by individual wind tur-bine controllers. These control systems increased wind farm efficiency by up to4% across simulations considering turbulence and time-varying wind velocities.These efficiency gains are substantially lower than the potential gains predicted bystatic optimization studies discussed in Section 2.4.1, which reached 25% and alsoemployed low and medium-fidelity wake models. Furthermore, Kim et al. [106]demonstrated that substantial levels of atmospheric turbulence could yield effi-ciency losses of up to 2.88%. Therefore, in addition to the shortcomings of powerde-rating reviewed in Section 2.4.1, the dynamic tracking of optimal set-points byindividual wind turbine controllers further deteriorates the potential of this windfarm control concept, even with the presumption that these optimal turbine set-points are known for all wind speeds and directions without the need for online37optimization.Alternatively, model-free controllers have also been implemented to computeoptimal wind farm set-points using real-time dynamic optimization. Marden etal. [122] developed a centralized game-theoretic dynamic optimization algorithmfor power de-rating. At each time-step, this algorithm perturbed the axial inductionfactors of individual wind turbines relative to preassigned baseline values usingsome probability distribution function. If the wind farm power output increased inresponse to these perturbations, then the baseline power output set-points were setequal to the perturbed values; otherwise, the baseline values were maintained. Adistributed game-theoretic algorithm was also implemented in which each controlunit had limited access to information from neighboring turbines only. This algo-rithm assigned moods of either content or discontent to each turbine, and optimumoperation was reached when all turbines were content. In order for any turbine tomaintain a content mood, it would have to sustain high power production, while itsneighbors also remained content. The centralized controller was evaluated by sim-ulating the Horns Rev wind farm (8×10) using the steady Park wake model. Un-der steady wind conditions and no turbulence, this controller increased the relativewind farm efficiency by 34.05%. At least 103 iterations were necessary however,which is indicative of the limitations that such an algorithm will encounter undertime-varying wind conditions. A wind-aligned row of three turbines was also op-timized using both controllers, and it was observed that the distributed controllerrequired on the order of 105 iterations, while the centralized algorithm requiredonly several hundred.Zhong and Wang [188] developed distributed discrete adaptive filtering algo-rithms for power de-rating. These algorithms were fundamentally similar to thegame-theoretic algorithms proposed by Marden et al. [122] in that they randomlyperturbed wind turbine axial induction factors based on some probability distribu-tion function and updated baseline axial induction factors only if the perturbationsincreased the power output of a cluster of turbines. The difference between thealgorithms was that Zhong and Wang [188] updated probability distribution func-tions using an adaptive filtering step that rejected noise and disturbances. For a sim-ulated 5× 5 wind farm subject to constant mean wind speed, wind directions thatvaried every 200 iterations, and added noise for modeling turbulence, the proposed38algorithm raised the wind farm efficiency by 3.87% in 200 iterations. Simulationswere conducted using the steady Park wake model and the effects of time-varyingwind speed and dynamic wake transport were therefore not considered. In com-parison, the game-theoretic algorithm proposed by Marden et al. [122] was alsoexamined by Zhong and Wang [188], and resulted in an efficiency loss of 5.92%,which was attributed to sensitivity of this algorithm to environmental disturbances.Gebraad and Wingerden [70] developed a distributed gradient-ascent maxi-mum power-point tracking controller for power de-rating. This controller opti-mized the axial induction factor of each turbine by considering only the power out-put fluctuations of the turbine itself plus that of its immediate downstream neigh-bor. Specifically, iterative measurements were used to numerically estimate thegradient of the power output of a pair of turbines with respect to the axial induc-tion factor of the leading machine. Appropriate adjustments in this axial inductionfactor were then calculated based on the estimated gradient. A distributed quasi-Newton maximum power-point tracking algorithm was additionally developed toimprove convergence. Simulations of the Princess Amalia wind farm (6×10) wereconducted using the dynamic Park wake model (i.e. the steady Park wake modelwith time delay added for wake transport effects). With mean wind speeds varyingbetween 8 and 10m/s, the gradient-ascent controller raised the wind farm effi-ciency by an average of 4%; the effect of turbulence was not considered however.Simulations were also conducted with constant mean wind speed to compare theproposed algorithms to the centralized game-theoretic controller presented by Mar-den et al. [122]. Results showed that the gradient-ascent and quasi-Newton con-trollers reached optimal solutions in approximately 30 minutes, while the game-theoretic approach required over 80 hours to achieve the same outcome. Thisfinding is unsurprising since the game-theoretic controller randomly varied axialinduction factors, while the maximum power-point tracking methods relied uponcomputed gradients.Ahmad et al. [3] presented a centralized multi-resolution simultaneous pertur-bation stochastic approximation control algorithm for power de-rating. A standardsimultaneous perturbation stochastic approximation algorithm randomly and si-multaneously perturbs all wind turbine axial induction factors to determine windfarm power output gradients, and then updates the axial induction factors using39a gradient ascent scheme. The proposed multi-resolution controller implementedthis standard algorithm repeatedly with wind turbines being clustered into smallergroups after each iteration. These groups were referred to as resolutions, withlarger groups being designated lower resolution values, and vice-versa. The solu-tion obtained from optimizing at a lower resolution then served as the initial guessfor optimization at a higher resolution. The performance of this controller was as-sessed by simulating the Horns Rev wind farm (8× 10) using the dynamic Parkwake model. Over the course of a ten-hour simulation, the mean wind speed anddirection were varied using a series of step functions. That is, the simulated windwas steady for hour-long intervals, and was instantaneously altered at the end ofeach interval. The proposed algorithm monotonically raised wind farm efficiencyby 32% over the first eight hours of the simulation. It is important to note how-ever that adjusting wind conditions in such a stepwise fashion is not tantamount tosimulating stochastic time-varying wind behavior that is encountered in the field.All algorithms reviewed thus far must therefore be evaluated under more realisticturbulent wind conditions with varying direction.Johnson and Fritsch [94] offered insight into the effects of stochastic time-varying wind conditions on the efficacy of wind farm control. They developed acentralized extremum seeking controller for power de-rating. The controller per-turbed wind turbine axial induction factors with a continuous periodic signal in or-der to observe changes in wind farm power production. Depending on whether thevariation in wind farm power output was in or out of phase with perturbation sig-nals, the mean values of the axial induction factors were raised or lowered towardtheir optimal values. This controller was evaluated by simulating a wind-alignedrow of three wind turbines using the dynamic Park wake model. Under steady meanwind conditions and a turbulence intensity of 2%, the extremum seeking controllerincreased wind farm efficiency by 3.8%. Increasing the turbulence intensity to8% then generated a 13.2% loss in wind farm efficiency. Yang et al. [187] im-plemented a nested-loop extremum seeking controller for power de-rating, whereoptimal generator torque values were calculated consecutively starting from themost downstream turbine to the leading machine. Simulations were conducted ona wind-aligned row of three wind turbines using the dynamic SimWindFarm simu-lation tool. Under steady mean wind conditions with a turbulence intensity of 5%,40the extremum seeking controller increased wind farm efficiency by 1.3%. Thesestudies did not consider the effects of time-varying mean wind speed and direc-tion. Nonetheless, the meager gains predicted in these studies highlight the factthat adapting to stochastic time-varying wind conditions is a major challenge forwind farm controllers.Ciri et al. [41] developed a nested extremum seeking controller for power de-rating similar to that implemented by Yang et al. [187]. Key differences betweenthe two studies were that Ciri et al. [41] used a discrete-time version of extremumseeking control with faster convergence, and they used the high-fidelity simulationtool UTD-WF with rotating actuator disc theory for controller evaluation. A wind-aligned row of three wind turbines was simulated under constant mean wind speedand a turbulence intensity of 8%, and a relative wind farm efficiency gain of 7.8%was reported. Ciri et al. [41] also tested an individual extremum seeking controllerwith the objective of maximizing turbine power outputs locally. Even though thisalgorithm did not constitute a wind farm controller, it interestingly raised wind farmefficiency by 7.6% relative to a greedy scenario wherein generator torques werefixed at ideal design values. It was explained that these ideal design torque valueswere intended for non-waked wind turbine rotors, and that individual extremumseeking control was able to tune generator torque values for optimal local operationin the presence of rotor wake overlap.A small number of studies have developed dynamic optimization algorithmsthat are model-based. Vali et al. [175] designed a centralized linear adjoint-basedmodel predictive controller7 for power de-rating that used WFSim to simulate wakeaerodynamics. The term adjoint-based refers to additional steps taken to efficientlyestimate the gradient of wind farm power output with respect to axial induction fac-7An explanation of model predictive control is provided for readers who are unfamiliar withthe topic. Traditional regulators such as proportional-integral-derivative control rely upon past andpresent information and measurements to determine control inputs at the current time-step. If onepossesses a model of the system that is being controlled, then the future behavior of this system inresponse to any hypothetical future control input may be simulated. With this capability, a dynamicoptimization problem may be solved over some future time horizon to determine the optimal futureresponse of the system, as well as the corresponding future control signals. This process describesthe basic principle of model predictive control, according to which a dynamic optimization problemis solved online at each time-step in order to determine optimal future control inputs. The objectiveof optimization may involve maximizing or minimizing an objective function with or without con-straints; hence the relevance of model predictive control in maximizing wind farm power production.41tors. Lagrange multipliers were used to remove constraints from the optimizationproblem, while WFSim was linearized around an appropriate operating point ateach time-step. The performance of the controller was assessed by simulating a2× 3 wind farm, again using WFSim. With constant mean wind speed and stepchanges in wind direction, the model predictive controller raised wind farm effi-ciency by up to 23.59% during major portions of the 30-minute simulation, eventhough there existed roughly a three-minute period directly after specific wind di-rection changes where wind farm efficiency was reduced by a maximum of 3.6%.This study did not consider the effects of turbulence nor varying wind speed.Shu et al. [164] developed a centralized nonlinear model predictive controllerfor power de-rating with dynamic wind farm aerodynamics modeled by combin-ing the Frandsen [64] and Larsen [110] wake models. Similar to the dynamicPark wake model, these wake models are based on integral relations from fluidmechanics; however they also consider wind shear effects and wake meanderingdue to atmospheric turbulence. A wind speed estimator was also designed to cor-rect for mismatch between measured power outputs and those predicted by thewake model. The model predictive controller was validated by simulating a wind-aligned row of ten wind turbines using the same combined wake model that wasused for model predictive control. Over the course of a four-minute simulation,during which the mean wind speed varied between 7 and 12m/s, the controllerraised wind farm efficiency by an average of 1.96%. This study did not considerthe effects of turbulence or time-varying wind direction. It is important to keepin mind the limitations of model predictive controllers that rely upon low fidelitywake models. As indicated in Section 2.4.1, these simulators tend to overestimateefficiency gains resulting from power de-rating due to inherent simplifications re-garding velocity distributions and rates of wake expansion. Such model predictivecontrollers must therefore be evaluated using higher-fidelity wake models in orderto ensure their effectiveness.Goit and Meyers [76], Goit et al. [77], and Munters and Meyers [138] devel-oped centralized model predictive controllers for power de-rating that solved dy-namic optimization problems using SP-Wind, an LES-based CFD tool that modelswind turbine rotors as actuator discs. The performance of their controllers wasalso evaluated using SP-Wind. These studies were unique in the sense that they42utilized a high-fidelity simulation tool to dynamically compute optimal wind farmset-points. All other studies cited in this chapter made use of either low or medium-fidelity simulators, or relied upon model-free dynamic optimization algorithms.Large wind farm efficiency gains of up to 21.19% were computed; however it wasindicated in all cases that the computational cost of running LES is impracticallyhigh for online dynamic optimization. Instead, the authors observed patterns in theoptimized flow field and proposed open-loop dynamic induction controllers thatwould replicate these patterns without the need for online dynamic optimization.Specifically, Munters and Meyers [138] proposed sinusoidal variations in the thrustcoefficients of upstream turbines in order to replicate vortex rings that were shed byrotors within the optimized flow field. This dynamic induction controller increasedpower output from a 4×4 wind farm subject to turbulent wind with constant meanspeed by a maximum of 5%.Van der Hoek et al. [176] recently conducted field tests of a power de-ratingcontroller on two wind-aligned rows of five wind turbines and one wind-alignedrow of six machines at the Goole Fields wind farm. Optimization studies were car-ried out offline using FarmFlow to determine optimal blade pitch angle set-pointsfor different wind directions. These set-points were tracked by the collective bladepitch controllers of the turbines during field experiments. Data was collected over ayear-long period during which optimal blade pitch settings were only implementedwhen the wind direction was near alignment with the rows of turbines. Upon ap-plying various filters and corrections, including the imposition of limits on winddirection, blade pitch angle, and yaw angle variability, efficiency gains rangingfrom −15.9 to 11.1% were reported for a single row of five turbines across a spec-trum of wind speeds. Using measured wind speed distribution data, an increase of3.3% in annual energy production was predicted.The results reported by van der Hoek et al. [176] indicate greater potentialfor power de-rating than was demonstrated in previous field tests by Schepers andvan der Pijl [157] and Boorsma [26]; however a few factors may have inflatedmeasured gains. First, wind turbines in a single row were not exactly aligned;therefore downstream wind turbines may have benefited from added kinetic powerconcentrated along the boundaries of the leading turbine’s wake. This phenomenonwas discussed when reviewing the work of Kazda et al. [99]. Second, the wind43turbines were spaced between 2.3 and 3.1 rotor diameters apart. These distancesare less that half the typical spacing of seven diameters that is used in numericalstudies. A shorter distance between adjacent wind turbines decreases the scale ofwake recovery; wind farm control strategies are therefore rendered relatively moreeffective. Important sources of error were reported by van der Hoek et al. [176];FarmFlow is less accurate at inter-turbine spacings of five rotor diameters; despitedata-filtering, wind turbines spent a substantial portion of ten-minute time intervalsoperating with yaw offset; and changes in turbulence intensity resulting from windfarm control affect wake recovery in ways that are not captured by wake models.All of these factors may have served to underestimate potential gains associatedwith power de-rating in field tests.2.4.3 Steady-state potential of yaw-based wake redirectionA list of studies that have evaluated the steady-state potential of yaw-based wakeredirection is provided in Table A.4. Several works have used parametric wakemodels (chiefly, modified versions of the FLORIS wake model) to perform staticoptimization studies that demonstrate the potential of this control concept. Thesestudies may be distinguished based on the optimization algorithms used, and thespecific modification made to the FLORIS wake model. The turbine parameterthat is varied in these studies has universally been the yaw angle γ . The resultingwind farm efficiency gains that have been predicted range from 2.85 to 17.7%,which represents a slight improvement relative to the potential of power de-ratingevaluated using low and medium-fidelity wake models. Unlike the case with powerde-rating however, higher-fidelity simulations and experiments have shown largepotential for yaw-based wake redirection.Fleming et al. [58] used SOWFA to simulate a two-turbine wind farm subjectto constant mean wind speed and direction, a hub-height turbulence intensity of4.9% [40], and wind shear. Upon modeling several scenarios with the yaw angleof the upstream turbine being varied by increments of 5deg, it was observed thata counter-clockwise yaw misalignment of 25deg generated a wind farm efficiencygain of 4.6%. Yawing the upstream nacelle clockwise by 25deg resulted in anefficiency loss of 4.0%. Gebraad et al. [72] attributed this asymmetry to the phe-44nomenon of wake rotation. A clockwise spinning rotor exerts a counter-clockwisetorque on the incoming wind, which causes the wake immediately downstream torotate counter-clockwise. Rotation of the wake causes lower speed wind closer tothe ground, which exists due to wind shear, to deflect in the crosswind direction.Hence, even in the absence of yaw misalignment, a spinning rotor generates asym-metric bias in the wind field that supplements wake deflection in one direction, andcurtails it in another.More recently, Fleming et al. [60] extended upon the above findings of Ge-braad et al. [72]. Using SOWFA simulations of a single wind turbine, they showedthat asymmetry in wake redirection is also substantially influenced by vortices gen-erated due to yaw misalignment. Moreover, when a second turbine was introducedfurther downstream, its wake was deflected even with zero yaw offset. This phe-nomenon was referred to as secondary steering, and was attributed to two pro-cesses; the interaction between the rotating wake of the second turbine and thevortex generated by the first turbine; and the fact that yawing the upstream tur-bine generated partial overlap along the rotor of the second machine. Simulationswere then conducted involving three rows of four wind turbines aligned with con-stant mean speed wind with a turbulence intensity of 6%. Implementing yaw off-set on the leading turbine in each row resulted in a wind farm efficiency gain of12.5%. This improvement relative to previous work by Fleming et al. [58] leadsto two major conclusions; wake redirection is relatively more beneficial in largerwind farms; and vortices generated by the leading turbines combine as they aretransported downstream to have profitable impact on trailing machines. This studyalso indicated that the low-fidelity FLORIS wake model actually underestimatedthe benefits of wake steering by not capturing vortex-related phenomena. This aninteresting distinction between wake redirection and power de-rating, for whichlow-fidelity wake models tend to overestimate efficiency gains.Churchfield et al. [39] used a dynamic LES-based CFD tool to simulate a singlerow of five turbines within the Fishermen’s Atlantic City Wind Farm. Constantmean wind speed and direction, wind shear, and a turbulence intensity of 5% weremodeled. In addition to a baseline scenario with no yaw offsets, three simulationswere performed during which the yaw angles of the four leading turbines were setto 15, 20, and 25deg. At the largest yaw offset, a relative efficiency gain of 13.77%45was observed, with diminishing returns apparent in the trend of reported data. Thisbenefit was accompanied with a penalty of increased blade-root flap-wise bendingmoment fluctuations however. While the leading turbine experienced a reductionof 50% in the root-mean-square (RMS) value of its blade bending moments, thisRMS value increased by 15 to 45% for the remaining turbines, with the largestvalue corresponding to the trailing turbine with no yaw-offset. These observationswere attributed to the fact that yaw-based wake redirection causes partial wakeoverlap along the rotors of downstream turbines, which generates fluctuations inaerodynamic loads as individual blades traverse the rotor swept area. It was alsosuggested that individual blade pitch control would be an effective approach formitigating this drawback.Miao et al. [130] used the commercial CFD package STAR-CCM+ to simu-late a two-turbine wind farm by solving the unsteady RANS equations with thek-ω turbulence model. Unlike CFD solvers designed specifically for wind farmapplications, which use either actuator disc or line theories to model rotor aerody-namic loads, this study considered three-dimensional wind turbine blade geome-tries within the fluid domain and solved for aerodynamic forces directly via theequations of fluid motion. The vicinity of each wind turbine rotor was thereforediscretized using a high-density rotating mesh, and a sliding mesh technique wasused to link data transfer between rotating and stationary mesh regions. With con-stant mean wind speed and direction, wind shear, and turbulence intensity rangingfrom approximately 3 to 23% across the fluid domain, yawing the upstream tur-bine by 30deg yielded a wind farm efficiency gain of 2.11%. The triviality of thisefficiency gain relative to predictions by other simulators was attributed to the ten-dency of the k-ω turbulence model to predict excessive turbulence dissipation. Thiscondition resulted in faster wake recovery, which would have reduced the relativebenefit of any wind farm control strategy.Adaramola and Krogstad [2] and Campagnolo et al. [31], whose wind tunnelexperimental works on power de-rating were reviewed in Section 2.4.1, also inves-tigated the potential of yaw-based wake redirection. Adaramola and Krogstad [2]used the same two scaled wind turbines with rotor diameters of 90cm. After yaw-ing the upstream rotor from 0 to 40deg, they measured a peak efficiency gain of12% at a yaw angle of 30deg. Campagnolo et al. [31] conducted their wake redi-46rection experiments on two scaled wind turbines with rotor diameters of 1.91msubject to constant mean wind speed and a turbulence intensity of 8%. The yawangle of the upstream turbine was varied from−20 to 20deg, while the downstreamturbine was simultaneously displaced in the crosswind direction. They observed awind farm efficiency gain of 7.0% when the upstream rotor was yawed by−20degand the two turbines were aligned with the wind. When the downstream turbinewas displaced by 0.45 rotor diameters, the efficiency gain increased to 25.5%,since repositioning the downstream turbine further contributed to reducing wakeoverlap. This result may be interpreted as a promotion for turbine repositioning.Bastankhah and Porte´-Agel [19] conducted wind tunnel experiments on wakeredirection wherein the objective was to examine the effects of yaw angle distribu-tion across a wind farm on overall power production. The wind farm consisted of awind-aligned row of five scaled devices with rotor diameters of 15cm. In a brute-force fashion, different combinations of yaw offsets ranging from 0 to 30deg wereimplemented, and a maximum wind farm efficiency gain of 17% was observed.This peak corresponded to a distribution in which the yaw offset was greatest atthe leading turbine, while gradually dropping to zero at the trailing machine. Ananalytical wake model was also used to ascertain the effects of turbine spacingand turbulence intensity on the benefits of wake redirection. The results were inagreement with others presented in this review. Increasing turbine spacing allowedfor more time and space for wakes to re-energize; thus reducing the relative gainsobtained from wake redirection. Greater turbulence intensity values yielded thesame outcome since larger turbulent kinetic energy quickened turbulent diffusionand wake recovery.Wagenaar et al. [181] conducted field tests on a single row of three 2.5MWwind turbines located at ECN’s experimental wind farm. Incremental clockwiseyaw offsets of the upstream turbine ranging from 0 to 16deg were introduced,while only measured data pertaining to the wind direction aligned with the row ofturbines were gathered and averaged. Analysis showed that the wind farm perfor-mance was highly scattered and sensitive to data selections, particularly to windspeed. At 4deg of yaw offset, wind farm efficiency gains of 32.9 and 11.1% wereobserved corresponding to wind speeds of 4 and 8m/s, respectively, while a 16degyaw offset resulted in efficiency losses of 26.3 and 11.7% at the same wind speeds,47respectively. Additionally, efficiency results corresponding to a reference row ofturbines without any yaw offset showed variability across different wind speedsand yaw angle offsets of the leading turbine. In light of these features of the col-lected data, no conclusions were drawn regarding the potential of yaw-based wakeredirection.Howland et al. [86] performed wake redirection field tests on a row includ-ing five 1.8 MW Vestas V80 wind turbines and one 2.0MW model. An analyticalwake steering model based on Prandtl’s lifting line theory [163] was calibratedusing five years of data obtained from the test site. This model was then used tosolve an analytical gradient ascent optimization problem to obtain the set of op-timal yaw angles for maximizing farm efficiency. Data collected over a ten-dayperiod showed a maximum relative efficiency gain of 47% at wind directions closeto alignment with the row of turbines. In these cases, operation without wind farmcontrol resulted in partial overlap along the rotors of downstream machines. Intro-ducing wake redirection then virtually eliminated the wake effect and resulted insubstantial power gains. Conversely, when the incoming wind was directly alignedwith the row of turbines, a smaller relative efficiency gain of 7% was reported.Implementation of the optimal yaw angles also served to decrease the variability(i.e. the standard deviation in one-minute averaged data) of power measurementsby 73%. This outcome was attributed to the fact that wake redirection reduced thelikelihood of downstream wind turbines operating below cut-in speeds as a resultof the wake effect. Reducing power output variability was noted to contribute toreduced intermittency and improved reliability of wind energy in the power grid.2.4.4 Controllers developed for yaw-based wake redirectionStudies that have developed wind farm controllers based on the concept of yaw-based wake redirection are summarized in Table A.5. Gebraad et al. [72] imple-mented a model-based version of the centralized game-theoretic algorithm pre-sented by Marden et al. [122]. The developed controller was model-based in thesense that the game-theoretic algorithm was not driven by measured data, but ratherby the steady FLORIS wake model. At each time-step, the algorithm used thesteady FLORIS model to determine the optimal set of nacelle yaw angles across48the wind farm for maximizing power production given the immediate wind speedand direction. This controller was evaluated using SOWFA by simulating two rowsof three turbines that were subject to constant mean speed wind with wind shearand a turbulence intensity of 6%. A relative wind farm efficiency gain of 13.03%was reported for a scenario in which the wind was aligned with the turbine rows.For this wind condition, it was also shown that fatigue damage, computed via dam-age equivalent loading (DEL), generally decreased for the leading two turbines ineach row (i.e. the turbines whose rotors were subject to yaw misalignment). Bladeflap-wise bending DEL was reduced by 11.2 to 18.4% for different machines, forinstance. Of the two trailing turbines, the machine toward which wakes were redi-rected experienced minor changes in DEL. The other trailing turbine experiencedsignificant rise in fatigue damage; 12.7% for blade flap-wise bending DEL, 18.0%for low-speed shaft torsional DEL, and 19.1% for yaw bearing moment DEL.Gebraad et al. [69] used the FLORIDyn wake model to develop a nonlinearmodel predictive controller for yaw-based wake redirection. A Kalman filter wasintroduced in order to account for sensor noise and power fluctuations resultingfrom turbulence; the Kalman filter would also correct mismatch between predic-tions from FLORIDyn and measured data. The controller was evaluated by sim-ulating a wind-aligned row of two wind turbines using FLORIDyn. Under steadywind conditions, the model predictive controller produced an increase of 0.19% inthe wind farm efficiency. Such a small relative power gain was associated with thelarge spacing between the modeled wind turbines; this spacing was ten rotor diam-eters, as opposed to the more commonly simulated seven rotor diameter spacing.Increasing turbine spacing allows for more time and space over which wakes arere-energized by the free stream wind [115], which diminishes the relative benefitsof any wind farm control strategy. The use of the Kalman filter generated sub-stantial reductions in RMSE values between turbine power outputs predicted bySOWFA and FLORIDyn. RMSE values ranged from 13.4 to 18.4% without thefilter, and from 7.6 to 11.5% upon inclusion. This result promotes the capability ofdynamic control-oriented wake models to adapt to time-varying wind conditions.Park and Law [146] developed a cooperative game-theoretic controller forcombined power de-rating and yaw-based wake redirection using Bayesian-ascentdynamic optimization. The Bayesian-ascent controller operated in a fashion sim-49ilar to previously described game-theoretic controllers, where axial induction fac-tors and yaw angles were perturbed, and the measured wind farm power outputinfluenced decisions at the next iteration. One major modification was that a Gaus-sian process (a non-parametric, probabilistic machine learning approach) was usedto construct an approximated objective function (i.e. an approximated model thatpredicted wind farm power production) in real-time based on measured data. Thisapproximated objective function was then used to solve a real-time optimizationproblem at each time-step to determine optimal axial induction factors and yawangles for future time-steps. The controller was evaluated by simulating a 10×10wind farm using a newly derived steady wake model8. Under steady wind con-ditions with measurement noise, the cooperative controller raised wind farm effi-ciency by 24.5% after 50 iterations.Park et al. [148] later tested this control algorithm in wind tunnel experimentsconsisting of a 2×3 layout of scaled turbines with rotor diameters of 1.5m. At con-stant mean wind speed, the controller monotonically increased the wind farm effi-ciency by 33.22% after 40 iterations. Campagnolo et al. [32] also performed exper-imental wind tunnel tests during which they implemented the gradient-ascent max-imum power-point tracking algorithm presented by Gebraad and Wingerden [70];except this algorithm was applied to the concept of yaw-based wake redirectionrather than power de-rating. Experiments were conducted on a wind-aligned rowof three scaled wind turbines with rotor diameters of 1.1m subject to constantmean wind speed. Implementation of the control algorithm increased the averagewind farm efficiency by approximately 20% after 200 seconds of operation. Thesetwo studies represent the only experimental evaluations of model-free wind farmcontrol systems to date, and their results are encouraging. Nonetheless, these algo-rithms must be evaluated using larger scaled models with Reynolds numbers basedon blade chord length that are representative of utility-scale wind turbines.Similar to their studies of power de-rating, Munters and Meyers [137, 139]developed centralized model predictive controllers for combined power de-rating8Similar to the Park wake model, this newly derived model relied upon momentum integral re-lations from fluid mechanics. However, rather than assuming uniform velocity distribution withinwakes, this new model computed the radial velocity distribution using a Gaussian exponential func-tion and performed numerical integration to determine the effective wind speed at some downstreamwind turbine rotor.50and yaw-based wake direction. They used the LES-based CFD tool SP-Wind toperform online dynamic optimization, which rendered their controller impracticalfor real-world implementation. As with their earlier work however, they utilizedobserved patterns in the optimized flow field to propose open-loop controllers thatwould replicate such flow patterns. Specifically, Munters and Meyers [139] pro-posed two simplified open-loop controllers; one involving steady yaw offset thatwould deliver mean deflections of wake centerlines, and one implementing dy-namic wake steering that would engender wake meandering. Upon implementingthese open-loop controllers on a 4× 4 wind farm subject to turbulent wind withconstant mean wind speed, a relative efficiency gain of 14% was obtained usingthe steady yaw offset open-loop controller, while a relative loss of 2% was ob-served using dynamic wake steering.Fleming et al. [59] conducted field tests of wake redirection at the LongyuanRudong Chaojiandai offshore wind farm, which consists of 25 4MW wind turbinesarranged in a gridded layout. For the purposes of this study however, wake steeringwas only implemented via a single turbine that shadowed one out of three other ma-chines at any given time depending on the immediate wind direction. The FLORISwake model was used in conjunction with an optimization algorithm to compute alookup table of optimal yaw angles as a function of wind direction. These lookuptables established the set-points that were tracked by the yaw controller of the regu-lated turbine. Based on collected data corresponding to the wind direction alignedwith a pair of two turbines that were spaced seven rotor diameters apart, a windfarm efficiency gain of 5.56% was reported. Meanwhile, over a 60deg range ofvariation in the wind direction, efficiency gains ranged from −12.16 to 19.96%,with losses occurring at a minority of wind directions. These are promising re-sults given the inconclusive outcomes that have typically been reported from fieldtests. Even more encouraging is the fact that the optimal yaw angles implementedin these field tests were calculated using an engineering wake model, the use ofwhich would lead to highly efficient model-based wind farm control.More recently, Fleming et al. [61] examined the performance of the above con-troller on two wind turbines within an unspecified commercial wind farm. Acrossan 80deg spectrum of wind directions surrounding alignment with the row, windfarm efficiency gains ranging from −7.9 to 10.2% were reported. The follow-51ing additional difficulties associated with field testing were pointed out; biases inyaw angle set-point tracking that could not be accounted for due to intellectualproperty protection; the inability of FLORIS to capture terrain effects, which ul-timately led to discrepancies relative to field measurements; the fact that FLORISwas not tuned to predict near-wake region behaviour, which was problematic dueto the close spacing of the wind turbines; and significant atmospheric instability,which resulted from data being collected during the summer season. All of thesefactors may justify the limited range of efficiency gains recorded in comparison tothe previous field test by Fleming et al. [59].2.4.5 Steady-state potential of turbine repositioningStudies that have examined the potential of turbine repositioning are listed in Ta-ble A.6. At the time of writing, no work has implemented a dynamic control al-gorithm involving this wind farm control concept. Fleming et al. [58] simulateda two-turbine wind farm using SOWFA and investigated the effect of incremen-tally relocating the downstream turbine in the crosswind direction. Constant meanwind speed and direction, with wind shear and a turbulence intensity of 4.9% [40]were modeled. Their results showed that shifting the downstream turbine by 130m(i.e. one rotor diameter) perpendicular to the wind direction increased the poweroutput from the pair by 41.0% relative to a baseline simulation with no crosswindshift. The computed trend in blade flap-wise bending DEL demonstrated that ac-cumulated fatigue damage rose as turbine relocation increased, and peaked at 50mof crosswind shift. The relative rise in bending DEL began to drop beyond thispoint, and was either negligible or negative (i.e. indicating load reduction) whenthe downstream machine had been shifted by an entire rotor diameter. Finally, aswith yaw-based wake redirection, it was shown that there exists a preferred direc-tion of turbine relocation due to the asymmetric bias in wake deflection that wasdiscussed in Section 2.4.3.Rodrigues et al. [153] proposed a mechanism for repositioning floating off-shore wind turbines in real-time which consisted of tensioned mooring lines feed-ing into winches mounted on-board floating platforms. By regulating the lengths ofthree mooring lines, each platform could be relocated within a movable range that52was outlined by an equilateral triangle with side lengths of approximately 330m.Multiple optimization studies with different wind directions were performed ona 25-turbine floating offshore wind farm using the steady Park wake model withconstant wind speed. The optimization goal was to determine the magnitude anddirection of turbine repositioning, within the aforementioned movable range sur-rounding the installation locations of the floating platforms, that would maximizewind farm power output. These optimization results were then validated usingFarmFlow simulations. For a wind farm with floating platforms installed in a grid-ded 5× 5 layout, turbine relocation yielded wind farm efficiency gains rangingfrom −11.49 to 18.11% across a 330deg spectrum of simulated wind speeds.A practical method for repositioning floating offshore wind turbines was pro-posed by Han et al. [81], and is illustrated in Figure 2.10. A floating offshorewind turbine implementing a greedy control algorithm, which aims to locally max-imize power production, will operate at its optimal axial induction factor with zeroyaw offset. This operating point corresponds to the faded turbine in Figure 2.10.Variations in the axial induction factor and nacelle yaw angle of this turbine alterthe effective magnitude and direction of the thrust force applied to its rotor. Theequilibrium established between this altered thrust force and the restoring forcesgenerated by the mooring lines permits relocation of the floating turbine anywherewithin a limited movable range. Unlike winch-based repositioning methods, thisproposed strategy for relocating floating wind turbines using yaw and axial induc-tion requires no additional hardware, while employing control degrees of freedomthat are available in currently operational wind turbines.In our recent work [101], the potential of the repositioning method proposedby Han et al. [81] was assessed. Optimization studies were carried out on a 3×6floating offshore wind farm with semi-submersible type platforms subject to con-stant wind speed and direction. FOWFSim was used for both optimization andevaluation. Optimization was performed for various mooring line lengths rangingfrom 835 to 925m in order to assess the potential of turbine repositioning as themovable range of the platforms was expanded. We reported that YITuR increasedwind farm efficiency by approximately 16.5% for mooring lines lengths beneath875m. Beyond this point, wind farm efficiency gains increased dramatically, withan increase of 53.5% corresponding to the largest cable length of 925m. It was53Wind directionThrust force vectorsAnchorsMooring linesFloating platformFigure 2.10: Schematic demonstrating the concept of yaw and induction-based turbine repositioning (YITuR).noted that despite such potential, the practicality of permitting a movable range ofnearly 90m, which was the case for the largest cable length examined, must becarefully assessed.2.5 Quantitative overview of published resultsThis section pieces together information that has been presented throughout theliterature review from Section 2.4 with the aim of drawing conclusions regardingthe impacts of controller properties, evaluation techniques, and simulated windconditions on the potential of wind farm control concepts and algorithms.2.5.1 Effects of evaluation techniques on predicted controllerpotentialThe influence of evaluation techniques on the predicted potential of each wind farmcontrol concept is assessed using the bivariate histograms shown in Figure 2.11–54Low-fidelity modelMedium-fidelity modelHigh-fidelity modelWind tunnel test0-202Histogram bins: Relative wind farm efficiencygains predicted by different studies (%)-10 Field test0 10 20430 40Histogram counts: Number of studies thathave predicted listed efficiency gains681012Figure 2.11: Bivariate histogram demonstrating the quantity of studies as-sessing power de-rating that have reported different wind efficiencygains using various evaluation models and techniques.Figure 2.13. These plots demonstrate the number of studies that have used a par-ticular evaluation method or tool to obtain relative wind farm efficiency gains cor-responding to 5% incremented bins. Focusing on Figure 2.11, which relates topower de-rating, the bulk of studies that have utilized low and medium-fidelitywake models have predicted efficiency gains ranging up to 25%, while few workshave attained values as high as 35%. As the fidelity of the evaluation method isincreased however, the range of predicted gains narrows to−10 to 10% in the caseof high-fidelity simulations, 0 to 5% for wind tunnel tests, and 0% or less for fieldtests.The few high-fidelity simulations that predicted gains up to the 20% bin corre-spond to the works of Goit and Meyers [76] and Munters and Meyers [138], whoused a high-fidelity LES-based solver to dynamically optimize power productionusing power de-rating. As discussed in Section 2.4.2, such a control approachis impractical given the computational cost of running CFD simulations for on-line optimization. Upon implementing a more practical control strategy involvingsinusoidal power de-rating, the wind farm efficiency gain predicted by Munters55and Meyers [138] reduced to 5%, which bears greater similarity to predictions bythe remaining studies that employed high-fidelity simulations. The field tests thatmeasured efficiency gains up to the 10% bin were performed by van der Hoek etal. [176], and we suggested that close turbine spacing and crosswind misalignmentamong rows of turbines resulted in inflated power de-rating potential. The lattercomment was also made regarding the work of Kazda et al. [99].The discrepancy between results obtained via low and high-fidelity simulationtools has been attributed to the concentration of kinetic power gain in the vicinity ofwake boundaries, and the effects of rotor thrust force changes on wake turbulence,neither of which are captured by lower-fidelity wake models. These phenomenamay also justify the negligible efficiency gains observed among field tests; althoughthe inconclusiveness of field test results has also been attributed to the difficulty ingathering consistent data for comparative assessment under stochastic time-varyingwind conditions. Finally, while wind tunnel tests have shown promising results incomparison to those obtained via field tests, the presence of low Reynolds numbersand tunnel effects in such tests may have inflated the measured potential benefits.Despite these conclusions, power de-rating should not be ruled out as an effectivewind farm control concept. Since the kinetic power gain from power de-ratingis concentrated along wake boundaries, potential benefits may arise in scenarioswhere neighboring wind turbine rotors are not directly aligned with the wind. Thisphenomenon may have caused the efficiency gains reported by Kazda et al. [99]and van der Hoek et al. [176]. Moreover, studies with inconclusive results concern-ing power gains have shown that power losses from upstream de-rated turbines areat the very least recovered by their downstream neighbors. Power de-rating maytherefore be employed for load mitigation without excessive power loss.Shifting focus to Figure 2.12, it is evident that the predicted potential of yaw-based wake redirection is less sensitive to the fidelity of the evaluation methodemployed. All simulation tools and wind tunnel tests have indicated potential ef-ficiency gains ranging up to the 35% bin. The reported value in the 45% bin forhigh-fidelity simulation once again corresponds to the work of Munters and Mey-ers [137], who employed an LES-based wake model for online optimization. Therange of efficiency gains obtained via field tests has been mostly limited to the−5 to 10% bins, which is foreseeable given the aforementioned challenges in han-56Low-fidelity modelMedium-fidelity modelHigh-fidelity modelWind tunnel test0-10 01Histogram bins: Relative wind farm efficiencygains predicted by different studies (%)Field test10 20 30240 503Histogram counts: Number of studies thathave predicted listed efficiency gains456Figure 2.12: Bivariate histogram demonstrating the quantity of studies as-sessing yaw-based wake redirection that have reported different windefficiency gains using various evaluation models and techniques.dling field test data. The large efficiency gain in the 45% bin for field testing wasreported by Howland et al. [86], who calibrated their analytical wake model usingfive years of historical site-specific wind farm data. Their promising results shouldarouse much interest in the use of historical data for wind farm control.On the whole, wake redirection field tests indicate a significant improvementrelative to those concerning power de-rating. Furthermore, it is worth observingthat significant losses in wind farm efficiency resulting from wake redirection havebeen seldom reported. There has also been general agreement between studiesemploying low-fidelity models and those relying upon high-fidelity simulationsor field tests. This consensus may be attributed to the tendency of low-fidelitymodels to actually underestimate the potential of wake redirection, as reported byFleming et al. [60]. Similarly positive results have been observed in the contextof turbine repositioning, for which the histogram of existing results is plotted inFigure 2.13. Although research concerning the potential of this control concept islimited, available data has indicated potential gains exceeding 20% regardless ofthe simulation tool employed.57Low-fidelity modelMedium-fidelity modelHigh-fidelity modelWind tunnel test020Histogram bins: Relative wind farm efficiencygains predicted by different studies (%)Field test30 40 50 60Histogram counts: Number of studies thathave predicted listed efficiency gains1Figure 2.13: Bivariate histogram demonstrating the quantity of studies as-sessing turbine repositioning that have reported different wind effi-ciency gains using various evaluation models and techniques.2.5.2 Effects of simulated wind conditions on predicted controllerpotentialMost studies that have been reviewed in this chapter have evaluated the potentialand performance of wind farm controllers under steady wind conditions; that is,constant mean wind speed and direction with zero turbulence intensity. The fewstudies that have considered more realistic wind behavior have shown that time-varying wind conditions significantly deteriorate the performance of any wind farmcontroller. This outcome has held true regardless of the model used for controllerevaluation, or the wind farm control technique in question. The explanation for thisobserved trend is that wind farm controllers, whether model-based or model-free,must adapt to time-varying wind conditions, which are stochastic and unpredictabledisturbances. Throughout the period of time over which this adaptation occurs, awind farm controller will be operating sub-optimally as it solves a dynamic opti-mization problem to compute an optimal setting. Furthermore, as wind conditionsvary, the optimal set-points that must be tracked by individual wind turbine con-trollers also fluctuate, which further increases the duration of sub-optimal operation58due to the time required for individual turbines to reach their optimal set-points.Particular attention has been attributed to the influence of turbulence, and allfindings indicate performance decay with rising turbulence intensity. Dynamiclow-fidelity simulations conducted by Johnson and Fritsch [94] demonstrated thatextremum seeking control for power de-rating resulted in efficiency gains drop-ping from 3.8 to −13.2% when the turbulence intensity of the incoming flow wasraised from 2 to 8%. Likewise, dynamic medium-fidelity simulations performedby Kim et al. [106] involving power de-rating predicted a decrease in wind farmefficiency gains from 2.92 to −2.88% when the turbulence intensity of the flowwas increased from 7.5 to 21%. Dynamic high-fidelity simulation results reportedby Munters and Meyers [139] presented a similar outcome. Their proposed sim-plified controllers for wake redirection were implemented on a wind farm subjectto both steady wind and airflow with a turbulence intensity of 8%. Wind farm ef-ficiency gains dropped from 97 to 14% as turbulence was introduced when testingtheir static controller, and 79 to −2% with dynamic wake steering.The effects of time-varying mean wind speed and direction on controller per-formance have not been investigated as thoroughly as the impacts of turbulence.Nonetheless, the tendency of controller performance to diminish under these windconditions holds. Studies by Gebraad and Wingerden [70] and Shu et al. [164] haveshown that both model-based and model-free controllers are capable of adapting totime-varying mean wind speeds, thereby increasing wind farm efficiency averagedover the simulation period by 4% in the former case, and 1.96% in the latter case.However, compared to other low-fidelity evaluations involving constant mean windspeeds, which have yielded efficiency gains as large as 25%, the time-varying re-sults reported by Gebraad and Wingerden [70] and Shu et al. [164] fall significantlyshort.To date, no study has evaluated the performance of a wind farm controller withthe wind direction varying continuously and stochastically over time. Several stud-ies have incrementally altered the wind direction using step functions, and thenallowed for their controllers to successfully adapt to the discrete changes [3, 175,188]; however such an analysis is not tantamount to gradual and continuous varia-tions in the direction of the wind. Simulation results presented by Vali et al. [175],who tested a model predictive controller for power de-rating, indicated efficiency59losses as large as 3.6% occurring at specific points in time corresponding to stepchanges in the wind direction. It is reasonable to conclude from this outcome thatcontinuous variations in wind direction that are of substantial magnitude and fre-quency would therefore engender similar losses throughout the operation of windfarm control.2.5.3 Effects of controller properties on performanceGiven the scarcity of studies that have evaluated wind farm controller performanceusing either high-fidelity simulation tools or experimental techniques, no decisiveconclusions may be drawn concerning the impacts of controller properties on per-formance. The following comments are nonetheless presented based on the limitedfindings that have been observed in this review.Effects of model dependencyWith regards to the model dependency of wind farm controllers, available resultsindicate that both model-based and model-free dynamic optimization algorithmsoffer potential to improve wind farm efficiency. The model-based game-theoreticwake redirection controller evaluated by Gebraad et al. [72], which performed on-line optimization of the FLORIS wake model, yielded an efficiency gain of 13.03%when subject to turbulent wind simulated by SOWFA. The model-based lookup ta-ble wake redirection controller tested in the field by Fleming et al. [59], which uti-lized optimal set-points determined via offline optimization of the FLORIS wakemodel, generated efficiency gains up to 19.96% across the majority of the spectrumof tested wind directions. The model-free gradient-ascent wake redirection con-troller evaluated by Campagnolo et al. [32] via wind tunnel experimentation pro-duced an efficiency gain of at least 20% after 200 seconds of operation. The game-theoretic controller tested via wind tunnel experimentation by Park et al. [148]also yielded impressive efficiency gains up to 33.22% after 40 iterations. Thiscontroller was model-based yet data-driven in the sense that it utilized a Gaussianmachine learning process for online dynamic optimization. Even the simplifiedopen-loop control algorithms for power de-rating and wake redirection proposed byMunters and Meyers [138, 139] led to efficiency gains of 5 and 14%, respectively.60These open-loop controllers may be categorized as model-free, yet not data-driven,since control actions are determined independently of any analytical or estimatedmodel, or power measurements.It is difficult to draw any deeper conclusions regarding the role of model de-pendency based on the magnitudes of efficiency gains reported in these studies.Each work utilized a different method of evaluation, and even studies involvingwind tunnel tests are not directly comparable due to the differences in Reynoldsnumbers that were effective during experimentation. Even so, data-driven con-trol approaches involving nonparametric machine learning or system identificationare, by definition, more robust and practical for field implementation. Such con-trollers are capable of adapting to time-varying wind conditions while computingoptimal wind farm set-points based on estimated models, rather than via randomor gradient-based probing. Examples include the game-theoretic controller pro-posed by Park et al. [148], which utilized a Gaussian process for machine learning,the game-theoretic controller implemented by Gebraad et al. [72], which tuned theparameters of the FLORIS wake model using measured data, and the model predic-tive controller implemented by Gebraad et al. [69], which used a Kalman filter tocorrect states of the FLORIDyn model based on wind farm power measurements.Effects of communication architectureThe effects of communication architecture on the performance of wind farm con-trollers also evade decisive conclusions due to the limited quantity of availableresults. Studies that have developed and evaluated both centralized and distributedwind farm controllers have yielded conflicting findings regarding the potenciesof their algorithms. One the one hand, the distributed game-theoretic power de-rating controller developed by Marden et al. [122] required an iteration countthat was three orders of magnitude greater than that of their centralized controlalgorithm to compute the optimal set-points of the simulated wind farm. Con-versely, the distributed adaptive filtering power de-rating controller developed byZhong and Wang [188] required 20 to 30 fewer iterations to determine optimalwind farm set-points compared to their centralized adaptive filtering algorithm.Both studies simulated wind farms of considerable size (an 8×10 wind farm in the61former case, and a 5× 5 wind farm in the latter work); thus the quantity of regu-lated wind turbines was likely not a contributing factor to the observed discrepancybetween the iteration counts of the two works.Gebraad and Wingerden [70] compared their distributed gradient-ascent con-trollers for power de-rating to the centralized game-theoretic controller proposedby Marden et al. [122]. The distributed gradient-ascent algorithms reached optimaloperation in 30 minutes, while the centralized game-theoretic controller required80 hours of simulation time to approach the same operational conditions. Cam-pagnolo et al. [32] implemented one of the distributed gradient-ascent controllersdeveloped by Gebraad and Wingerden [70] for wake redirection via wind tunnelexperimentation. Their three-turbine scaled wind farm was optimized within ap-proximately 3.3 minutes. In this case, wind farm size definitely contributed to therequired iteration count for optimal operation, since the wind farm simulated byGebraad and Wingerden [70] consisted of 60 wind turbines, which necessitated30 minutes of simulation time for optimal conditions to be reached.The results listed above show that the algorithmic operations of wind farm con-trollers contribute significantly to their required iteration counts for reaching op-timal operation. The quantity of wind turbines within a regulated wind farm alsoimpacts the time-efficiency of controllers. As a result, further research involvingdirect comparisons between centralized and distributed controllers is necessary forpinpointing the precise benefits and drawbacks of each communication architec-ture. The work by Gebraad and Wingerden [70] did validate a major preconceptionregarding communication architecture; in addition to optimizing a large number ofparameters, a centralized controller must allow sufficient time for changes in theaerodynamic field, which result from control actions, to transport across the en-tire wind farm [146]. Centralized algorithms are thereby rendered unpractical forimplementation in large wind farms from the standpoints of both computationalcost and time-efficiency [70]. Distributed controllers circumvent this concern sincetheir computational cost and required transport time remain unchanged irrespectiveof the wind farm size.622.6 Conclusions and recommendations for futureresearchThis chapter provided a review of three notable and prevalently examined windfarm control concepts for power maximization; power de-rating, yaw-based wakeredirection, and turbine repositioning. Discussions focused on the potential ofthese concepts as well as the performance of control algorithms in the context oftheir properties, the evaluation methods used, and the simulated wind conditions.The current section summarizes the major conclusions drawn from the findingsof published works, while offering recommendations on potential future researchgoals and practices that would be necessary for progress in the field of wind farmcontrol.With regards to the potential of each concept, power de-rating possesses theweakest capability of increasing wind farm power production. The added kineticpower generated via de-rating concentrates along the boundaries of wakes and doesnot sufficiently overlap with immediate downstream rotors. Power de-rating doesremain a viable control concept for load mitigation and power set-point trackinghowever. Both yaw-based wake redirection and turbine repositioning have demon-strated substantial potential for raising wind farm efficiency, and this conclusionhas been verified via high-fidelity simulations (and experimentation in the case ofwake redirection).In terms of evaluation techniques, lower-fidelity simulators do not capture ve-locity distributions and generated turbulence within wakes; both of which havecontributed to over-prediction of efficiency gains associated with power de-ratingcontrollers. On the other hand, predicted wind farm power output gains associatedwith wake redirection and turbine repositioning are less sensitive to these aerody-namic phenomena. This conclusion is verified by the commonality in simulationresults obtained from models of varying fidelities.On the experimental side, the low Reynolds numbers that are effective in windtunnel experiments may be inflating measured efficiency gains for all control con-cepts. Meanwhile, field tests have provided the most conservative evaluations dueto the stochastic time-varying nature of atmospheric wind, as well as the diffi-culties in extracting meaningful conclusions from the resulting measurements. In63general, it has been shown that added layers of realism in terms of simulated windconditions tend to deteriorate the performance of wind farm controllers. Addedmeasurement noise resulting from turbulence and time-varying mean wind speedand direction reduces the length of time over which wind farms operate at opti-mal conditions, since wind farm controllers are continuously searching for optimalset-points in an environment with stochastic and unpredictable disturbances.Further research is necessary for properly assessing the pros and cons of spe-cific wind farm controller properties such as model dependency and communica-tion architecture. Available data suggests that model-free, model-based, and evenopen-loop controllers all demonstrate potential for power maximization; howeverdata-driven controllers involving system identification, machine learning, and windfield estimation are best suited for adapting to time-varying wind conditions. Pub-lished results also indicate that distributed controllers offer greater time-efficiencyand computational expedience relative to centralized algorithms. This advantagestems from the lower wake transport periods and fewer optimization variables in-volved, respectively.Based on these conclusions, the following recommendations may assist in pro-pelling research within the wind farm control community:• Researching turbine repositioning: The limited quantity of work inves-tigating this control concept has demonstrated significant potential for windfarm power maximization. This field of research should receive considerableattention, particularly focusing on the practicality and safety concerns asso-ciated with relocating floating offshore wind turbines in real-time. Existingparametric wake models and LES-based CFD tools should also be extendedto capture turbine motion as a degree of freedom.• Simulating time-varying wind direction: This review has shown that nostudy has investigated the effects of continuously and stochastically time-varying wind direction on controller performance. This outcome has nodoubt resulted from the difficulty in simulating fluctuating wind directionusing models and wind tunnels. Nonetheless, both low-fidelity simulations(see Ahmad et al. [3] and Vali et al. [175]) and field tests have demon-strated indirectly that time-varying wind direction, much like unsteady mean64wind speed and turbulence, serves to diminish controller performance; ef-forts should thus be exerted into assessing control algorithms under realisticwind scenarios.• Incorporating wind field and state estimation techniques: Few of thestudies reviewed in this chapter utilized state estimators and wind field esti-mation techniques. These methods have been used obtain undistorted windfield measurements within experiments conducted by Annoni et al. [10] andSchreiber et al. [158], and to correct wind field predictions from control-oriented wake models within high-fidelity simulations performed by Ge-braad et al. [69] and Doekemeijer et al. [49]. Future wind farm controlresearch should therefore incorporate such estimation techniques, and prefer-ably assess their robustness experimentally.• Using machine learning for data-driven control: Machine learning tech-niques such as convolutional or recurrent neural networks have successfullybeen employed to model wake phenomena [132, 186]. Furthermore, the useof nonparametric Gaussian processes has proven to be highly successful inwind tunnel tests conducted by Park et al. [148]. The applicability of thesetechniques, along with classical forms of system identification, to wind farmcontrol should be further investigated.• Assessing joint wind farm and wind turbine-level control: One majordrawback associated with yaw-based wake redirection and turbine reposi-tioning is the increase in fatigue damage encountered by rotors that are par-tially shadowed. Fleming et al. [58] demonstrated via SOWFA simulationsthat individual blade pitch control could reduce such accumulated fatiguedamage. The performance of wind farm controllers operating in conjunc-tion with turbine-level control systems, which include individual blade pitchregulation, should therefore be examined in greater detail.• Evaluating dynamic induction and wake steering controllers: Works byMunters and Meyers [138, 139] have shown high potential for dynamic in-duction and wake steering controllers. Their open-loop regulators were de-signed based on detailed understanding of wind field phenomena that engen-65dered wind farm efficiency gains. As a result, they required neither wakemodels, nor model-free online dynamic optimizers. Future research shouldinvestigate other variations of such open-loop controllers, as well as evaluateexisting forms in wind tunnel experiments and field tests.• Using site-specific historical data for wind farm control: The recent promis-ing field test results by Howland et al. [86] on wake redirection have castlight on the merits of using site-specific historical data for wind farm con-trol. While this study utilized static set-points determined via offline op-timization, this approach should be further investigated in conjunction withonline dynamic optimization algorithms and state-estimators that are capableof adapting to time-varying wind conditions.• Understanding wake phenomena using CFD: The most significant leapstoward developing effective wind farm controllers have stemmed from CFDsimulations of wake aerodynamics. For instance, simulations conducted byAnnoni et al. [9] led to an understanding of the physical limitations of powerde-rating, while the findings of Fleming et al. [60] and Munters and Mey-ers [138, 139] shed light on aerodynamic phenomena that may be exploitedto further enhance wind farm control concepts. It is important that theseresearch objectives remain highly active.66Chapter 3Real-time Relocation of FloatingOffshore Wind Turbine Platformsfor Wind Farm EfficiencyMaximization: An Assessment ofFeasibility and Steady-statePotentialThis chapter examines the steady-state potential and feasibility of yaw and induction-based turbine repositioning (YITuR), which is a wind farm control concept thatpassively repositions floating offshore wind turbines using existing turbine con-trol degrees of freedom. To this end, the Floating Offshore Wind Farm Simula-tor (FOWFSim) is developed to model steady-state wind farm power productionwhile considering floating platform relocation. Optimization studies are carriedout with different floating wind farm design parameters and configurations. Theobjective is to determine sets of optimal wind turbine operating parameters that re-locate floating turbines such that wind farm efficiency is maximized. Results showthat the potential of YITuR is starkly limited by wind farm design parameters. In67particular, anchors should be placed adequately far from floating platform neutralpositions, mooring lines should be sufficiently long, and only specific mooringsystem orientations permit substantial gains in wind farm efficiency. With specificcombinations of these parameters, simulation results show that the efficiency ofa 7× 7 floating offshore wind farm may be raised by 42.7% when implementingYITuR in comparison to greedy operation.3.1 IntroductionClustering wind turbines in close proximity to form wind farms offers the pri-mary benefit of capital expense reduction resulting from shared grid connectionand project management costs [29]. From an operational standpoint however, clus-tering engenders aerodynamic interaction between individual wind turbines thatdiminishes the efficiency of wind farms as a whole [30]. As upstream machinesextract the wind’s kinetic energy, viscous interaction along their blades deceler-ates incoming flow and generates wakes with reduced average velocities [115].Downstream wind turbines that are aligned with these wakes produce less electri-cal power as a result [30]. This phenomenon is referred to as the wake effect andmay result in annual wind farm revenue losses ranging from 20 to 30% with tur-bines spaced seven to ten rotor diameters apart [30]. Further, instantaneous windfarm efficiencies may be reduced by as much as 54% [141].One approach to mitigating the wake effect involves appropriately siting in-dividual turbines within wind farms; that is, either increasing inter-turbine spac-ing [30], or utilizing wake models and historical wind velocity data to computeoptimal wind farm layouts [161]. Alternatively, wind farm control employs thedegrees of freedom of individual wind turbines to manipulate the aerodynamicfield such that the wake effect is mitigated and wind farm power output is maxi-mized [66]. Consequently, wind farm control requires no modification to turbinesiting, which enables wind farm developers to prioritize layout design around ter-rain, grid connectivity, installation, transportation, and land permitting consider-ations. Additionally, wind farm control strategies may be implemented in oper-ational wind farms, as demonstrated via recent efforts by the United Kingdom’sCarbon Trust [34].68Two wind farm control concepts that have received considerable attention overthe past decade include power de-rating (also referred to as axial induction-basedcontrol) and yaw-based wake redirection. The former concept de-rates upstreamturbines to reduce the net thrust force exerted upon the wind, which in theory in-creases the average velocity within generated wakes that influence downstream ma-chines [95]. Evaluations of power de-rating have reported differing results, withlarger efficiency gains corresponding to parametric wake models [3, 9, 47, 122,131, 160, 171], and negligible gains or losses associated with computational fluiddynamics (CFD) simulations [9, 48, 141] and field tests [26, 157]. Annoni et al. [9]identified the practical limitations of power de-rating by demonstrating via large-eddy simulations (LES) that the added kinetic power obtained from reducing thethrust force of an upstream turbine is primarily concentrated along the boundary ofthe generated wake, which occupies space beyond the swept-areas of rotors locateddirectly downstream.Yaw-based wake redirection operates upstream wind turbines with nacelle yawmisalignment, which generates unsteady and asymmetric blade loads that deflectgenerated wakes in the crosswind direction [93]. Vortices generated by these asym-metric loads also contribute to wake deflection farther downstream [60]. Theseprocesses result in reduced wake overlap along rotors that are located downstreamfrom turbines operating with yaw offset [58]. This control concept has been proveneffective via CFD simulations [39, 58, 72], wind tunnel experimentation [2, 31, 32,148], and field tests [59].A third wind farm control concept, broadly referred to here as turbine repo-sitioning, minimizes wake overlap by relocating wind turbines in real-time [24].This strategy is applicable to floating offshore wind turbines with non-taught moor-ing lines that allow for horizontal semi-submersible platform or spar buoy motionalong the ocean surface [81]. The few studies that have evaluated the potential ofturbine repositioning have reported promising results. Fleming et al. [58] simu-lated a two-turbine wind farm using the LES-based Simulator for Wind Farm Ap-plications (SOWFA). They incrementally shifted the downstream turbine in bothcrosswind directions and reported a relative wind farm efficiency gain of 41% atapproximately one rotor diameter of relocation relative to a baseline simulationwith no repositioning.69Rodrigues et al. [153] performed optimization studies using the parametricPark wake model [98] to maximize power production from a 5× 5 floating off-shore wind farm. The proposed mechanism for turbine repositioning consisted oftaught mooring lines whose lengths would be regulated by winches mounted on-board floating platforms. By regulating the lengths of its three mooring lines, asingle floating wind turbine could be repositioned within an equilateral-triangulararea with a side length of 330m surrounding its neutral position. Upon validat-ing their optimization results using FarmFlow (a three-dimensional parabolizedturbulent Navier-Stokes solver), a maximum relative wind farm efficiency gain of18.11% was reported across a 330deg span of simulated wind directions.Han et al. [81] proposed a passive mechanism that utilizes the aerodynamicforce acting on wind turbine rotors to reposition floating structures; this concept isdemonstrated in Figure 3.1. A floating offshore wind turbine that operates undergreedy settings (i.e. maximizing power production locally) maintains zero yaw off-set and optimal axial induction; the resulting position of such a turbine correspondsto the faded schematic in Figure 3.1. Introducing yaw misalignment and varyingaxial induction (i.e. by pitching blades or increasing/decreasing the tip-speed ra-tio) alter the magnitude and direction of the thrust force acting on the turbine’srotor. The result is platform motion until the altered aerodynamic force reachesequilibrium with restoring mooring lines forces. Appropriate manipulation of theyaw angle and axial induction factor of a floating wind turbine therefore permitsplatform relocation within a movable range that is limited by mooring line systemproperties. This relocation technique is referred to as yaw and induction-basedturbine repositioning, and will be abbreviated as YITuR for convenience.The objective of the current work is to evaluate the steady-state potential of YI-TuR by performing optimization studies under the assumption of idealized steadywind conditions. This task first requires the development of a floating offshorewind farm model that captures steady-state wake behaviour and turbine displace-ment. This tool is referred to as the Floating Offshore Wind Farm Simulator(FOWFSim). While assessing the effectiveness of YITuR, the influence of windfarm design parameters such as the quantity of turbines, inter-turbine spacing,mooring line length and orientation, and anchor location are investigated. A previ-ous conference publication [101] briefly introduced FOWFSim and presented effi-70Figure 3.1: Schematic demonstrating the concept of yaw and induction-basedturbine repositioning (YITuR). The faded diagram corresponds to afloating offshore wind turbine subject to greedy operation (i.e. agreedy =1/3 and γgreedy = 0deg).ciency gain results for a single wind farm configuration. The current chapter detailsthe mathematical model behind FOWFSim and presents comprehensive simulationresults and analyses concerning multiple wind farm configurations and properties.This chapter is organized as follows; Section 3.2 describes the structure andtheory behind FOWFSim; Section 3.3 defines the optimization problem that issolved to determine optimal turbine operating parameters for maximizing windfarm efficiency; Section 3.4 presents the results of optimization studies carriedout with different wind farm design parameters while providing detailed explana-tions regarding the observed phenomena; and Section 3.5 concludes the chapterby summarizing major findings and offering recommendations for future researchconcerning floating offshore wind farm control.3.2 Mathematical model - FOWFSimThe current section describes in detail the structure and theory behind FOWF-Sim, which is ultimately used to model steady-state floating offshore wind farmperformance. The FLOw Redirection In Steady-state (FLORIS) wake model pre-71sented by Gebraad et al. [72] is combined with a newly-derived steady-state two-dimensional platform displacement model. The result is a simulation tool that,for given mooring system, turbine, and platform properties, determines the posi-tion and power output of every turbine within a floating wind farm in response tosteady wind conditions and turbine operating parameters. It is assumed that allwind turbines within a given wind farm are identical; therefore rotor, platform,and mooring system dimensions are common to all machines and structures. Fur-ther, discussions of the model are focused on floating wind turbines with semi-submersible platforms; the model may be readily extended to simulate spar buoysby simply altering the locations of mooring line fairleads.3.2.1 Wind farm descriptionAs shown in Figure 3.2, a floating wind farm is modelled as a system of N par-ticles distributed along the two-dimensional ocean surface. Each particle repre-sents a floating wind turbine, and the turbines are numbered according to the setF = {1,2, · · · ,N} in an ascending order based on their downstream locations. Iftwo turbines are located at the same downstream distance, the structure with themost negative crosswind location is numbered first. The subscript i ∈F is used toindicate a particular wind turbine. The most upstream turbine is therefore labeledas i = 1, the most downstream machine is denoted by i = N, and all intermediaryturbines are numbered accordingly.Assuming steady wind conditions, the positive x direction is aligned with thefree stream wind (denoted as U∞ in Figure 3.2), and the y direction is chosen ac-cording to the right-hand rule such that the z axis points upward and normal to theocean surface. The resulting outputs of the system x and y represent the locationsof all turbines along the horizontal x-y plane as follows:x :=[x1 x2 · · · xN]T, (3.1)y :=[y1 y2 · · · yN]T, (3.2)where xi and yi denote the location of turbine i along the ocean surface. Also, the72Figure 3.2: Schematic demonstrating the representation of a floating off-shore wind farm as a system of N particles distributed along the two-dimensional ocean surface. The loads and local wind velocity Ui thatare incident on a single turbine i are also shown.position vector ri of turbine i is defined as follows:ri :=[xiyi]. (3.3)These positions are functions of the inputs to the system a and γ , which re-spectively symbolize the axial induction factors and yaw angles of all turbines asfollows:a :=[a1 a2 · · · aN]T, (3.4)γ :=[γ1 γ2 · · · γN]T, (3.5)where ai and γi denote the operating parameters of turbine i. All yaw angles aremeasured as positive counterclockwise from the x axis. At γi = 0deg, turbine i istherefore facing the incoming wind with zero yaw offset.Each turbine is subject to an aerodynamic thrust force vector Fthrust,i and threemooring line force vectors Fmoor,i, j for j ∈M , whereM = {1,2,3} represents the73set of indices that identify the three mooring lines attached to any given floatingplatform. The magnitude of Fthrust,i is a function of the effective wind speed Ui atturbine i, which is dependent upon the location of turbine i as well as the positionsand operating parameters of all machines that are situated upstream of turbine i.Conversely, Fmoor,i, j is independent of the aerodynamics of the wind farm and issolely a function of the position of turbine i.It is important to acknowledge the limitations and assumptions behind FOWF-Sim. First, it neglects floating platform motion in the z direction (i.e. platformheave), as well as rotations about the x and y axes (i.e. platform roll and pitch).Semi-submersible platforms are designed to exert large buoyancy and heave dragforces in order to limit these motions. Their effects on overall wind farm powerproduction are neglected. Second, platform rotation about the z axis (i.e. platformyaw) is neglected. It is assumed that platform yaw resulting from mooring linenonlinearity has negligible effect on steady-state platform displacement. Third,the impacts of ocean waves and currents are neglected. Rotor translational motionresulting from wave-induced platform oscillation is assumed not to significantlyaffect average wind power production. Steady ocean currents would induce a biasin floating turbine positions; however this effect is neglected in the present studysince it is a first examination of YITuR.3.2.2 Wind farm power output and efficiencyThe efficiency of a wind farm ηfarm as the ratio of wind farm power output Pfarm tohypothetical wind farm power output Pfarm,∞ in the absence of wakes is expressedas follows:ηfarm =PfarmPfarm,∞. (3.6)The power outputs Pfarm and Pfarm,∞ are calculated as follows:Pfarm = ∑i∈FPi, (3.7)Pfarm,∞ = N12CP,maxρAU3∞, (3.8)74where ρ is the density of air, A is the rotor swept area, and U∞ is the steady freestream wind speed. The variable CP,max is the optimal power coefficient of the sim-ulated wind turbines under greedy operation, which entails optimal axial inductionand no yaw offset (i.e. agreedy = 1/3 and γgreedy = 0deg) [121]. The power outputPi of turbine i is calculated as follows [29]:Pi =12CP,iρAU3i , (3.9)where Ui is the wind speed at the location of turbine i. The power coefficient CP,iof turbine i is calculated according to actuator disc theory with corrections applieddue to yaw offset as follows [128]:CP,i = 4ai (1−ai)2ηP (cosγi)pP , (3.10)where ηP is the power conversion efficiency and pP is a parameter required forfitting the power coefficient to experimental or numerical data. The values for theseparameters have been obtained from existing implementations of FLORIS [72].3.2.3 Wind farm aerodynamics - FLORISIn order to calculate effective wind speeds at all turbine locations within a windfarm, the FLORIS wake model as outlined by Gebraad et al. [72] is directly em-ployed. This parametric wind farm simulation tool utilizes integral momentumconservation relations from fluid mechanics to estimate centerline deflection andvelocity deficit within wakes [72]. Specifically, the Park wake model [91, 98] isused to compute velocity deficits, and an analytical derivation by Jime´nez et al. [93]serves as a basis for wake deflection calculations. The first strength of the FLORISwake model lies in the fact that its parameters have been estimated using simula-tion results from SOWFA. The second is that, rather than assuming uniform ve-locity distribution within wakes, FLORIS divides each wake into three concentriczones in order to better capture velocity gradients in relation to its predecessors.Recent releases of the FLORIS wake model have implemented Gaussian velocityprofiles within wakes [17] and curled wake effects [124] in order to better predictwind farm aerodynamic behaviour. The original version of FLORIS is used in the75current work due to the unavailability of these releases when simulations were con-ducted. Further, the wake model currently employed in FOWFSim may be readilyreplaced by an alternative simulation tool without altering other computational pro-cesses.The FLORIS model is implemented in a generalized manner wherein every tur-bine may be affected by the wakes generated by any of its upstream counterparts.This assumption is necessary since the locations of floating platforms are unknownvariables, and specific combinations of turbine operating parameters may bring anytwo turbines near alignment with the wind. The only simplifying assumption thatmay be imposed given the nature of movable platforms is that turbines operate in-dependently of their downstream neighbors. The resulting computational structureof FOWFSim is shown in Figure 3.3. When computing the local wind speed Ui atturbine i, the FLORIS wake model accepts as inputs the free stream wind speedU∞ and the operating parameters and positions of all turbines located upstream ofturbine i. In other words, Ui is a function of ak, γk, xk, and yk for k ∈ Ui, wherethe set Ui = {1,2, · · · , i−1} numbers all wind turbines that are located upstreamof turbine i. The turbine displacement and mooring line models then determinethe steady-state location of turbine i. This computation is repeated for each tur-bine starting from the most upstream machine and proceeding downstream. Thefollowing subsections provide greater detail on the remaining two functions shownin Figure 3.3.3.2.4 Floating turbine displacement modelThe steady-state position of turbine i is determined by the equilibrium establishedbetween all forces that act upon it. As shown in Figure 3.2, these entities arethe aerodynamic thrust force Fthrust,i and the net mooring line force Fmoor,i. Theequilibrium between these forces is described as follows:Fthrust,i+Fmoor,i = 0, (3.11)where each of these vectors contains the x and y components of the respectiveforces acting on turbine i. Since Fthrust,i is a function of Ui, and since both Uiand Fmoor,i are nonlinearly dependent upon floating platform positions, the above76Figure 3.3: Block diagram showing the computation modules of FOWFSimalong with information transfer routes.equation must be numerically solved with respect to xi and yi using a Newton-Raphson algorithm. Since the position and effective wind speed associated witha given turbine are independent of the positions and operating parameters of itsdownstream counterparts, Eq. (3.11) is solved using a sweeping algorithm startingfrom the leading turbine to the trailing machine.Similar to power production, aerodynamic thrust forces are also modeled ac-cording to actuator disc theory [121]. The thrust force vector acting the rotor ofturbine i is calculated as follows [72]:Fthrust,i =12Cthrust,iρA(Ui cosγi)2[cosγisinγi]. (3.12)The above equation computes the magnitude of the thrust force based on the windvelocity projected normal to a turbine’s rotor, and then computes the thrust forcevector components based on the yaw angle of the turbine’s nacelle. The thrustcoefficient Cthrust,i of turbine i is calculated as follows [72]:Cthrust,i = 4ai (1−ai) . (3.13)The net mooring line force acting on turbine i is calculated as a summation of77Figure 3.4: Schematic demonstrating the position vectors that are used to de-scribe the mooring system for turbine i.all three connected mooring line forces as follows:Fmoor,i = ∑j∈MFmoor,i, j, (3.14)where the force Fmoor,i, j exerted by mooring line j on turbine i (refer to Figure 3.2)is computed as follows:Fmoor,i, j =−HF,i, j rA→F,i, jLA→F,i, j . (3.15)The vectors that are relevant to mooring force calculations are described in Fig-ure 3.4. The position vector rA→F,i, j measures from the jth anchor of turbine i toits jth fairlead as follows:rA→F,i, j = ri+ rG→F,i, j− rA,i, j, (3.16)where rG→F,i, j is a constant vector that symbolizes the location of the jth fairleadof turbine i relative to the center of its floating platform, and rA,i, j is also a constantvector that measures the location of the jth anchor of turbine i relative to the originof the coordinate system.78The horizontal tension in the jth cable that is connected to turbine i is a func-tion of the horizontal distance LA→F,i, j between the cable’s respective anchor andfairlead, which is calculated as follows:LA→F,i, j = ‖rA→F,i, j‖, (3.17)where ‖ · ‖ represents the Euclidean norm. The magnitude of the horizontal cabletension HF,i, j is then obtained from the solution of the suspended cable problemdefined in Section 3.2.5. Since platform motion in the z direction is neglected,the vertical distance between the anchor and fairlead of any mooring line remainsunchanged. The only independent variable influencing HF,i, j is therefore LA→F,i, j.This characteristic is exploited and HF,i, j is computed using a lookup table functionas follows:HF,i, j = fmoor (LA→F,i, j) . (3.18)The lookup table function fmoor(·) is constructed for each new set of mooring lineproperties prior to running wind farm simulations or conducting optimization stud-ies.3.2.5 Mooring line tensionsThe horizontal tension generated within a mooring line is dependent upon the gen-eral shape of its cable, which is in turn a function of the location of its fairlead.Figure 3.5 identifies three different zones in which a fairlead may be located thatdetermine the general shape of a mooring line cable. Zone 1 is active when thehorizontal distance between the fairlead and anchor is small enough such that thecorresponding cable is partially resting on the seabed while its suspended portionis vertical with no catenary profile. Zone 2 is active when the fairlead is locatedbeyond the boundary of zone 1, where the cable is still partially resting on theseabed and its suspended portion possesses a catenary profile, up to a horizontalposition where no portion of the cable rests on the seabed and the slope of the cableprofile at the anchor is zero (i.e. no vertical force at the anchor). The blue line inFigure 3.5 is an example of a cable operating within zone 2. The green and orangelines represent the boundaries of zone 2. Zone 3 is then active when no portion79Figure 3.5: Schematic demonstrating the three zones in which mooring linefairleads may be located. The zone that is active determines the cableprofile and tension.of the cable rests on the seabed and the slope of the cable profile at the anchoris greater than zero, which results in vertical pull on the anchor. The red line inFigure 3.5 is an example of a cable operating within zone 3.The horizontal tension in the cable is equal to zero when the fairlead is locatedwithin zone 1. Finding the cable tension when zones 2 and 3 are active requiressolving the problem of an elastic cable suspended under its weight. Solutions to asimilar problem have been presented by Jonkman [97] with no derivation. Sincethe solutions contained in the current work differed from those of Jonkman [97],details concerning the setup and derived expressions are presented. While zone 2 isactive, the cable suspension problem is defined according to Figure 3.6a. The sus-pended cable portion (i.e. the blue line) is lifted by a vertical force VF at the fairleadwhich counteracts the weight of the cable Fw. The horizontal tension throughoutthe catenary profile is uniform and equal to the fairlead horizontal force HF. Thecable portion resting on the seabed (i.e the orange line) is subject to a horizontalanchor force HA, a static friction force Fs, and a horizontal reaction force of magni-tude HF exerted by the tension of the suspended portion of the cable. The resultingdifferential equations that describe the equilibrium of forces for this scenario are80formed as follows:dVds= 0 , s< Lsw1+ TAcE , s≥ Ls , (3.19)dHds=µsw1+ HAcE, s< Ls0 , s≥ Ls, (3.20)where H, V , and T are the horizontal, vertical, and total tensions at any point alongthe cable, s is a general descriptor of arc length along the cable measured from theorigin depicted in Figure 3.6a, w is the weight of the cable per unit length in oceanwater, µs is the coefficient of static friction between the cable and seabed, and Acand E are the cross-sectional area and elastic modulus of the cable, respectively.The total tension T is calculated as follows:T =√H2+V 2. (3.21)If the fairlead is located within zone 3, as shown in Figure 3.6b, the cableassumes a fully catenary profile with uniform horizontal tension HF throughout.The vertical fairlead force VF must then counteract the weight of the cable Fw andthe vertical anchor force VA. The resulting differential equation describing forceequilibrium in the vertical direction is therefore set up as follows:dVds=w1+ TAcE. (3.22)The solutions to these differential equations are obtainable analytically and,based on the zones defined in Figure 3.5, their solutions are grouped within fmoor(·)as follows:fmoor(xF) =0 , xF ≤ xF,1→2f1(xF) , xF,1→2 < xF ≤ xF,2→3f2(xF) , xF > xF,2→3, (3.23)where xF is the horizontal position of the fairlead relative to the anchor, whichis equal to LA→F,i, j, xF,1→2 and xF,2→3 are the fairlead locations that separate the81Figure 3.6: Schematic demonstrating the forces acting on different cable por-tions when the fairlead of the mooring line is located within (a) zone 2and (b) zone 3.different zones identified in Figure 3.5, and f1(·) and f2(·) are solutions to thesuspended cable problems corresponding to zones 2 and 3, respectively.First, the problems defined in Figure 3.6 are solved to yield the following ex-pressions for the transition locations between the different zones (refer back toFigure 3.5 for a description of these locations):xF,1→2 = Lc− zF, (3.24)xF,2→3 =HF,2→3w[wLcAcE+ sinh−1wLcHF,2→3], (3.25)where Lc is the length of the unstretched cable, zF is the vertical location of the82fairlead relative to the anchor, which is constant in all simulations since verticalplatform motion is neglected, and HF,2→3 is the horizontal cable tension when thefairlead is located at xF,2→3 and is calculated as follows:HF,2→3 =wLc2[1−(zFLc− wLc2AcE)2]( zFLc− wLc2AcE)−1. (3.26)The function f1(·), which outputs the horizontal fairlead force HF while zone 2is active, is obtained by solving the following system of equations for HF and VFusing a Newton-Raphson algorithm:xF−Ls = HFw(VFAcE+ sinh−1VFHF), (3.27)zF =1w V 2F2AcE −HF1−√1+(VFHF)2 , (3.28)where Ls is the stretched length of the cable portion that is resting on the seabed(i.e. the length of the orange line in Figure 3.6a) and is calculated as follows:Ls = Lc− VFw +(1+ HFAcE)3−[(1+ HFAcE)2− 2µswAcE xs]323µswAcE− xs. (3.29)The variable xs is the distance from the start of the catenary portion of the cable(i.e. where the orange and blue lines meet in Figure 3.6a) to the location along theflat cable portion where the horizontal tension is minimal. This minimal tensionpoint either corresponds to the position of the anchor, or some intermediate locationalong the flat cable portion where the tension reaches zero due to the opposition ofFs against HF. The distance xs is therefore calculated as follows:xs = min[Lc− VFw ,HFwµs(1+HF2AcE)](3.30)The function f2(·), which outputs the horizontal fairlead force HF while zone 3is active, is obtained by solving the following system of equations for HF and VF83using a Newton-Raphson algorithm:xF =HFw(wLcAcE+ sinh−1VFHF− sinh−1 VF−wLcHF), (3.31)zF =LcAcE(VF− wLc2)+HFw√1+(VFHF)2−√1+(VF−wLcHF)2 . (3.32)3.3 Optimization problemThe goal of optimization in the context of the current study is to determine thesets of axial induction factors a and yaw angles γ that maximize wind farm effi-ciency ηfarm. Obtaining ηfarm for given a and γ vectors requires solving Eq. (3.11)to determine the position of each floating turbine. This process involves 2N im-plementations of a Newton-Raphson algorithm to find the x and y positions of Nplatforms for each calculation of ηfarm. To improve computational efficiency, theNewton-Raphson algorithm is removed from the function ηfarm by instead defininga nonlinearly constrained optimization problem as follows:mina,γ,x,y1−ηfarm (a,γ,x,y) ,subject to0.2≤ ai ≤ 0.4,−20deg≤ γi ≤ 20deg,xi,min ≤ xi ≤ xi,max,yi,min ≤ yi ≤ yi,max,Fthrust,i+Fmoor,i = 0,(3.33)where the cost function 1−ηfarm represents the wind farm efficiency deficit, andxi,min, yi,min, xi,max, and yi,max are the bounds of the optimization variables xi andyi. These bounds are determined uniquely for each turbine assuming ±200m ofmovable range surrounding their neutral positions. This range is simply selectedto be large enough such that it exceeds the actual movable range of the floatingwind turbines. The bounds on γi are selected to conservatively maintain nacelle84yaw below ±25deg. Yaw offset beyond this range has been attributed to excessivefatigue loads [59]. The upper bound on ai is set to uphold the validity of actuatordisc theory, while the lower bound prevents excessive power de-rating.The minimization problem defined in Eq. (3.33) is solved using the MATLABoptimization toolbox based on Sequential Quadratic Programming (SQP). This op-timization framework forms a quadratic approximation of the objective function ateach iteration, while at the same time linearizing constraint equations and incor-porating them into the objective function using Lagrange multipliers [142]. Thisquadratic problem is then solved using an efficient matrix decomposition tech-nique [142]. Along with interior-point methods, SQP is considered the most ef-fective technique for solving nonlinearly constrained optimization problems [142].All axial induction factors are initialized according to greedy operation (i.e. ainit =agreedy = 1/3). To initialize the yaw angle vector γ init, a simple algorithm thatidentifies which pairs of turbines are strongly coupled aerodynamically, and thenensures that their nacelles are initially yawed in opposite directions, was developed.Initial turbine positions xinit and yinit are obtained by solving FOWFSim using ainitand γ init.3.4 Results and discussionThe current section first assesses the impacts of various floating wind farm designparameters on the potential of YITuR. These parameters include anchor locationand mooring line cable length (Section 3.4.1), mooring system orientation (Sec-tion 3.4.2), and wind farm size and inter-turbine spacing (Section 3.4.3). For theseanalyses, a wind-aligned row of floating wind farms is simulated and optimized.Then, the increase in wind farm efficiency obtained using YITuR is computed forvarious wind directions and wind farm sizes (Section 3.4.4). It is emphasized thatthe results presented in this section are obtained via optimal YITuR operation,which is implemented by solving Eq. (3.33) for each given set of wind farm prop-erties and wind conditions in order to acquire optimal a and γ values.The wind turbines and floating platforms that are simulated are based upondesigns outlined by Jonkman et al. [96] and Robertson et al. [152]; parametricdetails of these designs are listed in Table 3.1. One important characteristic of85Table 3.1: Properties of the floating wind turbine detailed by Jonkman etal. [96] and Robertson et al. [152].Baseline wind turbine propertiesRotor radius, R 63mPower conversion efficiency, ηP 0.768Power coefficient data fitting parameter, pP 1.88Baseline floating platform propertiesDistance from platform center to fairleads, LG→F 40.87mMooring line propertiesAnchor distance from neutral platform, LA→G,neutral 837.6mVertical distance between anchors and fairleads, zF 186mCable length of baseline mooring system, Lc 835mCable diameter, Dc 7.66cmCable weight per unit length in water, w 1065.7N/mCable tension per meter of elongation, AcE 753.6×106 NStatic friction coefficient between cable and seabed, µs 1.0the simulated floating platform is its baseline orientation relative to the incomingwind direction; this orientation is shown in Figure 3.7. Mooring line 1 is ori-ented 60deg counterclockwise from the positive x axis, and subsequent cables areoriented 120deg counterclockwise from their preceding line. This figure will be al-luded to in Section 3.4.2 when mooring system orientation changes are discussed.3.4.1 Effects of anchor location and cable lengthIt is first necessary to determine what length of mooring line cables and what an-chor locations will permit adequate efficiency gains from YITuR operation. Sincethe distance between an anchor and its respective floating platform influences thelength of the connecting mooring line, the cable length factor αcable is defined toparametrize mooring line length Lc as follows:Lc = Lc,min+αcable (Lc,max−Lc,min) , (3.34)where the smallest permissible cable length Lc,min for any anchor location is definedas follows:Lc,min =√x2F,neutral+ z2F, (3.35)86Figure 3.7: Schematic demonstrating the baseline orientation of a floatingplatform and mooring system relative to the incoming wind direction.The orientation of a mooring system that is rotated by a value θmoor isalso presented in the faded diagram.and the largest permissible cable length Lc,max for any anchor location is defined asfollows:Lc,max = xF,neutral+ zF. (3.36)The constant xF,neutral is the distance from anchor to fairlead when the respectivefloating platform is in its neutral position (i.e. no aerodynamic loads are present),and is calculated as follows:xF,neutral = LA→G,neutral−LG→F, (3.37)where LA→G,neutral is the distance between any anchor to the center of gravity ofits respective floating platform under neutral conditions, and LG→F is the distancebetween the center of gravity of a floating platform to any of its fairleads. Thelength Lc,min is small enough such that the unstretched cable follows a straight linefrom anchor to fairlead, while Lc,max is a large enough length such that the cable87is suspended vertically from the fairlead until it contacts the seabed. The cablelength factor therefore falls within the range 0 ≤ αcable ≤ 1 and linearly specifiesthe length of a cable such that Lc,min ≤ Lc ≤ Lc,max.Figure 3.8 plots the relative gain in the efficiency ∆ηfarm,rel of an optimized two-turbine floating wind farm as a function of different anchor placement distancesLA→G,neutral and cable length factors αcable. The general trends in this figure lead tothe following important point:Observation 1. Lengthening mooring line cables while keeping anchor locationsfixed increases the effectiveness of YITuR operation.For example, with LA→G,neutral = 800m, increasing αcable from 0.55 to 0.8 raises∆ηfarm,rel from 10.0 to 31.8%. This finding is explained by the simple fact that ifmooring lines are not sufficiently long, then floating platforms cannot be relocatedfar enough in the crosswind direction for YITuR to be fully effective in reducingwake overlap. Figure 3.9 shows the positions of the two floating wind turbinesfor two scenarios; αcable = 0.55 and αcable = 0.95, with LA→G,neutral = 900m inboth cases. When αcable = 0.55, the upstream turbines is shifted by 24.0m in thenegative y direction and the downstream machine is relocated by 39.4m in thepositive y direction. The total crosswind offset between their axes of rotation istherefore 63.4m, which is approximately equal to half of one rotor diameter of126m, and a relative wind farm efficiency gain of 10.0% is achieved. Raisingαcable to 0.95 increases this crosswind offset to 145.3m, which exceeds a singlerotor diameter and generates a larger relative efficiency gain of 34.6%.The following point may also be drawn from the trends depicted in Figure 3.8:Observation 2. Placing mooring system anchors farther from floating turbine neu-tral positions increases the effectiveness of YITuR operation.Fixing αcable = 0.8, increasing LA→G,neutral from 200 to 900m raises ∆ηfarm,rel from16.3 to 32.4%. This effect is attributed to the nonlinearity of mooring line tensions.The horizontal and vertical components of mooring line tension for a sample cableare plotted in Figure 3.10 as a function of the horizontal distance xF between its an-chor and fairlead. Notice that there exists a value of xF at which the cable tensionsbegin to rise dramatically; this point corresponds to xF,2→3, which is the fairlead88200 300 400 500 600 700 800 900Horizontal distance between the neutral position of a floatingplatform and its respective anchors, LA G,neutral (m) 0.050.10.150.20.250.30.350.4Relaive gain in wind farm efficiencyobtained using YITuR, farm,rel0.550.60.650.70.750.80.850.90.95Cable length factor, cableFigure 3.8: Relative gain in wind farm efficiency using YITuR operation withdifferent mooring line length factors αcable and anchor distances fromturbine neutral positions LA→G,neutral. Simulation properties: 1×2 windfarm size, free stream wind speed U∞ = 8m/s, inter-turbine spacing of7D, baseline mooring system orientation θmoor = 0deg.location that begins to fully lift the cable off of the seabed. For the cable plottedin Figure 3.10, xF,2→3 occurs at 902.9m. This dramatic rise in cable tension is thelimiting factor for YITuR, since it leads to a large restoring mooring line force thatcounters aerodynamic loads and prevents further platform displacement.Placing an anchor farther away from its respective platform while maintain-ing αcable constant allows for a larger range of platform motion before xF reachesxF,2→3. Figure 3.11 demonstrates this concept by showing the profiles of two moor-ing lines before and after platform displacement. Let the anchor in system a) beplaced closer to the neutral position of its respective platform in comparison to thatof system b). Let both of these turbines then be moved to the right by an equaldistance such that the mooring line from system a) reaches its xF,2→3 value (whichcorresponds to the orange line in Figure 3.11a). The turbine from system a) has890 200 400 600 800 1000 1200Downstream distance, x (m)-80-60-40-20020406080Crosswind distance, y (m)Neutral positionscable = 0.55cable = 0.952 = 5.8 dega2 = 0.331 = -8.3 dega1 = 0.271 = -4.9 dega1 = 0.332 = 17.2 dega2 = 0.36Figure 3.9: Floating wind turbine positions corresponding to optimization re-sults presented in Figure 3.8. Simulation properties: 1× 2 wind farmsize, free stream wind speed U∞ = 8m/s, inter-turbine spacing of 7D,baseline mooring system orientation θmoor = 0deg, distance from an-chors to turbine neutral positions LA→G,neutral = 900m.now reached the point where its cable is being fully lifted off of the seabed, andthus the restoring force that counteracts further displacement begins to rise dramat-ically. The cable from system b), however, still has a large portion of cable restingon the seabed (refer to the flat portion of the orange line in Figure 3.11b), and istherefore far from its xF,2→3 value. As a result, the turbine from system b) has ad-ditional range of motion toward the right before its cable tension begins to increasesharply. This turbine will therefore relocate farther from its neutral position beforemooring line forces equalize with aerodynamic loads.It may be concluded from the above findings that, in order for YITuR to be aneffective means of wind farm control, anchors must be placed sufficiently far fromthe neutral positions of their respective platforms and cables must be adequatelylong. These characteristics will ensure a range of platform motion that permits900 200 400 600 800 1000 1200Horizontal distance between anchor and fairlead, xF (m)020406080100120140160180Mooring line tension at fairlead, HF and V F (MN) Horizontal tensionVertical tensionFigure 3.10: Horizontal and vertical components of cable tension corre-sponding to a mooring line of length Lc = 925m with distance fromanchor to platform neutral position LA→G,neutral = 837.6m.significant reduction of wake overlap. However, this range of motion must notbe excessive since there may be safety and practicality concerns associated withmassive floating structures freely moving within large radii.3.4.2 Effects of mooring system orientationAnother question of interest is whether the nonlinearity of mooring line forcesalters the potential of YITuR if mooring systems are oriented differently. The angleθmoor is used to denote the counterclockwise rotation of a mooring system relativeto the baseline design described in Figure 3.7. The faded platform, cables, andanchors in this diagram represent the rotated mooring system. Since a rotation of120deg results in mooring system orientation that is identical to that of the baseline,only the range 0deg≤ θmoor ≤ 120deg is considered.Figure 3.12 plots the relative gain in wind farm efficiency ∆ηfarm,rel resultingfrom YITUR operation with different values of θmoor and αcable. Similar to Sec-91Figure 3.11: Schematic demonstrating the effect of increased anchor distancefrom floating platform neutral position.tion 3.4.1, the simulated wind farm consists of two floating wind turbines alignedwith the wind. It is observed that regardless of the value of αcable, YITuR is onlyeffective in significantly raising wind farm efficiency in the approximate rangesof 0deg ≤ θmoor ≤ 20deg and 100deg ≤ θmoor ≤ 120deg. For instance, withαcable = 0.8, rotating the baseline mooring system counterclockwise by 20deg re-duces ∆ηfarm,rel from 32.0 to 21.3%. This result implies a third important point:Observation 3. The baseline mooring system orientation shown in Figure 3.7 isoptimal for YITuR, and variations from this orientation exceeding ±20deg willyield significant reduction in wind farm efficiency gains.The trends in Figure 3.12 are the direct result of mooring line tension nonlinear-ity. Consider the floating wind turbines and mooring systems shown in Figure 3.13.With the wind blowing from the left, the mooring system in Figure 3.13a is orientedsimilar to the baseline from Figure 3.7 while the mooring system in Figure 3.13b isrotated 60deg counterclockwise relative to the baseline orientation. The platformlocations shown correspond to zero yaw offset in both cases. Under this setting,920 20 40 60 80 100 120Orientation of mooring line system relativeto baseline position, moor (deg) 00.050.10.150.20.250.30.35Relative gain in wind farm efficiency obtained using YITuR, farm,rel0.550.60.650.70.750.8Cable length factor, cableFigure 3.12: Relative gain in wind farm efficiency using YITuR operationwith different cable length factors αcable and mooring system orienta-tions θmoor. Simulation properties: 1× 2 wind farm size, free streamwind speed U∞ = 8m/s, inter-turbine spacing of 7D, distance fromanchors to platform neutral positions LA→G,neutral = 837.6m.mooring lines 1 and 3 of system a) are far from zone 3 operation (refer back toFigure 3.5 for discussions on mooring line zones) since their respective fairleadsand anchors are in close proximity. These cables therefore do not exert large restor-ing forces even if yaw offset were introduced and the platform were to shift in thecrosswind direction. Additionally, although mooring line 2 of system a) is close tozone 3 operation, it exerts zero force in the crosswind direction. As a result, if yawoffset were introduced, the floating turbine in system a) must traverse a significantdistance in the crosswind direction for the tension in cable 2 to equalize againstaerodynamic forces.The same characteristic does not hold true for the mooring system in Fig-ure 3.13b. Due to its altered orientation, cables 1 and 2 of system b) operate closerto zone 3 and their horizontal tension components will rise sharply as their fair-leads move farther away from their anchors. Furthermore, these two cables are93Figure 3.13: Schematic demonstrating the effects of mooring system orienta-tion on the crosswind mobility of floating offshore wind turbines.oriented such that they exert restoring forces in the crosswind direction even with-out yaw offset. Consequently, if yaw misalignment were introduced, the floatingturbine from system b) would have a smaller range of mobility in comparison tothat of system a) before mooring line forces equalize with aerodynamic loads. Fig-ure 3.14 plots the locations of the floating turbines simulated in Figure 3.12 atdifferent mooring system orientations. Increasing θmoor from 0 to 20deg reducesthe optimal crosswind offset between the two turbines from 139.6 to 97.7m (or1.1D to 0.8D). Further raising θmoor to 40deg decreases this crosswind offset to940 200 400 600 800 1000 1200Downstream distance, x (m)-80-60-40-20020406080Upstream distance, y (m)Neutral positionsmoor = 0 degmoor = 20 degmoor = 40 deg2 = 14.0 dega2 = 0.332 = 20 dega2 = 0.342 = 10.4 dega2 = 0.341 = 9.7 dega1 = 0.281 = -7.8 dega1 = 0.331 = 0.6 dega1 = 0.33Figure 3.14: Floating wind turbine positions corresponding to optimizationresults presented Figure 3.12. Simulation properties: 1×2 wind farmsize, free stream wind speed U∞ = 8m/s, inter-turbine spacing of7D, distance from anchors to platform neutral positions LA→G,neutral =837.6m, cable length factor αcable = 0.8 (i.e. cable length Lc = 950m).8.95m (or 0.1D).Another observation from Figure 3.14 is that bias appears in the crosswindpositioning of floating platforms as θmoor deviates from zero. At θmoor = 20degfor instance, turbine 1 maintains an insignificant yaw offset angle of 0.6deg yetis shifted by 65.7m in the −y direction. Additionally, with θmoor = 40deg, bothturbine nacelles are yawed counterclockwise yet they are displaced in the −y di-rection. Clearly, adjusting the mooring system orientation offsets the crosswindsteady-state positions of the floating platforms even with zero yaw misalignment.Introducing yaw offset then repositions turbines relative to these biased steady-statelocations. This outcome is also attributed to the nonlinearity of mooring line forces.Figure 3.15 shows the mooring system orientation corresponding to θmoor = 20deg.As wind drives the floating platform in the downstream direction, the tensions inboth mooring lines 1 and 3 decrease since their respective anchors move closer to95Figure 3.15: Schematic demonstrating the origin of bias in the steady-statedisplacements of a floating offshore wind turbines when mooring sys-tem orientations deviate from the baseline shown in Figure 3.7.their fairleads; thus, these mooring lines diverge from zone 3 operation. Mooringline 2, however, approaches zone 3 operation and experiences a significant increasein tension which, due to its orientation, exerts a force on the floating platform inthe −y direction. This force introduces the bias which is evident in Figure 3.14.The above observations indicate that care must be given to the orientation offloating turbine mooring systems if YITuR (or any turbine repositioning concept)is to be employed for wind farm control. The orientation of a mooring systemplaces limits on the mobility of floating platforms, which diminishes the potentialof any turbine repositioning concept to maximize wind farm efficiency. This pointis addressed quantitatively in Section 3.4.4, where the effect of wind direction onthe potential of YITuR is examined.3.4.3 Effects of turbine spacing and wind farm sizeThe effect of altering the spacing between wind turbines on the potential of YITuRis also examined. Varying inter-turbine spacing alone is uninteresting however.Increased spacing allows for more time and space over which wakes re-energize96and reduce their velocity deficits [115], hence raising wind farm efficiency andrendering any wind farm control concept redundant. Instead, the simultaneousimpacts of spacing and wind farm size (i.e. the number of installed wind turbines)are studied to answer the following question; how beneficial is YITuR in terms ofincreasing the turbine-density of a wind farm without relinquishing efficiency?Figure 3.16 plots the efficiency ηfarm of a wind-aligned row of N wind turbineswith various inter-turbine spacings under both greedy1 and YITuR operation. Thewind direction and mooring system orientation are defined according to the base-line system defined in Figure 3.7, and the mooring line length in these simulationsis 925m. The following significant conclusion may be derived from Figure 3.16:Observation 4. By introducing YITuR operation, a wind farm developer may in-crease the quantity of wind turbines within the same farm area by N − 1 whileapproximately maintaining wind farm efficiency.For instance, a wind farm consisting of a row of N = 5 turbines placed 8D apart andoperating with greedy settings has an efficiency ηfarm = 60.7%. If a single windturbine were installed between each existing pair, which would increase the windfarm size to N = 9 and decrease inter-turbine spacing to 4D without requiring ad-ditional wind farm area, the efficiency of the new wind farm with YITuR operationwould remain at 60.0%.To explain this outcome, wind farm layouts corresponding to N = 5 with 8Dspacing and N = 9 with 4D spacing are plotted in Figure 3.17. Since YITuR min-imizes wake overlap by yawing and relocating adjacent floating wind turbines inopposite directions, it effectively doubles the downstream spacing between any twomachines with significant wake interaction. Referring to Figure 3.17, the floatingplatforms corresponding to N = 9 are spaced 4D apart under neutral conditions.YITuR then shifts adjacent pairs in this set of turbines in opposite crosswind di-rections. The result is two new rows at approximately y = ±60m along whichadjacent turbines that are directly aligned with the wind are actually spaced 8Dapart. As a result, this wind farm will have the same effective inter-turbine spac-ing and efficiency as one with N = 5 turbines under greedy operation, since thesemachines will also be spaced 8D apart.1Refer to the discussion surrounding CP,max in Section 3.2.2 for a description of greedy operation.972 3 4 5 6 7 8 9 10Number of wind turbines in a single wind-aligned row, N0.20.30.40.50.60.70.80.91.0Wind farm efficiency, farm4D, YITuR4D, Greedy6D, YITuR6D, Greedy8D, YITuR8D, Greedy10D, YITuR10D, GreedyTurbine spacing, operationFigure 3.16: Efficiency ηfarm of a wind-aligned row of N turbines obtainedfrom greedy and YITuR operation for different inter-turbine spacingsand wind turbine quantities N. Simulation properties: free stream windspeed U∞ = 8m/s, mooring line cable length Lc = 925m, distancefrom anchors to platform neutral positions LA→G,neutral = 837.6m,baseline mooring system orientation θmoor = 0deg.3.4.4 Wind farm efficiency studiesAs a final assessment, the potential of YITuR for maximizing the efficiencies ofseveral wind farms of different sizes subject to different wind directions is exam-ined. All wind farms possess a grid layout with turbines spaced 7D apart in both xand y directions. All mooring systems are oriented according to the baseline designshown in Figure 3.7 with anchors located 837.6m away from the neutral positionsof their respective platforms and cables that are 925m long. The wind speed in allsimulations is 8m/s.Figure 3.18 shows polar plots of wind farm efficiencies under greedy and YI-TuR operation as functions of wind direction; each plot corresponds to a differ-ent farm size. For all wind farm sizes, greedy operation yields significantly low980 500 1000 1500 2000 2500 3000 3500 4000Downstream distance, x (m)-60-40-20020406080Upstream distance, y (m) Neutral positions, N = 5, 8D spacingNeutral positions, N = 9, 4D spacingGreedy, N = 5, 8D spacingYITuR, N = 9, 4D spacingFigure 3.17: Floating wind turbine positions corresponding to optimizationresults presented Figure 3.16. Simulation properties: free stream windspeed U∞ = 8m/s, mooring line cable length Lc = 925m, distancefrom anchors to platform neutral positions LA→G,neutral = 837.6m,baseline mooring system orientation θmoor = 0deg.wind farm efficiencies near wind directions that are aligned with or diagonal tothe farm’s grid layout (i.e. 0, 45, 90deg, etc.). At these angles, the distances be-tween turbines that are directly aligned with the wind are reduced; hence, wakes donot sufficiently re-energize via turbulent diffusion prior to interacting with down-stream rotors. When assessing the potential of YITuR operation for mitigating thisproblem, the following major point is established:Observation 5. YITuR operation is not equally effective at raising wind farm effi-ciency across all wind directions.At a wind direction of 0deg for example, YITuR operation is significantly bene-ficial as it increases ηfarm for a 2× 2 wind farm from 73.3 to 89.3%, which is a21.8% relative rise. At wind directions of 90, 180, and 270deg however, relativeefficiency gains obtained via YITuR operation are 7.1, 6.4, and 4.5%.99This limitation is directly attributed to discussions from Section 3.4.2 regardingthe effects of mooring system orientation on floating turbine mobility. Moreover, itis an extension of Observation 3 since changing the direction of the wind is tanta-mount to altering mooring system orientation relative to the incoming wind. Con-sequently, YITuR is only effective when the relative orientation between mooringsystems and the wind is close to the baseline scenario shown in Figure 3.7. Windfarm developers hoping to benefit from YITuR must therefore identify the winddirection at which efficiency is minimal, and then orient mooring systems appro-priately in order to maximize the potential of this wind farm control technique.Alternatively, mooring line orientation and YITuR operation could be consideredin optimization processes that are used to design wind farm layouts.As wind farm size is increased from 2× 2 to 7× 7, efficiencies under greedyoperation and at a wind direction of 0deg drop from 73.3 to 52.0%. This outcomeis expected; a larger quantity of wind turbines means that a higher portion of ma-chines experience aerodynamic coupling. Losses resulting from the wake effect aretherefore amplified. By the same logic, YITuR operation, or any wind farm controlconcept, must be more effective in raising efficiency for larger wind farm sizes;that is, a higher portion of machines experience the benefits of wind farm control.This outcome is observed in Figure 3.18. As wind farm size is increased from 2×2to 7× 7, the absolute gain in wind farm efficiency obtained via YITuR operationrises from 16.0 to 22.2% at a wind direction of 0deg. Relative gains in wind farmefficiency increase from 21.8 to 42.7%; although these values are inflated due tothe substantial drop in efficiency of the 7×7 wind farm under greedy operation.3.5 Conclusions and recommendations for futureresearchThe goal of this chapter has been to evaluate the steady-state potential of Yaw andInduction-based Turbine Repositioning (YITuR) as a control strategy for floatingoffshore wind farms. Optimization studies have shown that this potential is highlydependent upon wind farm design parameters. Mooring lines must be sufficientlylong to permit significant platform displacement such that wake overlap along tur-bine rotors may be minimized. Concurrently, due to the nonlinearity of mooring10003060901201501802102402703003300.40.50.60.70.80.91YITuRGreedyFarm operation03060901201501802102402703003300.40.50.60.70.8103060901201501802102402703003300.40.50.60.70.80.9103060901201501802102402703003300.40.50.60.70.80.9103060901201501802102402703003300.40.50.60.70.80.9103060901201501802102402703003300.40.50.60.70.80.913 32 24 4 5 56 6 7 7Figure 3.18: Efficiency ηfarm of wind farms with grid layouts and differentsizes under greedy and YITuR operation subject to different wind di-rections. The radial axes represent wind farm efficiency ηfarm and thecircumferential axes denote the wind direction with 0deg indicatingwind flow to the right. Simulation properties: free stream wind speedU∞ = 8m/s, inter-turbine spacing of 7D in x and y directions, mooringline cable length Lc = 925m, distance from anchors to turbine neutralpositions LA→G,neutral = 837.6m, baseline mooring system orientationθmoor = 0deg.101line tensions, anchors must be placed far enough from the neutral positions of float-ing turbines such that platforms may be adequately relocated before mooring linerestoring forces increase exorbitantly. Platform mobility was also heavily influ-enced by mooring system orientation. The nonlinearity of mooring line tensionsengendered unequal restoring forces at specific mooring system orientations which,in addition to limiting platform displacement, generated biases in the steady-statelocations of floating turbines.Studies of wind farm efficiency showed that, with specific mooring system ori-entations and anchor locations, and long enough mooring lines, YITuR operationcould increase the efficiency of a 7× 7 wind farm by 42.7% relative to greedyoperation. It was also shown that, due to mooring line tension nonlinearity, theeffectiveness of YITuR operation is highly dependent upon the relative orientationbetween mooring systems and the direction of incoming wind. Wind farm develop-ers considering any form of turbine repositioning as a wind farm control techniquemust therefore select appropriate mooring system orientations based on either com-prehensive optimization studies or a predetermined wind direction at which YITuRoperation is necessary.Future research on YITuR should attempt to solve such optimization problemswherein not only optimal turbine operating parameters are of interest, but also opti-mal wind farm design parameters. It is also necessary to move beyond steady-stateanalyses to investigate the efficacy of dynamic optimization algorithms and controlsystems in implementing YITuR in real-time under the presence of turbulence andtime-varying wind and wave conditions. Evaluating such controllers will requiredynamic floating offshore wind farm models that simulate turbine motion along theocean surface in addition to dynamic wake propagation.Further, aspects of safety and practicality should be investigated. Specifically,the effects of increasing mooring line length on floating wind turbine stabilityshould be examined quantitatively. Additionally, the effects of large platform dis-placement on maintenance procedures and crew safety should be investigated. Fi-nally, the efficacy of YITuR operation in raising wind farm efficiency should becompared against the use of thrusters for relocating floating platforms. This anal-ysis should consider the additional capital and maintenance costs of incorporatingthrusters into floating platforms.102Chapter 4A Low-fidelity Dynamic WindFarm Model for SimulatingTime-varying Wind Conditionsand Floating Platform MotionThis chapter introduces a dynamic parametric wind farm model that is capable ofsimulating floating wind turbine platform motion coupled with wake transport un-der time-varying wind conditions. The simulator is named FOWFSim-Dyn as itis a dynamic extension of the previously developed steady-state Floating OffshoreWind Farm Simulator (FOWFSim). One-dimensional momentum conservation isused to model dynamic propagation of wake centerline locations and average ve-locities, while momentum recovery is approximated with the assumption of a con-stant temporal wake expansion rate. Platform dynamics are captured by treating afloating offshore wind farm as a distribution of particles that are subject to aero-dynamic, hydrodynamic, and mooring line forces. The finite difference methodis used to discretize the momentum conservation equations to yield a nonlinearstate-space model. FOWFSim-Dyn is validated against steady-state experimentalwind tunnel data obtained from the literature along with simulation results acquiredfrom FAST. The model is shown to possess a satisfactory level of fidelity for en-103gineering applications. Finally, dynamic simulations are conducted to ensure thattime-varying predictions match physical expectations and intuition.4.1 IntroductionSince the introduction of parametric wake models by Jensen [91] and Katic´ etal. [98], such wind farm simulators have served as essential tools for enhancingwind farm performance. This enhancement has been achieved via two distinctfields of study. The older of the two is layout optimization, wherein the optimal in-stallation locations of wind turbines are computed with the objective of maximizingannual revenue [161]. Since such optimization problems are solved offline prior towind farm construction, steady-state wake models have sufficed for estimating an-nual energy production. The field of study that has more recently experienced asurge in interest is wind farm control, which involves real-time wind turbine actua-tion for the purpose of manipulating the wind field to achieve some wind farm-levelobjective [102]. This ultimate goal may be efficiency maximization, or power out-put tracking with turbine load alleviation [107]. In either case, since actuators areadjusted in real-time, dynamic wake phenomena such as turbulence, transport de-lay, time-varying mean wind speed and direction, and floating platform motion (fordeep-water offshore wind farms) are pertinent when evaluating controller perfor-mance.Steady parametric wake models have been used successfully to raise wind farmefficiency in large eddy simulations (LES) [72] and field tests [59]. Further, in oneinstance, Gebraad et al. [69] reported no significant performance gains when usinga dynamic wake model for wind farm control in contrast to using a steady wakemodel. Nonetheless, there are benefits associated with using dynamic wake mod-els. First, traditional state and parameter estimation techniques may be used toadapt such models to time-varying wind conditions [69]. Second, low-fidelity dy-namic wake models may be used to test controller robustness against time-varyingwind conditions prior to dedicating time and resources to conducting high-fidelitysimulations and field tests (as performed by Johnson and Fritsch [94] and Gebraadand van Wingerden [70]). Further research comparing wind farm control basedon dynamic versus steady parametric models may reveal additional benefits. Such104progress will only be possible, however, provided the availability of various dy-namic wake models.Reviews of wake modeling may be found in the works of Boersma et al. [24],Go¨c¸men et al. [75], Vermeer et al. [179], and in our previous review article [102].We will focus our current discussion on parametric dynamic wake models. Theearliest of such models found application in power de-rating wind farm control re-search conducted by Gebraad and van Windgerden [70], Johnson and Fritsch [94],and Ahmad et al. [3]. Power de-rating involves reducing the thrust force exertedonto the wind by upstream wind turbines as a means of increasing the fluid momen-tum available to downstream machines [95]. Since this application involves neitherwake deflection nor wind turbine relocation, wake dynamics in these studies weremodeled using time-delays in computed steady-state incident wind speeds. Thesetime-delays represented the duration required for changes in the wind field at someupstream turbine to propagate to downstream machines. This approach is valid aslong as the wind direction remains constant, and wake centerline deflection andturbine relocation are not pertinent.In order to account for transport delay of steered wakes, Gebraad and vanWingerden [67] developed the Flow Redirection and Induction Dynamics (FLORI-Dyn) model. Their approach involved representing flow within a wake using trans-lating points that were initialized at each turbine and then transported downstream.Each point contained information regarding its corresponding turbine’s operatingparameters at the instant in time at which the point was initialized. Using this in-formation, wake properties at the downstream location of the translating point wereobtained using the Flow Redirection and Induction in Steady-state (FLORIS) wakemodel [72], which utilizes integral forms of mass and momentum conservation tocompute downstream wake properties. More simply put, FLORIDyn transportssteady-state wake characteristics computed with FLORIS in the free stream winddirection. Time varying wind direction and floating platform motion are not con-sidered in FLORIDyn however.In an alternative approach, Shapiro et al. [162] used the differential forms ofmass and momentum conservation to simulate dynamic wake behavior. Local andconvective wake accelerations in the free stream wind direction were described bymaterial derivatives, and these accelerations were equated to force terms represent-105ing turbulent mixing and rotor thrust. The advantage of this modeling approachwas that wake transport would be inherently captured by convective accelerationterms, thus eliminating the need for the translating points employed by the FLORI-Dyn model. Instead, all wake characteristics were functions of a fixed grid in thedownstream direction. Shapiro et al. [163] later extended their model to capturewake redirection resulting from rotor yaw misalignment. Prandtl’s lifting line the-ory was used to compute transverse wake velocities, shed circulation, and vortexproperties immediately downstream of yawed rotors. Wake centerline deflection inthe free stream wind direction was then computed by equating the material deriva-tive of the centerline position to the transverse component of the wake velocity.These works do not capture floating platform motion or time-varying wind speedand direction.Finally, Boersma et al. [25] developed the Wind Farm Simulator (WFSim),which is a control-oriented dynamic wake model based on the two-dimensionalform of the unsteady turbulent Navier-Stokes equations. The major benefit of WF-Sim is that individual wake expansion and the interaction of multiple wakes areinherently captured by the mixing length turbulence model employed. In the pre-viously discussed models, the rate of linear wake expansion was either estimatedor assumed. Further, the previous models simulated flow behavior in regions withoverlapping wakes by assuming that the effective kinetic energy deficit in the windfield is equal to the sum of deficits corresponding to all pertinent wakes. De-spite its higher-fidelity, WFSim requires approximately 1000sec of computationfor 1000sec of simulation in comparison to previously discussed models (severalseconds of computation for 1000sec of simulation). Moreover, WFSim does notmodel floating platform motion or time-varying wind speed and direction.In the current chapter, we loosely follow the approach of Shapiro et al. [162],whereby partial differential equations are used to capture wake transport, and wedevelop a dynamic parametric wake model capable of simulating time-varyingwind speed and direction, along with platform motion for floating offshore windfarms. The novelty of this chapter therefore includes the following: (i) additionalterms in the wake momentum conservation equations to capture time-varying freestream wind velocity effects; and (ii) a coupled dynamic model that captures pla-nar floating wind turbine motion in the presence of aerodynamic interaction. Our106approach is physics-based with the rate of wake expansion as the only parametricassumption. This model serves as a dynamic extension of our previously developedsteady-state tool [103], which was named the Floating Offshore Wind Farm Sim-ulator (FOWFSim), and will henceforth be referred to as FOWFSim-Dyn. Fixed-foundation wind farms may also be modeled by simply deactivating turbine plat-form motion.The remainder of this chapter is organized as follows: Section 4.2 provides adetailed mathematical description of FOWFSim-Dyn along with a discussion ofits limitations. In Section 4.3, we perform a mesh convergence study and validateFOWFSim-Dyn’s ability to predict floating platform motion and steady and dy-namic wake transport. We finally conclude the chapter in Section 4.4 by listingpotential research directions for enhancing FOWFSim-Dyn.4.2 Mathematical model - FOWFSim-DynThis section details the mathematical formulation behind FOWFSim-Dyn. First,the problem setup, solver block diagram, and resulting equations of motion arepresented in Section 4.2.1–Section 4.2.7. Finally, important assumptions and limi-tations pertaining to FOWFSim-Dyn are discussed in Section 4.2.8.4.2.1 Wind farm descriptionFigure 4.1 shows a top view schematic of the general floating offshore wind farmthat we model in the current work. Floating wind turbines are treated as a system ofparticles that are distributed along the two-dimensional ocean surface. Throughoutthis chapter, we consider only three-cylinder semi-submersible floating platformsas per the baseline design presented by Robertson et al. [152]. Each floating struc-ture is therefore connected to three anchors via mooring lines for the purpose ofstation-keeping.We define the set F = {1,2, · · · ,N} to denote the indices of the N floatingwind turbines within the wind farm, and we refer to each individual turbine usingthe identifier i. We then number the wind turbines in ascending order based ontheir downstream location. That is to say, the most upstream turbine is numberedby i = 1, while the most downstream machine is identified by i = N.107Figure 4.1: Schematic of a general floating offshore wind farm with semi-submersible platforms used as a basis for FOWFSim-Dyn’s mathemati-cal model.The fixed global frame of reference is identified by the xˆ and yˆ axes. Eachwind turbine also possesses a local non-inertial translating (though not rotating)reference frame that is attached to its center of gravity. We identify the referenceframe that is fixed to turbine i as frame i. Further, the axes of frame i are referredto as xˆi and yˆi.We assume that a predominant wind direction exists, and that it is aligned withthe positive xˆ axis. The free stream wind velocity is then denoted by the vectorV∞(t), which contains xˆ and yˆ components U∞(t) and V∞(t) as follows:V∞(t) :=[U∞(t) V∞(t)]T. (4.1)U∞(t) therefore represents the free stream wind speed in the predominant winddirection, while V∞(t) accounts for fluctuations in the transverse free stream windspeed.1084.2.2 States and inputsUltimately, FOWFSim-Dyn takes the following nonlinear state-space form:x˙farm(t) := f (xfarm(t),ufarm(t),V∞(t)), (4.2)where the wind farm state vector xfarm(t) combines the floating wind turbine statevector x(t) with the wake state vector xw(t) as follows:xfarm(t) :=[xT(t) xTw(t)]T. (4.3)The wind turbine state vector x(t) comprises the position and velocity vectors ofall floating wind turbines within the wind farm as follows:x(t) :=[rT1 (t) rT2 (t) · · · rTN(t) vT1 (t) vT2 (t) · · · vTN(t)]T, (4.4)where ri(t) and vi(t) are vectors containing xˆ and yˆ components of the position andvelocity of turbine i as follows:ri(t) :=[xi(t) yi(t)]T, (4.5)vi(t) :=[vx,i(t) vy,i(t)]T. (4.6)The wake state vector xw(t) contains the states of the wakes generated by theN floating wind turbines as follows:xw(t) :=[xTw,1(t) xTw,2(t) · · · xTw,N(t)]T. (4.7)Assuming that the states of wake i are defined at Np,i discrete points along thedownstream direction, xw,i(t) comprises the states of wake i at each of these dis-crete points as follows:xw,i(t) :=[xTw,i,1(t) xTw,i,2(t) · · · xTw,i,Np,i(t)]T. (4.8)The state vector xw,i,p(t) at each point p along wake i then consists of the wake cen-terline location yw,i,p(t), wake velocity components uw,i,p(t) and vw,i,p(t), which109correspond to the xˆi and yˆi directions, and the wake diameter Dw,i,p(t) as follows:xw,i,p(t) :=[yw,i,p(t) uw,i,p(t) vw,i,p(t) Dw,i,p(t)]T. (4.9)These wake characteristics are portrayed in Figure 4.6 and discussed in Section 4.2.6.The wind farm input vector ufarm(t) contains the input vectors for the N floatingwind turbines as follows:ufarm(t) :=[uT1 (t) uT2 (t) · · · uTN(t)]T, (4.10)where ui(t) consists of the axial induction factor ai(t) and yaw angle γi(t) of tur-bine i as follows:ui(t) :=[ai(t) γi(t)]T, (4.11)with all yaw angles defined as positive counter-clockwise from the xˆ axis.4.2.3 Solver block diagramThe block diagram for FOWFSim-Dyn is shown in Figure 4.2. The simulator con-sists of two main modules. The aerodynamics module requires the states x(t) andinputs u(t) of all turbines, along with the free stream wind velocity and acceler-ation vectors V∞(t) and V˙∞(t) at time t. Its function is to compute the effectivewind velocity vector Vi(t) that is incident on the rotor of turbine i for all i ∈F .The Floating turbine dynamics module uses these incident wind velocity vec-tors, along with turbine states and inputs, to compute the rates of change of turbinestates x˙i(t) at time t. Using a standard ordinary differential equation solver, statederivatives are integrated to compute state trajectories over time. This module alsocomputes the power outputs of individual wind turbines as well as that of the entirewind farm Pfarm(t).110Figure 4.2: Block diagram showing the computation modules of FOWFSim-Dyn along with information transfer routes.4.2.4 Wind farm power outputThe total power output of the wind farm Pfarm(t) is computed as the sum of elec-tricity production from all wind turbines as follows:Pfarm(t) = ∑i∈FPi(t), (4.12)where Pi(t) is the power output of turbine i, and is estimated assuming steady-stateperformance as follows [121]:Pi(t) =18Cp,i(t)ρapiD2i ‖Vrel,i(t)‖3 . (4.13)Di is rotor diameter of turbine i, ρa is the density of air, and Vrel,i(t) is the wind ve-locity that is incident upon the rotor of turbine i from the perspective of an observerwho is fixed to turbine i. Referring to Figure 4.3, Vrel,i(t) is defined as follows:Vrel,i(t) = Vi(t)−vi(t), (4.14)where vi(t) is the velocity vector of turbine i, and Vi(t) is the wind velocity vector(in the global frame) that is incident upon the rotor of turbine i with the followingxˆ and yˆ components:Vi(t) :=[Ui(t) Vi(t)]T. (4.15)111Figure 4.3: Schematic of floating platform velocity vector vi(t), incidentwind velocity vector Vi(t), and the relative incident velocity vectorVrel,i(t) at the location of turbine i.Vi(t) is calculated using the wake interaction model discussed in Section 4.2.7.The power coefficient Cp,i(t) of turbine i is computed based on the vortex cylin-der model of a yawed actuator disc as follows [29]:Cp,i(t) = 4ai(t)(cosγrel,i(t)−ai(t))(cosγrel,i(t)+ tanχi(t)2sinγrel,i(t)−ai(t)sec2 χi(t)2), (4.16)where ai(t) is the axial induction factor of turbine i and, as per Figure 4.3, γrel,i(t)is the yaw misalignment of turbine i relative to Vrel,i(t) as follows:γrel,i(t) = γi(t)−θi(t). (4.17)In the above expression, γi(t) is the yaw angle of turbine i and θi(t) is the angle ofVrel,i(t) relative to the positive xˆ axis as follows:θi(t) = tan−1Vi(t)− vy,i(t)Ui(t)− vx,i(t) . (4.18)Finally, χi(t) is the wake skew angle immediately past the rotor of turbine i and is112approximated as follows [29]:χi(t) = (0.6ai(t)+1)γrel,i(t). (4.19)4.2.5 Floating wind turbine motionThe rates of change of the position and velocity of turbine i are expressed as fol-lows:r˙i(t) = vi(t), (4.20)v˙i(t) =Fi(t)mi+ma,i, (4.21)where mi is the mass of floating wind turbine i. The added mass1 ma,i associatedwith turbine i will be discussed along with the hydrodynamic drag force.As shown in Figure 4.4, the total force Fi(t) acting on turbine i is the sum ofits respective aerodynamic, hydrodynamic, and mooring line forces as follows:Fi(t) = Fa,i(t)+Fh,i(t)+Fm,i(t). (4.22)The aerodynamic thrust force Fa,i(t) acting on the rotor of turbine i is expressedas follows:Fa,i(t) =18Ct,i(t)ρapiD2i ‖Vrel,i(t)‖2 ni(t), (4.23)where the thrust coefficient Ct,i(t) is computed based on the vortex cylinder modelof a yawed actuator disc as follows [29]:Ct,i(t) = 4ai(t)(cosγrel,i(t)+ tanχi(t)2sinγrel,i(t)−ai(t)sec2 χi(t)2), (4.24)and ni(t) is a unit vector normal to the rotor of turbine i as follows:ni(t) =[cosγi(t) sinγi(t)]T. (4.25)1Added mass accounts for hydrodynamic loads that act upon an object that is accelerating withrespect to the surrounding fluid. It compounds with hydrodynamic drag forces, which are typicallymodeled as functions of instantaneous velocity only.113Figure 4.4: Schematic of aerodynamic thrust force Fa,i(t), hydrodynamicdrag force Fh,i(t), and mooring line forces Fm,i,k(t) acting on wind tur-bine i with a semi-submersible floating platform.Based on elementary fluid mechanics principles concerning immersed bodies,Fh,i(t) is approximated by summing the drag force contributions of all submergedcomponents of turbine i as follows:Fh,i(t) =12(∑j∈DiCd,i, jAd,i, j)ρw ‖w(t)−vi(t)‖(w(t)−vi(t)) , (4.26)where ρw is the density of ocean water, and w(t) is the ocean current velocity vector(which we assume to be w(t)= 0m/s in this work). Let the setDi = {1,2, · · · ,Nh,i}denote the indices of all submerged components that contribute to the hydrody-namic drag force acting on turbine i, with Nh,i being equal to the total number ofsubmerged components of turbine i. Cd,i, j and Ad,i, j are thereby the drag coefficientand reference area of the jth submerged component of turbine i.In a similar manner, the total added mass ma,i associated with turbine i is esti-mated by summing the added mass contributions of all submerged components ofturbine i as follows:ma,i = ρw ∑j∈DiCa,i, jAa,i, j, (4.27)where Ca,i, j is the added mass coefficient of the jth submerged component of tur-bine i, and Aa,i, j is the added mass reference area of the same component.114Let the set Mi = {1,2, · · · ,Nm,i} denote the indices of all mooring lines con-nected to turbine i, with Nm,i being equal to the total number of mooring linesattached to turbine i. Fm,i(t) may then be expressed as the sum of all mooringforce contributions acting on turbine i as follows:Fm,i(t) = ∑k∈MiFm,i,k(t), (4.28)where Fm,i,k(t) is the restoring force exerted on turbine i by its kth mooring line.This force is calculated by first finding the magnitude of the horizontal componentof tension within mooring line k of turbine i, and then projecting this tension in theappropriate direction as follows:Fm,i,k(t) =−HF,i,k(t)rF/A,i,k(t)∥∥rF/A,i,k(t)∥∥ . (4.29)The function HF,i,k(t) outputs the horizontal component of tension along the kthmooring line of turbine i. This function is generated by solving the static differen-tial equations describing a suspended cable which is either partially contacting orfully lifted above the seabed. The relevant solution was described in Section 3.2.5.As shown in Figure 4.5, the term rF/A,i,k(t) describes the position vector fromthe anchor of the kth mooring line of turbine i to the corresponding fairlead, and isexpressed as follows:rF/A,i,k(t) = ri(t)+ rF/G,i,k− rA,i,k, (4.30)where rF/G,i,k is a constant position vector from the center-of-gravity of turbine i tothe fairlead that connects to the kth mooring line of the same turbine, and rA,i,k is aconstant position vector representing the location of the anchor of the same moor-ing line. In Eq. (4.29), dividing rF/A,i,k(t) by its Euclidean norm therefore producesa unit vector that points from the anchor of the kth mooring line of turbine i to thecorresponding fairlead. The restoring force associated with this mooring line pullsthe turbine in the opposite direction.115Figure 4.5: Schematic of position vectors that are relevant for calculating theforce in mooring line k of turbine i.Figure 4.6: Schematic of characteristics necessary for modeling the wakegenerated by turbine i. The wake centerline position yw,i(xˆi, t), aver-age wake velocity vw,i(xˆi, t), and wake diameter Dw,i(xˆi, t) are definedwithin the reference frame that is fixed to turbine i.4.2.6 Single wake modelFigure 4.6 shows the characteristics of interest when modeling wake i, which isthe wake generated by the rotor of turbine i. These characteristics include thewake’s centerline position yw,i(xˆi, t) relative to the xˆi axis, its average velocity vec-tor vw,i(xˆi, t) measured in frame i, and its diameter Dw,i(xˆi, t).Two key assumptions are necessary for justifying the mathematical formula-tion presented in this section. First, if fluctuations in the wind direction relative tothe xˆi axes are presumed to be small, then all wake characteristics may be definedas smooth functions of only xˆi and t. Furthermore, wake cross-sections may beassumed to always remain normal to the predominant flow direction, which corre-116sponds to the positive xˆ and xˆi axes in our work.Second, if the free stream wind speed is presumed to be significantly largerthan the velocities of floating platforms, then the equations of motion describingany wake may be defined relative to a reference frame that is fixed to the wake-generating turbine. The frame of reference shown in Figure 4.6 is therefore non-inertial and translates with turbine i, while yw,i(xˆi, t), vw,i(xˆi, t), and Dw,i(xˆi, t) aredefined in this translating frame. This approach eliminates the need to model wakebehavior upstream of turbine i, while removing time-dependency from the wakecenterline boundary condition (i.e. yw,i(xˆi, t) is always equal to zero at xˆi = 0m).Granting these preliminaries, the equations of motion describing wake i maynow be derived. Specifically, we shall present partial differential equations thatmodel wake average velocities, wake centerline locations, and wake diameters overspace and time. Let the vector Li(xˆi, t) describe the linear momentum deficit ofwake i per unit length along the xˆi axis as follows:Li(xˆi, t) = ρapi4D2w,i(xˆi, t) [V∞(t)− (vi(t)+vw,i(xˆi, t))] . (4.31)As vw,i(xˆi, t) is measured in frame i, the term vi(t)+ vw,i(xˆi, t) redefines the ve-locity of wake i in the global frame. Since no external forces impact wake i, thetime-derivative of Li(xˆi, t) must equate to zero, which results in the following mo-mentum conservation equation:∂vw,i(xˆi, t)∂ t+(U∞(t)− vx,i(t)) ∂vw,i(xˆi, t)∂ xˆi =V˙∞(t)− v˙i(t)+ 2Dw,i(xˆi, t)dDw,i(xˆi, t)dt(V∞(t)−vi(t)−vw,i(xˆi, t)) . (4.32)The time-derivative of yw,i(xˆi, t) must equate to the yˆi component of vw,i(xˆi, t),which results in the following expression describing the wake centerline location:∂yw,i(xˆi, t)∂ t+(U∞(t)− vx,i(t)) ∂yw,i(xˆi, t)∂ xˆi = vw,i(xˆi, t). (4.33)In Eqs. (4.32) and (4.33), uw,i(xˆi, t) and vw,i(xˆi, t) are the xˆi and yˆi componentsof vw,i(xˆi, t), vx,i(t) is the velocity of turbine i in the xˆ direction, U∞(t) is the free117stream wind speed in the xˆ direction, and the term U∞(t)−vx,i(t) serves as the trans-port speed in the xˆi direction. When modeling fluids using the three-dimensionalNavier-Stokes equations, the transport and fluid velocities at any given point areequal. Following this logic, the transport speed in Eqs. (4.32) and (4.33) shouldsimply be uw,i(xˆi, t). However, when neglecting three-dimensional effects, it isdebatable exactly how the transport velocity should be defined. Our simulationsindicate that defining the transport speed as the free stream wind speed (definedin frame i) yields predictions closer to experimental observations than does settingthe transport speed to uw,i(xˆi, t).In steady-state parametric wake models, the wake diameter is typically as-sumed to grow at a constant spatial expansion rate kx along the downstream direc-tion. When modeling wakes dynamically, however, we assume that wake diametersgrow at a constant temporal expansion rate kt . In other words, the time-derivativeof Dw,i(xˆi, t) must equate to kt as follows:∂Dw,i(xˆi, t)∂ t+(U∞(t)− vx,i(t)) ∂Dw,i(xˆi, t)∂ xˆi = kt . (4.34)If the spatial expansion rate kx under steady-state conditions is known for somereference free stream wind speed U∞,ref, the temporal expansion rate at U∞,ref mustbe kt = kxU∞,ref. Assuming that the free stream wind speed ‖V∞(t)‖ does not varysignificantly from U∞,ref, then kt may be assumed to remain constant.In order to obtain the wake states employed in Eq. (4.9), the spatial gradientsin Eqs. (4.32), (4.33), and (4.34) must be discretized over some fixed downstreamdistance using the finite difference method, which would yield a system of non-linear ordinary differential equations that would be rearranged to state-space form.We will not present the discretized forms of these equations as the finite differencemethod is an elementary numerical technique.When implementing the above solution, we recommend the following initial118conditions:yw,i(xˆi,0) =V∞(0)U∞(0)xˆi, (4.35)vw,i(xˆi,0) = V∞(0)−vi(0), (4.36)Dw,i(xˆi,0) = Di+ kxxˆi, (4.37)which ensure, respectively, that all wake centerlines are initially aligned with thefree stream wind, wake velocities are initially equal to the free stream wind veloc-ity, and that wake diameters initially grow at a predefined spatial rate kx. Note thatDi is the diameter of turbine i. With regards to boundary conditions, the followingare necessary based on assumptions inherent to FOWFSim-Dyn:yw,i(0, t) = 0, (4.38)vw,i(0, t) = vw,init,i(t), (4.39)Dw,i(0, t) = Di, (4.40)Equation (4.38) states that the centerline of wake i at xˆi = 0m must always cor-respond to the location of turbine i, which is in fact the origin of frame i. Equa-tion (4.39) states that the velocity of wake i at xˆi = 0m must always be equal to thewake velocity vw,init,i(t) immediately downstream of the rotor of turbine i. Finally,Eq. (4.40) requires that the diameter of wake i at the location of turbine i is alwaysequal to the rotor diameter of this turbine.We calculate the velocity vector vw,init,i(t) based on simplifications made toGlauert’s momentum theory [29] by Bastankhah and Porte´ Agel [18] as follows:vw,init,i(t) = ‖Vrel,i(t)‖√1−Ct,i(t)[cos(ξw,init,i(t)+θi(t))sin(ξw,init,i(t)+θi(t))], (4.41)where ξw,init,i(t) is the initial wake skew angle, which is expressed as follows basedon a momentum conservation derivation reported by Jime´nez et al. [93]:ξw,init,i(t) =−Ct,i(t)2 cos2 γrel,i(t)sinγrel,i(t). (4.42)119The derivation by Bastankhah and Porte´ Agel [18] assumes that the free streamwind velocity is aligned with the xˆ axis. As a result, the addition of θi(t) toξw,init,i(t) in Eq. (4.41) accounts for the misalignment of V∞(t) relative to the xˆaxis.4.2.7 Wake interaction modelWhen a wind turbine rotor is influenced by wakes that are generated from multipleupstream turbines, a wake interaction model is necessary for approximating theresultant effective wind speed that is incident on the downstream rotor. The mostcommonly used wake interaction technique is based on the assumption that theeffective kinetic energy deficit at the location of the downstream rotor must beequal to the sum of kinetic energy deficits of all pertinent wakes [98]. As a result,the effective wind speed at the downstream rotor is a function of the root-sum-square of relevant wake velocity deficits. Further enhancement may be obtainedby approximating wake velocity profiles using Gaussian distributions [18]. Wecontinue to make use of this wake interaction methodology.Let the set Ui = {1,2, · · · , i−1} denote the indices of all turbines that are lo-cated upstream of turbine i. The effective wind velocity vector that is incident onthe rotor of turbine i may therefore be expressed as follows:Vi(t) =‖V∞(t)‖−√∑q∈Ui(‖V∞(t)‖−vw,q→i(t) ·n∞(t))2n∞(t). (4.43)where n∞(t) is a unit vector aligned with V∞(t) as follows:n∞(t) =V∞(t)‖V∞(t)‖ , (4.44)and vw,q→i(t) is the effective velocity of wake q that is incident upon the rotorof wake i. Equation (4.43) projects vw,q→i(t) along the free stream wind direction(hence the dot product operation with n∞(t)), and then computes the velocity deficitin this direction. Average wake velocities perpendicular to the free stream winddirection are assumed to be negligibly small far enough downstream; their effectsare therefore neglected.120We now describe our procedure for computing vw,q→i(t). Let vw,q→i(t) denotethe average velocity of wake q at the location of wake i as follows:vw,q→i(t) = vq(t)+vw,q(xi(t)− xq(t), t). (4.45)Since the average velocity vector of wake q is defined in frame q, the substitutionxˆq = xi(t)− xq(t) into vw,q(xˆq, t) is necessary for identifying the location of tur-bine i in frame q. The addition of the turbine velocity vector vq(t) then transformsvw,q(xi(t)− xq(t), t) to the global frame.The next step is to generate a Gaussian profile v˘w,q→i(r, t), where r is the radialdistance from the centerline of wake q, to approximate the continuous velocitydistribution of wake q at the location of wake i. Imposing a requirement that thetotal momentum deficit of V∞(t)− v˘w,q→i(r, t) per unit length must equate that ofa top-hat distribution with amplitude V∞(t)−vw,q→i(t) as follows:∫ ∞0ρa2pir (V∞(t)− v˘w,q→i(r, t))dr = ρapi4 Dw,q→i(t)(V∞(t)−vw,q→i(t)) , (4.46)the following Gaussian profile is then obtained:V∞(t)− v˘w,q→i(r, t) = 18(Dw,q→i(t)σ)2(V∞(t)−vw,q→i(t))exp−r22σ2, (4.47)where Dw,q→i(t) is the diameter of wake q at the location of turbine i as follows:Dw,q→i = Dw,q(xi(t)− xq(t), t). (4.48)The standard deviation σ in Eq. (4.47) may be estimated based on experimental orhigh-fidelity numerical data.Finally, the effective velocity vw,q→i(t) is obtained by averaging v˘w,q→i(r, t)along the rotor area Ai of turbine i. This task is achieved by numerically computingthe following integral at each time-step:vw,q→i(t) =4piD2i∫Aiv˘w,q→i(r, t)dA. (4.49)1214.2.8 Model limitationsSeveral assumptions have been made when developing FOWFSim-Dyn which im-pose limitations on its fidelity and applicability. The current subsection summa-rizes these limitations.Two-dimensional floating wind turbine dynamicsThe first and most crucial of these assumptions is that floating platform motionmay be adequately captured using a two-dimensional planar model. That is tosay, we neglect floating platform heave, yaw, pitch, and roll. In consequence,FOWFSim-Dyn fails to capture dynamic effects induced by ocean waves and oscil-latory wind conditions on platform rotation. FOWFSim-Dyn remains appropriatefor wind farm controller design and testing, since this application is primarily con-cerned with average rotor positions over extended periods of time. However, anyattempt to control or evaluate individual wind turbine dynamics requires the use ofthree-dimensional multi-body nonlinear modeling tools.Steady-state mooring line modelAlthough we present a dynamic model, mooring line tensions are found based onthe solution to a static suspended cable problem. It has been reported by Hall etal. [79] that such static models accurately predict mooring line loads and float-ing wind turbine motion; thus rendering them appropriate for wind farm control.However, Hall et al. [79] also mentioned that use of such models may lead to largeinaccuracies in turbine load predictions. Therefore, analysis and control of indi-vidual turbine motion must consider higher-fidelity modeling techniques such as alumped-mass dynamic mooring line model [78].Steady-state turbine aerodynamicsTurbine power outputs and thrust forces (Eqs. (4.13) and (4.23)), along with theirrespective coefficients (Eqs. (4.16) and (4.24)), are calculated based on steady-state actuator disc theory. This approach assumes ideal rotors and fails to captureunsteady aerodynamic effects and asymmetric rotor loadings. These phenomenasignificantly influence blade loads when yaw misalignment occurs; however, for122the purpose of wind farm control, our focus lies on the overall influence of rotoroperation on fully-developed wake regions. Nonetheless, any turbine-level analysisrequires more detailed fluid-structure interaction modeling.The computation of vw,init,i(t) in Eq. (4.41), which is the average wake velocityimmediately downstream of turbine i, relies on a steady-state momentum balanceon a control volume spanning across the rotor of turbine i. As a result, momentumfluxes into and out of this control volume are considered, while the rate-of-changeof momentum within the control volume is neglected. Given the low density ofair, these inertial effects may be neglected, although their significance should beinvestigated.Sources of wake deflectionFOWFSim-Dyn does not capture wake centerline deflection caused by rotor rota-tion. This phenomenon was first observed in high-fidelity simulations conductedby Gebraad et al. [72]; however, more recent work by Fleming et al. [60] showedthat the scale of this phenomenon is insignificant. Instead, Fleming et al. [60] ob-served that vortices generated by turbine rotors induce wake deflection past down-stream machines, even if their rotors are not operated with yaw offset. Additionalterms may be added to Eqs. (4.32) and (4.33) to account for such phenomena.Spatial-uniformity and consistency of the free stream windIn the current chapter, we have assumed that the free stream wind velocity is uni-form throughout the wind farm, which is why the variable V∞(t) is solely a functionof time. This variable may readily be expressed as V∞(xˆ, t) if spatial variations ofthe free stream wind velocity are known. Furthermore, in order to represent wakecharacteristics purely as a function of the downstream distance along the xˆi axes,while ignoring changes in the cross-sectional areas of wakes, we assumed that vari-ations in the free stream wind direction are small relative to the xˆ axis. The yˆ com-ponent of the free stream wind velocity must therefore remain small in comparisonto its xˆ component.1234.3 Model validationIn this section, we first perform a mesh sensitivity analysis to ascertain the de-pendency of model predictions upon the size of finite difference elements in Sec-tion 4.3.1. We then validate FOWFSim-Dyn’s capacity for predicting floating plat-form motion in Section 4.3.2, and wake transport in Section 4.3.3 and Section 4.3.4.4.3.1 Mesh sensitivity analysisFor a mesh sensitivity study, we simulate the experimental setup employed by Bas-tankhah and Porte´ Agel [18]. Namely, the wake of a single fixed-foundation tur-bine with diameter D = 15cm is simulated with a steady free stream wind speedof U∞ = 4.88m/s. The turbine’s axial induction factor is set to the optimal valueof a = 1/3 and a yaw angle of γ = 20deg is implemented to observe mesh effectson wake deflection. To approximate steady-state results, all simulations are run fora duration of 5sec and data is extracted from the final time-step. The Gaussianprofile standard deviation is set to σ = 0.025xˆ+ 0.396m based on experimentaldata2 reported by Bastankhah and Porte´ Agel [18]. The (diametrical) spatial wakeexpansion constant is set to kx = 0.08 as per the recommendation by Shakoor etal. [161].Simulated wake centerlines and normalized velocity deficit profiles at a down-stream distance of 7D are plotted in Figure 4.7 for different finite difference ele-ment sizes. Qualitatively, it is apparent that the evolution of the wake centerline isinsignificantly influenced by the mesh size. At xˆ/D = 16, the centerline deflectionobtained using an element size of 8D only differs by 5% relative to the value corre-sponding to an element size of 0.25D. As a result, we solely utilize the maximumnormalized velocity deficit as a convergence criterion.Table 4.1 lists the computation times corresponding to different element sizesfrom Figure 4.7 as well as predicted maximum normalized velocity deficits. Dy-namic simulations were performed using the MATLAB fourth-order Runge-Kuttasolver implemented on a laptop computer with a 2.80GHz Intel Core i7-7700HQprocessor. Table 4.1 also lists the convergence of the maximum normalized ve-2Velocity profiles corresponding to a yaw angle of γ = 0deg from Fig. 21 in the paper by Bas-tankhah and Porte´ Agel [18] were digitized and Gaussian function curve fitting was used to computethe standard deviation.1240 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1600.10.20.30.4-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.2500.10.20.3b)a)Figure 4.7: Effects of various finite difference mesh element sizes on a) thesteady-state wake velocity profile at a downstream distance of 7D, andb) the steady-state wake centerline evolution. Simulation parameters:D = 15cm, U∞ = 8m/s, a = 1/3, γ = 20deg, kx = 0.08, σ = 0.025xˆ+0.396m.locity deficit as the element size is decreased. We observe that mesh sensitivity issufficiently reduced at an element size of 1D since further reduction to 0.5D onlyresults in a 0.69% change in the predicted maximum normalized velocity deficit.An element size of 1D is also appropriate from the standpoint of time-efficiency asit requires 3.2sec of computation time to run a 5sec simulate.4.3.2 Validation of platform dynamicsTo validate the platform dynamics module of FOWFSim-Dyn, we compare its pre-dictions for a single floating wind turbine against those of the National RenewableEnergy Laboratory’s (NREL’s) wind turbine modeling software Fatigue, Aerody-namics, Structures, and Turbulence (FAST) in Figure 4.8. FAST is a widely-accepted tool for modeling three-dimensional floating offshore wind turbine dy-125Table 4.1: Computation times and maximum normalized velocity deficitscorresponding to different simulated mesh element sizes from Figure 4.7.The final column lists the convergence of the maximum normalized ve-locity deficit. In other words, it contains the relative difference in themaximum normalized velocity deficit that would be obtained if each ele-ment size was halved. For instance, if the element size were to be reducedfrom 8D to 4D, the predicted velocity deficit would change by 19.46%.The computation times correspond to 5sec long simulations.Elm. size (D) Comp. time (sec) Max. velocity deficit (-) Rel. diff. (%)8 0.397 0.261 19.464 0.703 0.219 7.192 1.496 0.204 2.001 3.237 0.200 0.690.5 8.948 0.199 0.170.25 24.035 0.198 -namics and has been experimentally validated in the context of semi-submersiblefloating platforms by Coulling et al. [44]. Results presented in Figure 4.8 wereobtained using a steady free stream wind speed of U∞ = 10m/s, optimal axial in-duction, a yaw angle of γ = 15deg, and mooring line cable lengths of L = 850m.In FAST simulations, all floating platform degrees-of-freedom were enabled. Thesimulated wind turbine is based on NREL’s 5MW baseline design presented byJonkman et al. [96], and the floating platform and mooring subsystem is modeledafter the design described by Robertson et al. [152]. Details corresponding to thesedesigns are listed in ??. Figure 4.8 also includes FOWFSim-Dyn predictions withthe static mooring line model replaced by the dynamic lumped-parameter cablemodel presented by Hall et al. [79]; this result is identified by FOWFSim-Dyn* inthe figure legend.From a qualitative standpoint, Figure 4.8 shows that all models predict the ex-pected platform motion in the downwind and crosswind directions. Since the tur-bine nacelle is yawed counterclockwise by γ = 15deg, aerodynamic forces displacethe floating platform in the positive crosswind direction. There is little discrepancybetween FAST and FOWFSim-Dyn predictions with regards to platform displace-ment in the downwind direction. The root-mean-square error (RMSE) betweenthese datasets is 1.37m, which is 8.5% of the steady-state downwind platform dis-placement of 16.1m predicted by FAST.126Table 4.2: List of floating wind farm properties used during dynamic simula-tions. All wind turbines are based on the NREL 5MW baseline designpresented by Jonkman et al. [96], and all floating platforms and moor-ing subsystems are modeled after the design described by Robertson etal. [152].External propertiesρa(kg/m3)1.225 Air densityρw(kg/m3)1028 Water densityFloating turbine propertiesmi (kg) 1.4×107 MassDi (m) 126 Rotor diameterAi(m2) pi4 D2i Rotor areaηp 0.786 Electrical power conversion efficiency [72]pp 1.88 Power coefficient tuning parameter [72]Floating platform hydrodynamic propertiesCd,i,1→3 0.61 Drag coefficients of three top cylinder portionsCd,i,4→6 0.68 Drag coefficients of three bottom cylinder portionsCd,i,7 0.56 Drag coefficient of middle cylinderDd,i,1→3 (m) 12 Diameters of three top cylinder portionsDd,i,4→6 (m) 24 Diameters of three bottom cylinder portionsDd,i,7 (m) 6.5 Diameter of middle cylinderLd,i,1→3 (m) 14 Submerged lengths of three top cylinder portionsLd,i,4→6 (m) 6 Submerged lengths of three bottom cylinder portionsLd,i,7 (m) 20 Submerged length of middle cylinderAd,i, j(m2)Ld,i, jDd,i, j Drag reference area of any cylinderCa,i, j 0.63 Added mass coefficients of any cylinderAa,i, j(m2) pi4 Ld,i, jD2d,i, j Added mass reference area of any cylinderMooring system propertiesrTF/G,i,1 (m)[20.4 35.4]Position vector from turbine center to first fairleadrTF/G,i,2 (m)[−40.9 0] Position vector from turbine center to second fairleadrTF/G,i,3 (m)[20.4 −35.4] Position vector from turbine center to third fairleadrTA,i,1 (m) rTneutral,i +[418.80 725.4]Location of first anchor of any turbinerTA,i,1 (m) rTneutral,i +[−837.6 0] Location of second anchor of any turbinerTA,i,1 (m) rTneutral,i +[418.80 −725.4] Location of third anchor of any turbinezF (m) 186 Fairlead distance above seabedL (m) 900 Cable lengthw (N/m) 1065.7 Cable weight per unit length in waterAmE (N) 753.6×106 Cable tension per unit strainµs 1 Coefficient of static friction between cable and seabed1270 200 400 600 800 1000 12000102030FASTFOWFSim-DynFOWFSim-Dyn*0 200 400 600 800 1000 1200051015Figure 4.8: Comparison between FOWFSim-Dyn and FAST predictions ofplatform motion for a single floating wind turbine based on the base-line designs described in ??. The curve identified by FOWFSim-Dyn*utilizes the dynamics lumped-parameter mooring line model proposedby Hall et al. [79] in place of the static catenary model described inChapter 3. Simulation parameters: U∞ = 10m/s, a = 1/3, γ = 15deg,L = 850m.When examining crosswind platform displacement, the RMSE value betweenFAST and FOWFSim-Dyn predictions rises to 2.74m. One method for assessingthe significance of this error is to compute the normalized overlap area (NOA)between turbines rotors from the two simulations. The NOA is the overlap areabetween the rotors of turbines normalized against the rotor area. Therefore, ifthe crosswind displacements of two turbines are identical, then the NOA betweentheir rotors must be unity. If, on the other hand, the crosswind displacements areso substantial that rotor overlap is completely eliminated, then the NOA will bezero. Since the ultimate purpose of FOWFSim-Dyn is to predict wind farm poweroutput, which, due to the wake effect, is primarily depended upon the projectedoverlap areas between the rotors of adjacent turbines, the NOA is a meaningfulmetric for evaluating the effectiveness of FOWFSim-Dyn in predicting crosswindplatform motion. The RMSE value of 2.74m corresponds to an NOA of 97.2%.128That is, when wind farm-level performance is of concern, FOWFSim-Dyn pre-dicts projected rotor overlap areas between adjacent turbines with a discrepancy of2.8%.One major source of error between FAST and FOWFSim-Dyn predictions isthat FAST uses actuator line theory to predict aerodynamic loads, while FOWFSim-Dyn utilizes actuator disc theory. This source of error may be mitigated by comput-ing aerodynamic loads in FOWFSim-Dyn using thrust coefficient values obtainedfrom FAST. Further, the presence of all six platform degrees-of-freedom in FASTcontributes to the discrepancy in oscillation periods observed in Figure 4.8. Finally,the mooring line modeling approach is a non-trivial contributing factor. FASTutilizes the lumped-parameter dynamic mooring line model proposed by Hall etal. [79], while FOWFSim-Dyn employs a static catenary model. The inclusionof FOWFSim-Dyn* curves in Figure 4.8 isolates the error introduced by switch-ing between static and dynamic mooring line models. Comparing FOWFSim-Dynto FOWFSim-Dyn* plots, a noticeable discrepancy in the oscillation period is in-troduced, while RMSE values between the two curves are 1.26 and 0.87m in thedownwind and crosswind directions.4.3.3 Steady-state validation of wake transportFOWFSim-Dyn predictions of steady-state3 wake centerlines and normalized ve-locity profiles are compared against experimental results reported by Bastankhahand Porte´ Agel [18] in Figure 4.9. Wake centerline evolutions are well-predictedfor all simulated yaw angles and downstream locations. For yaw angles of γ = 0,10, and 20deg, maximum discrepancies between predicted wake centerlines andexperimental measurements are 6.87, 7.60, and 8.19% of the rotor diameter, re-spectively.Simulated normalized velocity profiles deviate significantly from experimentalmeasurements at downstream locations closer than 7D. For instance, at a yawangle of γ = 0deg, the RMSE between experimental and predicted velocity profilesranges from 12.4% of the free stream wind speed at xˆ = 4D to 4.7% at xˆ = 7D.3Validating predictions of dynamic wake behaviour is not possible at this time due to the absenceof high-fidelity simulation tools capable of modeling floating offshore wind farms; we thus defer thisprocess to future work.1290 1 2 3 4 5 6 7 8 9 10 11 12-1-0.500.51-1-0.500.51-1-0.500.51Sim. wake centerlineSim. wake boundariesExp. wake centerlineSim. deficit profilesExp. deficit profilesa)b)c)Figure 4.9: Comparison between FOWFSim-Dyn predictions and experi-mental results reported by Bastankhah and Porte´ Agel [18]. Each fig-ure shows steady-state wake centerlines and normalized velocity pro-files corresponding to yaw angles of a) γ = 0deg, b) γ = 10deg, andc) γ = 20deg. Normalized velocity profiles range from zero to oneusing the same scaling as the xˆ/D axis, but have been shifted to thedownstream location where they are measured. Simulation parameters:D = 15cm, U∞ = 8m/s, a = 1/3, kx = 0.08, σ = 0.025xˆ+0.396m.Such inaccuracies at close downstream distances are expected since FOWFSim-Dyn does not consider the inviscid nature of flow within the near-wake region.Beyond xˆ= 7D, velocity profiles are well-predicted with RMSE values that remainbelow 3.87% of the free stream wind speed.130Figure 4.10: Schematic of the 1×3 wind farm with inter-turbine spacings of7D used for dynamic simulations. All wind turbines are based on theNREL 5MW baseline design presented by Jonkman et al. [96], andall floating platforms and mooring subsystems are modeled after thedesign described by Robertson et al. [152].4.3.4 Dynamic assessment of wake transportOur final tasks are to demonstrate the capability of FOWFSim-Dyn to capture theintended dynamic phenomena and to ensure that predicted turbine and wake be-haviors respect physical intuition. The wind farm configuration that is used fordynamic simulations is shown in Figure 4.10. This plant contains a single row ofthree floating offshore wind turbines that are aligned with the predominant freestream wind direction. The neutral positions of the floating turbines are spaced 7Dapart. All wind turbines are based on NREL’s 5MW baseline design presented byJonkman et al. [96], and all floating platforms and mooring subsystems are mod-eled after the design described by Robertson et al. [152]. Details corresponding tothese designs are listed in Table 4.2. In all simulations, we increase the lengths ofmooring lines from their baseline values (i.e. L = 835m) to L = 900m to renderfloating platform motion more notable. In all the following cases, less than 10secof computation time was required to complete simulations on a laptop computerwith a 2.80GHz Intel Core i7-7700HQ processor.Simulation scenario 1The first of three simulated scenarios maintains constant wind speed and direc-tion with U∞(t) = 8m/s and V∞(t) = 0m/s, while rotor yaw angles are fixed atγ1(t) = γ3(t) =−20deg and γ2(t) = +20deg. All axial induction factors are main-tained at a1(t) = a2(t) = a3(t) = 1/3. All floating platforms are locked at their131neutral positions for the first 1000sec of simulation, after which they are permittedto relocate. The aim of this scenario is to assess floating platform motion. Snap-shots of velocity contours for simulation scenario 1 are shown in Figure 4.11. Asexpected, the alternating assignment of yaw angles causes adjacent floating plat-forms to shift in opposite directions over time. Further, the leading turbine displaysthe greatest amount of relocation from its neutral position (i.e. the left-most white+ symbol) since its incident wind speed is the largest (i.e. its incident wind speedis the free stream wind speed uninhibited by upstream rotors). The trailing turbineundergoes the smallest amount of relocation over time since its incident wind speedis diminished by the velocity deficits of wakes 1 and 2.Simulation scenario 2The second simulation sinusoidally varies the yaw angles of the three turbines be-tween ±20deg with a period of 400sec. Specifically, the following yaw angleexpressions are used for t ≥ 1000sec:γ1(t) = γ3(t) = (−20deg)sin[2pi400(t−1000sec)], (4.50)γ2(t) = (+20deg)sin[2pi400(t−1000sec)]. (4.51)Velocity contours for this case are plotted in Figure 4.12. The sinusoidal yawangle fluctuations cause oscillations of floating platforms in the yˆ direction withthe expected 400sec excitation period. In terms of wake behaviour, the trans-port effect is clearly observed. As floating turbines shift in the yˆ direction, thecorresponding effects on their respective wakes are transported downstream at ap-proximately 8m/s. For instance, at t = 1400sec, the leading turbine is locatedat a peak value past its neutral position in the +yˆ direction (see the first ver-tical white dashed line in Figure 4.12). Given that U∞ = 8m/s , then 200seclater, the centerline of the leading turbine’s wake must peak in the +yˆ directionat xˆ= 8m/s×200sec= 1600m= 12.7D. Observing the velocity contours 200seclater at t = 1600sec, such a peak is observed at just under xˆ = 12D (see the secondvertical white dashed line in Figure 4.12).1320 2 4 6 8 10 12 14 16 18 20-2-1012012345678-2-1012-2-1012-2-1012Figure 4.11: Velocity contours at various time-steps of simulation scenario 1(i.e. fixed wind condition and turbine operating parameters, while plat-form motion is permitted). The white + symbols represent the neutralpositions of the floating platforms. All floating platforms are held fixedat their respective neutral positions for the first 1000sec of simulation.Simulation parameters: U∞(t) = 8m/s, V∞(t) = 0m/s, a1(t) = a2(t) =a3(t) = 1/3, γ1(t) = γ3(t) = −20deg and γ2(t) = +20deg, kx = 0.08,σ = 0.025xˆ+0.396m.1330 2 4 6 8 10 12 14 16 18 20-2-1012012345678-2-1012-2-1012-2-1012Figure 4.12: Velocity contours at various time-steps of simulation scenario 2(i.e. fixed wind condition and sinusoidally varying yaw angles, whileplatform motion is permitted). The white + symbols represent theneutral positions of the floating platforms. All floating platforms areheld fixed at their respective neutral positions for the first 1000sec ofsimulation. Simulation parameters: U∞(t) = 8m/s, V∞(t) = 0m/s,a1(t) = a2(t) = a3(t) = 1/3, γ1(t) and γ3(t) defined in Eq. (4.50) andγ2(t) defined in Eq. (4.51), kx = 0.08, σ = 0.025xˆ+0.396m.134Simulation scenario 3The third scenario assesses the impacts of time-varying wind direction, which ismodeled by maintaining U∞(t) = 8m/s and fluctuating V∞(t) sinusoidally between±2m/s with a period of 200sec. Specifically, V∞(t) is expressed as follows fort ≥ 1000sec:V∞(t) = (2m/s)sin[2pi200(t−1000sec)]. (4.52)All yaw angles in this scenario are maintained at γ1(t) = γ2(t) = γ3(t) = 0deg. Ve-locity contours for simulation case 3 are shown in Figure 4.13. The notable expec-tation here is that, as the wind direction changes, wake centerlines are transportedin tandem with the free stream wind in both xˆ and yˆ directions. For instance, att = 1000sec, the centerline of wake 1 is aligned with the xˆ axis since γ1(t) = 0degand V∞(t) had been equal to zero at all previous times. By t = 1050sec, the ef-fects of turbine 1 on the wind field should only be transported downstream by adistance of 8m/s× 50sec = 400m = 3.2D. Therefore, for xˆ < 3.2D, we expectvariations in the curvature of the centerline of wake 1 due to the presence of tur-bine 1, while for xˆ > 3.2D, this curvature should remain unchanged. Instead, forxˆ > 3.2D, the centerline of wake 1 should be shifted in the +yˆ direction as a re-sult of V∞(t) having held positive values for the past 50sec. Observing velocitycontours at t = 1050sec, it is evident that the centerline curvature of wake 1 re-mains flat at all downstream distances past approximately xˆ = 3D, while havingbeen shifted in the +yˆ direction.4.4 Conclusions and recommendations for futureresearchThis paper extended FOWFSim [103], which is a steady-state modeling tool thatmay be used for simulating and optimizing floating offshore wind farms, by addingcapabilities that captured time-varying free stream wind velocities and floatingplatform motion. In addition to presenting a mathematical formulation, we per-formed a mesh convergence study and validated FOWFSim-Dyn predictions re-lated to platform motion and steady wake transport. It was demonstrated that thelimited number of tunable parameters produced wake centerline deflection and ve-1350 2 4 6 8 10 12 14 16 18 20-2-1012012345678-2-1012-2-1012-2-1012Figure 4.13: Velocity contours at various time-steps of simulation scenario 3(i.e. fixed turbine operating conditions and fluctuating wind speed inthe yˆ direction, while platform motion is prohibited). The white +symbols represent the neutral positions of the floating platforms. Thewhite arrows denote the free stream wind direction. All floating plat-forms are held fixed throughout the simulation. Simulation param-eters: U∞(t) = 8m/s, V∞(t) defined in Eq. (4.52), a1(t) = a2(t) =a3(t) = 1/3, γ1(t) = γ2(t) = γ3(t) = 0deg, kx = 0.08, σ = 0.025xˆ+0.396m.136locity deficit results that matched experimental observations with reasonable simi-larity for engineering analysis. We then conducted simulations under various windand turbine operating conditions to assess the dynamic behavior of FOWFSim-Dyn. It was observed that FOWFSim-Dyn captures dynamic floating wind farmphenomena such as wake transport, time-varying wind speed and direction effects,and floating platform motion in line with physical reasoning and intuition.For the purposes of further developing and enhancing the current framework,several recommendations on potential research directions are made. First, to thisdate, no LES-based wind farm simulators are capable of capturing floating plat-form motion. Developing wind farm CFD tools that consider such dynamics wouldtherefore permit more comprehensive validation of FOWFSim-Dyn predictionspertaining to both platform motion and wake behaviour. Complementing this point,scaled wind tunnel experiments of floating wind turbines would also enable vali-dation of dynamic FOWFSim-Dyn predictions.Second, we did not model turbulence in the current framework. This featuremay be incorporated by adding measurement noise to model outputs, or by in-cluding temporally and spatially distributed turbulence acceleration terms in theequations of motion. Finally, additional force gradients may be included in theequations of motion to capture complex wake phenomena such as secondary steer-ing [60] and wake deflection due to rotor rotation [72].137Chapter 5Distributed Economic ModelPredictive Control – AddressingNon-convexity Using SocialHierarchiesThis chapter introduces a novel concept for addressing non-convexity in the costfunctions of distributed economic model predictive control (DEMPC) systems.Specifically, the proposed algorithm enables agents to self-organize into a hierar-chy which determines the order in which control decisions are made. This conceptis based on the formation of social hierarchies in nature. An additional feature ofthe algorithm is that it does not require stationary set-points that are known a pri-ori. Rather, agents negotiate these targets in a truly distributed and scalable manner.Upon providing a detailed description of the algorithm, guarantees of convergence,recursive feasibility, and bounded closed-loop stability are also provided. Finally,the proposed algorithm is compared against a basic parallel distributed economicmodel predictive controller using an academic numerical example.1385.1 Introduction5.1.1 BackgroundModel predictive control (MPC) entails recursively solving an optimization prob-lem over a finite prediction horizon to identify optimal future control input trajecto-ries. The popularity of MPC in academic and industrial environments is primarilyattributed to its capacity for handling constraints while computing control actionsthat minimize nonlinear performance criteria. The reader may refer to articles byMayne et al. [125] and Mayne [126] for reviews on MPC.In large-scale processes or multi-agent systems, implementation of MPC ina centralized manner may be impractical due to the computational complexity ofglobal optimization and the network infrastructure required for plant-wide com-munication. Distributed model predictive control (DMPC) surpasses these limita-tions by dispersing the burden of decision-making across a multitude of indepen-dent subsystems or agents. A trade-off that arises however, is that effective al-gorithms governing agent coordination are required to guarantee desirable closed-loop performance. The reader may refer to review articles by Al-Gherwi et al. [5],Christofides et al. [38], and Negenborn and Maestre [140] for further details on thesubject of DMPC.MPC has traditionally been utilized as a lower-level regulator and stabilizerthat tracks set-points determined by upper-level stationary optimizers. Economicmodel predictive control (EMPC) combines these upper- and lower-level roles byemploying cost functions that capture plant economics (e.g. power production oroperating cost over a finite time horizon). The effect is improved economic perfor-mance; however, additional measures for ensuring stability are required since theprimary control objective no longer involves regulation. The reader may refer toarticles by Ellis et al. [55] and Mu¨ller and Allgo¨wer [134] for reviews on EMPC.5.1.2 Distributed economic model predictive controlThis chapter addresses distributed economic model predictive control (DEMPC) ofsystems with non-convex objective functions and unknown stationary set-points.Applications with such characteristics include autonomous vehicle trajectory plan-139ning [54] and floating offshore wind farm control (see Chapter 6). DEMPC algo-rithms intended for such systems have been scarce in the literature as a result ofchallenges pertaining to stability and convergence. This subsection reviews rele-vant DEMPC and nonlinear DMPC algorithms as justification for the contributionsof the current work.Stabilizing DEMPC algorithmsAchieving stability in DEMPC requires first computing optimal stationary set-points for all agents, and then constraining state trajectories to approach these op-tima within the prediction horizon. If there exist feedback control laws that arethen capable of maintaining subsystems within specified bounds of their respectivesteady-states, stability may be guaranteed. To achieve such an outcome, theoret-ical studies focused on DEMPC have either treated these stationary set-points aspredefined references [184], or computed their values using centralized optimiza-tion [6, 36, 50, 111, 112, 185]. The latter group of algorithms are therefore nottruly distributed.To overcome this gap, Ko¨hler et al. [108] were the first to develop a DEMPCscheme without the requirement for centralized processing. They presumed thatoptimal stationary set-points were unattainable via centralized optimization, andinstead had to be negotiated online between agents in a distributed manner. Conse-quently, in tandem with solving their local EMPC problems and obtaining optimalinput trajectories, agents also performed one iterate of a distributed coordinationalgorithm at each sampling time to update their respective optimal steady-states.Nonetheless, this work focused on linear systems with convex cost functions andused a sequential coordination algorithm [109, 151]; thus suffering from lack ofscalability.Convergent DEMPC algorithmsIf DEMPC cost functions are non-convex, agents making decisions in parallel can-not guarantee convergence of their optimal input trajectories [116]. Several al-ternative classes of coordination algorithms within the nonlinear DMPC literatureaddress this convergence issue. Sequential methods first proposed by Kuwata et140al. [109] and Richards and How [151] represent the simplest solution. Agents solvetheir local optimization problems and exchange information with their neighbors insome predetermined order. The resulting advantage is that each subsequent agentcomputes its input trajectory based on updated and fixed information from its pre-decessors; guaranteeing convergence, stability, and feasibility is thus facilitated.The major drawback is lack of scalability to large interconnected systems, sinceagents at the tail-end of sequence must await decisions from all other subsystems.A secondary concern involves predetermining the sequence order, particularly insystems with time-varying interaction topologies.Coordination algorithms based on negotiation between agents were developedby Mu¨ller et al. [135], Maestre et al. [120], and Stewart et al. [170]. An agentreceives optimal decisions from its neighbors in the form of a proposal. Then,upon computing the corresponding effects of these decisions on its local objectivefunction, the agent may reject or approve proposals. These algorithms are capableof resolving conflict; however they face two limitations. The first is that, in orderto identify the impact of a specific agent’s control trajectory on neighboring costfunctions, this agent must not operate in parallel with others; thus limiting scala-bility. The second is that agents whose control actions are discarded at particulartime-steps remain idle. Finally, these algorithms possess no learning mechanismto ensure that, after a sufficient number of negotiations, proposals are guaranteedor more likely to be approved.Finally, group-based DMPC methods employ the connectivity information of aplant to identify the order in which agents should solve their local MPC problemsto resolve conflict. Pannek [144] proposed a covering algorithm that permittednon-interacting agent pairs to operate in parallel, while those that were coupledmade decisions sequentially according to some predetermined priority rule. Thisalgorithm eliminated the scalability issue of pure sequential DMPC; however itrequired a predetermined set of priority rules. Liu et al. [117] developed a clus-tering algorithm that assigned agents to dominant or connecting groups. Agents indominant clusters solved their local optimization problems first, thus eliminatingconflict with agents in connecting groups. The downside in this method was that asequential algorithm was required to determine clustering. Asadi and Richards [12]employed a slot allocation algorithm wherein each agent communicated with other141subsystems to randomly select an available space in the global sequential order.This method addressed the secondary drawback of sequential DMPC, which con-cerned determining an effective sequence order in systems with time-varying inter-action topologies. Nonetheless, the fully serial nature of the algorithm still sufferedfrom lack of scalability.5.1.3 ContributionsBased on the preceding literature review, we state, to the best of our knowledge,that a DEMPC algorithm that handles non-convex cost functions and unknown sta-tionary set-points in a scalable and truly distributed manner, with no predeterminedrules, has yet to be proposed. The existing method that meets all of these criteriaexcept for scalability and non-convexity is the algorithm of Ko¨hler et al. [108].The main contribution in this chapter is thus a DEMPC coordination algorithmthat is scalable, fully distributed, and that guarantees stability and convergencein the presence of non-convex cost functions and unknown stationary set-points.In brief, our approach borrows from the method of conflict resolution observedin nature. Namely, when it becomes apparent that agents operating in parallelgenerate conflicting decisions, a social hierarchy is established to yield resolution.Additionally, proofs of convergence, recursive feasibility, and bounded closed-loopstability are provided along with validation using a numerical example.5.1.4 Chapter organizationThe remainder of this chapter is organized as follows: Section 5.2 provides a de-scription of the nonlinear systems and cost functions that the proposed algorithmaddresses, along with an explanation of conflict and convergence issues arisingfrom non-convex objectives; Section 5.3 highlights the proposed DEMPC algo-rithm along with proofs of convergence, feasibility, and stability; Section 5.4 im-plements the proposed method on a numerical example with non-convex cost func-tions; and finally, Section 5.5 concludes the chapter with a summary of major find-ings, along with recommendations for future research directions.1425.2 Problem description5.2.1 NotationThis brief subsection introduces the reader to the set theory and other notationused in this work. The term Ia:b indicates a set of real integers ranging from a to b.The symbols x ∈ Rn state that x is a real-valued vector of dimensions n× 1. TheexpressionA \B denotes the difference between the setsA andB (i.e. the setAwith all elements of set B removed). The operation A ×B yields the Cartesianproduct of the sets A andB.5.2.2 Dynamic modelWe consider N agents that are dynamically decoupled and uninfluenced by distur-bances. The dynamics of each agent i ∈I = {1,2, · · · ,N} are represented by thefollowing discrete-time nonlinear state-space model:x+i = fi(xi,ui), (5.1)where xi ∈ Rni and ui ∈ Rmi denote vectors containing the ni states and mi inputsof agent i, and x+i represents xi at the subsequent sampling time-step. We considerthe case where xi and ui must be bounded within the convex sets Xi and Ui at alltimes, which results in the following state and input constraints:ui ∈ Ui, (5.2)xi ∈ Xi. (5.3)With these operational bounds defined, we make the following assumptionsconcerning controllability and continuity.Assumption 1. (Weak controllability) Let the setZ si comprise all feasible station-ary points of agent i as follows:Z si := {(xi,ui) ∈Xi×Ui | xi = fi(xi,ui)} . (5.4)All feasible stationary state vectors of agent i may then be collected within the set143X si , which is defined as follows:X si := {xi ∈Xi | ∃ui ∈Ui : (xi,ui) ∈Z si } . (5.5)Let the set Z 0→si contain all pairings of initial state vectors x0i and input tra-jectories ui =(u0i ,u1i , · · · ,uH−1i)that steer agent i to each feasible stationary pointxsi in H time-steps, while satisfying constraints. Z 0→si may therefore be defined asfollows:Z 0→si :={(x0i ,ui,xsi ) ∈Xi×U i×X si |∃x1i ,x2i , · · · ,xHi : xki = fi(xk−1i ,uk−1i ),xki ∈Xi,∀k ∈ I1:H ,xHi = xsi},(5.6)where U i = Ui× ·· · ×Ui = U Hi . All possible initial state vectors x0i that maybe steered to a feasible stationary point xsi , with constraint satisfaction, are thencontained within the setX 0→si defined as follows:X 0→si :={x0i ∈Xi | ∃ui ∈U i,xsi ∈X si :(x0i ,ui,xsi) ∈Z 0→si } . (5.7)For any agent i∈I , any initial state vector x0i ∈X 0→si , input vector trajectoryui ∈ U i, and stationary state vector xsi ∈X si such that(x0i ,ui,xsi) ∈ Z 0→si , andany stationary input vector usi ∈ Ui such that (xsi ,usi ) ∈ Z si , there exists a K∞function γ(·) that satisfies the following condition:H−1∑k=0‖uki −usi‖ ≤ γ(‖x0i −xsi‖). (5.8)Remark. The weak controllability assumption simply states that, for any feasiblestationary point, there exists some surrounding set from which an initial state vec-tor may be steered to the stationary point. This assumption is necessary for guar-anteeing feasibility of the optimization problem of the DEMPC algorithm sincereaching a stationary target is one of its constraints. Therefore, if an input trajec-tory exists that can steer an initial state vector to a stationary point, then a solutionto the optimization problem exists that satisfies its constraints.144Assumption 2. (Lipschitz continuous dynamics) For any agent i∈I , fi(·) satisfiesthe following condition for Lipschitz continuity for all (xai ,uai ),(xbi ,ubi ) ∈Xi×Ui:∥∥fi(xbi ,ubi )− fi(xai ,uai )∥∥≤ Λ fi ∥∥(xbi ,ubi )− (xai ,uai )∥∥ , (5.9)where the scalar Λ fi ≥ 0 is the Lipschitz constant of fi(·) on the setXi×Ui.Remark. Lipschitz continuity simply states that the function fi(·) must be contin-uous. In other words, there must exist no discontinuities along fi(·) that lead to anundefined gradient. This assumption is necessary for guaranteeing optimality inthe solution of an optimization problem. If gradients are defined, a local minimumof a cost function will be reached after a sufficient number of iterations.5.2.3 Control objectiveAt each time-step, the control objective of agent i is to minimize a cooperativeeconomic stage cost function Ji(·) over a finite prediction horizon H as follows:minH−1∑k=0Ji(xki ,uki ,xk−i|J,uk−i|J), (5.10)where the superscript k identifies the time-step number along the prediction horizonH, xki and uki denote the state and input vectors of agent i at time-step k alongthe prediction horizon, and xk−i|J and uk−i|J contain the state and input vectors attime-step k along the prediction horizon of all agents j ∈ I \ i that influence thecooperative cost function Ji(·) of agent i. We collect the indices of these agents intothe setN−i|J . Likewise, the indices of all agents j ∈I \ i whose cooperative stagecost functions J j(·) are influenced by xi and ui are collected into the setN+i|J .The objective function Ji(·) may be non-convex; however, it must adhere to thefollowing assumptions concerning cooperation, boundedness, and continuity.Assumption 3. (Neighborhood-cooperative objectives) Let each agent i ∈I pos-sess a stage cost function `i(·) that represents its local economic interests. Then,let the set N−i contain the indices of all agents j ∈ I \ i whose state and inputvectors x j and u j influence the local stage cost function `i(·). Likewise, let the set145N+i contain the indices of all agents j ∈ I \ i whose local stage cost functions` j(·) are influenced by xi and ui.The stage cost function Ji(·) for any agent i ∈I is neighborhood-cooperativein that it comprises the local interests of agent i and those of each downstreamneighbor j ∈N+i as follows:Ji(xi,ui,x−i|J,u−i|J) := `i(xi,ui,x−i,u−i)+ ∑j∈N+i` j(x j,u j,x− j,u− j). (5.11)The vectors x−i and u−i contain the states and inputs of all agents j ∈N−i. NotethatN−i|J =N−i∪N+i∪N− j∀ j ∈N+i.Remark. The purpose of the social hierarchy-based DEMPC algorithm presentedin this work is to enable coupled agents to compute optimal decisions that aremutually beneficial. A fundamental requirement for this algorithm is therefore thatcoupled agents share interests; hence the assumption of neighborhood-cooperativecost functions. If this assumption is not present, then the agents are competitive,and mutually beneficial decisions cannot be guaranteed.Assumption 4. (Bounded cost function minima) Let the sets X−i and U−i be de-fined as follows:X−i := ∏j∈N−iX j, (5.12)U−i := ∏j∈N−iU j, (5.13)For any agent i ∈ I , there exist state and input vectors (x∗i ,u∗i ,x∗−i,u∗−i) ∈Xi×Ui×X−i×U−i such that the following condition holds for all (xi,ui,x−i,u−i) ∈Xi×Ui×X−i×U−i:`i(x∗i ,u∗i ,x∗−i,u∗−i)≤ `i(xi,ui,x−i,u−i). (5.14)Remark. The bounded cost function minima assumption is necessary for guar-anteeing convergence of optimization problems. If the global minimum of a costfunction is finite, then an optimization algorithm that descends along the gradients146of the cost function is guaranteed to reach a point that satisfies optimality condi-tions after a sufficient number of iterations.Assumption 5. (Lipschitz continuous objectives) For any agent i ∈ I , the localcost function `i(·) satisfies the following condition for Lipschitz continuity for all(xai ,uai ,xa−i,ua−i),(xbi ,ubi ,xb−i,ub−i) ∈Xi×Ui×X−i×U−i:∥∥`i(xbi ,ubi ,xb−i,ub−i)− `i(xai ,uai ,xa−i,ua−i)∥∥≤Λ`i∥∥(xbi ,ubi ,xb−i,ub−i)− (xai ,uai ,xa−i,ua−i)∥∥ , (5.15)where the scalar Λ`i ≥ 0 is the Lipschitz constant of `i(·) on the set Xi×Ui×X−i×U−i.Remark. Refer to the remark of Assumption 2 for a simplified explanation of Lip-schitz continuity.5.2.4 Conflict under non-convexityIn this subsection, we elaborate further on the main challenge that is associatedwith non-convex cost functions in DEMPC. Consider a simple problem with onlytwo optimization variables z1 and z2, which are computed by agents 1 and 2, re-spectively. Further, let both agents share a common non-convex global objectivefunction with contours plotted in Figure 5.1.Assume initial values z01 and z02 obtained from a previous iteration or time-step. Under parallel and fully distributed operation, each agent must assume thatits neighbors optimization variable remains unchanged while locally minimizingthe global objective function. As a result, agent 1 assumes that z2 remains fixed atz02 and restricts its search path to the horizontal orange line shown in Figure 5.1.Likewise, agent 2 assumes that z1 is maintained at z01, which constrains its searchpath to the vertical orange line.Upon completion of its local optimization problem, agent 1 finds the local op-timum located at(z˘1,z02). Agent 2 achieves the same at(z01, z˘2). When the updatedoptimal variables z˘1 and z˘2 are combined however, the overall system operates atneither of the local optima identified by the individual agents. We refer to such147Figure 5.1: A visual explanation of conflict generated as a result of non-convexity in distributed parallel optimization.an outcome as conflict in the current work. Specifically, we define conflict andconflict-free operation as follows.Definition 1. (Conflict) Agent i encounters conflict when its economic performancedeteriorates upon considering the optimal control actions of its neighbors. Moreformally, consider Vˆ si and V˘si defined as follows:Vˆ si := Ji(x˘si , u˘si , xˆs−i|J, uˆs−i|J), (5.16)V˘ si := Ji(x˘si , u˘si , x˘s−i|J, u˘s−i|J), (5.17)where x˘si and u˘si denote the optimal stationary state and input vectors computedby agent i, xˆs−i|J and uˆs−i|J contain stationary state and input vectors that agent iassumes for all neighbors j ∈N−i|J , and x˘s−i|J and u˘s−i|J consist of optimal stateand input vectors computed by all agents j ∈N−i|J and communicated to agent i.The terms Vˆ si and V˘si represent naive and informed values of the stage cost functionof agent i at some stationary point, respectively. The term naive indicates that thecost function value is computed based on assumed values of neighboring agents’state and input vectors. Contrarily, the term informed is employed when agent iconsiders recently communicated updated optimal state and input vectors. Giventhese definitions, while attempting to negotiate an optimal stationary point (x˘si , u˘si ),148agent i operates in conflict with its neighbors if the following statement is true:V˘ si > Vˆsi . (5.18)Similarly, one may define naive and informed values of cost functions summedalong the prediction horizon as follows:Vˆi :=H−1∑k=0Ji(x˘ki , u˘ki , xˆk−i|J, uˆk−i|J), (5.19)V˘i :=H−1∑k=0Ji(x˘ki , u˘ki , x˘k−i|J, u˘k−i|J), (5.20)where x˘ki and u˘ki denote optimal state and input vectors computed by agent i attime-step k along the prediction horizon, xˆk−i|J and uˆk−i|J contain state and in-put vectors that agent i assumes for all neighbors j ∈ N−i|J at some time-stepk along the prediction horizon, and x˘k−i|J and u˘k−i|J consist of optimal state andinput vectors computed by all agents j ∈ N−i|J at some time-step k along theprediction horizon and communicated to agent i. Given this information, whileattempting to negotiate optimal state and input trajectories x∗i =(x˘0i , x˘1i , · · · , x˘Hi)and u∗i =(u˘0i , u˘1i , · · · , u˘H−1i), agent i operates in conflict with its neighbors if thefollowing statement is true:V˘i > Vˆi. (5.21)Definition 2. (Conflict-free operation) Agent i operates free of conflict when itseconomic performance improves or remains unchanged upon considering the opti-mal control actions of its neighbors. More formally, consider once again the valuesVˆ si , V˘si , Vˆi, and V˘i as previously defined. While attempting to negotiate an optimalstationary point (x˘si , u˘si ), agent i operates free of conflict with its neighbors if thefollowing statement is true:V˘ si ≤ Vˆ si . (5.22)While attempting to negotiate optimal trajectories x∗i and u∗i , agent i operates freeof conflict with its neighbors if the following statement is true:V˘i ≤ Vˆi. (5.23)149In the introduction, several algorithms based on sequential operation, agentnegotiation, and agent grouping that could resolve non-convex conflict were dis-cussed. The main disadvantages of these algorithms were lack of scalability, idle-ness of certain agents, and the requirement of predefined rules. In the next section,we propose a fully distributed and scalable solution to the problem of conflict thataddresses these drawbacks by establishing social hierarchies.5.3 Social hierarchy-based DEMPC algorithm5.3.1 Social hierarchy frameworkIn the current context, a social hierarchy consists of a finite number of levels thatestablish the sequence in which agents generate decisions in order to resolve con-flict. The concept is based loosely on social hierarchies that appear naturally amongliving organisms as a means to resolve conflict and establish which individuals’ de-cisions take priority over those of others. These hierarchies are often determined bythe evolutionary or cultural characteristics of the individuals, which are, in essence,randomly assigned. In a similar fashion, we propose a framework which permitsthe formation of hierarchies with elements of randomness in order to resolve con-flict resulting from non-convexity.For generality, we assume an iterative parallel coordination algorithm, the firstof which was presented by Du et al. [51]. Agents synchronously solve their localoptimization problems and communicate repeatedly within a single sampling time-step until some termination condition is satisfied or until a maximum number ofiterations have been implemented. The socially hierarchy framework is equallyapplicable to non-iterative parallel methods which were first investigated by Jiaand Krogh [92] and Dunbar and Murray [53].A visual representation of the social hierarchy framework is shown in Fig-ure 5.2. Within a single iteration, there exist Nq hierarchy levels which specifythe order in which agents make decisions. Each agent may solve for its optimalstationary point and control trajectory only once within an iteration; however, thiscomputation may take place within any hierarchy level. During each iteration,agents occupying hierarchy level q = 1 make decisions first and transfer relevant150Figure 5.2: Schematic demonstrating the organization structure of the socialhierarchy framework.information to their neighbors. Following this step, agents allocated to hierarchylevel q = 2 perform the same task. This trend continues until all hierarchy levelcomputations have been performed, at which point the entire process is repeatedduring the next iteration. It is important to note that multiple agents may occupythe same hierarchy level, and that Nq may be substantially smaller than N.Two fundamental questions now arise regarding (i) how agents should sortthemselves among the Nq hierarchy levels in order to resolve conflict, and (ii) howshould Nq be determined by the control system designer. The former concern isaddressed in Section 5.3.2, which describes and assesses a novel DEMPC coordi-nation algorithm that utilizes the concept of a social hierarchy. The latter questionis discussed in Section 5.3.4, which burrows elements from vertex coloring theoryto establish social hierarchy properties.5.3.2 DEMPC coordination algorithmThis subsection details a novel coordination algorithm for DEMPC with non-convexobjectives based on the social hierarchy framework described in Section 5.3.1. Ourapproach allows agents to resolve their conflicts in a truly distributed and scalablemanner without the requirement of predefined rules or access to the full system151interaction topology. Each agent achieves this outcome by occupying an appro-priate level along a social hierarchy when conflict arises. In brief, our algorithmfollows evolutionary principles. That is, if an agent occupying a particular hierar-chy level during some iteration experiences conflict, then its current hierarchy levelis detrimental to its performance and must be randomly mutated.While negotiating optimal trajectories x∗i and u∗i , agent i solves the followingoptimization problem during each iteration:minui,xiH−1∑k=0Ji(xki ,uki , xˆk−i|J, uˆk−i|J), (5.24)subject tox0i = xi, (5.25a)xk+1i = fi(xki ,uki ), (5.25b)xki ∈ Xi, (5.25c)uki ∈ Ui, (5.25d)xHi = xsi , (5.25e)where xki and uki denote the candidate state and input vectors of agent i at sometime-step k along the prediction horizon H, xi and ui represent candidate state andinput vector trajectories as follows:xi :=(x0i ,x1i , · · · ,xHi), (5.26)ui :=(u0i ,u1i , · · · ,uH−1i), (5.27)and xˆk−i|J and uˆk−i|J contain the state and input vectors that agent i assumes forall neighbors j ∈N−i|J at time-step k. For notational brevity in Algorithm 1, wecondense the assumed states and input of agent j into the trajectories x˜ j and u˜ j asfollows:x˜ j :=(xˆ0j , xˆ1j , · · · , xˆHj), (5.28)u˜ j :=(uˆ0j , uˆ1j , · · · , uˆH−1j). (5.29)152Solving Problem (5.24) yields the optimal state and input vector trajectories x∗iand u∗i defined as follows:x∗i :=(x˘0i , x˘1i , · · · , x˘Hi), (5.30)u∗i :=(u˘0i , u˘1i , · · · , u˘H−1i), (5.31)where x˘ki and u˘ki denote the optimal state and input vectors at time-step k alongthe prediction horizon. The difference between candidate and optimal solutions isthat the optimal solutions x∗i and u∗i are not immediately accepted by the DEMPCalgorithm. In other words, if it was required that DEMPC algorithm implement adecision immediately, the enacted action would follow the candidate trajectories xiand ui, not the optimal solutions x∗i and u∗i . As detailed in Algorithm 1, x∗i andu∗i are only accepted as candidate solutions if they do not yield conflict. If x∗i andu∗i do yield conflict, then the previous values of xi and ui remain as the candidatetrajectories to be implemented.Constraint (5.25a) serves as the initial condition of the prediction horizon bysetting the candidate state vector of agent i at k = 0 equal to the most recent statemeasurement xi. Constraint (5.25b) requires that the optimal input and state trajec-tories x∗i and u∗i computed along the prediction horizon satisfy the plant dynamicsof agent i. Constraints (5.25c) and (5.25d) state that the optimal input and statevectors x˘ki and u˘ki of agent i must remain within the process constraint setsXi andUi, respectively, at any time-step k along the prediction horizon. Finally, Con-straint (5.25e) ensures that, by the end of the prediction horizon H, the computedoptimal input trajectory u∗i leads the local state vector to a feasible candidate steadystate xsi .To negotiate the candidate stationary state vector xsi , agent i solves the follow-ing optimization problem during each iteration:minui,usi ,xi,xsiJi(xsi ,usi , xˆs−i|J, uˆs−i|J), (5.32)153subject tox0i = xi, (5.33a)xk+1i = fi(xki ,uki ),∀k ∈ I0:H−1, (5.33b)xsi = fi(xsi ,usi ), (5.33c)xsi = xHi , (5.33d)xki ∈ Xi,∀k ∈ I0:H , (5.33e)uki ∈ Ui,∀k ∈ I0:H−1, (5.33f)(xsi ,usi ) ∈ Xi×Ui, (5.33g)where xsi and usi denote the candidate stationary state and input vectors of agent i,and xˆs−i|J and uˆs−i|J contain similar information that agent i assumes for all neigh-bors j ∈N−i|J . The steady state and input vectors that agent i assumes for an indi-vidual neighbor j are denoted by xˆsj and uˆsj, respectively. Solving Problem (5.32)yields the optimal steady state and input vectors x˘si and u˘si . Similar to the optimaltrajectories x∗i and u∗i , the optimal stationary solution is only accepted if it yieldsconflict-free operation. Otherwise, the previous values of xsi and usi remain as thecandidate stationary terminal set-point.Constraint (5.33c) ensures that the computed optimum (x˘si , u˘si ) is a stationarypoint. Constraint (5.33d) then requires that the computed state vector trajectoryends at the optimal steady state x˘si . Finally, Constraint (5.33g) states that the com-puted optimal stationary point (x˘si , u˘si ) must lie within the process constraint setXi×Ui.The difference between Problems (5.24) and (5.32) is that the latter only con-siders the stage cost function Ji(·) and therefore computes a feasible optimal steadystate vector x˘si without minimizing Ji(·) over the prediction horizon. The reasonthat dynamics are considered in Problem (5.32) is to ensure that the computedoptimal steady state vector x˘si is reachable from the initial state vector x0i . Prob-lem (5.24) then computes optimal trajectories that minimize Ji(·) over the predic-tion horizon and steer the system to the reachable candidate steady state vectorxsi . A description of the proposed social hierarchy-based DEMPC algorithm nowfollows.154Algorithm 1. Social hierarchy-based DEMPC coordination scheme. Implementin parallel for all agents i ∈I .Communication protocol:• Send xi, ui, xsi , and usi to all agents j ∈N+i|J , receive x j, u j, xsj, and usj from allagents j ∈N−i|J , and set x˜ j = x j, u˜ j = u j, xˆsj = xsj, and uˆsj = usj for all j ∈N−i|J .Initialization:1. Specify Nq ≥ 1 and set qi = 1.2. Initialize xi, xi ∈X i =Xi×·· ·×Xi =X Hi , ui ∈U i, xsi ∈Xi, usi ∈Ui suchthat x0i = xi, xk+1i = fi(xki ,uki )∀k ∈ I0:H−1, xHi = xsi , and xsi = fi(xsi ,usi ).3. Implement communication protocol.Perform at each new time-step:1. Measure xi, and compute xi such that x0i = xi and xk+1i = fi(xki ,uki )∀k ∈ I0:H−1.2. Implement communication protocol.3. For iteration number p = 1,2, · · · ,Np, do:(a) For sequence slot number q = 1,2, · · · ,Nq, do:i. If q = qi, (i) solve Problem (5.32) to acquire x˘si and u˘si , (ii) compute Vˆ si ac-cording to Eq. (5.16), (iv) send x˘si and u˘si to all agents j ∈N+i|J , (v) receivex˘sj and u˘sj from all agents j ∈N−i|J , and update xˆsj = x˘sj and xˆsj = x˘sj.ii. Else, receive x˘sj and u˘sj from all agents j ∈ N−i|J , and update xˆsj = x˘sj andxˆsj = x˘sj.(b) Compute V˘ si according to Eq. (5.17). If V˘si > Vˆsi , randomly change qi withuniform probability and set x˘si = xsi and u˘si = usi . Else, if V˘ si ≤ Vˆ si , updatexsi = x˘si and usi = u˘si .(c) Implement communication protocol.4. For iteration number p = 1,2, · · · ,Np, do:155(a) For sequence slot number q = 1,2, · · · ,Nq, do:i. If q = qi, (i) solve Problem (5.24) to acquire x∗i and u∗i , (ii) compute Vˆi ac-cording to Eq. (5.20), (iv) send x∗i and u∗i to all agents j ∈N+i|J , (v) receivex∗j and u∗j from all agents j ∈N−i|J , and update x˜ j = x∗j and u˜ j = u∗j .ii. Else, receive x∗j and u∗j from all agents j ∈ N−i|J , and update x˜ j = x∗j andu˜ j = u∗j .(b) Compute V˘i according to Eq. (5.20). If V˘i > Vˆi, randomly change qi with uni-form probability and set x∗i = xi and u∗i = ui. Else, if V˘i ≤ Vˆi, update xi = x∗iand ui = u∗i .(c) Implement communication protocol.5. Apply candidate control input u0i to the system, update ui such that uki =uk+1i ∀k∈I0:H−2 and uH−1i = usi .During initialization, step 1 requires the control system designer to specify thequantity Nq of hierarchy levels and to allocate agent i to the first level. As a result,all agents initially solve their local optimization problems in parallel. Step 2 re-quires that feasible state and input trajectories xi and ui that satisfy the constraintsof Eq. (5.25), and steer the system to a reachable stationary point (xsi ,usi ), be speci-fied for agent i given the initial state vector xi. Finally, step 3 involves the exchangeof candidate trajectories x j and u j and stationary vectors xsj and usj between agent iand all of its neighbors j ∈N−i|J . Agent i then uses this incoming information toestablish the assumed values of its neighbors’ states and inputs x˜ j, u˜ j, xˆsj, and uˆsj forall j ∈N−i|J . These assumptions remain unchanged until future communication.Focusing on the recursive portion of the DEMPC algorithm, agent i first mea-sures its state vector xi and updates its local state trajectory xi using the most up-to-date candidate input sequence ui. In step 2, this updated information is communi-cated between neighbors. As a result, prior to step 3, agent i possesses the most up-to-date trajectories and stationary vectors x j,u j, xsj, and usj of neighbors j ∈N−i|J .The assumptions x˜ j, u˜ j, xˆsj, and uˆsj for all j ∈N−i|J are also updated as a result.Step 3 initiates an iterative process within the current sampling time-step withthe objective of identifying an appropriate candidate stationary point (xsi ,usi ) toserve as a terminal constraint in future steps. In step 3(a), agent i cycles through156all hierarchy levels sequentially and, during its allocated hierarchy level, solvesProblem (5.32) to obtain an optimal stationary point (x˘si , u˘si ). These vectors, alongwith the assumptions xˆsj and uˆsj for all j ∈ N−i|J , are then used to calculate thenaive stage cost function value Vˆ si to be used for comparison to V˘si later on. Fi-nally, agent i exchanges optimal stationary vector information with neighboringagents and updates its assumptions xˆsj and uˆsj for all j ∈N−i|J . Outside of its hier-archy level, agent i remains idle and only receives information from neighbors andupdates its assumptions.After all hierarchy levels have been cycled through, agent i will have receivedx˘sj and u˘sj from all neighbors j ∈ N−i|J . Step 3(b) requires the computation ofthe informed stage cost function value V˘ si . If V˘si > Vˆsi , then agent i is in conflictwith its neighbors while attempting to identify an optimal stationary point, and itis necessary for its hierarchy level qi to be randomly mutated. Further, if conflictis encountered, then agent i resets it optimal stationary vectors x˘si and u˘si to thecandidate stationary vectors xsi and usi . This reset essentially erases x˘si and u˘si sothat conflict-yielding optimal solutions are no longer communicated. If V˘ si ≤ Vˆ si ,then agent i is operating free of conflict, and its current hierarchy level may bemaintained. Further, since the recently computed stationary vectors x˘si and u˘si didnot yield conflict, they may replace the candidate stationary vectors xsi and usi . Asa result, agent i updates xsi and usi using x˘si and u˘si .Step 3 is terminated once the maximum number of iterations Np has beenreached. Step 4 essentially repeats step 3, except the candidate terminal state vec-tor xsi has now been updated, and the objective is to update candidate state andinput trajectories xi and ui. The maximum number of iterations available for thisstep is also Np. Step 5 requires that the control input vector corresponding to thefirst time-step along the prediction horizon be applied to the system. Additionally,the candidate input trajectory ui for the next sampling time-step is constructed byconcatenating it with the candidate stationary input vector usi .5.3.3 Closed-loop propertiesThe current subsection establishes closed-loop properties for Algorithm 1. We firstdemonstrate that, regardless of the interaction topology of a multi-agent system,157all conflicts may be resolved in a finite number of iterations with some probability.We then address convergence, feasibility, and stability.Theorem 1. (Conflict resolution) There exist a finite number of iterations afterwhich, with some probability greater than zero, Inequalities (5.22) and (5.23) areguaranteed to be satisfied at each iteration within a single time-step.Proof. This proof consists of two parts; (i) it is first necessary to prove that, for anyinterconnected system, at least one social hierarchy exists that will ensure system-wide conflict resolution; (ii) it is then proved that the probability of agents self-organizing according to such a social hierarchy is greater than zero during anyiteration.Part I: Consider a multi-agent system wherein all agents possess the same stagecost function J(·) defined as follows:J(x,u) := ∑i∈I`i(xi,ui,x−i,u−i), (5.34)where x ∈ R∑i∈I ni and u ∈ R∑i∈I mi contain the ni states and mi inputs of allagents i ∈ I , and J(·) is global-cooperative in that it considers the local inter-ests `i(·) of all agents i ∈I . The resulting dynamic optimization problem over theprediction horizon H for any agent i is therefore defined as follows:minui,xiH−1∑k=0J(xk,uk), (5.35)with constraints similar to those of Problem (5.24). The vectors xk and uk denotesystem-wide states and inputs at time-step k along the prediction horizon. Thestationary optimization problem for any agent i is defined as follows:minui,usi ,xi,xsiJ(xs,us), (5.36)with constraints similar to those of Problem (5.32). The vectors xs and us denote158system-wide steady states and inputs. We also define V (·) and V s(·) as follows:V (x,u) :=H−1∑k=0J(xk,uk), (5.37)V s(xs,us) := J(xs,us), (5.38)where x =(x0,x1, · · · ,xH) and u = (u0,u1, · · · ,xH−1) denote system-wide stateand input trajectories over the prediction horizon.As a reference case, let the agents operate in a fully sequential manner suchthat, at any given iteration, only one agent solves its local optimization problem.Focusing on the negotiation of a stationary point for now, only agent i updatesxsi and usi at some iteration p, then transmits this information to all other agents.Furthermore, let recursive feasibility, which is established independently in The-orem 3, be presumed, and let Assumptions 2, 4, and 5 concerning continuity andbounded minima hold. Under these conditions, upon solving Problem (5.36), eachagent i ∈ I is guaranteed to shift xs and us until the gradient of J(·) projectedalong the variable space (xsi ,usi ) satisfies optimality conditions. Since this processoccurs sequentially across all agents i ∈I , V s(·) is guaranteed to decrease or re-main unchanged after each subsequent agent’s update to xsi and usi , which ensuresconflict-free operation as per Definition 2.We now prove that the above result may be achieved using social hierarchiesthat are not necessarily fully sequential (i.e. a subset of agents may solve theirlocal optimization problems simultaneously) and also assuming neighborhood-cooperative stage cost functions as per Assumption 3. From the perspective ofany agent i ∈I , J(·) may be arranged as follows:J(x,u) = `i(xi,ui,x−i,u−i)+ ∑j∈N+i` j(x j,u j,x− j,u− j)+ ∑κ∈I \N+i,κ 6=i`κ(xκ ,uκ ,x−κ ,u−κ), (5.39)where the three terms on the right-hand-side represent, from left to right, the localinterests of agent i, the local interests of agents whose cost functions are influencedby agent i, and the local interests of all remaining agents whose cost functions are159uninfluenced by agent i. Taking the gradient of J(·) along the variable space (xi,ui)yields the following:∇xi,uiJ(x,u) = ∇xi,ui`i(xi,ui,x−i,u−i)+∇xi,ui ∑j∈N+i` j(x j,u j,x− j,u− j). (5.40)Note that gradient of `κ(·) for all κ ∈ I \N+i,κ 6= i is zero along (xi,ui) sincethese expressions have no dependency upon the operation of agent i. The shapeof J(·) along the variable space (xi,ui) is therefore only influenced by x j and u jfor all j ∈N−i|J , sinceN−i|J =N−i∪N+i∪N− j∀ j ∈N+i. This result yields twoimportant consequences.First, the local optimization problem of agent i is uninfluenced by the controlactions of agents κ ∈ I \N−i|J,κ 6= i. Agent i may therefore update its controlactions in parallel with agents κ ∈ I \N−i|J,κ 6= i, and the computed optimaltrajectories are guaranteed to decrease or preserve V s(·). Second, whether agent iemploys the global stage cost function J(·), or the neighborhood-cooperative stagecost function Ji(·) defined in Eq. (5.11), the computed optimal trajectories remainunchanged. This property is true since ∇xi,uiJ(·) = ∇xi,uiJi(·). Therefore, if As-sumption 3 concerning neighborhood-cooperative cost functions holds, and if notwo agents i and j such that j ∈N−i|J for all i, j ∈ I , j 6= i solve their local op-timization problems simultaneously, then at least one social hierarchy exists thatwill satisfy Inequality (5.22) after each iteration. The above logic may be extendedto step 4 in Algorithm 1 without modification. Part I of the proof is thus completed.Part II: Due to its distributed nature, Algorithm 1 may lead some agents to resolvetheir conflicts earlier than others. However, we prove that, even in a worst-casescenario in which all agents initially encounter conflict, the probability that allconflicts will be resolved within a single iteration is greater than zero. Let Nsdescribe the quantity of possible social hierarchies that will resolve conflict amongN agents. If agents must, with uniform probability, randomly choose among Nqhierarchy levels, the probability P that all conflicts are resolved within a singleiteration is defined as follows:P = Ns(1Nq)N. (5.41)160Thus, as long as Ns > 0, P > 0 must also be true. Part I of this proof demon-strated that Ns ≥ 1 for any interconnected system with neighborhood-cooperativecost functions.Theorem 2. (Convergence) Let xs,pi and upi denote the candidate steady state vec-tor and input trajectory xsi and ui held by agent i at iteration p. Given a sufficientlylarge number of iterations in Steps 3 and 4 in Algorithm 1, the following inequali-ties are guaranteed to be satisfied for all i ∈I :∥∥∥xs,pi −xs,p−1i ∥∥∥ ≤ α ∥∥∥xs,p−1i ∥∥∥ , (5.42)∥∥∥upi −up−1i ∥∥∥ ≤ β ∥∥∥up−1i ∥∥∥ , (5.43)where α > 0 and β > 0 represent fixed convergence tolerances.Proof. Theorem 1 guarantees that, with a large enough number of iterations, system-wide conflict-resolution may be achieved with some probability greater than zero.As per Definition 2, and focusing on step 3 from Algorithm 1, once system-wideconflict-free operation has been achieved, the inequality V˘ si ≤ Vˆ si is guaranteed tobe satisfied after each iteration for any agent i ∈ I . Let Vˆ s,pi and V˘ s,pi denote Vˆ siand V˘ si computed during iteration number p. Since Vˆpi is computed using optimalstate and input trajectories of agent i that have been updated during iteration p,then Vˆ s,pi ≤ V˘ s,p−1i must be true during conflict-free operation. Therefore, sinceV˘ s,pi ≤ Vˆ s,pi and Vˆ s,pi ≤ V˘ s,p−1i , then V˘ s,pi ≤ V˘ s,p−1i must be true after conflicts havebeen resolved, which indicates that the value of Ji(·) computed using updated sta-tionary points of all relevant agents is guaranteed to decrease with each subsequentiteration. If Assumption 4 concerning bounded minima holds, then (xsi ,usi ) is guar-anteed to approach a local minimizer of Ji(·), thus satisfying Inequality (5.42). Theabove logic may be extended to step 4 of Algorithm 1 without modification; thusconcluding the proof.Theorem 3. (Recursive feasibility) The constraints of Problems (5.32) and (5.24)are guaranteed to be satisfied during each iteration at any sampling time k≥ 0 forall agents i ∈I .161Proof. Provided that initial values of xi, ui, xsi , and usi established at the start ofsome time-step are feasible, then at least one viable set of values for xi, ui, xsi ,and usi exists that satisfies Constraints (5.25) and (5.33), thus ensuring recursivefeasibility at each subsequent iteration within the time-step. Further, since theabsence of disturbances restricts state progression to xi, then feasibility is preservedat subsequent time-steps. Guaranteeing recursive feasibility thus requires that xi,ui, xsi , and usi are initially feasible. Referring to Eq. (5.7), the existence of xi, ui,xsi , and usi such that Constraints (5.25) and (5.33) are satisfied requires that the setX 0→si is not empty. Assumption 1 concerning weak controllability bounds thecontrol input trajectory ui required to steer any initial state vector x0i ∈X 0→si to areachable stationary point (xsi ,usi ) such that (x0i ,ui,xsi )∈Z 0→si . As a result, the setX 0→si must be non-empty; thus concluding the proof.Theorem 4. (Closed-loop stability) For all agents i ∈I , as k→ ∞, the measuredstate vector xi will remain bounded within a setX ∗i ⊂X 0→si surrounding a fixedstationary point x∗i ∈X si . The setX ∗i is defined as follows:X ∗i :={xi ∈Xi | ∃ui ∈U i : (xi,ui,x∗i ) ∈Z 0→si}. (5.44)Proof. For all agents i∈I , while updating candidate state and input vector trajec-tories xi and ui, Constraint (5.25e) ensures that the state vector xi always remainswithin a reachable set surrounding some steady state vector xsi . Therefore, in orderto prove that this reachable set ultimately maintains a fixed value X ∗i , one mustprove that the terminal steady state vector xsi approaches a fixed value x∗i as k→ ∞for all agents i ∈I .We assume for the time being that, after a sufficient number of iterations, agentsestablish a social hierarchy that permanently resolves all conflicts. That is, Inequal-ity (5.22) is guaranteed to be satisfied at each iteration within all subsequent time-steps, and the social hierarchy therefore ceases to change. We shall refer to sucha social hierarchy as a universal social hierarchy. Let V˘ s,ki |A and V˘ s,ki |B denote V˘ sicomputed at the initial and final iterations, respectively, of time-step k. It is clearthat, after a universal social hierarchy has been established, V˘ s,ki |B ≤ V˘ s,ki |A is satis-fied at all subsequent time-steps. Since V˘ s,ki |A is computed using x˘si and u˘si obtainedby solving Problem (5.32) at the first iteration of time-step k, then V˘ s,ki |A≤ V˘ s,k−1i |B162must also be true. As a result, V˘ s,ki |B ≤ V˘ s,k−1i |B must be satisfied at all time-stepsfollowing the establishment of a universal social hierarchy. If Assumption 4 con-cerning bounded minima holds, (xsi ,usi ) is guaranteed to approach some fixed point(x∗i ,u∗i ) for all i ∈I as k→ ∞, where (x∗i ,u∗i ) is some local minimizer of Ji(·).The remaining task is to prove that a universal social hierarchy is in fact at-tainable after a sufficient number of iterations. The existence of at least one suchsocial hierarchy has already been established in the proof for Theorem 1. Namely,if Assumption 3 concerning neighborhood-cooperative cost functions holds, and ifno two agents i and j such that j ∈N−i|J for all i, j ∈I , j 6= i make decisions si-multaneously, then Inequality (5.22) is guaranteed to be satisfied at every iterationwithin any sampling time. If conflict persists, then agents will, after a sufficientnumber of iterations, self-organize according to a universal social hierarchy withsome probability greater than zero; thus concluding the proof.5.3.4 Determining social hierarchy propertiesOne final issue that must be addressed concerns the selection of Nq. In order toguarantee that a universal social hierarchy is attainable, Nq must be large enoughsuch that a social hierarchy wherein no neighboring pairs of agents operate in par-allel is permissible. This goal invokes the vertex coloring problem from graphtheory [35]. In brief, vertex coloring of a graph requires assigning colors to allnodes such that no two interconnected nodes share the same color. Returning tothe context of the current chapter, each node signifies an agent, each color repre-sents a specific level along a social hierarchy, and interconnection symbolizes costfunction coupling. In graph theory, the chromatic number refers to the minimumnumber of colors required to complete the vertex coloring problem. Therefore, inthe current context, Nq should be equal to or greater than the chromatic number ofthe system graph.163Figure 5.3: Schematic of the plate overlap problem used in the numerical ex-ample.5.4 Numerical example5.4.1 Problem descriptionConsider the mechanical system described in Figure 5.3. This setup consists of Nsquare plates that are supported by spring-damper systems and perfectly aligned atequilibrium. Each square plate has a mass m = 1.0kg, side length L = 0.25m, andis supported by stiffness and damping coefficients k = 1.0N/m and c = 1.0kg/s.Further, the vertical position of each plate i is controlled via an input force ui thatis regulated by agent i. The resulting continuous-time dynamics of each plate areexpressed in state-space form as follows:[x˙iv˙i]=[0 1− km − cm][xivi]+[0uim], (5.45)where xi and vi denote the vertical position and velocity of plate i. For the remain-der of the current section, we work with the discrete-time form of Eq. (5.45).The economic control objective of each agent i is to, without excessive actua-tion, minimize the overlap area between its respective plate and those of neighbor-ing agents i−1 and i+1. The resulting stage cost function of agent i is therefore164Figure 5.4: Interaction graph of the plate overlap problem with vertex color-ing used to identify a universal social hierarchy.expressed as follows:Ji(xi,ui,x−i) =1|N−i| ∑j∈N−iAi, j(xi,x j)+uTi ui, (5.46)where the setN−i contains the indices of all neighboring plates j that are physicallyadjacent to plate i, |N−i| is the cardinality of the set N−i, and Ai, j(·) defines theoverlap area between agents i and j as follows:Ai, j(xi,x j) ={0 ,∣∣xi− x j∣∣≥ L,L(L− ∣∣xi− x j∣∣) , ∣∣xi− x j∣∣< L. (5.47)Note that, although simple in terms of system dynamics, the numerical problemdescribed above entails nonlinear and non-convex cost functions. Physically, non-convexity stems from the property that any two adjacent plates may be relocated inmultiple ways to minimize their respective overlap areas.5.4.2 Social hierarchy-based DEMPC propertiesThe interaction graph for an example problem with ten plates is shown in Fig-ure 5.4. The solution to the vertex coloring problem for this example involvessimply alternating the color of each subsequent node, which yields a chromaticnumber of two regardless of the quantity of vertices. An appropriate choice for thenumber of hierarchy levels is thus Nq = 2.We implement prediction and control horizons of H = 5 time-steps and a sam-pling period of 1.0sec. Within a single sampling period, Np = 5 iterations are per-mitted for the negotiation of optimal stationary points (i.e. step 3 in Algorithm 1),followed by five more iterations for trajectory optimization (i.e. step 4 in Algo-rithm 1). Finally, the actuation force of any agent is bounded as −0.25N ≤ ui ≤1650.25N. These values were selected to limit the steady-state displacement of eachplate to a maximum of L = 0.25m.Finally, all plates are initially at rest with zero displacement from equilibrium(i.e. xi = 0 and ui = 0) and are therefore perfectly aligned with their neighbors.The state and input vector trajectories of all agents are initialized as xi = (0, · · · ,0)and ui = (0, · · · ,0).5.4.3 Simulation resultsSocial hierarchy-based DEMPC – ten platesThe first results we present pertain to five simulations involving ten plates. Theoutcomes in these simulations will differ due to the element of randomness in theproposed coordination algorithm. In Figure 5.5, we show the variation in socialhierarchy levels over the iteration number for all five simulations. Rather thandisplaying the actual hierarchy level of any particular agent, which would yield acluttered image, we instead present the cumulative number of social hierarchy levelchanges. That is, let some cumulative counter start at zero for each simulation.Then, each time an agent alters its social hierarchy level, the cumulative countincreases by one.What is observed in Figure 5.5 is that, in all five simulations, the cumulativecount of social hierarchy variations ultimately reaches a fixed value. This outcomeindicates that, with a sufficient number of iterations, the agents sort themselvesalong a social hierarchy that guarantees conflict-free operation according to Def-inition 2 in all future iterations, thus validating Theorem 1 concerning conflictresolution.Next, we plot the evolution of the global cost function V with respect to theiteration number in Figure 5.6 for all five simulations, with V computed as follows:V = ∑i∈IV˘i. (5.48)Note that the evolution of V differs in each case due to the element of stochasticityinherent to the proposed algorithm. However, in all five simulations, a reductionin V to some locally-optimal value is evident after conflicts have been resolved.1660 50 100 150 200051015202530Figure 5.5: Evolution of the cumulative number of social hierarchy changesfor five simulations of ten plates. Each color corresponds to a differentsimulation.Further, within each time-step (i.e. within each 20 iteration interval), the reductionor preservation of V is apparent after agents have settled on an appropriate socialhierarchy. This latter outcome validates Theorem 2 on convergence.As a reference in Figure 5.6, we plot (using a black dotted line) the globally-optimal evolution of V , which is obtained by initializing the agents’ hierarchy lev-els according to the universal social hierarchy from Figure 5.4. In this case, theagents do not alter their social hierarchy levels as conflict-free operation is guaran-teed from the beginning of the simulation. As a result, V is reduced or preservedwithin each time-step immediately from the start of the simulation. This resultvalidates the existence of a universal social hierarchy.In Figure 5.7, we plot the evolution of stationary target positions over the iter-ation number for five simulations. Rather than displaying the stationary targets ofindividual agents, which would yield a cluttered image, we plot the mean stationary1670 50 100 150 2000.511.522.533.54Simulation 1Simulation 2Simulation 3Simulation 4Simulation 5Universal social hierarchyFigure 5.6: Evolution of the global cost function for five simulations of tenplates.target xs of the entire plant, which is computed as follows:xs =1N ∑i∈Ix˘si . (5.49)It is evident from Figure 5.7 that values of x˘si for all agents i ∈I ultimately con-verge to fixed values in all simulations. This outcome validates Theorem 4 con-cerning bounded closed-loop stability, which requires fixed stationary targets.Finally, we have plotted the locations of all plates at the final sampling time ineach simulation in Figure 5.8. The color of each plate indicates its social hierarchylevel (i.e. blue denotes qi = 1, red denotes qi = 2). The globally-optimal con-figuration would involve all adjacent plates being relocated in opposite directions;however, the social hierarchy-based DEMPC algorithm is only capable of findinga locally-optimal layout wherein some plates (e.g. plate 8 in simulation 1) mustremain at the origin to minimize overlap with their neighbors.1680 50 100 150 200-0.1-0.0500.050.10.15Figure 5.7: Evolution of the mean system-wide stationary target position forfive different simulations involving ten plates.Parallel vs. social hierarchy-based DEMPCWe compare the performance of the proposed social hierarchy-based DEMPC to abasic parallel DEMPC algorithm wherein all agents solve their local EMPC prob-lems simultaneously and exchange stationary vectors and trajectories with theirneighbors. In essence, a parallel DEMPC algorithm is identical to Algorithm 1,except with Nq = 1.We first plot the evolution of V over the iteration number using a parallelDEMPC algorithm in Figure 5.9 for five simulations, each consisting of a differentnumber of plates. The parallel DEMPC algorithm is in fact able to naturally resolveconflicts and ultimately decrease V to some locally minimum value. However, asthe quantity of agents N increases, a greater number of iterations is required to de-crease V to a locally-minimum value. For instance, with N = 10, V reaches a localminimum in 20 iterations. If N = 80 however, over 100 iterations are required.In Figure 5.10, we plot the same information as Figure 5.9, except using theproposed social hierarchy-based DEMPC algorithm. The improvement is clear.The value of N has no discernible effect on the number of iterations necessary1691 2 3 4 5 6 7 8 9 10-0.200.21 2 3 4 5 6 7 8 9 10-0.200.21 2 3 4 5 6 7 8 9 10-0.200.21 2 3 4 5 6 7 8 9 10-0.200.21 2 3 4 5 6 7 8 9 10-0.200.2Figure 5.8: Plate locations at the final sampling time in (from top to bottom)simulations 1 to 5.1700 50 100 150 200102030405060708090Figure 5.9: Evolution of the global cost function V using a parallel DEMPCalgorithm for five simulations involving different quantities of plates.for reducing V to a locally-minimum value. In each simulation, approximately30 iterations are required for V to settle at some minimum value. This outcomeresults from the fact that the likelihood of any agent resolving conflict locally isdependent solely on its neighborhood interaction topology. Thus, raising N shouldnot impact the number of iterations required for system-wide conflict resolution.This outcome is further validation of the scalability of the social hierarchy-basedmethod.5.5 Conclusions and recommendations for futureresearchWe have presented a novel concept for addressing non-convexity in cost functionsof distributed economic model predictive control systems with unknown terminalstationary targets. This concept involves agents self-organizing into a finite hierar-chy using evolutionary principles, and ultimately enables agents to make decisionsthat are mutually beneficial with those of their neighbors. Theorems guarantee-1710 50 100 150 200102030405060Figure 5.10: Evolution of the global cost function V using the proposed so-cial hierarchy-based DEMPC algorithm for five simulations involvingdifferent quantities of plates.ing convergence, recursive feasibility, and bounded closed-loop stability have alsobeen provided for the proposed social hierarchy-based algorithm.These theorems were validated using a numerical example involving a seriesof suspended square plates wherein each agent attempted to minimize the overlaparea between its respective plate and those of its neighbors. Results showed that,across five simulations, the proposed algorithm was capable of establishing a socialhierarchy that reduced the system-wide cost function to some locally-minimumvalue. Another observation from numerical results was that increasing the sizeof the distributed system (i.e. the number of plates and agents) had no discernibleeffect on the number of iterations required to minimize cost function values to localminima. This behavior was not observed when using a parallel DEMPC algorithmwith no mechanism for addressing non-convexity.Several research directions exist for further enhancing the proposed DEMPCalgorithm; developing non-iterative algorithms using compatibility constraints asfirst proposed by Dunbar and Murray [53]; employing Lyapunov constraints to172guarantee asymptotic stability rather than bounded stability; guaranteeing conver-gence, feasibility, and stability under the effects of coupled dynamics and con-straints; ensuring robustness in the presence of bounded disturbances in the systemdynamics and cost functions; application of the proposed algorithm to distributedsystems with weak dynamic coupling such as autonomous vehicle trajectory plan-ning and wind farm control.173Chapter 6Social Hierarchy-basedDistributed Economic ModelPredictive Control of FloatingOffshore Wind FarmsThis chapter implements a recently developed social hierarchy-based distributedeconomic model predictive control (DEMPC) algorithm in floating offshore windfarms for the purpose of power maximization. The controller achieves this objec-tive using the concept of yaw and induction-based turbine repositioning (YITuR),which minimizes the overlap areas between adjacent floating wind turbine rotorsin real-time to minimize the wake effect. Floating wind farm dynamics and perfor-mance are predicted numerically using FOWFSim-Dyn. To ensure fast decision-making by the DEMPC algorithm, feed-forward neural networks are used to esti-mate floating wind turbine dynamics during the process of dynamic optimization.For simulated wind farms with layouts ranging from 1× 2 to 1× 5, an increaseof 20 % in energy production is predicted when using YITuR instead of greedyoperation. Increased variability in wind speed and direction is also studied and isshown to diminish controller performance due to rising errors in neural networkpredictions.1746.1 IntroductionThis chapter outlines the implementation of distributed economic model predictivecontrol (DEMPC) for the purpose of power maximization in floating offshore windfarms. This introductory section provides an overview of the control applicationalong with a description of the DEMPC algorithm.6.1.1 ApplicationWind turbines that are aligned with the free stream wind suffer from a phenomenontermed the wake effect. Viscous interaction along the blades of an upstream ma-chine generates a region downstream that is referred to as a wake in which the meanwind speed is reduced in comparison to that of the surrounding wind field. Anyturbine that is located downstream and aligned with this wake encounters a loweraverage wind velocity relative to its upstream neighbor, and may thus operate withup to a 60 % reduction in power production [141].One approach to mitigating the wake effect is wind farm control, which en-tails the operation of individual turbine actuators in a manner such that powerproduction from the collective is increased. Two conventional wind farm con-trol techniques have been widely investigated in the context of fixed-foundationwind farms; power de-rating, which involves sacrificing power production fromupstream turbines to increase the mean wind speeds to which downstream ma-chines are exposed; and wake steering, which requires operating upstream rotorswith deliberate yaw misalignment to deflect their generated wakes in the cross-wind direction. The reader may refer to our comprehensive review article [102] forfurther information on the wake effect and wind farm control.Floating wind farms render possible a third control strategy for power max-imization. Namely, the mooring systems that anchor floating platforms to theseabed permit limited mobility of the platforms along the ocean surface. By operat-ing a wind turbine with deliberate yaw misalignment, the direction of the net aero-dynamic thrust force acting on its rotor may be altered, hence shifting its positionalong the ocean surface. As shown in Figure 6.1, this mobility may be exploited toreduce the overlap area between the wake generated by an upstream machine andthe rotor of a downstream turbine. This wind farm control strategy is termed yaw175Wind directionThrust forceThrust forceAnchorsMooring line cablesThe wake produced bythe upstream turbineRepositioning the turbines reducesthe wake overlap at the location ofthe downstream machineFigure 6.1: Schematic describing the use of aerodynamic thrust forces to pas-sively relocate floating wind turbines in real-time.and induction-based turbine repositioning (YITuR).Our previous work [103] presented a steady-state model for predicting platformdisplacements and power production in floating offshore wind farms. Additionally,this work performed a stationary optimization study to evaluate the potential ben-efits of YITuR under steady-state conditions. To this end, we reported efficiencygains ranging from 21.8 to 42.7 % for wind farm sizes spanning from 2×2 to 7×7provided that the mooring lines were long enough to permit crosswind relocationof roughly one rotor radius. Given this potential for increasing energy production,the aim of the current work is to transcend a steady-state analysis and to implementYITuR in real-time using optimal control techniques.6.1.2 ControllerAt any given sampling time, model predictive control (MPC) utilizes a mathemat-ical model of a plant to solve a dynamic optimization problem over a finite predic-tion horizon. The output of this process is a sequence of optimal control inputs thatminimize or maximize a prescribed cost function over the prediction horizon whilesatisfying constraints. Solely the first step in this sequence is then implemented, atwhich point, based on updated state and disturbance measurements, the dynamicoptimization problem is recomputed at the next sampling time. More detail on176MPC is available in review articles by Mayne [125] and Mayne et al. [126].The turbine repositioning wind farm control problem addressed in this chapterdoes not simply involve set-point stabilization; the control objective is to minimizeoverlap areas between the rotors of adjacent floating wind turbines. Given thisproperty, economic model predictive control (EMPC) is a prime candidate for au-tomation. EMPC computes optimal control actions by minimizing a generalizedcost function that is a quantifier of plant performance over a finite prediction hori-zon. In comparison, traditional MPC minimizes a positive-definite cost functionthat simply penalizes deviation from a prescribed set-point. More detail on EMPCis available in the review article by Ellis et al. [55].In order to promote scalability (i.e. computation time must not be affectedby the size of the plant), it is necessary to distribute the optimal control problemamong the various operational turbines within the plant; hence the employment ofDEMPC in our approach. DEMPC eliminates the computational burden encoun-tered by centralized EMPC as each agent (i.e. floating wind turbine) computes itsown optimal control input sequence. In order to reach an optimal decision, ev-ery agent must make assumptions as to the state and control input trajectories ofneighbors with which it is coupled. A major component of any DEMPC schemeis therefore the coordination algorithm that is used to update the assumptions thateach agent holds regarding the future operation of its neighbors. More detail onDEMPC is available in the review article by Mu¨ller and Allgo¨wer [134].The turbine repositioning problem possesses two properties that pose chal-lenges in the implementation of DEMPC. The first is that objective functions arenon-convex. There exist multiple platform displacement paths that reduce the over-lap areas between the rotors of adjacent floating turbines. In standard parallelDEMPC algorithms, convergence of agent decisions cannot be guaranteed in thepresence of non-convex cost functions [116]; more elaborate sequential [109, 151],negotiation-based [120, 135, 170], and group-based [12, 117, 144] coordination al-gorithms are therefore required to establish convergence. These algorithms sufferfrom scalability issues however (see Chapter 5). The second property that posesa challenge is that steady-state terminal set-points are not known a priori in thecurrent application. In brief, terminal set-points are necessary for achieving stabil-ity in EMPC as they ensure that the computed optimal control actions always lead177the system to a stable steady-state [55]. Existing DEMPC methods either assumethat these set-points are known a priori [184], are determined using centralizedoptimization [6, 36, 50, 111, 112, 185], or are computed in a distributed mannerassuming convex cost functions [108]. Our previous work (see Chapter 5) over-came these drawbacks and presented a DEMPC algorithm that used the concept ofsocial hierarchies to guarantee convergence and stability in the presence of non-convex cost functions and unknown terminal set-points in a truly distributed andscalable manner (i.e. no centralized functionalities were required).6.1.3 ContributionsThe main contribution of this chapter is the first implementation of a social hierarchy-based DEMPC algorithm for the purpose of power maximization via YITuR infloating offshore wind farms. A secondary contribution includes the first use offeed-forward artificial neural networks to generate a computationally inexpensivecontrol-oriented model of floating wind turbine dynamics. These neural networksserve as the mathematical model used in the dynamic optimization process of theDEMPC algorithm. The final contribution is the first demonstration of the benefitsof YITuR implemented in real-time under time-varying wind conditions.6.1.4 Chapter organizationThe chapter is organized as follows: Section 6.2 describes FOWFSim-Dyn, whichis a simulator used to predict platform motion, wake aerodynamics, and wind farmpower production; Section 6.3 discusses the social hierarchy-based DEMPC objec-tive and algorithm, the use of feed-forward neural networks to approximate turbinedynamics, and the impacts of disturbances and model uncertainty on controllerperformance; Section 6.4 provides simulation results of controller implementationcorresponding to different wind conditions and turbine array sizes; and Section 6.5concludes the chapter with summaries of major findings and recommendations forfuture research.178Figure 6.2: Block diagram showing the computation modules of FOWFSim-Dyn along with information transfer routes.6.2 Simulation tool - FOWFSim-DynThis section briefly describes FOWFSim-Dym, which is a tool used to simulatefloating offshore wind farms in the current work. Full details of this software areavailable in Chapter 4. A system diagram of FOWFSim-Dyn is shown in Fig-ure 6.2. Two high-level modules are tasked with predicting floating platform mo-tion coupled with wake aerodynamics. The following subsections delineate eachof these modules.6.2.1 Floating turbine dynamics moduleThe purpose of the floating turbine dynamics module is to predict the positions,velocities, and power outputs of all floating wind turbines within the wind farm.As shown in Figure 6.3, a floating wind farm is treated as system of particles dis-tributed along the two-dimensional ocean surface. The predominant free streamwind direction is aligned with the global xˆ axis; however, FOWFSim-Dyn is ca-pable of simulating fluctuations in the free stream wind direction relative to the xˆaxis. The wind velocity vector is denoted as V∞.The variables xi and yi describe the positions of turbine i along the xˆ and yˆ axesover time. The velocities of the platform of turbine i in these directions are denotedby vx,i and vy,i. Finally, turbine i possesses two inputs; the rotor’s axial induction179Figure 6.3: Schematic of the general floating offshore wind farm used inFOWFSim-Dyn.factor ai and its yaw angle γi measured positive counter-clockwise relative to the xˆaxis.The motions of the floating platforms are determined by summing the aerody-namic thrust forces, hydrodynamic drag and added mass forces, and mooring linetensions acting on each turbine. Hydrodynamic drag and added mass forces arecalculated using Morison’s equation [152], wherein the drag and added mass con-tributions of all submerged components are summed to provide the total hydrody-namic force. Mooring line tensions are obtained by solving the problem of a staticcatenary that is either partially resting along or fully lifted above the seabed [103].Aerodynamic thrust forces are estimated by treating wind turbine rotors as ac-tuator discs and computing thrust coefficients according to vortex cylinder the-ory [29]. Computation of aerodynamic thrust forces also requires estimates of theeffective wind velocity that is incident upon the rotors of all turbines. The wakeaerodynamics module addresses this requirement.1806.2.2 Aerodynamics moduleThe wake aerodynamics module predicts characteristics of the wakes generated byeach turbine over time. These characteristics include the wake centerline position,wake diameter, and the average wake velocities in the xˆ and yˆ directions. Theevolution of these properties over space and time is captured using one-dimensionalmomentum conservation with an assumed wake expansion rate.The boundary conditions to this problem include the initial wake velocities atthe location of the turbine that generates each wake. The initial wake velocitiesin the downwind and crosswind direction are estimated using momentum balancemethods proposed by Bastankhah et al. [18] and Jime´nez et al. [93].The effective wind velocity that is incident upon the rotor of a downstreamturbine is computed using the principle of equivalent kinetic power deficit [98].Namely, the net kinetic power deficit at any downstream location is assumed toequate to the sum of kinetic power deficits contributed by all upstream wakes.Further, wake velocity profiles are modeled using Gaussian distributions as perexperimental results reported by Bastankhah et al. [18].6.3 Controller designThis section first describes the optimal control objective for maximizing power pro-duction in floating offshore wind farms. Then, the social hierarchy-based DEMPCalgorithm (see Chapter 5) is briefly described. Next, the use of feed-forward neu-ral networks for estimating floating platform motion during dynamic optimizationprocesses is discussed. These neural networks treat aerodynamic coupling betweenturbines as disturbances. The current section therefore also discusses the impact ofdisturbance on the stability and performance of EMPC.6.3.1 Controller objective functionsSimply put, the objective of DEMPC is to indirectly maximize power productionby minimizing the overlap areas between the rotors of adjacent turbines. Powerproduction is not directly maximized since this approach would require the iden-tification of functions that estimate the power output of each turbine using somemachine learning scheme. Identification of such functions is challenging since,181due to the transport phenomenon, the power produced by any turbine is stronglyinfluenced by the history of the states and inputs of its upstream neighbors. In otherwords, the states and inputs that upstream neighbors would have possessed in thepast influence the power produced by a downstream turbine at the present moment.Identification of a power function is therefore deferred to future work who’s pri-mary focus is dynamic system identification of floating offshore wind farms. In thecurrent work, the indirect approach of rotor area overlap minimization is taken.The resulting dynamic optimization problem to be solved by the local EMPCfunction of turbine i takes the following form:minxi,uiH−1∑k=0{∆uTi,kQ∆ui,k +1|Ni| ∑j∈Ni[AOL,i→ j(xi,k,x j,k)+∆uTj,kQ∆u j,k]}, (6.1)subject toxi,0 = xi, (6.2a)xi,k+1 = fi(xi,k,ui,k), (6.2b)ai,k =13, (6.2c)|γi,k| ≤ 10deg, (6.2d)xi,H = xi,s, (6.2e)where H is the length of the prediction horizon, and k is the time-step number alongthe prediction horizon. The vector xi contains the most recent state measurementsof turbine i as follows:xi :=[xi yi vx,i vy,i]T, (6.3)and the vectors xi,k and ui,k contain the candidate states and inputs of turbine i attime-step number k along the prediction horizon as follows:xi,k :=[xi,k yi,k vx,i,k vy,i,k]T, (6.4)ui,k :=[ai,k γi,k]T. (6.5)182The terms xi := (xi,0,xi,1, · · · ,xi,H) and ui := (ui,0,ui,1, · · · ,ui,H−1) denote the can-didate state and input trajectories of turbine i. The term ∆ui,k symbolizes the in-put vector of turbine i at time-step number k measured relative to the reference[13 0deg]T. This reference is used so that only deviation of the axial induction fac-tor relative to its optimal value of 13 is penalized in the cost function. Q is a weight-ing matrix that regulates the significance of this penalty. The set Ni contains theindices of all agents that are physically adjacent to turbine i. Finally, AOL,i→ j(·) isa function that computes the overlap area between the rotors of turbines i and j andthen normalizes this value based on the rotor swept area.Constraint (6.2a) simply states that the initial condition along the predictionhorizon must equate to the latest state measurement. Constraint (6.2b) relates thecandidate state and input vector trajectories via a model of the dynamics of tur-bine i. This dynamic model fi(·) is estimated using feed-forward neural networksin Section 6.3.2. Finally, constraints (6.2c) and (6.2d) limit variations of the axialinduction factor and yaw angle of turbine i. In the case of yaw angles, values arelimited to ±10 deg for operational safety. In the current work, the axial inductionfactors of all turbines are held fixed at the optimal operating point of 13 . In futureworks that attempt to directly maximize power production, the axial induction fac-tor may be permitted to vary since its impact is considered in the power estimationfunction.Constraint (6.2e) is the terminal set-point. It requires that the optimal statetrajectory xi ends at a stationary state vector xi,s, which is a candidate steady-state.The computation of xi,s requires solving the following optimization problem ateach time-step:minxi,xi,s,ui,ui,s∆uTi,sQ∆ui,s+1|Ni| ∑j∈Ni[AOL,i→ j(xi,s,x j,s)+∆uTj,sQ∆u j,s], (6.6)subject to constraints in Eq. (6.2) along withxi,s = fi(xi,s,ui,s), (6.7a)ai,s =13, (6.7b)|γi,s| ≤ 10deg, (6.7c)183where ui,s is the candidate optimal stationary input vector of turbine i, and ∆ui,s de-notes ui,s measured relative to the reference[13 0deg]T. Constraint (6.7a) requiresthat xi,s and ui,s correspond to a stationary point. Similar to Problem (6.1), con-straints (6.7b) and (6.7c) serve as limits on variations of the axial induction factorand yaw angle at steady-state.The difference between Problems (6.1) and (6.6) is that the latter does notminimize the stage cost function throughout the prediction horizon. Instead, itonly minimizes overlap areas and actuator deviation at steady-state. It thereforecomputes an optimal stationary point independent of the trajectory that leads to thispoint. However, since the constraints require that xi,H = xi,s, the computed optimalsteady-state is guaranteed to be reachable given the current state measurement. Thevector xi,s is therefore a candidate reachable steady-state.6.3.2 Feed-forward artificial neural networkIn order to hasten the dynamic optimization process of the DEMPC algorithm,feed-forward neural networks are used to approximate the dynamic model fi(·) forany turbine i. This estimated model is solely a function of the states and inputs ofturbine i and thus treats aerodynamic coupling as disturbance or model uncertainty.Each neural network takes the form shown in Figure 6.4. The six input neuronsrepresent the states xi,k and inputs ui,k of the corresponding turbine at some time-step number k along the prediction horizon. This input layer feeds into a singlehidden layer consisting of 20 neurons. The output layer contains four neuronsrepresenting the turbine states xi,k+1 at the subsequent time-step number k+1.To generate training data for tuning the neural networks, FOWFSim-Dyn sim-ulations consisting of 105 time-steps with sampling periods of 60 sec were com-pleted for each wind farm configuration that was examined. The free stream windvelocity was fixed at 8 m/s in the direction of the positive xˆ axis. The initial con-ditions of each simulation were computed by operating all turbines with an axialinduction factor of 13 and a yaw angle of 0 deg. Then, over the course of the sim-ulations, turbine inputs were randomly varied with uniform probability of 0.1 (i.e.there was a 10 % chance that the input values would change at each new time-step). The logic behind this variation approach was to allow enough time for the184++Figure 6.4: Schematic of the local neural network structure used to identifythe dynamic model of each floating wind turbine.turbine platforms to respond to each change in input values. Axial induction factorswere varied between 0.2 and 0.4, while yaw angles were varied between −20 and+20 deg.Sample validation data is provided for the two-turbine floating wind farm de-scribed in Figure 6.5. The turbine and platform designs are detailed in the worksof Jonkman et al. [96] and Robertson et al. [152], except mooring line lengthshave been increased from 835 to 950 m to increase platform range of motion. Intheir neutral positions, the platforms are spaced seven rotor diameters apart, whichequates to 882 m.For validation, FOWFSim-Dyn and the neural networks were given the samerandomly generated initial condition and turbine input sequences over 60 time-steps (i.e. 3,600 sec). Their predictions of the system response are compared forboth turbines 1 and 2 in Figure 6.6 and Figure 6.7, respectively. These plots cor-respond to a single run, and have been provided for qualitative inspection. It isapparent that neural network predictions for turbine 1 more closely match simula-tion results. This outcome is expected since turbine 1 is the leading machine and is185Figure 6.5: Schematic of a two-turbine floating offshore wind farm simulatedin the current work. Wind turbine and floating platform properties arebased on the National Renewable Energy Laboratory’s (NREL’s) base-line designs presented by Jonkman et al. [96] and Robertson et al. [152].The sole modification is that mooring line lengths have been increasedfrom 835 to 950m to permit greater platform displacement.thus uninfluenced by aerodynamic coupling. The behaviour of turbine 2 is affectedby the states and inputs of turbine 1 however, which induce greater prediction er-rors in the behavior of turbine 2 since its neural network does not consider dynamiccoupling.For a quantitative assessment of neural network performance, ten different runswere completed to acquire the average root-mean-square-error (RMSE) of eachneural network output. This information is listed in Table 6.1. Neural network pre-dictions of xi deviate from FOWFSim-Dyn results with RMSE values of 0.94 and5.79m for turbines 1 and 2, respectively. The greater RMSE associated with tur-bine 2 is attributed to the absence of dynamic coupling in the neural networks.This trend reverses in the case of yi predictions. Since turbine 1 is unaffected byany upstream wakes, it encounters higher wind speeds and thus displaces a greaterdistance in the crosswind direction relative to turbine 2 when operating with yawmisalignment. As a result, the RMSE value of y1 is inflated in comparison to thatof y2. These arguments apply equally well with regards to platform velocity RMSEvalues. Successful control of the current system therefore requires that local EMPCalgorithms are capable of rejecting these prediction errors. A description of the so-1860 20 40 608590950 20 40 60-500501000 20 40 60-0.1-0.0500.050.10 20 40 60-0.200.20.4Figure 6.6: Sample validation data for the neural network of the upstreammachine (i.e. turbine 1) in a two-turbine floating wind farm.0 20 40 609609659709759800 20 40 60-100-500501000 20 40 60-0.1-0.0500.050.10 20 40 60-0.200.20.40.6Figure 6.7: Sample validation data for the neural network of the upstreammachine (i.e. turbine 2) in a two-turbine floating wind farm.187Table 6.1: List of RMSE values between neural network predictions andFOWFSim-Dyn predictions.Turbine xi (m) yi (m) vx,i (m/s) vy,i (m/s)1 0.94 14.68 0.02 0.082 5.79 11.89 0.05 0.06cial hierarchy-based DEMPC scheme now follows.6.3.3 Social hierarchy-based DEMPC algorithmIf multiple optimal solutions exist in an optimal control problem (i.e. the costfunctions are non-convex), and decisions are made in a synchronous and distributedmanner by multiple agents, then there exists a potential for conflicting decisions (seeChapter 5). It is due to this logic that convergence of the decisions made bymultiple agents cannot be guaranteed in parallel distributed optimal control prob-lems [116]. The social hierarchy-based DEMPC algorithm used in this work wastherefore originally developed to address non-convexity in distributed optimal con-trol problems. The essential function of this algorithm is that all agents self-organize in a fully distributed manner into a social hierarchy that dictates the orderin which each agent makes a decision.The algorithm borrows from evolutionary principles. Namely, if an agent’scurrent level along a social hierarchy yields decisions that conflict with those of itsneighbors, then the agent’s current level is detrimental to performance and mustbe randomly varied. The method with which conflicting decisions are identified isnow described. At any iteration, each agent possesses assumed values of the stateand input trajectories of its neighbors. Based on these assumptions, and given someexisting social hierarchy that determines the order in which agents make decisions,the agent updates its own optimal trajectories and calculates the resulting value ofits neighborhood cost function. We refer to this value as the naive cost function.After receiving updated optimal state and input trajectories from its neighbors, theagent recomputes the value its neighborhood cost function. We refer to this updatedvalue as the informed cost function. If the informed cost function is improvedin comparison to the naive cost function, then the agent’s decisions are mutuallybeneficial with those of its neighbors. Otherwise, the decisions are conflicting and188the agent must randomly change its place along the social hierarchy. It was provedin our previous work that, for any interaction topology, at least one social hierarchyexists that enables the agents to avoid conflicting decisions; thus establishing aguarantee of convergence.Another unique property of this algorithm is that it was developed without thepresumption of terminal set-points that are known a priori. Instead, it was intendedfor the agents to determine these terminal targets in a fully distributed manner. Withmild similarity to the approach taken by Kohler et al. [108], the agents computetheir terminal set-points at each time-step by first solving an optimization problemwhich minimizes their cost functions at steady-state. This steady-state set-pointis utilized as a terminal constraint in the subsequent optimization problem thatyields optimal dynamic state and input trajectories. The computation of the steady-state set-points also makes use of the social hierarchy-based approach to eliminateconflicting decisions.6.3.4 Effects of dynamic couplingIn the current work, feed-forward neural networks are used to locally estimate thedynamic model of each floating wind turbine. These estimated models are bettersuited for dynamic optimization due to their low computation time requirements.Since these neural networks are tuned locally, they are solely functions of the in-puts and states of their corresponding turbines. They therefore do not consider theinputs and states of turbines with which aerodynamic coupling exists. Dynamiccoupling may thus be interpreted as a source of disturbance or model uncertaintyin our work.EMPC is inherently capable of rejecting disturbance or model uncertainty thatis appropriately bounded in magnitude given the dynamics of the plant. The impactof disturbances in EMPC is first demonstrated in the context of terminal equalityconstraint satisfaction in Figure 6.8. Let the set Xr contain all initial state vec-tors x0 from which a terminal set-point xs is reachable in H time-steps while sat-isfying process constraints. As long as x0 is contained within the set Xr, thenan input trajectory u = (u0,u1, · · · ,uH−1) exists that generates a state trajectoryx = (x0,x1, · · · ,xH) which ends at xH = xs. Disturbance or model uncertainty189Figure 6.8: Schematic showing the impacts of disturbance/model uncertaintyon terminal equality constraint satisfaction in EMPCcauses a deflection of the state trajectory from x to x′ = (x0,x′1, · · · ,x′H) as the sys-tem moves forward in time. Provided that the magnitude of disturbance or modeluncertainty is appropriately bounded, then x′H will not leave the setXr. As a result,even if the system follows the deflected path from x0 to x′H , then a path from x′Hto xs is guaranteed to exist. If, on the other hand, the magnitude of disturbance islarge enough to cause x′H to leave Xr, the satisfaction of the terminal constraintcannot be guaranteed and system redesign is recommended.Next, the impact of disturbance on the satisfaction of trajectory constraints isdiscussed. Let the set X contain all acceptable values of the state vector xk forall k ∈ I0:H . In other words, the state trajectory x must always be contained withinX . As before, disturbance or model uncertainty deflects x to x′ as the systemmoves forward in time. Provided that the magnitude of disturbance is appropriatelybounded, then x′ will not leave the set X and state constraints will be satisfied.This expectation is conservative however. If the magnitude of disturbance is notappropriately bounded, then the technique of constraint tightening [37] may beused to ensure constraint satisfaction.The mechanism of constraint tightening is demonstrated in Figure 6.9. Onceagain, it is required that x remains within the setX . To ensure this outcome, the setX may be duplicated and tightened to form the setX ′. The dynamic optimizationproblem must then be required to maintain x withinX ′. IfX ′ is sized accordingly,then, even with the worst-case disturbance scenario, the deflected state trajectoryx′ is guaranteed to remain within the original process constraint setX .190Figure 6.9: Schematic demonstrating the method of constraint tightening forprocess constraint satisfaction in EMPC.6.4 Results and discussionThe current section presents simulation results corresponding to the implemen-tation of the DEMPC scheme outlined in Section 6.3 on floating wind farms ofdifferent sizes. Each simulated wind farm possesses a row configuration similarto that shown in Figure 6.5. Inter-turbine neutral position spacings are seven rotordiameters and mooring line cable lengths are 950 m, while the sizes of the windfarms range from 1× 2 to 1× 5. First, the impact of wind speed and directionvariation on controller performance is assessed for a 1× 2 wind farm. Then, theperformance and behaviour of larger array sizes are investigated.6.4.1 Wind velocity variationThe neural networks described in Section 6.3.2 were tuned using data correspond-ing to 8 m/s free stream wind aligned with the positive xˆ axis. As the free streamwind speed and direction deviate from this condition, the error in neural networkoutputs will increase and cause detriment to controller performance. We thereforeassess in this subsection the impacts of free stream wind velocity fluctuations onplatform motion and optimal power production.To quantify velocity fluctuations, we define the parameter 0≤σ∞≤ 1 and use itto randomly perturb, with uniform probability, the ten-minute averaged free streamwind velocity vector. Therefore, at some ten-minute mark t, the free stream wind1910 500 1000 1500 2000 2500 3000 35007.67.888.28.40 500 1000 1500 2000 2500 3000 3500-0.4-0.200.20.4Figure 6.10: Sample randomly generated free stream wind speed evolutionsin the xˆ and yˆ directions corresponding to a perturbation parameter ofσ∞ = 5 %.velocity vector V∞(t) is computed as follows:V∞(t) = V∞+rand±σ∞‖V∞‖rand±σ∞‖V∞‖ , (6.8)where V∞ is the baseline free stream velocity vector of 8 m/s aligned with the posi-tive xˆ axis, and rand±σ∞‖V∞‖ is a function that generates a random number between−σ∞∥∥V∞∥∥ and +σ∞∥∥V∞∥∥. Spline interpolation is then used to generate smoothercurves with 0.1 sec resolution using these randomly generated ten-minute averages.Sample free stream wind velocity curves in the xˆ and yˆ directions corresponding toσ∞ = 5 % are plotted in Figure 6.10.6.4.2 Social hierarchy-based DEMPC propertiesThe relevant properties of the social hierarchy-based DEMPC algorithm are listedin Table 6.2. Only two social hierarchy levels are used since, in order to pre-vent conflicting decisions, each turbine must be given the opportunity to computeits optimal trajectory out of synchrony with its adjacent neighbors. Within each192Table 6.2: List of properties relevant to the social hierarchy-based DEMPCalgorithm.Property ValueNumber of social hierarchy levels 2Number of iterations per sampling period 3Number of prediction horizon time-steps, H 5Input weight matrix, Q[1 00 1]sampling period, each turbine is granted three iterations to compute its optimalstationary point by solving Problem (6.6), as well as three additional iterations tocompute its optimal state and input trajectories by solving Problem (6.1).The length of the prediction horizon is set to H = 5. Since the sampling periodof each controller is 60 sec, this value of H permits prediction of 5 min into future.Finally, the input weight matrix Q used in Problems (6.6) and (6.1) is set the iden-tity matrix. Since the overlap areas between turbine rotors are normalized in thecost functions of Problems (6.1) and (6.6), input weights of unity render the opti-mization problem approximately circular. An identity matrix was therefore used astarting point for Q and the corresponding results were deemed acceptable.Regardless of the wind farm size, the time required by the DEMPC algorithm tocompute optimal state and input trajectories per turbine ranged from 10 to 50 sec.This range of durations is smaller than the sampling period of 60 sec; the socialhierarchy-based DEMPC algorithm may therefore be implemented in real-time. Itis important to note that, if a sequential DEMPC algorithm [109, 151] had beenused, then the total computation time would be a summation of the durations re-quired by all turbines. This property exists since a sequential algorithm requiresthat individual turbines take turns to make decisions. In other words, even fora two-turbine wind farm, the total required computation time would range from20 to 100 sec, which exceeds the 60 sec sampling period.6.4.3 Effects of wind velocity variationsThe impact of time-varying free stream wind velocity on controller performance isassessed for a 1× 2 wind farm. Evolutions of turbine yˆ positions for different σ∞values are shown in Figure 6.11. As desired, the two floating wind turbines relocate1930 20 40 60-100-500501000 20 40 60-100-500501000 20 40 60-100-500501000 20 40 60-100-50050100Figure 6.11: Evolution of yˆ positions in a two-turbine floating wind farm thatis optimally controlled by the DEMPC from Section 6.3. The fourplots correspond to different wind velocity perturbations of a) σ∞ =5 %, b) σ∞ = 10 %, c) σ∞ = 15 %, and d) σ∞ = 20 %.in opposite directions in all cases to minimize their respective rotor overlap areas.Specifically, the turbines shift by approximately one rotor radius (i.e. 60 m), whichresults in a total clearance of one rotor diameter between their nacelle centerlines.The impact of neural network prediction error is also evident in two aspects.First, regardless of the wind velocity perturbation parameter σ∞, the downstreamturbine’s yˆ position shows greater variability over time. The reason is that thedynamics of the downstream turbine (i.e. turbine 2) are influenced by the states andinputs of the upstream machine. Since this coupling was not explicitly consideredin the machine learning process, the neural network of turbine 2 is subject to greaterprediction error. This error translates to deflections of optimal state trajectories asdiscussed in Section 6.3.4.Second, as the wind velocity perturbation σ∞ is increased from 5 to 20 %, theyˆ positions of both turbines show greater variability over time. This outcome isalso explained by neural network prediction error. The neural networks have beentuned using data corresponding to a free stream wind velocity of 8 m/s aligned with194the xˆ axis. As the wind velocity deviates from this reference, the neural networkuncertainties are exacerbated, once again leading to deflections in state trajectoriesrelative to computes optimal values.Evolutions of wind farm power production for different σ∞ values are shownin Figure 6.12. This figure displays power production trends corresponding toboth DEMPC and greedy1 operation subject to identical free stream wind condi-tions. The greedy performance curves demonstrate the amount of power producedwithout wind farm control. Any rise in power production in the greedy case isconsequently purely a result of free stream wind speed fluctuations. The benefitincurred by implementing YITuR is therefore quantified by the difference betweenthe DEMPC and greedy power trajectories. With 5 % variability in the wind veloc-ity, YITuR yields an 18.4 % gain in energy production relative to greedy operationthroughout the hour-long simulation.An additional observation from Figure 6.12 is that the increased variability ofthe free stream wind velocity diminishes wind farm controller performance. Asσ∞ is raised from 5 to 20 %, the relative gain in wind farm energy productionobtained by switching from greedy operation to optimal control decreases from18.4 to 7.3 %. This outcome is partly caused by the aforementioned rise in neuralnetwork prediction error that results from wind speed and direction deviations fromthe values used for neural network tuning. As seen in Figure 6.11, this error causesturbine yˆ displacements to decrease for brief periods, which increases rotor overlapareas. Another factor is that, as the free stream wind direction varies and misalignsfrom the row of turbines, wake overlap naturally subsides. As a result, wind farmcontrol fails to yield much benefit relative to greedy operation.Platform displacements in the xˆ direction are not considered in the DEMPCoptimization problems; their progressions are therefore entirely dictated by thefree stream wind velocity and aerodynamic coupling. Insight may nonethelessbe gained by examining their trends. Evolutions of turbine xˆ displacements (i.e.relative to neutral positions) are shown in Figure 6.13. The first observation is thatturbine 2 consistently displaces a smaller distance in the downwind direction in1Greedy operation implies the absence of wind farm control. That is to say, each wind turbineis operated with an axial induction factor and yaw angle that maximize its own power output; thesegreedy values are ai = 13 and γi = 0 deg.1950 20 40 6033.544.550 20 40 60234560 20 40 60234560 20 40 602468Figure 6.12: Evolution of power production in a two-turbine floating windfarm that is optimally controlled by the DEMPC from Section 6.3. Forcomparison, power production under greedy operation is also demon-strated. The four plots correspond to different wind velocity pertur-bations of a) σ∞ = 5 %, b) σ∞ = 10 %, c) σ∞ = 15 %, and d) σ∞ =20 %.comparison to turbine 1 for all wind velocity scenarios. The reason is simply that,as the downstream machine, turbine 2 encounters slower wind speeds on averagedue to the presence of the wake produced by its upstream neighbor.A second trend from Figure 6.13 shows that the downwind displacement ofturbine 2 is more volatile over time. This behaviour is once again the result of aero-dynamic coupling. As turbine 1 adjusts its yaw angle and relocates its platform,the resultant effects on the wind field are transported downwind to the location ofturbine 2. The downstream machine therefore encounters greater variability in theeffective wind velocity that is incident upon its rotor. Furthermore, larger variationsin the free stream wind velocity cause greater volatility in the downwind displace-ment of both turbines. In the case of σ∞ = 5 %, the xˆ displacement of turbine 2ranges from 85 to 100 m. This range expands to 75 to 105 m at σ∞ = 20 %. Thisoutcome indicates that some additional control function may be required to limitexcessive platform displacements in the downwind direction. This function may1960 20 40 607080901001100 20 40 607080901001100 20 40 607080901001100 20 40 60708090100110Figure 6.13: Evolution of xˆ positions in a two-turbine floating wind farm thatis optimally controlled by the DEMPC from Section 6.3. The fourplots correspond to different wind velocity perturbations of a) σ∞ =5 %, b) σ∞ = 10 %, c) σ∞ = 15 %, and d) σ∞ = 20 %.take the form of constraints in the DEMPC optimization process or, alternatively,downwind position regulation may be achieved via wind turbine-level control sys-tems.6.4.4 Performance of different wind farm sizesIn this subsection, the impact of increasing the wind farm array size is assessedwhile maintaining σ∞ = 5 %. Evolutions of wind farm power production for windfarm sizes ranging from 1× 2 to 1× 5 are shown in Figure 6.12 for both optimaloperation using DEMPC and greedy operation. In all cases, the energy producedby the wind farm over the simulated hour increases by approximately 20 % whenoptimal control is used in place of greedy operation. This result contradicts thefindings of our stationary optimization study [103], wherein the relative gain inwind farm efficiency increased from 21.8 to 42.7 % as the wind farm size wasvaried from 2×2 to 7×7.One possible explanation for this outcome is that, as the wind farm size in-1970 20 40 6033.544.550 20 40 6044.555.560 20 40 60456780 20 40 6056789Figure 6.14: Evolution of power production in floating wind farms of dif-ferent sizes that are optimally controlled by the DEMPC from Sec-tion 6.3. For comparison, power production under greedy operationis also demonstrated. The wind velocity perturbation is fixed at σ∞ =5 %. The four plots correspond to different wind farm configurationsof a) 1×2, b) 1×3, c) 1×4, and d) 1×5.creases, the trailing machines are subject to dynamic coupling from a larger numberof upstream counterparts. As a result, their neural networks are subject to greaterprediction error, which renders their optimal controllers less effective. This ex-planation is validated by plotting evolutions of turbine yˆ positions for the differentwind farm sizes in Figure 6.15. It is evident that, although σ∞ remains unchanged,the yˆ position of each subsequent downstream turbine displays greater variabil-ity in comparison to those of its upstream neighbors. The yˆ position trajectory ofturbine 5 in Figure 6.15d defies this trend; however, the yˆ position trajectory ofturbine 3 is consistently more volatile than those of turbines 1 and 2 in all relevantplots, and the yˆ position trajectory of turbine 4 is consistently more volatile thatthat of turbine 3 in all relevant plots. This increased volatility results in longermoments of increased overlap between turbine rotors.A final observation from Figure 6.15 that is worth point out is the fact that thesocial hierarchy-based DEMPC algorithm successfully prevents conflicting deci-1980 20 40 60-100-500501000 20 40 60-100-500501000 20 40 60-100-500501000 20 40 60-100-50050100Figure 6.15: Evolution of yˆ positions in floating wind farms of different sizesthat are optimally controlled by the DEMPC from Section 6.3. Thewind velocity perturbation is fixed at σ∞ = 5 %. The four plots cor-respond to different wind farm configurations of a) 1× 2, b) 1× 3, c)1×4, and d) 1×5.sions in all simulated cases. For each wind farm size, neighboring floating windturbines are relocated in opposite directions, which is the most effective modifiedlayout for minimizing adjacent overlap areas.Evolutions of turbine xˆ positions for the different wind farm sizes are plotted inFigure 6.16. As observed from xˆ displacements in the case of the two-turbine windfarm from Figure 6.13, downstream machines shift a smaller distance on averagein the downwind direction due to reduced incident wind speeds while also experi-encing greater volatility in xˆ displacements over time due to aerodynamic coupling.However, it is observed in Figure 6.16 that, with each subsequent downstream ma-chine, additional downwind displacements gradually diminish. For instance, overthe course of the 3,600 sec simulation from Figure 6.16d, the average downwinddisplacement of turbine 1 is the largest at 96.1 m. This value drops to 90.9 and86.0 m for turbines 2 and 3, and then to 82.8 m for both turbines 4 and 5. Clearly,the wind velocity deficit resulting from the compounding of overlapping wakes1990 20 40 607080901001100 20 40 607080901001100 20 40 607080901001100 20 40 60708090100110Figure 6.16: Evolution of xˆ positions in floating wind farms of different sizesthat are optimally controlled by the DEMPC from Section 6.3. Thewind velocity perturbation is fixed at σ∞ = 5 %. The four plots cor-respond to different wind farm configurations of a) 1× 2, b) 1× 3, c)1×4, and d) 1×5.ultimately converges to a finite value. This result is significant in the context ofturbine repositioning. It indicates that, regardless of the size of the wind farm, tur-bines that are located far downstream may still encounter large enough wind speedsto generate sufficient platform displacements for rotor overlap reduction.6.5 ConclusionThis chapter presented the first real-time implementation of wind farm control forpower maximization in floating offshore wind farms. The specific mechanism forpower maximization involved relocating wind turbines using aerodynamic forcesto minimize overlap areas between the rotors of adjacent turbines.Floating wind farms were simulated using FOWFSim-Dyn (see Chapter 4),which is a simulation tool designed specifically for capturing floating platform mo-tion coupled with wind farm aerodynamics. The wind farm controller was basedon a social hierarchy-based distributed model predictive control (DEMPC) algo-200rithm (see Chapter 5) developed to address convergence challenges in non-convexdistributed optimal control problems. Finally, feed-forward neural networks wereused to model floating turbine dynamics during the dynamic optimization processof the DEMPC algorithm.Conclusions and recommendations that have been derived from the results ofthis work are now discussed. Neural networks were effectively used to predictfloating wind turbine behaviour for wind farm rows with layouts of 1×3 or smallerwhile neglecting dynamic coupling. Future work should establish error bounds onneural network predictions while taking into consideration the effects of dynamiccoupling for improved performance in larger wind farms.Deviations in the free stream wind velocity from the baseline value at whichneural networks were tuned diminished controller performance. Future work mayconsider the free stream wind speed and direction as inputs to the neural networksor, alternatively, unique neural networks may be identified for multiple wind speedand direction ranges.In larger wind farms, downstream machines demonstrated high variability inplatform positions over time. This outcome resulted from increased neural networkprediction error due to dynamic coupling. Once the bounds on these uncertaintieshave been established, future work may implement constraint tightening methodsto limit the variability of floating platform positions.The current controller maximized wind farm power production indirectly byminimizing the overlap areas between the rotors of adjacent floating turbines. Fu-ture work should use some machine learning scheme to identify power output func-tions for each turbine to serve as cost functions in DEMPC, thus permitting directmaximization of wind farm power production.In the current work, a single row of wind turbines aligned with the predominantfree stream wind direction was examined. Identifying which pairs of turbines en-counter strong aerodynamic coupling was therefore a trivial task. In more complexwind farm configurations, such as gridded layouts, the pairings of turbines that in-teract aerodynamically is determined by the wind direction at any given instant intime. Future work may therefore extend the cost functions used by the DEMPCalgorithm to capture the effects of wind direction on turbine pairings.The current chapter was concerned solely with power maximization as the wind201farm control objective. Future work may assess alternative objectives. For instance,at sufficiently large wind speeds, the primary concern of a wind farm operator shiftsfrom power maximization to load reduction [107]. This shift may be accounted forvia adjustments to the cost functions used in optimal control algorithms. Specifi-cally, cost functions may be defined such that the turbines track power set-pointswhile minimizing blade and tower vibrations among their neighborhoods. The rolethat YITuR may play in such a control problem should be investigated.The optimal state and input trajectories computed using DEMPC in the currentwork should not be implemented directly. Instead, they should serve as set-pointsfor turbine-level controllers that consider additional dynamic phenomena such asplatform oscillation. Future work should therefore integrate wind farm-level con-trol with lower controllers capable of rejecting disturbances caused by waves andwind gusts with the aim of minimizing turbine loads and platform oscillation.Finally, all state information was assumed to be available in the current work.Future work should examine sensing and observer techniques for estimating float-ing platform positions and velocities.202Chapter 7ConclusionThis thesis assessed the potential of real-time turbine repositioning for increasingpower production in floating offshore wind farms. A thorough review of existingwind farm control literature was first presented in order to establish an understand-ing of the status of research as well as major gaps. This step was followed by thedevelopment of FOWFSim, which is a steady-state model for predicting platformdisplacement and power production in floating offshore wind farms. This modelwas used to conduct steady-state optimization studies to determine the potentialof the wind farm control method of yaw and induction-based turbine reposition-ing (YITuR) under idealized static wind conditions. Next, a parametric dynamicmodel of floating offshore wind farms named FOWFSim-Dyn was developed topermit real-time control system evaluations.The specific properties of the turbine repositioning control problems renderedexisting optimal control algorithms inadequate. As a result, the development ofa novel distributed economic model predictive control (DEMPC) algorithm thatovercame the shortcomings of existing theory was the next necessary step. Finally,the real-time performance this DEMPC algorithm was evaluated via implementa-tion using FOWFSim-Dyn. The following sections summarize the major findingsfrom these research steps and provide recommendations for future work. The fi-nal subsection directly answers the research question posed in the introduction ofthis thesis by addressing the potential and challenges associated with the practicalimplementation of YITuR wind farm control.2037.1 Steady-state assessment of YITuRThe primary conclusion from the steady-state assessment of YITuR was that thiswind farm control strategy has the potential to raise wind farm efficiency by morethan 20 %. However, its performance and viability were determined to be stronglydependent upon wind conditions and mooring system design parameters. Futurework in this area should therefore involve the development of optimization soft-ware that not only computes ideal turbine operating parameters, but also mooringsystem properties such as cable length and anchor locations in order to maximizeannual energy production. Further, such software should consider site-specificwind velocity data.7.2 Dynamic modeling of floating offshore wind farmsThe primary purpose for developing FOWFSim-Dyn was the absence of high-fidelity simulation tools with similar capabilities and the lack of access to scaledfloating wind farm experiments. Consequently, only steady-state predictions ofwake propagation were validated using published experimental results from windtunnel tests. Validation of dynamic wake transport and platform motion have notbeen performed. Future work in this area should therefore involve the developmentof high-fidelity floating wind farm simulators that utilize large eddy simulations(LES) to capture wake behaviour and couple this solver with a three-dimensionalfloating platform dynamics module. Such a simulation tool, along with scaled ex-perimental techniques may be used for further validation of FOWFSim-Dyn.7.3 Real-time YITuR controller implementationIt was demonstrated during the implementation of a YITuR controller that feed-forward neural networks may serve as appropriate surrogate models for predict-ing floating platform motion during the dynamic optimization process of DEMPC.However, the error in predictions was strongly influenced by aerodynamic cou-pling between turbines and time-varying wind conditions. As a result, future workon floating wind farm data-driven modeling should take into account several rec-ommendations. First, coupling effects between turbines should be considered in204the machine learning process. With this capability, each machine may better pre-dict its future trajectory once informed of the optimal decisions and actions of itsupstream neighbors. Alternatively, coupling may continue to be treated as a modeluncertainty; however, more extensive studies should be carried out to determinethe bounds on these uncertainties. Finally, the current condition of the free streamwind should factor into surrogate model outputs. This outcome may be achievedby directly including the free stream wind speed and direction as model inputs, orby tuning various neural networks for different ranges of wind velocity.The prediction error due to aerodynamic coupling resulted in significant plat-form motion and deterioration of performance among downstream machines. It istherefore recommended that the optimal control problem be augmented with con-straints that limit platform motion over time. Furthermore, satisfaction of theseconstraints should be guaranteed using constraint tightening approaches once errorbounds on neural network predictions have been established.The current implementation of YITuR control maximized wind farm powerproduction indirectly by minimizing rotor overlap areas between adjacent floatingwind turbines. Although this approach yielded gains in energy production exceed-ing 20 % in scenarios of low wind speed and direction variability, a more directpower maximization approach should be pursued. Namely, future research shouldfocus on the development of surrogate models that predict the power output of afloating wind turbine based on the inputs and states of the turbine itself, along withthose of its upstream neighbors. Once such a function is established, the distributedeconomic model predictive control theory developed in this work may be imple-mented without modification for the purpose of wind farm power maximization.The objective function used in Chapter 6 may be extended in two ways. First,the pairs of turbines that interact aerodynamically may be defined as a functionof the wind direction. In this manner, as the wind direction changes, any giventurbine may update the set of neighboring machines with which it should minimizerotor overlap. The current work only considered a single row of wind turbinesthat are aligned with the free stream wind. The modification to objective functionsmentioned above will generalize YITuR to any wind farm configuration. Second,the cost function used in DEMPC may be extended to accommodate the alternativewind farm control objective of power set-point tracking and load reduction. This205mode of wind farm control is relevant in cases where the wind speed is excessivelyhigh and power maximization is no longer of primary concern.The YITuR controller developed in this work is high-level and thus solely con-siders floating turbine states and inputs that are consequential at the scale of thewind farm. Motions such as platform oscillation and blade vibration are not fac-tored into the optimal control problem since their effects on wind farm power pro-duction are deemed to be insignificant. Nonetheless, platform oscillation and bladevibration are consequential in the context of individual turbine loads and fatigue.As a result, future work should develop turbine-level controllers that track the opti-mal trajectories computed by the higher-level YITuR controller while minimizingplatform oscillation and blade vibration.Finally, it was assumed throughout this thesis that floating wind turbine state in-formation, such as platform positions and velocities, is directly measurable. Physi-cal implementation of a YITuR controller will require appropriate sensing and stateestimation techniques that reliably deliver this information. Future work shouldtherefore assess the potential of different measurement and estimation technolo-gies using scaled floating wind farm experiments.7.4 Final comments on YITuRAlthough the analysis performed in this work indicates that YITuR has the po-tential to raise the efficiency of floating offshore wind farms by 20%, wind farmdevelopers may be hesitant to implement such a control technique in response toseveral challenges. First, due to the large range of motion of the floating platforms,gaining access to wind turbines will be a potentially unsafe and involved task formaintenance crews. Second, the large range of platform motion will render theocean surface area spanned by the wind farm unsafe for naval vessel travel. Third,lengthening mooring lines to expand movable ranges induces additional capital ex-penses and alters the oscillatory responses of the floating platforms in the presenceof waves, wind velocity fluctuations, and aerodynamic coupling effects betweenturbines. Finally, raising wind farm power production by reducing wake overlapmay exacerbate fatigue damage in turbines with partially overlapped rotors.Further investigation of these challenges will illuminate the practicality and206feasibility of YITuR operation. If the complications associated with large platformdisplacements are deemed to be a valid trade-off for the added revenue stemmingfrom increased efficiency, then a case could potentially be presented for YITuRwind farm control. Regardless of this outcome however, completion of the currentproject has been a worthwhile research endeavor. It yielded the first steady-stateand dynamic models of floating offshore wind farms, which will promote furtherdevelopment of floating offshore wind technology via design, optimization, andcontroller assessment studies. Further, tackling the repositioning problem necessi-tated the development of a novel distributed control system that will have implica-tions in a wide range of multi-agent systems such as autonomous vehicle platoons,traffic networks, and multi-robot systems.207Bibliography[1] M. Abbes and M. Allagui. Centralized control strategy for energymaximization of large array wind turbines. Sustainable Cities and Society,25:82–89, 2016. ISSN 22106707. doi:10.1016/j.scs.2015.11.007. →pages 37, 237[2] M. S. Adaramola and P. A˚. Krogstad. Experimental investigation of wakeeffects on wind turbine performance. Renewable Energy, 36(8):2078–2086, 2011. ISSN 09601481. doi:10.1016/j.renene.2011.01.024. →pages 36, 46, 69, 236, 242[3] M. A. Ahmad, S. Azuma, and T. Sugie. A model-free approach formaximizing power production of wind farm using multi-resolutionSimultaneous Perturbation Stochastic Approximation. Energies, 7:5624–5646, 2014. ISSN 19961073. doi:10.3390/en7095624. → pages39, 59, 64, 69, 105, 238[4] J. Ainslie. Calculating the flowfield in the wake of wind turbines. Journalof Wind Engineering and Industrial Aerodynamics, 24:213–224, 1988. →pages 31, 231[5] W. Al-Gherwi, H. Budman, and A. Elkamel. Robust distributed modelpredictive control: A review and recent developments. Canadian Journalof Chemical Engineering, 89(5):1176–1190, 2011. ISSN 00084034.doi:10.1002/cjce.20555. → page 139[6] F. Albalawi, H. Durand, and P. D. Christofides. Distributed economicmodel predictive control with Safeness-Index based constraints fornonlinear systems. Systems and Control Letters, 110:21–28, 2017. ISSN01676911. doi:10.1016/j.sysconle.2017.10.002. → pages 140, 178[7] D. Allaerts and J. Meyers. Large eddy simulation of a large wind-turbinearray in a conventionally neutral atmospheric boundary layer. Physics ofFluids, 27(6), 2015. ISSN 10897666. doi:10.1063/1.4922339. → page 30208[8] O. Anaya-Lara, J. O. Tande, K. Uhlen, and K. Merz. Offshore Wind EnergyTechnology. John Wiley & Sons Ltd, 2018. → page 10[9] J. Annoni, P. M. O. Gebraad, A. K. Scholbrock, P. A. Fleming, and J.-W.van Wingerden. Analysis of axial-induction-based wind plant control usingan engineering and a high-order wind plant model. Wind Energy, 19(6):1135–1150, 2016. doi:10.1002/we.1891. → pages33, 34, 35, 36, 66, 69, 235[10] J. Annoni, C. Bay, K. Johnson, E. Dall’Anese, E. Quon, T. Kemper, andP. Fleming. A Framework for Autonomous Wind Farms: Wind DirectionConsensus. Accepted for review in Wind Energy Science Discussions, n/a,2018. doi:10.5194/wes-2018-60. → pages 20, 22, 65[11] J. Annoni, C. Bay, T. Taylor, L. Pao, P. Fleming, and K. Johnson. Efficientoptimization of large wind farms for real-time control. In Proceedings ofthe 2018 American Control Conference, pages 6200–6205, 2018.doi:10.23919/ACC.2018.8430751. → page 241[12] E. F. Asadi and A. Richards. Scalable distributed model predictive controlfor constrained systems. Automatica, 93:407–414, 2018. ISSN 00051098.doi:10.1016/j.automatica.2018.03.050. → pages 141, 177[13] D. Aßmann, U. Laumanns, and D. Uh, editors. Renewable Energy: AGobal Review of Technologies, Policies and Markets. Earthscan, London,UK, 2006. → page 8[14] J. J. Barradas-Berglind and R. Wisniewski. Wind farm axial-inductionfactor optimization for power maximization and load alleviation. InProceedings of the 2016 European Control Conference, pages 891–896,2016. doi:10.1109/ECC.2016.7810402. → page 233[15] J. Barreiro-Gomez, C. Ocampo-Martinez, F. Bianchi, and N. Quijano.Model-free control for wind farms using a gradient estimation-basedalgorithm. In Proceedings of the 2015 European Control Conference,pages 1516–1521, 2015. doi:10.1109/ECC.2015.7330753. → page 237[16] J. Bartl and L. Sætran. Experimental testing of axial induction basedcontrol strategies for wake control and wind farm optimization. Journal ofPhysics: Conference Series, 753, 2016.doi:10.1088/1742-6596/753/3/032035. → pages 36, 236209[17] M. Bastankhah and F. Porte´-Agel. A new analytical model for wind-turbinewakes. Renewable Energy, 70:116–123, 2014.doi:10.1016/j.renene.2014.01.002. → pages 27, 75[18] M. Bastankhah and F. Porte´-Agel. Experimental and theoretical study ofwind turbine wakes in yawed conditions. Journal of Fluid Mechanics, 806:506–541, 2016. ISSN 14697645. doi:10.1017/jfm.2016.595. → pagesxx, 119, 120, 124, 129, 130, 181[19] M. Bastankhah and F. Porte´-Agel. Wind farm power optimization via yawangle control: A wind tunnel study. Journal of Renewable and SustainableEnergy, 11(2), 2019. ISSN 19417012. doi:10.1063/1.5077038. → pages47, 243[20] A. Behnood, H. Gharavi, B. Vahidi, and G. H. Riahy. Optimal outputpower of not properly designed wind farms, considering wake effects.International Journal of Electrical Power and Energy Systems, 63:44–50,2014. ISSN 01420615. doi:10.1016/j.ijepes.2014.05.052. → page 233[21] E. Bitar and P. Seiler. Coordinated control of a wind turbine array for powermaximization. In Proceedings of the 2013 American Control Conference,pages 2898–2904, 2013. doi:10.1109/ACC.2013.6580274. → page 233[22] G. Bo, L. Yongqian, Y. Jie, L. Li, and K. Shun. A wind farm optimalcontrol algorithm based on wake fast-calculation model. Journal of SolarEnergy Engineering, 138(2):024501, 2016. ISSN 0199-6231.doi:10.1115/1.4032297. → page 234[23] S. Boersma, P. M. Gebraad, M. Vali, B. M. Doekemeijer, and J. W. VanWingerden. A control-oriented dynamic wind farm flow model: WFSim.Journal of Physics: Conference Series, 753(3), 2016. ISSN 17426596.doi:10.1088/1742-6596/753/3/032005. → pages 30, 31, 231[24] S. Boersma, B. Doekemeijer, P. Gebraad, P. Fleming, J. Annoni,A. Scholbrock, J. Frederik, and J.-W. van Wingerden. A tutorial oncontrol-oriented modeling and control of wind farms. Proceedings of theAmerican Control Conference, pages 1–18, 2017.doi:10.23919/ACC.2017.7962923. → pages10, 12, 13, 17, 21, 24, 25, 29, 69, 105[25] S. Boersma, B. Doekemeijer, M. Vali, J. Meyers, and J.-W. van Wingerden.A control-oriented dynamic wind farm model: WFSim. Wind Energy210Science, 3(1):75–95, 2018. doi:10.5194/wes-3-75-2018. → pages30, 31, 106, 231[26] K. Boorsma. Heat and Flux - Analysis of Field Measurements. Technicalreport, Energy Research Center of the Netherlands, 2012. → pages36, 43, 69, 236[27] E. Bossanyi. Combining induction control and wake steering for wind farmenergy and fatigue loads optimisation. Journal of Physics: ConferenceSeries, 1037(3), 2018. ISSN 17426596.doi:10.1088/1742-6596/1037/3/032011. → pages 244, 245[28] C. L. Bottasso, S. Cacciola, and J. Schreiber. Local wind speed estimation,with application to wake impingement detection. Renewable Energy, 116:155–168, 2018. doi:10.1016/j.renene.2017.09.044. → page 22[29] T. Burton, N. Jenkins, D. Sharpe, and E. Bossanyi. Wind EnergyHandbook. John Wiley & Sons, Ltd, 2nd edition, 2011.doi:10.1002/9781119992714. → pages 16, 18, 68, 75, 112, 113, 119, 180[30] R. L. Busby. Wind Power: The Industry Grows Up. PennWell Corporation,Tulsa, Oklahoma, 2012. → pages 2, 10, 11, 12, 17, 68[31] F. Campagnolo, V. Petrovi, C. L. Bottasso, and A. Croce. Wind tunneltesting of wake control strategies. In Proceedings of the 2016 AmericanControl Conference, pages 513–518, 2016.doi:10.1109/ACC.2016.7524965. → pages 36, 46, 69, 236, 243[32] F. Campagnolo, V. Petrovic´, J. Schreiber, E. M. Nanos, A. Croce, and C. L.Bottasso. Wind tunnel testing of a closed-loop wake deflection controllerfor wind farm power maximization. Journal of Physics: Conference Series,753, 2016. doi:10.1088/1742-6596/753/3/032006. → pages50, 60, 62, 69, 245[33] F. Campagnolo, J. Schreiber, A. M. Garcia, and C. L. Bottasso. WindTunnel Validation of a Wind Observer for Wind Farm Control. InProceedings of the 2017 International Ocean and Polar EngineeringConference, pages 640–647, 2017. → page 22[34] Carbon Trust staff. Holistic offshore wind farm control strategies to be putto the test, 2017. URLhttps://www.carbontrust.com/news/2017/12/wind-farm-control-trials/. →pages 13, 68211[35] G. Chartrand and P. Zhang. Chromatic Graph Theory. CRC Press, Taylorand Francis Group, New York, NY, 2008. ISBN 978-1-58488-801-7. →page 163[36] X. Chen, M. Heidarinejad, J. Liu, and P. D. Christofides. Distributedeconomic MPC: Application to a nonlinear chemical process network.Journal of Process Control, 22(4):689–699, 2012. ISSN 09591524.doi:10.1016/j.jprocont.2012.01.016. → pages 140, 178[37] L. Chisci, J. A. Rossiter, and G. Zappa. Systems with persistentdisturbances: Predictive control with restricted constraints. Automatica, 37(7):1019–1028, 2001. ISSN 00051098.doi:10.1016/S0005-1098(01)00051-6. → page 190[38] P. D. Christofides, R. Scattolini, D. Mun˜oz de la Pen˜a, and J. Liu.Distributed model predictive control: A tutorial review and future researchdirections. Computers and Chemical Engineering, 51:21–41, 2013. ISSN00981354. doi:10.1016/j.compchemeng.2012.05.011. → page 139[39] M. Churchfield, P. Fleming, B. Bulder, and S. White. Wind TurbineWake-Redirection Control at the Fishermen’s Atlantic City Windfarm.Technical report, National Renewable Energy Laboratory, 2015. → pages45, 69, 242[40] M. J. Churchfield, S. Lee, J. Michalakes, and P. J. Moriarty. A numericalstudy of the effects of atmospheric and wake turbulence on wind turbinedynamics. Journal of Turbulence, 13(14):1–32, 2012. ISSN 1468-5248.doi:10.1080/14685248.2012.668191. → pages 29, 44, 52[41] U. Ciri, M. A. Rotea, and S. Leonardi. Model-free control of wind farms:A comparative study between individual and coordinated extremumseeking. Renewable Energy, 113:1033–1045, 2017.doi:10.1016/j.renene.2017.06.065. → pages 41, 240[42] T. A. Clevenhult and F. Himmelman. Added Turbulence and OptimalPower Distribution in Large Off-shore Wind Farms. MASc Thesis. LundUniversity, 2010. → page 234[43] G. Corten and P. Schaak. Heat and Flux - Increase of Wind FarmProduction by Reduction of the Axial Induction. Technical report, EnergyResearch Center of the Netherlands, 2003. → pages 35, 236212[44] A. J. Coulling, A. J. Goupee, A. N. Robertson, J. M. Jonkman, and H. J.Dagher. Validation of a FAST semi-submersible floating wind turbinenumerical model with DeepCwind test data. Journal of Renewable andSustainable Energy, 5(2), 2013. ISSN 19417012. doi:10.1063/1.4796197.→ page 126[45] A. Crespo, J. Hernandez, E. Fraga, and C. Andreu. Experimental validationof the UPM computer code to calculate wind turbine wakes andcomparison with other models. Journal of Wind Engineering and IndustrialAerodynamics, 27(1-3):77–88, 1988. ISSN 01676105.doi:10.1016/0167-6105(88)90025-6. → pages 31, 231[46] J. Dahlberg and D. Medici. Potential improvement of wind turbine arrayefficiency by active wake control (AWC). In Proceedings of the 2003European Wind Energy Conference, 2003. → page 241[47] M. De-Prada-Gil, C. G. Alı´as, O. Gomis-Bellmunt, and A. Sumper.Maximum wind power plant generation by reducing the wake effect.Energy Conversion and Management, 101:73–84, 2015. ISSN 01968904.doi:10.1016/j.enconman.2015.05.035. → pages 69, 233[48] D. Dilip and F. Porte´-Agel. Wind turbine wake mitigation through badepitch offset. Energies, 10(757), 2017. doi:10.3390/en10060757. → pages33, 69, 235[49] B. M. Doekemeijer, S. Boersma, L. Y. Pao, T. Knudsen, and J.-W. vanWingerden. Online model calibration for a simplified LES model in pursuitof real-time closed-loop wind farm control. Wind Energy Science, 3(2):749–765, 2018. doi:10.5194/wes-3-749-2018. → pages 20, 23, 65[50] P. A. Driessen, R. M. Hermans, and P. P. Van Den Bosch. Distributedeconomic model predictive control of networks in competitiveenvironments. In Proceedings of the IEEE Conference on Decision andControl, pages 266–271, 2012. ISBN 9781467320665.doi:10.1109/CDC.2012.6426492. → pages 140, 178[51] X. Du, U. Xi, and S. Li. Distributed model predictive control forlarge-scale systems. In Proceedings of the American Control Conference,pages 3142–3143, 2001. → page 150[52] T. Duc, O. Coupiac, N. Girard, G. Giebel, and T. Go¨c¸men. Localturbulence parameterization improves the Jensen wake model and itsimplementation for power optimization of an operating wind farm.213Accepted for review in Wind Energy Science Discussions, n/a, 2018.doi:10.5194/wes-2018-62. → page 234[53] W. B. Dunbar and R. M. Murray. Distributed receding horizon control formulti-vehicle formation stabilization. Automatica, 42(4):549–558, 2006.ISSN 00051098. doi:10.1016/j.automatica.2005.12.008. → pages 150, 172[54] J. Eilbrecht and O. Stursberg. Hierarchical solution of non-convex optimalcontrol problems with application to autonomous driving. EuropeanJournal of Control, 50:188–197, 2019. ISSN 09473580.doi:10.1016/j.ejcon.2019.05.006. → page 140[55] M. Ellis, H. Durand, and P. D. Christofides. A tutorial review of economicmodel predictive control methods. Journal of Process Control, 24(8):1156–1178, 2014. ISSN 09591524. doi:10.1016/j.jprocont.2014.03.010.→ pages 139, 177, 178[56] G. Emanuel. Analytical Fluid Dynamics. Taylor & Francis Group, BocaRaton, Florida, 3rd edition, 2016. → pages 25, 29[57] Environmental Protection Agency staff. Avoided emissions and generationtool (AVERT), 2018. URL https://www.epa.gov/statelocalenergy/avoided-emissions-and-generation-tool-avert. → page 8[58] P. Fleming, P. M. Gebraad, S. Lee, J.-W. van Wingerden, K. Johnson,M. Churchfield, J. Michalakes, P. Spalart, and P. Moriarty. Simulationcomparison of wake mitigation control strategies for a two-turbine case.Wind Energy, 18(12):2135–2143, 2015. doi:10.1002/we.1810. → pages16, 17, 18, 44, 45, 52, 65, 69, 242, 247[59] P. Fleming, J. Annoni, J. J. Shah, L. Wang, S. Ananthan, Z. Zhang,K. Hutchings, P. Wang, W. Chen, and L. Chen. Field test of wake steeringat an offshore wind farm. Wind Energy Science, 2:229–239, 2017.doi:10.5194/wes-2017-4. → pages 51, 52, 60, 69, 85, 104, 246[60] P. Fleming, J. Annoni, M. Churchfield, L. A. Martinez-Tossas,K. Gruchalla, M. Lawson, and P. Moriarty. A simulation studydemonstrating the importance of large-scale trailing vortices in wakesteering. Wind Energy Science, 3(1):243–255, 2018.doi:10.5194/wes-3-243-2018. → pages16, 31, 45, 57, 66, 69, 123, 137, 242214[61] P. Fleming, J. King, K. Dykes, E. Simley, J. Roadman, A. Scholbrock,P. Murphy, J. K. Lundquist, P. Moriarty, K. Fleming, J. V. Dam, C. Bay,R. Mudafort, H. Lopez, J. Skopek, M. Scott, B. Ryan, and D. Brake. InitialResults From a Field Campaign of Wake Steering Applied at a CommercialWind Farm : Part 1. Accepted for review in Wind Energy ScienceDiscussions, n/a, 2019. doi:10.5194/wes-2019-5. → pages 51, 246[62] P. A. Fleming, P. M. Gebraad, S. Lee, J. W. van Wingerden, K. Johnson,M. Churchfield, J. Michalakes, P. Spalart, and P. Moriarty. Evaluatingtechniques for redirecting turbine wakes using SOWFA. RenewableEnergy, 70:211–218, 2014. ISSN 09601481.doi:10.1016/j.renene.2014.02.015. → pages 18, 242, 247[63] P. A. Fleming, A. Ning, P. M. O. Gebraad, and K. Dykes. Wind plantsystem engineering through optimization of layout and yaw control. WindEnergy, 19(2):329–344, feb 2016. doi:10.1002/we.1836. → page 241[64] S. T. Frandsen, H. E. Jørgensen, O. Rathmann, J. Badger, S. Ott, P. E.Rethore, S. E. Larsen, R. Barthelmie, K. Hansen, and L. E. Jensen. Themaking of a second-generation wind farm efficiency model complex. WindEnergy, 12(5):445–458, 2009. ISSN 10954244. doi:10.1002/we.351. →page 42[65] P. Gebraad, J. J. Thomas, A. Ning, P. Fleming, and K. Dykes.Maximization of the annual energy production of wind power plants byoptimization of layout and yaw-based wake control. Wind Energy, 20(1):97–107, jan 2017. doi:10.1002/we.1993. → page 242[66] P. M. Gebraad. Data-driven Wind Plant Control. PhD Thesis. DelftUniversity of Technology, 2014. → pages 68, 238, 242, 245, 247[67] P. M. Gebraad and J. W. Van Wingerden. A control-oriented dynamicmodel for wakes in wind plants. Journal of Physics: Conference Series,524(1), 2014. ISSN 17426596. doi:10.1088/1742-6596/524/1/012186. →pages 27, 29, 105, 231[68] P. M. Gebraad, F. W. Teeuwisse, J. W. Van Wingerden, P. A. Fleming, S. D.Ruben, J. R. Marden, and L. Y. Pao. A data-driven model for wind plantpower optimization by yaw control. In Proceedings of the 2014 AmericanControl Conference, pages 3128–3134, 2014.doi:10.1109/ACC.2014.6859118. → page 245215[69] P. M. Gebraad, P. A. Fleming, and J. W. Van Wingerden. Wind turbinewake estimation and control using FLORIDyn, a control-oriented dynamicwind plant model. Proceedings of the American Control Conference, pages1702–1708, 2015. doi:10.1109/ACC.2015.7170978. → pages20, 23, 49, 61, 65, 104, 244[70] P. M. O. Gebraad and J. W. van Wingerden. Maximum power-pointtracking control for wind farms. Wind Energy, 18(3):429–447, mar 2015.doi:10.1002/we.1706. → pages 23, 39, 50, 59, 62, 104, 105, 238[71] P. M. O. Gebraad, F. C. van Dam, and J. W. van Wingerden. A model-freedistributed approach for wind plant control. In Proceedings of the 2013American Control Conference, pages 628–633, 2013.doi:10.1109/ACC.2013.6579907. → page 238[72] P. M. O. Gebraad, F. W. Teeuwisse, J. W. van Wingerden, P. A. Fleming,S. D. Ruben, J. R. Marden, and L. Y. Pao. Wind plant power optimizationthrough yaw control using a parametric model for wake effects - A CFDsimulation study. Wind Energy, 19(1):95–114, 2016. doi:10.1002/we.1822.→ pages20, 22, 26, 27, 31, 44, 45, 48, 60, 61, 69, 72, 75, 77, 104, 105, 123, 127, 137, 231, 245[73] N. Gionfra, H. Siguerdidjane, G. Sandou, and D. Faille. Hierarchicalcontrol of a wind farm for wake interaction minimization. InternationalFederation of Automatic Control, 49(27):330–335, 2016.doi:10.1016/j.ifacol.2016.10.713. → pages 37, 237[74] T. Go¨c¸men, G. Giebel, P. E. Sørensen, and N. K. Poulsen. Possible PowerEstimation of Down-Regulated Offshore Wind Power Plants. PhD Thesis.Technical University of Denmark, 2016. ISBN 9788793278530. → page239[75] T. Go¨c¸men, P. V. D. Laan, P. E. Re´thore´, A. P. Diaz, G. C. Larsen, andS. Ott. Wind turbine wake models developed at the technical university ofDenmark: A review. Renewable and Sustainable Energy Reviews, 60:752–769, 2016. ISSN 18790690. doi:10.1016/j.rser.2016.01.113. → pages25, 105[76] J. P. Goit and J. Meyers. Optimal control of energy extraction in wind-farmboundary layers. Journal of Fluid Mechanics, 768:5–50, 2015. ISSN14697645. doi:10.1017/jfm.2015.70. → pages 42, 55, 240216[77] J. P. Goit, W. Munters, and J. Meyers. Optimal coordinated control ofpower extraction in LES of a wind farm with entrance effects. Energies, 9(29), 2016. ISSN 19961073. doi:10.3390/en9010029. → pages 21, 42, 240[78] M. Hall and A. Goupee. Validation of a lumped-mass mooring line modelwith DeepCwind semisubmersible model test data. Ocean Engineering,104:590–603, 2015. doi:10.1016/j.oceaneng.2015.05.035. → page 122[79] M. Hall, B. Buckham, and C. Crawford. Evaluating the importance ofmooring line model fidelity in floating offshore wind turbine simulations.Wind Energy, 17(12):1835–1853, 2014. doi:10.1002/we.1669. → pagesxx, 122, 126, 128, 129[80] C. Han. Position Control of a Floating Offshore Wind Turbine SystemUsing Aerodynamic Force. MASc Thesis. The University of BritishColumbia, 2017. → page 17[81] C. Han, J. R. Homer, and R. Nagamune. Movable range and positioncontrol of an offshore wind turbine with a semi-submersible floatingplatform. In Proceedings of the 2017 American Control Conference, pages1389–1394, 2017. doi:10.23919/ACC.2017.7963147. → pages17, 53, 69, 70[82] E. Hau. Wind Turbines: Fundamentals, Technologies, Application,Economics. Springer Berlin Heidelberg, Berlin, Heidelberg, 3rd ed edition,2013. ISBN 978-3-642-27150-2. doi:10.1007/978-3-642-27151-9. →page 22[83] F. Heer, P. M. Esfahani, M. Kamgarpour, and J. Lygeros. Model basedpower optimisation of wind farms. In Proceedings of the 2014 EuropeanControl Conference, pages 1145–1150, 2014.doi:10.1109/ECC.2014.6862365. → pages 37, 239[84] J. Herp, U. V. Poulsen, and M. Greiner. Wind farm power optimizationincluding flow variability. Renewable Energy, 81:173–181, 2015. ISSN18790682. doi:10.1016/j.renene.2015.03.034. → page 233[85] T. Horvat, V. Spudic, and M. Baotic. Quasi-stationary optimal control forwind farm with closely spaced turbines. Proceedings of the 2012International Convention on Information and Communication Technology,Electronics, and Microelectronics, pages 829–834, 2012. → page 234217[86] M. Howland, S. K. Lele, and J. O. Dabiri. Wind farm power optimizationthrough wake steering. ArXiv : Submit / 2588307 [ Physics . Flu-Dyn ],n/a, 2019. doi:10.13140/RG.2.2.16116.40322. → pages 48, 57, 66, 243[87] International Renewable Energy Agency staff. Statistics time series, 2018.URL http://resourceirena.irena.org/gateway/dashboard/?topic=4{&}subTopic=16.→ pages xv, 8, 9[88] International Renewable Energy Agency staff. Global trends in renewableenergy costs, 2018. URL http://resourceirena.irena.org/gateway/dashboard/?topic=3{&}subTopic=1066.→ pages xv, 10, 11[89] International Renewable Energy Agency staff. Employment time series,2018. URL http://resourceirena.irena.org/gateway/dashboard/?topic=7{&}subTopic=53.→ page 8[90] International Renewable Energy Agency staff. Investment trends, 2018.URL http://resourceirena.irena.org/gateway/dashboard/?topic=6{&}subTopic=11.→ pages xv, 9, 10[91] N. O. Jensen. A Note on Wind Generator Interaction. Technical report,Risø National Laboratory. Report number: Risø-M-2411, 1983. → pages25, 75, 104[92] D. Jia and B. H. Krogh. Dsitributed model predictive control. InProceedings of the American Control Conference, pages 2767–2772, 2001.→ page 150[93] A´. Jime´nez, A. Crespo, and E. Migoya. Application of a LES technique tocharacterize the wake deflection of a wind turbine in yaw. Wind Energy, 13(6):559–572, 2009. doi:10.1002/we.380. → pages 16, 26, 69, 75, 119, 181[94] K. E. Johnson and G. Fritsch. Assessment of extremum seeking control forwind farm energy production. Wind Engineering, 36(6):701–715, 2012.ISSN 0309-524X. doi:10.1260/0309-524X.36.6.701. → pages40, 59, 104, 105, 238[95] K. E. Johnson and N. Thomas. Wind farm control: Addressing theaerodynamic interaction among wind turbines. Proceedings of the218American Control Conference, pages 2104–2109, 2009.doi:10.1109/ACC.2009.5160152. → pages 16, 69, 105, 238[96] J. Jonkman, S. Butterfield, W. Musial, and G. Scott. Definition of a 5-MWReference Wind Turbine for Offshore System Development. Technicalreport, National Renewable Energy Laboratory. Report number:NREL/TP-500-38060, 2009. → pagesxiii, xx, xxii, 85, 86, 126, 127, 131, 185, 186[97] J. M. Jonkman. Dynamics modeling and loads analysis of an offshorefloating wind turbine. Technical report, National Renewable EnergyLaboratory, 2007. → page 80[98] I. Katic´, J. Højstrup, and N. Jensen. A simple model for cluster efficiency.Proceedings of the European Wind Energy Association Conference andExhibition, pages 407–410, 1986. → pages20, 26, 70, 75, 104, 120, 181, 231[99] J. Kazda, M. Zendehbad, S. Jafari, N. Chokani, and R. S. Abhari.Mitigating adverse wake effects in a wind farm using non-optimumoperational conditions. Journal of Wind Engineering and IndustrialAerodynamics, 154:76–83, 2016. ISSN 0167-6105.doi:10.1016/j.jweia.2016.04.004. → pages 35, 43, 56, 235[100] J. Kazda, M. Mirzaei, and N. A. Cutululis. On the architecture of windturbine control required for induction-based optimal wind farm control. InProceedings of the 2018 American Control Conference, pages 3074–3079,2018. → pages 37, 239[101] A. C. Kheirabadi and R. Nagamune. Modeling and power optimization offloating offshore wind farms with yaw and induction-based turbinerepositioning. Proceedings of the American Control Conference, pages5458–5463, 2019. → pages 29, 53, 70, 247[102] A. C. Kheirabadi and R. Nagamune. A quantitative review of wind farmcontrol with the objective of wind farm power maximization. Journal ofWind Engineering & Industrial Aerodynamics, 192(May):45–73, 2019.doi:10.1016/j.jweia.2019.06.015. → pages 4, 104, 105, 175[103] A. C. Kheirabadi and R. Nagamune. Real-time relocation of floatingoffshore wind turbine platforms for wind farm efficiency maximization: Anassessment of feasibility and steady-state potential. Ocean Engineering,219208(May), 2020. ISSN 00298018. doi:10.1016/j.oceaneng.2020.107445.→ pages 5, 107, 135, 176, 180, 197[104] H. Kim, K. Kim, I. Paek, and N. Yoo. Development of a time-domainsimulation tool for offshore wind farms. Journal of Power Electronics, 15(4):1047–1053, 2015. ISSN 15982092. doi:10.6113/JPE.2015.15.4.1047.→ page 239[105] H. Kim, K. Kim, and I. Paek. Power regulation of upstream wind turbinesfor power increase in a wind farm. International Journal of PrecisionEngineering and Manufacturing, 17(5):665–670, 2016. ISSN 20054602.doi:10.1007/s12541-016-0081-1. → page 234[106] H. Kim, K. Kim, and I. Paek. Model based open-loop wind farm controlusing active power for power increase and load reduction. AppliedSciences, 7(10):1068, 2017. ISSN 2076-3417. doi:10.3390/app7101068.→ pages 37, 59, 239[107] T. Knudsen, T. Bak, and M. Svenstrup. Survey of wind farm control-powerand fatigue optimization. Wind Energy, 18(8):1333–1351, 2015.doi:10.1002/we.1760. → pages 13, 104, 202[108] P. N. Ko¨hler, M. A. Mu¨ller, and F. Allgo¨wer. A distributed economic MPCframework for cooperative control under conflicting objectives.Automatica, 96:368–379, 2018. ISSN 00051098.doi:10.1016/j.automatica.2018.07.001. → pages 140, 142, 178, 189[109] Y. Kuwata, A. Richards, T. Schouwenaars, and J. P. How. Distributedrobust receding horizon control for multivehicle guidance. IEEETransactions on Control Systems Technology, 15(4):627–641, 2007. ISSN10636536. doi:10.1109/TCST.2007.899152. → pages 140, 141, 177, 193[110] G. C. Larsen, H. A. Madsen, K. Thomsen, and T. J. Larsen. Wakemeandering: A pragmatic approach. Wind Energy, 11(4):377–395, 2008.ISSN 10954244. doi:10.1002/we.267. → page 42[111] J. Lee and D. Angeli. Cooperative distributed model predictive control forlinear plants subject to convex economic objectives. In Proceedings of theIEEE Conference on Decision and Control, number 6, pages 3434–3439.IEEE, 2011. ISBN 9781612848006. doi:10.1109/CDC.2011.6160694. →pages 140, 178220[112] J. Lee and D. Angeli. Distributed cooperative nonlinear economic MPC. InProceedings of the 20th International Symposium on Mathematical Theoryof Networks and Systems (MTNS), 2012. → pages 140, 178[113] J. Lee, E. Son, B. Hwang, and S. Lee. Blade pitch angle control foraerodynamic performance optimization of a wind farm. Renewable Energy,54:124–130, 2013. ISSN 09601481. doi:10.1016/j.renene.2012.08.048. →page 234[114] Y. Li and Z. Wu. Stabilization of floating offshore wind turbines byartificial muscle based active mooring line force control. In Proceedings ofthe 2016 American Control Conference, pages 2277–2282, 2016.doi:10.1109/ACC.2016.7525257. → page 17[115] P. B. S. Lissaman. Energy effectiveness of arbitrary arrays of wind turbines.Journal of Energy, 3(6):323–328, 1979. → pages 10, 11, 25, 34, 49, 68, 97[116] J. Liu, X. Chen, D. M. De La Pen˜a, and P. D. Christofides. Iterativedistributed model predictive control of nonlinear systems: Handlingasynchronous, delayed measurements. IEEE Transactions on AutomaticControl, 57(2):528–534, 2012. ISSN 00189286.doi:10.1109/TAC.2011.2164729. → pages 140, 177, 188[117] P. Liu, A. Kurt, and U. Ozguner. Distributed Model Predictive Control forCooperative and Flexible Vehicle Platooning. IEEE Transactions onControl Systems Technology, 27(3):1115–1128, 2019. ISSN 1558-0865.doi:10.1109/TCST.2018.2808911. → pages 141, 177[118] L. Machielse, S. Barth, E. Bot, H. Hendriks, and G. Schepers. Evaluationof “Heat and Flux” Farm Control. Technical report, Energy ResearchCentre of the Netherlands, 2007. → pages 36, 236[119] D. Madjidian and A. Rantzer. A stationary turbine interaction model forcontrol of wind farms. In Proceedings of the 18th World Congress, TheInternational Federation of Automatic Control, pages 4921–4926, 2011.doi:10.3182/20110828-6-IT-1002.00267. → page 234[120] J. M. Maestre, D. Mun˜oz De La Pen˜a, E. F. Camacho, and T. Alamo.Distributed model predictive control based on agent negotiation. Journal ofProcess Control, 21(5):685–697, 2011.doi:10.1016/j.jprocont.2010.12.006. → pages 141, 177221[121] J. Manwell, J. McGowen, and A. Rogers. Wind Energy Explained: Theory,Design, and Application. John Wiley & Sons, Ltd, 2nd edition, 2009. →pages 9, 11, 25, 75, 77, 111[122] J. R. Marden, S. D. Ruben, and L. Y. Pao. A model-free approach to windfarm control using game theoretic methods. IEEE Transactions on ControlSystems Technology, 21(4):1207–1214, 2013. ISSN 10636536.doi:10.1109/TCST.2013.2257780. → pages21, 23, 38, 39, 48, 61, 62, 69, 237[123] L. A. Martı´nez-Tossas, M. J. Churchfield, and S. Leonardi. Large eddysimulations of the flow past wind turbines: actuator line and disk modeling.Wind Energy, 18(6):1047–1060, jun 2015. ISSN 10954244.doi:10.1002/we.1747. → page 30[124] L. A. Martı´nez-Tossas, J. Annoni, P. A. Fleming, and M. J. Churchfield.The aerodynamics of the curled wake: a simplified model in view of flowcontrol. Wind Energy Science, 4(1):127–138, 2019.doi:10.5194/wes-4-127-2019. → pages 27, 75[125] D. Mayne, J. Rawlings, C. Rao, and P. Scokaert. Constrained modelpredictive control: Stability and optimality. Automatica, 36(6):789–814,2000. doi:10.1016/S0005-1098(99)00214-9. → pages 139, 177[126] D. Q. Mayne. Model predictive control: Recent developments and futurepromise. Automatica, 50(12):2967–2986, 2014.doi:10.1016/j.automatica.2014.10.128. → pages 139, 177[127] S. McTavish, D. Feszty, and F. Nitzsche. Evaluating Reynolds numbereffects in small-scale wind turbine experiments. Journal of WindEngineering and Industrial Aerodynamics, 120:81–90, 2013. ISSN01676105. doi:10.1016/j.jweia.2013.07.006. → page 35[128] D. Medici. Experimental Studies of Wind Turbine Wakes - PowerOptimisation and Meandering. PhD Thesis. KTH Royal Institute ofTechnology, 2005. → pages 75, 241[129] J. Meyers, W. Munters, and J. Goit. A framework for optimization ofturbulent wind-farm boundary layers and application to optimal control ofwind-farm energy extraction. In Proceedings of the 2016 American ControlConference, pages 519–524, 2016. → page 240222[130] W. Miao, C. Li, J. Yang, and X. Xie. Numerical investigation of the yawedwake and its effects on the downstream wind turbine. Journal ofRenewable and Sustainable Energy, 8(3), 2016. doi:10.1063/1.4953791.→ pages 46, 242[131] M. Mirzaei, T. Go¨c¸men, G. Giebel, P. E. Sørensen, and N. K. Poulsen.Turbine control strategies for wind farm power Optimization. InProceedings of the 2015 American Control Conference, pages 1709–1714,2015. → pages 69, 234[132] T. P. Miyanawala and R. K. Jaiman. A low-dimensional learning model viaconvolutional neural networks for unsteady wake-body interaction.Accepted for review in the Journal of Fluid Mechanics, n/a, 2018.doi:arXiv:1807.09591v1. → pages 20, 65[133] G. Mosetti, C. Poloni, and D. Diviacco. Optimization of wind turbinepositioning in large wind farms by means of a Genetic algorithm. J WindEng Ind Aerody 51:105–116. Journal of Wind Engineering and IndustrialAerodynamics, 51(51):105–116, 1994. → page 12[134] M. A. Muller and F. Allgower. Economic and Distributed Model PredictiveControl : Recent Developments in Optimization-Based Control. SICEJournal of Control, Measurement, and System Integration, 10(2):39–52,2017. → pages 139, 177[135] M. A. Mu¨ller, M. Reble, and F. Allgo¨wer. Cooperative control ofdynamically decoupled systems via distributed model predictive control.International Journal of Robust and Nonlinear Control, 22:1376–1397,2012. → pages 141, 177[136] W. Munters and J. Meyers. Effect of wind turbine response time on optimaldynamic induction control of wind farms. Journal of Physics: ConferenceSeries, 753, 2016. ISSN 17426596. doi:10.1088/1742-6596/753/5/052007.→ page 240[137] W. Munters and J. Meyers. Optimal dynamic induction and yaw control ofwind farms: Effects of turbine spacing and layout. Journal of Physics:Conference Series, 1037, jun 2018. ISSN 1742-6588.doi:10.1088/1742-6596/1037/3/032015. → pages 50, 56, 245[138] W. Munters and J. Meyers. Towards practical dynamic induction control ofwind farms : analysis of optimally controlled wind-farm boundary layers223and sinusoidal induction control of first-row turbines. Wind EnergyScience, 3:409–425, 2018. → pages 42, 43, 55, 56, 60, 65, 66, 240[139] W. Munters and J. Meyers. Dynamic strategies for yaw and inductioncontrol of wind farms based on large-eddy simulation and optimization.Energies, 11(177), 2018. ISSN 1996-1073. doi:10.3390/en11010177. →pages 50, 51, 59, 60, 65, 66, 245[140] R. R. Negenborn and J. M. Maestre. Distributed model predictive control:An overview and roadmap of future research opportunities. IEEE ControlSystems Magazine, 34(4):87–97, 2014. ISSN 1066033X.doi:10.1109/MCS.2014.2320397. → page 139[141] K. Nilsson, S. Ivanell, K. S. Hansen, R. Mikkelsen, J. N. Sørensen, S.-P.Breton, and D. Henningson. Large-eddy simulations of the Lillgrund windfarm. Wind Energy, 18(3):449–467, 2015. doi:10.1002/we.1707. → pages2, 10, 34, 68, 69, 175, 235[142] J. Nocedal and S. J. Wright. Numerical Optimization. Springer Science +Business Media, New York, NY, 2nd edition, 2006. → page 85[143] J. G. J. Olivier, G. Janssens-Maenhout, M. Muntean, and J. A. H. W.Peters. Trends in Global CO2 Emissions: 2016 Report. Technical report,PBL Netherlands Environmental Assessment Agency, 2016. → page 8[144] J. Pannek. Parallelizing a state exchange strategy for noncooperativedistributed NMPC. Systems and Control Letters, 62(1):29–36, 2013. ISSN01676911. doi:10.1016/j.sysconle.2012.10.015. → pages 141, 177[145] J. Park and K. H. Law. Cooperative wind turbine control for maximizingwind farm power using sequential convex programming. EnergyConversion and Management, 101:295–316, 2015.doi:10.1016/j.enconman.2015.05.031. → page 244[146] J. Park and K. H. Law. A data-driven, cooperative wind farm control tomaximize the total power production. Applied Energy, 165:151–165, 2016.ISSN 03062619. doi:10.1016/j.apenergy.2015.11.064. → pages49, 62, 236, 244[147] J. Park, S. Kwon, and K. H. Law. Wind farm power maximization based ona cooperative static game approach. In Proceedings of the 2013 SPIESmart Structures and Materials + Nondestructive Evaluation and HealthMonitoring, 2013. doi:10.1117/12.2009618. → page 244224[148] J. Park, S.-D. Kwon, and K. Law. A data-driven, cooperative approach forwind farm control: A wind tunnel experimentation. Energies, 10(852),2017. ISSN 1996-1073. doi:10.3390/en10070852. → pages50, 60, 61, 65, 69, 245[149] J. Quick, J. Annoni, R. King, K. Dykes, and P. Fleming. OptimizationUnder Uncertainty for Wake Steering Strategies. Technical report, NationalRenewable Energy Agency, 2017. → page 241[150] S. Raach, D. Schlipf, F. Beyer, and P. W. Cheng. Wake redirecting usingfeedback control to improve the power output of wind farms. InProceedings of the 2016 American Control Conference, pages 1387–1392,2016. → page 244[151] A. Richards and J. P. How. Robust distributed model predictive control.International Journal of Control, 80(9):1517–1531, 2007. → pages140, 141, 177, 193[152] A. Robertson, J. Jonkman, and M. Masciola. Definition of theSemisubmersible Floating System for Phase II of OC4. Technical report,National Renewable Energy Laboratory. Report number:NREL/TP-5000-60601, 2014. → pagesxiii, xx, xxii, 85, 86, 107, 126, 127, 131, 180, 185, 186[153] S. F. Rodrigues, R. Teixeira Pinto, M. Soleimanzadeh, P. A. Bosman, andP. Bauer. Wake losses optimization of offshore wind farms with moveablefloating wind turbines. Energy Conversion and Management, 89:933–941,2015. doi:10.1016/j.enconman.2014.11.005. → pages 17, 52, 70, 247[154] S. G. Rubin. A review of marching procedures for parabolizedNavier-Stokes equations. In T. Cebeci, editor, Numerical and PhysicalAspects of Aerodynamic Flows, chapter 11, pages 171–185. SpringerScience + Business Media, New York, 1st edition, 1982. → page 31[155] A. Sanfelice Bazanella, L. Campestrini, and D. Eckhard. Data-drivenController Design: The H2 Approach. Springer Netherlands, 1st edition,2012. ISBN 978-94-007-2300-9. doi:10.1007/978-94-007-2300-9. →page 22[156] C. Santoni, U. Ciri, M. Rotea, and S. Leonardi. Development of a highfidelity CFD code for wind farm control. In Proceedings of the 2015American Control Conference, pages 1715–1720, 2015. → pages 34, 235225[157] J. G. Schepers and S. P. Van Der Pijl. Improved modelling of wakeaerodynamics and assessment of new farm control strategies. Journal ofPhysics: Conference Series, 75, 2007. ISSN 17426588.doi:10.1088/1742-6596/75/1/012039. → pages 36, 43, 69, 236[158] J. Schreiber, S. Cacciola, F. Campagnolo, V. Petrovic´, D. Mourembles, andC. L. Bottasso. Wind shear estimation and wake detection by rotor loads -First wind tunnel verification. Journal of Physics: Conference Series, 753(3), 2016. doi:10.1088/1742-6596/753/3/032027. → pages 22, 65[159] J. Serrano Gonza´lez, M. B. Paya´n, and J. R. Santos. Optimum windturbines operation for minimizing wake effect losses in offshore windfarms. In Proceedings of the 2013 International Conference onEnvironment and Electrical Engineering, 2013.doi:10.1109/EEEIC-2.2013.6737905. → page 233[160] J. Serrano Gonza´lez, M. Burgos Paya´n, J. Riquelme Santos, and A´. G.Gonza´lez Rodrı´guez. Maximizing the overall production of wind farms bysetting the individual operating point of wind turbines. Renewable Energy,80:219–229, 2015. ISSN 18790682. doi:10.1016/j.renene.2015.02.009. →pages 69, 233[161] R. Shakoor, M. Y. Hassan, A. Raheem, and Y. K. Wu. Wake effectmodeling: A review of wind farm layout optimization using Jensen’smodel. Renewable and Sustainable Energy Reviews, 58:1048–1059, 2016.doi:10.1016/j.rser.2015.12.229. → pages 10, 12, 68, 104, 124[162] C. R. Shapiro, P. Bauweraerts, J. Meyers, C. Meneveau, and D. F. Gayme.Model-based receding horizon control of wind farms for secondaryfrequency regulation. Wind Energy, 20(7):1261–1275, 2017. ISSN10991824. doi:10.1002/we.2093. → pages 105, 106[163] C. R. Shapiro, D. F. Gayme, and C. Meneveau. Modelling yawed windturbine wakes: a lifting line approach. Journal of Fluid Mechanics, 841:1–12, 2018. ISSN 0022-1120. doi:10.1017/jfm.2018.75. → pages48, 106, 243[164] J. Shu, B. H. Zhang, and Z. Q. Bo. A wind farm coordinated controller forpower optimization. In Proceedings of the 2011 IEEE Power and EnergySociety General Meeting, 2011. doi:10.1109/PES.2011.6039552. → pages42, 59, 238226[165] A. J. Sørensen. A survey of dynamic positioning control systems. AnnualReviews in Control, 35(1):123–136, 2011. ISSN 13675788.doi:10.1016/j.arcontrol.2011.03.008. → page 17[166] N. N. Sørensen. General Purpose Flow Solver Applied to Flow Over Hills.PhD Thesis. Technical University of Denmark, 1995. → page 30[167] E. staff. Electricity production, 2018. URL https://yearbook.enerdata.net/electricity/world-electricity-production-statistics.html. → page 8[168] S. staff. STAR-CCM+, 2018. URLhttps://mdx.plm.automation.siemens.com/star-ccm-plus. → page 30[169] M. Steinbuch, W. D. Boer, and O. Bosgra. Optimal control of wind powerplants. Journal of Wind Engineering and Industrial Aerodynamics, 27:237–246, 1988. ISSN 01676105. doi:10.1016/0167-6105(88)90039-6. →pages 14, 234[170] B. T. Stewart, S. J. Wright, and J. B. Rawlings. Cooperative distributedmodel predictive control for nonlinear systems. Journal of Process Control,21(5):698–704, 2011. ISSN 09591524.doi:10.1016/j.jprocont.2010.11.004. → pages 141, 177[171] J. Tian, C. Su, M. Soltani, and Z. Chen. Active power dispatch method fora wind farm central controller considering wake effect. In Proceedings ofthe 2014 Industrial Electronics Conference, pages 5450–5456, 2014.doi:10.1109/IECON.2014.7049333. → pages 69, 233[172] J. Tian, D. Zhou, C. Su, F. Blaabjerg, and Z. Chen. Optimal control toincrease energy production of wind farm considering wake effect andlifetime estimation. Applied Sciences, 7(65), 2017.doi:10.3390/app7010065. → page 234[173] U.S Department of Energy staff. U.S. Energy and Employment Report.Technical report, U.S. Department of Energy, 2017. → page 9[174] U.S. Energy Information Administration staff. Assumptions to the AnnualEnergy Outlook 2018: Electricity Market Module. Technical report, U.S.Energy Information Administration, 2018. → page 10[175] M. Vali, V. Petrovic´, B. Sjoerd, J.-W. van Wingerden, and M. Ku¨hn.Adjoint-based model predictive control of wind farms: Beyond the quasisteady-state power maximization. International Federation of Automatic227Control, 50(1):4510–4515, 2017. doi:10.1016/j.ifacol.2017.08.382. →pages 41, 59, 64, 239[176] D. van der Hoek, S. Kanev, J. Allin, D. Bieniek, and N. Mittelmeier.Effects of axial induction control on wind farm energy production - A fieldtest. Renewable Energy, 140:994–1003, 2019.doi:10.1016/j.renene.2019.03.117. → pages 43, 44, 56, 240[177] M. T. van Dijk, J.-W. van Wingerden, T. Ashuri, Y. Li, and M. A. Rotea.Yaw-misalignment and its impact on wind turbine loads and wind farmpower output. Journal of Physics: Conference Series, 753, 2016. ISSN1742-6588. doi:10.1088/1742-6596/753/6/062013. → page 241[178] M. T. van Dijk, J. W. van Wingerden, T. Ashuri, and Y. Li. Wind farmmulti-objective wake redirection for optimizing power production andloads. Energy, 121:561–569, 2017. ISSN 03605442.doi:10.1016/j.energy.2017.01.051. → page 241[179] L. J. Vermeer, J. N. Sørensen, and A. Crespo. Wind turbine wakeaerodynamics. Progress in Aerospace Sciences, 39(6-7):467–510, 2003.ISSN 03760421. doi:10.1016/S0376-0421(03)00078-2. → pages 25, 105[180] P. Vermeulen and P. Builtjes. Mathematical Modelling of Wake Interactionin Wind Turbine Arrays. Technical report, Netherlands Organization forApplied Scientific Research, Division of Technology for Society, 1981. →page 234[181] J. Wagenaar, L. Machielse, and J. Schepers. Controlling Wind in ECN’sScaled Wind Farm. Technical report, Energy Research Centre of theNetherlands, 2012. → pages 16, 47, 243[182] C. Wang, J. Wang, F. Campagnolo, D. B. Carrao´n, and C. L. Bottasso.Validation of large-eddy simulation of scaled waked wind turbines indifferent yaw misalignment conditions. Journal of Physics: ConferenceSeries, 1037, jun 2018. ISSN 1742-6588.doi:10.1088/1742-6596/1037/6/062007. → pages 30, 231[183] L. Wang, A. Tan, and Y. Gu. A novel control strategy approach tooptimally design a wind farm layout. Renewable Energy, 95:10–21, 2016.ISSN 18790682. doi:10.1016/j.renene.2016.03.104. → page 233[184] R. Wang, I. R. Manchester, and J. Bao. Distributed economic MPC withseparable control contraction metrics. IEEE Control Systems Letters, 1(1):228104–109, 2017. ISSN 24751456. doi:10.1109/LCSYS.2017.2708660. →pages 140, 178[185] I. J. Wolf, H. Scheu, and W. Marquardt. A hierarchical distributedeconomic NMPC architecture based on neighboring-extremal updates. InProceedings of the American Control Conference, pages 4155–4160. IEEE,2012. ISBN 9781457710957. doi:10.1109/acc.2012.6315353. → pages140, 178[186] C. Yan. Wind Turbine Wakes: From Numerical Modeling to MachineLearning. PhD Thesis. University of Delaware, 2018. → pages 20, 22, 65[187] Z. Yang, Y. Li, and J. E. Seem. Optimizing energy capture of cascadedwind turbine array with nested-loop extremum seeking control. Journal ofDynamic Systems, Measurement, and Control, 137(12), 2015.doi:10.1115/1.4031593. → pages 40, 41, 238[188] S. Zhong and X. Wang. Decentralized model-free wind farm control viadiscrete adaptive filtering methods. IEEE Transactions on Smart Grid, 9(4):2529–2540, 2018. ISSN 19493053. doi:10.1109/TSG.2016.2614434.→ pages 38, 39, 59, 61, 237229Appendix AWind Farm Control ReviewTables230Table A.1: Summary of wake models and simulation tools used within wind farm control studies that have been re-viewed in the current chapter.Model Type FidelityTime-dependencyWake phenomena Relative errorPark Parametric Low Steady Momentum deficit 0.1 to 15.9 %1FLORIS Parametric Low SteadyMomentum deficitWake redirection0.1 to 5.3 %2FLORIDyn Parametric Low Quasi-dynamicMomentum deficitWake redirection7.9 to 11.7 %3FOWFSim Parametric Low SteadyMomentum deficitWake redirectionFloating turbine motion-WFSim Turbulent 2D NS Medium DynamicMomentum deficitWake redirection82.1 % VAF4Ainslie Turbulent axisymmetric NS Medium Steady Momentum deficit 0.5 to 8.4 %5FarmFlow Turbulent 3D parabolized NS MediumSteadyQuasi-dynamicMomentum deficit 0.2 to 26.9 %6SOWFA 3D LES + Actuator line theory High DynamicMomentum deficitWake redirection0.2 to 14 %7SP-Wind 3D LES + Actuator disc theory High DynamicMomentum deficitWake redirection-1Relative error results were obtained by digitizing Figure 3 and calculating the percent deviation of the Park model results relative to the TNO windtunnel results [98].2Relative error results were obtained from Table II [72].3RMSE values were obtained from Figure 5a [67]4Variance accounted for (VAF) result was obtained from Table 3 [23, 25]5Relative error results were obtained by digitizing Figure 2 [4]6Relative error results were obtained by digitizing Figures 2 to 4 and comparing UPM Code results with TNO experimental results [45].7Relative error results were obtained from Tables 1 and 2 [182].231Table A.1 continued from previous pageModel Type FidelityTime-dependencyWake phenomena Relative errorUTD-WF 3D LES + Actuator disc/line theory High DynamicMomentum deficitWake redirection-EllipSys3D 3D RANS High DynamicMomentum deficitWake redirection-STAR-CCM+ 3D RANS/LES High DynamicMomentum deficitWake redirection-232Table A.2: Summary of studies that have assessed the steady-state potential of power de-rating.Ref.Optimization/analysismethodInputs Optimization model Evaluation modelSimulated windconditionsRelativeefficiency gain[20]Centralized particle swarmoptimizationλ , β Steady Park model Steady Park model Const. speed 10.57%8[21]Centralized dynamicprogramminga Steady Park model Steady Park model Const. speed 8.33%9[47] Manual optimization λ Steady Park model Steady Park model Const. speed 12.80%10[159] Centralized genetic algorithm λ , β Steady Park model Steady Park model Const. speed 10.94%11[160] Centralized genetic algorithm λ , β Steady Park model Steady Park model Const. speed 7.55%12[171]Centralized particle swarmoptimizationλ , β Steady Park model Steady Park model Const. speed 26.77%13[84]Centralized sequentialoptimizationa Steady Park model Steady Park model Const. speed 5.40%14[183] Distributed genetic algorithm a Steady Park model Steady Park model Const. speed 1.52%15[14]Centralized sequentialquadratic programminga Steady Park model Steady Park model Const. speed -168Relative wind farm efficiency gain corresponds to simulation case 1, and was obtained from discussions in the section entitled Case Study [20].9Relative wind farm efficiency gain was obtained from discussions in Section III [21].10Relative wind farm efficiency gain was obtained by digitizing Figure 18 at a wind direction of 90deg [47].11Relative wind farm efficiency gain was obtained from discussions in Section IV [159].12Relative wind farm efficiency gain was obtained from discussions in Section 4 [160].13Relative wind farm efficiency gain was obtained from discussions in Section VII [171].14Relative wind farm efficiency gain was obtained from Table 1 [84].15Relative wind farm efficiency gain was obtained from Table 5 [183].16The wind farm power output corresponding to an uncontrolled wind farm is not listed; therefore wind farm efficiency gains cannot be calcu-lated [14].233Table A.2 continued from previous pageRef.Optimization/analysismethodInputs Optimization model Evaluation modelSimulated windconditionsRelativeefficiency gain[22]Centralized particle swarmoptimizationa Steady Park model Steady Park model Const. speed 9.96%17[172] Manual optimization λ , β Steady Park model Steady Park model Const. speed 5.41%18[131] Centralized genetic algorithm P Steady Park model Steady Park model Const. speed 13.34%19[52] Manual optimization CPModified steady parkmodelModified steady park modelConst. speedTurbulence2.6%20[85]Centralized sequentialquadratic programmingω Steady empirical model Steady empirical model Const. speed 2.85%21[42]Centralized gradient descentoptimizationPSteady parametric wakemodel [119]Steady parametric wakemodel [119]Const. speed 3.23%22[169] Manual optimization λSteady Millymodel [180]Steady Milly model [180] Const. speed 4%23[113] Centralized genetic algorithm βSteady eddy viscositymodelSteady eddy viscosity modelConst. speedTurbulence4.50%24[105] Numerical simulation P No optimization Dynamic Ainslie modelConst. speedTurbulence4.10%2517Relative wind farm efficiency gain was obtained from discussions in Section 5 [22].18Relative wind farm efficiency gain was obtained from Figure 8 [172].19Relative wind farm efficiency gain was obtained from Table I and corresponds to the gain obtained from using the constant rotor speed controlmethod relative to constant reference control at 5MW [131].20Relative wind farm efficiency gain was obtained from Figure 12a.21Relative wind farm efficiency gain was obtained from Table I [85].22Relative wind farm efficiency gain was obtained from Tables 4.1 and 4.2 [42].23Relative wind farm efficiency gain was obtained from discussions under the heading entitled Maximizing energy capture in Section B [169].24Relative wind farm efficiency gain was obtained from discussions in Section V [113].25Relative wind farm efficiency gain was obtained from discussions in Section 3.1 [105].234Table A.2 continued from previous pageRef.Optimization/analysismethodInputs Optimization model Evaluation modelSimulated windconditionsRelativeefficiency gain[141] Numerical simulation β No optimization EllipSys3D (dynamic CFD)Const. speedTurbulenceWind shear−7.70%26[9] Centralized grid search β Steady Park modelSteady FLORIS modelSOWFA27Const. speedTurbulenceWind shear24.82%28−9.35%29[48] Numerical simulation β No optimization Dynamic LESConst. speedTurbulenceWind shear2.80%30[99] Numerical simulation λ , β No optimizationMULTI3 (Dynamic k-ωRANS model)Const. speedTurbulenceWind shear2.5%319.7%32[156]Centralized dynamicprogrammingλ Steady Park model UTD-WF (Dynamic LES)Const. speedTurbulenceWind shear5.7%330.2%3426Relative wind farm efficiency gain was obtained from Table III [141].27The listed simulated wind conditions correspond to the dynamic SOWFA CFD tool. Constant wind speeds were simulated when using the FLORISwake model.28Relative wind farm efficiency gain corresponds to the steady FLORIS wake model, and was obtained by digitizing Figure 8[9].29Relative wind farm efficiency gain corresponds to the dynamic SOWFA CFD tool, and was obtained by digitizing Figure 8[9].30Relative wind farm efficiency gain was obtained from Figure 8 [48].31Relative wind farm efficiency gain corresponds to only WTG 4 and 11, and was obtained from Figure 13b and discussions in Section 3.2.2 [99].32Relative wind farm efficiency gain corresponds to all four wind turbines, and was obtained from Figure 13b and discussions in Section 3.2.2 [99].33Relative wind farm efficiency gain was obtained from Table III and corresponds to evaluation using the Jensen wake model [156].34Relative wind farm efficiency gain was obtained from Table III and corresponds to evaluation using UTD-WF [156].235Table A.2 continued from previous pageRef.Optimization/analysismethodInputs Optimization model Evaluation modelSimulated windconditionsRelativeefficiency gain[43]35 Wind tunnel experimentation β No optimization Scaled wind farm - 4.60%36[31] Wind tunnel experimentation P No optimization Scaled wind farm - 0.90%37[16] Wind tunnel experimentation λ , β No optimization Scaled wind farm - 0%38[118] Wind tunnel experimentation β No optimization Scaled wind farm - 3.46%39[2] Wind tunnel experimentation λ , βNo wind farmoptimizationScaled wind farm -3.26%403.81%41[157]Field experimentation + Noisereduced wind turbine operationβ No optimization ECN’s EWTW wind farm - Inconclusive42[26] Field experimentation β No optimization ECN’s EWTW wind farm - Inconclusive4335Additional information is available in a companion study [146].36Relative wind farm efficiency gain was obtained from discussions in Section 5.2 [43].37Relative wind farm efficiency gain was obtained from Figure 7 [31].38The zero gain in wind farm efficiency is evident from Figures 4a and 4b [16]; namely, the combinations of turbine operating parameters thatmaximize wind farm performance are those that maximize individual turbine performance.39Relative wind farm efficiency gain corresponds to a wind speed of 9m/s, and was obtained by digitizing Figure 5.3 [118].40Relative wind farm efficiency gain corresponds to a scenario where the upstream turbine operates at a tip-speed ratio λ that was higher than itsoptimal value, and was obtained from by digitizing Figure 10b [2].41Relative wind farm efficiency gain corresponds to the gain in wind farm power output observed when the upstream turbine’s collective blade pitchangle β was varied from 0deg to 2deg at a spacing of six rotor diameters, and was obtained by digitizing Figure 10a [2].42There is no clear discrepancy between the wind farm power output results in Figure 10, which correspond to wind farm operation under differentcontrol strategies [157]. It is mentioned by the authors that the concept of axial induction control is validated however, since little change in the windfarm power output indicates that the loss in power output from the upstream turbine is compensated by a gain in power output from the remainingturbines.43There is no discernible rise in the wind farm power coefficient in Figure 4.1.f [26]. A wind power output increase is observed at a wind speed of8m/s; however the upstream turbine also experiences a power output increase at this operating point. Since the upstream turbine is undisturbed, thisdata point is deemed erroneous. This point is confirmed by the authors in Section 4.1.236Table A.3: Summary of studies that have developed wind farm controllers based on the concept of power de-rating.Ref. Control method Inputs Control model Evaluation modelSimulated windconditionsRelativeefficiency gain[1]Centralized artificial beeoptimization + Maximumpower-point tracking windturbine controlλ Steady Park model Steady Park modelConst. speedVar. direction4.03%44[73]Centralized sequential quadraticprogramming optimization +Feedback linearization and modelpredictive wind turbine controla Steady Park model Steady Park model Var. speed 4%45[122]Centralized/distributedgame-theoretic controla Model-free Steady Park model Const. speed 34.05%46[15]Centralized/distributed gradientestimation-based controla Model-free Steady Park model Const. speed -47[188]Distributed discrete adaptivefiltering optimizationa Model-free Steady Park modelConst. speedVar. directionTurbulence483.87%4944Relative wind farm efficiency gain was obtained by digitizing Figure 12g and corresponds to a wind direction of 180deg [1].45Relative wind farm efficiency gain was obtained from discussions in Section 5[73].46Relative wind farm efficiency gain was calculated based on results presented in Figure 5b [122]. This figure presents the wind farm power outputnormalized relative to the optimal wind farm power output. The normalized power output was converted to the relative gain in wind farm efficiency viasimple algebraic manipulation.47Power production from the suboptimal wind farm is not presented; therefore the efficiency gain cannot be calculated [15].48The authors mention that a random variable is added to the power output of each turbine at each iteration to simulate environmental distur-bances [188]. No statistical information is provided regarding this random signal.49Relative wind farm efficiency gain corresponds to Algorithm 2, and was obtained by digitizing Figure 13 [188].237Table A.3 continued from previous pageRef. Optimization/analysis method Inputs Control model Evaluation modelSimulated windconditionsRelativeefficiency gain[94]50Centralized extremum seekingcontrola Model-free Dynamic Park modelVar. speedTurbulence3.8%51−13.2%52[70]53Distributed maximumpower-point trackinga Model-free Dynamic Park model Var. speed 4%54[3]Centralized multi-resolutionsimultaneous perturbationstochastic approx.a Model-free Dynamic Park modelVar. speedVar. direction32%55[187]Centralized nested-loopextremum seeking controlTgen Model-freeDynamicSimWindFarmConst. speedTurbulence1.30%56[164]Centralized nonlinear modelpredictive control + Maximumpower-point tracking windturbine controlωDynamicmosaic-tile modelDynamic mosaic-tilemodelVar. speed 1.96%5750Additional information is available in a companion publication [95].51Relative wind farm efficiency gain corresponds to a low turbulence intensity setting of 2%, and was obtained from discussions in Section 5 [94].52Relative wind farm efficiency gain corresponds to a high turbulence intensity setting of 8%, and was obtained from discussions in Section 5 [94].53Additional information is available in companion publications [66, 71].54Relative wind farm efficiency gain was obtained from Figure 8 and from discussions in Section 5.4 [70].55Relative wind farm efficiency gain was obtained from Figure 10 [3].56Relative wind farm energy production gain was obtained from discussions in the section entitled Simulation study: Simulation for turbulentwind [187].57Relative wind farm efficiency gain corresponds to a turbine spacing of 560m (i.e. seven rotor diameters), and was obtained from Figure 9 [164].238Table A.3 continued from previous pageRef. Optimization/analysis method Inputs Control model Evaluation modelSimulated windconditionsRelativeefficiency gain[83]Centralized brute-forceoptimization + Model predictivewind turbine controlλ , β Steady Park modelDynamicSimWindFarmVar. speedTurbulence0.41%581.42%59[106]Centralized Nelder-Meadsimplex optimization +Demanded power-point trackingwind turbine controlPSteady Ainsliemodel [104]Dynamic Ainsliemodel [104]Const. speedTurbulence2.92%60−2.88%61[100]DTU wind farm controller +Maximum power-point trackingwind turbine controlPPossPOW wakemodel [74]PossPOW wakemodel [74]Const. speedTurbulence1%62[175]Centralized adjoint-based modelpredictive controla Dynamic WFSim Dynamic WFSimConst. speedVar. direction23.59%63−3.6%6458Relative energy production gain corresponds to a 1×3 wind-aligned row wind farm modeled using FAST, and was obtained by manipulating theEtot,greedy/E∗tot ratio described in Section V.A [83]. This ratio was manipulated to determine the energy production increase resulting from the proposedMPC controller, operating at optimal conditions dictated by the wind farm optimization algorithm, relative to a traditional greedy controller.59Relative energy production gain corresponds to a 1×10 wind-aligned row wind farm modeled using Aeolus SimWindFarm, and was obtained bymanipulating the Etot,greedy/E∗tot ratio described in Section V.B [83]. This ratio was manipulated to determine the energy production increase resultingfrom the proposed MPC controller, operating at optimal conditions dictated by the wind farm optimization algorithm, relative to a traditional greedycontroller.60Relative wind farm efficiency gain corresponds to a turbulence intensity of 7.5%, a mean wind speed of 8m/s, a controller sampling frequency of0.1Hz, and a turbine spacing of seven rotor diameters, and was obtained from Figure 10 [106].61Relative wind farm efficiency gain corresponds to a turbulence intensity of 21%, a mean wind speed of 10.5m/s, a controller sampling frequencyof 0.01Hz, and a turbine spacing of seven rotor diameters, and was obtained from Figure 11 [106].62Relative wind farm efficiency gain was obtained from discussions in Section III.B [100].63Relative wind farm efficiency gain was obtained by digitizing Figure 4 [175]. The value corresponds to the maximum relative gain in wind farmpower output over the course of the simulation.64Relative wind farm efficiency gain was obtained by digitizing Figure 4 [175]. The value corresponds to the maximum relative loss in wind farm239Table A.3 continued from previous pageRef. Optimization/analysis method Inputs Control model Evaluation modelSimulated windconditionsRelativeefficiency gain[41]Centralized nestedextremum-seeking controlTgen Model-free UTD-WFConst. speedTurbulence7.8%65[76]Centralized model predictivecontrolCT Dynamic LES Dynamic LESConst. speedTurbulenceWind shear15.80%66[138]67Centralized model predictivecontrolCT Dynamic LES Dynamic LESConst. speedTurbulenceWind shear21.19%68[77]69Centralized model predictivecontrolCT Dynamic LES Dynamic LESConst. speedTurbulenceWind shear7%70[176]Lookup table for optimal bladepitch angle settingsβ FarmFlowGoole Fields windfarm-−15.9 to11.1%71power output over the course of the simulation.65Relative wind farm efficiency gain was obtained by from Table 3 [41].66Relative wind farm efficiency gain corresponds to a simulation with no penalization of turbulent dissipation, and was obtained from Table 3 [76].67Additional information is available in a companion publication [136].68Relative wind farm efficiency gain corresponds to the least restrictive C3t0 controller setting, and was obtained by digitizing Figure 3a [138].69This study expands on previous work by considering wind farm entrance effects [76]. Additional information is available in a companionstudy [129].70Relative wind farm efficiency gain was obtained from discussions in Section 3.1 [77].71Range of relative wind farm efficiency gains was obtained by digitizing Figure 12, and corresponds to different wind speeds while the wind wasaligned with the row of turbines [176].240Table A.4: Summary of studies that have assessed the steady-state potential of yaw-based wake redirection.Ref. Optimization/analysis method Inputs Optimization model Evaluation modelSimulated windconditionsRelativeefficiency gain[128]72 Manual optimization γInterpolated wind tunnel testdataInterpolated wind tunneltest data- 8.9%73[11]Distributed alternating directionmethod of multipliers algorithmγ Steady wake model Steady wake model Const. speed 17.7%74[149]Centralized constrainedoptimization by using a linearapproximationγ Steady FLORIS model Steady FLORIS model Const. speed 3.29%75[63]Centralized sequential quadraticprogrammingγ Steady FLORIS model Steady FLORIS model Const. speed 7.7%76[178]Centralized quasi-NewtonalgorithmγModified steady FLORISmodelModified steady FLORISmodelConst. speed 2.85%77[177]Centralized game-theoreticalgorithmγModified steady FLORISmodelModified steady FLORISmodelConst. speed 5.49%7872Additional information is available in a companion publication [46].73Relative wind farm efficiency gain corresponds to a turbine spacing of three rotor diameters, and was obtained from Table 2 in Paper 2 [128].74Relative wind farm efficiency gain corresponds to the proposed distributed algorithm, and was obtained from Table II [11].75Relative wind farm efficiency gain was obtained from Table 1 and corresponds to the stochastic average power production increase obtained fromusing the OUU solution relative to no yaw offsets [149].76Relative wind farm efficiency gain corresponds to the original wind farm layout with yaw optimization, and was obtained from discussions inSection 5.2 [63].77Relative wind farm efficiency gain corresponds to the single-objective optimization of power case (λ = 1), and was obtained from Table 1 [178].78Relative wind farm efficiency gain was obtained from Table 1 and corresponds to the single-objective optimization of power case (γ = 1) for awind direction of 5deg [177].241Table A.4 continued from previous pageRef. Optimization/analysis method Inputs Optimization model Evaluation modelSimulated windconditionsRelativeefficiency gain[65]Centralized gradient-basedsequential quadraticprogrammingx, y, γModified steady FLORISmodelModified steady FLORISmodelConst. speed 5.26%79[130] Numerical simulation γ No wind farm optimizationSTAR-CCM+ (dynamicCFD)Const. speedTurbulenceWind shear2.11%80[39] Numerical simulation γ No wind farm optimization Dynamic LESConst. speedTurbulenceWinds shear13.77%81[58]82 Numerical simulation γ No wind farm optimization SOWFA (dynamic CFD)Const. speedTurbulenceWind shear4.6%83[60] Numerical simulation γ No wind farm optimization SOWFA (dynamic CFD)Const. speedTurbulenceWind shear12.5%84[2] Wind tunnel experimentation γ No wind farm optimization Scaled wind farm - 12%8579Relative wind farm annual energy production (AEP) gain corresponds to the case with both layout and wake steering optimization, and wasobtained from Table II [65].80Relative wind farm efficiency gain was obtained from Figure 7b and discussions in Section III.A [130].81Relative wind farm efficiency gain was obtained by digitizing Figure 8a and corresponds to a yaw offset of 25deg [39].82Additional information is available in companion publications [62, 66].83Relative wind farm efficiency gain was obtained from Figure 4 [58].84Relative wind farm efficiency gain was obtained from Figure 4 [58].85Relative wind farm efficiency gain was obtained from discussions in Section 3.2.4 and Figure 10c, and corresponds to the highest wind farm poweroutput gain observed in response to varying the nacelle yaw angle γ of the upstream turbine [2].242Table A.4 continued from previous pageRef. Optimization/analysis method Inputs Optimization model Evaluation modelSimulated windconditionsRelativeefficiency gain[31] Wind tunnel experimentation γ No optimization Scaled wind farm -25.2% %867.0%87[19] Wind tunnel experimentation γ No optimization Scaled wind farm - 17%88[181] Field experimentation γ No wind farm optimization ECN’s scaled wind farm - Inconclusive89[86] Field experimentation γ Lifting line model [163]1×6 wind farm inAlberta, CA- 7 to 47%9086Relative wind farm efficiency gain corresponds to a lateral downstream wind turbine displacement of −0.45 rotor diameters, and was obtainedfrom Figure 11 [31].87Relative wind farm efficiency gain corresponds to no lateral downstream wind turbine displacement, and was obtained from Figure 11 [31].88Relative wind farm efficiency gain was obtained from Figure 5, and corresponds to the right-most column of colored dots that represent turbineyaw angles [19].89There is no discernible pattern that relates the yaw angles of individual turbines to the wind farm power output; this point is discussed in Section 3.3and is evident in Figure 6 [181].90Range of relative wind farm efficiency gains was obtained from Table 1, and corresponds to different wind speeds and direction deviations fromperfect wind-alignment [86].243Table A.5: Summary of studies that have developed wind farm controllers based on the concept of yaw-based wakeredirection.Ref. Control method Inputs Control model Evaluation modelSimulated windconditionsRelative efficiencygain[147]Centralized cooperativegame-theoretic controla, γ Steady wake model Steady wake model Const. speed 25.51%91[145]Centralized cooperativegame-theoretic controla, γ Steady wake model Steady wake model Const. speed 7.14%92[146]Centralized/distributedcooperative game-theoreticcontrola, γ Gaussian process learning Steady wake model Const. speed 24.5%93[69]Nonlinear model predictivecontrol using extensive gridsearchγ FLORIDyn FLORIDyn Const. speed 0.19%94[150]Predefined optimal yaw angles +PI wind turbine wake steeringcontroller + Gain scheduled PIwind turbine power controllerγ No wind farm controllerDynamic modifiedSimWindFarmConst. speedTurbulence4.5%95[27]Lookup table of optimalset-points + Turbine yawcontrollera, γSteady wake model(similar to FLORISmodel)Dynamic wake model(similar to SimWindFarm)Const. speedVar. directionTurbulence-9691Relative wind farm efficiency gain corresponds to the highest percent gain in wind farm power output across different wind directions, and wasobtained by digitizing Figure 7 [147].92Relative wind farm efficiency gain was obtained from discussions in Section 6.3 [145].93Relative wind farm efficiency gain corresponds to a distributed optimization scheme, and was obtained by digitizing Figure 13c and calculatingthe final rise in mean power efficiency relative to its initial value [146].94Relative wind farm energy production gain was obtained from discussions in Section IV [69].95Relative wind farm efficiency gain was obtained from discussions in Section VI.B [150].96It was not possible to obtain relative efficiency gain data from this study; however, relative energy production gain over three hours of simulation244Table A.5 continued from previous pageRef. Control method Inputs Control model Evaluation modelSimulated windconditionsRelative efficiencygain[72]97Centralized game-theoreticadaptive controlγ Steady FLORIS model SOWFA (dynamic CFD)Const. speedTurbulenceWind shear13.03%98[139]Centralized model predictivecontrolCT , γ˙ Dynamic LES Dynamic LESConst. speedTurbulenceWind shear34.48%99[137]Centralized model predictivecontrolCT , γ˙ Dynamic LES Dynamic LESConst. speedTurbulenceWind shear46%100[148]Centralized cooperativegame-theoretic controlTg, β , γ Gaussian process learning Scaled wind farm - 33.22%101[32]Distributed extremum seekingcontrol + PI wind turbine wakesteering controller + Gainscheduled PI wind turbine powercontrollerγ Model-free Scaled wind farm - 20%102was 2.06% [27].97Additional information is available in companion publications [66, 68].98Relative wind farm efficiency gain was obtained from Table II [72].99Relative wind farm efficiency gain was achieved using combined yaw-based wake redirection and power de-rating wind farm control methods, andcorresponds to turbulent inflow wind. The result was obtained from Figure 12b [139].100Relative wind farm efficiency gain corresponds to the gridded layout with a wide wind turbine spacing of six rotor diameters, and was obtainedfrom Table 1 [137].101Relative wind farm efficiency gain was obtained from Figure 14 [148].102Relative wind farm efficiency gain was obtained by digitizing the right plot in Figure 5 and averaging all data points past 200sec [32].245Table A.5 continued from previous pageRef. Control method Inputs Control model Evaluation modelSimulated windconditionsRelative efficiencygain[59]Lookup table for optimal yawsettings + Turbine yaw offsetcontrollerγ Steady FLORIS modelLongyuan RudongChaojiandai wind farm-5.56%103−12.16 to19.96%104[61]Lookup table for optimal yawsettings + Turbine yaw offsetcontrollerγ Steady FLORIS model1×2 commercial windfarm- −7.9 to 10.2%105103Relative wind farm efficiency gain corresponds to a scenario in which the wind is aligned with turbines C1 and D1, and was obtained by digitizingdata from Figure 6c [59]. This figure was selected for analysis since the turbine spacing between turbines C1 and D1 was seven rotor diameters.104Range of relative wind farm efficiency gains corresponds to the entire range of wind directions presented in Figure 6c [59].105Range of relative wind farm efficiencies was obtained by digitizing Figure 14, and corresponds to different wind directions.246Table A.6: Summary of studies that have assessed the steady-state potential of turbine repositioning.Ref. Optimization/analysis method Inputs Optimization model Evaluation modelSimulated windconditionsRelative efficiencygain[153]Centralized covariance matrixadaption evolutionary strategyx, y Steady Park model FamFlow (steady CFD) Const. speed−11.49 to18.11%106−4.04 to 4.77%107[58]108 Numerical simulation y No optimization SOWFA (dynamic CFD)Const. speedTurbulenceWind shear41.0%109[101]Centralized sequential quadraticprogramminga, γ FOWFSim FOWFSim Const. speed 16.5 to 53.0%110106Range of relative wind farm efficiency gains corresponds to turbine relocation within a suboptimal gridded wind farm layout, and was obtained bydigitizing Figure 12b [153].107Range of relative wind farm efficiency gains corresponds to turbine relocation within an optimized wind farm layout, and was obtained by digitizingFigure 12b [153].108Additional information is available in companion publications [62, 66].109Relative wind farm efficiency gain corresponds to lateral position control, and was obtained from Figure 4 [58].110Range of relative wind farm efficiency gains corresponds to different mooring line lengths, and was obtained from Figure 3 and discussions inSection IV.a [101].247
Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Theoretical foundations for optimal control of floating offshore wind farms
Text