@prefix vivo: .
@prefix edm: .
@prefix ns0: .
@prefix dcterms: .
@prefix skos: .
vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Mechanical Engineering, Department of"@en ;
edm:dataProvider "DSpace"@en ;
ns0:degreeCampus "UBCV"@en ;
dcterms:creator "Homer, Jeffrey R."@en ;
dcterms:issued "2017-02-28T00:00:00"@en, "2015"@en ;
vivo:relatedDegree "Master of Applied Science - MASc"@en ;
ns0:degreeGrantor "University of British Columbia"@en ;
dcterms:description """As offshore wind technology advances, floating wind turbines are becoming larger and moving further offshore, where wind is stronger and more consistent. Despite the increased potential for energy capture, wind turbines in these environments are susceptible to large platform motions, which in turn can lead to fatigue loading and shortened life, as well as harmful power fluctuations. To minimize these ill effects, it is possible to use advanced, multi-objective control schemes to minimize harmful motions, reject disturbances, and maximize power capture. Synthesis of such controllers requires simple but accurate models that reflect all of the pertinent dynamics of the system, while maintaining a reasonably low degree of complexity.
In this thesis, we present a simplified, control-oriented model for floating offshore wind turbines that contains as many as six platform degrees of freedom, and two drivetrain degrees of freedom. The model is derived from first principles and, as such, can be manipulated by its real physical parameters while maintaining accuracy across the highly non-linear operating range of floating wind turbine systems. We validate the proposed model against advanced simulation software FAST, and show that it is extremely accurate at predicting major dynamics of the floating wind turbine system.
Furthermore, the proposed model can be used to generate equilibrium points and linear state-space models at any operating point. Included in the linear model is the wave disturbance matrix, which can be used to accommodate for wave disturbance in advanced control schemes either through disturbance rejection or feedforward techniques. The linear model is compared to other available linear models and shows drastically improved accuracy, due to the presence of the wave disturbance matrix.
Finally, using the linear model, we develop four different controllers of increasing complexity, including a multi-objective PID controller, an LQR controller, a disturbance-rejecting H∞ controller, and a feedforward H∞ controller. We show through simulation that the controllers that use the wave disturbance information reduce harmful motions and regulate power better than those that do not, and reinforce the notion that multi-objective control is necessary for the success of floating offshore wind turbines."""@en ;
edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/54891?expand=metadata"@en ;
skos:note "Physics-Based Control-OrientedModelling for Floating Offshore WindTurbinesbyJeffrey R. HomerB.A.Sc., The University of British Columbia, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2015c© Jeffrey R. Homer 2015AbstractAs offshore wind technology advances, floating wind turbines are becominglarger and moving further offshore, where wind is stronger and more con-sistent. Despite the increased potential for energy capture, wind turbinesin these environments are susceptible to large platform motions, which inturn can lead to fatigue loading and shortened life, as well as harmful powerfluctuations. To minimize these ill effects, it is possible to use advanced,multi-objective control schemes to minimize harmful motions, reject distur-bances, and maximize power capture. Synthesis of such controllers requiressimple but accurate models that reflect all of the pertinent dynamics of thesystem, while maintaining a reasonably low degree of complexity.In this thesis, we present a simplified, control-oriented model for floatingoffshore wind turbines that contains as many as six platform degrees of free-dom, and two drivetrain degrees of freedom. The model is derived from firstprinciples and, as such, can be manipulated by its real physical parameterswhile maintaining accuracy across the highly non-linear operating range offloating wind turbine systems. We validate the proposed model against ad-vanced simulation software FAST, and show that it is extremely accurate atpredicting major dynamics of the floating wind turbine system.Furthermore, the proposed model can be used to generate equilibriumpoints and linear state-space models at any operating point. Included in thelinear model is the wave disturbance matrix, which can be used to accommo-date for wave disturbance in advanced control schemes either through dis-turbance rejection or feedforward techniques. The linear model is comparedto other available linear models and shows drastically improved accuracy,due to the presence of the wave disturbance matrix.Finally, using the linear model, we develop four different controllers ofiiAbstractincreasing complexity, including a multi-objective PID controller, an LQRcontroller, a disturbance-rejecting H∞ controller, and a feedforward H∞controller. We show through simulation that the controllers that use thewave disturbance information reduce harmful motions and regulate powerbetter than those that do not, and reinforce the notion that multi-objectivecontrol is necessary for the success of floating offshore wind turbines.iiiPrefaceThis dissertation is original intellectual property of the author, Jeffrey Homer,under supervision of Dr. Ryozo Nagamune.A version of Chapters 3 and 4 has been published in:• J. R. Homer and R. Nagamune, Control-oriented physics-based modelsfor floating offshore wind turbines, in the Proceedings of 2015 Ameri-can Control Conference, Chicago, IL, USA, July 1–3, pp. 3696–3701.I performed this work independently and wrote all of the manuscript, underguidance of Dr. Ryozo Nagamune.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAbbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . 41.3 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . 41.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Baseline Offshore Wind Turbine . . . . . . . . . . . . . . . . 72.2 Physical Modelling of Floating Offshore Wind Turbines . . . 92.3 Multivariable Control for Reduction of Fatigue Loads . . . . 132.3.1 Multi-Objective Control . . . . . . . . . . . . . . . . 13vTable of Contents2.3.2 Utilization of Wave Disturbance . . . . . . . . . . . . 152.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Non-Linear Modelling . . . . . . . . . . . . . . . . . . . . . . . 183.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Generalized Non-Linear Model . . . . . . . . . . . . . . . . . 183.2.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . 193.2.2 States . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.3 Control Inputs . . . . . . . . . . . . . . . . . . . . . . 233.2.4 Disturbances . . . . . . . . . . . . . . . . . . . . . . . 253.2.5 Equations of Motion . . . . . . . . . . . . . . . . . . . 253.3 Forces and Torques for the Baseline Turbine . . . . . . . . . 273.3.1 Hydrodynamics . . . . . . . . . . . . . . . . . . . . . 283.3.2 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . 343.3.3 Mooring Line . . . . . . . . . . . . . . . . . . . . . . 383.4 Parameter Identification . . . . . . . . . . . . . . . . . . . . 393.4.1 Cp and Ct Curves . . . . . . . . . . . . . . . . . . . . 393.4.2 Added Inertia and Drag Constants . . . . . . . . . . 403.5 Validation of Non-Linear Model . . . . . . . . . . . . . . . . 413.5.1 Perturbation Test . . . . . . . . . . . . . . . . . . . . 423.5.2 Gust Test . . . . . . . . . . . . . . . . . . . . . . . . . 433.5.3 Regular Wave Test . . . . . . . . . . . . . . . . . . . 453.5.4 Realistic Open-Loop Test . . . . . . . . . . . . . . . . 473.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Derivation of Linear Model and Equilibrium Points . . . . . 594.2.1 Selection of an Operating Point . . . . . . . . . . . . 604.2.2 Derivation of the State-Space Model . . . . . . . . . . 624.3 Manipulation of the Linear Model . . . . . . . . . . . . . . . 634.3.1 Quadratic Drag . . . . . . . . . . . . . . . . . . . . . 634.4 Validation of the Linear Model . . . . . . . . . . . . . . . . . 64viTable of Contents4.4.1 Linearization Point - NLM . . . . . . . . . . . . . . . 644.4.2 Linearization Point - FAST . . . . . . . . . . . . . . . 654.4.3 Open-Loop Test Results . . . . . . . . . . . . . . . . 664.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . 705.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.2 Test Conditions and Equilibrium Point . . . . . . . . . . . . 705.2.1 Disturbance . . . . . . . . . . . . . . . . . . . . . . . 715.2.2 Linearization Point . . . . . . . . . . . . . . . . . . . 725.2.3 Control Objectives . . . . . . . . . . . . . . . . . . . 735.2.4 Performance Metrics . . . . . . . . . . . . . . . . . . 745.3 Design of the PID Controller . . . . . . . . . . . . . . . . . . 745.4 Design of the LQR Controller with Integrator . . . . . . . . 765.4.1 LQR with an Integrator . . . . . . . . . . . . . . . . . 775.4.2 Reduced-order Observer . . . . . . . . . . . . . . . . 795.5 Design of the H∞ with Integrator . . . . . . . . . . . . . . . 805.5.1 Integrator . . . . . . . . . . . . . . . . . . . . . . . . 825.5.2 Weighting Selection . . . . . . . . . . . . . . . . . . . 825.5.3 Final Structure . . . . . . . . . . . . . . . . . . . . . 845.6 Design of the H∞ Controller with Feedforward Terms . . . . 865.7 Weighting Selection . . . . . . . . . . . . . . . . . . . . . . . 865.7.1 Identification of the Feedforward Terms . . . . . . . . 865.7.2 Final Structure . . . . . . . . . . . . . . . . . . . . . 885.8 Controller Comparison Results . . . . . . . . . . . . . . . . . 885.8.1 PID vs Baseline GSPI . . . . . . . . . . . . . . . . . . 885.8.2 LQR vs PID . . . . . . . . . . . . . . . . . . . . . . . 905.8.3 H∞ vs LQR . . . . . . . . . . . . . . . . . . . . . . . 915.8.4 H∞ with Feedforward vs H∞ . . . . . . . . . . . . . . 925.8.5 Overall Controller Comparison . . . . . . . . . . . . . 925.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94viiTable of Contents6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.1 Summary Remarks . . . . . . . . . . . . . . . . . . . . . . . 966.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 976.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101AppendicesA Physical Constants for the 5MW Sample WTS . . . . . . . 107B Controller Parameters . . . . . . . . . . . . . . . . . . . . . . . 111B.1 PID Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 111B.2 LQR Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 111B.3 Parameters for LQR Observer and H∞ Controllers . . . . . . 112viiiList of Tables2.1 Complexity and Functionality Comparison of Models. . . . . 122.2 Summary of Control Contributions of Floating Offshore WindTurbines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1 Perturbation Response Results . . . . . . . . . . . . . . . . . 423.2 Aerodynamic Response Results . . . . . . . . . . . . . . . . . 453.3 Realistic Open-Loop Test Results for Non-Linear Models . . . 554.1 Realistic Open-Loop Test for Linear Models. . . . . . . . . . 665.1 Performance Results from Controller Validation Tests. . . . . 95A.1 General Properties . . . . . . . . . . . . . . . . . . . . . . . . 107A.2 Aerodynamic Properties . . . . . . . . . . . . . . . . . . . . . 107A.3 Mooring Line Properties . . . . . . . . . . . . . . . . . . . . . 108A.4 Hydrodynamic Properties . . . . . . . . . . . . . . . . . . . . 109A.5 Hydrodynamic Properties (cont’d) . . . . . . . . . . . . . . . 110B.1 PID Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111ixList of Figures1.1 Evolution of Offshore Wind Turbines (Source: Principle Power) 22.1 5MW Baseline Wind Turbine on the Semi-Submersible Plat-form (Taken from [1]). . . . . . . . . . . . . . . . . . . . . . . 83.1 Coordinate Frames for the General Non-Linear Model. . . . . 203.2 Position and Orientation States. . . . . . . . . . . . . . . . . 213.3 Rotor and Generator Angle States. . . . . . . . . . . . . . . . 223.4 Blade Pitch Angle and Generator Torque Inputs. . . . . . . . 243.5 Yaw Angle Control Input. . . . . . . . . . . . . . . . . . . . . 243.6 Overall Force Diagram of the Non-Linear Model. . . . . . . . 283.7 Separated Cylinder Segments. . . . . . . . . . . . . . . . . . . 293.8 Buoyancy Force Diagram. . . . . . . . . . . . . . . . . . . . . 293.9 Drag and Inertial Force Diagram. . . . . . . . . . . . . . . . . 333.10 Aerodynamic Force Diagram. . . . . . . . . . . . . . . . . . . 343.11 Rotor-Generator Shaft Torque Diagram. . . . . . . . . . . . . 363.12 2-D Catenary Line Model . . . . . . . . . . . . . . . . . . . . 383.13 Simulation Conditions for Obtaining Cp and Ct Surfaces. . . 403.14 Cp and Ct Surfaces. . . . . . . . . . . . . . . . . . . . . . . . 413.15 Perturbation Test Results. . . . . . . . . . . . . . . . . . . . . 433.16 Gust Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.17 Time Response Comparison for the Regular Waves Test. . . . 473.18 Overview of the Realistic Open-Loop Test Wind and WaveDisturbances. . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.19 Wind and Wave Profiles for Realistic Open-Loop Test. . . . . 49xList of Figures3.20 Pierson-Moskowitz Energy Density Spectrum for Given Cri-terion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.21 Nacelle Yaw Angle Trajectory. . . . . . . . . . . . . . . . . . 513.22 Blade Pitch Angle Trajectory for Realistic Open-Loop Test. . 523.23 Generator Torque Trajectory. . . . . . . . . . . . . . . . . . . 533.24 Realistic open-loop Test Results. . . . . . . . . . . . . . . . . 543.25 Frequency Response for Translational States. . . . . . . . . . 553.26 Frequency Response for Rotational States. . . . . . . . . . . . 563.27 Frequency Response for Shaft States. . . . . . . . . . . . . . . 574.1 Realistic Open-Loop Test Results for the Linear Models. . . . 675.1 Construction of a Wave Profile from the Pierson-MoskowitzSpectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2 PID Block Diagram. . . . . . . . . . . . . . . . . . . . . . . . 755.3 LQR Block Diagram. . . . . . . . . . . . . . . . . . . . . . . . 775.4 Reduced Row Observer Block Diagram. . . . . . . . . . . . . 805.5 H∞ Block Diagram. . . . . . . . . . . . . . . . . . . . . . . . 815.6 H∞ Integrator Block Diagram. . . . . . . . . . . . . . . . . . 835.7 Closed-loop Transfer Function Between Wave Accelerationand Platform Motion. . . . . . . . . . . . . . . . . . . . . . . 855.8 GSPI vs MOPID . . . . . . . . . . . . . . . . . . . . . . . . . 895.9 MOPID vs LQR . . . . . . . . . . . . . . . . . . . . . . . . . 905.10 LQR vs H∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.11 H∞ vs H∞ FF . . . . . . . . . . . . . . . . . . . . . . . . . . 935.12 Overall Controller Comparison . . . . . . . . . . . . . . . . . 94xiNomenclatureGeneral Non-Linear Model Variablesf The general non-linear model functionx The state vector for the proposed non-linear modelu The input vector for the proposed non-linear modelv The wind disturbance vector for the proposed non-linear modelw The wave disturbance vector for the proposed non-linear modelF0 The world frame, also can be written without superscriptFb The body coordinate frameFn The nacelle coordinate frameFs The shaft coordinate frameFr The rotor coordinate frameFg The generator coordinate frameeˆi The unit vector about axis i~xg The displacement vector, also the vector between the origins ofthe world and body frame, in the world coordinate systemxg The first component of ~xgyg The second component of ~xgzg The third component of ~xgR¯The rotation matrix that represents the rotation from the worldframe to the body frame~θ The orientation quasi-vector whose components make up the ro-tation matrix R¯θx The first component of ~θθy The second component of ~θθz The third component of ~θθr The azimuth angle of the rotor, also the rotation between theshaft frame and the rotor frameωr The rotor speedθg The azimuth angle of the generator, also the rotation betweenthe shaft frame and the generator framexiiNomenclatureGeneral Non-Linear Model Variables (cont’d)ωg The generator speedβ The collective blade pitch angleTg The generator torqueγ The yaw angle, also the rotation between the body frame andnacelle framevx The first component of the wind disturbance vectorvy The second component of the wind disturbance vectorvz The third component of the wind disturbance vector~wv,i The wave disturbance velocity vector at column i~wa,i The wave disturbance acceleration vector at column iwp,i The wave disturbance dynamic pressure at column iwh,i The wave disturbance height at column i~fF The force model for the non-linear model function~fT The torque model for the non-linear model functionfQ The shaft model for the non-linear model function~FA The total aerodynamic force~FB The total buoyancy and gravity force~FC The total catenary line force~FD The total drag forceTr The aerodynamic rotor torque~Hr The angular momentum vector of the rotor about the rotor axis~Tgyro The gyroscopic torque that the rotor induces on the WTSR¯x(a) A rotation of angle a about the x axisR¯ y(a) A rotation of angle a about the y axisR¯ z(a) A rotation of angle a about the z axisGeneral Constants for the Sample WTSg The gravitational constantmg The physical mass of the wind turbine system~mba The hydrodynamic added mass vectorI¯bg The moment of inertia of the wind turbine system, given in thebody frameJr The rotor inertiaJg The generator inertiaxiiiNomenclatureAerodynamic Constants for the Sample WTSρ The air densityAr The swept area of the rotorCt The thrust coefficientCp The power coefficientλ The tip-speed ratio, given by the speed of the blades at the tipdivided by the wind speed~vn The wind speed projected onto the rotor axisθtilt The tilt angle of the rotor shaft with respect to eˆb2~rbgt The displacement vector between the center of gravity of theWTS and the center of thrust, given in the body frameP The aerodynamic powerNGR The drivetrain gear ratiok The drivetrain stiffness on the rotor sideb The drivetrain damping constant on the rotor sideHydrodynamic Constants for the Sample WTSρw The water densityAi The cross-sectional area of cylinder ili The length of cylinder il0,i The submerged length of uppermost cylinder i in the undisplacedstate of the wind turbine system~rbgi The distance from the center of gravity of the wind turbine sys-tem to the center of volume of cylinder i, in the body framecoordinate system~rbgi,0 The distance from the center of gravity of the undisplaced windturbine system to the center of submerged volume of cylinder i,in the body frame coordinate systemKd,i The drag constant from the Morison equation for cylinder iKa,i The inertial constant from the Morison equation for cylinder iKdh,i The heave drag constant from the Morison equation for cylinderiKah,i The heave inertial constant from the Morison equation for cylin-der ixivNomenclatureMooring Line Constants for the Sample WTSL The mooring line lengthx The horizontal position of the attachment point relative to theanchor of the 2-D mooring line representationy The vertical position in the attachment point relative to the an-chor in the 2-D mooring line representationFx The horizontal force of the 2-D mooring line representation atthe attachment pointFy The vertical force of the 2-D mooring line representation at theattachment pointw The equivalent weight of the mooring line in water~xa,i The position of the anchor point for mooring line i~xt,i The displacement vector between the anchor point and attach-ment point of mooring line i~rbgci The distance between the center of gravity of the WTS and theattachment point of mooring line i, in the body frameWave Equation Parametersh The height of a regular wave. For irregular waves, the height ofeach monochromatic frequency is distinguished by the subscriptiA The amplitude of a regular wave. For irregular waves, the am-plitude of each monochromatic frequency is distinguished by thesubscript iω The temporal frequency of a regular wave. For irregular waves,each of the frequencies is distinguished by the subscript iφ The phase angle of a regular wave. For irregular waves, the phaseangle of each monochromatic frequency is distinguished by thesubscript iζ The summation of the temporal, spatial and phase angle. Forirregular waves, this summation is distinguished by the subscripti for each monochromatic frequencyα The angle indicating the wave direction about the eˆe axis~xw The displacement vector indicating the location in the eˆ1-eˆ2 planethat we would like the wave informationz The vertical distance from the mean water level to the locationwe would like the wave information~v The total wave velocity vector~a The total wave acceleration vectorPd The total dynamic pressure of the wavexvNomenclatureWave Equation Parameters (cont’d)S(ω) The Pierson-Moskowitz spectrum magnitude at frequency ωsβ, sα The constants used in the Pierson-Moskowitz spectrum fitUx The mean wind velocity at an elevation x mhx The height of x mκ The wind power exponent for the wind power lawLinearization Parametersp The operating point for linearizationxop The state vector at the operating point puop The control input vector at the operating point pvop The wind disturbance vector at the operating point pwop The wave disturbance vector at the operating point pA¯|p The linear state matrix for the linearization at pB¯|p The linear input matrix for the linearization at pB¯v|p The linear wind disturbance matrix for the linearization at pB¯w|p The linear wave disturbance matrix for the linearization at pC¯|p The linear output state matrix for the linearization at pD¯|p The linear input matrix for the outputs of a linearization at pD¯ v|p The linear output wind disturbance matrix for the linearizationat pD¯w|p The linear output wave disturbance matrix for the linearizationat pnx The number of states at the equilibrium pointnu The number of inputs at the equilibrium pointnv The number of wind disturbances at the equilibrium pointnw The number of wave disturbances at the equilibrium pointpu The parameters to-be-determined by the equilibrium point solverT¯pThe transformation from the operating point p to pufu The parts of the non-linear function necessary that describe theto-be-determined parameters onlyT¯fThe transformation from the non-linear function f to fugo The non-linear output functionxviNomenclatureController Design ParametersGi The PID system iJ¯The input weighting matrix for LQR designQ¯The combined state-output weighting function for LQR designQ¯ xThe state weighting function for LQR designQ¯ yThe output weighting function for LQR designz The combined state-output vector for LQR designP¯The observer transformation matrixB¯yThe observer output B matrixB¯uThe observer input B matrixL¯The observer gainA¯ obsThe observer state matrixA˜¯ 11The first block of the transformed state matrix used for observerdesignA˜¯ 12The second block of the transformed state matrix used for ob-server designA˜¯ 21The third block of the transformed state matrix used for observerdesignA˜¯ 22The fourth block of the transformed state matrix used for ob-server designB˜¯1The first block of the transformed input matrix used for observerdesignB˜¯2The second block of the transformed input matrix used for ob-server designxaug The augmented state vector for H∞ designe The output vector for H∞ designe˜ The output minimization vector for H∞ designu˜ The input minimization vector for H∞ designw˜ The weighted wave disturbance vector for H∞ designuk The controller output for H∞ designP¯augThe augmented plant for H∞ designuP The proportional control input for H∞ designuI The integral control input for H∞ designxI The integral state vector for H∞ designxw The wave disturbance state vector for H∞ designxviiNomenclatureController Parameters (cont’d)W¯ uThe input weighting function for H∞ designW¯ eThe output weighting function for H∞ designW¯ vThe wind disturbance weighting function for H∞ designW¯ wThe wave disturbance weighting function for H∞ designW¯ w,azThe transfer function used to reject wave acceleration in theheave directionA¯W,wA part of the state-space system representing the wave distur-bance weighting functionB¯W,wA part of the state-space system representing the wave distur-bance weighting functionC¯W,wA part of the state-space system representing the wave distur-bance weighting functionD¯W,wA part of the state-space system representing the wave distur-bance weighting functionPz,i The measured pressure at the bottom of heave plate iC¯P,wThe linear output matrix describing the effect of states on theheave pressureD¯P,wThe linear output matrix describing the effect of control inputson the heave pressurevr The measured relative wind velocity at the center of thrust iC¯vrThe linear output matrix describing the effect of states on therelative wind velocityD¯ vrThe linear output matrix describing the effect of control inputson the relative wind velocityD¯ vr,vThe linear output matrix describing the effect of wind distur-bance on the relative wind velocityxviiiList of AbbreviationsDAC Disturbance accommodating controllerDOF Degree of freedomFAST Fatigue, aerodynamics, structures and turbulence (simulationsoftware)GSPI Baseline gain-scheduled PI controllerH∞ H-Infinity controlIBP Individual blade pitchLQR Linear quadratic regulatorMBC Multi-blade coordinate transformationMOPID Proposed multi-objective PID controllerPID Proportional-Integral-Derivative (control)RMS Root-mean-squareTSR Tip-speed-ratio, or the ratio of the velocity of the blade tip tothe mean wind speedWTS The wind turbine system (wind turbine and platform)xixAcknowledgementsI would like to thank my supervisor Prof. Ryozo Nagamune for his incrediblesupport throughout the duration of my degree. The amount of hours hespent teaching me, discussing problems and solutions, and reviewing mywork, are far and beyond what I could have asked for.I am very grateful to my committee members Prof. Jon Mikkelsen andProf. Juri Jatskevich for taking the time to review my thesis, and for pro-viding me with invaluable feedback.I would also like to thank Dr. Jason Jonkman and Dr. Alan Wright fromthe National Renewable Energy Laboratory in the United States for answer-ing all my questions related to wind turbine simulation software FAST, andproviding me with important information on its usage.Finally, this thesis was supported by the National Sciences and Engi-neering Research Council of Canada and the Institute for Computing, In-formation and Cognitive Systems at the University of British Columbia.xxDedicationTo my friends and familyxxiChapter 1Introduction1.1 MotivationWind power has grown enormously over recent years. Over the 2007-2013period, the United States saw an average 7.1 GW increase in wind powercapacity per year. If the same trend were to continue over the next tenyears, approximately 40% of the entire country’s electricity generation couldbe supplied via wind power [2]. As wind turbine technology advances, re-searchers are beginning to look toward offshore environments where wind isstronger and more consistent than wind on land. By the end of 2012, 1,662offshore wind turbines had been constructed across 10 European countriesand generated enough electricity to power five million houses [3]. A furthertrend for offshore wind turbines is that as the industry evolves, larger windturbines are being constructed further from the coast in deeper waters, whereenergy resources are plentiful and floating platforms are necessary. For ex-ample, the average water depth for offshore wind farms went from 22m in2012 to 215m for projects announced for 2013 and beyond. Likewise, averagedistance from shore went from 29km to 200km [3].An important factor, particularly for deep water floating offshore windturbines, is the design of the base, or floating platform. Typically, a bal-ance must be made between capital cost and stability. An expensive baseis undesirable from an economic perspective, however a low-cost platformthat is sensitive to wind and wave effects can allow harmful fatigue loads tosignificantly reduce the operating life of the wind turbine. Furthermore, thedynamic response of a particular platform will be affected not only by itsphysical characteristics, but also by its control scheme [3]. Generally, the op-timal platform design is a tightly coupled problem involving the water depth,11.1. Motivationthe site’s typical wind and wave characteristics, the physical characteristicsof the mounted wind turbine, and the control scheme used. Figure 1.1 showsseveral offshore wind turbine concepts for various water depths.Figure 1.1: Evolution of Offshore Wind Turbines (Source: Principle Power)It is possible to reduce the high capital costs associated with the floatingplatform via intelligent, multi-objective control schemes. That is, insteadof simply maximizing power capture, the many available actuators can beutilized to achieve secondary objectives of regulating platform motions anddrivetrain loads. Many researchers have shown that such advanced controlcan significantly increase fatigue life of deep water floating wind turbinesystems at a small deficit or sometimes even positive effect to capturedpower. For example, the multi-objective LQR controller used in [4] was ableto reduce platform pitch motions by 18%, compared to a simple baseline PIgain-scheduled controller, with a negligible effect to power capture. Since thefloating wind turbine is subject to large wind and wave disturbances, it is nosurprise that there is huge potential in advanced controllers that predict andaccommodate for such disturbances. In particular, one study showed thatusing wind disturbance estimation, a 38% reduction in tower side-to-sidefatigue loading could be made while reducing power error by 44%, relative21.1. Motivationto a simple baseline controller [5]. Controllers that utilize wave disturbanceinformation are also very promising, but less common in research due tothe difficulty in acquiring a linear wave disturbance matrix with existingmodelling methods. In general, control of floating offshore wind turbinesis trending toward multi-objective, disturbance-accommodating control; thesuccess of which depends on effective mathematical models.Another important consideration for both onshore and offshore floatingwind turbines is wind farm optimization. Because wind changes in bothmagnitude and direction, and because wind turbines that are aligned in thedirection of the wind will interfere with the power capture of one another, itis important to place the wind turbines such that cost of energy is minimized[6]. The layout problem is especially complicated in the case of offshore windturbines because, while available wind energy increases with water depthand turbine spacing, so do the cost of the support platform required andthe cost of electrical interconnection [7]. Furthermore, while in operation,it is inevitable that some turbines will be operating in the wake of otherturbines, resulting in a decrease in power capture of the farm. It is possible tominimize this power loss by developing a combined feedback control strategyfor all the wind turbines that are aligned in each other’s wake. Generally,highly simplified models of the individual turbines are used to estimate thetotal power capture of the farm and from them design control strategies,such as in [8] and [9]. In both of these models, relative motion of the windturbines and feedback control of nacelle yaw have been neglected. With aslightly more advanced model, both the relative motion of the turbines andnacelle yaw control may be utilized to further optimize wind farm control.Mathematical modelling of key dynamics represents an additional chal-lenge for the success in analysis and control of floating offshore wind turbines.Because of the mobility of the floating platform, and the flexibility of such amassive structure, a high fidelity model requires a massive amount of states.On the more complex end of the modelling spectrum, there exists a high fi-delity simulator built by the National Renewable Energy Laboratory calledFAST (Fatigue, Aerodynamics, Structures, Turbulence). FAST, which isdescribed in [10], has 44 states in its full form, which is far too many for31.2. Problem Statementmodel-based controller design. On the other hand, simpler control-orientedmodels have been shown to accurately represent the important dynamics ofthe system and to be suitable for controller design, such as the 2-D model in[11]. In this way, there is a need for a control-oriented model with adequatedegrees of freedom to represent all the important dynamics of the floatingwind turbine system, but simple enough for controller design.1.2 Problem StatementDue to the size and mobility of floating platforms, effective control schemesfor offshore wind turbines must incorporate both power production and fa-tigue minimization as control objectives to maximize the cost effectivenessof offshore wind energy. The construction of controllers depends on effectivecontrol-oriented models that contain all the relevant dynamics of the sys-tem, including the effects of wind and wave disturbances, while containinga minimal number of states. Existing models either lack the wave distur-bance relationship [10], or the number of degrees of freedom to adequatelypredict the highly coupled motions of the non-linear system [11]. In thisthesis we propose a simple control-oriented, physics-based model that sat-isfies the above requirements, and using it we design several controllers toprove the potential improvements that are possible for both power captureand reduction of fatigue loads.1.3 Research ObjectivesThe objectives of this thesis can be summarized as follows• Create and validate a simplified control-oriented model that– is modular and easily adjustable for various systems,– includes effects of wind and wave disturbance,– can be used to generate equilibrium points for linearization, and– can be used to create linear state-space models including the windand wave disturbance information.41.4. Thesis Outline• Use linear state-space models to construct advanced controllers that– utilize wind and wave disturbance information, and– perform better than existing controllers due to consideration ofdisturbance in their design.1.4 Thesis OutlineThe thesis has been organized as follows. In Chapter 2, a literature reviewof existing research relevant to this thesis is conducted. First, we describea baseline offshore wind turbine that has been used extensively in research,and that will be used for validation purposes in this thesis. Next, existingmodelling methods for floating offshore wind turbines are described and com-pared. Finally, advanced control techniques based on these mathematicalmodels are explored. Following this preliminary chapter, in Chapter 3, wepropose, describe, and validate our own simplified, physics-based non-linearmodel. This proposed model is given in both a general form, which can eas-ily be extended to different floating offshore wind turbine systems, and alsomore specifically for the sample wind turbine. For validation we performmultiple tests that cover a wide range of possible operating conditions, andcompare the non-linear model results with those from the high-fidelity sim-ulator FAST. Next, Chapter 4 presents how we obtain a linear model fromthe proposed non-linear model. This chapter also covers the generation ofequilibrium points, where linearization is possible. Finally, we validate thelinear model with an all-encompassing realistic open-loop test, comparedto both the non-linear model and FAST. In Chapter 5, we design severaladvanced controllers based on the linear model in order to verify severalfacts. First, we confirm the importance of multi-objective control in floatingoffshore wind turbine systems. After this confirmation, we show the ad-vantages of considering wind and wave disturbance in the controller designprocess, both in the form of disturbance rejection, and feed-forward control.Finally, we summarize and conclude the thesis in Chapter 6, and revisit theresearch objectives. We will also make some concluding remarks and outline51.4. Thesis Outlinefuture work.A preliminary version of this thesis has been published in [12].6Chapter 2Literature ReviewThis chapter provides a summary of the modelling and control techniquesemployed by other researchers on horizontal-axis floating wind turbines. Anintroduction to the fundamentals of simple wind turbine modelling and con-trol strategies can be found in [13]. Offshore wind turbine control presentsfurther challenges, which were introduced and explained in [14]. We be-gin the literature review by detailing a baseline wind turbine that is usedcopiously in research, as well as in this thesis. In the next section, twoexisting mathematical models for floating wind turbines are described withtheir advantages and disadvantages. One such model is a highly complexsimulation-oriented model, and the other, a simplified, control-oriented 2-D model. Finally, following the model descriptions, a thorough survey ofexisting wind turbine control strategies is carried out.2.1 Baseline Offshore Wind TurbineThe baseline offshore wind turbine is a virtual wind turbine created by theNational Renewable Energy Laboratory (NREL) in the United States. Itis by far the most commonly used wind turbine by control researchers forrealistic simulation purposes [4, 5, 15–17], as it is publicly available in opensource simulation software FAST [10], and more importantly it provides acommon ground for comparison purposes. The combined wind turbine sys-tem, which we call the WTS throughout this thesis, consists of both thewind turbine and the floating platform upon which the turbine sits. In thisthesis, whenever validations are conducted for proposed modelling and con-trol methods, we consider the semi-submersible platform specifically, whichis taken from [1] and shown with the attached wind turbine in Figure 2.1.72.1. Baseline Offshore Wind TurbineFigure 2.1: 5MW Baseline Wind Turbine on the Semi-Submersible Platform(Taken from [1]).The wind turbine is extensively described in [18], including its struc-tural, geometric, aerodynamic, and drivetrain properties. It also includes abaseline PI-gain-scheduling controller for collective blade pitch control aboverated wind speed that is commonly employed in literature as a comparisonobject for other controllers. The tower is 87.7m tall, with a blade swept di-ameter of 126m. Its generator is rated for 5MW at 1173.7rpm, which makesthe rotor side rated speed 12.1rpm, after a 97 times gear reduction. Thebaseline wind turbine was initially designed as an onshore wind turbine, butit was eventually extended for offshore use, with most properties remainingthe same. For offshore applications, various baseline platforms have beendesigned and detailed by NREL. These platforms include the barge [19],spar-buoy [20], tension-leg platform [21], and finally the semi-submersibleplatforms [1]. Even though the modelling and control results in this thesisare applicable to all these platforms, wherever we show the applicability, wewill focus on the semi-submersible platform due to its recently increasing82.2. Physical Modelling of Floating Offshore Wind Turbinespopularity.The semi-submersible platform that was presented in [1] consists of threeouter buoyant columns arranged in an equilateral triangular fashion, in ad-dition to the central column that supports the tower. The platform has amass of 13.5 ktonnes, with a center of mass at 13.46m below sea level. Itsstability comes from a combination of its low center of gravity and largelyspaced out columns. Compared to other platforms, the experienced fatigueloading for given wind and wave disturbances is very low, with the exceptionof tower loading [22]. Since tower loads can be reduced by limiting platformmotions, the semi-submersible platform is a good test subject for platformmotion control. From an economic perspective, the levelized cost of energyhas been compared between various types of floating and bottom-fixed windturbines [23]. In this study, all aspects of the manufacturing, installation andmaintenance were considered for the platforms, assuming the same averagewind speed and years of operation for all concepts. The semi-submersiblewas found to have very low installation and mooring line costs, compara-ble grid, development and turbine costs, but high structural costs. Overall,the semi-submersible was deemed slightly more expensive than its competi-tors, however it also experiences minimal increase in cost with increasingdepth. Since wind power is more available further offshore, at increasingwater depths, it is expected that the cost for the semi-submersible platformin this analysis is overestimated for deep-water locations.The modelling for floating wind turbines plays an essential role in subse-quent controller design. Existing modelling techniques that can be appliedto the baseline floating offshore wind turbines, including the baseline semi-submersible wind turbine, are explained in the following section.2.2 Physical Modelling of Floating OffshoreWind TurbinesMulti-objective control is a necessity for floating offshore wind turbines dueto the significant structural fatigue damage that occurs when using simple92.2. Physical Modelling of Floating Offshore Wind Turbinespower-optimization control strategies [16]. This presents a need for math-ematical models of the floating wind turbine system that accurately reflectall the important dynamics of the complex system. In particular, the math-ematical model must accurately predict the power extraction from wind, aswell as the major platform motions which cause the harmful tower bendingloads [5]. Several general and publicly available mathematical models existfor floating wind turbine systems, including the highly-complex but accuratesimulator FAST developed by Jonkman and Buhl [10], and the simplified,control-oriented model proposed by Betti et al. [11]. Other researchers haveused their own mathematical models for floating wind turbine systems, asin [24] and [25]. However, the complex model in [24] is not open-source, andthe model in [25] does not contain platform motion beyond a 1-D horizontaldisplacement of the nacelle. The latter model was also not validated eitherexperimentally or in simulation. Therefore, the review of existing modelswill focus on FAST, as well as the model in [11] which we call Betti modelin this thesis.FAST is an open-source, highly complex non-linear simulator for onshoreand offshore wind turbines, developed at NREL in the United States [10]. Itis based on a mixture of both fundamental physical laws and empirical rela-tionships, and has been validated by Germanischer Lloyd, the world’s largestrenewable energy consultancy, for several onshore wind turbines. FAST cal-culates the aerodynamic loads on the wind turbine using full field in-flowdata, solving for the the rotor-wake effects and blade-element forces alongeach blade. Hydrodynamic loads include hydrostatic, radiation, diffraction,and viscous forces, calculated over the entire platform structure, includingsupport beams. All actuators and electronic devices associated with thecontrol are also modelled. Finally, the structural dynamics are modelledusing first or second order flexibility modes for all three blades, about twoaxes, and the tower, about two axes. Although the non-linear FAST modelis much too complex for controller design, it is capable of generating linearstate-space models numerically at static equilibrium points. These gener-ated state-space models have been used quite readily in wind turbine controlresearch, including in [26], [27], and [5]. As we will see, disturbance accom-102.2. Physical Modelling of Floating Offshore Wind Turbinesmodation has been very important in the control of floating wind turbines,and although FAST is able to produce a linear wind disturbance matrix, todate, it is not able to produce a wave disturbance matrix. This means thatthe FAST model cannot be used for advanced control design techniques thattake into account rejection of wave disturbance. Furthermore, the complex-ity of building a FAST project may not be available for the control engineer,as it requires a highly detailed knowledge of the geometric, inertial, elastic,aerodynamic and hydrodynamic properties of the floating wind turbine. Onthe other hand, the Betti model, for instance, is able to accurately reproducethe same major platform motions as FAST, while requiring far fewer inputparameters.The Betti model (BM) is a simplified 2-D control-oriented model of thefloating wind turbine. In contrast to FAST which has 44 states in full form,BM only has 7. The simplicity of the model is achieved by constructing anequivalent 2-D wind turbine and assuming all wind and wave disturbancesact only in the 2-D plane. The authors further simplified the non-linear sys-tem by using lumped forces and assuming no flexibility in the wind turbinestructure. Despite being far less complex, it was shown in [11] and [28] thatsuch large simplifications had very little consequence to the accuracy of themajor platform motions and power capture model under 2-D disturbancesfor their tests on the 5MW baseline wind turbine on a tension-leg type plat-form. The authors of this paper validated their model by running identicalopen- and closed-loop tests in the simulator FAST, and in their own model.To measure the similarity, they calculated the mean and standard deviationsof the time series data, for both FAST and BM. With all degrees of freedomactivated in FAST, the differences between the performance metrics for bothmodels were within 10%. Like FAST, BM can be used to generate linearstate-space models at specific equilibrium points, with the unique advantageof being able to generate the wave disturbance matrix. This enabled the au-thors to design an H∞ controller for the entirety of the high speed windregion, called Region 3, while rejecting both wind and wave disturbances.Unfortunately, despite the accuracy of the 2-D motions, the platform mo-tions out-of-plane are also very important for fatigue damage reduction [5].112.2. Physical Modelling of Floating Offshore Wind TurbinesIn fact, in [4], it was shown that 2-D platform pitch control could actuallyinduce undesired platform roll motions.In summary, in this section, we have reviewed two existing models forfloating offshore wind turbines. FAST, which was designed as a high-fidelitysimulation tool, is capable of producing linear state-space models for con-troller design, but falls short in that it cannot produce the wave disturbancematrix. It is also extremely complex and requires a high-level knowledge ofthe WTS. BM addresses many of the issues with the FAST model by sim-plifying the equations of motion and relying on accurate estimation of 2-Dplatform motions to mitigate structural fatigue loading. Unfortunately the2-D limitation is not realistic because of coupling of motion in and out of theassumed 2-D plane. Table 2.1 summarizes the differences between FAST,BM and a control-oriented model proposed in this thesis. The proposedmodel covers the drawbacks of the FAST and BM models, by predicting3-D platform motions, and by including the wave disturbance matrix fordisturbance rejection and feedforward control.FAST BM ProposedStates 44 8 15Flexibility Y N NPlatform Motions 3-D 2-D only 3-DLinear model with wind Y Y Ydisturbance matrixLinear model with wave N Y Ydisturbance matrixTable 2.1: Complexity and Functionality Comparison of Models.In the next section, we consider the existing control research that hasbeen done using the existing models.122.3. Multivariable Control for Reduction of Fatigue Loads2.3 Multivariable Control for Reduction ofFatigue LoadsThe main control objective for all wind turbines, both on- and offshore, is tomaximize the cost-effectiveness of wind energy. For onshore wind turbines,this typically translates to maximizing the power capture within the capacityof the on-board generator. A good example of a controller with such a controlobjective is described in [18]. The designed PI gain-scheduled controllermaximizes power capture using blade pitch angle actuation in the operatingregion where extracted power is less than the maximum capacity of thegenerator (Region 2). Above the transitional wind speed (Region 3), wherethe potential power extraction exceeds the generator limit, the controllersimply regulates power at the maximum capacity. On the other hand, forfloating offshore wind turbines, the same control strategies are not optimaldue to the fatigue life impact of uncontrolled platform motions. It wasshown in [16] that traditional onshore control strategies can actually lead tonegative damping of platform motions, exacerbating the lifetime reductionof floating wind turbines due to wind and wave disturbances. Therefore,multivariable control strategies which target both power maximization andplatform motion reduction, as in [4, 5, 11, 15, 17, 24–27, 29–31], are necessaryfor the success of floating offshore wind turbines. This subsection reviewsthe existing advanced control techniques for dealing with this multi-objectiveproblem.2.3.1 Multi-Objective ControlIn [4], the authors developed a full-state-feedback controller for Region 3operation of the 5MW baseline wind turbine on a barge platform, usingthe LQR method. The control objectives for this controller were to regulateboth rotor speed and platform pitch. The linear model required for controllersynthesis was created using FAST’s linearization feature, and was chosen tohave four states including rotor speed, platform pitch and their respectivetime derivatives. By tuning the Q and R gains, specifically to put heavy132.3. Multivariable Control for Reduction of Fatigue Loadsweight on the platform pitch speed, the authors were able to achieve a21% reduction in rotor speed root-mean-square error, an 18% reductionin platform pitch oscillations, and a 23% reduction in platform pitch ratecompared to the baseline PI controller described in [18] for a full DOF FASTsimulation. The authors note that the improvements come at a cost to bladepitch actuation and blade flapping modes, and they do not quantify eitherthe improvement or deterioration of power regulation relative to the baselinecontroller. In the same paper, the authors describe and design a rudimentaryindividual blade pitching controller with further potential for improvement,however a more complete account of this is described in another paper,described next.It was described in [5] a disturbance-accommodating, individual bladepitch (IBP) controller for the 5MW baseline wind turbine on a tension-legplatform. The purpose of IBP is to utilize each blade individually to createasymmetric loads that can mitigate platform motions and blade bendingloads even more than collective blade pitch controllers. For example, plat-form pitch can be controlled by adjusting the pitch angles of top-most bladeand bottom-most blade, oppositely, without affecting the net power captureor net thrust force. Because the blades are constantly rotating, IBP is in-herently a periodic problem. However, to avoid the complexity of periodiccontrol, the authors employed a a method called Multi-blade coordinate(MBC) transformation. MBC effectively captures the periodic properties ofa system and transforms them into a linear time-invariant model, which ismuch simpler to deal with from a controls perspective. The other methodthat the authors make use of is that of Disturbance Accommodating Con-trol (DAC). In this method, persistent disturbances that are not measurableare estimated by assuming a specific disturbance waveform and by the useof measurable states of the system. The authors use this method to rejectthe periodic wind disturbance of the system. The final proposed controllerwas based on a 6 DOFs linearized state-space model created using FAST,including platform roll, pitch and yaw, tower side-side bending, rotor speedand drivetrain twist. Following the synthesis of the IBP DAC, a full DOFRegion 3 simulation was carried out using FAST for both the proposed con-142.3. Multivariable Control for Reduction of Fatigue Loadstroller and a modified version of the baseline controller made for the offshoretension-leg system. The simulation results showed 44% reduction in powererror 73% reduction in rotor speed error, 13% reduction in platform pitchingand 38% reduction in tower side-side fatigue loading relative the baseline PIcontroller.Another example of the use of wind estimation for floating wind turbinecontrol can be found in [25]. In this paper, the authors a simplified 8-statemodel was adopted that includes the drivetrain dynamics, the tower-topmotion, and the wind turbulence to develop an optimal LQR controller.The authors suggested the use of an extended Kalman filter to estimatethe non-periodic relative wind speed at the wind turbine hub using therealistic measurements of rotor velocity, blade pitch angle and generatortorque. An observer was also designed to estimate what the authors deemas unmeasurable or unrealistically measurable states. Similar to the previousworks, the proposed controller was compared to the baseline gain-scheduledPI controller by running Region 3 simulations with stochastic wind and waveconditions. The proposed controller achieved a 19% reduction in platformpitch moments, a 6% reduction in tower base foreaft moments, and a 24%reduction in generator speed standard deviation, compared to the baselinecontroller. A compromise was made however, as the proposed controllersuffered a 5% increase in damage-equivalent-loading of the drivetrain due totorsion, compared to the baseline controller.2.3.2 Utilization of Wave DisturbanceDespite the effectiveness of wind disturbance accommodation in research,the accommodation of waves has not been explored in very much detail.This is largely because most existing modelling techniques do not offer away to construct the wave disturbance matrix in linear state-space models.Some attempts have been made, however, and the results are described inthis subsection.In [27], the authors developed a linear-parameter-varying model for thefloating wind turbine system that depends on the frequency of the propagat-152.3. Multivariable Control for Reduction of Fatigue Loadsing ocean wave. A wave interaction analysis tool called WAMIT was utilizedto generate the linear hydrodynamic damping matrix at various wave fre-quencies within the expected range for ocean waves. Using the set of linearmodels for various wind speeds and wave frequencies, the authors generateda set of LQR controllers assuming full state feedback, including the waveperiod and wind speed. The authors then proposed a method for estimatingthe peak period of the waves based on measured wave height, and a methodfor estimating wind speed, based on the extended Kalman filter describedin [25]. Finally, they applied a gain-scheduling method to combine their setof controllers, and validate it using a realistic simulation test, compared toa baseline controller. The simulation study demonstrated that accountingfor both wind and wave disturbance results in a 50% improvement in plat-form pitch motions, a 7-20% improvement in power capture, and a 5-17%improvement in fore-aft deflection. This comes at a cost of 150% to bladepitch activity.In [11], the authors used their developed non-linear 2-D physics-basedmodel to create a linear model of their system, including both the windand wave disturbance matrices. Using this linear model, the authors devel-oped an H∞ controller for the entirety of Region 3 that rejects wind andwave disturbances in their expected frequency range. They compared theirH∞ regulator to the baseline GSPI controller in a realistic simulation withstochastic wind and wave disturbances. In this test, they showed a largeimprovement in rotor speed regulation and torque variation, at a slight costto power regulation, platform pitch regulation, and blade pitch actuation.The major control contributions for floating offshore wind turbines aresummarized in Table 2.2, below. Here, we can clearly see the benefit ofincreasing the amount of information used in controller design on both powerregulation and platform motion. It is expected, then, that by utilizing amodel with an adequate amount of states, as well as the wind and wavedisturbance matrices, we will see yet greater improvements in controllerperformance.162.4. SummarySource States ConsidersWindConsidersWaveRotorSpeedRegu-lationPlatformMotionReduc-tionPowerRegu-lation[18] 1 No No 0% 0% 0%[4] 4 No No -21% -18% -[5] 12 Yes No -73% -13% -44%[27] 24 Yes Partial - -50% -14%[11] 8 Yes Yes -66% +37% +22%Table 2.2: Summary of Control Contributions of Floating Offshore WindTurbines.2.4 SummaryIn this section, a commonly used baseline 5MW wind turbine and floatingplatform were outlined, existing models and their advantages and disad-vantages were described, and advanced control techniques that have shownpromise in maximizing the cost effectiveness of wind energy were detailed.More precisely, it was shown that simple onshore control strategies do notmaximize the cost effectiveness of offshore wind power due to the neglectof structural fatigue loading and its effect on turbine lifetime, and thatadvanced control strategies are necessary for further reducing fatigue load.Many researchers have shown the potential in wind disturbance accommoda-tion for fatigue reduction in floating wind turbines, but the accommodationof wave disturbance is yet to be explored. This is largely due to the inabilityin existing modelling techniques to generate the wave disturbance matrix, ormathematical relationship between waves and important states of the float-ing wind turbine system. It should be clear, then, that a control-orientedfloating wind turbine model would include the ability to generate such a dis-turbance matrix. The following chapter will describe the process in creatingsuch a model.17Chapter 3Non-Linear Modelling3.1 IntroductionThe objective of this chapter is to prove that highly simplified, control-oriented models can accurately predict the dynamics of floating wind turbinesystems. This is important, because it allows for the use of complex, high-information control strategies to push both the performance, and lifespan offloating wind turbine systems beyond what is possible with existing models.The first part of this chapter will formulate a higher level, generalized model,that can be extended to any floating wind turbine system with known forcesand torques. The next section will detail the force and torque models for theBaseline 5MW Semi-Submersible system, and apply them to the generalizedmodel. Finally, the performance will be measured in four distinct testsby comparing the dynamic response of the simplified model to the FASTequivalent model.3.2 Generalized Non-Linear ModelThe purpose of the control-oriented non-linear model is to accurately predictthe relationship between the relevant states of the system (x), the controlinputs (u) and the environmental wind and wave disturbances (v and w).The proposed model follows the formx˙ = f(x,u,v,w), (3.1)where f denotes the non-linear function relating the states to the inputsand disturbances. The simplicity of the proposed model depends on several183.2. Generalized Non-Linear Modelimportant assumptions. First, it is assumed that the wind turbine andplatform behave as a single rigid body, which we call the wind turbine system(WTS). With this assumption, we inherently neglect all flexibility and effectsof flexibility of the wind turbine tower structure. Even without flexibility inthe model, it is still possible to take into account fatigue load attenuationindirectly in controller design because internal fatigue loads are related toplatform motion [17]. Furthermore, we assume all platform angular motionsare relatively small (≤ 10◦). Finally, we assume that the inertia tensor of thewind turbine system is constant with respect to a coordinate frame rigidlyattached to its center of gravity. This last assumption is generally true ifthe inertia of the rotor is negligible compared to that of the platform andtower.3.2.1 Coordinate SystemsIn order to properly define the states and disturbances of the WTS, it isnecessary to define a set of coordinate frames. In particular, five coordinateframes will be defined. These are illustrated in Figure 3.1.The world frame is an inertial reference frame denoted F0. It is definedby its orthogonal unit vectors eˆ10, eˆ20 and eˆ30. With respect to the world,the unit vector eˆ30 points vertically upward, opposite to gravity, while unitvectors eˆ10 and eˆ20 point horizontally, parallel to the surface of the earth. Inthis thesis, any vector that does not have an attached superscript is definedin this frame.The body frame is attached to the center of gravity of the model WTS,and is denoted Fb. The body frame can be defined by orthogonal unitvectors eˆ1b, eˆ2b and eˆ3b, which make up the principal axes of the rigid body.The unit vector eˆ1b points in the nominal wind direction for an un-yawedstate. The unit vector eˆ3b, points parallel to the tower of the wind turbine,vertically upward, from platform to nacelle. Finally, the unit vector eˆ2bpoints orthogonally to eˆ1b and eˆ3b such as to satisfy the criteria of a righthanded coordinate system. When the WTS is not under the influence ofwind or wave disturbance, the body frame aligns with the world frame.193.2. Generalized Non-Linear ModelFigure 3.1: Coordinate Frames for the General Non-Linear Model.The nacelle frame is attached to the nacelle of the WTS, at the pointwhere the yaw axis meets the rotor axis. It is denoted Fn. This frame canbe defined by orthogonal unit vectors eˆ1n, eˆ2n and eˆ3n, where eˆ3n is alwayspointing in the same direction as eˆ3b. Unit vectors eˆ1n and eˆ2n rotate aboutthe yaw axis of the WTS with the nacelle.The shaft frame, Fs, shares its origin with the nacelle frame, but isrotated with respect to Fn about its second axis by the constant angle θtilt.Fs and is composed of the orthogonal unit vectors eˆ1s, eˆ2s and eˆ3s.The rotor and generator frames align their first axes with the first axis ofthe shaft frame, and are denoted Fr and Fg, respectively. These two framesrotate independently about eˆ1s with the rotor and generator, respectively.3.2.2 StatesFor controller design, it is important for the model to be as minimalisticas possible, while still maintaining adequate accuracy. Following the above203.2. Generalized Non-Linear Modelassumptions, a general floating wind turbine with six platform DOFs, and aflexible-drivetrain, will have as many as 16 states. This includes three statesfor both position and orientation of the platform, one azimuth angle statefor each the rotor and generator, and the time derivatives of all of the above.The three position states make up the displacement vector ~xg = (xg, yg, zg)[m] between the origin of the world frame and the origin of the body frame,given in world frame coordinates. These three states are shown in Figure 3.2,below. Likewise, the time derivatives of the three position states are alsostates ~˙xg = (x˙g, y˙g, z˙g) [m/s].Next, if we let R¯be the rotation matrix that specifies the orientation ofthe body frame Fb with respect to the world frame F0, then by consequenceof the small angle assumption, we can formulate R¯with three distinct anglessuch thatR¯=1 −θz θyθz 1 −θx−θy θx 1 . (3.2)The three orientation states of the WTS, then, are the three angles (θx, θy, θz)[rad] that form the rotation matrix. We can also write these states morecompactly as the pseudo-vector ~θ. The three orientation states are shownin Figure 3.2, below. Three additional time derivative states exist, and are~˙θ = (θ˙x, θ˙y, θ˙z)[rad/s].Figure 3.2: Position and Orientation States.213.2. Generalized Non-Linear ModelThe rotor azimuth angle, θr[rad], is defined as the angle between thethird axis of the rotor frame, and the third axis of the shaft frame. Like-wise, the generator azimuth angle, θg[rad] is defined similarly between thegenerator frame and the shaft frame. If our model depends on one or bothof these angles explicitly, they make up the azimuth states of the WTS.However, it is possible that only the difference between these two angles isimportant. In this case, we can define the azimuth state to simply be thedifference between the rotor angle, and the equivalent generator angle on thelow-speed side, or θr− 1NGR θg, where NGR is the gear ratio between the rotorand generator side shafts. In either case, it is necessary to define two deriva-tive states, as the aerodynamic equations of motion in Section 3.3.2 dependon the rotor speed explicitly. These derivative states can be written as ωr[rad/s] and ωg [rad/s], for rotor speed and generator speed, respectively.The rotor and generator azimuth angles are illustrated in Figure 3.3.The three position and orientation states of the WTS are illustrated inFigure 3.2.Figure 3.3: Rotor and Generator Angle States.223.2. Generalized Non-Linear ModelTo recap, the state vector of the WTS can be written as follows:x =~xg~θ(θrθg)or(θr − 1NGR θg)~˙xg~˙θωrωg. (3.3)3.2.3 Control InputsOnly three inputs are considered for this model, which are collective bladepitch angle β [deg], generator torque, Tg [Nm], and nacelle yaw angle γ[deg].Each blade rotates about an axis that travels through its long dimension.When all blades rotate together, we call the unified angle the collective bladepitch angle β. Individual blade-pitch-actuation, on the other hand, hasbeen shown to be quite effective for control of floating wind turbines [5, 17]and is possible using our model. However, doing so introduces periodiceffects which must be dealt with using advanced control techniques. Forsimplicity, we will only present collective blade pitch control in this thesis.Our simplified model neglects actuator dynamics for this rotation, whichresults in the unrealistic ability to instantaneously change the angle. Wecan prevent such unrealistic motions in the non-linear model by limiting therate at which this angle is allowed to move. Furthermore, we must ensurethat designed controllers obey this limit.Generator torque Tg can be controlled via the electric current appliedto the generator. This allows for the manipulation of instantaneous powerextraction, as well as the the control of equilibrium rotor speed.Nacelle yaw angle is the angle between the unit vectors eˆn1 and eˆb1, in thebody frame. As in the blade pitch case, we ignore actuator dynamics. Again,we can prevent unrealistic rotations in the non-linear model by limiting the233.2. Generalized Non-Linear ModelFigure 3.4: Blade Pitch Angle and Generator Torque Inputs.rate at which this angle is allowed to change. As with the blade pitch rate,we must ensure that any designed controllers obey this limit.Figure 3.5: Yaw Angle Control Input.243.2. Generalized Non-Linear ModelThus, the input vector of the WTS can be written as followed:u =βTgγ (3.4).3.2.4 DisturbancesThe wind disturbance is simplified as the point wind velocity vector relativeto the world frame, located at the center of thrust of the rotor. Thus, it canbe written as:v =vxvyvz [m/s] (3.5).The wave disturbance can be simplified as a collection of n wave veloc-ity vectors ~wv,1 . . . ~wv,n − [m/s], n wave acceleration vectors ~wa,1 . . . ~wa,n −[m/s2], n wave heights wh,n . . . wh,n − [m] and n dynamic pressure termswp,n . . . wp,n − [Pa], all relative to the world frame. The number of suchterms is highly dependent on the geometry of the platform, and will bediscussed in more detail in Section 3.3.1. For now, let us write the wavedisturbance vector as:w =[~wv,1 . . . ~wv,n, ~wa,1 . . . ~wa,n, wh,n . . . wh,n, wp,n . . . wp,n]T(3.6).3.2.5 Equations of MotionWith the states x, control inputs u and disturbances v and w identified,we can return to (3.1) and derive the equations of motion. The non-linear253.2. Generalized Non-Linear Modelfunction f can be written asf(x,u,v,w) =~˙xg~˙θ(ωrωg)or(ωr − 1NGRωg)~fF (x,u,v,w)~fT (x,u,v,w)fQ(x,u,v), (3.7)where ~fF − [m/s2], ~fT − [rad/s2], and fQ − [rad/s2] are obtained from theforce, torque and shaft torque models, respectively.The force model is a summation of all relevant forces that act on thewind turbine system and takes the form~fF (x,u,v,w) = (mgI¯3x3+ diag[~ma])−1∑j~Fj(x,u,v,w), (3.8)where mg is the total mass of the turbine platform, I¯3x3is a 3x3 identitymatrix, and ~ma is the hydrodynamic added mass vector, which is describedin Section 3.3.1. The summation of forces ~Fj contains all the relevant forceson the system such as aerodynamic forces ~FA, buoyancy forces ~FB, catenaryforces ~FC , and hydrodynamic forces ~FD, as depicted in Figure 3.6. Theseforces are described in detail in Section 3.3. Forces can easily be added,modified or dropped from the summation for each specific wind turbinesystem.Like its force counterpart, the torque model is a summation of all therelevant torques acting on the wind turbine about its center of gravity. Thistakes the form~fT (x,u,v,w) = (R¯I¯−1g R¯T )∑j~Tj(x,u,v,w). (3.9)Here, I¯gis the inertia tensor of the wind turbine in its upright orientation,R¯is the rotation matrix that represents the orientation of the wind turbine263.3. Forces and Torques for the Baseline Turbinewith respect to the fixed world frame, and the summation of torques ~Tjcontains all the torques on the system. Most torques take the form~Tj(x,u,v,w) = R¯~rgj × ~Fj(x,u,v,w), (3.10)where ~rgj is the position vector from the turbine center of gravity to thelocation of the corresponding force in the fixed body frame. Finally, therotor torque model is given by the torque balancefQ(x,u,v) =∑kr1JrQkr(x,u,v)∑kg1JgQkg(x,u,v) , (3.11)where Jr and Jg represent the inertia about the rotor-side shaft and generator-side shaft, respectively, and Qkr and Qkg represent the krth and kgth torqueabout each respective shaft.So far, we described the structure of the proposed simplified model fora general floating wind turbine system. Depending on the turbine type,mooring type, and platform type, the forces and torques that make up theforce and torque models, and the locations of the forces may be different.In the next section, we derive the force and torque models for the baseline5MW semi-submersible wind turbine. We then validate it using simulationsoftware FAST in Section 3.5.3.3 Forces and Torques for the Baseline TurbineThis section describes the force and torque models used to construct theexample 5-MW semi-submersible wind turbine [1, 18]. Forces are depictedin Figure 3.6 and include the the aerodynamic force ~FA, buoyancy force~FB, catenary line forces ~FC and hydrodynamic drag/inertial force ~FD. Foreach of these forces, there is an accompanying torque (~TA, ~TB, ~TC and ~TD),however these are not shown for clarity. The rotor torque Tr, however, whichis a part of the shaft model and does not directly affect the wind turbine273.3. Forces and Torques for the Baseline Turbinesystem, but instead acts directly on the rotor is also shown in the diagram.Figure 3.6: Overall Force Diagram of the Non-Linear Model.3.3.1 HydrodynamicsThe hydrodynamic model includes all forces related to the interaction be-tween the platform and the surrounding water. These forces act continu-ously and unevenly over the submerged surface area of the platform, mak-ing it extremely difficult to integrate them analytically. Instead, we cansplit the platform up into small cylinders, inside which we assume thatthe variables affecting the hydrodynamic forces are constant. We can thenapply to each cylinder Archimedes’ principle to determine the buoyancyforce, and Morison’s equation to obtain the drag and inertial forces. Forthe semi-submersible platform, we can achieve good results by consideringthree cylinders for each column. These are shown graphically in Figure 3.7.Here, only the four top-most cylinders are partially submerged, and thustheir submerged lengths change. The rest always remain fully submerged.283.3. Forces and Torques for the Baseline TurbineFigure 3.7: Separated Cylinder Segments.Buoyancy Archimedes’ principle states that the buoyancy force on anobject is equal to the weight of fluid that it displaces [32]. A depiction ofthe buoyancy model is given in Figure 3.8. Here, ~xg and ~xi represent thewind turbine center of mass and the cylinder center of volume in the worldframe, respectively. The force on a submerged cylinder i is given byFigure 3.8: Buoyancy Force Diagram.~FB,i(x) = ρwgAilieˆ3, (3.12)293.3. Forces and Torques for the Baseline Turbinewhere ρ is the density of the fluid, g is the gravitational constant, and Ai andli are the cross sectional area and length of the cylinder, respectively. Forthe top-most cylinders, the length will change based on platform positionand orientation. This relationship can be approximated by the expression:li(~xg, ~θ) =l0,i2− wh,i −[θy θx 1]~rbgi,0 (3.13)where ~rbgi,0 is the constant vector that points from the center of gravity ofthe wind turbine to the initial center of volume of the cylinder, given inthe body-fixed frame. Note that we do not include changes in water leveldue to waves or other factors in this expression. Instead, we deal with theresulting effects from wave height via dynamic pressure which is consideredin the drag and inertial forces.Given the force on the cylinder, the torque that it transmits to the windturbine is given by~TB,i(x) = R¯~rbgi × ~FB,i, (3.14)where ~rbgi is the vector that points from the center of gravity of the WTSto the center of volume of the cylinder, in the body frame. For all butthe top-most cylinders, this vector is equivalent to ~rbgi,0. For the top-mostcylinders, the location of the force changes with platform position, and canbe expressed as~rbgi = ~rbgi,0 +[0 0 1]T ( li − li,02). (3.15)Lastly, we must include the force of gravity acting at the center of gravityof the wind turbine system, and given by~FG = −mggeˆ03. (3.16)This does not produce a torque on the WTS because it is acting at the centerof gravity, and thus the cross product is zero. Finally, the total buoyancy303.3. Forces and Torques for the Baseline Turbineforce is give by~FB(x) =∑i~FB,i + ~FG, (3.17)and the torque by~TB(x) =∑i~TB,i. (3.18)Drag and Inertial Forces Drag is the dissipative force that resists rela-tive motion between a body and a fluid. A depiction of the force model isgiven in Figure 3.9. For a submerged cylinder in transverse flow, Morison’sequation gives a simple approximation for the drag force on the cylinderface in the relative direction of flow [1]. For the 3-D case, this force can berealized using the following equation:~FDt,i(x,w) = Kd,i||~vt,i||~vt,i +Ka,i~at,i, (3.19)where the drag and inertia constants Kd,i and Ka,i are the constants fromMorison’s equation, and ~vt,i and ~at,i are the equivalent transverse velocityand acceleration vectors for the calculation. The transverse velocity andacceleration vectors can be calculated via the series of transformations~vt = R¯110R¯T ~wrel, (3.20)~at = R¯110R¯T ~˙wrel, (3.21)and the magnitude of the transverse velocity can be simplified to||~vt|| =~wTrelR¯110R¯T ~wrel12. (3.22)313.3. Forces and Torques for the Baseline TurbineIn the previous three equations, the relative wave velocity, ~wrel, and its timederivative ~˙wrel are given by~wrel = ~w − ~˙xg − R˙¯~rbgi (3.23)~˙wrel = ~˙w. (3.24)Here, ~w is the undisturbed wave velocity at the center of volume of thecylinder. Note that the inertial force due to the platform’s acceleration hasnot been included here as it is included in the overall force model in theform of added mass.For the bottom-most cylinders, there is also a drag force and inertialforce in the heave direction. This force is described in 1-D in [1], and canbe extended to 3-D with the following equation:~FDh,i(x,w) = Kdh,i||~vh,i||~vh,i +Kah,i~ah,i + PiAieˆh, (3.25)Again, Kdh,i and Kah,i are the constants from the heave plate equation in [1],and ~vh,i and ~ah,i are the equivalent heave velocity and acceleration vectorsfor the calculation. These vectors can be calculated via a similar series oftransformations as for the transverse case, but with a different projectionmatrix. This is shown for the heave velocity calculation below:~vh = R¯001R¯T ~wrel. (3.26)Finally, we write the total drag and inertia force on each cylinder as~FD,i(x,w) = ~FDt,i + ~FDh,i. (3.27)Again, the induced torques on the wind turbine are given by crossing thevector travelling from the turbine center of mass to the cylinder center ofvolume with the forces at the cylinder center. Then the torque can be323.3. Forces and Torques for the Baseline Turbinewritten as~TD,i(x,w) = R¯~rbgi × ~FD,i. (3.28)Figure 3.9: Drag and Inertial Force Diagram.Added Mass Added mass is the effective mass increase experienced by asubmerged body in motion due to the liquid that accelerates with the object.In the proposed model, we assume that this is constant in the body frame(i.e., it does not get recomputed as the lengths of the submerged cylinderschange). Added mass is calculated as it appears in Morison’s equation foreach cylindrical face exposed to the water. For the lateral and bottom faces,the added mass is given respectively by ma,i = Ka,i and ma,i = Kaz,i. Thetotal added mass then is given by the sum of all these parts, or:~ma =∑iR¯Ka,iKa,iKaz,i . (3.29)Inertia is also affected by this effect, however because the inertia tensor isdefined in the body frame, it remains constant, even in platform rotations.Therefore, the modified inertia tensor we use in our model represents boththe physical inertia of the system, as well as the additional inertia fromadded mass.333.3. Forces and Torques for the Baseline Turbine3.3.2 AerodynamicsThe aerodynamic model is concerned with the interaction between the windand the turbine. This includes the forces acting on the blades and theresulting power transferred to the rotor. In the proposed model, a separatethrust and drag force is calculated for each blade in order to accuratelypredict individual blade pitch actuation.Thrust and Drag The thrust force is the wind force in a direction parallelto the rotation axis of the rotor, whereas the drag force is the wind force inthe direction of motion of a point on the blade. In reality, these forces actcontinuously over the entire blade; however, for simplicity, we localize thenet thrust force for all three blades at a location we call the center of thrust.A depiction of the thrust and drag forces is given in Figure 3.11. The netFigure 3.10: Aerodynamic Force Diagram.thrust force vector for is given by the relationship~FA(x,u,v) =12ρArCt(λ, β)||~vn||~vn, (3.30)343.3. Forces and Torques for the Baseline Turbinewhere Ct is the thrust coefficient which is a function of the tip-speed-ratio(TSR) λ, and the blade pitch angle β [33]. Parameters ρ and Ar representthe air density and the swept area of the rotor, respectively. The equivalentvelocity vector normal to the face of the rotor blades, ~vn, can be calculatedby~vn = R¯ eq100R¯Teq~vrel, (3.31)and the magnitude simplified to||~vn|| =[1 0 0]R¯Teq~vrel (3.32)whereR¯ eq= R¯ y(θtilt)R¯ z(γ)R¯. (3.33)The last inclusion is the gyroscopic torque that is transferred to the platformwhen we utilize the generator torque input to accelerate or decelerate therotor. The magnitude of this torque is simply the product of the generatortorque and the gear ratio, and thus the total aerodynamic torque can thenbe written as~TA(x,u,v) = (R¯~rbgt × ~FA) + R¯ eqNGRTg eˆ1, (3.34)where NGR is the gear ratio between the two shafts.Power Finally, the aerodynamic power is given byP =12ρArCp(λ, β)||~vn||3. (3.35)The torque balance about the rotor shaft and generator shaft are then givenby~˙ωr =1Jr(Pωr− k(θr − 1NGRθg)− b(ωr − 1NGRωg)), (3.36)353.3. Forces and Torques for the Baseline TurbineFigure 3.11: Rotor-Generator Shaft Torque Diagram.and~˙ωg =1Jg(−Tg + kNGR(θr − 1NGRθg)+bNGR(ωr − 1NGRωg)), (3.37)respectively. Finally, we can write the shaft torque model fQ from Section3.2 asfQ =[~˙ωr~˙ωg]. (3.38)Gyroscopic Effect Because we consider the entire WTS as a single rigidbody, we are neglecting the gyroscopic effect that results from the rotorframe rotating with respect to the world frame. It turns out that for thesemi-submersible platform, this effect is negligible, and for simplicity it is notincluded in the proposed model. However, for platforms with less inertia,this may not be the case. For these platforms, we can account for thegyroscopic torque on the WTS by enforcing the conservation of angularmomentum about the entire system. That is, when the angular momentumof the shaft changes, the angular momentum of the platform must change inan equal and opposite way. We can quantify this effect by considering the363.3. Forces and Torques for the Baseline Turbinerate of change of the angular momentum of the rotord ~Hrdt=ddtR¯ eqJrωr00 , (3.39)where ~Hr is the angular momentum of the rotor. This can be further sim-plified toR¯ tilt(R˙¯γR¯+ R¯γR˙¯)Jrωr00+ R¯ tiltR¯γR¯Jrω˙r00 . (3.40)The left-most term represents the change in angular momentum of the shaftas the shaft frame rotates in the world frame, and the right-most term repre-sents the change in angular momentum of the shaft due to acceleration in itsown frame, which has already been accounted for in 3.3.2. The gyroscopictorque, then, is an equal and opposite torque to the left-most term in (3.40),or.~Tgyro = −R¯ tilt(R˙¯γR¯+ R¯γR˙¯)Jrωr00 . (3.41)Relative Tip-Speed Effect The wind power and thrust models used inthe proposed model assume that the wind travels in a direction perpen-dicular to the rotation path of the blades. In reality, this is not the case,and when the wind vector is not aligned in such a fashion, there will beazimuth-dependent changes in the aerodynamic force for each blade as itmoves relative to the wind vector. In [34], the authors accounted for this byconsidering an aerodynamic force at the midpoint of each blade, instead ofat a single location. This adds periodicity to the model, which makes lin-earization at static operating points not possible. As it turns out, however,the complexity of adding this effect does not provide a substantial increasein overall accuracy of the major dynamics of the semi-submersible system.373.3. Forces and Torques for the Baseline TurbineThe effect is more important for vibrational analysis of the blades.3.3.3 Mooring LineThe mooring system consists of a series of cables attaching the wind turbineto the sea bed. These cables provide a restoring force in response to platformdisplacements caused by wind, waves and other disturbances. The quasi-static model for a single cable in 2-D is described in [35] and consists of twocoupled, non-linear equations that relate horizontal and vertical distancebetween cable ends with the 2-D force at the wind turbine attachment point.It is important to note that these equations change depending on whether aportion of the line rests on the sea bed, or whether it is fully ungrounded. Forsimplicity, we will only consider the former case. This follows the assumptionthat the wind turbine does not move a large distance from its equilibriumposition. A depiction of the catenary force model is shown in Figure 3.12.Figure 3.12: 2-D Catenary Line ModelBy using a multi-dimensional Newton-Raphson iteration, the unique 2-Dforces Fx(x, y) and Fy(x, y) are determined for a range of relevant values ofx and y. If we define ~xt,i as the total 3-D distance vector between the anchorpoint of the i’th mooring line and the i’th attachment point on the turbine,then x is the magnitude of the projection of ~xt,i on the x-y plane. Similarly,y is the projection of ~xt,i on the z-axis. Vector ~xt,i then is given by~xt,i = ~xa,i − ~xg − R¯~rbgci, (3.42)where ~rgci is the distance vector between the turbine center of gravity and383.4. Parameter Identificationthe attachment point in the BF. Finally, the total catenary force at theattachment point of mooring line i is given by~FC,i(x) =Fx(~xt,i)||projeˆ1(~xt,i)||,Fx(~xt,i)||projeˆ2(~xt,i)||Fy(~xt,i) , (3.43)It follows that the total force and torque are given by~FC(x) =∑i~FC,i(x) (3.44)~TC(x) =∑i(R¯~rbgci × ~FC,i), (3.45)respectively.3.4 Parameter IdentificationAll of the parameters used in the non-linear model are listed in AppendixA. The majority of said parameters, such as the inertial and geometricinformation for the WTS structure and platform, and for the flexibilityinformation for the shaft, were found in [18] and [1]. Some parameters,however, had to be collected from FAST simulations. These include the Cpand Ct curves, the modified inertia tensor, and small tunings to the dragconstants. The process of obtaining this additional information is explainedin this section.3.4.1 Cp and Ct CurvesThe power and thrust curves are a function of blade-pitch angle, β, andTSR, λ, which is the ratio of blade tip speed and normal wind velocity, orλ =ωrrv(3.46)The Cp and Ct surfaces were obtained by running a series of FAST simula-tions (i = 1 : n) under the following conditions:393.4. Parameter Identification• All DOFs disabled except aerodynamic force• Rotor shaft tilt brought down to 0◦• Perfectly horizontal wind speed that steps by ∆v every T seconds• Constant rotor speed ωi• Sinusoidal blade pitch angle with half period equal to TThe time series information for the simulation input conditions is shown inFigure 3.13. Following the simulations, the Cp and Ct values were extractedfrom the simulation data at one hundred equally spaced values of ωr and vand the surfaces, shown in Figure 3.14 were created.Figure 3.13: Simulation Conditions for Obtaining Cp and Ct Surfaces.3.4.2 Added Inertia and Drag ConstantsBoth the added inertia and the drag constants were found by tuning theplatform natural frequencies and damping characteristics in a test similar to403.5. Validation of Non-Linear ModelFigure 3.14: Cp and Ct Surfaces.the perturbation test, described in Section 3.5.1. The resulting values aregiven in Appendix A3.5 Validation of Non-Linear ModelIn order to validate the model derived for the Baseline 5MW wind turbine,several tests were performed targeting specific effects of the non-linear modelboth in isolation and all together. First, a series of simple 2-D perturbationtests were performed to examine the inertial-elastic response of the WTS toinitial displacements of the states. Next, a series of tests were performed toanalyze the effect of wind disturbance at various WTS operating conditions.Following these tests without wave excitation, a simple wave test was per-formed to analyze the hydrodynamic response of the WTS and the effect ofwave disturbances. Finally, an overall realistic open-loop response test wasperformed to validate the WTS under realistic wind, wave and control input413.5. Validation of Non-Linear Modeltrajectories.3.5.1 Perturbation TestAs a means to validate the inertial and elastic response of the non-linearmodel to perturbations in platform position, the perturbation response testwas conducted. In this test, the platform was given an initial displacement ineach of the six DOFs under no wind or wave disturbance, and the time seriesresponses were collected. The responses are shown for both the non-linearmodel and for FAST in Figure 3.15.To quantify the test results, the RMS errors between the time series ofFAST and the proposed non-linear model were calculated, as well as themajor oscillation frequency differences between them. It should be notedthat the rotor and generator speed degrees of freedom were excluded fromthe simulation. The results are shown in Table 3.1, with initial displacementsin parentheses.Table 3.1: Perturbation Response ResultsDirection RMS error Frequency errorSurge (9m) 0.48m 0.93%Sway (9m) 0.45m 1.97%Heave (9m) 0.20m 3.19%Roll (9◦) 0.34◦ 2.34%Pitch (9◦) 0.35◦ 1.85%Yaw (9◦) 0.536◦ 2.57%With all the translational RMS errors less than 0.5m and the rotationalRMS errors less than 0.6 degrees, the undisturbed inertial-elastic responseof the WTS is clearly very close to that of FAST. Furthermore, all the majornatural frequencies of the WTS are within 3.2% of those measured in FAST,which effectively validates the mooring line and buoyancy models, and theinertial properties of the wind turbine system over a large operating range(10m in translational displacement and 10deg in rotational displacement islarger than we would expect to see in typical operation).423.5. Validation of Non-Linear ModelFigure 3.15: Perturbation Test Results.3.5.2 Gust TestNext, we consider the effect of wind disturbance. As a means to validatethe effect of wind disturbance on the WTS model, several tests involving theinstantaneous application of wind were performed. Because the aerodynamic433.5. Validation of Non-Linear Modelforces are a function of both the wind disturbance, and the rotor speed,the test was run at various tip-speed-ratios and wind speeds. However, toexclude the effects of rotor shaft dynamics, the rotor speed was held fixed forthe duration of each test. Blade pitch angle was also fixed at zero degrees.Again, the time series response data was collected in FAST and the proposedmodel, which is illustrated in Figure 3.16 for the affected states.Figure 3.16: Wind Gust Test ResultsRMS errors were calculated for the affected states, as shown in Table 3.2.The average state values over the course of the simulation are also given inthe table as a reference point for the RMS error.Once again, the RMS values for the non-linear model compared to FAST443.5. Validation of Non-Linear ModelTable 3.2: Aerodynamic Response ResultsTSR Wind Speed Surge RMS Pitch RMS1.32 25m/s 0.51m 0.11◦(average 5.64m) (average 1.10◦)5.05 17m/s 1.46m 0.33◦(average 11.75m) (average 2.80◦)17.15 5m/s 0.33m 0.05◦(average 2.95m) (average 0.52◦)are very similar over a wide operating range. For both translation androtation, the RMS error is at the highest 12.4% of the average, and eventhat is likely caused by a small difference in the steady state value of thetwo curves. Once again we consider this test a success, which effectivelyvalidates the aerodynamic force model and effect of wind disturbance onthe WTS.3.5.3 Regular Wave TestWith the inertial-elastic, and aerodynamic responses validated, we turn tothe effect of wave disturbance on the WTS. The next test looked at the plat-form response to wave disturbance. Unfortunately, due to the complexity ofirregular waves, and their spatial and time dependence, the multi-frequencywave profile generated by FAST could not be duplicated for the non-linearmodel. Instead, a regular wave profile, or single frequency sinusoid profilewas obtained using linear wave theory [36], using a wave elevation of 1.75 mand a single spectral period of 12 s. In linear wave theory, we assume the3-D wave height in space and time is given byh(~xw, t, α) = A sin(ζ(~xw, t, α)), (3.47)where A is the amplitude of the wave, t is time, α is the wave directionabout the world frame z axis, ~xw is the spatial location of the wave height,453.5. Validation of Non-Linear Modeland the space/time dependent parameter ζi is given byζ(~xw, t, α) =−ω2g(eˆ1R¯Tz (α)~xw)+ ωt+ φ. (3.48)Here, ω represents the frequency of the wave, R¯Tz (α) is the z-rotation ofmagnitude α, g is the gravitational constant, and φ is the phase angle.Unfortunately, as FAST only allows for the specification of period and am-plitude, the phase parameter and direction of the spatial term had to befitted. With the frequency, amplitude and phase known, we can calculatethe remaining hydrodynamic disturbances as follows~v(~xw, t, α) = ω exp(−ω2gz)cos(α) sin(ζ)sin(α) sin(ζ)cos(ζ), (3.49)~a(~xw, t, α) = ω2 exp(−ω2gz)cos(α) cos(ζ)cos(α) cos(ζ)− sin(ζ) , (3.50)Pd(~xw, t, α) = ρg exp(−ω2gz)sin(ζ), (3.51)where parameter z is given byz = eˆ3R¯Tz (α)~xw. (3.52)After collecting all the necessary parameters to recreate the FAST waveprofile, an identical simulation was run using the WTS, and the responsesfor the relevant states are shown in Figure 3.17.Once again, the time responses of FAST and the proposed model arevery similar. The biggest discrepancy seems to be a slight difference in thedamping, however it is minimal. At this point, we say that all the individualpieces of the non-linear model have been validated. However, it is importantto validate not only each piece individually, but also the entire system asa whole, with realistic wind and wave disturbances, varying control inputs,463.5. Validation of Non-Linear ModelFigure 3.17: Time Response Comparison for the Regular Waves Test.and varying states.3.5.4 Realistic Open-Loop TestThe purpose of the realistic open test is to compare the time response of thenon-linear model to the FAST model with all degrees of freedom activated,under realistic disturbance and control input. For the comparison, we firstexplain specific wind and wave disturbances that we employed in the test.Disturbances The realistic wind disturbance profile was created by pro-gram TurbSim [37] with a mean wind speed of 18 m/s at a mean angle of 20◦about the eˆ3 axis, as shown in Figure 3.18. Using this program, we are able473.5. Validation of Non-Linear Modelto produce similar wind profiles for FAST, which is capable of using Turb-Sim output files, and the proposed model, which takes a single 3-D windvelocity vector. This wind vector is given relative to the world frame at ahub height of 90 m (the height of the center of thrust for the undisplacedWTS) and is shown in Figure 3.19. We make the assumption that this windvector is always acting at the center of thrust of the WTS.Figure 3.18: Overview of the Realistic Open-Loop Test Wind and WaveDisturbances.Again, due to the limitation of collecting wave information in FAST,only a single frequency wave was used. The frequency and amplitude of thewave were determined by considering the Pierson-Moskowitz ocean wavespectrum [38]. This wave energy spectrum assumes a fully developed waveprofile caused by steadily blowing wind over an infinitely long amount oftime, and can be calculated byS(ω) =sαg2ω5exp(sβ(ω0ω)4), (3.53)483.5. Validation of Non-Linear ModelFigure 3.19: Wind and Wave Profiles for Realistic Open-Loop Test.where parameters sα, g and sβ are constants, and ω0 is given byω0 =gU19.5. (3.54)Here, U19.5 is the wind velocity at a height of 19.5 m above sea level. Sinceour reference height is much higher, we can estimate U19.5 by using the493.5. Validation of Non-Linear Modelpower law approximationU19.5 = U90(h19.5h90)κ, (3.55)where U90 is the wind speed at height 90 m, h19.5 and h90 represent referenceheights of 19.5 m and 90 m, respectively, and κ is the wind power exponentwhich has the value 0.11 over open water environments [39]. The Pierson-Moskowitz spectrum is shown in Figure 3.20 for the above conditions. Fromhere we extract the significant wave height and peak spectral period for theabove wind conditions, which we can then use as inputs to FAST.Figure 3.20: Pierson-Moskowitz Energy Density Spectrum for Given Crite-rion.Control Inputs The open-loop control input trajectories were chosen tomimic realistic responses to the wind disturbance with the objective of sim-ple power capture, similar to that of the 5MW baseline controller. First,the yaw angle was created with the objective of tracking the wind direction.However, given the limited speed of the yaw actuator, the one-to-one trajec-tory was smoothed using a locally weighted scatter plot smoothing method503.5. Validation of Non-Linear Modelsuch that its maximum time derivative was less than the actuator limitationof 0.3 deg/s. The resulting yaw trajectory, overlaid on the wind direction,is shown in Figure 3.21.Figure 3.21: Nacelle Yaw Angle Trajectory.The blade pitch angle trajectory was then created with the goal of main-taining constant aerodynamic power. This is accomplished by setting theinstantaneous change in power to zero for all time, orδP (t) =∂P (t)∂xδx +∂P (t)∂uδu +∂P (t)∂vδv = 0. (3.56)As this is an open-loop simulation, and thus we are not interested in in-tegrating the equations of motion and solving for the states over time, weignore the state vector x. Removing the state term, and inserting the rele-vant inputs, this simplifies to∂P (t)∂βδβ +∂P (t)∂γδγ +∂P (t)∂vδv = 0. (3.57)Finally, we solve for the blade pitch angleδβ = −(∂P (t)∂γδγ +∂P (t)∂vδv)/(∂P (t)∂β). (3.58)513.5. Validation of Non-Linear ModelAs in the case for the yaw angle, the resulting trajectory invalidates theactuator limits. To resolve this issue, we again apply locally weighted scatterplot smoothing, leading to the final trajectory, shown in Figure 3.22.Figure 3.22: Blade Pitch Angle Trajectory for Realistic Open-Loop Test.In region 3, generator torque is typically either held fixed at the ratedvalue, letting the blade pitch actuation maintain constant power, or it isadjusted inversely to generator speed to help maintain constant power. Ne-glecting state changes, both these options result in constant torque for theopen-loop simulation. However, for the purpose of validating the effects ofall control inputs and disturbances, we elect to use a slightly more advancedgenerator torque control strategy. That is, we attempt to hold rotor speedconstant with generator torque. This can be done by ignoring the shaftdynamics between the rotor and generator side shafts, and translating thegenerator torque directly to the rotor sideω˙r =1Jr(Pωr− (NGR)Tg)= 0. (3.59)Following similar steps to the blade pitch angle case, we can simplify this523.5. Validation of Non-Linear Modelequation toδTg =1NGRωr(∂P (t)∂βδβ +∂P (t)∂γδγ +∂P (t)∂vδv)(3.60)The final trajectory is shown in Figure 3.23.Figure 3.23: Generator Torque Trajectory.Results The time series results for the simulations done in both FASTand the proposed non-linear model are shown in Figure 3.24, and the meanand standard deviations of the time series’ are tabulated in Table 3.3.Evidently, from Figure 3.24, the proposed control-oriented model veryaccurately predicts the major motions of the full 44-state FAST model, evenin the presence of wind and wave disturbance, actuator effects, and coupled3-D motion. There is minor discrepancy in the platform yaw degree offreedom as well as the two drivetrain velocities. In all three cases, this canbe attributed to the periodic bending motions of the blades as they rotateabout the hub due to periodic gravitational effects, and the tip-speed effectdiscussed in Section 3.3.2. Because the effect is minor, however, and becauseintroducing periodicity to the non-linear model would make generating lineartime-invariant plants impossible, we continue in our decision not to model533.5. Validation of Non-Linear ModelFigure 3.24: Realistic open-loop Test Results.it.As an additional source of validation, we looked at the frequency responseof the time series responses of the realistic open-loop test. These are shown inFigures 3.25, 3.26 and 3.27. In Figure 3.25, the large amplitude spikes at 0.5543.5. Validation of Non-Linear ModelTable 3.3: Realistic Open-Loop Test Results for Non-Linear ModelsDOF MeanCOMMeanFASTRMS COM RMSFASTSurge (m) 3.4363 3.8418 1.1789 1.152Sway (m) 1.8477 1.7045 0.55266 0.53453Heave (m) -9.9328 -9.9155 0.407 0.46666Roll (deg) -0.34107 -0.33334 0.27595 0.26428Pitch (deg) 1.6481 1.5351 0.76216 0.5842Yaw (deg) -0.15391 -0.48659 0.17994 0.32669Rotor Speed(rpm)11.9842 11.9191 0.14502 0.161GeneratorSpeed (rpm)1162.4665 1156.1478 15.1428 15.6932rad/s correspond to the effect of the wave disturbance on the translationalstates. The lower frequency excitation can be attributed to the naturalFigure 3.25: Frequency Response for Translational States.frequencies of the the platform, and the actuator effects. Figure 3.26 showsthe frequency response of the rotational states of the system. Again, we553.6. SummaryFigure 3.26: Frequency Response for Rotational States.see the effect of wave disturbance, as well as several other modes. Theadditional peaks are a result of the fact that the rotational reactions areaffected by both the temporal frequency of the waves (like the displacementstates) and the spatial frequency. Finally, in Figure 3.25, we see that thedrivetrain degrees of freedom are relatively unaffected by the wave effects,and dominated by the wind and actuator effects.3.6 SummaryIn this chapter, we defined a general, control-oriented, non-linear model forfloating offshore wind turbines that captures the major dynamics of a WTS.By considering the 5MW baseline wind turbine, we described the detailsof generating this non-linear model and acquiring the necessary parametersfrom existing software. Finally, we subjected the proposed model to fourvalidation tests spanning a multitude of operating conditions to verify itseffectiveness in the presence of wind and wave disturbances, actuator dy-namics, and strongly coupled 3-D motions. More importantly, we confirmed563.6. SummaryFigure 3.27: Frequency Response for Shaft States.that a simple non-linear model could very accurately predict the major dy-namics of a floating wind turbine, while maintaining reasonable simplicitysuch that is a good fit for controller design.The major contributions of this chapter stem from two key facts. Thefirst is simply that the proposed model offers an optimized balance of sim-plicity and accuracy. Simplicity allows us to maintain a fundamental under-standing of its workings and perform operations like analytical linearization,while accuracy is necessary for controller design purposes. More specifi-cally, the model remains accurate throughout the highly non-linear operat-ing range of the wind turbine. This means that this one model is alwaysvalid, and does not need to be re-derived or regenerated like many of thesimple control models used in research [11, 25, 28, 31]. On a more fun-damental level, though, because the model is physics based, it means thatwe can make intuitive changes to the parameters for testing design changesor replaceable components. For example we can experiment with differentlevels of mooring line stiffness, tower heights, mass properties, or aerody-namic properties. The allowance for these mechanical design changes means573.6. Summarythat the non-linear model could even be used as a first step in collaborat-ing mechanical design with the design of the control system. Traditionally,mechanical design would precede controller design, and the processes wouldbe independent.58Chapter 4Linear Model4.1 IntroductionIn the previous chapter, we derived the simplified non-linear model from firstprinciples. For controller design, however, it is generally more advantageousto have a time-invariant linear model. The purpose of this section is toshow how to use the physically-derived non-linear model from Chapter 3to generate a linear model. We begin by defining the operating point for asuch a linear model, and describing a method to generate it. Next, we usethe generated operating point to derive the time-invariant linear model, andperform some simple manipulations to it to improve its accuracy. Finally,we validate it by comparing it to FAST and a linearized model generatedby FAST, in a time-series simulation.4.2 Derivation of Linear Model and EquilibriumPointsIn order to create a linear model from a non-linear model, we first need todefine an operating point, p. The operating point can be either constant, orperiodic, and comprises the states, inputs and disturbances for the particulartrajectory. More precisely, we define the operating point asp =xopuopvopwop , (4.1)594.2. Derivation of Linear Model and Equilibrium Pointswhere the subscript ”op” indicates the operating point. Given such anoperating point, any small perturbation from it can be approximated bya linear state space modelδx˙ = A¯|p δx + B¯ |p δu + B¯v|p δv + B¯w|p δw (4.2)δy = C¯|p δx + D¯ |p δu + D¯ v|p δv + D¯w|p δw, (4.3)where the perturbation state vector is defined by δx := x− xop, and otherperturbation vectors δu, δv and δw are defined analogously. The followingsubsections will explain the process of acquiring a valid operating pointfrom the non-linear model, and deriving the linear state space model at theacquired operating point.4.2.1 Selection of an Operating PointDue to the non-linearity of the WTS, the selection of an operating pointcan make a large difference in the accuracy of the model. For example, theeffect of a positively increasing control input β and the produced power ispositive in one part of the operating region of the wind turbine, and negativein another. Hence, a controller synthesized at a far operating point wouldnot only be ineffective, but almost surely unstable.The only criterion for the validity of an operating point or trajectoryis that the non-linear equation of motion for the WTS from Chapter 3 issatisfied, that isx˙op(t) = f(xop(t),uop(t),vop(t),wop(t)). (4.4)Generally, a time-varying operating trajectory should be periodic. If theoperating point is time-invariant, this restriction can be simplified tof(xop,uop,vop,wop) = 0. (4.5)It is important to note that we cannot solve for an independent rotor andgenerator azimuth angle using this method. That is, we must use the com-604.2. Derivation of Linear Model and Equilibrium Pointsbined state θr − θgNGR . The method of finding the operating point differsdepending on whether or not we are interested in a fixed operating point, ora periodic operating trajectory. While the proposed model can be linearizedabout either, for simplicity we will focus on only the fixed operating point.Fixed operating point Fixed operating points can easily be solved forusing iterative methods, once the problem has been properly defined. First,consider the non-linear function f from (4.5). Let the variables xop,uop,vopand wop have nx, nu, nv and nw elements, respectively. Then, f must havenx rows. Furthermore, in order for the operating point to be time-invariant,several states are predetermined. This can also be shown as a consequenceof (3.7) and (4.5), or~˙xg = ~0~˙θ = ~0ωr =1NGRωg. (4.6)Through this simplification, both the number of undetermined state vari-ables and order of the non-linear equations drop from nx to a certain numberwhich we call nxu. Let us also define a new non-linear function that doesnot include the redundant informationfu(x,u,v,w) :=~fF (x,u,v,w)~fT (x,u,v,w)fQ(x,u,v) . (4.7)The order of the function fu represents the maximum number of variablesthat can be solved for using iteration methods. In other words, nxu of thenxu + nu + nv + nw operating point variables can be explicitly solved for,the others must be chosen. If we denote the to-be-determined variables pu,then there exists a linear transformation T¯psuch thatpu = T¯pp, (4.8)614.2. Derivation of Linear Model and Equilibrium Pointsand another linear transformation T¯fsuch thatfu = T¯ff (4.9)Using a multi-variable form of Newton’s method [40], we can iteratively solvefor pu using the expressionpk+1u = pku −[∂fu∂pku]−1fu(pk), (4.10)where the superscript k represents the kth iteration, and the inverse termcan be found by solving the equation[∂fu∂pku]T¯p= T¯fg[A¯|pk B¯ |pk B¯v|pk B¯w|pk]. (4.11)Equation (4.11) represents a very important result, as it lets us solve forthe equilibrium point very quickly using only the partial derivatives of thenon-linear function at each iteration step. On the other hand, in FAST,equilibrium points are determined by running long simulations in the timedomain. Depending on the initial conditions, the time it takes to reachequilibrium often exceeds the maximum allowable runtime, which makesthe equilibrium point unobtainable.4.2.2 Derivation of the State-Space ModelConsider a small perturbation about the equilibrium trajectory from (4.4).From general linearization theory, as long as the perturbation is sufficientlysmall, we can write a linear system asδx˙ =∂f∂x∣∣∣∣p(t)δx +∂f∂u∣∣∣∣p(t)δu +∂f∂v∣∣∣∣p(t)δv +∂f∂w∣∣∣∣p(t)δw. (4.12)624.3. Manipulation of the Linear ModelIt is clear, then, that the linear model matrices from (4.2) are simply thepartial derivative matrices from (4.12), orA¯|p(t) = ∂f∂x∣∣∣p(t), B¯|p(t) = ∂f∂u∣∣∣p(t)B¯v|p(t) = ∂f∂v∣∣∣p(t), B¯w|p(t) = ∂f∂w∣∣∣p(t). (4.13)Similarly, if we are interested in outputs y, given by the equationy = go(x,u,v,w), (4.14)then the output matrices from (4.2) can be written asC¯|p(t) = ∂go∂x∣∣∣p(t), D¯|p(t) = ∂go∂u∣∣∣p(t)D¯ v|p(t) = ∂go∂v∣∣∣p(t), D¯w|p(t) = ∂go∂w∣∣∣p(t). (4.15)There are some effects, however, that the partial derivative model does notaccurately represent. Therefore, some slight manipulations can be made tothe linearization process. These are explained in the following section.4.3 Manipulation of the Linear Model4.3.1 Quadratic DragFrom Chapter 3, Morison’s equation contains the hydrodynamic drag forceexpression:~Fd = C||~vt||~vt. (4.16)The partial derivative of this, with respect to one of the states, i, can bewritten as∂ ~Fd∂xi(∂||~vt||∂xi~vt + ||~vt||∂~vt∂xi). (4.17)For the WTS, we usually linearize the model about a point with zero meanvelocity, which renders all terms in the above equation zero. This is notideal, however, because although we expect zero mean wave velocity, we doexpect oscillatory movement about the equilibrium point. For this reason,634.4. Validation of the Linear Modelwe have applied a correction to the above equation, to account for the actualdrag force on the system. This is done by replacing the absolute value ||~vt||in the right-most term by a new, constant value ||~vt,nom||. This does notalter the desired directional properties of the force, as the corrected termmust always be positive. The corrected velocity is determined by equatingthe energy dissipated in a 1-D linear drag model and a 1-D quadratic dragmodels over half an oscillation period. That is,−C∫ piω0||v(t)||v(t)(v(t)dt) = −C||v||nom∫ piω0v(t)(v(t)dt). (4.18)If we assume v(t) follows a sinusoidal profile v(t) = v0sin(ωt), the aboveintegrals can be solved analytically. Switching back to the 3-D model, thisleaves the relationship||~v||nom =(83pi)v0. (4.19)To obtain v0, we can excite our non-linear model with a regular wave profilewhose frequency and amplitude corresponds to the expected ocean wavespectrum, and find the resulting velocity amplitude v0 such thatv0 = max(||~v(t)||) (4.20)4.4 Validation of the Linear ModelTo validate the linear model, we can compare the simulation results for alinear model created using the above process, with the full non-linear FASTsimulation that was done in Section 3.5.4. Furthermore, to highlight theadvantages of the proposed linear model, we compare it to a 42-state linearmodel generated by FAST.4.4.1 Linearization Point - NLMThe equilibrium or linearization point was generated using the static equi-librium point finding process described in Section 4.2. Because of the staticcondition, all the velocity states were set to zero. The wind velocity vector644.4. Validation of the Linear Modelwas set to constant 18 m/s at 20◦ about the eˆ3-axis. All of the componentsin the wave disturbance signal were set to zero. Finally, the rotor speedand generator torque were set to their rated values, and the yaw was set to20◦. That left all the displacement states to be determined, as well as therotor flexibility and the blade pitch angle by the equilibrium point solvingmethod. The full equilibrium point then isx ={3.90m, 1.76m,−9.91m,−0.50◦, 1.60◦,−0.06◦,0.26◦,0, . . . 0,12.1rpm, 1173.7rpm}, (4.21)u = {15.16◦, 40.68kNm, 20◦}, (4.22)v = {16.91m/s, 6.16m/s0m/s}, (4.23)w = {0, . . . , 0}. (4.24)4.4.2 Linearization Point - FASTThe linearization point was found by using FAST’s linearization feature withall degrees of freedom enabled except the nacelle yaw bearing stiffness. Thisprocess is explained in more detail in [10]. The linearization process tookjust less than 1500 simulated seconds to achieve a convergence toleranceof less than 10−3 rad and 10−3 rad/s for the 2-norm of displacements andvelocities, respectively. For brevity, the full 42-state equilibrium point willnot be shown, but the values corresponding to the states and control inputsof the proposed non-linear model arex ={3.66m, 1.85m,−9.90m,−0.35◦, 1.56◦,−0.70◦,0.26◦,0, . . . 0,12.13rpm, 1176.4rpm}, (4.25)654.4. Validation of the Linear Modelu = {15.11◦, 40.65kNm, 20◦}, (4.26)v = {16.91m/s, 6.16m/s0m/s}, (4.27)All of these values are very close to those obtained using the non-linear modelwith one exception, the platform yaw. This deviation can be explained bythe relative tip-speed effect which was mentioned in Section 3.3.2. Because ofthe tilt of the rotor about the eˆ2-axis, both due to the steady state platformpitch angle, and the physical tilt of the rotor, the blades alternate betweenmoving toward the incoming wind and away from it, as they perform a fullrevolution. This effect causes a constant moment about the eˆ3-axis due tothe changing thrust forces on the blades, which leads to the steady-stateplatform yaw angle. As explained in 3.3.2, however, the effect is small andthus it is neglected in the proposed model.4.4.3 Open-Loop Test ResultsThe test conditions remain exactly the same as those described in Section3.5.4, and each of the two linear models begin the simulation at all statesequal to zero. The time series results for the full-DOF FAST model, pro-posed linear model, and the linear FAST model are shown in Figure 4.1,and the RMS errors, given in percent, for the time series difference betweeneach linear model and FAST are given in Table 4.1.Table 4.1: Realistic Open-Loop Test for Linear Models.DOF RMS Error % NLM RMS Error % FASTSurge 12.3103 98.0504Sway 33.5927 142.1242Heave 31.552 100.0169Roll 57.3344 105.7494Pitch 43.4443 88.5098Yaw 89.4817 96.0532Rotor Speed 108.1024 134.5055Generator Speed 108.1649 134.0284As we expect, the lack of a Bw disturbance matrix in the FAST linear664.4. Validation of the Linear ModelFigure 4.1: Realistic Open-Loop Test Results for the Linear Models.model results in an inability to predict the wave-induced platform motions.Contrarily, the proposed non-linear model shows very accurate results for the674.5. Summaryplatform motion DOFs, resulting in massive improvements in RMS error %.For the drivetrain degrees of freedom, the proposed non-linear model is ableto predict the major motions caused by the balanced aerodynamic forces(i.e., changing wind disturbance and effects of control input), however it isunable to predict the periodic motions caused by the bending of the bladesdue to gravity, the tip-speed-effect, and inertial forces. Because this is aperiodic effect, it cannot be captured by any time-invariant linear model,including the 42-state FAST linear model. As a result, there is a minorimprovement in the rotor and generator RMS error % for the proposedmodel compared to the FAST linear model.Another important note to make is, because of the complexity of theFAST model, the equilibrium point that was found was not accurate enough,despite a very lengthy linearization process. As a result, we start to seewandering of the states by the end of the linear simulation. This is analogousto linearizing a non-linear model by taking partial derivatives at an operatingpoint where (4.4) is not satisfied. To combat this, either less states couldbe chosen, or tolerance restrictions could be raised. The former solutionwould result in a lower accuracy linear model, and the latter would requireexponentially longer linearization time. Furthermore, FAST restricts thetotal allowable linearization time to 9999s, which is only slightly higherthan the time used to find the above equilibrium point.4.5 SummaryIn this chapter, we formulated an explicit procedure for determining time-invariant operating points from the proposed non-linear model, describedin Chapter 3. Using this operating point, we then derived the state-spacelinearized model for use in controller design. Because of the intuitive natureof the proposed model, we were also able to perform manipulations to thelinearized model to improve accuracy where the behavior of the WTS wasparticularly non-linear. Finally, we validated the linear model using a realis-tic open-loop test, involving changing wind and wave disturbances, actuatordynamics, and fully coupled 3-D motion.684.5. SummaryThis chapter highlights several important contributions of our work. Themost obvious of such being the generation of accurate linear models whichare required for many types of controller design. Another major achievementis the proposed method for determining operating points which remains fastand effective even when initial condition selection is poor. In contrast, thelinearization feature in FAST is slow and highly sensitive to initial condi-tion selection. Often the linearization point is not solvable to a reasonabledegree of accuracy in the maximum allowable runtime of the software, aswas indicated by the states drifting in the realistic open-loop test. Fur-thermore, other interesting things can be done using the proposed methodfor determining operating points, such as finding the moveable range of thewind turbine at various power levels. This could prove to be very useful inwind farm control to minimize the wake effects of upwind wind turbines ondownwind wind turbines.69Chapter 5Controller Design5.1 IntroductionIn the previous chapters, it was shown that a simplified non-linear modelcould be used to generate accurate linear state-space models that includethe wave disturbance matrix Bw. Based on the effectiveness of disturbanceaccommodation in other floating wind turbine research [11, 24, 27], it is pre-dicted that controllers that consider this disturbance relationship in theirdesign, will provide significant improvement over control techniques that donot. In this chapter, we explore this thesis by comparing four types of con-trollers; a non-model-based PID controller that is tuned using the non-linearmodel; a model-based linear-quadratic-regulator (LQR) with integrator andobserver; a disturbance-rejecting model-based robust H∞ controller; andfinally a disturbance-accommodating feedforward H∞ controller.5.2 Test Conditions and Equilibrium PointIn this section, the test conditions for the comparison of the three controllersare defined, and the equilibrium point and linear model is established. Thegeneral conditions are similar to the validation tests for both the linear andnon-linear models, however as we are no longer comparing with FAST, it ispossible to use more realistic irregular waves, which will be explained in thefollowing subsection.705.2. Test Conditions and Equilibrium Point5.2.1 DisturbanceThe wind disturbance remains the same as in the non-linear and linear vali-dation tests. That is, 18m/s mean at a 20◦ angle about the z-axis. The wavedisturbance was built using the Pierson-Moskowitz spectrum with the samepeak spectral period, significant wave height, and direction as before, butthe single wave frequency was replaced by a more realistic fully developedwave profile that fits the Pierson-Moskowitz energy density spectrum. Thewave profile was constructed using the following method. First, the energydensity spectrum is split into n = ωco∆ω frequency components, as shown inFigure 5.1. Here, ωco is the cut-off frequency we choose to approximatethe highest frequency in the realistic wave profile. Then, an irregular waveFigure 5.1: Construction of a Wave Profile from the Pierson-MoskowitzSpectrum.profile can be constructed by the summation of n monochromatic waves, orone for each frequency band. The amplitude of each wave is given byAi =√2S(ωi)∆ω, (5.1)and the phase byφi = rand(0 : 2pi). (5.2)715.2. Test Conditions and Equilibrium PointFinally, we can calculate the full wave disturbance vector at any spatialand temporal location with the following formulaeζi =−ω2igx+ ωit+ φi, (5.3)h(x, t) =∑iAi sin(ζi) (5.4)~v(x, t) =∑iωi exp(−ω2igz)cos(α) sin(ζi)sin(α) sin(ζi)cos(ζi) , (5.5)~a(x, t) =∑iω2i exp(−ω2igz)cos(α) cos(ζi)cos(α) cos(ζi)− sin(ζi) , (5.6)P (x, t) =∑iρg exp(−ω2igz)sin(ζi). (5.7)5.2.2 Linearization PointThe equilibrium or linearization point was generated using the static equi-librium point finding process described in Section 4.2, and is the same as theone derived in Section 4.4.1. To reiterate, the wind velocity vector was set toconstant 18m/s at 20◦ about the z-axis, the wave disturbance signals wereset to zero, the rotor speed and generator torque were set to their ratedvalues, and the yaw was set to 20◦. The other displacement states weredetermined, as well as the rotor flexibility and the blade pitch angle weredetermined using the equilibrium point solving method. The full equilibriumpoint then isx ={3.90m, 1.76m,−9.91m,−0.50◦, 1.60◦,−0.06◦,0.26◦,0, . . . 0,12.1rpm, 1173.7rpm}, (5.8)725.2. Test Conditions and Equilibrium Pointu = {15.16◦, 40.68kNm, 20◦}, (5.9)v = {16.91m/s, 6.16m/s0m/s}, (5.10)w = {0, . . . , 0}. (5.11)5.2.3 Control ObjectivesFor region 3 operating conditions, the controller objective should be to main-tain a constant 5MW power capture, while minimizing platform and rotortwisting motions. More specifically, the controller should minimize platformroll and pitch accelerations, and rotor twist acceleration, while maintain-ing rotor speed and power captured. It will be assumed that there aremultiple sensors on the wind turbine, with the ability to measure all thepositional states of the turbine, the rotor twist angle, the platform roll andpitch accelerations, and rotor twist accelerations, the generator speed andthe power. All these measurements are realistically obtainable, however itis also possible to use a state-estimator to reduce the number of sensorsrequired. Furthermore, because the design of such a state estimator can bedecoupled from the controller design, it is justified to assume access to allthe above output information. The output vector y can then be written asy =~xg~θθr − 1NGR θgθ¨xθ¨yωgθ¨r − 1NGR θ¨gωgTg. (5.12)Note that not all the controllers use all the available outputs, but the abovevector y represents the potential outputs available. With all of the testconditions set, and the controller objectives defined, we can design the PID,LQR and H∞ controllers for comparison purposes. The following sections735.3. Design of the PID Controllerwill explain the design of each controller, and finally the comparison will beperformed.5.2.4 Performance MetricsSeveral performance metrics will be used to compare the relative perfor-mance of the various controllers. Specifically, similar to [5] and [17], wewill use the RMS of the following signals: platform roll, pitch and their timederivatives, rotor twist and its time derivative, blade pitch rate, power error,and rotor speed error. The platform motion indices give an estimate of thetower structural fatigue loading, the rotor twist indices give an estimate ofthe drivetrain fatigue loading, and finally the rotor speed and power errorsgive an estimate of both the power output and the generator loads. Often,we will normalize these performance metrics for a particular controller withrespect to a baseline controller by dividing the former by the latter.5.3 Design of the PID ControllerGain-scheduling PID controllers are often used in onshore wind turbine con-trol due to the simplicity of their design, analysis and implementation. How-ever, it was shown in Chapter 1 that the floating offshore wind turbine con-trol problem is much more complex than the onshore problem because ofthe additional aspects to be considered, such as platform motion, auxiliarysensors and wave disturbance. For PID design, this means that for everyinput/output combination, a distinct PID controller must be developed andtuned in parallel with every other PID controller. It is also important tocarefully select a number of such controllers that balances design complexityand controllability. Intuitively, we elect to design six distinct PID controllers(G1(s) . . . G6(s)), to regulate the platform roll and pitch accelerations, thegenerator speed, the rotor shaft torsion, and the captured power. The blade745.3. Design of the PID Controllerpitch angle, then, is given byβ(s) =[G1(s) G2(s) G3(s) G4(s)]θ¨x(s)θ¨y(s)δωg(s)θ¨r(s)− 1NGR θ¨g(s) , (5.13)and the generator torque, byTg(s) =[G5(s) G6(s)] [ θ¨r(s)− 1NGR θ¨g(s)δ(ωg(s)Tg(s))]. (5.14)Because the complexity of the PID design process is highly dependent onthe number of PID controllers used, we elect not to design any controllersfor the nacelle yaw input, as its restrictive actuator limitations make it arelatively ineffective choice for control. The block diagram for the overallPID feedback structure is shown in Figure 5.2. The six controllers weredesigned by following a three step process.Figure 5.2: PID Block Diagram.Step 1: Independent Design The first step was to design each con-troller independently, using the linear model and a process similar to theZiegler-Nichols tuning formula [41]. Since each output requires regulation,the controllers were tuned for the response to a 1m/s step increase in windspeed, such that the following conditions were met:755.4. Design of the LQR Controller with Integrator• Settling time as fast as possible• Zero steady state error• Actuator limits obeyedOnce each controller is tuned, we move on to Step 2.Step 2: Collective Design With each controller set, the step responseis repeated for the linear model with all controllers enabled. Since the con-trollers are either competing or collaborating for each control input, theindependent controller criterion from Step 1 are no longer satisfied. Eachcontroller, then, is tuned once again such that the collection satisfies all thecriterion.Step 3: Non-Linear Implementation Finally, the controller is appliedto the full non-linear simulation, as described in Section 5.2. Once again,the conditions are closely monitored and any tunings necessary are made tothe controllers such that the criterion are met.This concludes the design process for the PID controller. The final valuesobtained after following this process are shown in Appendix B. Next, theLQR controller is examined.5.4 Design of the LQR Controller with IntegratorThe LQR state-feedback controller, which is introduced and described in[42], is very common in literature for the control of floating offshore windturbines [5],[27],[24],[25]. It is a natural choice because, unlike the PIDwhere each input/output combination is tuned independently by the op-erator, in the LQR all such combinations are tuned simultaneously. Fur-thermore, the process is automated and guaranteed to be optimal, withrespect to the weighting scheme chosen by the operator. Since the LQRis a state-feedback controller, a state-observer is necessary to acquire thenon-measurable states. In the next subsections, the LQR controller and theobserver will be presented in order.765.4. Design of the LQR Controller with Integrator5.4.1 LQR with an IntegratorTraditionally, LQR controllers regulate only the states of the plant. Inorder to also regulate the outputs and eliminate steady-state-error, an inte-grator is added to the customary state-feedback structure. The full feedbackstructure, that is state-feedback plus output error integration is shown inFigure 5.3. Here, K¯ aand K¯are the integrator and state-feedback gains, re-Figure 5.3: LQR Block Diagram.spectively, and xh is the full estimated state vector from the state-observer.The matrices K¯ aand K¯can be solved for simultaneously using a traditionalLQR cost function by stacking the usual state vector with the output vector.The cost function to be minimized, then, isJ =∫ ∞0(zTQ¯z + uTR¯u)dt, (5.15)where the augmented state vector z is given byz =[xyLQR], (5.16)775.4. Design of the LQR Controller with Integratorand the augmented state weighting matrix Q is the diagonal matrixQ¯= Q¯ xQ¯ y . (5.17)Here, Q¯ xand Q¯ yrepresent the weighting matrices for the states and outputs,respectively. Furthermore, because the position states of y are included inx, the LQR output vector yLQR consists of only the latter five componentsof y. The cost function in (5.15) can be minimized using the function LQI.min Matlab. The initial weighting values were intuitively selected with thegoal of normalization with respect to an arbitrarily selected state, state 1.For example, we say one meter of surge is equally weighted to one degreeof platform pitch rotation, which is equivalent to 1 degree of blade pitch.Then, the initial weightings for surge, pitch and blade pitch are 1, 180/pi,and 1, respectively. Finally, through simulations, the weighting values areiteratively adjusted until we have minimized fatigue-related loads and powerfluctuation as much as possible, while staying within our actuator limits.The diagonal values of the final state and output weighting functions, and785.4. Design of the LQR Controller with Integratorinput weighting function areQ¯ x= diag1101.05e11.05e101.75e-21e-11e-101.75e-31.75e-301e-11.03e-3, Q¯ y= diag1.75e-41.75e-41.8e-51.19e81e-6, R¯= diag1e31e−11e6 ,(5.18)and the final controller gains are given in Appendix B.5.4.2 Reduced-order ObserverFor the LQR controller, it is necessary to have a real-time estimate xhfor all the states x of the system. This can be accomplished either witha full-order state-observer, which estimates the measured and unmeasuredstates, or a reduced-row observer for estimation of only the unmeasuredstates [43]. It was clear, after implementing both, that the reduced-rowobserver was more accurate in predicting the unmeasured states. Therefore,the reduced-row observer was selected. The block diagram of the observeris shown in Figure 5.4. Here, P¯is the transformation matrix that separatesthe measurable states xm and unmeasurable states xu, such that[xmxu]= P¯x. (5.19)795.5. Design of the H∞ with IntegratorFigure 5.4: Reduced Row Observer Block Diagram.P¯is chosen such that xm consists of all the platform position and orientationstates, the rotor flexibility state and the generator speed, and xu consists ofthe platform velocity and angular velocity states, as well as the rotor speed.L¯is the observer gain, and the remaining matrices B¯y, B¯u, and A¯ obscanbe found by following the procedure for a continuous-time reduced orderobserver in [44]. Using the transformation P¯, we can re-express the linearstate-space model asddt[xmxu]= P¯A¯P¯−1[xmxu]+ P¯B¯u, (5.20)or equivalentlyddt[xmxu]=[A˜¯ 11A˜¯ 12A˜¯ 21A˜¯ 22] [xmxu]+[B˜¯1B˜¯2]u. (5.21)Finally, L¯is designed by pole placement of the system A˜¯ 22− L¯A˜¯ 12. For theproposed observer, we use poles {−10,−10.5,−11,−11.5,−12,−12.5,−13},.The final observer matrix L¯is given in Appendix B.5.5 Design of the H∞ with IntegratorThe H∞ method is well suited for multi-objective control problems and dis-turbance rejection, particularly when the disturbance frequency properties805.5. Design of the H∞ with Integratorare well known. Although H∞ control has been applied to onshore windturbines [45, 46], it remains far less explored for offshore turbines. For anintroduction and overview of H∞ control, see [47], and for a good introduc-tion on selecting weighting functions see [48]. The authors of [11] were ableto utilize the known frequency properties of wind and wave for disturbancerejection in their H∞ controller for a 2-D wind turbine system with success,however as was explained in Chapter 2, this could only be done for unreal-istic 2-D motion. This section will explain the methods used to derive theproposed disturbance rejecting H∞ controller.A general block diagram for the H∞ feedback loop is shown in Figure 5.5.Here, K¯∞is the dynamic H∞ controller, and W¯ v, W¯ w, W¯ uand W¯ eare theweighting functions for wind, wave, control inputs and outputs, respectively.All the weighting functions can be either constant or frequency dependent,but generally we select W¯ uand W¯ eas constant gains to balance actuatorusage and reference tracking, and W¯ vand W¯ was frequency dependent, fordisturbance rejection. We also define the augmented plant P¯augas the linearFigure 5.5: H∞ Block Diagram.815.5. Design of the H∞ with Integratortransformation x˙auge˜u˜e = P¯augxaugvwuk , (5.22)where uk represents the controller outputs, e the measured outputs, u˜ and e˜the minimization variables, and xaug the augmented state vector. As we willsee, it is convenient to decompose this augmented plant into a state-spacesystem asP¯aug= A¯ aug B¯augC¯augD¯ aug . (5.23)Identifying the components of this matrix P¯augwill be key to synthesizingan H∞ controller, however before we can do so we must first define theweighting functions.5.5.1 IntegratorTo eliminate steady-state-error, we can force the augmented plant to havean integrator. We do this by splitting the actuator input u into two parts,the proportional part uP and the integrated part uI such that u = uP+1suI.We also need to define a new state xI for the augmented plant which is givenbyx˙I = (−I)xI + uI. (5.24)Here, is simply a very small scalar which is necessary for numerical stabil-ity of the augmented plant, and I¯is the identity matrix. Graphically, theintegrator system is shown in Figure 5.6.5.5.2 Weighting SelectionMany weighting function combinations were considered in the design of theH∞ controller. Constant weighting functions W¯ uand W¯ ewere determinediteratively to give a good balance between output tracking and control input.825.5. Design of the H∞ with IntegratorFigure 5.6: H∞ Integrator Block Diagram.In the end, the values we used wereW¯ e= diag[0, . . . , 0, 0.00396, 0.00726, 8.0617, 0.199, 9.698], (5.25)W¯ u= diag[10, 0.2, 2]. (5.26)For the frequency-dependent weighting functions W¯ vand W¯ w, we werecareful not to use too many different transfer functions, as each one adds anadditional state to the augmented plant. It was determined through experi-mentation that the most significant wave disturbance was the z-componentof wave acceleration, which caused large platform roll and pitch motions.Thus, we designed three identical weighting functions to reject the verticalwave acceleration at each of the outer columns. Using the ocean wave spec-trum from Figure 3.20, and with the goal of attenuating frequencies belowthe apparent cut-off of 1 rad/s, we came to the weighting functionW¯ w,az=1.1s+ 0.01146s+ 11.46. (5.27)The overall wave disturbance weighting matrix W¯ w, then, is a 36x36 diagonalmatrix where all the elements have value 1 except those in rows 6, 15, and 24,which have value W¯ w,az. These three rows correspond to the z-componentwave acceleration disturbances at each of the outer columns. Finally, we835.5. Design of the H∞ with Integratorcan convert the weighting matrix W¯ wto state space form for compatibilitywith our augmented plant. This adds three additional states to the model,given byx˙w = A¯W,wxw + B¯W,ww, (5.28)and the new weighted wave disturbance signal, which is given byw˜ = C¯W,wxw + D¯W,ww. (5.29)The values of these four matrices have been omitted for brevity. The result-ing closed-loop transfer functions between the first column’s z-componentwave acceleration and the platform roll and pitch outputs are shown inFigure 5.7. It is clear that attenuation is achieved below 1 rad/s. It isimportant to note that we can achieve further attenuation by using higherorder weighting functions, however that comes at the cost of increasing thenumber of states of our augmented plant.For the wind disturbance weighting function W¯ v, we choose not to addany additional states. That is, W¯ v= I3×3. This decision was made forthe purposes of this thesis to highlight the benefits of rejecting wave dis-turbance. In reality, wind spectrum frequency profiles are well understood,and rejecting major frequencies via the wind disturbance weighting functioncould prove to be very useful.5.5.3 Final StructureAfter applying all of the above-described weighting functions, we can com-bine the states from the wind turbine system with the integrator and wavedisturbance weighting function states to form the augmented state vectorxaug = [x,xI,xw]T. Similarly, we can form the control output vector withthe proportional and integrator terms, or uk = [uP,uI]T. It is easy to show845.5. Design of the H∞ with IntegratorFigure 5.7: Closed-loop Transfer Function Between Wave Acceleration andPlatform Motion.that the four components of the augmented plant Paug are given byA¯ aug=A¯B¯B¯wC¯W,w0 − 00 0 A¯W,w (5.30)B¯aug=B¯vB¯wD¯W,wB¯00 0 0 I¯0 B¯W,w0 0 (5.31)C¯aug=−W¯ eC¯−W¯ eD¯−W¯ eD¯wC¯W,w0 W¯ u0−C¯−D¯−D¯wC¯W,w (5.32)855.6. Design of the H∞ Controller with Feedforward TermsD¯ aug=−W¯ eD¯ v−W¯ eD¯wD¯W,w−W¯ eD¯00 0 W¯ u0−D¯ v−D¯wD¯Ww−D¯0 (5.33)Finally, with the defined augmented plant, we can use the Matlab functionhinfsyn.m, to find the controller system K¯∞that both stabilizes the closed-loop system from Figure 5.5 and minimizes the infinity norm of the resultingtransformation from inputs to outputs of the system. The ensuing controllerK¯∞is given in Appendix B.5.6 Design of the H∞ Controller withFeedforward TermsThe design process for the feedforward H∞ controller is largely the sameas the disturbance rejecting H∞ controller, but with two major differences.First, we elect not to use the wave disturbance weighting function thatwas used in the alternate H∞ controller. Second, we add two disturbancemeasurements to the augmented plant outputs.5.7 Weighting SelectionThe constant weighting functions W¯ uand W¯ ewere once again determinediteratively to balance output tracking and control input. The final valuesused wereW¯ e= diag[0, . . . , 0, 0.32, 0.61, 1.18e4, 25.73, 9.32], (5.34)W¯ u= diag[10000, 200, 2000]. (5.35)5.7.1 Identification of the Feedforward TermsFor feedforward control, we make use of two disturbance measurements;relative wind velocity, and heave pressure. Heave pressure can be measuredby placing a pressure sensor at the bottom of each column of the semi-865.7. Weighting Selectionsubmersible platform, and wind velocity can be either measured with lidartechnology [49], or estimated with a disturbance estimator [24].The heave pressure is the total pressure acting on the heave surface,which we assume is the total hydrodynamic force acting perpendicular tothe heave plate, divided by the plate area. From Chapter 3, we can formulatethe mathematical expression asPi =1Ai[ 0 0 1 ]R¯T ~FDh,i, (5.36)where the heave force ~FDh,i on each plate i is given by~FDh,i = Kdh,i||~vh,i||~vh,i +Kah,i~ah,i + PiAieˆh. (5.37)Using the linearization process described in the Chapter 4, we can generatea linear expression for the pressure P as a function of the system states andthe wave disturbance, orδP = C¯Px + D¯P,ww (5.38)The relative wind velocity is simply the relative speed between the windand the rotor, in a direction perpendicular to the rotor plane. This is givenbyvr = ||~vn||, (5.39)where ||~vn|| is described in Chapter 3. Again, we linearize the formulationat the equilibrium point, which gives the expressionδvr = C¯vrx + D¯ vru + D¯ vr,vv. (5.40)875.8. Controller Comparison Results5.7.2 Final StructureThe final structure for the augmented plant is given byx˙x˙ie˜u˜ePvr=A¯B¯B¯vB¯wB¯00 − 0 0 0 I¯−W¯ eC¯−W¯ eD¯−W¯ eD¯ v−W¯ eD¯w−W¯ eD¯00 W¯ u0 0 W¯ u0−C¯−D¯−D¯ v−D¯w−D¯0C¯P0 0 D¯P,w0 0C¯vrD¯ vrD¯ vr,v0 D¯ vr0xxivwuPuI.(5.41)Once again, we use the Matlab function hinfsyn.m to solve for a controllerK¯∞, which is given in Appendix B.5.8 Controller Comparison ResultsIn this section, the results are shown and explained for each controller indi-vidually, and then all together at the end.5.8.1 PID vs Baseline GSPIThe multi-objective PID (MOPID) results can be compared against thebaseline GSPI described in [18] and updated for the semi-submersible plat-form in [1]. This GSPI controller is a constant torque, collective blade pitchcontroller whose sole objective is rotor speed regulation (and by consequence,power regulation) via blade pitch actuation. The authors do not use plat-form motion for feedback, instead they choose their controller gains suchthat the controller does not excite the natural frequencies of the platformmotions.The normalized performance metrics, for the MOPID compared to theGSPI are shown in Figure 5.8. It is clear that the MOPID achieves dras-tically reduced drivetrain loads and improved power regulation comparedto the baseline GSPI, with a 50.1% and 60.4% improvement in rotor speed885.8. Controller Comparison ResultsFigure 5.8: GSPI vs MOPIDand power error, respectively, and a 81.6% and 80.8% improvement in rotortwist acceleration and velocity, respectively. The improvements come at a2995% increase, however, in blade pitch rate. It is explained in [16] that thebaseline GSPI, which was originally designed for an onshore wind turbine,had its gains significantly reduced for its offshore implementation because ofexcitation of the platform roll and pitch natural frequencies. In other words,the desired, much higher, gain values for the baseline GSPI were not possi-ble because of the single-input, single-output design choice. In the MOPID,this is not a problem, as minimization of these platform motions are controlobjectives and as can be seen in Figure 5.8, the platform motions for theMOPID controller are similar to those for the reduced-gain GSPI, even withthe increased blade pitch rate. Furthermore, we verify that the RMS bladepitch rate of 6.6 deg/s is well below the actuator limit of 8 deg/s, whichcorresponds to the blade pitch actuator spending about 5.8% of the total895.8. Controller Comparison Resultssimulation time at its maximum rate.5.8.2 LQR vs PIDWhile the input-output relationships have to be carefully selected by theuser to balance design complexity and controller performance in the PIDdesign process, the LQR process is an automated approach that utilizes allpossible input-output relationships. Of course this requires a linear state-space model containing all such relationships. The designed LQR controlleris compared to the MOPID, and the results are shown in Figure 5.9. TheFigure 5.9: MOPID vs LQRresults suggest a 80.0% and 77.7% improvement in rotor speed and powererror, as well as a 45.8% decrease in blade pitch rate. Similarly, we alsosee improvements in platform motion, with a 19% and 19.2% reduction inroll and pitch, respectively, and a 5.8% and 4.3% reduction in roll and pitchrates, respectively. Finally we see a 14.2 and 12.7% reduction in rotor twist905.8. Controller Comparison Resultsrate and acceleration, respectively. This drastic increase in performance canlikely be attributed to the complex, coupled relationships between the statesand inputs of the wind turbine system. In the PID case, we had to restrictfeedback to only the most obvious input-output relationships. As a result,we probably missed some important coupled phenomena, such as the effectof generator torque on platform motion.5.8.3 H∞ vs LQRLike the LQR controller, the H∞ controller is designed based on all input-output relationships. It also takes into account the disturbance relation-ships, and through loop-shaping, can reject said disturbances based on theirexpected frequency profiles. The designed H∞ controller is compared to theLQR, and the results are shown in Figure 5.10. Here, we see further improve-Figure 5.10: LQR vs H∞ments in fatigue minimization performance metrics, however this comes at915.8. Controller Comparison Resultsa cost to power and rotor speed error and blade pitch usage. Specifically,we see a 16.0% and 21.0% improvement in platform roll and pitch, and a9.4% and 6.9% improvement in platform roll and pitch rates. We also seea 14.3% improvement in rotor twist rate, but a 1.1% increase in rotor twistacceleration. Finally, we see a 15.6% increase in blade pitch usage, and a166% and 151% increase in rotor speed and power error. It should be noted,however, that the rotor speed and power error for the H∞ controller is still76% and 75% lower than in the baseline GSPI controller, and 40% and 49%lower than the MOPID controller, respectively. The overall comparisons willbe further described in Subsection 5.8.5. The improvements shown here forplatform motions and drivetrain loads can be attributed to the disturbancerejection capabilities of the H∞ controller.5.8.4 H∞ with Feedforward vs H∞Under the H∞ framework, it is very easy to add feed-forward signals tothe feedback loop, provided that such signals are measurable, and increasethe performance capabilities of the controller. The results for the H∞ feed-forward controller compared to the disturbance rejecting H∞ controller areshown in Figure 5.11. Using such a controller, we achieve a 53% decrease inpower error and a 16% decrease in rotor speed, at a 3% cost to blade pitchusage. These power and rotor side improvements can largely be attributedto the preview information about the wind speed, which allows the controllerto react to changing wind speeds before the rotor speed does. Platform pitchand roll rates, and the rotor twist metrics remain largely unchanged com-pared to the H∞ controller, however platform roll and pitch metrics increaseby 10% and 17%, respectively. Overall, the H∞ feed-forward controller al-lows for similar improvements in fatigue minimization as the H∞ controller,while recovering much of the power regulation capability.5.8.5 Overall Controller ComparisonThe results for all of the above-described controllers compared to the base-line GSPI controller are shown in Figure 5.12. Here, a value less than one925.8. Controller Comparison ResultsFigure 5.11: H∞ vs H∞ FFrepresents an improvement in the respective performance metric comparedto the baseline PID. Channels 1 through 9 are the same as above or, inorder: platform roll (1) and pitch (2), platform roll and pitch rate (3 and4), rotor twist rate (5) and acceleration (6), blade pitch rate (7), powererror (8) and rotor speed error (9). Non-normalized magnitudes of the per-formance parameters for each controller validation test are also given inTable 5.1, below. It is clear that the floating wind turbine system benefitsgreatly from model-based, multi-objective controllers such as the LQR, H∞and H∞ feed-forward. Furthermore, methods that use the wind and wavedisturbance information, either through loop-shaping or feed-forward, cansignificantly reduce platform motions and drivetrain loads, compared evento other multi-objective methods.935.9. SummaryFigure 5.12: Overall Controller Comparison5.9 SummaryIn this chapter, we used the linear model derived in Chapter 4 to constructseveral multi-objective controllers with the goal of regulating power captureand major platform motions. The controllers include a PID controller, anoptimal LQR controller, with integrator and observer, and two H∞ con-trollers, one with wave disturbance rejection and the other with disturbancefeedforward. After designing each of the controllers, we compared their per-formance with respect to one another, and to a baseline gain-schedulingcontroller created by the makers of advanced simulation software FAST.The goal of this section was both to reinforce the notion that multi-objective control is necessary for the success of floating offshore wind turbinesystems and, more importantly, that consideration of disturbance can signif-icantly enhance controller performance. Through closed-loop simulations ofthe four increasingly complex controllers, we showed these two statements945.9. SummaryPerformanceParameterGSPI MOPID LQRI H∞ H∞ FFPlatformRoll (rad)7.16e-03 7.34e-03 5.94e-03 4.99e-03 5.75e-03PlatformPitch (rad)1.48e-02 1.48e-02 1.19e-02 9.43e-03 1.11e-02PlatformRoll Rate(rad/s)2.38e-03 2.38e-03 2.24e-03 2.03e-03 2.09e-03PlatformPitch Rate(rad/s)5.52e-03 5.42e-03 5.18e-03 4.83e-03 4.71e-03Rotor TwistRate (rad/s)7.72e-03 1.48e-03 1.27e-03 1.09e-03 1.10e-03Rotor TwistAcc (rad/s2)1.02e-01 1.87e-02 1.63e-02 1.65e-02 1.72e-02Blade PitchRate (deg/s)2.13e-01 6.58e+00 3.57e+00 4.12e+00 4.57e+00Power Error(MW)4.50e-01 1.78e-01 3.99e-02 1.06e-01 5.00e-02Rotor SpeedError (rad/s)1.11e+01 5.51e+00 1.11e+00 2.79e+00 2.31e+00Table 5.1: Performance Results from Controller Validation Tests.to be true. Specifically, we showed that controllers that utilize the derivedwave disturbance matrix, either to reject the wave disturbance, or to predictthe effects of wave disturbance through feedforward loops, are superior interms of minimizing harmful motions and regulating power.95Chapter 6Conclusion6.1 Summary RemarksThis thesis proposes a simple physics-based control-oriented model for float-ing offshore wind turbines that aims to facilitate the design of advancedcontrollers. Using such a model provides several key advantages existingmethods, which were highlighted in each of the chapters of this text.In Chapter 3, we presented the simplified non-linear model for the float-ing wind turbine, which was derived from first principles. The general modelhas up to 15 or 16 states, 3 control inputs, 3 wind disturbance signals, and 9wave disturbance signals for each major buoyant column or platform. Thisgeneral model was then applied to a sample floating wind turbine that hadbeen well defined in research, and the specific force and torque models forthe aerodynamics, the hydrodynamics and the mooring lines were described.Finally, the model was validated in four distinct tests that covered a largerange of operating conditions for the highly non-linear system.As a control-oriented model, it is of course very important to be able togenerate accurate linear state-space models from the non-linear plant. InChapter 4 we described how to solve for equilibrium points of the WTS usinga simple iteration scheme and the partial derivatives of the non-linear func-tion. Using said equilibrium points, a procedure for linearizing the modelwas given, and some minor, intuitive adjustments were made to increase itsaccuracy. Finally, we used the realistic open-loop test defined in Chapter 3to compare the linear model to the complex simulator FAST, as well as ahigh state FAST-linearized model.Next, in Chapter 5, we designed several controllers using the proposedmodel. The controllers included a multi-objective PID controller, an LQR966.2. Contributionscontroller with an integrator and reduced-order observer, a wave disturbancerejecting H∞ controller, and a wave disturbance feedforward H∞ controller.These controllers were then compared to one another, and to a baselinegain-scheduled PID controller developed by the makers of FAST. In the firstcomparison, we showed that the proposed PID controller greatly outper-formed the baseline GSPI controller in rotor speed and power errors, and indrivetrain loads, but had relatively minor impact on platform motions. Thisincrease in performance was attributed to the fact that the GSPI controller,which only uses generator speed for feedback, had to be detuned to avoid ex-citing resonances in the drivetrain and tower. The next comparison showedthat the proposed LQR controller outperformed the proposed PID controllerin every performance metric. Most importantly, it was able to greatly reduceplatform motions, reducing tower fatigue loading. The wave disturbance re-jecting H∞ controller was able to further reduce drivetrain and platformmotions, but at the cost of higher rotor speed and power fluctuation com-pared to the LQR controller. These fluctuation were still considerably lessthan those of the PID and baseline PI controllers. Finally, the H∞ feedfor-ward controller performed in between the other H∞ controller and the LQRcontroller. From these results, we concluded that because of the complexcoupling of the WTS, controllers with more information about the dynamicsof the system, such as LQR and H∞ controllers, are better suited for thisparticular problem. More importantly, controllers that can utilize informa-tion about the wind and wave disturbances, are superior. The non-linearmodel proposed in this thesis offers the ability to incorporate wave distur-bance both directly (through feedforward), or indirectly (through rejection)thanks to the ability to generate the disturbance matrix in the linear model.6.2 ContributionsThe main contribution of this thesis is the development of the simplified,physics-based, control-oriented model of a floating wind turbine system.This model facilitates the creation of advanced controllers that are necessaryfor improving fatigue life of floating wind turbines while maximizing power976.2. Contributionscapture. More specific realizations of the contributions are listed below:• An accurate non-linear model that is based on physical parametersthat can easily be adjusted for different wind turbine systems, andfor experimenting with mechanical design. Higher complexity simula-tors, which consider high order dynamics at a cost to complexity, alsorequire considerable effort in changing mechanical parameters.• A model that maintains its accuracy throughout the highly non-linearoperating range for wind turbine systems, and can be linearized an-alytically at any point within this range for controller design. Mostsimple models used in existing research are based on parameter iden-tification of complex simulators. This type of fitting must be done foreach operating point, and is subject to high-order dynamical influence.• A method to solve for equilibrium points of the system, which can be avery long and tedious process when using high complexity simulators.This method can also be extended to other useful tasks such as findingthe moveable range of the wind turbine at a particular power capturelevel for wind farm optimization• A method of obtaining the Bw disturbance matrix, which is currentlynot available in research. This disturbance matrix can be used fordisturbance rejection or for feedforward control to greatly improvecontroller performance• The generation of several advanced controllers for the baseline 5MWwind turbine described by the National Renewable Energy Labora-tory. These controllers include a multi-objective PID controller, anLQR controller with an integrator and reduced-order observer, an H∞controller that rejects waves within its estimated energy spectrum, andfinally a feedforward H∞ controller that uses available wind and waveinformation to improve performance986.3. Future Work6.3 Future WorkControl of offshore wind turbines continues to be an important role in re-ducing the cost of wind energy and making it a top competitor in the energymarket. The research presented in this thesis can be extended to the follow-ing areas:Combined Mechanical and Control Design of Floating Wind Tur-bines Since the proposed model is based on physical parameters, and sinceit can accurately predict the major motions of floating wind turbines sys-tems, it is possible to collaboratively adjust the mechanical and controlsproperties of the system to optimize coupled parameters such as closed loopnatural frequencies and stability.Non-Linear Model Extensions Several possibilities for improvementof the non-linear model exist, including adding additional states such asblade or tower flexibility, adding additional inputs such as individual bladepitch control, and adding more complex behavior to the existing modelsuch as skewed wake correction and other aerodynamic effects, or actuatordynamics. It was found in [50], for example, that the inclusion of actuatordynamics in the linear model greatly improved drivetrain performance fora simple PI controller. It should be deeply considered whether or not theincrease in model complexity is worth the added accuracy, however.Moveable Range and Wind Farm Optimization Because of the equi-librium point generation procedure outlined in this thesis, it would be possi-ble to explore the effects of moving the individual floating wind turbines ina wind farm to optimize the combined power capture. This could also po-tentially reduce the large distances between turbines in the farm, decreasingthe electrical interconnectivity costs.LPV Control Finally, the nature of the simplified model, and the ease ofgenerating equilibrium points makes it a strong candidate for LPV control.996.3. Future WorkAlthough LPV control is common for onshore wind turbines [51–53], it is amore difficult problem for offshore wind turbines because of platform motion.Still, some work has been done on LPV control for floating offshore windturbines such as in [30] and [29]. 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Brath, “Linear parameter vary-ing control of wind turbines covering both partial load and full loadconditions,” International Journal of Robust and Nonlinear Control,vol. 19, no. 1, pp. 92–116, 2009.[52] C. Sloth, T. Esbensen, and J. Stoustrup, “Active and passive fault-tolerant lpv control of wind turbines,” in Proceedings of American Con-trol Conference, Baltimore, MD, pp. 4640–4646, 2010.[53] C. Sloth, T. Esbensen, and J. Stoustrup, “Robust and fault-tolerant lin-ear parameter-varying control of wind turbines,” Mechatronics, vol. 21,no. 4, pp. 645–659, 2011.106Appendix APhysical Constants for the5MW Sample WTSThis appendix lists the physical constants used for the sample WTS in Chap-ters 3, 4 and 5. Surfaces such as the Cp and Ct curves are not given, butthe method for obtaining them is explained in the thesis body.Property Variable Value UnitWater density ρ 1025 kg/m3Physical mass mg 14, 072, 718 kgTotal inertia about x-axis of body frame Ixx 1.695e10 kg ·m2Total inertia about y-axis of body frame Iyy 1.695e10 kg ·m2Total inertia about z-axis of body frame Izz 1.845e10 kg ·m2Table A.1: General PropertiesProperty Variable Value UnitAir density ρa 1.225 kg/m3Effective rotor radius Rr 62.94 mDistance vector from COM to centerof thrust~rgt −5099.889 mRotor Inertia Jr 3.5444e7 kg ·m2Generator Inertia Jg 5.34116e2 kg ·m2Driveshaft stiffness on rotor side kr 8.676e8 Nm/radDriveshaft damping on rotor side br 6.215e6 Nm · s/radGear ratio Ngr 97 [−]Table A.2: Aerodynamic Properties107Appendix A. Physical Constants for the 5MW Sample WTSProperty Variable Value UnitSubmerged density of the line ρc 113.35 kg/mLength of line lcat 835.5 mPretension of line EA 7.5363e8 NDistance vector from COM to line 1attachment point~rbgp1 −40.8680−4.11 mDistance vector from COM to line 2attachment point~rbgp2 20.434−35.3927−4.11 mDistance vector from COM to line 3attachment point~rbgp3 20.43435.3927−4.11 mLocation of line 1 attachment point ~xanc1 −837.60−200 mLocation of line 2 attachment point ~xanc2 418.8−725.3829−200 mLocation of line 3 attachment point ~xanc3 418.8725.3829−200 mTable A.3: Mooring Line Properties108Appendix A. Physical Constants for the 5MW Sample WTSProperty Variable Value UnitDamping coefficient for columns ca1,ca2, ca3Cd,a 0.68 [−]Damping coefficient for columns cb1,cb2, cb3, cc1, cc2, cc3Cd,bc 0.61 [−]Damping coefficient for innermostcolumns ca4,cb4, cc4Cd,4 0.56 [−]Damping coefficient for heave plates Cdz 4.8 [−]Added mass coefficient for outercolumnsCa 0.63 [−]Added mass coefficient for heaveplatesCaz 0.67 [−]Length of columns ca1, ca2, ca3 la 6 mLength of columns cb1, cb2, cb3 lb 4.11 mLength of columns cc1, cc2, cc3 lc 9.889 mDiameter of columns ca1, ca2, ca3 Dd,a 24 mDiameter of columns cb1, cb2, cb3, cc1,cc2, cc3Dd,bc 12 mDiameter of innermost columnsca4,cb4, cc4Dd,4 8.054 mDistance vector from COM to columnca1 center (undisplaced)~rbga1 −28.870−7.11 mDistance vector from COM to columncb1 center (undisplaced)~rbgb1 −28.870−2.06 mDistance vector from COM to columncc1 center (undisplaced)~rbgc1 −28.8704.94 mDistance vector from COM to columnca2 center (undisplaced)~rbga2 14.43−25−7.11 mDistance vector from COM to columncb2 center (undisplaced)~rbgb2 14.43−25−2.06 mDistance vector from COM to columncc2 center (undisplaced)~rbgc2 14.43−254.94 mTable A.4: Hydrodynamic Properties109Appendix A. Physical Constants for the 5MW Sample WTSDistance vector from COM to columnca3 center (undisplaced)~rbga3 14.4325−7.11 mDistance vector from COM to columncb3 center (undisplaced)~rbgb3 14.4325−2.06 mDistance vector from COM to columncc3 center (undisplaced)~rbgc3 14.43254.94 mDistance vector from COM to columnca1 center (undisplaced)~rbga4 00−7.11 mDistance vector from COM to columncb1 center (undisplaced)~rbgb4 00−2.06 mDistance vector from COM to columncc1 center (undisplaced)~rbgc4 004.94 mTable A.5: Hydrodynamic Properties (cont’d)110Appendix BController ParametersThis appendix lists the final controller parameters used for the validationtests in Chapter 5.B.1 PID ParametersThe PID parameters are shown in Table B.1.Table B.1: PID ValuesGain PID 1 PID 2 PID 3 PID 4 PID 5 PID 6P 250 100 -0.2 -10 50 0.002I 200 1 -0.05 -2.5 5 0.0002D 100 10 0 0 0 0B.2 LQR ParametersThe LQR matrices K¯ aand K¯are given in B.1 and B.2, respectively.K¯Ta =−2.25e−05 −5.40e−03 4.16e−10−2.29e−05 −5.55e−03 5.14e−105.15e−05 1.24e−02 3.24e−112.84e−03 6.83e−01 2.72e−072.92e−05 −1.21e−03 −8.60e−12, (B.1)111B.3. Parameters for LQR Observer and H∞ ControllersK¯T =−3.81e−03 −4.14e−01 −2.16e−07−8.47e−04 −9.63e−02 1.72e−086.43e−03 1.66e−01 2.97e−077.27e−01 1.63e+02 5.21e−05−2.15e+00 −4.43e+02 −2.23e−041.09e−02 1.04e+00 1.41e−06−5.77e+02 1.30e+04 2.23e−041.35e+00 −1.41e+00 2.59e−064.89e−01 −4.08e−01 3.84e−06−1.66e−01 9.07e−01 4.39e−06−5.19e+01 2.23e+02 −1.59e−031.42e+02 −7.06e+02 4.28e−03−1.11e−02 1.99e+00 7.22e−08−3.68e+02 5.33e+03 4.63e−052.54e+00 −2.91e+01 −1.28e−06. (B.2)B.3 Parameters for LQR Observer and H∞ControllersBecause of the large size of the observer matrix and the H∞ systems, thevalues have not been tabulated in this thesis. Instead, they are available athttp://www.sites.mech.ubc.ca/ cel/files.php.112"@en ;
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