"Applied Science, Faculty of"@en . "Mechanical Engineering, Department of"@en . "DSpace"@en . "UBCV"@en . "Bagherieh, Omid"@en . "2013-08-23T15:14:04Z"@en . "2013"@en . "Master of Applied Science - MASc"@en . "University of British Columbia"@en . "This thesis studies the application of gain-scheduling (GS) control techniques to floating offshore wind turbines on barge platforms. Modelling, control objectives, controller design and performance evaluations are presented for both low wind speed and high wind speed cases. Special emphasis is placed on the dynamics variation of the wind turbine system caused by plant nonlinearity with respect to wind speed.\nThe dynamics variation is represented by a linear parameter-varying (LPV) model. The LPV model for wind turbines is derived by linearizing the nonlinear dynamics at various operating wind speeds and by interpolating the linearized models.\nIn low wind speed, to achieve control objectives of maximizing power capture and minimizing platform movements, for the LPV model, the LPV GS design\ntechnique is explored. In this region, the advantage of making use of blade pitch angle as a control input is also investigated. In high wind speed, to achieve control objectives of regulating power capture and minimizing platform movements, both LQR and LPV GS design techniques are explored.\nTo evaluate the designed controllers, simulation studies are conducted with a realistic 5 MW wind turbine model developed at National Renewable Energy Laboratory, and realistic wind and wave profiles. The average and root mean square values of power capture and platform pitch movement are adopted as performance measures, and compared among designed GS controllers and conventional controllers. The comparisons demonstrate the performance improvement achieved by GS control\ntechniques."@en . "https://circle.library.ubc.ca/rest/handle/2429/44879?expand=metadata"@en . "Gain-Scheduling Control of Floating Offshore WindTurbines on Barge PlatformsbyOmid BagheriehB.Sc., Sharif University of Technology, 2011A 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 ColumbiaAugust 2013c? Omid Bagherieh, 2013AbstractThis thesis studies the application of gain-scheduling (GS) control techniques tofloating offshore wind turbines on barge platforms. Modelling, control objectives,controller design and performance evaluations are presented for both low wind speedand high wind speed cases. Special emphasis is placed on the dynamics variation ofthe wind turbine system caused by plant nonlinearity with respect to wind speed.The dynamics variation is represented by a linear parameter-varying (LPV)model. The LPV model for wind turbines is derived by linearizing the nonlinear dy-namics at various operating wind speeds and by interpolating the linearized models.In low wind speed, to achieve control objectives of maximizing power cap-ture and minimizing platform movements, for the LPV model, the LPV GS designtechnique is explored. In this region, the advantage of making use of blade pitchangle as a control input is also investigated. In high wind speed, to achieve controlobjectives of regulating power capture and minimizing platform movements, bothLQR and LPV GS design techniques are explored.To evaluate the designed controllers, simulation studies are conducted with arealistic 5 MW wind turbine model developed at National Renewable Energy Labo-ratory, and realistic wind and wave profiles. The average and root mean square val-ues of power capture and platform pitch movement are adopted as performance mea-sures, and compared among designed GS controllers and conventional controllers.The comparisons demonstrate the performance improvement achieved by GS controltechniques.iiPrefaceThis dissertation is original intellectual property of the author, Omid Bagherieh,under supervision of Dr. Ryozo Nagamune. This work has been completed in theControl Engineering Laboratory at the University of British Columbia. The resultspresented in Chapters 5 and 6 are going to be submitted for publications.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Wind Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Wind Turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Onshore and offshore wind turbines . . . . . . . . . . . . . . 21.2.2 Platforms for floating offshore wind turbines . . . . . . . . . 21.2.3 Mechanisms of wind turbines . . . . . . . . . . . . . . . . . . 41.2.4 Operating regions of wind turbines . . . . . . . . . . . . . . . 51.3 Wind Turbine Control . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7iv1.5 Wind Turbine Control Goals . . . . . . . . . . . . . . . . . . . . . . 91.6 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6.1 Wind turbine modelling . . . . . . . . . . . . . . . . . . . . . 101.6.2 Wind turbine controllers . . . . . . . . . . . . . . . . . . . . . 111.6.3 Control design theory . . . . . . . . . . . . . . . . . . . . . . 131.6.4 Simulation software . . . . . . . . . . . . . . . . . . . . . . . 141.7 Research Objectives and Methodology . . . . . . . . . . . . . . . . . 141.8 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Wind Turbine Modelling . . . . . . . . . . . . . . . . . . . . . . . . . 172.1 Non-linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Model Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 General Control Strategy and Performance Measures . . . . . . . 253.1 Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.1 Control objectives for Region II . . . . . . . . . . . . . . . . . 263.1.2 Control objectives for Region III . . . . . . . . . . . . . . . . 273.2 Feedback Structure for Controller Design . . . . . . . . . . . . . . . 293.3 Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.4 Performance Measures for Comparisons . . . . . . . . . . . . . . . . 363.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 Baseline Controllers in Region II and Region III . . . . . . . . . . 384.1 Low Wind Speed (Region II ) . . . . . . . . . . . . . . . . . . . . . . 384.1.1 Controller structure . . . . . . . . . . . . . . . . . . . . . . . 394.1.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . 394.2 High Wind Speed (Region III ) . . . . . . . . . . . . . . . . . . . . . 414.2.1 Controller structure . . . . . . . . . . . . . . . . . . . . . . . 414.2.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . 42v4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Gain-scheduling Control in Region II . . . . . . . . . . . . . . . . . 455.1 Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 LPV Gain-scheduling Controller . . . . . . . . . . . . . . . . . . . . 465.2.1 LPV gain-scheduling controller structure . . . . . . . . . . . . 465.2.2 LPV gain-scheduling controller design problem . . . . . . . . 495.2.3 LPV gain-scheduling controller tuning . . . . . . . . . . . . . 505.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 Gain-scheduling Control in Region III . . . . . . . . . . . . . . . . 566.1 Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.2 LQR Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.2.1 LQR controller structure . . . . . . . . . . . . . . . . . . . . 596.2.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 606.3 LPV Gain-scheduling Controllers . . . . . . . . . . . . . . . . . . . . 616.3.1 LPV controller structures . . . . . . . . . . . . . . . . . . . . 616.3.2 LPV gain-scheduling controller tuning . . . . . . . . . . . . . 646.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.1 Summary of Achievements . . . . . . . . . . . . . . . . . . . . . . . . 757.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Appendix A LPV Model of the Wind Turbine . . . . . . . . . . . . . 84A.1 LPV Model for Region II . . . . . . . . . . . . . . . . . . . . . . . . 84viA.2 LPV Model for Region III . . . . . . . . . . . . . . . . . . . . . . . . 86Appendix B Performance Measures for the Designed Controllers inRegion II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Appendix C Design Parameters for LPV Gain-scheduling Controllersin Region III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Appendix D Performance Measures for the Designed Controllers inRegion III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Appendix E Time Domain Simulations for the Selected Controllersin Region III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98viiList of Tables2.1 Specified parameter values of equilibrium points at a given distur-bance input ud,0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Meteorological conditions in Region III. . . . . . . . . . . . . . . . . 323.2 Wind turbine performance measures . . . . . . . . . . . . . . . . . . 363.3 Platform performance measures not considered in the controller design 374.1 Evaluared performance measures for the baseline controller in Region II 414.2 Evaluated performance measures for the baseline controller in Region III 445.1 Weighting functions for LPV gain-scheduling controllers in Region II 505.2 Evaluated performance measures and their percent variation com-pared to BaseR2 in Region II . . . . . . . . . . . . . . . . . . . . . . 516.1 Controller properties in Region III . . . . . . . . . . . . . . . . . . . 576.2 Evaluated performance measures for selected controllers in Region III(Cond. I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3 Evaluated performance measures for selected controllers in Region III(Cond. II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70B.1 Evaluated performance measures for power capture and platformmovements in Region II. . . . . . . . . . . . . . . . . . . . . . . . . . 88viiiB.2 Evaluated performance measures for derivative of platform move-ments of the baseline controller in Region II. . . . . . . . . . . . . . 89C.1 Weighting functions for LPV GS OF controllers in Region III . . . . 91C.2 Weighting functions for LPV GS SF controllers in Region III . . . . 91D.1 Evaluated performance measures for power capture and platformmovements of the baseline controller in Region III. . . . . . . . . . . 97D.2 Evaluated performance measures for derivative of platform move-ments of the baseline controller in Region III. . . . . . . . . . . . . . 97E.1 Controller parameters for selected controllers in Region III . . . . . . 98ixList of Figures1.1 Floating platform concepts for offshore wind turbines. (Taken from [23]with the author?s permission.) . . . . . . . . . . . . . . . . . . . . . . 31.2 An illustration of a floating offshore wind turbine system and itsplatform degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . 51.3 The power curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Thesis organization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1 The block diagram representation of the non-linear model (2.1) inFAST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 The block diagram representation of the linearized model in FAST. . 213.1 A general feedback structure for controller design. . . . . . . . . . . 293.2 A general feedback structure for simulations. . . . . . . . . . . . . . 303.3 Wind speed profiles for simulations. . . . . . . . . . . . . . . . . . . 333.4 Wave elevation profiles for simulations. . . . . . . . . . . . . . . . . . 344.1 The baseline controller structure in Region II. . . . . . . . . . . . . . 394.2 Control inputs for the baseline controller in Region II. . . . . . . . . 404.3 System responses for the baseline controller in Region II. . . . . . . . 404.4 The baseline controller structure in Region III. . . . . . . . . . . . . 414.5 Control inputs for the baseline controller in Region III under Cond. I. 434.6 Control inputs for the baseline controller in Region III under Cond. II. 43x4.7 System responses for the baseline controller in Region III under Cond. I. 444.8 System responses for the baseline controller in Region III under Cond. II. 445.1 An LPV gain-scheduling controller structure in Region II. . . . . . . 475.2 A feedback control structure . . . . . . . . . . . . . . . . . . . . . . . 485.3 A feedback structure with a generalized plant. . . . . . . . . . . . . . 495.4 Control inputs for simulations. . . . . . . . . . . . . . . . . . . . . . 535.5 System responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.1 LQR feedback structure for Region III . . . . . . . . . . . . . . . . . 596.2 LPV GS feedback structures for Region III. . . . . . . . . . . . . . . 626.3 LPV GS OF controller for Region III. . . . . . . . . . . . . . . . . . 636.4 LPV GS SF controller for Region III. . . . . . . . . . . . . . . . . . . 646.5 Error in power versus platform pitch response for different controllers(Cond. I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.6 Error in power versus platform pitch response for different controllers(Cond. II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.7 System responses for time-domain simulations for selected controllersin Region III (Cond. I) . . . . . . . . . . . . . . . . . . . . . . . . . . 726.8 System responses for time-domain simulations for selected controllersin Region III (Cond. II) . . . . . . . . . . . . . . . . . . . . . . . . . 73D.1 Platform angle response for different controllers (Cond. I) . . . . . . 93D.2 Platform angular rate response for different controllers (Cond. I) . . 93D.3 Platform displacement response for different controllers (Cond. I) . . 94D.4 Platform translational rate response for different controllers (Cond. I) 94D.5 Platform angle response under for different controllers (Cond. II) . . 95D.6 Platform angular rate response for different controllers (Cond. II) . . 95D.7 Platform displacement response for different controllers (Cond. II) . 96D.8 Platform translational rate response for different controllers (Cond. II) 96xiE.1 Control inputs for time-domain simulations for selected controllers inRegion III (Cond. I) . . . . . . . . . . . . . . . . . . . . . . . . . . . 99E.2 Control inputs for time-domain simulations for selected controllers inRegion III (Cond. II) . . . . . . . . . . . . . . . . . . . . . . . . . . . 100xiiNomenclatureSymbol Description? Blade pitch angle (degree)?max Tip speed ratio for maximum efficiency?g Generator speed (rpm)?g,ref Reference for generator speed (rpm)?r Rotor speed (rpm)A State matrix of the linearized plantAK State matrix of the controllerB Input matrix of the linearized plantBK Input matrix of the controllerBd Disturbance matrix of the linearized plantC Output matrix of the linearized plantCK Output matrix of the controllerDK Feed-forward matrix of the controllere Error in the feedback signal from the reference valuef Non-linear plant modelJ Cost function for designing the linear quadratic regulatorK Gain for the baseline controller in low wind speedK(v) Controller state-space modelKI Integral gain for the baseline controller in high wind speedKP Proportional gain for the baseline controller in high wind speedxiiiKe Gain for the error in the state-feedback controllerKv Ratio of reference generator speed signal to wind speed in lowwind speedKx Gain for the states in the state-feedback controllerN Gear-box ratioP Power capture (W)P (v) Plant state-space modelPavg Average of power capture (W)Plyp Lyapunov functionPrated Rated power capture (W)Prms Root mean square of power capture (W)phv Platform heave displacement (m)phvrms Root mean square of platform heave displacement (m)ppt Platform pitch angle (degree)pptrms Root mean square of platform pitch angle (degree)prl Platform roll angle (degree)prlrms Root mean square of platform roll angle (degree)psg Platform surge displacement (m)psgrms Root mean square of platform surge displacement (m)psw Platform sway displacement (m)pswrms Root mean square of platform sway displacement (m)pyw Platform yaw angle (degree)pywrms Root mean square of platform yaw angle (degree)Q Weighting matrix on states for the linear quadratic regulatorR Weighting matrix on states for the linear quadratic regulatorr Blade radius (m)Tg Generator torque (N.m)t Time (s)xivu Control inputs to the plantud Disturbance inputs to the plantv Wind speed (m/s)vRegionII The low wind speed operating range (Region II )vRegionIII The high wind speed operating range (Region III )We Weighting function for error in LPV gain-scheduling controllerWppt Weighting function for platform pitch angle in LPV gain-scheduling controllerWu Weighting function for control inputs in LPV gain-schedulingcontrollerWu? Weighting function for control inputs? rate-of-change in LPVgain-scheduling controllerx States of the linearized planty Plant outputsy?g Generator speed output from the plant (rpm)y?g,0 Generator speed at an equilibrium condition (rpm)y?g,ref Reference for generator speed (rpm)yppt Platform pitch angle output from the plant (degree)yref Reference for the feedback signalxvAcronymSymbol DescriptionBaseR2 Baseline controller in low wind speedBaseR3 Baseline controller in high wind speedGS Gain-schedulingICICS Institute for Computing, Information and Cognitive SystemsLMI Linear matrix inequalityLPV Linear parameter-varyingLPV GS Linear parameter-varying gain-scheduling controllerLPV GS OF Linear parameter-varying gain-scheduling controller withoutput-feedbackLPV GS SF Linear parameter-varying gain-scheduling controller with state-feedbackLQR Linear quadratic regulatorLQR F Linear quadratic regulator controller with constant gainsLQR GS Gain-scheduling linear quadratic regulator controllerNREL National Renewable Energy LaboratoryNSERC National Sciences and Engineering Research CouncilRMS Root mean squarexviAcknowledgementsThis work was supported by the Natural Sciences and Engineering Research Councilof Canada (NSERC) and the Institute for Computing, Information and CognitiveSystems (ICICS) at the University of British Columbia. I would like to thankmy supervisor Dr. Ryozo Nagamune for his support, keeping me going when timeswere tough, asking insightful questions, and offering invaluable advice. It is a plea-sure to thank Dr. Jason M. Jonkman at the National Renewable Energy Laboratory(NREL) in US for his helpful comments and advices on the usage of the computer-aided engineering software FAST. I also would like to thank Mr. Jeff Homer for hisvaluable comments on writing the thesis.Omid BagheriehThe University of British ColumbiaAugust 2013xviiThis thesis is dedicated to my parents.For their endless love, support and encouragement.xviiiChapter 1Introduction1.1 Wind EnergyThe earth is unevenly heated; the poles get less heat from the sun than the equatordoes, and dry land heats up and cools down more quickly than the sea. Theseconditions make a global convection system on the earth; the movement of airbecause of this convection system is called wind.The energy that is available in the wind is significantly higher than the totalenergy which is used by human beings. At the present time, only a small portionof the energy used by human beings is produced by wind. Nowadays, because ofconcerns about the limited amount of fossil fuels and their adverse impact on en-vironment, as well as clean and renewable nature of wind energy, there is a trendworldwide that aims at increasing the wind energy production. For example, duringthe period 2005?2009, the installed wind energy capacity has increased approxi-mately 27 percent per year. In 2009, Denmark wind energy production accountedfor approximately 20 percent of total energy usage [41]. Now, European Union isaiming at producing 20 percent of its electrical energy from wind by 2020 [17].Although the price of wind energy is approaching to that of electricity gen-erated by fossil fuels, it is still a key to reduce further the wind energy cost for the1increased usage of wind energy.1.2 Wind TurbinesThe energy available in the wind is captured by a device called a wind turbine. Awind turbine converts the kinetic energy in the wind into the mechanical energyby rotating the turbine blades. Then, this rotational energy is transformed intoelectrical energy through a generator.1.2.1 Onshore and offshore wind turbinesWind turbines can be installed on land or at sea which are respectively called onshoreand offshore wind turbines. Although most of the currently installed wind turbinesare onshore, offshore wind turbines have several advantages over onshore ones. Firstof all, the wind is usually steadier and less turbulent above the sea due to the evensurface of the sea in comparison with the land. In addition, the noise made byturbines due to the rotation of blades is not an issue for the wind turbines that areinstalled away from the land. Furthermore, the scenery disturbed by wind turbineson the sea is not an issue, because no one is living there.In this thesis, offshore wind turbines are studied.1.2.2 Platforms for floating offshore wind turbinesOffshore wind turbines are categorized into two major groups based on the depth ofwater; shallow or deep. In shallow water, they are connected by piles to the seabed,i.e., non-floating. In this case, they are similar to the onshore wind turbines, butthe tower vibrations excited by the waves need to be accounted for in turbines?performance analysis. On the other hand, in deep water which is more than about60m depth [37], it is not economical to use the long pile and fix the turbine to theseabed. Rather, a turbine with a platform floating on the water is an economically2viable solution in deep water sea. Three different types of platforms were presentedin [23] to be used in deep water, as shown in Figure 1.1. The turbines installed ontop of these platforms are called Floating Offshore Turbines.Figure 1.1: Floating platform concepts for offshore wind turbines. (Taken from [23]with the author?s permission.)In the Ballast Stabilized Platform which is also called Spar-buoy, sand isadded at the bottom of the platform to pull down the center of gravity of theturbine. This added mass stabilizes the system against the disturbances induced bythe sea waves and wind. Moreover, this platform is restricted by cables in order toprevent collision between turbines in the wind farm.In the Mooring Line Stabilized Platform which is also called Tension LegPlatform (TLP), the platform is fixed by cables. These cables are under tensionand fixed to the anchors which are installed on the seabed. It is relatively expensive3to make the anchors that can tolerate the tension of these cables on the seabed.In the Buoyancy Stabilized Platform which is also called Barge Platform, theplatform is floated on the sea and restricted by cables to prevent it from collidingwith other turbines. This platform is cheaper compared to other platforms, becauseit is floating on the water. Since part of this platform is above sea level, waves caninduce large loads on the platform. These loads will cause fatigue in the structureand components of the turbine, as well as the reduction in the captured energy.In this thesis, the Barge Platform is studied.1.2.3 Mechanisms of wind turbinesThe overall structure of a wind turbine and its platform degrees of freedom areshown in Figure 1.2. The wind passes through the rotational plane of the blades.The interaction between the blades and wind produces drag and lift forces, whichrotates the blades. This rotation is transmitted to the generator via a gear-box.The reasons for using a gear-box are that the blades which are large mechanicalparts cannot rotate with high rotational speed, and that generator speed shouldbe relatively high in order to have lower torque on the generator side due to itstorque limitation. Therefore, the gear-box increases the speed and reduces theforces transmitted to the generator.Each wind turbine has three kinds of actuators, in order to change generatortorque, blade pitch angles and nacelle yaw angle. Both generator torque and bladepitch angles are used to control the rotational speed of the shaft and the powercapture. The three blades used in a wind turbine can be controlled individually ortogether which are called individual blade pitch and collective blade pitch control,respectively. To design a less complex controller, we will use the collective bladepitch method. For offshore wind turbines, blade pitch angles can also affect theplatform movements. On the other hand, the actuator for the nacelle yaw angleis used in order to face the turbine in the direction of wind. Here, it is assumed4that the wind direction is not changing and the turbine is always facing the winddirection. Therefore, nacelle yaw angle actuator is not used for this work.In this thesis, generator torque and collective blade pitch angle areused as control inputs.Figure 1.2: An illustration of a floating offshore wind turbine system and its platformdegrees of freedom.1.2.4 Operating regions of wind turbinesFigure 1.3 illustrates an example of the operating region for the specific wind turbineconsidered in [23] and in this thesis. The available energy in wind and also the energywhich should be captured by the turbine is shown in this figure. If the wind speedis lower or higher than a limit, the turbine is not in operation. These two limitsare called cut-in and cut-out wind speeds. The operating region of wind turbines5is between these two speeds. If the wind speed is lower than the cut-in wind speed(Region I ), the energy in the wind is not enough to rotate the blades. Also, if thewind speed is higher than the cut-out wind speed, the turbine will be shut down toavoid excessive mechanical and electrical loads.0 5 10 15 20 2501234567x 106Wind speed (m/s)Power capture (W) Turbine power captureWind powerRated wind speedCut?out wind speedCut?in wind speedRegion I Region IIRegion I?II Region II?IIIRegion IIIFigure 1.3: The power curve.The rated wind speed is the lowest wind speed at which the rated power of aturbine is produced. This wind speed divides the operating region of a wind turbineinto low wind speed region (Region II ) and high wind speed region (Region III ).For simplicity, the transition regions between Regions I and Regions II, as well asRegions II and Regions III, are not studied in this thesis.The low wind speed operating range (Region II ) and high wind speed op-erating range (Region III ) for the specific turbine considered in [23] and here arerespectively given byvRegionII := {v : 7.8 ? v ? 10.5} , (1.1)vRegionIII := {v : 11.4 ? v ? 25} , (1.2)where v (m/s) denotes wind speed.6In this thesis, Region II and Region III operating regions are studied forthe controller design.1.3 Wind Turbine ControlWind turbines need to be controlled in order to achieve its objectives. There aremultiple levels of control for a wind turbine; supervisory, operational and subsystem.These control levels are explained as follows [41]:? Supervisory control level: This control level, based on the wind speed, deter-mines whether the turbine should be in operation or not. In addition, theturbine needs to be shut down whenever faults are detected in the system.? Operational control level: This control level is used in the operating regionof wind turbines, and determines the proper values for each control input inorder to satisfy the control objectives.? Subsystem control level: This control level is used inside each actuator andfollows the commands given to the actuator at the operational control level.In this thesis, only the operational control level is considered. Thiscontrol level will be realized using the actuators for generator torque and blade pitchangle.1.4 MotivationTo make wind energy competitive with fossil fuel energy, the price of electricityproduced by wind turbines should be reduced. Some of the means to reduce thewind energy price are to increase energy efficiency, to increase lifetime of the windturbine, and to reduce maintenance costs. The last two means can be grouped underthe structural loads reduction objective.7One of the most economical ways for achieving these objectives, withoutinvolving any additional hardware or modifications to the structure, is to use aneffective feedback control algorithm which controls the actuators in the turbines.Therefore, we seek to design a controller which can achieve the mentioned objectives.The current state of the art in controlling the wind turbine can get thepower efficiency of less than 50 percent [41]. However, the Betz law [41] states thatmaximum achievable efficiency is 59.3 percent. This gap exists because of mechanicallimitations such as friction in the system, and also because of underdeveloped controltechniques. By implementing a proper controller, this gap can be reduced. Evenone percent increase in energy efficiency of a wind farm of 100 MW can increase theincome more than $120, 000 annually [41]. On the other hand, better regulation ofpower capture in high wind speed can reduce the power fluctuation of the producedelectricity fed into the electrical grid.Since the wind turbine is an expensive device, the equipment and mainte-nance cost play a substantial role in the end-price of the produced electricity. Thiscost will be influenced by large structural loads in the system, because these loadswill produce fatigue, leading to the decreased lifetime of the turbine and the in-creased maintenance costs. Therefore, reduction in the structural loads is one of themost important objectives in designing the controller for the turbines.In the floating wind turbines, the base of the structure is not fixed to theseabed, and can move in water. These movements are rotations and displacementsin any direction which will cause structural loads on the system in addition toreduction in the amount of energy capture. These structural loads can producefatigue, decrease the lifetime of the turbine, and also increase maintenance costs.Therefore, considering the floating platform movements reduction in the controllerdesign is important.81.5 Wind Turbine Control GoalsConsidering the objectives mentioned in the previous section, the problem is inves-tigated in two wind speed regions: Region II and Region III. In Region II, the goalsare to maximize power capture and simultaneously reduce structural loads on thesystem. In Region III, the power capture goal is changed to regulating the power ofturbines to the rated power, and at the same time trying to reduce structural loadson the system.These problems are for both onshore and offshore wind turbines. However, inthe case of offshore wind turbine, the platform movements can affect energy captureas well as structural loads on the system. Therefore, platform movements reductionneeds to be considered in the problem.Also, waves introduce another disturbance which exists at sea. This distur-bance mainly affects platform movements which in turn affects the energy capture.The effects of waves need to be reduced using proper control algorithms.The control problem is stated mathematically for each region in the subse-quent chapters.1.6 Literature ReviewThe European Union targets 20 percent of total electrical energy production to comefrom wind by 2020 [20, 17]. This decision shows the importance and growth of windenergy. The main problem for development of wind energy is its cost in comparisonwith other sources of energy. The National Renewable Energy Laboratorys (NREL)National Wind Technology Center estimated the cost for onshore and offshore windturbines in [18]. This estimation can help to investigate the feasibility of developingwind farms for each specific location. The installation cost for offshore wind turbinesis much higher than onshore ones, but there is more potential for power capture inoffshore sites.9Offshore wind turbines are still in the development process. The basic con-cepts used in offshore wind turbines and the different platforms used for shallowand deep water were investigated in [33, 37, 42]. Also, the engineering challengesin offshore wind turbines were presented in [14]. In June 2009, the world?s firstfloating offshore wind turbine was installed in the North Sea [10]. This turbine usesa spar-buoy platform.In this section, the current state of the art in modelling and control of onshoreand offshore wind turbines are explained. Then, the control theories needed todesign the controllers are explained. Also, software used for running simulations areintroduced.1.6.1 Wind turbine modellingA model which describes system behaviours is necessary to design a controller.The wind turbine models for both onshore and offshore wind turbines have beendeveloped. In [7], the model was presented for an onshore wind turbine as aninterconnection between mechanical subsystems such as rotor, blades and generator.Since this model is non-linear, it should be linearized for controller design based onlinear control theories [46].The model of floating offshore wind turbines is also non-linear. This non-linear model of floating offshore wind turbines was thoroughly explained by Jonkmanin [26]. He implemented this non-linear model in software FAST. This softwareis capable of not only running simulations using the non-linear model, but alsoobtaining the linearized model of the turbine at each operating point. On the otherhand, an overview of offshore wind turbine modelling was presented by [36]. Thisoverview mainly focuses on modelling for the controller design purposes. Also, acontrol oriented model of offshore wind turbines were explained in [6].The linearized models obtained in these papers are only accurate around thelinearization point and the linearized models will vary by the change in operating10points. Therefore, a linear model which can take into account this variation shouldbe found. The linear parameter-varying (LPV) model of the system is one solutionto this problem, because it will take into account the plant variation and at thesame time offers a linear model for the system. The LPV model of onshore windturbines is developed in [49]. Also, the LPV model for offshore wind turbines isobtained in [4]. This LPV model only takes into account the blade rotation angle asthe varying parameter. Therefore, a more general LPV model which considers thechange in operating conditions of the wind turbine as the varying parameter shouldbe developed.1.6.2 Wind turbine controllersThe wind turbine control is necessary for the operation of a wind turbine from start-up to shut-down. The controllers for starting up and shutting down the turbineare developed in [23] and [32]. The operation of a turbine between start-up andshut-down moments are called the operating region of a turbine. As mentioned,this operating region is divided into Region II and Region III. In [9] and [40], onecontroller is designed for the entire operating region. However, the controllers areusually designed separately for these two regions [5, 1, 21, 23, 24, 38, 39] and acontroller for the transition between these regions is employed [8, 23]. In this thesis,we only consider the design of separate controllers for Region II and Region III; thetransition controller is beyond the scope of this thesis.Several different control methods have been applied to onshore wind tur-bines for both Region II and Region III. In Region II, an output feedback controllerwas presented by [23]. To improve the system performance, LPV gain-schedulingcontrollers were designed in [9, 40]. In Region III, model predictive control andreceding-horizon control were designed for onshore wind turbines in [46] and [47],respectively. These controllers were designed based on the model of a turbine atone operating point. On the other hand, the PI gain-scheduling controller which11considers the model variation is presented in [19]. This controller has a PI structureand its gains are scheduled by the change in operating conditions. Other advancedcontrollers which consider the LPV model of the system in the controller designwere developed in [7, 8, 9, 1, 34, 35, 40, 43, 44, 45].Control of floating offshore wind turbines has been an active research topicrecently; An overview for the controller design in offshore wind turbines was givenin [48]. As far as wind turbine control in Region II is concerned, it is conventionalthat generator torque is manipulated to regulate generator speed by fixing the bladepitch at its optimal angle, in order to achieve the optimal power efficiency [23]. Thisstrategy is optimal when platform movements do not exist. However, for floatingwind turbines, since the tower and platform oscillation induced by wind and waveswill increase fatigue loadings to the structure, it is questionable whether the controlstrategy with fixed blade pitch is still optimal. Perhaps a more complete conceptof optimality would incorporate power efficiency, maintenance, and lifespan. Oneof the chapters in this thesis investigates a potential advantage of employing bladepitch control using LPV gain-scheduling technique for Region II. This chapter usesLPV gain-scheduling technique in order to consider plant dynamics variation in thecontroller design.In order to control offshore wind turbines in Region III, different controlmethods such as state-feedback, loop shaping and model predictive control [27, 28,38, 39] are considered. These controllers, however, do not address the variation inplant dynamics. In [4, 5], constant controllers were designed based on the LPVmodel of floating offshore wind turbines. Also, similar to onshore wind turbines,a PI gain-scheduling controller which deals with the plant dynamics variation wasproposed in [23]. The constant and PI controllers may encounter some performancelimitations, because they have only one or two tuning parameters. If more complexcontrol structures are adapted, there will be potentials for performance improve-ment. On the other hand, it is shown in [38, 39] that the state-feedback structure12can improve the turbine performance compared to the output-feedback structure.Therefore, in this thesis, gain-scheduling controllers considering plant dynamics vari-ation are investigated by using both output and state-feedback structures.The performance of these controllers should be investigated under the stan-dard meteorological conditions such as wind and waves. International Electro-technical Commission (IEC) has developed the standards [13] for wind and wavesprofiles used in performance evaluation as well as standards for designing onshoreand offshore wind turbines.1.6.3 Control design theoryThere are two control theories used in this paper; the linear quadratic regulator(LQR) and LPV gain-scheduling control. These control theories are explained se-quentially.The application of LQR control to wind turbines was proposed in [38]. TheLQR control uses state-feedback structure; the plant?s states are fed back and mul-tiplied by the designed gains for the state-feedback controller. These gains can beobtained using pole placement or LQR methods, see, e.g., [11, 50]. The pole place-ment methods determine gains based on the given poles of the closed loop system,while the LQR method with an integrator finds these gains in order to minimize theobjective function given byJ =? ?0[(yref ? y(t))tQ(yref ? y(t)) + u?t(t)Ru?(t)] dt, (1.3)where Q and R are the design parameters which are positive definite matrices. Also,yref , y and u are the reference, output and input signals to the plant.In this thesis, the LQR method is used and the control objectives can beconsidered in controller design using (1.3). Minimization of this objective functionattempts to reduce control efforts as well as the error on the output signal. Param-eters Q and R will determine the relative importance of the control effort reductionand the error.13The LQR controller, designed based on the system model in one operatingpoint, cannot guarantee performance for the entire operating region. Therefore,in this thesis, an LQR gain-scheduling (LQR GS) method, which designs differentLQR controllers for different operating points, is proposed. The LQR GS controlleris obtained using linear interpolation between the designed controllers. The stabilityof this controller is not guaranteed in the design stage, but the quadratic stabilityis analyzed after the design.The gain scheduling LPV technique is developed in [3]. In this method,an LPV controller is designed based on the LPV model of the system. Therefore,the designed controller depends on the parameter which characterizes the dynamicsvariation of the system. This parameter is measured in real time and used forcontroller update. Closed-loop stability and performance are guaranteed over theentire parameter region. Another advantage of this technique is the capability ofsolving multi-objectives control problems systematically.1.6.4 Simulation softwareThe main computer software which is widely used for wind turbine modelling andsimulation is FAST [23, 34, 35, 38, 39]. This software was developed specifically forwind turbines by the NREL?s National Wind Technology Center [25], and validatedin [12, 23, 31]. This software can be linked with other software such as MATLAB orADAMS in order to take advantages of these numerical and graphical environments.1.7 Research Objectives and MethodologyWe will investigate the usefulness of gain-scheduling control techniques for controlproblems in offshore floating wind turbines with barge platforms. Moreover, utilizingblade pitch angle actuator in low wind speed and using state-feedback structure inhigh wind speed are investigated. To apply these kind of controllers, we would liketo:14? obtain LPV models of the system,? define the control objectives and performance measures based on these controlobjectives,? propose proper feedback structures,? apply existing advanced GS techniques,? do the closed-loop simulations using FAST,? compute performance measures and compare the designed controllers with theexisting ones.1.8 Organization of ThesisThe organization of this thesis is illustrated in Figure 1.4. In Chapter 2, modellingof floating offshore wind turbines for both Region II and Region III is explained.In Chapter 3, first control objectives and feedback structures are given. Then,the general settings needed for running the simulations and performance indicesevaluated after running the simulations are introduced. Chapter 4 addresses thebaseline controllers used to compare our controllers? results. The designed controllersfor Region II and Region III are given in Chapter 5 and Chapter 6, respectively.Lastly, a chapter summary is given at the end of each chapter, and the thesis?conclusion is given in Chapter 7.15Figure 1.4: Thesis organization.16Chapter 2Wind Turbine ModellingTo design a feedback controller, it is important to acquire a mathematical modelaccurately for the plant to be controlled. If the derived model represents the dynam-ics of the plant accurately, it is expected that the controller designed based on themodel performs well. An accurate mathematical model for a floating offshore windturbine is described in [23]. This non-linear model was implemented in softwareFAST [25] by Jonkman.The non-linear floating offshore wind turbine model in FAST is an aero-hydro-servo-elastic model. The aero-hydro-elastic model of the turbine representsthe open-loop non-linear model of the turbine. The term aero refers to the interfacebetween the turbine and wind, while the term hydro refers to the turbine?s interfacewith waves, sea currents and mooring lines. The dynamics of different parts such asthe rotor, drive train and blades are contained in this model which corresponds tothe term elastic. On the other hand, the term servo represents a feedback structurebuilt inside FAST for running simulations in FAST. In this thesis, instead of usingthe servo model in FAST, the feedback structure is realized with MATLAB byimporting the aero-hydro-elastic model of the turbine and embedding the designedcontrol law into Simulink.In this chapter, a method to derive an LPV model numerically for offshore17floating wind turbines is presented, by using the software FAST. The model is usedin subsequent chapters for controller design. This chapter first reviews a non-linearmodel realized in FAST briefly, and then presents a method for the LPV modellingfrom linearized models at various operating points of the non-linear model. TheLPV modelling method is applied to a specific wind turbine, Offshore NREL 5.0MW Baseline Wind Turbine, which is a combination of the Onshore NREL 5.0 MWBaseline Wind Turbine and the ITI Energy Barge Platform [23]. This turbine iscalled NREL 5 MW1 throughout this thesis.2.1 Non-linear ModelThe non-linear model of an offshore wind turbine realized in FAST is representedin a state-space form as???x?(t) = f(x(t), u(t), ud(t)),y(t) = Cx(t).(2.1)Here, the control input vector u consists of generator torque, blade pitch angleand nacelle yaw angle, and the disturbance input vector ud is comprised of windand waves disturbance inputs. The vectors x and y are state and output vectors,respectively, and their components depend on the system complexity that can bespecified by FAST users. The block diagram representation of (2.1) is shown inFigure 2.1.The non-linear model (2.1) can be a complex model with a number of degreesof freedom and outputs. Depending on the purpose of modelling, we can simplifythe model by enabling only relevant degrees of freedom. Also, inputs and outputscan be selected based on the specific problem to be dealt with. In this thesis, thefollowing simplified model is considered;1The basic properties of this wind turbine are as follows; the total mass is about 6000 tons, thetower height about 87 m, the blade length about 63 m and the platform diameter about 45 m.See [23].18Nonlinear Modelin FASTFigure 2.1: The block diagram representation of the non-linear model (2.1) in FAST.? States (x): Platform pitch angle (ppt [rad]), platform pitch rate ( ?ppt [rad/s])and rotor speed (?r [rad/s]). The states of the system are divided into platformstates and wind turbine states. For the platform states, platform pitch angleand its derivative are considered, because platform pitch movement is themost significant among all the platform movements, under the assumptionthat wind is normal to the rotational plane of the blades. On the other hand,only rotor speed is considered as a wind turbine state. Flexibility of the drivetrain, blades and tower are ignored for the purposes of the simplified model.? Control inputs (u): Generator torque and blade pitch angles. The yaw angleis fixed with the assumption of a fixed wind direction.? In Region II, blade pitch angles (? [rad]) and generator torque (Tg [N.m])are used as control inputs. However, blade pitch angles are fixed for someof the designed controllers in this region.? In Region III, generator torque (Tg [N.m]) is fixed and blade pitch angles(? [rad]) are used as control inputs.? Disturbance inputs (ud): Wind speed (v [m/s]) and wave elevation (w [m]).? Outputs (y): Platform pitch angle (ppt [rad]) and generator speed (?g [rpm]).19These signals are summarized as follows2:???????????????????????????x = [ppt, ?ppt, ?r]T ,u =???[?, Tg]T , for Region II,?, for Region IIIud = [v,w]T ,yppt = ppt,y?g = ?g.(2.2)Platform pitch angle and generator speed, which are considered output sig-nals, can be calculated using the states and inputs of the system by substituting thefollowing equation into (2.1).Cppt =[1 0 0], (2.3)C?g =[0 0 60N2pi]. (2.4)In this equation, N denotes the gear-box ratio. This number represents the rota-tional speed ratio between the two sides of the gear-box.2.2 Model LinearizationUsing the non-linear model (2.1), we will derive the linearized model of the floatingoffshore wind turbine. The FAST linearization process does not have the capabilityof handling waves as disturbance inputs. Instead, the model is linearized in stillwater. Study of the wave effects on the model is future work, which may help tofurther reduce wave effects on the system.Figure 2.2 shows the inputs and the outputs of the linearized model. Sincethe linearized wind turbine model varies by the change in operating points, the LPVmodel is obtained for this system. This linearized model is represented by?x?(t) = A(x0, u0, ud,0)?x(t) +B(x0, u0, ud,0)?u(t) +Bd(x0, u0, ud,0)?v(t), (2.5)2Waves are only used as disturbance inputs for running simulations and they are not consideredin model linearization for controller design purposes.20Linearized Modelin FASTFigure 2.2: The block diagram representation of the linearized model in FAST.where signals x0, u0 and ud,0 determine the equilibrium points. We will show thatthe equilibrium points are functions of the disturbance input which is wind speed vhere. By having the equilibrium points and using (2.3) and (2.4), signals yppt,0 andy?g,0 can be calculated. The signals ?x, ?u, ?ud, ?yppt and ??g represent deviationfrom these equilibrium points;?x := x? x0(v), (2.6)?u := u? u0(v), (2.7)?ud := ud ? ud0(v), (2.8)?yppt := yppt ? yppt0(v), (2.9)?y?g := y?g ? y?g0(v), (2.10)and the system matrices are calculated analytically as:A(x0, u0, ud,0) :=?f?x????x=x0,u=u0,ud=ud,0, (2.11)B(x0, u0, ud,0) :=?f?u????x=x0,u=u0,ud=ud,0, (2.12)Bd(x0, u0, ud,0) :=?f?ud????x=x0,u=u0,ud=ud,0. (2.13)To obtain the LPV model (2.5) of the system, first equilibrium points andstate space matrices in (2.11)-(2.13) are calculated for each operating point. Then,interpolation is used to find the LPV model between the points.21In order to determine the equilibrium points, signals x0, u0 and ud,0 shouldbe obtained. Based on the definition of equilibrium points, the rate of change forthe states should be zero. In other words, according to (2.1), we have to solve thenon-linear equationf(x0, u0, ud,0) = 0. (2.14)FAST has an ability to solve the equation (2.14) when a part of parameter values inx0, u0 and ud,0 are specified. The specified and unspecified parts of these parametervalues at a given disturbance input ud,0 (wind speed) are presented in Table 2.1by using check-marks and question-marks, respectively. The unspecified parametervalues will be determined by solving (2.14) using FAST.The platform pitch rate is one of the specified parameter values and accordingto (2.14), its value is zero. Also, other specified parameter values for Region II andRegion III are explained here;? In Region II, generator speed ?g is a function of wind speed v (see (3.1)), andblade pitch angle ? is set to zero.? In Region III, generator speed and torque are set to their rated values.Table 2.1: Specified parameter values of equilibrium points at a given disturbanceinput ud,0.Wind speed x uoperating region ppt ?ppt ?r ? TgRegion II ? X X X ?Region III ? X X ? XAs one can see in Table 2.1, the unspecified parameters for determiningequilibrium points include one of the control inputs and the platform pitch angle.22These parameters are obtained in each operating point by substituting specifiedparameters for a given wind speed into (2.14) and using FAST to slove this equation.Therefore, equilibrium points for each operating point can be completely determinedusing wind speed at that point. In other words, the states of the system and controlinputs are functions of wind speed (v). Accordingly, system matrices can be shownas;A(x0, u0, ud,0) =: A(v), (2.15)B(x0, u0, ud,0) =: B(v), (2.16)Bd(x0, u0, ud,0) =: Bd(v). (2.17)For each equilibrium point corresponding to wind speed v, to obtain thematrices in (2.15)-(2.17) numerically, A(v), B(v) and Bd(v) are parametrized interms of v, using the following procedure.1. Take samples v(k), k = 1, 2, 3, . . . , from the operating wind speed range.2. For each sample v(k), calculate system matrices using FAST asA(k) = A(v(k)),B(k) = B(v(k)), (2.18)B(k)d = Bd(v(k)).3. Parameterize A(v) and B(v), e.g.,A(v) =: A0 +A1v,B(v) =: B0 +B1v, (2.19)Bd(v) =: Bd0 +Bd1v.4. Using curve-fitting techniques, obtain the coefficient matrices in (A.2) suchthat the equalities in (2.18) are satisfied within a certain accuracy.23The coefficient matrices of A(v), B(v) and Bd(v) are calculated for the NREL5 MW in Region II and Region III considering the difference in control inputs forthese regions. These matrices are shown in Appendix A.2.3 SummaryThe non-linear model realized in the software FAST was explained first. Then, forcontroller design based on linear control theory, we provided a procedure to deriveLPV models in terms of wind speed from the non-linear model.24Chapter 3General Control Strategy andPerformance MeasuresThe model of the floating offshore wind turbine system was obtained in the previ-ous chapter. Using this model, various controllers are designed in the subsequentchapters, and their performances are compared by simulations.This chapter first presents control objectives for floating offshore wind tur-bines. To achieve these objectives, typical structures of the feedback control systemsfor both Region II and Region III are reviewed. Then, the settings needed to runsimulations, using these feedback structures, are explained. At the end, performancemeasures to be evaluated are introduced.3.1 Control ObjectivesThe ultimate goal in the wind energy industry is to reduce the cost of producedelectricity. From a control engineering viewpoint, this goal is tackled by definingproper objectives on power capture and fatigue loads. These control objectives aredefined separately in Region II and Region III.253.1.1 Control objectives for Region IIIn the low wind speed operating range (Region II ), the power which can be capturedby the turbine is less than the rated power of generator. Therefore, we try tomaximize energy capture. Also, platform movements, which cause large structuralloads and degradation of energy capture, need to be minimized. Therefore, thecontrol objectives in Region II can be summarized as:(OL1) maximization of power capture,(OL2) minimization of platform movements.Energy capture is maximized by regulating the generator speed ?g so that ittracks the desired speed. This desired speed is determined by wind speed v as?g,ref(v) := Kv ? v, (3.1)where the gain Kv is defined byKv :=?maxr ?60N2pi . (3.2)Equation (3.1) shows dependence of reference generator speed on wind speed forRegion II. In this equation, ?max is the tip speed ratio associated with the maximumefficiency path for the generator speed. The tip speed ratio is defined as the ratioof the blade tip speed over the wind speed. N and r denote the gear-box ratioand radius of turbine blades, respectively. For the NREL 5 MW [23], the followingvalues are used: ?max = 7.55, r = 63 (m) and N = 97.For the case of onshore wind turbines and offshore wind turbines with fixedfoundations, we do not have the control objective (OL2), whereas in the case offloating offshore turbines, platform movements need to be minimized in controllerdesign. As described in Section 2.1, since this thesis considers only platform pitchmovement among all the platform movements in the controller design phase, itsminimization needs to be achieved.In summary, the control objectives for Region II are:26(OL1) min |?g ? ?g,ref |,(OL2) min |yppt|.3.1.2 Control objectives for Region IIIIn the high wind speed operating range (Region III ), the power which can becaptured by the turbine is more than the rated power of the generator. Therefore,we need to regulate power capture at the rated power. As with Region II, reductionof platform movement is again one of the control objectives. These movementsinduce structural loads on the system and hinder power regulation. Thus, the controlobjectives in Region III can be presented as:(OH1) regulation of power capture,(OH2) minimization of platform movements.Note that the main difference between control objectives for Region II and Region IIIis seen by the difference between (OL1) and (OH1). In OL1, power capture ismaximized, while in OH1, power capture is regulated to the rated power.The captured power, denoted by P , is obtained using the following equation:P = Tg ? ?g. (3.3)There are two standard methods to regulate the power capture to the rated power(Prated) using (3.3):? Fixing generator torque (Tg) to the rated torque (Trated), and regulating gener-ator speed (?g) to the rated generator speed (?g,rated) using blade pitch angle(?).? Directly regulating power capture (P ) to the rated power (Prated) using gen-erator torque (Tg) and blade pitch angle (?) simultaneously.27The first method simplifies the controller and reduces loads on the turbine,but increases power fluctuations. The second method regulates power more effec-tively, but adds complexity to the controller and increases loads on the turbine [38].In this thesis, to make the controller less complex and to reduce turbine loads,the first method is studied. However, the second method is used for the design of thebaseline controller, because there was an instability issue in the baseline controllersimulation using the first method. The reason for this instability was most likely thefact that the simulation contained more degrees of freedom than were incorporatedin the design of the controller [51].To regulate generator speed in the first method, the torque applied by thewind must be manipulated, because generator torque is fixed. The torque appliedby the wind is a function of wind speed and blade pitch angle. This torque can becontrolled by modifying blade pitch angle in order to compensate for the variationin wind speed. In other words, the objective (OH1) is to regulate generator speed tothe rated generator speed (min|?g ? ?g,rated|) by controlling the blade pitch angle.This objective is exactly similar to regulation of power (min|P ? Prated|), becausegenerator torque is assumed to be constant.Objective (OH2) is the minimization of platform movements. As it was men-tioned in the previous section, the only platform movement that will be consideredfor design of the controller is pitch movement (min(|yppt|)). However, other platformmovements are compared after running simulations.In summary, the control objectives for Region III are:(OH1) min |P ? Prated|,(OH2) min |yppt|.283.2 Feedback Structure for Controller DesignIn order to design controllers, the linearized model depicted in Figure 2.2 is used.The states and output are fed back to the controller for making the general feedbackstructure illustrated in Figure 3.1. Using this feedback structure, different controllersare designed for Region II and Region III. The inner structure of each controller isexplained in detail in later chapters. In the general feedback structure in Figure 3.1,the term ? is used to show the deviation of signals from equilibrium conditions.These equilibrium conditions are obtained in Chapter 2.LinearizedModelControllerStateFeedbackOutputFeedbackFigure 3.1: A general feedback structure for controller design.In Figure 3.1, ud represents disturbances to the system. These disturbancesare wind and waves for floating offshore wind turbines. However, only wind speed isconsidered in the controller design. Blade pitch angle (?) and generator torque (Tg)are the control inputs. Depending on the operating region of the wind turbine andthe type of the controller, blade pitch angle or generator torque may be consideredconstant and the non-constant term is used as a control input. The feedback signalsare output and state signals. The output signal is generator speed (y?g) and thestate signals are rotor speed (?r), platform pitch angle (ppt) and platform pitch rate( ?ppt). In this thesis, whenever state-feedback is used, we assume that all the statesare directly measurable. Also, the reference signal is the reference for generator29speed.Using this feedback structure, different controllers are designed for Region IIand Region III. The inner structure of each controller is explained in detail in laterchapters.3.3 Simulation SettingsTo evaluate the designed controllers, simulations are conducted for the closed-loopsystem depicted in Figure 3.1. The simulation block diagram used in MATLAB isshown in Figure 3.2. Since the controllers are designed for the linearized plant, theinput and output of the designed controllers are the deviation from the equilibriumpoints. This deviation is calculated in the simulation by adding or subtracting theequilibrium values to the related signals wherever needed. These equilibrium valuesfor reference signal and output signal are not illustrated in this figure, because theywill cancel out each other. The elements of this figure are explained in this section.Non-linear Modelin FASTController SaturationblockStateFeedback ( )OutputFeedbackFigure 3.2: A general feedback structure for simulations.30Non-linear Model in FASTThe non-linear model (2.1) has been realized with an interconnection between Simulinkand FAST [25] as a Simulink block. Since the block contains a complete model witha number of inputs and outputs which are irrelevant to this thesis, unnecessarysignals are eliminated from the block.Sensor measurements y?g and xAlthough noise exists in any measurement in real implementation, for simplicity,measurements are assumed to be noise-free in this thesis. If the noise is significantin this application, the noise effect can be reduced by inserting a low-pass filter inthe feedback path.Saturation blockThere are constraints on system inputs and their rates of changes due to the physicallimitations of actuators. These constraints for the NREL 5 MW are given as:? ? 90 (degree), (3.4)?? ? [?8, 8] (degree/s), (3.5)Tg ? 47402.91 (Nm), (3.6)T?g ? [?15000, 15000] (Nm/s). (3.7)These constraints are realized in the saturation blocks.ControllerThe designed controllers are plugged into the ?Controller? block in order to compareclosed-loop performances.31Disturbance inputsWind and wave are disturbance inputs to the system. The properties of thesewind and wave profiles for Region II and Region III are shown in Table 3.1. Thewave properties are selected for each wind speed profile using IEC Standards [13].Figures 3.3 show the wind profiles for these different meteorological conditions.These wind profiles are generated by software TurbSim [22]. Also, wave profileswhich are generated by FAST are shown in Figures 3.4.Table 3.1: Meteorological conditions in Region III.Meteorological Average Significant Peak spectralconditions wind speed wave height period(m/s) (m) (s)Region II 9.3 1.08 5.8Region III/Cond. I 18 3.25 9.7Region III/Cond. II 20 3.67 13.4Reference profileThe reference ?g,ref for generator speed is different in Region II and Region III.In Region II, this reference signal is generated using (3.1) and wind profile in Fig-ure 3.3(a). In Region III, the reference ?g,ref for generator speed is set to the ratedgenerated speed. This reference speed will regulate the power capture to the ratedpower. These reference signals are given by?g,ref =???Kv ? v for Region II,?g,rated for Region III,(3.8)where Kv is obtained in (3.2).320 50 100 150 200 250 300789101112Time (s)Wind speed (m/s) (a) Region II (Average speed=9.3 m/s)0 50 100 150 200 250 30010152025Time (s)Wind speed (m/s)(b) Region III (Average speed=18 m/s)0 50 100 150 200 250 3001015202530Time (s)Wind speed (m/s)(c) Region III (Average speed=20 m/s)Figure 3.3: Wind speed profiles for simulations.330 50 100 150 200 250 300?1?0.500.51Time (s)Wave elevation (m)(a) Region II (Significant wave height=1.08 m, Peak spectral period=5.8 s)0 50 100 150 200 250 300?2?10123Time (s)Wave elevation (m)(b) Region III (Significant wave height=3.25 m, Peak spectral period=9.7 s)0 50 100 150 200 250 300?2?1012Time (s)Wave elevation (m)(c) Region III (Significant wave height=3.67 m, Peak spectral period=13.4 s)Figure 3.4: Wave elevation profiles for simulations.34Initial operating conditionsThe controllers in this thesis are not designed for starting the turbine and therefore,the initial condition of the system should be defined. Here, the initial condition isconsidered to be the equilibrium condition for the given initial wind speed distur-bance input1. The procedure for obtaining the equilibrium point at a given windspeed disturbance input is presented in Section 2.2.First, the parameters specified with check-mark in Table 2.1 should be spec-ified. These parameters are as follows:? Region II: ?r, ?ppt, ?,? Region III: ?r, ?ppt, Tg.In Region II, initial rotor speed is calculated based on the reference signal for gen-erator speed (3.1) using initial wind speed (vint). In Region III, initial generatorspeed is set to the rated generator speed. Accordingly, the initial rotor speed inRegion II and Region III are given by?r,int =???Kv ? vint/N, for Region II,?g,rated/N. for Region III.(3.9)Also, platform pitch rate, which is the derivative of one of the states, is zero, becauseinitial condition is assumed to be an equilibrium condition. On the other hand, oneof the control inputs in each region is already specified. In Region II, initial bladepitch angle is set to zero for getting maximum efficiency. In Region III, generatortorque is also set to the rated generator torque.By considering these specified parameters and running the open-loop simu-lation, the equilibrium point is found. In this way, the initial platform pitch angle,generator torque for Region II and blade pitch angle for Region III, which are theunspecified parameters in Table 2.1, are determined.1For simplicity, water is assumed to be still in determining the equilibrium condition.353.4 Performance Measures for ComparisonsBased on the control objectives presented in Section 3.1.1 and 3.1.2, several perfor-mance measures are defined in order to evaluate performance of designed controllers.These measures are summarized in Table 3.2.Table 3.2: Wind turbine performance measuresWind speed Control Evaluatedregion objectives parametersRegion II(OL1) maximization of power capture Pavg(OL2) minimization of platform movements pptrms, ?pptrmsRegion III(OH1) regulation of power capture Prms(OH2) minimization of platform movements pptrms, ?pptrmsIn Region II, the achievement of objective (OL1) is evaluated by comparingthe average power capture, denoted Pavg , among designed controllers. In this region,the second objective (OL2) is minimization of platform pitch movement. Therefore,the root mean square (RMS) of platform pitch angle (pptrms) and its derivative?pptrms, which is directly affected by platform pitch movement, are calculated asperformance measures.In Region III, the objective (OH1) is regulation of power capture. This isevaluated by taking the RMS value of the difference between the power captureand rated power. This parameter is denoted by Prms. Similar to Region II, theobjective (OH2) is evaluated by the RMS value of platform pitch angle pptrms andits derivative ?pptrms.Furthermore, the RMS value of other platform movements and their deriva-tive are calculated as performance measures. These performance measures are dis-played in Table 3.3. Since the reduction of these movements are not considered in thecontroller design, we cannot predict their behaviour. However, their performance36can be compared based on the simulation results.Table 3.3: Platform performance measures not considered in the controller designPlatform movements Evaluated parametersRoll angle prlrmsRoll rate ?prlrmsYaw angle pywrmsYaw rate ?pywrmsSway displacement pswrmsSway rate ?pswrmsSurge displacement psgrmsSurge rate ?psgrmsHeave displacement phvrmsHeave rate ?phvrms3.5 SummaryThe general control strategy has been explained in this chapter by first presentingcontrol objectives for Region II and Region III. Then, the general feedback structurefor designing controllers which can achieve these objectives has been introduced.This feedback structure will be used for running the simulations and the requiredsettings for running these simulations have been explained. In order to comparedifferent controllers, performance measures have been defined and will be used toevaluate the behaviour of the controllers in simulations.37Chapter 4Baseline Controllers inRegion II and Region IIIThe basic controllers for Region II and Region III developed in [23] are reviewedin this chapter. These controllers are popular among wind turbine researchers forcomparison purposes [4, 5, 15, 16, 27, 38, 39]. In subsequent chapters, performancemeasures for designed controllers will be compared with those of the baseline con-trollers.Since this baseline controller was designed for the onshore wind turbine and itwas applied to the offshore wind turbine by [23], the platform movements reductionobjective was not considered in the controller design. Using this controller may notbe optimal and therefore in this thesis, we seek to design a controller which can alsoconsider platform movements reduction in the controller design.4.1 Low Wind Speed (Region II )The baseline controller for Region II called BaseR2 is explained in this section. First,its controller structure is introduced. Then, simulation results are presented.384.1.1 Controller structureThe general feedback structure was given in Figure 3.2. The baseline controllerfor Region II which was presented by [23] is substituted in the controller block, asshown in Figure 4.11.Non-linear Modelin FASTSaturationblockStateFeedback ( )OutputFeedbackFigure 4.1: The baseline controller structure in Region II.This baseline controller employs generator speed as a feedback signal, andspecify generator torque byTg = K?2g , K = 2.55764 ? 10?2N.m/rpm2. (4.1)This generator torque regulate the generator speed in order to follow the maximumefficiency path given in (3.1). Also, the blade pitch angle is fixed with ?eq = 0,because the maximum power is captured at this angle.4.1.2 Simulation resultsSimulation results for NREL 5 MW are calculated based on the inputs given inFigures 3.3 and 3.4. The control input (generator torque) and system responses1The output for generator torque of the baseline controller is an absolute value and we do notneed to add an equilibrium value.39(platform pitch angle and generator speed) are given in Figures 4.2 and 4.3. Also,the performance measures for the parameters listed in Table 3.2 are calculated inTable 4.1. The performance measures for other platform movements are given inTable B.1 and B.2.0 100 200 3001.522.533.5x 104Time (s)Generator torque (N.m)(a) Generator torque (N.m)0 100 200 300?1?0.500.51Time (s)Blade pitch angle (degree)(b) Blade pitch angle (degree)Figure 4.2: Control inputs for the baseline controller in Region II.0 100 200 3000.511.522.5Time (s)Platform pitch angle (degree)(a) Platform pitch angle (degree)0 100 200 300900100011001200Time (s)Generator speed (rpm)(b) Generator speed (rpm)Figure 4.3: System responses for the baseline controller in Region II.40Table 4.1: Evaluared performance measures for the baseline controller in Region IIPavg pptrms ?pptrms(MW) (degree) (degree/s)2.937 1.403 0.1264.2 High Wind Speed (Region III )The baseline controller for Region III called BaseR3 is presented by first introducingthe control structure. Then simulation results are given for this controller.4.2.1 Controller structureThe baseline controller for Region III presented by [23] is substituted in the con-troller block in Figure 3.2. The detailed structure of this baseline controller isillustrated in Figure 4.4.Non-linear Modelin FASTSaturationblockStateFeedback ( )OutputFeedbackFigure 4.4: The baseline controller structure in Region III.41The control inputs for NREL 5 MW are given in [23]:Tg =Prated?g, Prated = 5.2966 MW, (4.2)? = KP (?)??g +KI(?)???g, (4.3)whereKP (?) = 0.01882681/(1 + ?/6.302336), (4.4)KI(?) = 0.008068634/(1 + ?/6.302336). (4.5)As one can see, the controller gains for the blade pitch input are not fixed.These controller gains vary over time and this controller is called a PI-gain schedul-ing controller. This gain-scheduling controller is considered to take into accountdynamics variation according to the change of operating conditions2.4.2.2 Simulation resultsSimulations for NREL 5 MW with the baseline controller are conducted, with thedisturbance inputs given in Figures 3.3 and 3.4. The control inputs (blade pitchangle and generator torque) and system responses (platform pitch angle and gener-ator speed deviation) for meteorological conditions Cond. I and Cond. II are shownin Figures 4.5?4.8. Also, the performance measures for the parameters listed inTable 3.2 are calculated from simulation results and given in Table 4.2. The per-formance measures for other platform movements are presented in Tables D.1 andD.2.4.3 SummaryThe baseline controllers which are used to compare the performance of our designedcontrollers in Region II and Region III were reviewed in this chapter. These baseline2The right hand side of (4.3) will determine ? for the next time step.420 100 200 3000246810x 104Time (s)Generator torque (N.m)(a) Generator torque (N.m)0 100 200 300510152025Time (s)Blade pitch angle (degree)(b) Blade pitch angle (degree)Figure 4.5: Control inputs for the baseline controller in Region III under Cond. I.0 100 200 3000246810x 104Time (s)Generator torque (N.m)(a) Generator torque (N.m)0 100 200 30010152025Time (s)Blade pitch angle (degree)(b) Blade pitch angle (degree)Figure 4.6: Control inputs for the baseline controller in Region III under Cond. II.controllers were designed for onshore wind turbines and have been applied to offshorewind turbines in [23]. In the design of these controllers, power maximization forRegion II and power regulation for Region III are considered. However, platformmovements reduction was not considered. The controller for Region II fixes bladepitch angle and controls generator torque, while the controller for Region III controlsboth blade pitch angle and generator torque.430 100 200 300?10?50510Time (s)Platform pitch angle (degree)(a) Platform pitch angle (degree)0 100 200 300?100?50050100150Time (s)Generator speed deviation (rpm)(b) Generator speed deviation (rpm)Figure 4.7: System responses for the baseline controller in Region III under Cond. I.0 100 200 300?10?5051015Time (s)Platform pitch angle (degree)(a) Platform pitch angle (degree)0 100 200 300?200?1000100200Time (s)Generator speed deviation (rpm)(b) Generator speed deviation (rpm)Figure 4.8: System responses for the baseline controller in Region III under Cond. II.Table 4.2: Evaluated performance measures for the baseline controller in Region IIIMeteorological Prms pptrms ?pptrmsConditions (MW) (degree) (degree)Cond. I 1.752 2.605 1.336Cond. II 2.518 4.110 2.10344Chapter 5Gain-scheduling Control inRegion IIThe LPV gain-scheduling controller for Region II is developed in this chapter. InRegion II, it is conventional that only generator torque is used as the control input,by fixing the blade pitch angle to zero. However, the possibility of utilizing bladepitch angle as a control input in addition to generator torque for the reduction ofplatform movements is also studied. Then, closed-loop simulations will be carriedout, and the performance measures are obtained and compared with those of thebaseline controller.5.1 Control ObjectivesAs mentioned in Chapter 3, the general control objectives for Region II are summa-rized as:(OL1) min |?g ? ?g,ref |,(OL2) min |yppt|,45where ?g,ref is the speed that offers maximum power capture, and is calculated by(3.1).5.2 LPV Gain-scheduling ControllerWind turbine dynamics change with the variation in operating conditions. There-fore, the corresponding linearized model is different at each operating point. To ac-count for this model variation in controller design, a well-known LPV gain-schedulingcontrol method [3] is utilized. This method takes into account the plant variationby designing a varying controller.To achieve the control objectives (OL1) and (OL2), the following two con-trollers are designed and their performances are compared.(GS I ) The LPV gain-scheduling controller with fixed blade pitch angle,(GS II ) The LPV gain-scheduling controller with varying blade pitch angle.In the controller (GS I ), generator torque is the only control input, while in thecontroller (GS II ), blade pitch angle is added to reduce the platform pitch move-ment.In this section, first the controller structure used for the controller design isexplained. Then, by having the complete closed-loop structure, the control designproblem is defined and controllers are tuned. Following this, simulation results arediscussed for the designed controllers. Finally, the chapter is summarized.5.2.1 LPV gain-scheduling controller structureTwo different structures for the LPV gain-scheduling controller in Region II aresubstituted into the general feedback structure shown in Figure 3.2. Figures 5.1(a) and (b) show the feedback structures for controllers (GS I ) and (GS II ),respectively.46Non-linear Modelin FASTSaturationblockStateFeedback ( )OutputFeedbackGain-scheduling LPV Controller(a) Controller structure for controller (GS I )Non-linear Modelin FASTSaturationblockStateFeedback ( )OutputFeedbackGain-scheduling LPV Controller(b) Controller structure for controller (GS II )Figure 5.1: An LPV gain-scheduling controller structure in Region II.To mathematically formulate a controller design problem reflecting the con-trol objectives (OL1) and (OL2), we will consider the more detailed feedback struc-ture depicted in Figure 5.2, where reference and error signals are introduced asfollows:47y?g,ref(v) := ?g,ref(v), (5.1)e := y?g,ref (v)? y?g , (5.2)and the LPV gain-scheduling controller K(v) is represented byK(v) :???x?K = AK(v)xK +BK(v)e,?u = CK(v)xK +DK(v)e.(5.3)Figure 5.2: A feedback control structureAccording to Figure 5.1, the control inputs u for controllers (GS I ) and(GS II ) are generator torque and the combination of generator torque and bladepitch angle, respectively.The closed-loop system in Figure 5.2 can be reconfigured into a form witha generalized plant G(v) and a controller K(v) as drawn in Figure 5.3. In thisfigure, the signals pptW := Wppt?yppt, eW := Wee , ?uW := Wu?u and ?u?W :=Wu??u? are respectively the weighted platform pitch angle, the weighted error, theweighted input deviation from the operating point u0, and the weighted input ratedeviation. The purpose of weighting functions Wppt, We, Wu and Wu? is to takethe trade-off among the regulation of generator speed, the minimization of platform48pitch movement, and the fulfilment of the control inputs and control inputs rateconstraints.Figure 5.3: A feedback structure with a generalized plant.The constraints on control inputs and their rates are due to the physicallimitation of actuators. For the NREL 5 MW [23], these constraints are given in(3.4).5.2.2 LPV gain-scheduling controller design problemUsing the feedback system in Figure 5.3, the controller design problem can be for-mulated as follows: Given G(v) and specified weighting functions Wppt, We, Wu andWu?, design a gain-scheduling controller K(v) such that, for any trajectory of v(?) inthe operating range with constraints on rate-of-changes1|v?| ? 1 (m/s2), (5.4)1The constraints on rate-of-changes are imposed to design less conservative gain-schedulingcontrollers.49the closed-loop system is asymptotically stable, and the worst-case L2-gain2 from theexogenous inputs[?v, yT?g,ref (v), yT?g,0(v)]Tto the performance outputs[eTW , ?uTW , ?u?TW , pptTW]Tis minimized.To design gain-scheduling controllers that solve the formulated problem, thewell-known method in [3] is used. This method is based on convex optimizationwith linear matrix inequality constraints.5.2.3 LPV gain-scheduling controller tuningThe controller parameters are tuned by selecting proper weighting functions. Thereare four weighting functions in this control problem; Wppt, We, Wu and Wu?. Therole of Wu and Wu? is to prevent the violation of constraints on control inputs givenin (3.4). Weighting functions Wppt and We relatively emphasize the regulation ofgenerator speed and the reduction of platform pitch angle.The appropriate weighting functions are searched for by trial and error. Theselected weighting functions are given for both controllers (GS I ) and (GS II ) inTable 5.1.Table 5.1: Weighting functions for LPV gain-scheduling controllers in Region IIController Wppt We Wu Wu?GS I 1 0.001s + 0.5656s+ 5.527 0.03981 0.003162GS II 0.1 0.0003162s + 1.491s+ 46.06??0.003162 00 5012????0.3162 00 3972??2L2-gain of a system with the input u and the output y is defined by???0 yT (t)y(t) dt??0 uT (t)u(t) dt .505.3 Simulation ResultsTo compare the performance of the GS I and GS II with the baseline controller,performance measures introduced in Section 3.4 are calculated from time domainsimulations3. As explained in Chapter 3, average absorbed power (Pavg) is comparedamong different controllers to compare energy efficiency. Moreover, for comparingthe fatigue load, the RMS value of platform pitch angle and its rate (pptrms and?pptrms) are computed. According to the defined objectives in Section 5.1, the in-crease in the value of Pavg and decrease in the values of pptrms and ?pptrms are desired.The values of computed measures for the controllers are shown in Table 5.2. Theperformance variation in percentage compared to the controller (BaseR2) is alsogiven in this table.Table 5.2: Evaluated performance measures and their percent variation comparedto BaseR2 in Region IIPavg pptrms ?pptrmsController(MW) (degree) (degree)BaseR2 2.937 1.403 0.126GS I2.911 1.366 0.119(-0.89%) (-2.64%) (-5.56%)GS II2.886 1.303 0.118(-1.74%) (-7.13%) (-6.35%)Using only generator torque as a control input in (GS I ) has reduced platformpitch angle and its rate by about 2.64 and 5.56 percent, respectively. However,power capture has decreased by about 0.89 percent. The main reason for this powerreduction is that a fraction of control effort for the generator torque has been used3The performance measures for other platform movements are given in Table B.1 and B.2.51for the reduction of platform pitch movement in addition to maximization of power.In controller (GS II ), utilization of both generator torque and blade pitchangle as control inputs have decreased the average platform pitch angle and itsrate respectively by 7.13 and 6.35 percent, with 1.74 percent loss of energy capturecompared to the (BaseR2). In this way, the platform pitch movement is decreased bysacrificing a small amount of energy capture. This loss of power capture is expectedbecause of deviating blade pitch from its maximum efficiency angle.Therefore, compared to the BaseR2 controller, the platform pitch movementcan be reduced by using multi-objective gain-scheduling controller (GS I ) at thecost of losing small portion of power capture. If the user wants to reduce platformpitch movement further, one can use the controller (GS II ) to utilize slight changesin blade pitch angle. This will improve platform pitch movement at the cost of asmall drop in power capture.Control inputs (blade pitch angle and generator torque) for these controllersare shown in Figure 5.4, while the platform pitch angle and generator speed errorfrom reference signal ?g,ref are drawn in Figure 5.5. Figure 5.4(a) indicates thatblade pitch angle for the controller GS II is relatively small. Therefore, the selec-tion of ?g,ref of zero blade pitch angle for non-zero blade pitch application can bejustified. Blade pitch angle is fixed to zero for BaseR2 and GS I controllers. On theother hand, generator torque is used as a control input for all the controllers andshown in Figure 5.4(b). Figure 5.5(a) demonstrates that (GS II ) has the smallestplatform pitch movement, while (BaseR2) has the largest platform movement. Also,generator speed error is the smallest for BaseR2, which can be seen in Figure 5.5(b).5.4 SummaryIn this chapter, we investigated application of the LPV GS control and also the uti-lization of blade pitch angle for reducing the platform oscillation caused by wind andwaves in Region II. Two LPV controllers were designed by considering maximization520 50 100 150 200 250 30000.10.20.30.40.50.60.70.8Time (s)Blade pitch angle (degree) BaselineGS IGS II(a) Blade pitch angle (degree).0 50 100 150 200 250 30015202530354045Time (s)Generator torque (KN.m) BaselineGS IGS II(b) Generator torque (N.m).Figure 5.4: Control inputs for simulations.of power capture and minimization of platform pitch movement as a multi-objectiveproblem. Generator torque is a control input for both of the controllers, while bladepitch angle is only utilized in one of these controllers. Their simulation results arecompared with the baseline controller presented in Chapter 4. Generated power530 50 100 150 200 250 3000.511.522.5Time (s)Platform pitch angle (degree) BaselineGS IGS II(a) Platform pitch angle (degree)0 50 100 150 200 250 300?300?200?1000100200Time (s)Generator speed deviation(rpm) BaselineGS IGS II(b) Generator speed error (rpm)Figure 5.5: System responses.and platform oscillation are the parameters which are considered for comparisonpurposes.The important results for this chapter is summarized as follows:54? Multi-objective controllers are designed which can reduce platform pitch move-ment at the cost of losing a small amount of power capture.? Utilization of blade pitch angle as a control input will reduce the platformpitch movement.? In the sense of platform pitch movement reduction, the LPV gain-schedulingcontroller with varying blade pitch was the best, and the BaseR2 was theworst.? In the sense of power capture increase, the BaseR2 was the best, and the LPVgain-scheduling controller with varying blade pitch was the worst.? The percentage of improvement in the platform pitch movement reduction isgreater than the percentage of reduction in power capture.55Chapter 6Gain-scheduling Control inRegion IIIThe basic method used to control the floating offshore wind turbines in Region IIIis explained in Chapter 4. In order to improve the performance, the LQR controlmethod using state-feedback structure was proposed by [38]. This method designsan LQR controller based on one wind speed and applies this controller to the wholeRegion III.Since turbine dynamics is changing over different operating points, design-ing a controller based on one operating point cannot guarantee performance overthe whole operating region. Therefore, gain-scheduling LQR control, which consid-ers variation in plant dynamics, is selected in this thesis. The gains in the gain-scheduling LQR controller vary with changes in operating conditions.For both fixed and gain-scheduling LQR controllers, stability is a concern,because the fixed LQR controller is not effective far away from its design point, andalso transitioning from one controller to the next using the gain-scheduling LQRmethod can make the system unstable. Therefore, these LQR controllers cannotguarantee stability over the whole operating region, and the stability analysis isnecessary. The stability is analysed using the quadratic stability method [2].56In order to consider both stability and dynamics variation in the controllerdesign, the LPV gain-scheduling controller is designed. The process for designing anLPV gain-scheduling controller is well established in [3]. The dynamics variation ofthe plant is considered in the controller design and a varying controller is designed.This gain-scheduling LPV controller is designed for both output-feedback and state-feedback structures. The advantage of state-feedback structure over the output-feedback structure is that more information is fed back which can improve thecontroller performance.In this thesis, the LQR fixed controller designed for one wind speed andLQR gain-scheduling controller designed for the entire Region III are called LQRF and LQR GS, respectively. Also, the LPV gain-scheduling (LPV GS) controllerswith output-feedback and state-feedback are called LPV GS OF and LPV GS SF,respectively. Table 6.1 summarizes some of the properties of these controllers aswell as the baseline controller (BaseR3) [23]. As one can see in this table, stabilityonly can be guaranteed in the controller design using the LPV GS method. In otherwords, for the case of PI GS and LQR, the stability will be analyzed after controllerdesign and we cannot guarantee the stability in the design stage.Table 6.1: Controller properties in Region IIIControllersFeedback DesignStabilityStructure MethodBaseR3 Output feedback PI GS AnalyzedLQR F State feedback LQR AnalyzedLQR GS State feedback LQR GS AnalyzedLPV GS OF Output feedback LPV GS GuaranteedLPV GS SF State feedback LPV GS GuaranteedIn this chapter, control objectives are presented first. Then, controllers are57designed based on LQR and LPV GS methods. At the end, simulation results ofthese controllers are compared.6.1 Control ObjectivesAs mentioned in Chapter 3, the general control objectives for Region III are sum-marized as:(OH1) min |P ? Prated|,(OH2) min |yppt|.where Prated is the rated power of the turbine. As mentioned in Section 3.1.2, theobjective (OH1) is simplified to the regulation of generator speed (min|?g??g,rated|),because generator torque is assumed to be constant for the design of controllers inRegion III.The control objective (OH1) which is the regulation of power is only consid-ered in the design of LQR controllers. However, the LPV gain-scheduling controllersare designed considering both objectives simultaneously.6.2 LQR ControllersIn this section, fixed and gain-scheduling controllers using the LQR method aredesigned. The structure for these controllers is a full state-feedback structure withan integrator. The states, as mentioned in Chapter 2, are rotor speed, platformpitch angle and platform pitch rate; all of which can be measured by the availablesensors in the plant. Therefore, similar to [38, 39], the full state-feedback structureis considered. On the other hand, an integrator is used inside the structure in orderto reduce the steady state error for generator speed regulation. This integrator alsohelps to compensate for the constant disturbance in the system such as wind andwaves.58Non-linear Modelin FASTSaturationblockStateFeedback ( )OutputFeedbackFigure 6.1: LQR feedback structure for Region IIIWhile the LQR F controller is designed by considering the turbine model atone operating point, the LQR GS controller is designed by considering the variationof plant dynamics over the entire operating region. Moreover, quadratic stabilityfor these LQR controllers is analyzed.6.2.1 LQR controller structureThe LQR control structure is substituted into Figure 3.2. This new feedback struc-ture is presented in Figure 6.1, where Kx and Ke are control gains for state anderror signals, respectively.Using this feedback structure, LQR F and LQR GS controllers are designed.The controller gains for the LQR F controller are constant, while these gains arefunctions of wind speed for the LQR GS controller.LQR F controllerThe LQR F controller is designed based on the plant model at the middle point of theoperating range (v = (11.4 + 25)/2 = 18.2 m/s). The obtained controller gains are59constant, and are applied to the entire operating region. The LQRmethod places theclosed-loop poles at proper positions in order to minimize the defined cost function.This cost function (J) for an infinite horizon LQR problem with an integrator isdefined in (1.3), where R is set to 104 and Q = 10?6, 10?5.5, 10?5.0, ? ? ? , 10?2.LQR GS controllerTo design an LQR GS controller, the operating region of wind turbines (wind speed)is gridded, and an LQR F controller is designed at each gridded point using thesame design parameters as for the LQR F case. Then, a varying controller at eachoperating point (wind speed) is obtained by linear interpolation of the controllergains at the two adjacent gridded points.6.2.2 Stability analysisThe LQR F controllers are designed based on the model of the plant at one operatingpoint, while they have been applied to the entire operating region. For this reason,stability is not guaranteed for all operating points. On the other hand, stability ofLQR GS control is not guaranteed because transition between the controllers wasnot considered in the controller design. Therefore, stability should be analyzed forboth fixed and gain-scheduling LQR controllers.To analyze the stability of the controllers, quadratic stability using the Lya-punov stability criteria is considered. In the Lyapunov method, if a positive definitematrix Plyp satisfiesATclPlyp + PlypAcl < 0, (6.1)the system is quadratic stable at that point. Therefore, the operating region isgridded into several points and a common positive definite Plyp should be found bysolving (6.1) for all of the gridded points using linear matrix inequality (LMI). Ifsuch a Plyp exists, quadratic stability is approved; otherwise, the designed controllermay or may not be stable. In this case, another controller is designed by increasing60the number of points at which LQR F are designed. Then, the stability is analyzedagain.6.3 LPV Gain-scheduling ControllersThe procedure for designing LPV gain-scheduling controllers in Region II and RegionIII is similar. However, the differences in control objectives and control inputsshould be taken into account. This procedure is presented in Chapter 5 for windspeed in Region II. As it was explained, a well-known LPV gain-scheduling controlmethod [3] is used to take into account the plant variation by designing a varyingcontroller.6.3.1 LPV controller structuresA general feedback structure is given in Figure 3.2. Two feedback structures forLPV gain-scheduling controllers are substituted into said figure and are given inFigure 6.2. Figure 6.2 (a) is an output feedback structure used for designing LPVGS OF, and Figure 6.2 (b) is a state feedback structure used for designing LPVGS SF. In these structures, generator torque is constant and blade pitch angle isconsidered a control input.LPV GS OF controllerThis controller uses an output feedback structure with generator speed as its outputfeedback signal. To mathematically formulate a control design problem reflectingthe control objectives (OH1) and (OH2), we will consider the feedback structuredepicted in Figure 6.3 (a), where the control input u is collective blade pitch angleand controller K(v) is given in (5.3).In order to use the well-known gain-scheduling technique presented in [3], theclosed-loop system in Figure 6.3 (a) can be reconfigured with a generalized plantG(v) as drawn in Figure 6.3 (b). In the figure, the weighting functions Wppt, We61Non-linear Modelin FASTSaturationblockStateFeedback ( )OutputFeedbackGain-scheduling LPV Controller(a) LPV GS OF structureNon-linear Modelin FASTSaturationblockStateFeedback ( )OutputFeedbackGain-scheduling LPV Controller(b) LPV GS SF structureFigure 6.2: LPV GS feedback structures for Region III.62(a) A Feedback control structure(b) A feedback structure with a generalized plantFigure 6.3: LPV GS OF controller for Region III.and Wu are used to compromise between the regulation of generator speed, theminimization of platform pitch movement, and the fulfilment of the control inputconstraints.The definition of the gain-scheduling control design problem is similar toChapter 5. However, the control objectives for Region III, (OH1) and (OH2) , areconsidered in the design process. Also, the control input u is collective blade pitchangle.63LPV GS SF controllerThe procedure for designing an LPV GS SF controller is similar to that of the LPVGS OF. However, instead of feeding back generator speed, all the states are fed back.The feedback structure used for controller design and also the generalized plant aregiven in Figure 6.4 (a) and (b), respectively. Matrix C?g is given in (2.3).(a) A Feedback control structure(b) A feedback structure with a generalized plantFigure 6.4: LPV GS SF controller for Region III.6.3.2 LPV gain-scheduling controller tuningThe controller parameters are tuned by selecting proper weighting functions. Theweighting functions used in this control problem are: Wppt, We and Wu. By increas-64ing degree of weighting functions, the controller structure becomes more complex.This means that the controller will have more freedom and theoretically it can im-prove the performance.The weighting functions are selected by comparing several simulation results.The weighting functions for LPV GS OF and LPV GS SF are given in Appendix C.6.4 Simulation ResultsFive controllers have been designed for Region III; BaseR3, LQR F, LQR GS, LPVGS OF and LPV GS SF controllers. Time-domain simulations are carried out forthese controllers, with simulation settings as explained in Section 3.3, under twometeorological conditions given in Table 3.1. Figures 3.3 and 3.4 show their windand wave profiles over time. These meteorological conditions have also been usedin [38].In order to compare the performance of the designed controllers, the perfor-mance measures introduced in Table 3.2 are evaluated. These performance measuresare the RMS value of power regulation error, platform pitch angle and its derivative.According to the defined objectives in Section 6.1, the decrease in the values of allthe parameters Prms, pptrms and ?pptrms is desired.For each controller, simulations are run for various design parameter valuesunder the mentioned meteorological conditions. The evaluated performance mea-sures for meteorological conditions (Cond. I and Cond. II) are shown in Figures 6.5and 6.6, respectively. These figures depict the relations between the RMS value ofpower regulation error and the RMS values of platform pitch angle and its rate ofchange. Also, the performance measures evaluated for the BaseR3 controller aregiven in Table 4.2. Using these performance measures, different controller struc-tures are compared. Then, one controller from each structure is selected in orderto compare the performance parameters over time.The performance measures forthe BaseR3 are given in Table 4.2. By comparing these results with the evaluated65performance measures for other designed controllers given in Figures 6.5 and 6.6,one can see that all the presented controller types can improve the performancemeasures compared to the BaseR3 by reducing both power regulation error andplatform pitch movement at the same time. Specifically, power regulation has beenimproved substantially compared to the BaseR3.Figures 6.5 and 6.6 imply that there is a trade-off between reduction of powerregulation error and platform pitch movement for the LQR controllers. However asone can see in these figures, reduction of the platform pitch movement is restricted.By decreasing Q less than approximately Q = 10?4.0 in these simulations, powerregulation will increase without improvement in the platform pitch movement. Also,improvement in power regulation is limited due to the stability guarantee issue. Byincreasing Q more than approximately Q = 10?2.5 in these simulations, which leadto high improvement in power regulation, the stability cannot be guaranteed usingquadratic stability, even though the closed-loop system turned out to be stable insimulations. The LQR F and LQR GS controllers which can guarantee the stabilityare denoted by LQR F Stable and LQR GS Stable. Another important result inferredfrom these figures is that the LQR GS improves both power regulation and platformpitch compared to the LQR F. The main reason for this improvement is that thecontroller LQR GS is designed as a time-varying controller, with the considerationof plant dynamics variations.To guarantee the stability at the design stage, the LPV GS controllers aredesigned. In Figures 6.5 and 6.6, the trade-off between the reduction of powerregulation error and platform pitch movement can also be seen for the LPV GScontrollers. In these figures, performance of the LPV GS OF controller is betterthan that of the LQR F for the small platform pitch movement. However, byincreasing platform movements, the performance of LQR F controller outperformsLPV GS OF controller. The main reason for the less favorable performance ofLPV GS OF controller compared to the LQR controller is that the LPV GS OF661 2 3 400.20.40.60.8pptrms (degree)Prms(MW) LQR FLQR F stableLQR GSLQR GS stableLPV GS OFLPV GS SF(a) Error in power versus platform pitch angle0 0.5 1 1.5 200.20.40.60.8?pptrms (degree/s)Prms(MW)(b) Error in power versus platform pitch rateFigure 6.5: Error in power versus platform pitch response for different controllers(Cond. I)1 2 3 4 500.20.40.60.8pptrms (degree)Prms(MW) LQR FLQR F stableLQR GSLQR GS stableLPV GS OFLPV GS SF(a) Error in power versus platform pitch angle0 1 2 300.20.40.60.8?pptrms (degree/s)Prms(MW)(b) Error in power versus platform pitch rateFigure 6.6: Error in power versus platform pitch response for different controllers(Cond. II)67controller uses only the output as the feedback signals, while LQR controllers feedback more information and use all the states as the feedback signal. On the otherhand, performance of the LPV GS SF controller which uses all the states as thefeedback signal is better than all other controllers in the sense of reducing powerregulation error and platform pitch movement. The performance of the LPV GSSF controller is close to that of the LQR GS controller for the range of platformmovements achieved by the latter. The great advantage of the LQR GS SF controlleris that it can reduce platform pitch movement substantially at the cost of increasingerror in power regulation compared to other controllers.Platform pitch movement is only considered in the design of LPV GS con-trollers. Also, other platform movements were not considered in the controller de-sign. Therefore, performance of only platform pitch movement for LPV GS con-trollers can be predicted. Performance of platform pitch movement for LQR con-trollers and all other platform movements are compared based on the simulationresults. These simulation results are given in Appendix D. In these simulations, theLPV GS SF controller can generally improve platform roll, pitch and sway move-ments by sacrificing power regulation and at the same time increasing the yaw andsurge movements. Also, heave movement is almost constant.In order to compare the response of different controller structures over time,one controller from each structure is selected. The selected controller is the onewhich have achieved high reduction in platform pitch movement in that structure.The parameters for the selected controllers are given in Table E.1. Also, the controlinputs are presented in Figures E.1 and E.2. Finally, the performance measures arecompared in Tables 6.2 and 6.3As one can see from Tables 6.2 and 6.3, and also time domain simulationsin Figures 6.7 and 6.8, power regulation for all of the designed controllers has beenimproved substantially compared to the baseline controller. In the sense of platformmovements reduction, LPV GS SF is the best, while in the sense of power regulation,68Table 6.2: Evaluated performance measures for selected controllers in Region III(Cond. I)ControllerPrms pptrms ?pptrms(MW) (degree) (degree)BaseR3 1.752 2.605 1.336LQR F0.376 2.101 1.071(-78.54%) (-19.35%) (-19.84%)LQR GS0.247 1.981 0.982(-85.90%) (-23.95%) (-26.50%)LPV GS OF0.298 2.072 1.031(-82.99%) (-20.46%) (-22.83%)LPV GS SF0.423 1.222 0.472(-75.86%) (-53.09%) (-64.67%)69Table 6.3: Evaluated performance measures for selected controllers in Region III(Cond. II)ControllerPrms pptrms ?pptrms(MW) (degree) (degree)BaseR3 2.518 4.110 2.103LQR F0.483 3.223 1.636(-80.82%) (-21.58%) (-22.21%)LQR GS0.410 3.141 1.579(-83.72%) (-23.58%) (-24.92%)LPV GS OF0.433 3.438 1.730(-82.80%) (-16.35%) (-17.74%)LPV GS SF0.622 1.275 0.492(-75.30%) (-68.98%) (-76.60%)70LQR GS is the best. However, the LPV GS SF structure can achieve almost thesame power regulation and platform pitch movement as the LQR GS by changingits weighting functions, see Figures 6.7 (a) and 6.8 (a). In LPV GS SF, weightingfunctions are selected to improve platform pitch angle substantially compared toother controllers at the cost of increasing power regulation error.6.5 SummaryIn Region III, regulation of power capture and reduction of platform movements arethe original objectives that have been simplified to regulation of generator speedand reduction of platform pitch movement.In this chapter, we have utilized the advantages of state-feedback structureand gain-scheduling controller technique. The state-feedback structure providesmore information than the output-feedback structure. Also, the gain-schedulingtechnique considers the plant dynamics variation and designs a varying controller.The important simulation results for this chapter are summarized as follows;? Improvement of power regulation and platform pitch movement in all of thedesigned controllers compared to the baseline controller (Substantial improve-ment of power regulation),? Improvement of power regulation and platform pitch movement in LQR GScompared to LQR F because of considering plant dynamics in the controllerdesign,? Improvement of power regulation and platform pitch movement in LQR GScompared to LPV GS OF because of using a state-feedback structure,? Improvement of power regulation and platform pitch movement in LPV GSSF compared to all other controller because of considering plant dynamics in710 50 100 150 200 250 300?6?4?202468Time (s)Platform pitch angle (Degree)(a) Platform pitch angle0 50 100 150 200 250 300?4?20246x 106Time (s)Power regulation error (W) BaseR3LQR FLQR GSLPV GS OFLPV GS SF(b) Power regulation errorFigure 6.7: System responses for time-domain simulations for selected controllers inRegion III (Cond. I)720 50 100 150 200 250 300?10?5051015Time (s)Platform pitch angle (Degree) BaseR3LQR FLQR GSLPV GS OFLPV GS SF(a) Platform pitch angle0 50 100 150 200 250 300?50510x 106Time (s)Power regulation error (W)(b) Power regulation errorFigure 6.8: System responses for time-domain simulations for selected controllers inRegion III (Cond. II)73the controller design, using a state-feedback structure and also utilizing theLPV GS technique,? Guaranteeing stability in the design stage for LPV GS controllers rather thananalyzing the stability in LQR controllers after controller design,? Capability of LPV GS SF controller for substantial improvement of platformpitch movement compared to all other controllers.74Chapter 7ConclusionThe important achievements accomplished in this thesis are first summarized. Then,the contributions of this thesis and potential future work follow.7.1 Summary of AchievementsThe operating region of wind turbines is divided into Region II and Region III.Separate controllers were designed for these two regions and compared with theconventional controllers presented in [23, 38].In Region II, LPV gain-scheduling controllers were designed for both fixedand varying blade pitch angle. The advantages of using an LPV gain-schedulingcontroller are consideration for plant dynamics and consideration for the multi-objectives problem in the controller design. The simulation results show that re-duction of platform pitch movement is possible with a slight loss in energy capture,and that the varying blade pitch strategy is more effective in platform movementsreduction than the fixed blade pitch strategy.In Region III, LQR gain-scheduling and LPV gain-scheduling controllers withboth state feedback and output feedback structures were designed. These controllerswere compared with the conventional controllers such as baseline and fixed LQR75controllers presented in [23, 38]. The simulation results show that all the designedcontrollers have better performance compared to the conventional controllers in thesense of better power regulation and platform pitch movement reduction. Among thethree designed controllers, the LPV gain-scheduling controller with state-feedbacksurpassed the other two controllers, especially in the sense of platform pitch reduc-tion.While stability can be guaranteed in the design stage for the LPV gain-scheduling controllers, this is not the case for the LQR controllers. For LQR con-trollers, stability can be analyzed after controller design using quadratic stability.7.2 ContributionsThe main contributions addressed in this thesis for the control of floating offshorewind turbines can be categorized as follows;? LPV model of the system was developed using interpolation.? Gain-scheduling controllers were demonstrated to be efficient in improvingenergy capture and platform pitch movement in simulations.? Blade pitch angle in Region II was utilized in order to reduce platform pitchmovement with a slight loss in energy capture.? The LPV gain-scheduling controller with state feedback was found to performbetter than other gain-scheduling controllers in Region III.7.3 Future WorkControl of floating offshore wind turbines is a field that has a great deal of poten-tial for improvement. Some of the future work which can be done in this area issummarized below;76? Considering other platform movements in the controller design: Inthe designed controllers, only reduction of platform pitch was considered. How-ever, other platform movements also induce structural loads on the system.Therefore, reduction of these movements can be considered in the controllerdesign.? Modelling the wave effects on the system: Wave affects platform move-ments substantially and can diminish performance of wind turbines. As afuture work, the effects of waves on the system should be modelled and incor-porated into controller design.? Designing a switching gain-scheduling controller for both state-feedbackand output feedback structures: If the LPV gain-scheduling controller isdesigned for a smaller operating range, there is a possibility for performanceimprovement. Therefore, switching gain-scheduling controllers [29, 30] can beconsidered as future work. In this method, different gain-scheduling controllersare designed for different operating regions and they are switched between theregions based on the operating points.77Bibliography[1] Fabiano D. Adegas and Jakob Stoustrup. Structured control of LPV systemswith application to wind turbines. In Proceedings of American Control Confer-ence (ACC), pages 756?761, 2012.[2] Francesco Amato. 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NationalRenewable Energy Laboratory, 2004.83Appendix ALPV Model of the WindTurbineThe state-space matrices, A(v), B(v) and Bd,wind(v), are calculated for the NREL5 MW in both Region II and Region III. These matrices are obtained using samplesof wind speed for every 0.5m/s and parameterizing according to Section 2.2.A.1 LPV Model for Region IIThe LPV model of the wind turbine for Region II is parameterized asA(v) =: A0 +A1v,B(v) =: B0 +B1v, (A.1)Bd(v) =: Bd0 +Bd1v.84These coefficents are given here for the condition that generator torque is the onlycontrol input.A0 =?????0 1 0?2.909 ? 10?1 ?4.353 ? 10?3 1.811 ? 10?43.075 ? 10?2 ?5.76 ? 10?3 9.266 ? 10?5?????,A1 =?????0 0 0?5.348 ? 10?4 ?6.525 ? 10?3 4.741 ? 10?4?5.531 ? 10?3 ?9.395 ? 10?2 ?5.735 ? 10?3?????.[B0, B1] =?????0 01.348 ? 10?12 ?2.17 ? 10?13?2.215 ? 10?6 0?????[Bd0, Bd1] =?????0 02.075 ? 10?6 1.334 ? 10?42.644 ? 10?4 2.062 ? 10?3?????Also, the coefficents of state-space matrices are obtained for the condition thatcontrol inputs are generator torque and blade pitch angle.A0 =?????0 1 0?2.909 ? 10?1 ?6.688 ? 10?3 ?5.761 ? 10?43.048 ? 10?2 ?1.354 ? 10?2 8.465 ? 10?1?????,A1 =?????0 0 0?5.229 ? 10?4 ?6.537 ? 10?3 5.090 ? 10?4?5.434 ? 10?3 ?9.584 ? 10?2 ?1.146 ? 10?1?????.[B0, B1] =?????0 0 0 0?9.008 ? 10?13 2.842 ? 10?2 6.692 ? 10?14 ?6.578 ? 10?3?2.215 ? 10?6 ?1.493 ? 10?2 0 ?4.928 ? 10?3?????85[Bd0, Bd1] =?????0 0?1.541 ? 10?5 1.331 ? 10?4?5.413 ? 10?4 2.091 ? 10?3?????A.2 LPV Model for Region IIIThe LPV model of the wind turbine for Region III is parameterized asA(v) =: A0 +A1v +A2v2 +A3v3 +A4v4,B(v) =: B0 +B1v +B2v2 +B3v3 +B4v4, (A.2)Bd(v) =: Bd0 +Bd1v +Bd2v2 +Bd3v3 +Bd4v4.These coefficents are given here for the condition that blade pitch angle is the onlycontrol input.86A0 =?????0 1 0?3.341 ? 10?1 8.874 ? 10?1 1.499 ? 10?1?3.842 ? 10?2 1.238 ? 101 ?7.296 ? 10?1?????,A1 =?????0 1 08.372 ? 10?3 ?2.051 ? 10?1 ?2.793 ? 10?28.079 ? 10?2 ?2.754 ? 100 1.761 ? 10?1?????,A2 =?????0 1 0?6.714 ? 10?4 1.574 ? 10?2 1.975 ? 10?3?6.499 ? 10?3 ?7.062 ? 10?1 ?1.478 ? 10?2?????,A3 =?????0 1 02.336 ? 10?5 ?5.357 ? 10?4 ?6.525 ? 10?52.176 ? 10?4 ?7.062 ? 10?3 4.65? 10?4?????,A4 =?????0 1 0?3.015 ? 10?7 6.803 ? 10?6 8.123 ? 10?7?2.754 ? 10?6 9.071 ? 10?5 ?5.712 ? 10?6?????.[B0, B1, B2, B3, B4] =?????0 0 0 0 03.283 ? 10?1 ?7.799 ? 10?2 5.918 ? 10?3 ?2.012 ? 10?4 2.548 ? 10?61.112 ? 101 ?2.266 ? 100 1.674 ? 10?1 ?5.717 ? 10?3 7.268 ? 10?5?????[Bd0, Bd1, Bd2, Bd3, Bd4] =?????0 0 0 0 04.633 ? 10?3 ?7.126 ? 10?4 5.553 ? 10?5 ?1.898 ? 10?6 2.397 ? 10?81.102 ? 10?1 ?2.037 ? 10?2 1.662 ? 10?3 ?5.638 ? 10?5 7.065 ? 10?7?????87Appendix BPerformance Measures for theDesigned Controllers inRegion IIThe performance measures for the BaseR2 and the two gain-scheduling controllers((GS1) and (GS2)) are presented in Tables B.1 and B.2.Table B.1: Evaluated performance measures for power capture and platform move-ments in Region II.Performance Pavg prlrms pptrms pywrms psgrms pswrms phvrmsMeasures (MW ) (degree) (degree) (degree) (m) (m) (m)BaseR2 2.937 0.127 1.403 0.718 25.867 0.329 0.409GS1 2.911 0.141 1.366 0.710 25.442 0.332 0.409GS2 2.886 0.140 1.303 0.607 24.680 0.340 0.40988Table B.2: Evaluated performance measures for derivative of platform movementsof the baseline controller in Region II.Performance ?prlrms ?pptrms ?pywrms ?psgrms ?pswrms ?phvrmsMeasures (degree) (degree) (degree) (m) (m) (m)BaseR2 0.036 0.126 0.107 0.488 0.017 0.314GS1 0.048 0.119 0.105 0.453 0.017 0.314GS2 0.047 0.118 0.089 0.445 0.0164 0.31489Appendix CDesign Parameters for LPVGain-scheduling Controllers inRegion IIIThe weighting functions for the LPV GS OF and LPV GS SF controllers are pre-sented here. The values of We1, Wu1 and Wppt1 used in Table C.1 are:We1(s) =0.003162s + 0.1491s+ 0.4606 , (C.1)Wu1(s) =1? 105s2 + 1.257 ? 105s+ 3.952 ? 104s2 + 4.513s + 5.091, (C.2)Wppt1(s) =15.85s + 74.33s+ 0.4583 . (C.3)Also, the values of We2, Wu2 and Wppt2 used in Table C.2 are:We2(s) = 100, (C.4)Wu2(s) = 102.5, (C.5)Wppt2(s) = 103.5. (C.6)90Table C.1: Weighting functions for LPV GS OF controllers in Region IIIWe Wu WpptWe1 Wu1 Wppt1We1 10?0.2 ?Wu1 Wppt1We1 10?0.4 ?Wu1 Wppt1We1 10?0.6 ?Wu1 Wppt1We1 10?0.8 ?Wu1 Wppt1We1 10?1.0 ?Wu1 Wppt1We1 10?1.2 ?Wu1 Wppt1We1 10?1.4 ?Wu1 Wppt1100.5 ?We1 10?1.4 ?Wu1 Wppt1101.0 ?We1 10?1.4 ?Wu1 Wppt1Table C.2: Weighting functions for LPV GS SF controllers in Region IIIWe Wu WpptWe,2 Wu,2 Wppt,2100.25 ?We,2 Wu,2 Wppt,2100.50 ?We,2 Wu,2 Wppt,2100.75 ?We,2 Wu,2 Wppt,2101.0 ?We,2 Wu,2 Wppt,2101.25 ?We,2 Wu,2 Wppt,2101.50 ?We,2 Wu,2 Wppt,2101.75 ?We,2 Wu,2 Wppt,2102.00 ?We,2 Wu,2 Wppt,291Appendix DPerformance Measures for theDesigned Controllers inRegion IIIThe performance measures are computed for different controllers designed in Region III.These simulations are carried out under two meteorological conditions (Cond. I andCond. II). Figures D.1-D.8 show the performance measures for the designed con-trollers, while Tables D.1 and D.2 show these performance measures for the baselinecontroller.920 0.5 1 1.500.20.40.60.8prlrms (degree)P rms(MW)(a) Error in power versus platform roll angle1 2 3 400.20.40.60.8pptrms (degree)P rms(MW)(b) Error in power versus platform pitch angle1.4 1.6 1.8 200.20.40.60.8pywrms (degree)P rms(MW)(c) Error in power versus platform yaw angleFigure D.1: Platform angle response fordifferent controllers (Cond. I)0 0.5 100.20.40.60.8?prlrms (degree/s)P rms(MW)(a) Error in power versus platform roll rate0 1 200.20.40.60.8?pptrms (degree/s)P rms(MW)(b) Error in power versus platform pitch rate0.1 0.15 0.2 0.2500.20.40.60.8?pywrms (degree/s)P rms(MW) LQR FLQR F stableLQR GSLQR GS stableLPV GS OFLPV GS SF(c) Error in power versus platform yaw rateFigure D.2: Platform angular rate re-sponse for different controllers (Cond. I)931.6 1.7 1.800.20.40.60.8pswrms (m)P rms(MW)(a) Error in power versus platform sway displacement19.5 20 20.5 2100.20.40.60.8psgrms (m)P rms(MW)(b) Error in power versus platform surge displacement2.8 2.85 2.9 2.9500.20.40.60.8phvrms (m)P rms(MW)(c) Error in power versus platform heave displacementFigure D.3: Platform displacement re-sponse for different controllers (Cond. I)0.04 0.06 0.08 0.100.20.40.60.8?pswrms (m/s)P rms(MW)(a) Error in power versus platform sway rate0.4 0.5 0.6 0.700.20.40.60.8?psgrms (m/s)P rms(MW)(b) Error in power versus platform surge rate2.29 2.3 2.3100.20.40.60.8?phvrms (m/s)P rms(MW) LQR FLQR F stableLQR GSLQR GS stableLPV GS OFLPV GS SF(c) Error in power versus platform heave rateFigure D.4: Platform translational rateresponse for different controllers (Cond. I)940.5 1 1.5 200.20.40.60.8prlrms (degree)P rms(MW)(a) Error in power versus platform roll angle0 2 4 600.20.40.60.8pptrms (degree)P rms(MW)(b) Error in power versus platform pitch angle1.5 2 2.500.20.40.60.8pywrms (degree)P rms(MW)(c) Error in power versus platform yaw angleFigure D.5: Platform angle response un-der for different controllers (Cond. II)0 0.5 100.20.40.60.8?prlrms (degree/s)P rms(MW)(a) Error in power versus platform roll rate0 1 2 300.20.40.60.8?pptrms (degree/s)P rms(MW)(b) Error in power versus platform pitch rate0.16 0.18 0.2 0.22 0.2400.20.40.60.8?pywrms (degree/s)P rms(MW) LQR FLQR F stableLQR GSLQR GS stableLPV GS OFLPV GS SF(c) Error in power versus platform yaw rateFigure D.6: Platform angular rate re-sponse for different controllers (Cond. II)951.5 2 2.500.20.40.60.8pswrms (m)P rms(MW)(a) Error in power versus platform sway displacement18 19 20 2100.20.40.60.8psgrms (m)P rms(MW)(b) Error in power versus platform surge displacement2.04 2.05 2.06 2.0700.20.40.60.8phvrms (m)P rms(MW) LQR FLQR F stableLQR GSLQR GS stableLPV GS OFLPV GS SF(c) Error in power versus platform heave displacementFigure D.7: Platform displacement re-sponse for different controllers (Cond. II)0 0.1 0.200.20.40.60.8?pswrms (m/s)P rms(MW)(a) Error in power versus platform sway rate0.4 0.6 0.800.20.40.60.8?psgrms (m/s)P rms(MW)(b) Error in power versus platform surge rate1.57 1.575 1.58 1.58500.20.40.60.8?phvrms (m/s)P rms(MW)(c) Error in power versus platform heave rateFigure D.8: Platform translational rate re-sponse for different controllers (Cond. II)96Table D.1: Evaluated performance measures for power capture and platform move-ments of the baseline controller in Region III.Performance Prms prlrms pptrms pywrms psgrms pswrms phvrmsMeasures (MW ) (degree) (degree) (degree) (m) (m) (m)Cond. I 1.752 1.081 2.605 1.761 20.344 1.800 2.866Cond. II 2.517 1.504 4.110 2.114 19.512 1.977 2.053Table D.2: Evaluated performance measures for derivative of platform movementsof the baseline controller in Region III.Performance ?prlrms ?pptrms ?pywrms ?psgrms ?pswrms ?phvrmsMeasures (degree) (degree) (degree) (m) (m) (m)Cond. I 0.592 1.336 0.189 0.523 0.097 2.289Cond. II 0.827 2.103 0.236 0.693 0.118 1.58197Appendix ETime Domain Simulations forthe Selected Controllers inRegion IIIThe controller parameters for the controllers which have been selected for compar-ison of time-domain simulations are given in Table E.1. The values of parametersWe1, Wu1, Wppt1, We2, Wu2 and Wppt2 are given in (C.1)-(C.6). Also, Figures E.1and E.2 show the control inputs of these selected controllers.Table E.1: Controller parameters for selected controllers in Region IIIController parameters Q RLQR F 10?4 104LQR GS 10?4 104Controller parameters We Wu WpptLPV GS OF We1 Wu1 Wppt1LPV GS SF 100.25 ?We2 Wu2 Wppt2980 50 100 150 200 250 300123456789x 104Time (s)Generator torque (N.m) BaseR3LQR FLQR GSLPV GS OFLPV GS SF(a) Generator torque0 50 100 150 200 250 3000510152025Time (s)Blade pitch angle (Degree)(b) Blade pitch angleFigure E.1: Control inputs for time-domain simulations for selected controllers inRegion III (Cond. I)990 50 100 150 200 250 3000246810x 104Time (s)Generator torque (N.m) BaseR3LQR FLQR GSLPV GS OFLPV GS SF(a) Generator torque0 50 100 150 200 250 300051015202530Time (s)Blade pitch angle (Degree)(b) Blade pitch angleFigure E.2: Control inputs for time-domain simulations for selected controllers inRegion III (Cond. II)100"@en . "Thesis/Dissertation"@en . "2013-11"@en . "10.14288/1.0074126"@en . "eng"@en . "Mechanical Engineering"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "Attribution-NonCommercial-NoDerivatives 4.0 International"@en . "http://creativecommons.org/licenses/by-nc-nd/4.0/"@en . "Graduate"@en . "Gain-scheduling control of floating offshore wind turbines on barge platforms"@en . "Text"@en . "http://hdl.handle.net/2429/44879"@en .