Fault Tolerant Wind TurbineControl

Christian Dobrila Rasmus Stefansens021800 s021789August 31, 2007

iiiFault Tolerant Wind Turbine ControlFejl-Tolerant Vind Turbine ReguleringThis report is written by:MSc student Christian Dobrila MSc student Rasmus StefansenStudent ID: s021800 and Student ID: s021789Kollegiebakken 9, vær 3406 Asminderødgade 7, 3th2800 Kgs. Lyngby 2200 København NSupervisor (DTU):Professor Mogens Blanke Associate Professor Henrik NiemannØrsted•DTU Ørsted•DTUAutomation AutomationTechnical University of Denmark and Technical University of DenmarkElektrovej, bldg. 326 Elektrovej, bldg. 326DK-2800 Kgs. Lyngby DK-2800 Kgs. LyngbySupervisor (Siemens):Kenneth ThomsenSiemens Wind Power A/SStructural dynamics, PG R317SCION-DTUDiplomvej 372DK-2800 Kgs. LyngbyØrsted•DTUAutomationTechnical University of DenmarkElektrovejBuilding 326DK-2800 Kgs. LyngbyDenmarkwww.oersted.dtu.dk/english/research/au.aspxTel: (+45) 4525 3550Fax: (+45) 4588 1295Date: August 30, 2007Classi�cation: Free to distributeComments: This report is submitted in partial ful�llment of the require-ments for the Master degree at the Technical University ofDenmark. This report represents 35 ECTS points of work.

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SummaryWind turbines are subdued to highly varying wind velocities across its ro-tor diameter as a result of atmospheric turbulence and wind shear e�ects.These varying wind velocities causes highly varying forces on the blades mak-ing both the blades themselves and the entire structure of the tower bend.This thesis will introduce the basic dynamics describing a horizontal axiswind turbine in form of a nonlinear model. A collective pitch control schememaintaining a constant output e�ect of the model is outlined, and a indi-vidual pitch controller minimizing the tilt and yaw moments, thus also theblade loads, is designed. The design of the individual is such that its controlsignal is added to the control signal of the power scheme controller, providinga easy way to remove the individual controller.Band limited white noise are then implemented in measurements taken fromthe wind turbine model and a fault detection unit based on a set of systemresiduals is designed. Faults are implemented on the blade load measure-ments and are detected through CUSUM test on the residuals.Sensitivity study of the system toward fault tolerances is examined and basedon these a proper fault action can be taken by a fault decision unit.

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ResumeVindmøller er udsat for stærkt varierende vindhastigheder på tværs af ro-torarealet. Disse variationer skyldes atmosfærisk turbulens og vindgradiente�ekter. De varierende vindhastigheder forårsager stærkt varierende kræfterpå møllevingerne, som bøjer i vinden og får hele tårnet til at bøje.Dette afgangsprojekt introducerer en ulineær model, som beskriver den basaledynamik af en horisontal akse vind turbine. En kollektiv pitch reguler-ingsstrategi til at holde en konstant outpute�ekt for modellen bliver beskrevet,og en individuel regulator til at minimere belastningerne i tilt og yaw ret-ningen, og dermed belastningen på møllevingerne, bliver designet. Designetaf den individuelle pitchregulator sker således, at kontrolsignalet fra dennetillægges kontrolsignalet fra den kollektive pitchregulator, hvorved der opnåsen simpel metode til at frakoble den individuelle regulator løkke.Båndbegrænset hvid støj implementeres på målinger taget fra vindmølle-modellen, og der designes en fejldetekteringsenhed udfra et sæt systemresid-ualer. Herefter bliver der implementeret fejl på vingemålingerne, som bliverdetekteret via en CUSUM test på residualerne.Der designes en fejlbestemmelsesenhed og foretages en fejlsensitivitetsundersøgelseaf systemet således, at der afhængig af fejlen, foretages en passende handling

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PrefaceThis Master's Thesis was written at DTU in collaboration with SiemensWind Power, Brande, Denmark as a requirement for acquiring the Mastersdegree in Electrical Engineering at the Institute Oersted, at the TechnicalUniversity of Denmark.This thesis deals with a robust and fault tolerant control approach to re-duce fatigue of a horizontal axis wind turbine caused by varying blade loads.Both have participated in the documental work described in this thesis.Lyngby, August 2007Christian Ioan Dobrila & Rasmus Stefansen

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AcknowledgementsWe would like to thank our contact person from Siemens Wind TechnologyKenneth Thomsen for valuable feedback regarding the design and under-standing of wind turbine dynamics.Also a thank goes to Martin O.L. Hansen for help on implementing an un-steady BEM model and providing airfoil data.A special thanks goes to our two supervisors, Assoc. Prof. Hans HenrikNiemann and Professor Mogens Blanke, Oersted, DTU, for providing quali-�ed guidance throughout the project period.

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Contents1 Introduction 11.1 Wind turbine design . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis design . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 HAWT data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.2 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Wind Turbine Modeling 72.1 Wind Turbine Components . . . . . . . . . . . . . . . . . . . 72.2 Wind model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 Uniform wind �eld . . . . . . . . . . . . . . . . . . . . 132.3.2 Unsteady BEM Model . . . . . . . . . . . . . . . . . . 152.4 Drive train . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.6 Pitch actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.7 Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.8 Unmodeled dynamics . . . . . . . . . . . . . . . . . . . . . . . 332.9 Complete nonlinear model . . . . . . . . . . . . . . . . . . . . 352.10 Nonlinear model summary . . . . . . . . . . . . . . . . . . . . 363 Model linearization 374 Wind Turbine Control 414.1 Existing work on HAWT control . . . . . . . . . . . . . . . . 414.2 De�ning the control object . . . . . . . . . . . . . . . . . . . . 414.2.1 An operational wind turbine . . . . . . . . . . . . . . . 424.3 Collective pitch control . . . . . . . . . . . . . . . . . . . . . . 454.3.1 Design of LQI controller . . . . . . . . . . . . . . . . . 464.3.2 Implemented LQI controller . . . . . . . . . . . . . . . 484.3.3 LQI Dynamics . . . . . . . . . . . . . . . . . . . . . . 49

xiv CONTENTS4.3.4 Simulations with LQI . . . . . . . . . . . . . . . . . . 494.3.5 Limitation of collective pitch control . . . . . . . . . . 544.4 Individual pitch control . . . . . . . . . . . . . . . . . . . . . 564.4.1 Previous work . . . . . . . . . . . . . . . . . . . . . . . 564.4.2 Individual pitch control . . . . . . . . . . . . . . . . . 564.4.3 De�ning the control object . . . . . . . . . . . . . . . 594.4.4 Design of individual pitch controller . . . . . . . . . . 594.4.5 Implementation of I-compensator . . . . . . . . . . . . 604.4.6 Simulation with I-compensator . . . . . . . . . . . . . 614.5 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . 634.5.1 Power spectrum density . . . . . . . . . . . . . . . . . 634.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665 Fault Tolerant Control 675.1 Fault tolerant pitch control . . . . . . . . . . . . . . . . . . . 675.2 Sensor system . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3 Structural model . . . . . . . . . . . . . . . . . . . . . . . . . 685.3.1 Residuals . . . . . . . . . . . . . . . . . . . . . . . . . 685.4 Dynamics a�ecting the residuals . . . . . . . . . . . . . . . . 735.4.1 Pitch actuator time constants . . . . . . . . . . . . . . 745.4.2 BEM time constant . . . . . . . . . . . . . . . . . . . 755.5 Fault detection unit . . . . . . . . . . . . . . . . . . . . . . . 765.5.1 Noisy measurements . . . . . . . . . . . . . . . . . . . 765.5.2 CUSUM change detection . . . . . . . . . . . . . . . . 775.6 Implemented fault detection . . . . . . . . . . . . . . . . . . . 815.7 Recon�guration of controller . . . . . . . . . . . . . . . . . . . 825.8 Simulation with fault . . . . . . . . . . . . . . . . . . . . . . . 826 Sensitivity analysis 856.1 Controller performance concerns . . . . . . . . . . . . . . . . 856.2 Fatigue as a measure . . . . . . . . . . . . . . . . . . . . . . . 866.3 Single blade measurement fault . . . . . . . . . . . . . . . . . 876.4 Two blade measurement faults . . . . . . . . . . . . . . . . . . 896.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897 Conclusions 937.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 937.2 Suggestions for future work . . . . . . . . . . . . . . . . . . . 947.3 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94A Controller simulations 97B Sensitivity analysis 101C Airfoil data 103

CONTENTS xvD Selected matlab scripts 105D.1 Unsteady BEM model . . . . . . . . . . . . . . . . . . . . . . 105E Simulink models 113

xvi CONTENTS

List of Figures2.1 A horizontal axis wind turbine (HAWT). . . . . . . . . . . . . 82.2 Block diagram of full model. . . . . . . . . . . . . . . . . . . . 82.3 Wind model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Parameters in the turbulence model as functions of the meanwind speed v. As the mean wind speed increases so does thedc-gain k but the time constants τ1 and τ2 are decreasing. . . 112.5 An increase in the mean wind speed v contributes to an in-crease in the turbulent spectrum, (this corresponds to the de-crease in the time constant τ1 and τ2 seen in �gure 2.4). Anincrease in the mean wind speed v also causes an increase inthe variance of the turbulence. . . . . . . . . . . . . . . . . . 122.6 Aerodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 Power e�ciency coe�cient CP given the tip-speed-ratio λ andthe pitch angle β. . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 Thrust coe�cient CT given the tip-speed-ratio λ and the pitchangle β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.9 Windmill represented by four describing coordinate systems. . 162.10 Rotor disc seen from downstream. . . . . . . . . . . . . . . . . 182.11 A point on a blade described by system transformation vectors. 192.12 Velocity seen by the rotor plane . . . . . . . . . . . . . . . . . 202.13 Local e�ect seen on a blade. . . . . . . . . . . . . . . . . . . . 212.14 Rotor disc yawed towards the incoming wind. . . . . . . . . . 242.15 Drive train block description. . . . . . . . . . . . . . . . . . . 262.16 Drive train description. . . . . . . . . . . . . . . . . . . . . . . 272.17 Drive train response to a 106 Nm step in the rotor torque Qr.The resonance frequency at 1.65Hz is easily identi�ed. . . . . 282.18 Tower block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.19 Bode plot of the tower system. A resonance top is seen at thefrequency 0.3Hz. . . . . . . . . . . . . . . . . . . . . . . . . . 302.20 Pitch block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.21 Bodeplot of the transfer function from βref to β. The band-width can be read to approximately 1Hz. . . . . . . . . . . . . 322.22 Generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

xviii LIST OF FIGURES2.23 Generator model with variation in generator shaft speed - re-sulting in a positive feedback loop when having a �xed powerreference Pref . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1 Block diagram of linearized state space model . . . . . . . . . 374.1 A general control setup for the wind turbine. . . . . . . . . . 424.2 Pitch action of a wind turbine as a function of the wind speed. 434.3 State space model of the LQ control loop. . . . . . . . . . . . 464.4 G - Transferfunction from βref to P . . . . . . . . . . . . . . . 504.5 System dynamics identi�cation given a 1 m/s step in windinput at t = 5s. The step is performed from the linearizationpoint of mean wind speed of 15 m/s. Simulation is performedwith no turbulence or tower shade e�ects present to show theHAWT dynamic e�ects on the power production. . . . . . . . 514.6 LQI closed loop system response to a 1 m/s step in the turbu-lent wind input at t=20s. The plot shows a step in the meanwind velocity from 15 m/s to 16 m/s. The power is seen stablewithin a range of approximately ±250W of its nominal outpute�ect of 2MW. . . . . . . . . . . . . . . . . . . . . . . . . . . 524.7 LQI closed loop system response to a 1 m/s step in the tur-bulent wind input at t=20s with a performance weight on thetower position represented by state z. The plot shows a stepin the mean wind velocity from 15 m/s to 16 m/s. The posi-tion of the tower is seen more stable but greater variations inthe output e�ect as a result of the changed performance weight. 534.8 Trend curve for life expectancy of a blade given the fatigueexposure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.9 Collective- and individual pitch controller joined as a com-bined pitch controller for the HAWT model. . . . . . . . . . . 574.10 Layout of control loops for individual pitch control. . . . . . . 604.11 Integrator gain Ki determined by the inverse of the tangentialincrease inMflap as a result of changing the pitch angle of theblade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.12 I-compensator closed loop system response to a 1 m/s step inthe turbulent wind input at t=20s. The plot shows a step inthe mean wind velocity from 15 m/s to 16 m/s. The poweris seen stable within a range of approximately ±250W of itsnominal output e�ect of 2MW. . . . . . . . . . . . . . . . . . 624.13 I-compensator closed loop system response with gain Ki in-creased by a factor 4.2. . . . . . . . . . . . . . . . . . . . . . . 63

LIST OF FIGURES xix4.14 Power spectrum density of tilt and yaw moments with a tur-bulent wind input. The PSD from the collective controller hasbeen shifted 0.15Hz for easy comparison. It is seen that theDC gain has been removed with individual pitch control and1P frequency content has been greatly reduced. . . . . . . . . 644.15 Power spectrum density of a blade with a turbulent wind in-put. Using individual control removes the power of 1P har-monics, thus the periodic load variations due to tower shadeand wind shear are removed by the pitch scheme. . . . . . . . 655.1 Structural graph of subsystem to the full HAWT model. Sub-system is based on dynamic e�ects having a close relation tothe blade loads and the pitch actuator. . . . . . . . . . . . . . 695.2 Residual generator designed from the unmatched constraints. 735.3 Step input of the pitch signal at a wind velocity of 15 m/s.Due to dynamic wake e�ects there is a time constant fromchange of pitch angle, to the resulting torque has settled. . . . 755.4 Simulink model of implemented band limited white noisegenerator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.5 Noise propagation from Mflap readings to coherent residuals.Note the signal size of r0 − r4 is of 10−7 magnitude and less,thus considered as zero. . . . . . . . . . . . . . . . . . . . . . 785.6 Time histories of residual r9 and its amplitude distributionprior and during a change in mean value. . . . . . . . . . . . . 795.7 For residual r9 is seen a even power density distribution andthus white noise can be assumed. . . . . . . . . . . . . . . . . 805.8 Fault detection time and time expected between false alarmreadings given a threshold h derived from the ARL functionwith µ0 = 0Nm, µ1 = 29000Nm and σ = 24172Nm2. . . . . . 815.9 A 2% fault on blade load measurement y6 is implemented att=20s. Fault is immediately detected in residual r6 and r9. . 835.10 System behavior at a fault occurring in measurement y6. Con-troller is switched from the more advanced based individualpitch control, to simple collective pitch control. . . . . . . . . 846.1 Blade measurement fault causes a rise in the 1P e�ect of theindividual pitch control, while the amplitude of the 3P e�ectremains less than the 3P amplitude of the collective pitchcontrol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.2 Power spectrum density graphs when running the HAWTmodelwith a 12% deviation in a singleMflap measurement. It is seenthat the 2P e�ect on the blade PSD then is of same magni-tude for both individual and collective pitch control. Systemremains within its 10◦/s pitch velocity limit at all times. . . . 88

xx LIST OF FIGURES6.3 Power spectrum density graphs when running the HAWTmodelwith a 12% deviation in two Mflap measurement. At 12% o�-set in two measurements the fatigue of both controllers is thesame in terms of tilt-wise moments. . . . . . . . . . . . . . . . 906.4 Power spectrum density graphs when running the HAWTmodelwith a ±2% deviation in two Mflap measurement. For two±2% o�set faults the fatigue of both controllers is the samein terms of yaw-wise moments. . . . . . . . . . . . . . . . . . 91A.1 Collective pitch controller: Steady state of the system at thepoint of linearization. -No wind shear e�ects and no turbulence. 97A.2 Collective pitch controller: Steady state with turbulent inputand wind shear e�ects applied. . . . . . . . . . . . . . . . . . 98A.3 Cyclic controller: Steady state of the system at the point oflinearization. -No wind shear e�ects and no turbulence. . . . 99A.4 Cyclic pitch controller: Steady state with turbulent input andwind shear e�ects applied. . . . . . . . . . . . . . . . . . . . . 100B.1 Power spectrum density graphs when running the HAWTmodelwith a three o�set faults of same magnitude. A 12% devia-tion in all three Mflap measurement is implemented and thethe energy of the varying moments is seen identical as whenhaving no o�set fault. . . . . . . . . . . . . . . . . . . . . . . 102E.1 Main simulink �gure. . . . . . . . . . . . . . . . . . . . . . . . 113E.2 Implemented Coleman transformations . . . . . . . . . . . . . 114E.3 Nonlinear HAWT model. . . . . . . . . . . . . . . . . . . . . . 115E.4 Cusum test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116E.5 Residual generator. . . . . . . . . . . . . . . . . . . . . . . . . 117E.6 Linear HAWT model. . . . . . . . . . . . . . . . . . . . . . . 118

Chapter 1IntroductionThroughout the last decade there has been a political interest in sustain-able energy and the commercial industry of wind turbine manufacturing haslargely increased.In Denmark alone there is approximately 5.200 operational horizontal axiswind turbines maintaining 20% of the entire electrical energy production.This is quite a large fraction compared to other european countries, but thisdoes not limit the vision of the manufacturers, that expects this fraction toincrease to 50% in year 2025 with only 1.700 operational wind turbines.It is not easy to convince power companies that they should invest in amore expensive wind energy power plant because of environmental bene�ts,thus in order to increase further growth the wind turbines must be able tocompete with other suppliers of electrical energy such as conventional powerplants. This implies a further optimization for pro�t of the wind turbines.1.1 Wind turbine designAn optimization for pro�t is basically governed by considerations of trade-o�sin the wind turbine design: low production costs to maximize contributionmargin for the manufacturer. This implies low-weight and low-cost struc-tures and components. On the other hand reliability is major concern andthe need for a constant power production at all time is a top priority issuewhich advocates for advanced and more sturdy components.Better aerodynamics and material research have made it possible for thecommercial wind turbines to increase their rotor span dramatically over thelast couple of years. Due to the fact that the wind changes in time and space,wind turbines are exposed to highly varying loads and with an increase inthe size of the area swept by the rotor, these varying loads becomes more

2 1 Introductionapparent for each blade. These varying loads are fatiguing for the bladesand the structure of the turbine and reduces overall durability.A di�erent approach to increase durability, other than using heavier andmore robust components, is via the design of the control scheme. Through amore advanced control scheme it is possible to both prolong life expectancyand have a potential increased operating range. Individual pitching of thesingle blades to minimize load variations is an e�ective way to reduce fatigueof blades and structure, but requires a higher dependency of the sensors used.This introduces a new problem since these sensors have questionablereliability and in many cases a fault is poorly tolerated and may severelydamage the wind turbine. Adding the more advanced individual pitch con-troller scheme introduce higher complexity and requires a scheme for faultdetection and fault handling.By using a robust and fault tolerant control approach it is possible to detectfaults and thus prevent the controller from acting on faulty sensor signals.It also introduces a measure of the reliability of the control action taken,giving valuable feedback about the reliability of the wind turbine.1.2 Thesis designThis thesis will concentrate on the design of a fault tolerant load reductioncontrol scheme for a three bladed horizontal axis wind turbine (HAWT). Thedesign of the HAWT control will be described in two parts consisting of, acollective pitch control scheme which is designed for the purpose of main-taining a constant electrical power output and a individual pitch controllerto reduce the varying blade loads which is causing the unwanted structuralfatigue.A model of the HAWT will be designed to describe the dynamics of thesystem to be controlled. The design model will contain the HAWT dynam-ics and will be kept simple for controller design purpose. The HAWT modelis used for both designing the controllers, but also for simulations and thusveri�cation of the control of the wind turbine. Simulations of a HAWT how-ever requires as much dynamic as possible and thus the model will be acompromise between complexity and simplicity.Based on the HAWT model dynamics a collective pitch control strategywith a purpose to maintain a constant electrical power output is designed.The controller will be implemented as a full state feedback LQI controllerthat through pitch action and controlling the torque exerted by the genera-tor maintains a steady electrical power output.

1.2 Thesis design 3A individual pitch controller based on local blade load measurements willbe designed for the purpose of reducing the �ap wise moments occurring atthe blades due to the varying loads. The purpose for the individual controllerwill be reducing the stochastic load variations caused by the turbulent wind,but also reducing cyclic variations due to a inhomogeneous wind �eld andtower shade e�ects. The controller will be implemented such that the col-lective pitch controller is independent of a control signal from the individualcontroller.For performance measure in sense of fatigue reduction similar simulationsperformed with both controllers will be compared.Introducing the more advanced control scheme as the individual pitch controladds a high dependency of the sensor feedback used for blade load measure-ments. The blade load measurements are implemented as strain guages fromwhich a reliability is vital for the controller. A fault tolerant controller isdesigned to prevent faulty blade load measurements from a�ecting the indi-vidual pitch controller in such a way, that the fatigue actually is increased.Fault occurring from blade measurements will be detected and diagnosedthrough a residual analysis and a recon�guration of the controller is per-formed to secure that no vital damages is done to the HAWT structure.Finally an analysis of how faulty o�set measurements a�ects the systemis investigated. The performance of the individual pitch control will be com-pared to the una�ected collective pitch control to obtain a description of howmuch o�set in a blade measurement can be tolerated.

4 1 Introduction1.3 HAWT dataThe horizontal axis wind turbine model presented in this thesis has beenmodeled from the physical properties described table 1.3 and airfoil data�les listed in appendix C.Symbol Unit Descriptionpnom 2 [MW] Nominal output e�ect.wg,nom 195,7 [rad/s] Nominal angular speed of generator.τg 0.1 [s] Generator time constantNg 1:85 Gear ratio from rotorshaft to generator.ht 60 [m] Tower height.Rb 30,56 [m] Blade length.Ir 8.7·106 [kgm2] Inertial moment of rotor.Ig 150 [kgm2] Inertial moment of generator.rt,top 2.125 [m] Tower radius - top.rt,bot 3.625 [m] Tower radius - bottom.dt 7 [m] Distance from rotor to center of tower.mt 250·106 [kg] Mass of tower.wn 8.88 [Hz] Undamped frequency of pitch actuator.ξ 0.9 Damping ratio of pitch actuator.β 10 [deg/s] Pitching speed limitation.1.4 Nomenclature1.4.1 NotationFor the mathematical expressions used throughout this thesis, unless statedotherwise, the following conventions has been used:• Scalars � constants aswell as variables � italic font is used: x1, x2, Av,kv.

• Vectors are typed with bold font: x, u0, y. Vectors are column vectorsunless otherwise is stated.• Matrices is typed in uppercase and bold letters: A, B, Q.• Components in vectors/matrices is written as scalars with an index:

x = [x1 x2 . . . xn

]T.Function names from Matlab-functions and Simulink-models is writtenwith a type- writer font to clearly indicate that this is the name of afunction/model.

1.4 Nomenclature 51.4.2 SymbolsIn table 1.4.2 is a subset of the used symbols in this thesis shown.Symbol Unit Descriptionα [◦] Angle of attackβ [◦] Blade pitch angleβref [◦] Blade pitch angle referenceβ [◦/s] Pitch angle velocityΨ [◦] Azimuthal angleλ − Tip speed ratioCD − Drag coe�cientCL − Lift coe�cientCP − Power coe�cientCT − Thrust coe�cientFt [N ] Thrust force of the rotorMflap Nm Flap wise moment of bladeP [MW ] Electrical output e�ect of the turbinePref [MW ] Generator power referenceQg [Nm] Generator shaft torqueQg_ref [Nm] Generator torque referenceQr [Nm] rotor shaft torquev [m/s] Wind inputvr [m/s] Wind seen by the rotorw [m/s] Induced wind velocitywr [rad/s] Angular speed of rotor shaftwg [rad/s] Angular speed of generator shaftz [m] Position of top of towerz [m/s] Velocity of top of tower

6 1 Introduction1.5 AbbreviationsFollowing abbreviations of terms will be used throughout this thesis:ARL - Average run length.BEM - Blade element momentum.CUSUM - Cumulative summationHAWT - Horizontal axis wind turbine.LQ - Linear Quadratic.MIMO - Multiple input multiple output.PSD - Power spectrum density.SISO - Single input single output.

Chapter 2Wind Turbine ModelingAnalyzing the dynamics of the horizontal axis wind turbine is essential toobtain a mathematical model, from which control theory can be applied to.This chapter will give a mathematical description of the dynamics repre-senting the design model which will be used throughout this thesis. Theresult is a unstable nonlinear model consisting of ten di�erential equations.When describing how the aerodynamics transform the kinetic energy fromthe wind there will be taken two approaches. First a uniform aerodynamicdescription of how a wind �eld at the entire area of the rotor disc transformsinto a rotor torque and second a detailed blade element momentum (BEM)model description of how each of the blades are a�ected by the passing windvelocities.Latter description is �rst used during design of an individual pitch con-troller.Finally neglected dynamics due to simpli�cations are brie�y discussed.2.1 Wind Turbine ComponentsThe structure of a classic HAWT can be described by:• Tower• Nacelle• Hub• Blades.The tower provides elevation for the rotor and with increased height comesmore stable wind �ows and height allows rotor dimensions to increase. On

8 2 Wind Turbine Modelingtop of the tower the nacelle is placed. The nacelle is the housing of thegenerator and e�ect electronics. At the end of the nacelle is the hub. Thehub is where the blades are fastened to a pitch servo allowing blade rotationaround its own axis. The blades of the rotor is a�ected by the passing windand kinetic energy is transferred via the hub to the rotor shaft. The rotorshaft is connected via the gearbox to the high-speed shaft on the generator,that exerts a torque on the rotating shaft..................................................................... ......... ......... ......... Nacelle

TowerBlade-vHub -

6-

y z

R

��

Tilt Yaw

Figure 2.1: A horizontal axis wind turbine (HAWT).These HAWT dynamics can be represented as a joined block model describ-ing the interaction of the signals of the wind turbine model. Each of theWindCharacteristics AerodynamicsPitch

Tower Drive train Generator-

?-

� -� -

6?

6

v vr

βref

β

Ft z

Qr

wr

Qg

wg

Pref

P- -

Figure 2.2: Block diagram of full model.di�erent blocks in the block diagram of the full model in �gure 2.2 will beexplained in details, beginning with the wind model.

2.2 Wind model 92.2 Wind modelDescribing wind behaviour is a troublesome task. Wind is chaotic of natureand is varying in a wide spectra, ranging from seasonal wind changes ona yearly scale, to diurnal winds on daily basis and to the fast changingvariations measured in minutes and seconds, referred to as turbulence.WindCharacteristics- -v vrFigure 2.3: Wind model.The seasonal and diurnal wind change is very slow changing compared tothe turbulence and thus we can express the wind speed v, by a mean windspeed v and a turbulent wind speed v as in equation 2.1.v = v + v (2.1)The turbulent �uctuations corresponding to the highest frequency spec-tral peaks in wind model has a mean value of zero, when averaged overa timescale.The chaotic nature of the wind is not easily harnessed into deterministicequations, but given its rather random behaviour a suitable description canbe done in terms of statistical properties.In general, since the turbulence intensity is a measure of the overall levelof turbulence, then the turbulence intensity It can be de�ned asIt =

σt

v, (2.2)where σt is the standard deviation of the wind about the mean wind velocity

v.The turbulence intensity also clearly depends on the surface structure andthe height above ground. Cities and forest areas funnels the wind and con-tributes to the turbulence intensity measured at tree and building level.When height is increased the turbulence intensity declines. This observationcan be formulated into a turbulence intensity formulation depending on thesurface structure and the height above ground. Danish standard (DS472,1992) speci�es turbulence intensity asIt =

1

ln(

zz0

) , (2.3)

10 2 Wind Turbine Modelingwhere z is the height above ground and z0 is the roughness length. Theroughness length is an estimated parameter which is adjusted through ob-servations. Typical roughness lengths is shown in table 2.1. The modeling ofType of terrain Roughness length z0(m)Cities, forests 0.7Wooded countryside 0.3Village, countryside with some trees 0.1Open farmland 0.03Flat green areas 0.01Desert, rough sea 0.001Table 2.1: Typical surface roughness lengthsthe turbulent part of the wind itself can be described by an irrational spec-tral model. This is not easily usable and it is therefore much desirable toapproximate the irrational spectral model with a linear stochastic descrip-tion of the approximated rational spectrum. A possible approximation ofthe turbulence spectrum Sv, proposed by Højstrup (1982), isSv(w) =

k2

(1 + τ21w

2)(1 + τ22w

2)(2.4)This turbulence description can be modeled as a unity intensity white noiseprocess �ltered by the stable �lter

H(s) =k

(1 + τ1s)(1 + τ2s). (2.5)Rewriting equation 2.5 into

H(s) =k

τ1τ2

s2 + τ1+τ2τ1τ2

s+ 1τ1τ2and the turbulence description is intuitively written in state space form as

[˙v¨v

]

=

[0 1

− 1τ1τ2

− τ1+τ2τ1τ2

] [v˙v

]

+

[0k

τ1τ2

]

εv, (2.6)where εv is a zero mean white noise with unit variance. When assuming thatxv = [v˙v] then we can express

xv = Avxv + Bvεv. (2.7)

2.2 Wind model 11The parameters k, τ1 and τ2 are determined when minimizing the cost func-tion of the approximated turbulence spectrum and the wind experienced bythe rotorJ =

∫ w2

w1

[

log (Sv(w)) − log

(1

4πSef (

w

2π)

)]2

dw. (2.8)The wind experienced by the rotor is considered as an average of the spatialturbulence and according to Xin (1997), it can be modeled by dynamically�ltering the point wind, Sef ( w2π

). The calculated parameters k, τ1 and τ2depends on the mean wind speed, v and is shown in �gure �gure 2.410 20 30

0

5

Vm [m/s]

k

10 20 300

10

20

Vm [m/s]

τ1

10 20 300

0.5

Vm [m/s]

τ2

Figure 2.4: Parameters in the turbulence model as functions of the meanwind speed v. As the mean wind speed increases so does the dc-gain k butthe time constants τ1 and τ2 are decreasing.

12 2 Wind Turbine Modeling

10−2

100

102

−200

−150

−100

−50

0

50

Frequency [rad/s]

Sv [d

B]

Turbulence spectrum (Vintensitet

=0.132)

v = 5 m/sv = 10 m/sv = 20 m/sFigure 2.5: An increase in the mean wind speed v contributes to an increasein the turbulent spectrum, (this corresponds to the decrease in the timeconstant τ1 and τ2 seen in �gure 2.4). An increase in the mean wind speed

v also causes an increase in the variance of the turbulence.

2.3 Aerodynamics 132.3 AerodynamicsThe aerodynamics description is the part of the wind turbine that containsthe information on how the kinetic energy of the wind is passed on to therotating shaft through the blades. The aerodynamics of the windmill will bedescribed in two parts.The �rst part is an introduction to a simple model from which a powercontrol scheme is easy to derive. This model is based on the assumption ofa uniform wind �eld and thus a equal contribution of energy from each blade.In the second part of the aerodynamic description a Blade Element Methodis introduced. While the �rst described model only gave a description ofthe energy transformed by the entire disc area of the rotor, the BEM modelyield a far more detailed description on how the wind a�ect each section ofa blade. It is possible to perform a lot of modi�cations to the �rst describedmodel, to make it more realistic, but when wanting a precise description ofthe forces along the blades, this is considered an inadequate model. TheBEM model description is �rst used when designing a cyclic pitch controllerfor the wind turbine.-vr Aerodynamics?

-�

?6

β

Ft z

Qr

wrFigure 2.6: Aerodynamics.2.3.1 Uniform wind �eldAt �rst the rotor is considered as a uniform wind extractor transformingkinetic energy from the wind passing through the rotor plane, to mechanicalenergy at the shaft and through the generator to electrical power. The powerPr which is obtained by the rotor can be described by following relation

Pr =1

2ρArv

3rCP (λ, β), (2.9)where Ar=πR2 is the swept area by the rotor, ρ the density of air, and CP isthe power coe�cient depending on the tip-speed ratio λ and the pitch angle

β. The tip-speed ratio will here be de�ned asλ ≡ wrR

vr(2.10)

14 2 Wind Turbine ModelingThe cubic relation between the wind speed seen by the rotor plane vr andthe obtained power Pr readily verify the high increase in energy which canbe extracted from even a small increase in mean wind speed, whereas anincrease in area or density only results in a linear increase.The relation between the power obtained by the rotor Pr and the result-ing torque Qr is considered ideal and is related to the angular speed of therotor by Pr=Qrwr. Thus yieldingQr =

1

wr

1

2ρArv

3rCP (λ, β). (2.11)Furthermore the wind induces a thrust on the rotor. This thrust is describedas a thrust force FT that a�ects the nacelle in the fore-aft direction and isgiven as

FT =1

2ρArv

2rCT (λ, β), (2.12)where CT , much like CP , is a function of the tip-speed-ratio λ and the pitchangle β.

05

1015

−50

510

150

0.2

0.4

0.6

λ [rad]β [°]

CP

Figure 2.7: Power e�ciency coe�cient CP given the tip-speed-ratio λ andthe pitch angle β.This description of the aerodynamics will be used when designing the speedcontroller for the windmill. It has the advantage of simplifying the torqueenforced on the rotor as a equal contribution from each of the blades.The disadvantage by using this approach is that a description on the forces

2.3 Aerodynamics 15

05

1015

−50

510

150

1

2

3

λ [rad]β [°]

CT

Figure 2.8: Thrust coe�cient CT given the tip-speed-ratio λ and the pitchangle β.a�ecting each blade is not included, thus rendering useless when wanting toequally distribute the loads on the blades by pitch control. Therefore whenwanting a thorough description of the forces on each blade a Blade ElementMomentum model is used.2.3.2 Unsteady BEM ModelThe BEM method is a more realistic approach when describing the energy ex-traction from the wind, and it does not suggest a uniform wind distributionacross the entire rotor disc area. This model description of the transfor-mation of wind energy is �rst implemented when designing a cyclic pitchcontroller that reduces fatigue damage to the blades.BEM is a shortening of Blade Element Momentum which is a speci�c wayof calculating thrust, lift and drag forces on a blade. In short, when aerofoilcharacteristics, wind speed and angle of attack is known, it is possible toelement-wise calculate the forces on the blade thus giving a measure for theresulting torque at the rotor shaft.Due to the unsteady behaviour of the wind seen by the rotor from atmo-spheric turbulence, wind shear and the presence of the tower it is necessaryto use a unsteady BEM method to compute a more realistic behaviour of howthe energy is obtained from the wind. This unsteady BEM model, unlikethe �rst simple aerodynamic model, requires a complete structural model of

16 2 Wind Turbine Modelingthe wind turbine. More speci�c it requires a model for the movement of thetower, nacelle and the drive train.Describing coordinate systemsAs a result of the turbulent nature of wind, which changes in time and space,it is important at any time to know the position relative to a �xed coordinatesystem, of any section along a blade. In order to do so a number of coordi-nate systems is introduced to illustrate the movement in the complexity ofthe wind turbine.This BEM model description operates with four di�erent coordinate systemsas shown in �gure 2.9, which describes the position of the various movingparts of the wind turbine.

Figure 2.9: Windmill represented by four describing coordinate systems.1. First coordinate system is a �xed coordinate system which is placed atthe bottom of the tower.2. Second system is placed in the nacelle and is non-rotating.3. Third system is a rotating coordinate system �xed to the rotor shaft.4. Fourth coordinate system is aligned with one of the blades.

2.3 Aerodynamics 17By introducing these four coordinate systems it is easier to express how ex-actly the di�erent parts in�uences each other in the describing model. Notethat the describing coordinate system 1 for the tower does not match thecoordinate system for the tower given in �gure 2.1.By applying a transformations matrix AAB, a vector xA = (xA, yA, zA) inone coordinate system can now be expressed in another coordinate systemxB = (xB , yB , zB)

xB = AABxA, (2.13)Where the coloumns in the transformation matrix AAB express the unit vec-tors of system A in system B. When the tranformation matrix from systemA to system B is AAB then according to linear algebra the transformationmatrix from system B to system A is given by AAB = ATAB .These simple algebraic rules can be applied to the describing coordinatesystems for the windmill yielding three di�erent transformation matrices.First transformation matrix A12 is contructed as following:Initially system 1 and system 2 are identical but with an displacement of theorigos. System 2 is then rotated at the x-axis with the angle θyaw, represent-ing the turning of the nacelle. This gives the tranformation matrix

A1 =

1 0 00 cosθyaw sinθyaw

0 −sinθyaw cosθyaw

(2.14)Then system 2 is rotated along the y-axis with the angle θtilt which resultsin the transformation matrixA2 =

cosθtilt 0 −sinθtilt

0 1 0sinθtilt 0 cosθtilt

(2.15)System 2 is not rotated about the z-axis thus the tranformation matrix isgiven asA3 =

1 0 00 1 00 0 1

. (2.16)Given these three partial transformation matrices a total tranformation ma-trix A12 is found by coupling the partial tranformation matrices yieldingA12 = A1 · A2 ·A3.The transformation matrix A23, that describes a vector in system 2 in sys-tem 3 coordinates, is found as a rotation about the z-axis since the shaftis considered sti�. So much like the �rst partial transformation matrix was

18 2 Wind Turbine Modelingfound when going from system 1 to system 2, so is the transformation matrixfrom system 2 to system 3 generatedA23 =

cosΨwing sinΨwing 0−sinΨwing cosΨwing 0

0 0 1

. (2.17)Ψ

Blade 1Blade 2

Blade 3Figure 2.10: Rotor disc seen from downstream.For the transformation matrix A23, Ψwing is de�ned as the rotation of blade1 as shown in �gure 2.10.System 4 is only rotated, θcone, about the y-axis which results in the fol-lowing transformation matrix

A34 =

cosθcone 0 −sinθcone

0 1 0sinθcone 0 cosθcone

. (2.18)• Note that in order to have the blades to cone at the angle as shown i�gure 2.9 on page 16, the angle must be negative.A point along blade 1 is described as a vector in coordinate system 4 as

r4 = (x, 0, 0), where x is the radial distance from the rotational center to thepoint on the blade.Transformation through coordinate systemsNow it is possible to transform vectors into any given system in our windturbine model, thus transforming a vector r4 = (x, 0, 0) from system 4 intosystem 1 yields following transformation:r3 = A43 · r4

r2 = A32 · r3

r1 = A21 · r2,

2.3 Aerodynamics 19written shortly asr1 = A21 · A32 ·A43 · r4 = AT

12 ·AT23 ·AT

34 · r4

6

6x zy -

p

r12

r23

r34

r

Figure 2.11: A point on a blade described by system transformation vectors.With the four di�erent coordinate systems it is possible to express everysingle point on the turbine in each of the coordinate systems. It is particularimportant to be able to express a point p at a blade, in the 1. coordinatesystem where the wind is represented.Given the four describing coordinate systems, a point p on the blade, isrepresented in coordinate system 1, as the vectorr =

xp

yp

zp

= r12 = r23 = r34, (2.19)where the vectors r12, r23 and r34 all are given in coordinate system 1.Calculating the forces on a bladeBasically what is needed to calculate the forces on a blade, is informationsabout the velocity of the relative wind vrel and its angle of attack towardsthe blade.The vector vrel is the sum of three other vectors: The wind free �ow orundisturbed wind velocity v∞, the rotational speed of the point on the bladevrot and the induced wind velocity w. See �gure 2.12 on the following page

vrel = v∞ + vrot + w (2.20)All vectors are calculated in coordinate system 4 which represents the blades,where

20 2 Wind Turbine ModelingFigure 2.12: Velocity seen by the rotor plane

v∞ =

vx

vy

vz

= a34 · a23 · a12 · v1 (2.21)and vrot, which is allready in system 4, isvrot =

0−x · w · cosθcone

0

(2.22)and �nally w can be stated as a vector with two composantsw =

0wy

wz

. (2.23)Since the blades are assumed sti�, there will be no θcone angle, thus cosθcone =1. This simpli�es the vrot to the angular velocity at the current radiusvrot = −x · w.When w is known at every segment of the blade, we can then calculatethe velocity of the relative wind vrel seen by the blade and the angle ofattack α, which is the angle of attack of the relative wind.The angle of attack α is given from �gure 2.12 as α = φ − β, where theangle φ is

tanφ =vrel,z

−vrel,y

. (2.24)With the angle of attack determined, the lift and drag (CL & CD) coe�cientscan be found from a blade speci�c aerofoil characteristic lookup table, andthe forces on the blades can then be calculated. So basically what needs tobe done, is calculating the local induced velocities at every blade segments,and then it is possible to determine the reulting lift and drag force on eachblade.

2.3 Aerodynamics 21Calculating the induced velocitiesTo calculate the induced velocity w it is necessary to understand how mo-mentum theory works on the rotor. From a global consideration the rotoracts as a disc with a discontinuous pressure drop through it. This generatesa thrust which induces a velocity wn normal to the rotorplane, that de�ectsthe wake.Simple momentum theory states that this induced velocity in the far wake istwice the induced velocity in the rotorplane. Glauerts relation between thethrust and the induced velocity iswn = n ·w =

T

2ρA |v′| , (2.25)where |v′| = |v + fgn(n ·w)| and n is the unit vector of the direction of thethrust. fg is the Glauert correction term which is dependant of the size ofthe axial induction factor a.In case of zero yaw misalignment, thus a 90◦ angle between the incomingwind and the rotor disc, equation 2.25 reduces to classical BEM theory.φ

vrel w

L6� xzy.............................................Figure 2.13: Local e�ect seen on a blade.When looking at the local e�ect close to the blade, it is assumed that onlythe lift L contributes to the induced velocity w, and that the lift is oppositedirected of the induced velocity. The local e�ect at a blade at radial position

r a�ects the air in the area dA = 1B

2πrdr, in a way such that all blades Bcovers the entire rotor disc area at the radius r.Using the description of the local e�ect on the blade shown in �gure 2.13,and the information about the induced velocity given in equation 2.25, thenit is possible to express the induced velocity by two composants: one normal

22 2 Wind Turbine Modelingto the rotor plane and one tangential composant to the annular rotor disc.wn = wz =

−Lcos(φ)dr

2ρ2πrdrB

F |v + fgn(n ·w)|=

−B · Lcos(φ)dr

4ρπrF |v + fgn(n ·w)| (2.26a)wt = wy =

−B · Lsin(φ)dr

4ρπrF |v + fgn(n · w)| , (2.26b)with F being Prandtl's tip loss factor.It is important to note that the equations 2.26a and 2.26b must be solvediteratively due to the fact that the �ow angle, thus the angle of attack, de-pends on the induced velocity w itself. Furthermore equation 2.26a and2.26b yields the values for a state of the wake being in equilibrium with thedisc load. This is only true when the load is static. If the load changes atthe disc, it takes time before the wake and the load is in equilibrium. Thistime is proportional to the diameter of the rotor divided by the free windspeed.In order to get a description of this delay before equilibrium occurs, theinduced velocities should be relaxed through the use of a dynamic in�owmodel. When operating a pitch controlled wind turbine, it is essential toknow the settling time of the new equilibrium, when changing the loads asa result of a pitching of the blades.Dynamic in�ow modelThe time constant arising from a change in the in�ow at the rotor disc toan equilibrium of the induced velocities wz and wy has settled, is importantwhen modeling load as a result of pitch action. A model to account for thistime constant suggested by Øye (1991), is a �ltering of the induced velocities.The �lter proposed consists of two �rst order di�erential equationswint + τ1

dwint

dt= wqs + kτ1

dwqs

dt(2.27a)

w + τ2dw

dt= wint, (2.27b)where wqs is the quasi static value of the induced velocities found estimatedin equation 2.26a and 2.26b, wint is an intermediate value and w is the�ltered value containing the induced velocities at equilibrium between thedisc load and the dynamic wake.The two time constants are estimated through the use of a vortex method

2.3 Aerodynamics 23yieldingτ1 =

1.1

1 − 1.3a· R|v| (2.28a)

τ2 = (0.39 − 0.26( r

R

)2)τ1, (2.28b)with R being the radius of the rotor, k a constant k=0.60 and a the axialinduction factor with the axial induction factor is de�ned as a = wn

|v| . Whenusing this simple vortex method to estimate the �lter time constants, theaxial induction faction is not allowed to exceed 0.50.Another important e�ect to account for when determining time constantsin the BEM model, is that the angle of attack α towards the wind changesdynamically at every revolution of the rotor. E�ects as wind shear, towershade and the stochastic turbulence contained in the wind causes constantvariations in the actual angle of attack. These changes in the angle of attackdo not a�ect the loads instantaneously, but with a time constant proportionalto the blade chord c divided by the relative velocity vrel. The response onthe aerodynamic loads depends on whether the boundary layers are attachedat the trailing edge of the blade or if they are separated. If a separation ofthe boundary layers starts at the trailing edge of the blade, then the settlingtime is changed due to a change in the aerodynamical loads, caused by in�u-ences a�ecting the static data of the aerofoil. A modi�cation to the aerofoildata is therefore introduced through a dynamic stall model.Dynamic stall modelTrailing edge separation of the boundary layer is important a importantdynamic to model, since it has a strong e�ect on the aerofoil data, thusactively changing the aerodynamic behaviour of the blade. To get a moreprecise model of the blades then a correction term of the lift CL and drag CDcoe�cients should be implemented. This model however only implementsa correction to the lift coe�cient CL, which has a direct in�uence on theinduced velocity as seen in equation 2.26a and 2.26b.According to Øye (1991) it is important for stability reasons to include adynamic stall model, to prevent potential calculations of non-existing �ap-wise vibrations. For a trailing edge stall, the degree of stall is describedthrough the use of a separation function fs such thatCL(α) = fsCL,inv(α) + (1 − fs)CL,fs(α), (2.29)where CL,inv is the lift coe�cient for inviscid �ow without any boundaryseparation and CL,fs denotes the lift coe�cient for a fully separated �ow.The separation function fs is represented as a �rst order �lter

24 2 Wind Turbine Modelingdfs

ds=f st

s − fs

τ, (2.30)where it is assumed that fs always tries to get back to the static value f st

sof the airfoil data. Integration of equation 2.30 yieldsfs = (t+ δt) = f st

s + (fs(t) − f sts )e−

δtτ , (2.31)with t being a time constant approximated as A·c

vrel, c is the local chord and

vrel is the relative wind velocity at the current section of the blade. A is aconstant where A = 4.What is obtained so far from the BEM model, is a way of calculating theupdated induced velocities for each of the blades. What is still needed in theunsteady BEM description, is a description of the e�ect of the rotor havinga yaw angle towards the incoming wind and the e�ect caused by the verypresence of the tower.w

w

n

v

v′

χθyaw

- -Rotor discFigure 2.14: Rotor disc yawed towards the incoming wind.Yaw modelWhen the rotor is tilted og yawed towards the wind direction, there will bean azimuthal variation of the induced velocities. The induced velocities atthe blade pointing upstream will be smaller than the induced velocities atthe same blade, half a revolution later, pointing downstream. The reason forthis e�ect is, that the blade pointing downstream is deeper into the wake,thus the blade upstream is a�ected by a higher wind velocity and thereby ahigher bladeload.This di�erence in blade loads contributes to a beni�cial yaw moment,which has a positive e�ect on the angle towards the wind, trying to stabilizeitself.Through a yaw model a description of the distribution of the induced veloc-ities is given, when the incoming wind �ow towards the rotor is skewed withan angle θyaw. A yaw model proposed by Glauert is

2.3 Aerodynamics 25w = w0

(

1 +r

R· tanχ

2· (Ψwing − θ0)

)

, (2.32)where θ0 is the angle where the blade is deepest into the wake, w0 is themean induced velocities estimated bywz =

1

B(wz,1 + wz,2 + . . .+ wz,B) (2.33a)

wy =1

B(wy,1 + wy,2 + . . .+ wy,B) , (2.33b)and χ is the wake skew angle, de�ned as the angle between the wind velocityin the wake v′ and the rotational axis of the rotor w. The skew angle canbe found from simple algebra ascosχ =

n · v′

|n| |v′| , (2.34)with n being the unit vector in the direction of the rotational axis w. See�gure 2.14 on the facing page.With the description from the yaw model it is now possible to determinethe induced velocities when the rotor is yawed towards the wind. What hasnot been taking into account by the BEM model so far, is the e�ect of havingthe presence of the tower.Tower shadeThe tower impose an obstacle for the free wind �ow v∞, de�ecting the wind�ow around the tower structure. This de�ection of the wind can be describedby expressing v∞ into two composants in semi-polar coordinates as: radialvelocity vr and tangential velocity vθ

vr = v∞

(

1 +(a

r

)2)

cosθ (2.35a)vθ = −v∞

(

1 +(a

r

)2)

sinθ, (2.35b)with a being the the radius of the tower. Transforming the velocities intothe cartesian coordinate system 1 yieldsvz = vrcosθ − vθsinθ (2.36a)vy = −vrsinθ − vθcosθ, (2.36b)

26 2 Wind Turbine Modelingwith the relations between the two coordinate system described bycosθ =

z

r

sinθ = −yr

r =√

z2 + y2.The normal Pz and the tangential Py loads a�ecting the blade elementscan now be calculated fromPz = L · cosφ+D · sinφ (2.37a)Py = L · sinφ+D · cosφ (2.37b)andL =

1

2ρ |vrel|2 c · CL (2.38a)

D =1

2ρ |vrel|2 c · CD. (2.38b)2.4 Drive trainThe structural model of the drive train used is shown in �gure 2.15, yields aDrive train-

� -�

Qr

wr

Qg

wgFigure 2.15: Drive train block description.rather simpli�ed model consisting of, from left to right• The inertia of the rotor and low-speed shaft, Ir.• A massless, viscously damped rotational spring with viscosity Ds andspring coe�cient Ks. Thus there is a rotational deformation in thelow-speed shaft de�ned as θtorsion = (θr − θg

Ng).

• A gearbox with ratio Ng.• The inertia of the gearbox, high-speed shaft and generator, Ig.

2.4 Drive train 27Ir

Ig

Ng

U U�

U

wr Qr

wg QgRotor Gear GeneratorKs −Ds

Figure 2.16: Drive train description.The dynamics of the drive train model can be described by the di�erentialequations:Irwr = Qr −Ksθtorsion −Ds

(

wr −wg

NG

) (2.39)IgwgNg = −QgNg +Ksθtorsion +Ds

(

wr −wg

NG

) (2.40)˙θtorsion = wr −

wg

Ng

(2.41)whereθtorsion = θr −

θg

Ngis the torsion in the shaft on the rotor side, Qr the torque imposed at thelow-speed shaft by the rotor and Qg is the generator torque. Note the de�-nitions in turning directions of wr and wg from �gure 2.16.When put into state space form is

wr

wg

θ

=

−Ds

IrDs

IrNg−Ks

IrDs

IgNg− Ds

IgNgKs

IgNg

1 − 1Ng

0

·

wr

wg

θ

+

1Ir

0

0 − 1Ig

0 0

·[Qr

Qg

]

. (2.42)The constants used for the drive train isIr = 8.7 ∗ 106Nms2 Ks = 1.039 ∗ 108Nm Ng = 85

Ig = 1.5 ∗ 102Nms2 Ds = Ks

100Nms.When these are inserted in the system model then the eigenvalues for thedrivetrain is, 0.0000-0.5391 +10.3693i-0.5391 -10.3693i. (2.43)From the complex part of the eigenvalues for system it is seen that thedamped eigenfrequency of the drive train is approximately 1.65 Hz.

28 2 Wind Turbine ModelingIt should be noted that the drive train system has a pole in zero which can beseen when simplifying the system byIr = 1Ig = 1Ng = 1.The poles of the system is then given as,

det(sI − A) = det

s+Ds −Ds Ks

−Ds s+Ds −Ks

−1 1 s

= −DsKs −DsKs + (s+Ds)(s +Ds)s−D2

ss+Ks(s+Ds) +Ks(s +Ds)

= s(s2 + 2Dss+ 2Ks) = 0

s = 0 ∨ s = ±(√

D2s − 2Ks +Ds

)

(2.44)This can o�er some problems when numerical solving the equations, and maycause instability to the drive train system.From the eigenvalues it is seen that the drive train system has a poorlydamped mode at approximately 1.65Hz which is veri�ed when looking at astep response depicted in �gure 2.17.

0 1 2 3 4 50

0.5

1

1.5

2x 10

−3 Step Response

time [s]

tors

io [r

ad]

Figure 2.17: Drive train response to a 106 Nm step in the rotor torque Qr.The resonance frequency at 1.65Hz is easily identi�ed.2.5 TowerThe wind passing through the swept area by the rotor causes a thrust and thisthrust force causes the tower to bend in the fore-aft direction (horizontal).

2.6 Pitch actuator 29This e�ect can be described by a linear displacement of the nacelle which isTower?6Ft zFigure 2.18: Tower block.approximated by a simple model that doesn't take in consideration the slightmovement of the nacelle in the vertical direction. The dynamic is modeledas

mtz = −Dtz−Ktz + Ft, (2.45)where Dt is the tower dampener coe�cient, Kt the tower spring coe�cientand mt the mass of the tower. When put in State Space form dynamics isexpressed as[z

z

]

=

[0 1

−Kt

mt−Dt

mt

] [zz

]

+

[01

mt

]

Ft, (2.46)where z denotes the position, z the velocity and z the acceleration of thenacelle at the top of the tower. The spring constant and the dampeningcoe�cient incorporated in the structure plus the mass of the tower isKt = 8.88 · 105NmDt = 296 · 102Nmsmt = 250 · 103Kg.When these are inserted into the model we obtain the system eigenvalues-0.0592 + 1.8837i-0.0592 - 1.8837i (2.47)From the system eigenvalues it is seen that the damped eigenfrequency forthe tower is 0.30 Hz which is veri�ed by looking at the bode plot for thetower depicted in �gure 2.19 on the next page2.6 Pitch actuatorTo change the angle of the blade towards the wind in the model, a pitchactuator is used for every blade. The pitch actuators are placed in the hubof the wind turbine and consist of a hydraulic driven servo system. Thedynamic of the hydraulic servo system can be described by a linear secondorder model as stated in equation 2.48.

β = w2nβref − β2wnξ − βw2

n (2.48)

30 2 Wind Turbine Modeling

10−1

100

−150

−140

−130

−120

−110

−100

−90

System: systowerFrequency (Hz): 0.3

Magnitude (dB): −94.9M

agni

tude

(dB

)

Frequency (Hz)Figure 2.19: Bode plot of the tower system. A resonance top is seen at thefrequency 0.3Hz.

Pitch-

?

βref

βFigure 2.20: Pitch block.

2.7 Generator 31When put in State Space form becomes[β

β

]

=

[0 1

−w2n −2wnξ

] [β

β

]

+

[0w2

n

]

βref . (2.49)For the used pitch servo system the undampened eigenfrequency and thedamping ratio iswn = 8.88[rad/s]

ξ = 0.9.Furthermore it is stated from the pitch actuator speci�cations that the fastestpitching speed is β = 10 [degree/s].From the pitching system model we obtain following eigenvalues−7.9920 + 3.8707i−7.9920 − 3.8707i,from which a damped eigenfrequency of approximately 0.62 Hz is obtained.It can be shown that the transfer function from βref to β is

β

βref

=w2

n

s2 + 2w2nζs+ w2

n

, (2.50)yielding the bodeplot shown in �gure 2.21. From the bodeplot the frequencyband of the pitch actuator is estimated to approximately 1Hz. This has thephysical e�ect that when changing the pitch reference βref faster than thefrequency band of 1Hz, the signal amplitude is then limited due to actuatordynamics.2.7 GeneratorThe generator used in this HAWT model is an asynchronous generator thatexerts a torque Qg on the generator side of the drive train. The generator isfor this purpose considered lossless and thus Pg = Qg ·wg. Note that in thedescription given from the drive train that Qg and wg are de�ned in oppositedirections.The power produced by the generator Pg is controlled by adjusting the�ow of current in the rotor of the generator thus adjusting the wanted powerreference Pref .Physically it is not possible to change the torque in the generator model

32 2 Wind Turbine Modeling

10−1

100

−6

−5

−4

−3

−2

−1

0

System: syspitchFrequency (Hz): 1.04

Magnitude (dB): −3Mag

nitu

de (

dB)

Frequency (Hz)Figure 2.21: Bodeplot of the transfer function from βref to β. The bandwidthcan be read to approximately 1Hz.

Generator-� -

6

Qg

wg

Pref

P

Figure 2.22: Generator.

2.8 Unmodeled dynamics 33instantaneously, and the generator is therefore modeled as a �rst order sys-tem with a time constant τgQg =

1

τg

(Pref

wg−Qg

)

, (2.51)where Pref is a input signal relating to the torque as Pref = Qgref · wg.This is a fairly simpli�ed model but has the advantage of easily adjust-ing the exerted torque by controlling the power reference. Since the windturbine should be operating at full speed it is expected that the generatorproduces its maximum output at all time thus the power reference is �xedto Pref = 2MW.An important note for the generator model should be made. When look-ing at the relation between the torque exerted by the generator Qg and theideally �xed desired power output Pref

Qg =Pref

wg. (2.52)It is seen that if there is an increase in the rotational speed of the generatorshaft, then, because of the �xed power reference, there will be a drop in thetorque exerted on the gearbox Qload and thus the rotor shaft. This reducedtorque then results in a further increase in the rotational speed wg. This is apositive feedback loop which causes the generator model to be unstable. Thisunstable part can be veri�ed when looking at the generator model insertedin the system in �gure 2.23.

Out11

P_ref / (w_g0)^2P_ref

P_ref

Integrator

1s

1 / I_g

1/I_gQ_r1

delta−w_g

Q_gFigure 2.23: Generator model with variation in generator shaft speed - re-sulting in a positive feedback loop when having a �xed power reference Pref2.8 Unmodeled dynamicsThe model which has been described in this chapter, is as stated before asimpli�ed model that represents the major describing dynamic parts of a

34 2 Wind Turbine Modelinghorizontal axis wind turbine. Dynamics which has been left out is all veryimportant when practically designing a wind turbine, but in simulations ithas lesser in�uence on the dominating dynamics. Such to mention is the leftout modeling of the bearings in the hub and the blade pitch system. Thesehas been stated as ideal and lossless. The rotor itself has also been consid-ered perfect with no deviation from its center position. It is assumed thata complete blade symmetry exits thus making the three blades completelyidentical. These e�ects are normally referred to as 1P e�ects and occur oncein every rotation of the rotor. Furthermore no gravitational or gyroscopice�ect has been modeled. A more detailed description of P e�ects will begiven in section 4.4.An important dynamic which has been left out in the model descriptionis the e�ect of the tower shade. This e�ect occur with a frequency of num-ber of blades pr. rotation. Most HAWT use three blades and thus this e�ectis known as a 3P e�ect. The tower shade is an important 3P e�ect whichcauses a drop in wind pressure on the blade passing the tower. This e�ectcauses a severe part of the fatigue damage delivered to the blades and willbe accounted for in section 4.4 when using the unsteady BEM descriptionfor designing a cyclic pitch controller.Finally the rotor has been considered as a sti� inertia which is a poor as-sumption. But for the purpose of designing a speed controller for the windturbine to maintain a steady power output, it is suitable simpli�cation.

2.9 Complete nonlinear model 352.9 Complete nonlinear modelWith the state vector x, the input vector u, the output vector y and thedisturbance vector d de�ned asx ≡

wr

wg

θzzQg

β

βv˙v

, u =

(βref

Pref

)

, y =

(wg

P

)

, d =

(vεv

)

, (2.53)then the complete model described in this chapter can be summarized by

x =f(x,u,d) (2.54)y =g(x,u), (2.55)where the function f is a vector function containing the describing dynamicsof the wind turbine given a mean wind speed v

f(x,u,d) =

1Ir

(

Qr −Ksθtorsion −Ds

(

wr − wgNG

))

1IgNg

(

−QgNg +Ksθtorsion +Ds

(

wr − wgNG

))

wr − wgNg

z1

mt(−Dtz −Ktz + Ft)1τg

(Prefwg

−Qg

)

β

w2nβref − β2wnξ − βw2

n˙v

− ˙v τ1+τ2τ1τ2

− 1τ1τ2

v + kτ1τ2

εv

(2.56)where from the dynamics we remember that

Qr = 1wr

12ρπR

2v3rCp(λ, β)

vr = v + v − z

λ = wrRvr

Ft = 12ρπR

2v2rCp(λ, β).

(2.57)

36 2 Wind Turbine ModelingThe output function g depends on the system outputs from the generatorwhich is the power generated Pg = Qgwg and the axle speed wg and is givenbyg(x,u) =

(wg

wgQg

) (2.58)2.10 Nonlinear model summaryThe model description of the HAWT structure has now been con�ned to thedynamics represented by the ten di�erential equations in 2.56. These arenonlinear relations of the physical system from which valuable behaviouraldescription can be derived. It is noted from the drive train model that thedrive train is poorly damped at a frequency about 1.65Hz and taking intoconsideration a large inertia of the rotor, it should be expected to o�er sometrade o�s if a controller should e�ectively cancel out the eigenfrequency.At a closer look at the individual block descriptions, it is seen that thegenerator model has an unstable behaviour when looking at the transferfunction from wg to Qg in equation 2.52, and thus the jointed nonlinearmodel becomes unstable.This o�ers some complexity issues for the requirements of a controller thatkeeps the output of the system stable.

Chapter 3Model linearizationThe model described until now consists of a set of nonlinear physical equa-tions thus yielding a nonlinear model. Since the controller design that wewill apply in the following chapters is based upon a linear model description,it is needed to perform a linearization of the model summarized in section 2.9.The linearized model can be described as a state space model in the usualfor as seen in �gure 3.1BdB ∫A C+ ++ Σ

?

-u x x y-

?- - -

6�

d

Figure 3.1: Block diagram of linearized state space modelWhen linearizing we want to describe the behavior of the system in closevicinity of the nominal trajectory. For the state, input output and distur-bance vectors it is therefore assumed, that they consist of a function rep-resenting the nominal value of the linearization point, marked with a bar,and a disturbance function indicating the deviation from the linearizationpoint, marked with a tilde. Thus yielding a new state, input, output anddisturbance vectorx = x + x, u = u + u, y = y + y, d = d + d. (3.1)Furthermore it is assumed that x is close to x and that the function f listedin equation 2.56 on page 35 is partially di�erentiable. The derivatives are

38 3 Model linearizationthenx = ˙x + ˙x = f(x + x, u + u, d + d). (3.2)The model can now be linearized by approximating the system with a 1storder Taylor expansion of f(x,u,d) around the nominal values (x, u, d) whichyields

˙x ≈ ∂f(x, u, d)

∂xx +

∂f(x, u, d)

∂uu +

∂f(x, u, d)

∂dd. (3.3)From an emperical point of view equation 3.3 is simply written as

˙x = Ax + Bx + Bdd, (3.4)where A, B and Bd represents the Jacobian matricesA = ∂f(x,u,d)

∂x

∣∣∣ x=xu=u

B = ∂f(x,u,d)∂u

∣∣∣ x=xu=u

Bd = ∂f(x,u,d)∂d

∣∣∣ x=xu=u

. (3.5)Same procedure follows when linearizing the system outputs yieldingy =

∂g(x, u)

∂xx +

∂g(x, u)

∂uu, (3.6)written in short form as

y = Cx + Du, (3.7)with C and D representing the Jacobian matricesC = ∂g(x,u)

∂x

∣∣∣x=xu=u

, D = ∂g(x,u)∂u

∣∣∣ x=xu=u

. (3.8)Calculating the Jacobian matrices yield the following system-, input-, disturbance-and output matrices:A =

∂Qr∂wr

−Ds

Ir

Ds

IrNg−Ks

Ir0

∂Qr∂z

Ir0

∂Qr∂β

Ir0

∂Qr∂v

Ir0

Ds

IgNg− Ds

IgN2g

Ks

IgNg0 0 − 1

Ig0 0 0 0

1 − 1Ng

0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0∂Ft∂wr

mt0 0 −Kt

mt

∂Ft∂z

−Dt

mt0

∂Ft∂β

mt0

∂Ft∂v

mt0

0 − Pref

τgw2n

0 0 0 − 1τg

0 0 0 0

0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 −w2

n −2wnξ 0 00 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 − 1

τ1τ2− τ1+τ2

τ1τ2

where the partial derivatives

∂Qr

∂wr, ∂Qr

∂z, ∂Qr

∂β, ∂Qr

∂v,

∂Ft

∂wr, ∂Ft

∂z, ∂Ft

∂β, ∂Ft

∂v

(3.9)

39are calculated directly from the Simulink model, shown in appendix E, by usingthe matlab function linmod.B =

0 00 00 00 00 1

τgwg

0 00 0w2

g 00 00 0

Bd =

∂Qr

∂v0

0 00 00 0

∂Ft

∂v0

0 00 00 00 00 k

τ1τ2

C =

[0 1 0 0 0 0 0 0 0 00 Qg 0 0 0 wg 0 0 0 0

]

D =[0]With the Jacobians determined the linearized system is then fully formulated.For the physical HAWT model the system is linearized around a mean wind velocity

v of 15 m/s, an angular rotor speed wr of 2.295 rad/s, the output e�ect P of 2 MWand the pitch angle β of 5.27◦.The eigenvalues for the linearized system are then−0.46114 + 10.292i−0.46114− 10.292i−10.1960.024871

−0.09019 + 1.8832i−0.09019− 1.8832i−7.992 + 3.8707i−7.992− 3.8707i−0.17657−6.3015.

(3.10)As mentioned before it is seen to be an unstable system due to an unstable systempole of 0.024871 from the generator.

40 3 Model linearization

Chapter 4Wind Turbine ControlIn this chapter a control scheme for the wind turbine will be outlined and the areaof focus for the controller will be given.At �rst, the wind turbine is de�ned as a control object. A closer look at theoperation range for the wind turbine shows a number of physical constraints thatleads to a number of operation points for the model.By narrowing the area of focus a LQ controller with integral action is designedto maintain maximum power production. The LQI controller is then tested to ver-ify wind turbine design trade o�s.Finally an individual blade pitch control scheme to reduce varying blade loadswill be introduced and implemented and a comparison of the two controllers bladeload variances are performed.4.1 Existing work on HAWT controlIn Engelen (2002) it is suggested that one could use a Linear Quadratic controllerwith integral action (LQI) to maintain a desired power output. Results from use ofa LQI controller in speed control of a wind turbine, yields satisfying results of thecontroller performance to maintain the desired output e�ect.In Larsen and Mogensen (2006) a scheme for implementing individual pitch con-trol based on a stationary BEM model description is designed and veri�ed throughHAWC1 comparison.4.2 De�ning the control objectThe model setup described earlier in chapter 2 is a dynamic system with four inputsignals and and two system outputs:1Horizontal axis wind turbine code. A simulation tool developed at research centerRisoe representing a operational turbine.

42 4 Wind Turbine Controlinputs Mean wind velocity v and turbulent wind v both described by the distur-bance vector d, the power reference Pref and pitch reference angle βref bothdescribed by the input vector u.outputs Angular speed of the generator wg and the electrical output e�ect P forthe HAWT model.When looking at the model the pitch angle reference signal βref and the powerreference signal Pref are both considered controllable inputs, but the wind velocityv and v are uncontrollable disturbance input from which the controller must acton. Full state knowledge of the system is assumed, and thus the information madeavailable to the controller, ym, is a function of the system states x and also thecurrent control signal u. With this in mind a general control setup of the windturbine can be shown as in �gure 4.1.

Controllerx = f(x, u, d)

y = g(x, u)

ym = h(x, u)u=(βref

Pref

)

-

�

y=(wg

P

)

- -

d=(vv

)

Figure 4.1: A general control setup for the wind turbine.4.2.1 An operational wind turbineThe very purpose for a wind turbine is very naturally to produce its maximumgenerator power output at all time. This is of course not always possible since thewind that drives the wind turbine is a changing factor ranging from strong and highvarying to zero wind at all. Furthermore practical limitations prevents the windturbine from always operating at maximum power e�ciency.This leads to a number of operation points that requires di�erent control strategiesto uphold the physical limitation of the wind turbine. The generator speed wg mustbe kept within operating range wg_min ≤ wg ≤ wg_max and the produced powerP must not exceed Pmax.With this in mind four operational modes for the wind turbine is formulated asa function of the mean wind speed:

4.2 De�ning the control object 431. Active pitch actuators keep the wind turbine slowly rotatinguntil the wind speed is strong enough to have the generatorspeed exceed its minimum operational speed wg > wg_min.2. Constant pitch angle at maximum power e�ciency is main-tained until max generator speed is reached wg = wg_max.Power reference is adjusted according to the nonlinear be-havior of the generator mentioned in the model description.3. When maximum generator speed wg_max is reached, thisspeed is held constant. The pitch angle is kept at maximumpower e�ciency while acting on the power reference.4. When maximum power output P = Pmax is obtained thenthe generator speed is held constant by acting on the pitchangle reference signal.This is a classical approach when designing controllers for a wind turbine and adi�erent controller will be used for each of the operating modes. These controllersare then joined together to form a multiple state controller.Controllers for the �rst three operating modes have been designed to a satisfyingextend and thus we will keep our focus on the fourth and last described operatingmode, where the pitch action is the dominating factor to maintain the desired out-put power P, and fatigue is a genuine concern.Figure 4.2 shows the relationship between the wind speed and the pitch actionand how this is related to the generator speed and power output.

PowerPitchangle

Generator speedWind speed

wg−nomwg−min

Pmax

Figure 4.2: Pitch action of a wind turbine as a function of the wind speed.

44 4 Wind Turbine ControlWhen the wind speed exceeds the limit where maximum output power is gener-ated, it is necessary to somehow brake the generator speed, so it doesn't exceed itsrotational speed limit. This is done by pitching the blades to a less power e�ciencyangle. This type of pitching is done by applying a collective pitch controller, thatvaries the angle of all three blades at the same time, thus keeping the rotor speedat the nominal value.

4.3 Collective pitch control 454.3 Collective pitch controlThe de�ned control object is a multiple input multiple output system (MIMO Sys-tem). It should be reminded that the nonlinear system description is unstable dueto the generator model, which adds to the requirements of the controller.A possible solution for a controller that keeps the system stable and the output ef-fect of the wind turbine at a desired 2MW, is through the use of a Linear Quadraticcontroller.Linear Quadratic control technique is control based on an optimization of a quadraticcost-function J , that minimizes the error between the ideal response and the actualresponse and through its de�nition always yields a stable closed loop system.J =

∞∫

0

(r(t) − y(t))2dt, (4.1)where r(t) is the ideal response and y(t) is the controller response. The optimalsolution for this cost function would be r(t) = y(t). But in order to do so, thecontroller has to follow variations in the ideal response in�nitely fast which againrequires control signals of in�nite size.Signals of in�nite size requires in�nite energy. This is of course not possible andwhen taking into account, that this is a cost function to be applied to a physicalsystem, equation 4.1 can be rewritten to equation 4.2J =

t2∫

t1

(xT Q1x + uTQ2u)dt, (4.2)which is a weighted cost function with t1 = 0 and t2 < ∞. In this weighted costfunction x is the states of the system, u is the control signal and Q1 & Q2 are theweight matrices that can be adjusted given to a desired performance behaviour.One disadvantage that follows the use of LQ control is, that full state knowledgeis required at all time which is only possible when reliable instantaneous measure-ments are �xed to every state of the system at all time or when a reliable stateestimate is known.The reason for its use is that it is relatively simple to design, and it has strongcontrol theory possibilities which can be realized through the use of an observer thatestimates the states of the system. The observer based controller is obviously moreadvanced and for the purpose of showing control performance, both are equallyadequate. In �gure 4.3 a state space model of the general LQ controller is shown.As mentioned the controller should maintain a steady electrical power output for theHAWT system at a constant 2MW under any circumstances. The LQ controlleritself provides a controller that stabilizes the system, but in order to prevent astationary error an integrator is added thus yielding a LQI controller.

46 4 Wind Turbine ControlBd

B∫

A

C

+ ++ Σ

?

u x x y-

?- - -

6�

d

K

Σ

6- -

�Figure 4.3: State space model of the LQ control loop.4.3.1 Design of LQI controllerLet's consider the linearized state space model of the system as described in chapter3, in terms of the deviations from the linearization point:˙x = Ax + Bu + Bdd

y = Cx.The purpose of the Linear Quadratic controller is to maintain a desired outputpower P and keep the generator speed, wg at its nominal value. In this setup bothmust be controlled without having any stationary errors which is done by addingtwo new integral action states xi based on P and wg, that ensures correct referencefollowing:xi =

[PIwgI

]withxi = −Cx.With the added integral action states, the system expands from a ten state systemto a twelve state system and the new augmented state vector can be expressed asxa =

[x

xi

]

. (4.3)The expansion of the state vector also causes an expansion of the system. Thissystem expansion leads to the augmented system description[

˙x

xi

]

=

[A 0

−C 0

] [x

xi

]

+

[B

0

]

u +

[Bd

0

]

d

y =[C 0

][x

xi

]

4.3 Collective pitch control 47also written as˙x = Aaxa + Bau + Bd,ad (4.4a)y = Caxa. (4.4b)The augmentation of the system by adding the integral states follows the principalsdescribed in Hendricks et al. (2006) in chapter 4.With the system expressed in terms of deviation from the linearization point, onecould consider the cost functionJ = E

{∫

εTQεdt

}

, (4.5)where the error vector ε is de�ned as:ε =

(

wg P Qg β wgI PI βref Pref

) (4.6)It is then possible to express the cost J as a wighted sum of the variances of all theelements in the error vector ε:J =

∫(yT xT xT

i uT)

Qy 0 0 00 Qx 0 00 0 Qxi

0 00 0 0 Qu

˜y˜xxi

˜u

dt, (4.7)where the weights are the weights individually placed on the input-, output- andcontrol signals, and the integral states and the states of the system:

Qy =

(Qwg

00 QP

)

, Qxi=

(QwgI 0

0 QPI

)

, Qu =

(

Qβref0

0 QPref

) (4.8)andQx =

0 0 0 0 0 0 0 0 0 00 Qwg

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 QQg

0 0 0 0

0 0 0 0 0 0 Qβ 0 0 0

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

. (4.9)

48 4 Wind Turbine ControlBy combining the equation 4.4b and 4.3, with the equation describing the cost J ,then a new expression for the cost J is obtained:J =

∫(CT xT

a xTa uT

)

Qy 0 00 Qxa

00 0 Qu

xa

xa

xu

dt

=

∫(xT

a uT)(CT

a QyCa + Qxa0

0 Qu

)(xa

u

)

dt

=

∫

xTa (CT

a QyCa +Qxa)

︸ ︷︷ ︸

Q1

xa + uT Qu︸︷︷︸

Q2

u

dt (4.10)whereQxa

=

(Qx 00 Qxi

)

.It can be shown that the control signal u that minimizes the performance indexgoverned by the cost function J in 4.10 regarding the the system in 4.4, is given bythe control law u = −Kxa.This however only holds if all the uncontrollable states in the system is stable.The only uncontrollable states belongs to the wind model v and ˙v. These can beshown stable and thus the system remains stable.Solving the Ricatti equation yields the optimal controller gain K. The Ri-catti equation is solved in Matlab by the use of the function lqr, based on theinformations speci�ed in the performance weights Q1 and Q2.4.3.2 Implemented LQI controllerFirst thing to do when implementing the collective controller described in the pre-vious sections, is to determine the performance weights Q1 and Q2 based on thedesign criteria's. As mentioned the weights in the LQ design represents a weightingof the variances and since the design criteria of the collective pitch controller, isde�ned as maintaining a constant e�ect output for the wind turbine at a 2MWlevel, then weights are placed upon the integral states. Weights are also places onthe input and output of the system and the system states wg, Qg and β.A common starting point when calibrating weights in LQ design is to use weightsequal to the inverse of the square of the nominal value. This corresponds to a equalweighting of variances in the weighted variables. Hereafter the weights are adjustedthrough a series of performance tests.Qy =

(1

w2g

0

0 1w2

g· 0.1

Q2g

)

, Qxi=

(100w2

g0

0 3000P 2

ref

)

, Qu =

(0.15β2 0

0 0.5P 2

ref

) (4.11)

4.3 Collective pitch control 49andQx =

0 0 0 0 0 0 0 0 0 00 1

w2g

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0.1

Q2g

0 0 0 0

0 0 0 0 0 0 0.15β2

0 0 0

0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0

. (4.12)It is common in other control strategies to include weights on the system statesrepresenting the deformation of the shaft θtors and the movement of the tower z.These have been left out in this control scheme to keep focus on the electrical outpute�ect and limit the pitch action for now.4.3.3 LQI DynamicsAdding the LQI controller to the linearized MIMO system is seen to stabilize thesystem and the unstable pole from the generator model is now stable.

−20.479 + 18.86i−20.479− 18.86i−0.38994 + 10.372i−0.38994− 10.372i−8.9556 + 5.5959i−8.9556− 5.5959i−0.27555 + 0.27455i−0.27555− 0.27455i−0.090103 + 1.8832i−0.090103− 1.8832i−0.17657−6.3015

(4.13)The bode plot for the transfer function from βref to P is shown in �gure 4.4. Fromthe gain it is seen that the eigenfrequency of the drive train has remained undampedwhich corresponds with the design proporties. Only the gain is depicted since theLQI controller operates on a MIMO system.4.3.4 Simulations with LQITo verify the behavior of the system dynamics a set of simulations is performed.Steady state responses for the closed loop LQI system is shown in appendix A.DynamicsThe �rst simulation in �gure 4.5 shows a step response in the closed loop system.The simulation is performed with a 1 m/s step in the wind input with the windhaving no turbulence, and tower shade e�ects have been left out. The purpose of

50 4 Wind Turbine Control

10−2

100

102

10−1

101

−150

−100

−50

0

50

100

150From: Bref,in To: Pg,out

Mag

nitu

de (

dB)

Frequency (Hz)

open Gopen GKclosed GKFigure 4.4: G - Transferfunction from βref to P .this simpli�ed simulation is to clear out the resonance frequencies from the dynam-ics and their general e�ect on the power output P .From �gure 4.5 it is seen that a step in the wind input causes some poorly dampedoscillations on the power output. These oscillations occur at a frequency of about1.65Hz and are easily identi�ed as the spring torsio e�ect of the drive train. Thesettling time of the power overshoot is approximately 10s which is caused by thelarge inertia of the rotor. Furthermore the step input forces a change in pitch angleand thus changing the thrust force FT of the rotor, causes a oscillation of the towerat its eigenfrequency at 0.3Hz. One might �nd it peculiar that the position of thetower is closer to its relaxed position after an increase in the mean wind speed.This oddity is due to the fact that the increased wind speed causes a pitching ofthe blades to a less power e�ciency angle, which also decreases the thrust FT of therotor and thus the only described dynamic a�ecting the movement of the tower.LQI systemThe step response of the full closed loop LQI system is now investigated. Having astep from 15 m/s to 16 m/s in the stochastic wind input with a turbulence intensityof 13.16%, which corresponds to the expected turbulence of the used wind modelat open farmlands, results in the dynamic system behaviour as depicted in �gure 4.6.The step response shows a steady power output of the HAWT model with a sta-tionary error close to zero. As a result of the 1 m/s increase in the mean windvelocity, the pitch angle changes approximately 3◦ to maintain the nominal poweroutput. At all times it is seen that the pitch velocity is kept below the limit of 10◦/s.This behaviour corresponds to the desired behaviour for the closed loop system

4.3 Collective pitch control 51

0 5 10 1514

16

18

Windspeed[m/s]

0 5 10 151999.5

2000

2000.5

Poweroutput[kW]

0 5 10 15

8.2

8.4

8.6

x 10−3

Shafttorsio[rad]

0 5 10 1510

15

20

25

Towerposition

[cm]

time [s]Figure 4.5: System dynamics identi�cation given a 1 m/s step in wind inputat t = 5s. The step is performed from the linearization point of mean windspeed of 15 m/s. Simulation is performed with no turbulence or tower shadee�ects present to show the HAWT dynamic e�ects on the power production.

52 4 Wind Turbine Controlthus the collective pitch controller maintains a stable output power close to itsnominal value at 2MW.0 10 20 30 40 50 60

14

16

18Windspeed[m/s]

0 10 20 30 40 50 60194

195

196

197Generator

speed[rad/s]

0 10 20 30 40 50 601999.5

2000

2000.5Poweroutput[kW]

0 10 20 30 40 50 6010.1

10.2

10.3Rotortorque[kNm]

0 10 20 30 40 50 600

5

10

15Pitchangle

[°]

0 10 20 30 40 50 60−10

0

10Pitch

velocity[°/s]

0 10 20 30 40 50 6010

15

20

25Tower

position[cm]

time [s]Figure 4.6: LQI closed loop system response to a 1 m/s step in the turbulentwind input at t=20s. The plot shows a step in the mean wind velocity from15 m/s to 16 m/s. The power is seen stable within a range of approximately±250W of its nominal output e�ect of 2MW.In �gure 4.7 it is demonstrated how the trade-o� in the performance weightinga�ects the system. When wanting to have more stable tower behaviour then aperformance weight in the tower position state z could be implemented. This isseen in �gure 4.7 which results in more steady tower movement but at the cost ofpitch action.

4.3 Collective pitch control 53

0 10 20 30 40 50 6014

16

18

Windspeed[m/s]

0 10 20 30 40 50 60194

195

196

197Generator

speed[rad/s]

0 10 20 30 40 50 601999.5

2000

2000.5Poweroutput[kW]

0 10 20 30 40 50 6010.1

10.2

10.3

Rotortorque[kNm]

0 10 20 30 40 50 600

5

10

15Pitchangle

[°]

0 10 20 30 40 50 60−10

0

10Pitch

velocity[°/s]

0 10 20 30 40 50 6010

15

20

25Tower

position[cm]

time [s]Figure 4.7: LQI closed loop system response to a 1 m/s step in the turbulentwind input at t=20s with a performance weight on the tower position rep-resented by state z. The plot shows a step in the mean wind velocity from15 m/s to 16 m/s. The position of the tower is seen more stable but greatervariations in the output e�ect as a result of the changed performance weight.

54 4 Wind Turbine Control4.3.5 Limitation of collective pitch controlIn the previous section, a LQI controller for the wind turbine to maintain a desiredpower output was designed. The main design criteria was to keep the generatorspeed at a nominal value wg and thereby the power at Pnom. This type of HAWTcontroller is often referred to as a speed controller. The LQI speed controller has agood performance in terms of maintaining the generator speed by pitching all threeblade angles at the same time and at the same angle.This type of pitching is only optimal if the turbine is operating in a uniformdistributed wind �eld with no periodic deviations in the wind �ow through the discarea.In reality wind turbines are exposed to highly varying wind �elds due to turbu-lence and wind shear. Furthermore the wind a�ecting the rotor disc area are alsosubject to e�ects occurring at rotational frequency, with tower shade being themost dominant factor. These e�ects together imposes a highly varying wind �owthrough the area swept by the rotor, and causes di�erent loads a�ecting each of theblades.The varying loads cause a �ap wise bending torque which stresses the structureof the blades and causes unwanted fatigue damage.Remembering from the wind model in �gure 2.5 on page 12 that when the meanwind speed v increases by a factor then the turbulent part of the wind v increaseswith larger factor. With this in mind one may realize that the loads on each ofthe blades become more and more varying at increasing wind speeds, and the needfor action to prevent these increasing load variances becomes more apparent. Ifno action is taken, the gradually increasing load variances would then in�ict anincrease in fatigue damage, which greatly reduces life expectancy of the blades.6

-

FatigueLife expectancy Glass�berSteel

Figure 4.8: Trend curve for life expectancy of a blade given the fatigueexposure.Fatigue of a HAWT structure is a large topic and not the focus of this thesis. Itis though very interesting when measuring performance estimates, and for this rea-son a simpli�cation of fatigue damage will be used. Fatigue damage will be directlyreferred to as the varying of the loads measured at the blades. Thus a reduction inthe load variances will be considered as a reduction of fatigue damage.With the still increasing rotor diameters and thus the swept area by the blades,wind shear and turbulence is becoming a more dominant factor when measuring

4.3 Collective pitch control 55di�erence in the loads a�ecting each of the three blades. Combined with the manu-facturers urge to increase the operating range of the HAWT in terms of wind speeds,it is far from optimal to use same pitch angles at all three blades. An individualblade pitch control scheme is therefore introduced and will be implemented in thefollowing section.

56 4 Wind Turbine Control4.4 Individual pitch controlOne major aspect of wind turbine control concerns the load variations causing bladebending moments which may be considered as fatigue. These blade load variationsare contributed from two types of e�ects: Cyclic- and stochastic variations.The cyclic load variations occur due to a rotational sampling of a non-uniformwind �eld. The wind �eld is considered inhomogeneous, thus wind shear and towershade are dominant factors, that give rise to periodic variations in the bendingmoments at the blades. These bending moments occur at a frequency equal to amultiple of the angular rotor speed wr, and are generally referred to as nP e�ects,with n being a integer.The stochastic load variations are caused by the rotational sampling of the tur-bulent part of the wind, which results in stochastic blade variations with a broadfrequency content centered in peaks around nP .The resulting rotor torque experienced is the sum of the three blade moments,which for a 1P stochastic loading of each blade, transforms into a n3P harmonic.It is the periodic 1P harmonics of the blades the individual pitch controller shouldminimize.4.4.1 Previous workReduction of cyclic and stochastic blade loads is a very present concern of commer-cial wind turbine manufacturers and several reports and thesis's have been writtenabout this topic. The most common approach chosen is transforming the threeblade moments into a tilt and yaw moment, which then is tried minimized. Thisis an indirect way of minimizing the blade loads but has shown to yield good results.Using this approach has especially been contributed by works of T.G. van En-gelen and E.A. Bossanyi, which is documented in Engelen (2002), Engelen (2006)and Bossanyi (2003).4.4.2 Individual pitch controlAn intuitive approach of decreasing the cyclic load variances is implementing a con-troller that individually determines the blade angles in such a way, that all varyingloads are minimized. This however introduces an issue concerning the pitch actionfrom the individual pitch controller which could e�ectively minimize all loads, andwould bring the rotor to a unwanted stop. An unfortunate event since the generalpurpose of the wind turbine is to keep its output e�ect at a constant nominal value.To prevent above from occurring the individual pitch controller is implementedas a separate control loop that adds a pitch signal ∆βindv to the pitch signal βcollof the collective pitch controller. Shown in �gure 4.9 on the next page. This waythe collective controller maintains the desired output e�ect, and the individual pitchcontroller minimizes the blade load variations.

4.4 Individual pitch control 57HAWTColl. CIndv. C Σ

6

βcoll

∆βindvMflap

˙x6Pref

βref ++�

--

-Figure 4.9: Collective- and individual pitch controller joined as a combinedpitch controller for the HAWT model.It should be noted that with the chosen structure of the jointed controller design itis possible to perform an online removal of the added control loop from the indi-vidual pitch controller, such that control action is reduced to the previous describedcollective pitch control.Updated model descriptionIn order to be able to use blade loads as a measure of control, it is needed to re-con�gure the aerodynamics of the model used for the collective pitch controller.For a thorough description of how the dynamic wind model a�ects the rotor andthus the individual blades, the description of the aerodynamics is changed from"`uniform distribution"' to "`dynamic BEM model"' as described in section 2.3.2.With this change, the model now describes the behaviour of the blades, and it ispossible to simulate and calculate the loads at each blade element. This measurehowever is only available internally in the model, since the wind is still considereda uncontrollable input. Thus to obtain the magnitudes and frequency of the bladeload variations a direct measure is needed.A direct measure of the blade loads are implemented as sensors in form of strainguages, from which the bending of the blades are estimated and a measure of blade�ap moment or blade load, can be calculated. The three strain guages are repre-sented by measurement signals: y4,y5 and y6.y4: Mflap1y5: Mflap2y6: Mflap3With the implemented measures of the �ap wise forces through use of the strainguages, it is now possible to perform some types of blade load reduction control.One approach is to use the mean of the three blade loads as a reference, and thentrying to minimize the di�erence between the individual blade loads and the refer-ence. This however introduces issues concerning stability of the output e�ect andnegative feedback loops.Another approach is to divide the rotor into sections. Sections describing howeach blade load can be described by two components: A tilt- and a yaw component

58 4 Wind Turbine Controlwhich depending on the position of the blades, adds a negative or positive contri-bution to the tilt/yaw moments. This way of transforming the three blade loadsinto a tilt and yaw moment is also known as a Coleman representation. The controlstrategy, when using Coleman representation, is to minimize the di�erence betweenthe negative and the positive tilt and yaw contributions in such a way, that thereis a load equilibrium across the disc.Coleman representationTransforming the three blade loads or �ap moments into the two component tilt/yawsystem is done through the inverse Coleman transformation matrix P−1 :P−1 =

13

13

13

23sinψ1

23sinψ2

23sinψ3

23cosψ1

23cosψ2

23cosψ3

(4.14)When applied on the blade root moments yields

∆Mcm1

∆Mcm2

∆Mcm3

= P−1 ·

Mflap1

Mflap2

Mflap3

, (4.15)where the tilt and yaw moments are given by ∆Mcm2 and ∆Mcm3. ∆Mcm1 containsthe average �ap wise moments for the three blades, which is not used. The azimuthangle ψ is de�ned as being zero at the top point of the disc, thus the sinus componentindicates the force in the yaw direction, and the cosine indicates the tilt.It should be noted that the change in the frequency band when transformingthe 1P e�ects of the blade load variations to the rotor torque, also applies whentransforming the blade loads into a tilt and yaw component. Thus the frequencyof the blade loads centered around 1P e�ect transforms into tilt and yaw variationscentered around n3P frequencies.When transforming the other way around, thus when a proper pitch angle correctingthe tilt and yaw moments has been determined by the individual pitch controller,it is possible to remodulate this into three individual blade angles through theColeman transformation P:P =

1 sinψ1 cosψ1

1 sinψ2 cosψ2

1 sinψ3 cosψ3,

(4.16)applied to the pitch angles determined for the tilt and yaw direction yields the threeblade pitch angles.

∆βindv,1

∆βindv,2

∆βindv,3

= P ·

βcm1

∆βcm2

∆βcm3

(4.17)

4.4 Individual pitch control 594.4.3 De�ning the control objectThe control strategy is to minimize the variance of the blade loads measured iny4, y5 and y6, by minimizing the resulting tilt Mtilt and yaw Myaw moments, ob-tained from the blade moments through a Coleman transformation.The control object is still described by the dynamic model of the wind turbineoutlined in chapter 2, but is expanded by three blade loads measurements. Withthe control objective now being to minimize the varying blade loads measured bythe three strain guages, the control object can be de�ned by eight inputs and twooutputs:inputs The disturbance d in form of the mean wind velocity v and turbulent wind

v, the three pitch angle references βref1, βref2 and βref3, and the three bladeload measurements Mflap1, Mflap2 and Mflap3.outputs The resulting moment contributions in the tilt and yaw direction Mtiltand Myaw.Referring to the model description, the pitch angle reference signals βref1−3 andthe blade load measurementsMflap1−3 are considered controllable inputs, while thedisturbance d is a uncontrollable input of the varying wind velocities.4.4.4 Design of individual pitch controllerWhen using Coleman transformations of the three blade load measurements, adecoupling of the moments in the tilt and yaw axis is obtained. Based on this de-coupling of the system can be represented as two independent SISO systems, fromwhich a two di�erent controllers can be designed. One controller for each axis. Thecontrol output for each of the controllers can then be remodulated into the threepitch angles.An important note to make about using the Coleman transformation is that thedescribed Coleman transformation assumes that the rotor disc is a�ected by a ho-mogeneous wind �eld. This is a rather rough assumption, but for the simplicity ofthe design of making a fast controller, it yields good results. A small justi�cationof this approach is that since the controller uses blade load measurements takenin the real inhomogeneous wind �eld, then the controller is still considered a realindividual pitch controller.The two controllers for the decoupled tilt and yaw moments will be designed asI-compensators. This is a fast and e�ective controller when operating with a de-layed proportional system such as the pitch vs. load system. The I-compensatorhas the transfer function G(s),G(s) =

Ki

s. (4.18)As previously described, the 1P harmonic content of the blades transforms inton3P harmonics in the tilt and yaw description. The postulate is that reducing

60 4 Wind Turbine Controlthis 3P harmonic will in turn reduce the 1P harmonics experienced by the blade.Harmonics higher than the 3P content, is considered multiples of the 1P and 3Pharmonics, thus reducing these will also reduce harmonics higher than the 3P. Toprevent the controller from acting on higher harmonics than the 3P, a low pass�lter around and beyond the content of the 3P harmonic is applied. The low pass�lter used is a Butterworth �lter having a general transfer function H(s):H(s) =

B(s)

A(s)=b(1)sn + b(2)sn−1 + · · · + b(n+ 1)

sn + a(2)sn−1 + · · · + a(n+ 1)(4.19)4.4.5 Implementation of I-compensatorThe I-compensators are implemented as two separate controllers for the SISO sys-tems represented by the demodulated blade moments in axis of tilt and yaw. Thegeneral setup used for the control feedback loops of the individual pitch controlleris shown in �gure 4.10

Ki

s

Ki

s

�

�

demodulation

sinψ1 sinψ2 sinψ3

cosψ1 cosψ2 cosψ3

Low-passw3P

∆Mcm2

∆Mcm3

��

-

-

βtilt

βyaw

+ + +++--

-+βcoll

??

?

βref1

βref2

βref3

6

?���

ψ

∆Mz1

∆Mz2

∆Mz3

HAWTColl. C

˙x?

Individual Pitch Controller

Collective Pitch Controller---

Pref

∆βindv1

∆βindv2

∆βindv3

remodulation

sinψ1 cosψ1

sinψ2 cosψ2

sinψ3 cosψ3

Figure 4.10: Layout of control loops for individual pitch control.The three blade load measurements are demodulated by the use of Coleman trans-formation into two separate signals. A signal containing the tilt axis describing theresulting tilt moment ∆Mcm2, and a signal of the yaw axis containing the resultingyaw moment ∆Mcm3. ∆Mcm2 and ∆Mcm3 are then low pass �ltered, such thatcontent from harmonics higher than 3P are not seen by the controllers. The uselow pass �lter is a 4th order Butterworth �lter with a cuto� frequency at 3P =(3wr)(2π) = 1.0958Hz, where wr is the angular velocity of the rotor having a steady

4.4 Individual pitch control 61value from the collective pitch controller at 2.295 rad/s.The integrator gain Ki of the controller, is determined by the negative inversepartial derivative of the �ap-wise moment Mflap in the point of linearization, withregards to an increase in the pitch angle β.Ki = − ∂β

∂Mflap

. (4.20)The gain Ki is calculated from the BEM model by changing the pitch angle ∆βfrom the linearized point, waiting until the �ap wise moments have settled, andthen calculate the change in the Mflap. This is repeated for a number of pointsnear the linearized point of the pitch angle, yielding the graph shown in 4.11.

4.8 4.9 5 5.1 5.2 5.3 5.4 5.5 5.6 5.73.3

3.35

3.4

3.45

3.5

3.55

3.6

3.65

Mfla

p [MN

m]

β [deg]Figure 4.11: Integrator gain Ki determined by the inverse of the tangentialincrease in Mflap as a result of changing the pitch angle of the blade.The negative of the inclination of the graph is equal to the magnitude of the in-tegrator gain Ki used for the individual pitch controller for both the tilt and yawcontrol feedback loops. From 4.11 the integral gain is estimated to Ki = 4.14 ·10−6[◦/Nm]. This may seem as an extremely small gain, but when taking into consid-eration that it is an expression of the gain from the pitch angle to the blade loadmeasured at the entire blade, it is a reasonable gain.4.4.6 Simulation with I-compensatorMuch like the simulations performed with the collective pitch controller, simu-lations of the closed loop system with the added individual pitch controller arecarried out. Steady state responses for the cyclic control loop system is shown inappendix A on page 97.The closed loop response to a 1 m/s step in the wind input at t=20s is depicted in

62 4 Wind Turbine Control�gure 4.12, showing a system behaviour much like the behaviour of the collectivepitch control system. The output power is kept close to its nominal value and thesettling time is approximately the same 10s. Pitch velocity is kept below 10◦/s.The only visible di�erence between this simulation and the simulation performedwith LQI is that now three pitch signals are modeled and there is a slight increasein the pitch action. The e�ect of this increase in pitch action can be seen whenperforming a spectral analysis.0 10 20 30 40 50 60

14

16

18

Windspeed[m/s]

0 10 20 30 40 50 60194

195

196

197Generator

speed[rad/s]

0 10 20 30 40 50 601999.5

2000

2000.5Poweroutput[kW]

0 10 20 30 40 50 6010.1

10.2

10.3

Rotortorque[kNm]

0 10 20 30 40 50 600

5

10

15Pitchangle

[°]

0 10 20 30 40 50 60−10

0

10Pitch

velocity[°/s]

0 10 20 30 40 50 6010

15

20

25Tower

position[cm]

time [s]Figure 4.12: I-compensator closed loop system response to a 1 m/s step inthe turbulent wind input at t=20s. The plot shows a step in the mean windvelocity from 15 m/s to 16 m/s. The power is seen stable within a range ofapproximately ±250W of its nominal output e�ect of 2MW.When increasing the gain Ki of the I-compensator then the pitch velocity activitygets more aggressive and at 4.2 times the designed gain the pitch velocity is limitedby the pitch actuators and the system is not controlled properly. In �gure 4.13 isdepicted the unstable pitch velocity behaviour due to increasing the I-compensatorgain Ki by 4.2 times.

4.5 Spectral analysis 630 10 20 30 40 50 60

1999.5

2000

2000.5

Poweroutput[kW]

0 10 20 30 40 50 600

5

10

15

Pitchangle

[°]

0 10 20 30 40 50 60−10

0

10

Pitchvelocity

[°/s]

time [s]Figure 4.13: I-compensator closed loop system response with gain Ki in-creased by a factor 4.2.4.5 Spectral analysisIt is not obvious from the simulation results of each of the controllers, to verify theexact e�ects of implementing the cyclic pitch controller to the control loop of thepower scheme. Both simulations show a steady power output and that pitch actionis provided to counteract the variations in the stochastic wind.The only apparent di�erence between the two simulated controllers is the amountof applied pitch action. The pitch action when having added the cyclic pitch con-trol scheme on top of the collective controller has increased compared to when onlyusing the collective pitch controller. The e�ect of this extra pitch action have nogreater e�ect on the power output or the rotor torque of the turbine, nor the move-ment of the tower.The cyclic pitch controller was designed to minimize blade load variances in termsof tilt and yaw moments and thus it does not make sense to look at output e�ectP to see the result. Instead, to see a change in the �ap wise moments of the bladeswhen adding a cyclic pitch controller, a spectral analysis is performed to see howthe power of the varying moments are distributed.4.5.1 Power spectrum densityBy looking at the power spectrum density (PSD) graph of the tilt and yaw momentit is expected for both control systems to see a reduction in the frequency containingthe power of the 3P variations. (Remembering that the 1P stochastic blade loadvariations transforms into a 3P variation in the tilt and yaw moments)The PSD graphs described in this section are all performed at a mean wind ve-locity of 15 m/s with wind shear, turbulence and tower shade e�ect. Note that in

64 4 Wind Turbine Controlthe PSD graphs the spectral density of the collective controller have been shifted0.15Hz for the purpose of easy comparison.Figure 4.14 shows a comparison of the PSD in the tilt and yaw wise moment whenusing collective- and cyclic pitch control. It is seen that when using collective pitchcontrol there is a dominating DC gain in both the tilt and the yaw moments whichis due to the wind shear e�ect. This DC gain is completely removed when switchingto the cyclic pitch controller and thus the constant load on the nacelle structure isremoved. This is a very positive result that releases some of the constant stress onthe HAWT structure. This however does not have a direct e�ect on the blade loads.But the power of the 3P frequency (3wr/2π = 1.10Hz) containing the variationsfrom the blade loads, is seen to decrease with approximately 13% in the tilt momentand 20% in the yaw moment, when adding the cyclic pitch control and thus it isexpected that the power of the 1P blade load harmonic is decreased.

0 1 2 3 4 50

1

2

3

4

5x 10

10 PSD of Mtilt

Frequency [Hz]

Pow

er [(

Nm

)2 /Hz]

CollIndv

0 1 2 3 4 50

1

2

3

4

5

6

7

8

9x 10

8 PSD of Myaw

Frequency [Hz]

Pow

er [(

Nm

)2 /Hz]

CollIndv

Figure 4.14: Power spectrum density of tilt and yaw moments with a tur-bulent wind input. The PSD from the collective controller has been shifted0.15Hz for easy comparison. It is seen that the DC gain has been removedwith individual pitch control and 1P frequency content has been greatlyreduced.Looking at the PSD of a single blade in 4.15 it is seen that the power density ofthe 1P harmonics are almost canceled out when implementing the individual pitchcontroller. This corresponds well to the reduction of the power density of the 3Pharmonics as seen for the tilt and yaw moments. For the collective pitch controllerthe power of the 1P harmonic is seen to be of great magnitude compared to theindividual control. This 1P harmonic is mainly due to the wind shear e�ect andthe presence of tower shade with wind shear being the dominating contribution.

4.5 Spectral analysis 65

0 0.5 1 1.5 2 2.5 30

5

10

15x 10

10 PSD of Mblade

Frequency [Hz]

Pow

er [(

Nm

)2 /Hz]

CollIndv

Figure 4.15: Power spectrum density of a blade with a turbulent wind in-put. Using individual control removes the power of 1P harmonics, thus theperiodic load variations due to tower shade and wind shear are removed bythe pitch scheme.

66 4 Wind Turbine Control4.6 ConclusionBased on a simple wind turbine model a MIMO control strategy in form of a LQIstate feedback controller has been applied and is seen to stabilize the system. TheLQI controller can be tuned through the cost function and is optimal in the senseof designers speci�cations of the performance weights. The LQI controller showsa steady performance of controlling the system outputs, the electrical producedpower P and the generator speed wg trough pitching of the blades, such that asteady electrical production is maintained.Secondly an extra control loop was added to the system to reduce blade loads.Based on three blade load measurements a decomposition into a tilt and yaw mo-ment reduces the system into two SISO systems. These two SISO systems can beidenti�ed as proportional systems and thus it is su�cient to use a I-compensatoras controller.The e�ect of adding the individual pitch controller is veri�ed when looking at thepower density spectrum for both collective and individual pitch control. The PSDfor individual pitch control shows a complete removal of the DC gain in both tiltand yaw wise moments, which was of a magnitude +1010 [Nm2/Hz]. Furthermorethe blade load around the 1P harmonic is seen to reduce to less than 1/10 of thecorresponding magnitude when using only collective pitch control. Overall addingthe individual pitch controller signi�cantly reduces the blade load variations andthe structural variation contributed through the tilt and yaw wise moments.

Chapter 5Fault Tolerant ControlThis chapter will describe how a fault occurring in the blade load measurementswill be processed.Residuals generated from a structural model are used to detect and isolate faults onthe blade load measurements which are critical feedback signals for the individualpitch control.A cumulative sum (CUSUM) test on the residuals is implemented to detect faultsin the presence of measurement noise.Finally a sensitivity analysis of the system is performed and based on this a re-con�guration of the HAWT control is made available.5.1 Fault tolerant pitch controlPitching of the blades is an e�cient way of keeping the generator speed at a de-sired level. But as the mean wind speed increases, an even further increase ofthe turbulent part of the wind occurs. This results in a great variance of the loadse�ecting each blade and the cost of a sensor fault may be fatal for the wind turbine.When operating a wind turbine, in very turbulent wind �elds, a controller is highlydependent on the feedback from the sensor signals.5.2 Sensor systemThe sensor system in a HAWT is a complex system with many variables beingmonitored continuously to make sure that the wind turbine is within its operatingrange and thus it is safe to continue its energy production, without any risk ofdamaging vital parts of the turbine.Our main focus will be the monitoring of the blades and their status when individ-ually pitching them to reduce the variance of the loads. Thus the initially chosenfeedback signals used on the HAWT model consist of 12 measurements y1 − y12

68 5 Fault Tolerant Controllisted in table 5.1. It should be noted that three more blade load measurementshave been added to the system. These are used for fault diagnosis and isolation.y1: β1 y7: ψy2: β2 y8: Py3: β3 y9: vry4: Mflap1 y10: Mflap1y5: Mflap2 y11: Mflap2y6: Mflap3 y12: Mflap3Table 5.1: Measurements performed on the HAWT model.From the complete dynamic system model of the HAWT model as described insection 2.9 on page 35 it is possible to simulate and calculate system behavior tovariable changes and their various e�ect. This is tedious work that requires manycalculations, and the relation between the components in the wind turbine and theire�ect on other parts may not be obvious.For the purpose of fault diagnosis, the dynamic system is instead represented by anabstraction of its behavioural model. This abstraction shows merely the existenceof links between variables and parameters through a set of constraints. This is acoarse model of the system behaviour but yields important information about thesystem properties.Through a structural model of the dynamic system, it is possible to identify com-ponents in the system that are monitorable and those that are not. Furthermorethrough structural analysis it is possible to generate residuals from which a failurescheme for a possible recon�guration of the system can be extracted.5.3 Structural modelThe structural model for the complete system is rather complicated and sincethe main focus will be on the fault detection in the pitch actuator system andthe e�ect on signals in its close vicinity, a subsystem model representing theseparts interaction will be used. This sub-structural model is depicted in �gure �g-ure 5.1 on the facing page.5.3.1 ResidualsFrom the structural model, it is possible to generate a set of residuals describingthe interrelation between the known and unknown variables, through the systemconstraints. The approach for designing a set of residuals, is �rst to verify theknown and unknown variables. In this particular setup this results in:

5.3 Structural model 69

βref1

β1

C11 − 13

Mflap1

BEM1 − 3

Ψ

v

zvr

Qr

P

FT

C10

BEM4

C6

BEM5

C9?

6

?

?

? ?

?

? ??

6

? ??

No measurement Direct measurement

Figure 5.1: Structural graph of subsystem to the full HAWT model. Sub-system is based on dynamic e�ects having a close relation to the blade loadsand the pitch actuator.

70 5 Fault Tolerant Controlknown variables: [βref1, βref2, βref3, y1, y2, y3, y4,y5, y6, y7, y8, y9, y10, y11, y12]unknown variables: [Ψ, v, vr, FT , z, Qr, P , β1, β2,β3, Mflap1, Mflap2, Mflap3]When all the known and unknown variables are identi�ed, it is possible to �nd aset of unmatched constraints that only depends on known signals trough the rela-tions between the variables given in the system constraints in the structural model.These unmatched constraints are also referred to as residuals.The residuals are generated through the use of SA-Tool v1.31. This small Mat-lab program uses a straight forward process which through the use of RankingAlgorithm (Blanke et al. (2006)) on the incidence matrix, calculates a set of un-matched constraints. These resulting unmatched constraints are re-written to aform where they are expressed by known variables (input and measured output),and by constraints in the particular matching. In this way, an unmatched constraintis used to test the validity of available information in both known signals and theconstraints that are used to evaluate the particular unmatched constraint. Thesere-written unmatched constraints are referred to as residuals and are listed below:

r0 = C10(BEM4(M9(y9),M1(y1),M2(y2),M3(y3),M7(y7)),M8(y8))

r1 = C11(βref1,M1(y1))

r2 = C12(βref2,M2(y2))

r3 = C13(βref3,M3(y3))

r4 = M4(y4,M10(y10))

r5 = M5(y5,M11(y11))

r6 = M6(y6,M12(y12))

r7 = BEM1(M9(y9),M1(y1),M10(y10),M7(y7))

r8 = BEM2(M9(y9),M2(y2),M11(y11),M7(y7))

r9 = BEM3(M9(y9),M3(y3),M12(y12),M7(y7))From the residuals it is possible to identify system faults. When a fault occurs inthe system model, this will cause an e�ect which can be seen in the residuals, andby implementing a fault detection unit, many di�erent faults can be detected andidenti�ed so proper action can be taken.Transforming the residuals from the symbolic description obtained from the be-havioural model to the analytic form represented by the HAWT model yields theactual residuals from which fault detection can be performed.

5.3 Structural model 71r0 = BEM (ψ, vr, β1, β2, β3) ∗ (HAWT )− P = 0

r1 =

(

−β 1

w2n

− β2ζ

wn

+ βref

)

βref1 − (β1) = 0

r2 =

(

−β 1

w2n

− β2ζ

wn

+ βref

)

βref2 − (β2) = 0

r3 =

(

−β 1

w2n

− β2ζ

wn

+ βref

)

βref3 − (β3) = 0

r4 = 1Mflap1 − 2Mflap1 = 0

r5 = 1Mflap2 − 2Mflap2 = 0

r6 = 1Mflap3 − 2Mflap3 = 0

r7 = BEM (ψ, vr, β1) − (2Mflap1) = 0

r8 = BEM (ψ, vr, β2) − (2Mflap2) = 0

r9 = BEM (ψ, vr, β3) − (2Mflap3) = 0,with BEM being the entire BEM dynamic description given in section 2.3.2. Itshould be noted that the BEM description uses a vr reading from which, whenhaving a known wind model, the blade loads can be calculated. On a real HAWTthe actual wind model is not known and thus instead a direct load measurement ora estimate of the wind velocity taken on the wing can be used.Rewriting the residuals to the form of a dependency matrix makes it easier to com-prehend the e�ect of faults and how these could be identi�ed from the residuals.By adding a binary interpretation to the dependency matrix it is easy to clarifythose constraint where faults are isolable. Constraints with a unique binary valueare isolable constraints.From the dependency matrix it is seen, that the constraints describing the rela-tions of the signals from the sensors measuring the pitch angle and also the bladeloads, all are isolable.This is a fortunate feature of the residuals that allows individual fault analysison the measured and calculated blade loads and also the measured pitch angles.Implementing residualsThe residuals r1 to r6 can all be generated as the di�erence between either twomeasurements or the di�erence between a measurement and a direct input signal.Generating residuals r7 to r9 and r0 is a bit more complicated since these requireinformations about the BEM constraints -see �gure 5.1. The BEM constraints arethe description of the signal route through the BEM model which is not easily inter-preted as an equation and therefore a full copy of the BEM model is used instead.It should further be noted that the complexity in residual r0 is even greater, since italso requires signal relation given constraint C10, which states the relation betweenthe rotor torque Qr and electrical output P . This is when implementing the resid-ual realized by adding a copy of the model containing the rotor-shaft, tower-modeland generator dynamics. The residual generator is depicted in �gure 5.2

72 5 Fault Tolerant Control

r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 2rC6 0C9 0C10 1 1C11 1 2C12 1 4C13 1 8BEM1 1 128BEM2 1 256BEM3 1 512BEM4 1 1BEM5 0M1 1 1 1 131M2 1 1 1 261M3 1 1 1 521M4 1 16M5 1 32M6 1 64M7 1 1 1 1 897M8 1 1M9 1 1 1 1 897M10 1 1 144M11 1 1 288M12 1 1 576Table 5.2: Dependency matrix generated from the residuals with a addedbinary interpretation pattern so isolable faults are easy identi�ed as havinga unique binary value.

5.4 Dynamics a�ecting the residuals 73It is seen that implementing noise on the six blade load measurements y4−6 andy10−12 has an e�ect on the residuals generated from these and the same noise willappear on r4 to r9. It can be shown, based on the assumption that both set ofblade measurements are a�ected with noise of the same magnitude, that the noiseappearing on residuals r4− r6 due to noise from two measurements have twice thevariance of the noise seen in residuals r7 − r9.

Σ

Σ

Σ

-

-

6

6

6

+-+-+-

Σ+--βref1−3y1-y3y4-y6y10-y12-

C11-C13BEM-

-

-

-

-

y7y96y8

-

-

-

-

r1-r3Dynamics

r4-r6r7-r9r0Mflap

Qr

(βref1−3)(β1−3)(1Mflap1−3)(2Mflap1−3)(ψ)(vr)(P )Figure 5.2: Residual generator designed from the unmatched constraints.Before it is possible to fully design a fault detection unit it is important to have acloser look on the faults and their e�ects on the residuals. Since the structural modeldescribes the physical system and its properties, there are physical limitations andparameters in the system that causes time constants, from fault to fault e�ect isseen in the residuals.5.4 Dynamics a�ecting the residualsAs mentioned, certain time constants are expected throughout the system describedin the structural model. These time constants can be calculated and then an es-timate of the propagation time of the signal fault-to-residual is obtained. Thisis important to know when designing the fault detection unit, since when dealingwith faults and their e�ect seen on the residuals, the propagation time constantmay imply that same fault is seen at di�erent time value thus same fault may a�ectdi�erent residuals at di�erent time.When looking at the structural model one can divide it into a subset of modelseach containing its own expected time constant.• From βref to β (through C11-13)

74 5 Fault Tolerant Control• and from β to Qr (Through BEM)Starting with the simple subsystem representing the pitch actuator. It is needed toknow the expected time constant from when the pitch reference to the actual pitchaction.5.4.1 Pitch actuator time constantsWhen looking at the model description of the pitch actuator in section 2.6 on page 29it is modeled as a second order system, and the transfer function from βref to βcan be stated as:

β = −β2wnζ − βw2n + βrefw

2n ⇒

d2β

dt=dβ

dt2wnζ − βw2

n + βrefw2n,which when written in the Laplace domain becomes

F (s) = L ⇒s2β = −sβ2wnζ − βw2

n + βrefw2n ⇔

βrefw2n = s2β + sβ2w2

nζ + βw2n ⇔

β

βref

=w2

n

s2 + 2w2nζs+ w2

n

. (5.1)From the transfer function it is possible to calculate the settling time which is thesame as the expected time constant. But before we can calculate the time constantthe transfer function in equation 5.1 must �rst be transformed into the time domainthrough the Inverse Laplace transform.f(t) = L−1 {F (s)}

=wn

√

s− ζ2eζwntsinwn

√

1 − ζ2 · t, (5.2)Model reductionIt is a tedious task to obtain the exact analytical expression for the settling timetsett, and it is suggested by Jannerup and Sørensen (2000) that an approximationof equation 5.2 is used, yielding a settling time within 5% �nal value as

t5%,settl∼= 3

ζwn

.Applied to the system this produced a expected settling time oft5%,settl

∼= 0.38s. (5.3)This model reduction is introduced due to the fact that the signal representing thefault is unknown. A fault may be a sudden change or gradually increasing and thusa very detailed dynamic modeling is not necessarily worth the e�ort.

5.4 Dynamics a�ecting the residuals 755.4.2 BEM time constantWhen looking at the time constants arising from the use of the BEM model thereare two. These two time constants arise from the settling time of the dynamic stalldescription and from the dynamic in�ow model. Both time constants are caused bya change in the pitch angle and occur simultaneously thus overlapping each other.It can be shown that the largest time constant is described by the dynamic in�owmodel and are directly in�uenced by the settling time of the dynamic wake andthus the induced wind speed w.The longest time constant from the dynamic in�ow model can be represented bythe exponential functionstwake1 =exp−∆t

τ1 (5.4)twake2 =exp−∆t

τ2 (5.5)where τ1 and τ2 are found in 2.28a and equation 2.28b on page 23. The �nal timeconstant is then given as tbem = twake15%settl+ twake25%settl

.This time constant can also be estimated when performing a step input of the pitchand measure the settling time of the rotor torque. In 5.3 a step in the pitch of 1◦is performed at a wind velocity of 15 m/s yielding a settling time of approximately10s.

−5 0 5 10 15 20 259.3

9.4

9.5

9.6

9.7

9.8

9.9

10

10.1x 10

5 BEM−model Pitch Stepresponse

Figure 5.3: Step input of the pitch signal at a wind velocity of 15 m/s. Dueto dynamic wake e�ects there is a time constant from change of pitch angle,to the resulting torque has settled.With the time constants determined for both the BEMmodel and the pitch actuatorsystem, it is possible to determine the worst case time constant for a fault a�ectingmore than one residuals, and thus take this into account in the fault determination.

76 5 Fault Tolerant Control5.5 Fault detection unitThe base for a fault detection unit has now been outlined and from the residualgenerator it is possible to detect and isolate a number of possible system faults.The focus will however be placed on fault occurrences in the six �ap wise momentmeasurements. These fault can through the residuals be detected as a change inthe mean value of the residual but �rst it must be taken into consideration that the�ap moment measurements are taken from a physical system and deviations fromthe actual value and signal noise must be expected. Thus noise is implemented onthe six blade �ap moment measurements.5.5.1 Noisy measurementsTypical noise implemented in electrical systems have some form of correlation tothe surrounding system which when known can be �ltered. For the �ap momentmeasurements this correlation is assumed unknown and thus band limited whitenoise is implemented as noise on the blade load measurements.Band limited white noiseThe band limited white noise used in this chapter follows the description given inBlanke (2006) from which a dynamic �lter is determined.A stationary random process w(t) can be described by the autocorrelation functionRww(τ) = σ2

w exp−κ|τ |, (5.6)where κ is a positive real constant. The generator for such a signal is a stochasticdi�erential equationdw(t) = −κw(t)dt +

√2κσwdv(t), (5.7)where v(t) is a Wiener process. w(t) is generated by the dynamic �lter

H(s) =

√2κ

s+ κσw, (5.8)which has the random signal v(t) as input and w(t) as output. The signal v(t) hasthe intensity 1 which means that the total power of the signal is equal to 1.Implemented white noiseThe band limited white noise are implemented in each of the six blade load mea-surementsMflap as a Simulink block model containing the transfer function of thedynamic �lter 5.8 with a stochastic input signal. The stochastic input signals usedfor each of the noise generators use di�erent seeds to avoid correlation.From factory design it is stated that the strain guages may deviate 0.5 percentof the full measuring range. Assuming normal distribution of measuring deviation,

5.5 Fault detection unit 77then 99.6 percent of the signal is contained within three times the standard devi-ation σ. Thus the standard deviation of the noise on the blade measurements arederived asσblade =

0.5% of max Mflap

3= 2417Nm.The band limit of the white noise process is set to ten times the bandwidth of thepitch actuator yielding 10 Hz. The Simulink model is depicted in �gure 5.4.

1

Out1

sigma*sqrt(2*beta)

s+beta

Transfer Fcn1

Band−LimitedWhite Noise1

Figure 5.4: Simulink model of implemented band limited white noise gen-erator.From the residual generator it is possible to detect blade load faults as a change inthe mean value of the residual. But due to the noise from measurements it can bedi�cult to determine small fault occurrences in r4 − r9 by using simple thresholdmethods and instead a cumulative sum CUSUM change detection algorithm is usedfor these.A prerequisite for using such a detection of a mean value change in a residualis that the residual must be Gaussian distributed and that the presence of a meanvalue change in a fault signal is equivalent with a change in mean value of theresidual.When looking at the amplitude distribution of the time histories of residual r9in �gure 5.6 it is seen that the residual is reasonable Gaussian distributed bothprior and during fault and thus a CUSUM test can be used.5.5.2 CUSUM change detectionIn the case of fault detection on the blade loads it is needed to detect both negativeand positive changes in the mean of the noisy residuals and thus a two sided CUSUMalgorithm as described in Blanke et al. (2006) is used. In short the CUSUM algo-rithm detects a known change in the mean of a Gaussian sequence and by integratingthe change in the mean a positive or negative drift is derived as long as the changein the mean of the signal is present. This is summed up over time and thus eventhe smallest faults can be detected.The CUSUM algorithm relies on the concept of the log-likelihood ratio of an ob-servation z de�ned ass(z) = ln

pµ1(z)

pµ0(z), (5.9)where p(z) is a probability density function, µ0 and µ1 the mean of that particularGaussian distribution. Based on a Gaussian probability density function for a

78 5 Fault Tolerant Control0 10 20 30 40 50 60

−1

0

1

R0

0 10 20 30 40 50 60−1

0

1

R1

0 10 20 30 40 50 60−1

0

1

R2

0 10 20 30 40 50 60−1

0

1

R3

0 10 20 30 40 50 60−2

0

2x 10

4

R4

0 10 20 30 40 50 60−2

0

2x 10

4

R5

0 10 20 30 40 50 60−2

0

2x 10

4

R6

0 10 20 30 40 50 60−1

0

1x 10

4

R7

0 10 20 30 40 50 60−1

0

1x 10

4

R8

0 10 20 30 40 50 60−1

0

1x 10

4

R9

time [s]Figure 5.5: Noise propagation from Mflap readings to coherent residuals.Note the signal size of r0−r4 is of 10−7 magnitude and less, thus consideredas zero.

5.5 Fault detection unit 790 10 20 30 40 50 60

−1

−0.5

0

0.5

1x 10

4

R9

time [s]

−1

−0.5

0

0.5

1x 10

4

0 200 400

0 10 20 30 40 50 60−1

0

1

2

3

4x 10

4

R9

time [s]

−1

0

1

2

3

4x 10

4

0 100 200Figure 5.6: Time histories of residual r9 and its amplitude distribution priorand during a change in mean value.random variable with a mean µ and a variance σ which can be formulated aspµ(z) =

1√2πσ

exp

(

− (z − µ)2

2σ2

)

, (5.10)the resulting log-likelihood ratio for detecting a change in the mean from µ0 to µ1is thens(z) = ln

pµ1(z)

pµ0(z)=µ1 − µ0

σ2

(

z − µ1 − µ0

2

) (5.11)from which an cumulative sum of the random variable when expressed as z = z(i)can be obtained yieldingS(k) =

k∑

i=1

s(z(i)) = lnpµ1(z(i))

pµ0(z(i))(5.12)with k being the present time constant.If the mean of the change is unknown, then the CUSUM test is no longer us-able and instead a generalized log likely-hood-ratio GLR test should be used. Thisis not the case since faults of certain magnitude is to be detected.To specify a certain minimum detection time tdet for the CUSUM test it is neededto �nd a proper threshold h.A way of choosing this h level is through the use of a average run length (ARL)function (Blanke et al. (2006)) which requires a known mean µs and variance σs ofthe signal the CUSUM algorithm is applied to.

80 5 Fault Tolerant ControlL(µs) = (exp

[

−2

(µsh

σ2s

+µs

σ1.166

)]

− 1+

2

(µsh

σ2s

+µs

σ1.166

)

)

(σ2

s

2µ2s

)

| µs 6= 0, (5.13)From the ARL function the average detection time is estimated byˆτ = L

((µ1 − µ0)

2

2σ2

) (5.14)and the average run time between false alarmsˆT = L

(

− (µ1 − µ0)2

2σ2

)

, (5.15)where µ0 is the mean of the residual and µ1 is the size of the fault to be detectedand σ2 is the variance of the residual.The ARL is strictly valid only if the residual is white noise. This can be examinedby looking at the power spectrum density of a noisy residual. For a white noisedistribution the power must be equally distributed throughout the entire spectra.This is not possible to reproduce since it would require a graph of in�nite size, andwas also the reason why instead band limited white noise was implemented. Thewhite noise was generated in a spectra up to 10Hz and thus the power spectrumdensity is only expected to act as white noise in this spectra. From �gure 5.7 it isseen that the power density is evenly distributed and that it is reasonable to assumethat the residual is white noise.

0 2 4 6 8 100

2

4

6

8

10

12

14x 10

5 PSD of R9

Frequency [Hz]

Pow

er [(

Nm

)2 /Hz]

Figure 5.7: For residual r9 is seen a even power density distribution and thuswhite noise can be assumed.

5.6 Implemented fault detection 815.6 Implemented fault detectionThe fault detection unit is now implemented into the system as a simple thresholdtest on residual r0 − r3 which are una�ected by the measurement noise, and as aCUSUM test on residuals r4− r9 due to the fact that these are generated from thenoisy blade load measurements. The fault detection based on the residuals is thenpassed on to a fault isolation unit, that based on the dependency matrix seen intable 5.2 and the expected time constants described in section 5.4, determines thetype of fault.The fault detection unit is for r4 − r9 implemented with a noise variance σ2blade= 24172Nm2, mean residual value under normal conditions as µ0 = 0, the size ofthe fault to be detected is estimated as 2% of the max range of the �ap measure-ment, thus µ1 = maxMflap ∗ 0.02 = 29000Nm.Inserting the variance of the noise and the size of the fault into equation 5.14and 5.15 produces the graphs in �gure 5.8 from which a threshold h = 50 for theCUSUM test is chosen.

0 20 40 60 80 1000

1

2

sam

ples

Time to detect

0 20 40 60 80 10010

0

1050

sam

ples

Time between false alarms

hFigure 5.8: Fault detection time and time expected between false alarmreadings given a threshold h derived from the ARL function with µ0 = 0Nm,µ1 = 29000Nm and σ = 24172Nm2.When a blade measuring fault is detected and isolated, then the fault decision unitwill determine an appropriate action. In case of the fault on the blade measure-ments is of such magnitude that it is causing the load reduction performance of theindividual pitch controller to perform less e�ective that the collective pitch control,then the faulty control signal from the individual pitch controller should be removedthrough a recon�guration of the controller.

82 5 Fault Tolerant Control5.7 Recon�guration of controllerWhen the fault of a speci�c magnitude is detected in one of the blade load measure-ments then the controller terminates the controller feedback from this measurementand only uses the remaining. Immediately after the individual pitch control loop isremoved. The controller would be able to work properly with only the remainingblade load measurement but if a second fault occurs on the remaining load mea-surement, then the system, as it is, cannot properly isolate the fault.Recon�guration of the control of the HAWT must be done without having anabrupt passing between the control signals. The design of the jointed controllerallows removal of the more advanced individual control loop but an immediate re-moval can cause a sudden change in the pitch angle which may cause unwantedstructural fatigue. To prevent this from happening the recon�guration is done byremoving the individual control signal contribution to the pitch angle in a expo-nentially manner. This way over a short time the pitch angle contribution from theindividual control is zeroed and only the collective pitch controller is controllingthe HAWT.f(t) = e−t, (5.16)where t = 0 until fault is detected.Based on a detection of a fault in one of the blade load measurements a change inthe advance level of the control loop can be performed in the running system.5.8 Simulation with faultFrom HAWT simulations with the implemented fault detection and recon�gura-tion scheme a fault occurring in a blade load measurement is detected from theresiduals and the HAWT control is recon�gured so the individual controller contri-bution is removed over a short period of time. Removal of individual pitch controlloop of the system is seen in �gure 5.10 as a result of fault detected via the residuals.In �gure 5.9 a blade load fault of a 2% deviation of the maxMflap moment isintroduced at blade load measurement y6 at t=20s. The fault is seen a�ectingresiduals r6 and r9 and can be correctly identi�ed by the detection unit.A measurement fault in sense of a 2% deviation is considered a large fault andthus a removal of the individual pitch control is performed. In �gure 5.10 selectedstates in the HAWT model is shown depicted as a result of the fault occurrenceat t=20s. Approximately three seconds after the arised fault the pitch signal fromthe individual control loop is removed, and the system behaviour transforms intothe system behaviour as seen from the collective pitch control scheme thus a smalldescrease in pitch action.In order to tune the fault handling unit a closer look on how faults of di�erentmagnitude a�ects the system will be performed in the following chapter.

5.8 Simulation with fault 830 10 20 30 40 50 60

−1

0

1

R0

0 10 20 30 40 50 60−1

0

1

R1

0 10 20 30 40 50 60−1

0

1

R2

0 10 20 30 40 50 60−1

0

1

R3

0 10 20 30 40 50 60−2

0

2x 10

4

R4

0 10 20 30 40 50 60−2

0

2x 10

4

R5

0 10 20 30 40 50 60−5

0

5x 10

4

R6

0 10 20 30 40 50 60−1

0

1x 10

4

R7

0 10 20 30 40 50 60−1

0

1x 10

4

R8

0 10 20 30 40 50 60−5

0

5x 10

4

R9

time [s]Figure 5.9: A 2% fault on blade load measurement y6 is implemented att=20s. Fault is immediately detected in residual r6 and r9.

84 5 Fault Tolerant Control

0 10 20 30 40 50 6014

16

18Windspeed[m/s]

0 10 20 30 40 50 601999.5

2000

2000.5Poweroutput[kW]

0 10 20 30 40 50 6005

1015

Pitchangle

[°]

0 10 20 30 40 50 60−10

0

10Pitch

velocity[°/s]

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time [s]Figure 5.10: System behavior at a fault occurring in measurement y6. Con-troller is switched from the more advanced based individual pitch control, tosimple collective pitch control.

Chapter 6Sensitivity analysisIn this chapter an analysis of having o�set faults in the blade load measurementsfor the individual pitch controller is investigated.Performance of the individual pitch control will be compared, when having o�-set faults, to the performance of the una�ected collective pitch control.At �rst simulations of only a single faulty blade load measurement with an ad-ditive fault is simulated. The magnitude of the fault is increased and the powerspectrum density of both the tilt- and yaw moments and the faulty blade are ana-lyzed.Same simulations are performed with two simultaneously faulty blade load mea-surements.This sensitivity analysis will strictly evaluate controller performance in terms ofthe magnitude of the power around nP harmonics, with n = (1, 2, 3), which havethe frequencies1P - 0.370Hz2P - 0.740Hz3P - 1.110Hz.Fault will be implemented as a percentage of the full measuring range of the sensorsmaxMflap.6.1 Controller performance concernsWhen implementing a small o�set fault on a single blade measurement it will in-troduce a 1P harmonic in the PSD of the tilt and yaw moments of the individualpitch controller, which does not occur when operating at collective pitch control.At the same time the amplitude of the 3P harmonic will be less than the amplitudeof the 3P harmonics produced with the collective pitch controller. This behavior isseen for the tilt moment in �gure 6.1 and imposes some di�culties when trying toestimate the actual fatigue of the controller.

86 6 Sensitivity analysis

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Figure 6.1: Blade measurement fault causes a rise in the 1P e�ect of theindividual pitch control, while the amplitude of the 3P e�ect remains lessthan the 3P amplitude of the collective pitch control.6.2 Fatigue as a measureTo evaluate the performance of the controllers such that a comparison is possible,fatigue will be considered as the sum of the harmonic times its max amplitudes.Fatigue =

∞∑

n=1

nAnP , (6.1)with AnP being the max amplitude to the nP harmonic.To get a precise estimate of fatigue one should implement a di�erent and moreaccurate measure. Rain�ow counting would be a interesting fatigue measure to im-plement.Simulations of both the individual pitch controller and the collective pitch controllershows, that the PSD of harmonics higher than 3P are of same magnitude even whenoperating with measurement faults. With this in mind the measure of fatigue asintroduced in equation 6.1 reduces toFatigue =

3∑

n=1

nAnP ,and the focus of the PSD graphs will mainly be in this spectra.The PSD graphs described in this section are all performed at a mean wind ve-locity of 15 m/s with wind shear, turbulence and tower shade e�ect. Note that inthe PSD graphs the spectral density of the collective controller have been shifted0.15Hz for the purpose of easy comparison.

6.3 Single blade measurement fault 87We want to investigate for how large a o�set fault in the blade measurements canbe allowed before the individual pitch controller, will provide a less fatigue measurethan the collective controller.Faults will be investigated for �ve di�erent fault measurement scenarios.• Having 1 positive o�set fault• Having 1 negative o�set fault• Having 2 positive o�set fault• Having 2 negative o�set fault• Having both 1 positive and 1 negative o�set fault6.3 Single blade measurement faultSimulations with faulty single blade load measurements will be performed to inves-tigate the systems tolerance level towards faulty measurements. It is particularlyinteresting to see how much fault is allowed on a single blade load before the perfor-mance of the individual pitch controller is equal, in sense of blade load reduction,to the collective pitch controller.When looking at the PSD graphs from the blades the DC gain is representingthe magnitude of the constant blade load which in this thesis is not consideredfatiguing. It is the varying Mflap moments that are causing blade fatigue and thusthe focus of the PSD graphs of the blades will be on 1P and higher harmonics.It is seen that when having a constant deviation in the measurement of a bladeload there is an increase in the power density around the 2P frequency for the indi-vidual pitch controller. The reason for this increase around the 2P harmonic is dueto the behaviour of the faulty blade performing cyclic rotations in a non-uniformwind �eld where wind shear is present. The wind shear e�ect and the measurementfault causes a change in pitch angle twice per rotation and thus a 2P harmonic thatincreases when the measurement deviation increases.At a measurement deviation of 8% on one blade the magnitude of the 2P e�ecthas increased for the individual pitch controller to the same magnitude as the col-lective pitch controller and at this point the individual pitch control will start toperform less than the collective controller in terms of 2P harmonics.When looking at 1P harmonics of the blade load the individual pitch controlleris still performing better than the collective controller and likewise for the 3P har-monics of the tilt and yaw wise moments. But it is seen that implementing a faultis immediately resulting in a 1P harmonic in both the tilt and yaw wise momentswhich is not occurring when using collective pitch control.

88 6 Sensitivity analysisUtilizing the fatigue estimate given in equation 6.1, the analysis of a single blademeasurement fault shows that a 12% deviation is where the overall varying �apwise moments of the individual pitch controller, starts to perform less than thecollective pitch control, in sense of reducing varying loads.

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Figure 6.2: Power spectrum density graphs when running the HAWT modelwith a 12% deviation in a single Mflap measurement. It is seen that the 2Pe�ect on the blade PSD then is of same magnitude for both individual andcollective pitch control. System remains within its 10◦/s pitch velocity limitat all times.

6.4 Two blade measurement faults 896.4 Two blade measurement faultsIt should be stressed that the individual pitch control acts on the tilt and yaw mo-ments, thus implementing a fault on a single blade a�ects the control of all threeblades and thus a�ects their �ap wise momentsMflap. Implementing measurementfaults on two subsequent blades enforces this control e�ect, and it is not necessar-ily the blades having faulty measurements which have the most varying �ap wisemoments. In light of this, when operating with two or more blade measurementfaults, the mean of the three Mflap moments are used as a measure instead of theindividual moments.Simulations with two positive measurement faults shows a fault tolerance of 12%on both measurements is allowed for Mflap, before the amount of fatigue on theblade is equal to what is achieved by the collective controller. Figure 6.3. It shouldbe noted that in spite of operating with two measurement faults, the individualpitch controller still has a much better performance around 1P, when looking atthe PSD of the blade, which is seen almost cancelled out when comparing to thecollective pitch controller.When having two measurement o�set faults, with one being positive and the otherbeing negative, it is determined that only 7% fault is allowed on each measure-ment, before the amplitude of the power in 2P and higher harmonics, increasesto the level seen for the collective pitch controller. When instead looking at theyaw-wise performance only a ±2% o�set fault is allowed which is depicted in �gure6.4.6.5 ResultsFrom the sensitivity analysis a set of blade load measurement fault tolerances canbe listed. This list can be used when designing the fault tolerant controller andwanting to determine under which circumstances a recon�guration of the controllerwould be advised given the design parameters.In table 6.1 the tolerance levels are put up against the magnitude of allowed mea-surement o�set, and is based on the assumption that performance is estimated byequation 6.1. When comparing the varying blade moments it is seen, that themagnitude of the allowed o�set fault before the individual pitch controller startsto perform less than the collective pitch control, is of great size. The reason forthis high tolerance is due to the fact that the individual pitch controller, even dur-ing o�set fault, manages to reduce 1P blade load variations. For the tilt and yawmoments this is not the case and a much less tolerance to o�sets in the blade mea-surements exists.From simulations it is seen that having one measurement fault yields the sameperformance as when having two blade faults with the same o�set value. Thiscorresponds well to what could be expected with the used individual pitch controlscheme based on the Coleman decoupling of the tilt and yaw axis.

90 6 Sensitivity analysis

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Figure 6.3: Power spectrum density graphs when running the HAWT modelwith a 12% deviation in two Mflap measurement. At 12% o�set in twomeasurements the fatigue of both controllers is the same in terms of tilt-wisemoments.

6.5 Results 91

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Figure 6.4: Power spectrum density graphs when running the HAWT modelwith a ±2% deviation in two Mflap measurement. For two ±2% o�set faultsthe fatigue of both controllers is the same in terms of yaw-wise moments.

92 6 Sensitivity analysisO�set fault toleranceMeasurement fault Blade Tilt Yaw1 positive blade fault 60% 12% 3%1 negative blade fault 60% 12% 4%2 positive blade fault 60% 12% 4%2 positive blade fault 60% 12% 3%1 positive blade fault and1 negative blade fault 35% 7% 2%Table 6.1: Measurement o�set tolerances. The performance is considered atthree di�erent measures: when fatigue is equal for collective and individualpitch control at respectively 1-blade, 2-tilt axis and 3-yaw axis. Tolerancelevel is given as a percentage of the max Mflap measuring range.The sensitivity analysis shows a that the individual pitch controller has reason-able tolerance to blade load measurement o�set faults. Due to the setup of theindividual pitch control scheme it is seen that the actual di�culties with o�setfaults is when they occur out of symmetry. When having three simultaneous o�setfaults, one on each blade, with the same magnitude, then controller performanceremains intact, see appendix B. This is a fortunate feature when remembering thatthe sensors used for blade load measurements are strain guages having o�set faultswith a correlation to the temperature and similar o�set fault on all three bladescan be expected.

Chapter 7Conclusions7.1 ResultsThe work described in chapter 2 yielded a nonlinear unstable ten state wind turbinemodel. A linearization of the model in a working point was performed so traditionallinear control theory could be applied. An analysis of the physical horizontal axiswind turbine (HAWT) system led to a number of operation modes given generatorspeed and electrical output e�ect, from which di�erent control schemes could bedesigned.From the linearized model a linearized quadratic controller with integral action(LQI) was designed for the purpose of maintaining a stable electrical power outputgiven a turbulent wind �eld. The LQI controller design showed good results interms of a stable power output and some of the trade o�s when tuning a wind tur-bine controllers were demonstrated. The trade o� by damping the tower movementwas an increased pitch activity and a more varying generator angular velocity andoutput power.An individual pitch control scheme was then designed to reduce the varying loads oneach of the blades thus reducing fatigue. This required a change in the aerodynamicdescription and a unsteady blade element momentum (BEM) model to calculateforces along the blades was introduced. Furthermore a set of blade load measure-ments was implemented from which a decomposition of the �ap moments into atilt and a yaw moment was performed. The postulate for the control scheme wasthat a reduction in the tilt and yaw wise moments would reduce the �ap momentsand thus the varying loads on the blades. With the three blade loads transformedinto a tilt and a yaw axis the system was reduced to two SISO system, allowing theindividual pitch controller to be implemented as two I-compensators.The individual pitch controller shows that with a slight increase in the pitchaction a remarkable reduction in the tilt and yaw wise moments is obtained. Sim-ulations with a comparisons of the power spectrum density, made between thecollective pitch controller and the individual pitch controller, shows an almost com-plete removal of DC gain for both the tilt and yaw spectrum. These DC gains wasof magnitude 108Nm2/Hz with a wind velocity of 15m/s. Furthermore a notice-able reduction is seen for the power of the 3P harmonics in the tilt and yaw axis

94 7 Conclusionswhich corresponds to a reduction of the 1P harmonics of the blade moments. Thereduction in the 1P moments on the blade was seen reduced by 1/10.The individual pitch controller is based on direct blade load measurements whichmakes the controller highly dependent on correct readings. To prevent the systemfrom acting on faulty signals a fault tolerant controller was designed. This providedmeans for detecting and determining faults occurring on the blade load measure-ments and recon�guring the HAWT control. Based on a sensitivity analysis of faulttolerance, it was possible to determine the percentage of allowed measurement devi-ation in blade load measurements, before the blade load reduction of the individualpitch controller dropped to the level of the simpler collective pitch control scheme,and thus for a further increase in the fault and a recon�guring of the controllerwould be advised.7.2 Suggestions for future workImplement controller based on nacelle accelerations thus controlling tilt and yawaxis movements .Add control loops with more speci�c control abilities to the jointed controller andthrough the fault tolerant controller terminated in faulty situations.Expand the model to contain di�erent measures of blade load- or �ap momentreadings. This way a more advanced fault tolerant system can be designed with anadvanced way of detecting individual measurement faults.Investigate a more thorough sensitivity analysis, a complete fault scheme for faulttype and size versus load reduction performance, for purpose of individual controldesign.7.3 PerspectivesIt should be stressed that the method presented when designing a fault tolerantcontroller for the HAWT model holds valid for any control system relying on sen-sor feedback and that the means for detecting faults are true for any Gaussiandistribution.In e�ect, we will conclude this thesis by stating that the main result of thework is the implementation of an unsteady BEM model from which an individualpitch control scheme was designed and implemented. Through fault tolerant designmethods the control system was designed such that measurement faults of certainmagnitudes were acceptable.

BibliographyBlanke, M. (2006). Frequently asked questions on noise and covariance, Technicalreport, Technical University of Denmark, DTU.Blanke, M., Kinnaert, M., Marcel, J. L. and Staroswiecki (2006). Diagnosis andFault-Tolerant Control, second edn, Springer.Bossanyi, E. A. (2003). Wind turbine control for load reduction., Wind Energy6: 229�244.Engelen, T. V. (2002). Morphological study of aerolastic control concepts for windturbines, EU-Contract ENK5-CT-2002-00627 .Engelen, T. V. (2006). Design model and load reduction assesment for multi-rotational mode individual pitch control (higher harmonics control), Technicalreport, Energy research centre of the Netherlands (ECN).Hendricks, E., Jannerup, O. and Sørensen, P. H. (2006). Linear Systems Control:Deterministic and stochastic methods, Ørsted DTU.Højstrup, J. (1982). Velocity spectra in the unstable planetary boundary layer.Jannerup, O. and Sørensen, P. H. (2000). Introduktion til Reguleringsteknik, 2 edn,Polyteknisk Forlag.Larsen, A. J. and Mogensen, T. S. (2006). Individuel pitchregulering af vindmølle,Master's thesis, Informatics and Mathematical Modelling, Technical University ofDenmark, DTU, Richard Petersens Plads, Building 321, DK-2800 Kgs. Lyngby.Supervised by Assoc. Prof. Niels Kjølstad Poulsen, IMM.URL: http://www2.imm.dtu.dk/pubdb/p.php?4462Øye, S. (1991). Dynamic stall, simulated as time lag of separation, ETSU-N-118 .Xin, M. (1997). Adaptive Extremum Control and Wind Turbine Control, PhD the-sis, Informatics and Mathematical Modelling, Technical University of Denmark,DTU, Richard Petersens Plads, Building 321, DK-2800 Kgs. Lyngby. Supervisor:Niels K. Poulsen.URL: http://www2.imm.dtu.dk/pubdb/p.php?2441

96 BIBLIOGRAPHY

Appendix AController simulations0 10 20 30 40 50 60

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98 A Controller simulations

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99

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time [s]Figure A.3: Cyclic controller: Steady state of the system at the point oflinearization. -No wind shear e�ects and no turbulence.

100 A Controller simulations

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Appendix BSensitivity analysis

102 B Sensitivity analysis

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Figure B.1: Power spectrum density graphs when running the HAWT modelwith a three o�set faults of same magnitude. A 12% deviation in all threeMflap measurement is implemented and the the energy of the varying mo-ments is seen identical as when having no o�set fault.

Appendix CAirfoil dataThe airfoil data from each element of the blades is listed in the data �les on theCD-rom.tjaere04_ds.dattjaere05_ds.dattjaere06_ds.dattjaere07_ds.dattjaere08_ds.dattjaere09_ds.dattjaere10_ds.dattjaere11_ds.dattjaere12_ds.dattjaere13_ds.dattjaere14_ds.dat

104 C Airfoil data

Appendix DSelected matlab scriptsD.1 Unsteady BEM modelf unc t i on [ output_args ] = BEM( input_args )% Var iab l e s that are saved f o r each i t e r a t i o np e r s i s t e n t Wqs Wqsold Wold W Winterm Wav f s theta_skew theta_wake% inputs are :%% 1 . V% 2−4. Beta_1−3% 5 . Wr% 6 . phi% 7 . s imu la t i on time% 8 . s imu la t i on step% 9 . wind_shear% ge t t i n g a l l inputsve l = input_args ( 1 ) ;p i t ch = input_args ( 2 : 4 ) ;omegashaft = input_args ( 5 ) ;theta_wing = input_args (6)−pi /2 ;n = input_args ( 7 ) ;% BEM f l a g sdynwake = true ; % Dynamic in f l owdyn s t a l l = true ; % Dynamic s t a l lwindmodel = true ; % Wind−modeltower_shadow = true ; % Tower shadowt i p l o s s = true ; % Prandtl ' s t i p l o s s f a c t o rg l au e r t = true ; % Glauertskew_model = true ; % Skew modelyawmodel = true ; % Yaw model

106 D Selected matlab scripts% i n i t windturbineBEM_initf s t a t = ze ro s (1 , npoint ) ;c l i n v = zero s (1 , npoint ) ;c l f s = ze ro s (1 , npoint ) ;cd = zero s (1 , npoint ) ;% r e s e t s a l l va lue s at time 0i f (n == 0) ,d i sp ( 'Wqs i s empty ' )Wqs = zero s (nb , npoint , 3 ) ;Wqsold = zero s (nb , npoint , 3 ) ;Wold = zero s (nb , npoint , 3 ) ;W = zero s (nb , npoint , 3 ) ;Winterm = zero s (nb , npoint , 3 ) ;Wav = zero s ( npoint , 3 ) ;f s = ze ro s (nb , npoint ) ;theta_skew = 0 ;theta_wake = 0 ;i f ( sim_pitch == 1)load BEM_intern1 ;e l s eload BEM_intern ;endlambda = omegashaft∗ rad/ ve lendtorque_blade = zero s (1 , nb ) ;thrust_blade = zero s (1 , nb ) ;power_blade = zero s (1 , nb ) ;mflap_blade = zero s (1 , nb ) ;% loop f o r each bladef o r b = 1 : nb% f i n d s the b lades azimuth ang letheta_wing_actual=theta_wing+(b−1.d0 )∗2∗ pi /3 . d0 ;theta_wing_actual=mod( theta_wing_actual , 2∗ pi ) ;cos_thetawingactual = cos ( theta_wing_actual ) ;s in_thetawingactua l = s i n ( theta_wing_actual ) ;% c a l c u l a t e s the new coord ina t e systemsa23 = [ cos_thetawingactual s in_thetawingactua l 0 ; . . .−s in_thetawingactua l cos_thetawingactual 0 ; 0 0 1 ] ;a14 = a34∗a23∗a12 ;a41 = a14 ' ;

D.1 Unsteady BEM model 107% ge t t i n g windspeed from the r e a l wind modeli f windmodel == truepos = a41∗X4+X40∗ones (1 , npoint ) ;% wind speed with wind shearvwind (Z , : ) = ve l ∗( pos (X, : ) / tower_height ) .^ wind_shear ;i f ( tower_shadow == f a l s e ) ,tower_rad = 0 ;e l s ei f ( pos (X) < tower_height ) ,tower_rad=tower_radbottom+pos (X, : ) ∗ tower_slope ;e l s etower_rad = 0 ;endend% ca l c tower shadowradd i s = sq r t ( pos (Z , : ) . ^ 2 + pos (Y, : ) . ^ 2 ) ;co s the ta = pos (Z , : ) . / radd i s ;s i n th e t a = −pos (Y, : ) . / radd i s ;vr = vwind (Z , : ) .∗ ( 1 − ( tower_rad . / radd i s ) . ^ 2 ) . ∗ co s the ta ;vtheta = −vwind (Z , : ) . ∗ ( 1+ ( tower_rad . / radd i s ) . ^ 2 ) . ∗ s i n th e t a ;vwind (Y, : ) = −vr .∗ s in the ta−vtheta .∗ co s the ta ;vwind (Z , : ) = vr .∗ costheta−vtheta .∗ s i n th e t a ;vwind = a14∗vwind ;e l s evwind = repmat ( [ 0 0 ve l ] ' , 1 , npoint ) ;end% r e l a t i v e wind v e l o c i t yv r e l = vwind + . . .[ z e ro s (1 , npoint ) ; −X4(X, : ) ∗ omegashaft∗ cos_thetacone + W(b , : ,Y) ; . . .W(b , : , Z ) ] ;% f i n d s the ang le o f at tack and c a l c u l a t e s the f o r c e s on each elementf la_rad = atan(− v r e l (Z , : ) . / v r e l (Y , : ) ) ;f low_angle = fla_rad ∗180/ pi ;ang l e o f a t t a ck = flow_angle − twis t ' − p i t ch (b ) ;vre l_sqr = v r e l (Y, : ) .^2+ v r e l (Z , : ) . ^ 2 ;% get l i f t and drag with dynamic s t a l li f ( dyn s t a l l == true ) ,f s o l d = f s (b , : ) ;tau=4.0∗ twis t ' . / s q r t ( vre l_sqr ) ;

108 D Selected matlab scripts% in t e r p o l a t ef o r i = 1 : npoint ,j = f i nd ( [ a tabe l (: ,1)>= ang l e o f a t t a ck ( i ) ] , 1 ) ;dx =( ang l e o f a t t a ck ( i )−atabe l ( j −1 ,1 ) ) . / ( a tabe l ( j ,1)− atabe l ( j −1 ,1) ) ;f s t a t ( i ) = fs tat_tab ( j −1, i )+dx∗( f s tat_tab ( j , i )− f s tat_tab ( j −1, i ) ) ;c l i n v ( i ) = cl inv_tab ( j −1, i )+dx∗( c l inv_tab ( j , i )−cl inv_tab ( j −1, i ) ) ;c l f s ( i ) = c l f s_tab ( j −1, i )+dx∗( c l f s_tab ( j , i )− c l f s_tab ( j −1, i ) ) ;cd ( i ) = cdtabe l ( j −1, i )+dx∗( cd tabe l ( j , i )− cdtabe l ( j −1, i ) ) ;endf s (b , : )= f s t a t +( f so ld−f s t a t ) . ∗ exp(− de l t a t . / tau ) ;c l=c l i n v .∗ f s (b , : )+ c l f s .∗(1− f s (b , : ) ) ;e l s ec l = in t e rp2 ( 1 : npoint , a tabe l ( : , 1 ) , c l t ab e l , 1 : npoint , ang l e o f a t t a ck ) ;cd = inte rp2 ( 1 : npoint , a tabe l ( : , 1 ) , cdtabe l , 1 : npoint , ang l e o f a t t a ck ) ;endkoe f = 0.5∗ den∗ vre l_sqr .∗ chord ' ;l i f t 6 = koe f .∗ c l ;drag6 = koe f .∗ cd ;s ind_f lowangle6 = max( s i n ( f la_rad ) , 0 . 0 0 1 ) ;cosd_flowangle6 = cos ( f la_rad ) ;p6 (b , : ,Y) = l i f t 6 .∗ s ind_f lowangle6 − drag6 .∗ cosd_flowangle6 ;p6 (b , : , Z) = l i f t 6 .∗ cosd_flowangle6 + drag6 .∗ s ind_f lowangle6 ;% app l i e s Glaurt c o r r e c t i o n and t i p l o s s f a c t o rvprime = vwind + [ ze ro s (2 , npoint ) ;W(b , : , Z ) ] ;absvprime = sum( abs ( vprime ) ) ;a6 = ( abs ( ve l ) − absvprime )/ abs ( ve l ) ;i f ( g l au e r t == true )ac = 0 . 2 ;f g = ones (1 , npoint ) ;i = f i nd ( ac < a6 ) ;f g ( i ) = ( ac . / a6 ( i ) ) .∗ (2− ( ac . / a6 ( i ) ) ) ;vprime = vwind + [ ze ro s (2 , npoint ) ; f g .∗W(b , : , Z ) ] ;absvprime = sum( abs ( vprime ) ) ;end%∗∗∗∗∗∗∗∗∗∗ Prandtl ' s t i p l o s s f a c t o ri f t i p l o s s == truesma l l f =0.5∗nb∗( rad−X4(X, : ) ) . / ( X4(X, : ) . ∗ s ind_f lowangle6 ) ;

D.1 Unsteady BEM model 109sma l l f = max(0 .0001 , sma l l f ) ;f a c6=min (1 ,2∗ acos ( exp(− sma l l f ) ) / p i ) ;e l s ef a c6=ones (1 , npoint ) ;end% loop f o r each element on the bladef o r i = 1 : npoints ind_f lowangle = s ind_f lowangle6 ( i ) ;cosd_flowangle = cosd_flowangle6 ( i ) ;l i f t = l i f t 6 ( i ) ;drag = drag6 ( i ) ;a = a6 ( i ) ;f a c = fac6 ( i ) ;p (Y, i , b) = p6 (b , i ,Y) ;p (Z , i , b) = p6 (b , i , Z ) ;% c a l c u l a t e s the dynamic in f l owi f dynwake == truea=min ( 0 . 5 , abs ( a ) ) ;tau1=1.1∗ rad /( ve l ∗(1−1.3∗a ) ) ;tau2=(0.39−0.26∗(X4(X, i )/ rad )^2)∗ tau1 ;konst =0.6;r ad r a t i o = X4(X, i )/ rad ;fyaw = rad ra t i o +0.4∗ r ad r a t i o ^3+0.4∗ r ad r a t i o ^5;% c a l c u l a t e s f o r Y d i r e c t i o nWqs(b , i ,Y) = −nb∗( l i f t ∗ s ind_f lowangle ) / . . .(4∗den∗pi ∗X4(X, i )∗ cos_thetacone∗ f a c ∗absvprime ( i ) ) ;dxdt = (Wqs(b , i ,Y) − Wqsold (b , i ,Y) ) / d e l t a t ;Wqsold(b , i ,Y) = Wqs(b , i ,Y) ;Wtilde=Wqs(b , i ,Y) + konst∗ tau1∗dxdt ;Winterm(b , i ,Y) = Wtilde + (Winterm(b , i ,Y)−Wtilde )∗ exp(−de l t a t / tau1 ) ;W(b , i ,Y) =Winterm(b , i ,Y)+(Wold(b , i ,Y)−Winterm(b , i ,Y) )∗ exp(− de l t a t / tau2 ) ;Wold(b , i ,Y) = W(b , i ,Y) ;% c a l c u l a t e s f o r Z d i r e c t i o nWqs(b , i , Z) = −nb∗( l i f t ∗ cosd_flowangle ) / . . .(4∗den∗pi ∗X4(X, i )∗ cos_thetacone∗ f a c ∗absvprime ( i ) ) ;dxdt = (Wqs(b , i , Z) − Wqsold (b , i , Z ) ) / d e l t a t ;Wqsold(b , i , Z) = Wqs(b , i , Z ) ;Wtilde=Wqs(b , i , Z) + konst∗ tau1∗dxdt ;Winterm(b , i , Z) = Wtilde + (Winterm(b , i , Z)−Wtilde )∗ exp(−de l t a t / tau1 ) ;W(b , i , Z) =Winterm(b , i , Z)+(Wold(b , i , Z)−Winterm(b , i , Z ) )∗ exp(− de l t a t / tau2 ) ;Wold(b , i , Z) = W(b , i , Z ) ;

110 D Selected matlab scripts%%% Yaw model . . . .i f ( yawmodel == true )yaw = (1+fyaw∗ tan ( theta_skew /2)∗ cos ( theta_wing_actual−theta_wake ) ) ;W(b , i ,Y)=W(b , i ,Y)∗yaw ;W(b , i , Z)=W(b , i , Z)∗yaw ;ende l s e% no dynamic in f l ow modelr e l a x = 0 . 5 ;W(b , i , Z) = −nb∗( l i f t ∗ cosd_flowangle ) / . . .(4∗den∗pi ∗X4(X, i )∗ cos_thetacone∗ f a c ∗absvprime ) ;W(b , i ,Y) = −nb∗( l i f t ∗ s ind_f lowangle ) / . . .(4∗den∗pi ∗X4(X, i )∗ cos_thetacone∗ f a c ∗absvprime ) ;W(b , i , Z) = r e l ax ∗ W(b , i , Z) + (1 − r e l a x ) ∗ Wold(b , i , Z ) ;W(b , i ,Y) = r e l ax ∗ W(b , i ,Y) + (1 − r e l a x ) ∗ Wold(b , i ,Y) ;Wold(b , i , Z) = W(b , i , Z ) ;Wold(b , i ,Y) = W(b , i ,Y) ;end% f i n d s element torque and thrus ti f ( i > 1)Ai = (p(Y, i , b)−p(Y, i −1,b ) ) / (X4(X, i )−X4(X, i −1)) ;Bi = ( ( p (Y, i −1,b)∗X4(X, i )−p(Y, i , b )∗X4(X, i −1))/(X4(X, i )−X4(X, i −1)) ) ;Ci = (p(Z , i , b)−p(Z , i −1,b ) ) / (X4(X, i )−X4(X, i −1)) ;Di = ( ( p (Z , i −1,b)∗X4(X, i )−p(Z , i , b )∗X4(X, i −1))/(X4(X, i )−X4(X, i −1)) ) ;torque_blade (b ) = torque_blade (b) . . .+ (Ai ∗(X4(X, i )^3−X4(X, i −1)^3)/3+Bi ∗(X4(X, i )^2−X4(X, i −1)^2)/2) ;thrust_blade (b ) = thrust_blade (b) . . .+ ( Ci ∗(X4(X, i )^2−X4(X, i −1)^2)/2+Di ∗(X4(X, i )^1−X4(X, i −1)^1)/1) ;mflap_blade (b ) = mflap_blade (b ) . . .+ ( Ci ∗(X4(X, i )^3−X4(X, i −1)^3)/3+Di ∗(X4(X, i )^2−X4(X, i −1)^2)/2) ;endendendpower_blade = torque_blade∗omegashaft ;% t o t a l thrust , blade torque and f l a p torquetorque = sum( torque_blade ) ;th rus t = sum( thrust_blade ) ;power = sum( power_blade ) ;mflap = sum(mflap_blade ) ;mflaps = mflap_blade ;% Ca l cu la t e average induced−v e l o c i t y

D.1 Unsteady BEM model 111Wav( : , : ) = sum(W)/3 ;% Ca l cu la t e skew and wake ang lei f ( skew_model == true )n70 = 7 ;n3=[0 0 1 ] ' ;vwind1 = [0 0 ve l ] ' ;vwind2 = a12∗vwind1 ;vwind3 = a23∗vwind2 ;vp3 = vwind3+a43∗Wav(n70 , : ) ' ;cosskew = (n3 '∗ vp3 )/ sq r t ( vp3 '∗ vp3 ) ;theta_skew = acos ( cosskew ) ;theta_wake = atan2 ( vwind2 (Y) , ( vwind2 (X) ) ) ;i f ( theta_wake < 0)theta_wake=theta_wake+2∗pi ;endend% i f mod( input_args (6 ) , 2∗ pi ) < Wr∗ de l ta t ,% s p r i n t f ( '%d %f %f ' , n , input_args ( 6 ) ,mod( input_args (6 ) , 2∗ pi ) )% save BEM_intern1 Wqs Wqsold Wold W Winterm Wav f s theta_skew theta_wake ;% endoutput_args = [ torque thrus t mflaps ] ;

112 D Selected matlab scripts

Appendix ESimulink models

sim_exp

Wind turbine

V

Beta_ref

P_ref

turbulence

psi

Output

Full_States

beta

psi_out

M_flap

sim_inv

sim_beta

Step

Scope3

Indv. Pitch Control

theta_azimuth

M_flap

beta_inv

[P_col]

[beta_ref]

[beta_inv]

[Full_States]

[M_flap]

[psi]

[beta_col]

[v_turb]

[v_m]

[Full_States]

[M_flap]

[psi]

[P_col]

[beta_ref]

[psi]

[Full_States]

sim_pitch

[P_col]

[beta_inv]

[beta_ref]

[beta_col]

[v_turb]

[v_m]

sim_turb

sim_pitch

Pref

pitch0

Col. Pitch Control

−K* u

Band−limited noise

Out1

Wr

Wg

torsio

z

z’

Qg

beta

beta’

Wg

P

Figure E.1: Main simulink �gure.

114 E Simulink models

beta1cos

sin

Product

MatrixMultiply

MatrixConcatenation1

4*pi/3

2*pi/3

beta_cm

2

theta_azimuth1

Mflap_cm

1

cos

sin

Product

MatrixMultiply

MatrixConcatenation1

2/3

2/3

4*pi/3

2*pi/31/3

Mflap

2

theta_azimuth

1

beta_inv

1

−1/MflapBeta

s

−2/MflapBeta

s

LP 3p filter1

LPyaw_num(s)

LPyaw_den(s)

LP 3p filter

LPtilt_num(s)

LPtilt_den(s)

1p. invColeman

theta_azimuth

Mflap

Mflap_cm

1p. Coleman

theta_azimuth

beta_cm

betaM_flap

2

theta_azimuth

1

Ftilt

FyawFigure E.2: Implemented Coleman transformations

115

w_torsion

M_flap

4

beta

3

Full_States

2

Output

1

z’

1s

z

1s

wr1s

k/(tau1*tau2)

1/(tau1*tau2)

wn^2

(tau1+tau2)/(tau1*tau2)

wg1s

vt’

1s

vt

1s

torsio1s

beta1

1s

beta’

1s

sim_out

Wind

sim_wind

Wg,int

1s

sim_betadot

Qg

1s

Product P,int

1s

Measurements & Fault tolerant module

Mflap_in

states

Mflap

MATLAB Fcn

MATLABFunction

Ks

[Vtdot]

[Vt]

[beta_dot]

beta

[Qg]

[zdot]

[z]

Pref

wr_out

[Mflap]

[torsio]

[V_in]

psi

beta_ref

V_r

[betad_col]

[beta_col]

[wg_out]

P

[P_int]

[wg_int]

[wg][wr]

1/mt

Dt

Kt[beta_dot]

beta

[Qg]

[zdot]

[z]

[torsio]

[wg]

Pref

[Mflap]

psi

V_r

beta_ref

[wr]

beta

[Vtdot]

[Vt]

[betad_col]

[beta_col]

beta

[P_int]

[wg_int]

wr_out

P

P

[wg_out]

Divide

Ds

Wr

z0

pitch0

Pref

torsio0

Wg

pitch0

Qg

Clock

2*wn*xi

wn^2

1/taug

1/nb

1/nb

−1

−1

1/Ng

1/Ng

1/I_rotor1/Ir 1/Ig

psi

5

turbulence

4

P_ref

3

Beta_ref

2

V

1

Power

Q_rotor

Q_rotor

z

z’

beta

beta

Q_g

Q_g

Q_g

Q_g

Q_gQ_load

Q_shaft

Q_shaft

Theta_torsion

w_g

w_r

w_r

Ft

v_rFigureE.3:NonlinearHAWTmodel.

116 E Simulink models

H12

g(k)

1

zero

0

scale

invert

1

u

half

1/2

Zero−OrderHold

Unit Delay

z

1

Relay

Max(o,g+s)

max

Gain1

1

Gain −1

−1

sigma^24

mu13

mu0

2

z1

z

z

−z

Figure E.4: Cusum test.

117

w_torsion

res910

res89

res78

res67

res56

res45

res34

res23

res12

res01

wr1s

wn^2

wg1storsio

1s

beta’

1s

beta

1s

Qg

1s

Product

Ks

Divide

Ds

wind_shear

deltat

Pref

Clock

BEM model

MATLABFunction

2*wn*xi

wn^2

1/taug

1/Ng

1/Ng

1/I_rotor1/Ir

1/Ig

Pref9

Wr8

y10 (Mflap 1−3)7

y9 V_r6

y8 Power5

y7 Psi4

y4 (Mflap1−3)3

b_ref1−32

y1 (beta1−3)1

beta

beta

V_r

W_r

psi

Power

Q_rotor

Q_rotor

Q_g

Q_g

Q_g

Q_g

Q_g

Q_load

Q_shaft

Q_shaft

Theta_torsion

w_gw_r

w_r

P_ref

FigureE.5:Residualgenerator.

118ESimulinkmodels

w_torsion

PowerGenerator

2

w_g1

z’

1s

z

1s

wr1s

k/(tau1*tau2)

1/(tau1*tau2)wn^2

(tau1+tau2)/(tau1*tau2)

wg1s

vt’

1s

vt

1s

torsio1s

beta’

1s

beta

1s

Radius

rad

Qg

1s

Product

Ks

1/mt

Dt

Kt

Fcn1

0.5*den*pi*rad^2*u(1)^2

Fcn

0.5*den*pi*rad^2*u(1)^3

Divide

Ds

Ct−curve

Cp−curve

2*wn*xi

wn^2

1/taug

1/Ng

1/Ng

1/I_rotor 1/Ir 1/Ig

turbulens4

P_ref3

beta_ref2

v1

w_r

w_r

w_r w_g

Theta_torsion

Q_shaft

Q_shaft

Power

Q_load

Q_g

Q_g

Q_g

Q_g

Q_g

lambda

beta

beta

z’

z

Ft

v_rQrP_rotor

FigureE.6:LinearHAWTmodel.