A Robust LMI-Based Pitch Controller for Large Wind Turbines

This paper utilizes the linear matrix inequalities’ techniques (LMI) for designing a robust collective pitch controller (CPC) for large wind turbines. CPC operates during up rated wind speeds to regulate the generator speed in order to harvest the
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  A robust LMI-based pitch controller for large wind turbines H.M. Hassan * , A.L. ElShafei, W.A. Farag, M.S. Saad Electrical Power and Machines Department, Faculty of Engineering, Cairo University, El-gamea Street, Giza 12613, Egypt  a r t i c l e i n f o  Article history: Received 8 July 2011Accepted 30 December 2011Available online 31 January 2012 Keywords: Pitch controlLMI H  N  problem H  2  problemPolytopic systemPole clustering a b s t r a c t This paper utilizes the linear matrix inequalities ’  techniques (LMI) for designing a robust collective pitchcontroller (CPC) for large wind turbines. CPC operates during up rated wind speeds to regulate thegenerator speed in order to harvest the rated electrical power. The proposed design takes into accountmodel uncertainties by designing a controller based on a polytopic model. The LMI-based approachallows additional constraints to be included in the design (e.g.  H  N problem,  H  2  problem,  H  N / H  2  trade-off criteria, and pole clustering). These constraints are exploited to include requirements for perfect regu-lation, ef  fi cient disturbance rejection, and permissible actuator usage. The proposed controller iscombined with individual pitch controller (IPC) that reduces the periodic blade ’ s load by alleviating onceper revolution (1P) frequency fatigue loads. FAST (Fatigue, Aero-dynamics, Structures, and Turbulence)software code developed at the US National Renewable Energy Laboratory (NREL) is used to verify theresults.   2012 Elsevier Ltd. All rights reserved. 1. Introduction The use of wind power is increasing rapidly. At the same timethe need for better cost effectiveness of wind power plants hasstimulatedgrowthinwindturbines ’ sizeandpower.Inabove-ratedwind conditions, the goals for turbine operation change fromcontrol of generator torque for maximum power tracking to thoseof regulating power at rated levels with mitigating fatigue loadingon the turbine structure. An ordinary PI pitch controller regulatesthe generator speed without taking into consideration theunstructured dynamics of the blades, the drivetrain nor the tower.The nonlinear variation of rotor torque with wind speed and thepitchanglearetypicallynotconsideredindesign. Further,thepitchactuator also has restricted limits on pitch angle and pitch rate [1].Other challenging problems are the presence of nonlinearities inthe system dynamics, and the continuous change of the operatingpoints during operation. All previous reasonsmotivate the need forrobust pitch controller that provides an accepted performance, anddisturbance rejection at different operating points within theallowed actuator constrains. In this paper, a multi-objectivecollective pitch controller will be designed using LMI techniquesfor generator speed regulation.Another objective is to reduce the structural mechanical loadsby using IPC. This should be ful fi lled within the permissible rangeand rate of the pitch angle of the actuator. The importance of loadreduction becomes vital as turbines become larger and more fl exible. When the turbine blade sweeps, it experiences changes inwind speed due to wind shear, tower shadow, yaw misalignmentand turbulence. These variations lead to (1P) large component inthe blade loads, it ’ s essential to design (IPC) to cancel this compo-nent [2].Pitch controller is designed using  H  N technique in [3,4]. In thesepapers, the controller main objective is to regulate the speed byimproving disturbance rejection. The required control effort isn ’ tconsidered in the design. In [5], it is proposed to design gain scheduled feedback/feed forward CPC for speed regulationcombined with IPC for load reduction. Also in [6], optimal LQGfeedback/feed forward CPC is proposed for speed regulationcombined with IPC for load reduction. Combined CPC with IPC isproposed in [7] both as PI controllers. In [5 e 7], all the proposedcontrollers isbasedonasinglelinearizedmodel,whichonlyre fl ectsone single operating point. A multi-objective ( H  2 / H  N ) pitchcontrollerisproposedin[8],butitdoesn ’ tprovide( H  2 / H  N )trade-off criteria. It also doesn ’ t consider improving the transient response atdifferent operating points. In our proposed work in this paper, anLMI-based CPC is considered. The controller design constraintsinclude  H  N problemfor betterspeed regulation, and H  2  problemforoptimizing control action with performance. The design alsoaddresses  H  N / H  2  trade-off criteria for the optimization of the twopreviousproblems.Pole clusteringforimproving transientresponseis also considered. The controller is based on a polytopic model toovercome model uncertainty at different operating points. CPC iscombined with IPC to mitigate mechanical fatigue loads. *  Corresponding author. Tel.:  þ 20 102241739. E-mail address: (H.M. Hassan). Contents lists available at SciVerse ScienceDirect Renewable Energy journal homepage: 0960-1481/$  e  see front matter    2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.renene.2011.12.016 Renewable Energy 44 (2012) 63 e 71  In Section 2, the turbine model speci fi cations plus the turbinelinearized models are discussed. In Section 3, the proposed CPCdesign, and the controller objectives are shown. The designconsiders two cases; single operating point-based model, anda polytopic-based model. In Section 4, IPC design is discussed. Thesimulation results showing a comparison between the proposedcontroller and a conventional PI controller are shown in Section 5.Finally the conclusions are stated in Section 6. 2. Model description Simulations are performed on a full nonlinear turbine modelprovided by the FAST (Fatigue, Aero-dynamics, Structures, andTurbulence)softwarecodedevelopedattheUSNationalRenewableEnergy Laboratory (NREL) [9]. The model used is a 3-bladed, variable-speed 5 MW wind turbine model with the speci fi cationsgiven in Table 1.More speci fi cations could be found in [10, pp. 26]. The pitchactuator, represented as a second order model, has a pitch anglerange from 0 to 90  with maximum rate of 8  /s.FAST provides many degrees of freedom re fl ecting whether ornot different turbine parts ’  dynamics are considered. The followingdegrees of freedom (DOF) are considered in our study:(a) Generator DOF ( q 1 ).(b) Drivetrain rotational- fl exibility DOF ( q 2 ).(c) First fore-aft tower bending-mode DOF ( q 3 ).(d) First  fl apwise blade mode for each blade DOF ( q 4 ,  q 5 ,  q 6 ).where( q I  )denotes thedisplacementof the I  thDOF.EachDOF couldbe presented as a linearized model around certain operating pointaccording to: M  D € q I   þ  C  D _ q I   þ  K  D q I   ¼  F  * u  þ  F  d * u d  (1) where  M  ,  C  ,  K  ,  F F  d ,  u ,  and u d  denote mass matrix, stiffness matrix,damping matrix, control input matrix, wind input disturbancematrix, control input vector, and disturbance input vector, respec-tively. Assume  D  x  ¼ ½ D q I  ; D _ q I   T  , the linearized model takes theform: P  ð S  Þ  :  D _  x  ¼  A D  x  þ  B D u  þ  B d D u d D  y  ¼  C  D  x  þ  D D u  þ  D d D u d (2) where  ∆  x ,  D u ,  D u d , and  D  y  are the state vector perturbation,perturbation in the control action, perturbation in input distur-bance,andperturbationintheoutput,respectively.Fig.1showsthesynthesis of FAST model used in simulation.The generator torque has four control regions: 1, 2, 2.5, and 3.Region 1 is a control region before cut inwind speed ( v ci ) with zerogenerator torque so no power is extracted from the wind. Instead,the wind is used to accelerate the rotor for start-up. The main taskin Region 2 is optimizing power capture by maintaining a constant(optimal) tip-speed ratio; ( l ¼ l 0 ), while the pitch angle is keptzero. In Region 3, the wind speed is above-rated speed. In thisregion, the generator controller task is to hold the generator torqueconstant. In the same time, the pitch controller regulates thegenerator speed at the rated value in order to capture the ratedpower. Region 2½ is a linear transition between Regions 2 and 3usedtolimittipspeed(forlessnoiseemissions)atratedpower.Thetorque speed response of the model is shown in Fig. 2. 3. Designing an LMI-based collective pitch controller  The proposed technique is to design state feedback, LMI-basedcollective pitch controller (CPC) to regulate the generator speedin region 3. This controller is combined with IPC that mitigates the fl apwise moment by canceling (1P) frequency. The proposedcontrol strategy is shown in Fig. 3. M  1,2,3  are the blade tip  fl apwise moments of each blade.  u gen  isthe generator speed. The total control action ( b ) is calculated asfollows: b  ¼  b ipc  þ b cpc  þ b  (3) where  ð b Þ  is the pitch angle operating point. It is calculated bychanging operating point with wind speed through a look up table.The generator speed is regulated by the control action ( b cpc ), andthe  fl apwise moment is reduced by the control action ( b ipc ).In this design, we are looking for a solution that addresses thecombination of the following objectives:1. Ef  fi cient disturbance rejection for better speed regulation ( H  N problem) [11]. This could be achieved by keeping the RMS gainof   T  ( s ) N ( H  N norm) belowa prede fi ned value  g 0 : ( g 0 > 0).  T  ( s ) N is the closed loop transfer function from  W   to  Z  N , where  Z  N ¼ [ D u gen ]representstheregulationerrorduetodisturbance( W  ).2. To minimize a cost function (  J  ) that re fl ects a weighted sum of the control effort and states ’  perturbations. Trade-off betweenthe control effort and the performance is represented as a  H  2 problem. [11]. The minimization is carried by keeping the  H  2  Table 1 Wind turbine speci fi cations.Hub height 90 mRotor diameter 126 mCut in, rated, cut out wind speed 3 m/s, 11.4 m/s, 25 m/sCut in, rated rotor speed 6.9 rpm, 12.1 rpmGear box ratio 97Rated generator speed 1173.7 rpmRotor, Tower, nacelle mass 110 ton, 347.4 ton, 240 ton Fig. 1.  FAST model components. H.M. Hassan et al. / Renewable Energy 44 (2012) 63 e 71 64  norm (LQG Cost) of   T  ( s ) 2  below a prede fi ned value  v 0 : ( v 0 > 0). T  ( s ) 2  is the closed loop transfer function from  W   to  Z  2 .  Z  2  isde fi ned as: ð  Z  2  ¼  Q  * D  X   þ  R * D u Þ where the square matrix  Q  ¼ diag { Q  1 ,  Q  2 ,  Q  3 } and vector  R  repre-sent the weighting terms of the LQG cost function (  J  ) given in Eq.(4).  Z  2  represents the trade-off criteria between the perturbationsin states and control action according to the following objectivefunction:  J   ¼ Z  N 0  Q  2 D  X  2 þ  R 2 D u 2  dt (4) The LQG minimization problem could be used as a tool forcomparison between performance and control action.3. Achieving a desired transient response by maintaining theclosedlooppolesinsideaparticularregion( D ):(poleclusteringproblem) [12].Two design cases will be considered; a single operating point-based case and a polytopic-based model case.  3.1. Designing a CPC for a single operating point-based model FASTcanprovide a linearized model in the form given in Eq. (2).This linearized model is calculated at certain operating point ð u gen ;  b ;  v w Þ  [9], where  ð $ Þ  represents the operating point value of the variable ( $ ).  v w  is the hub height wind speed. In our designmodel, the enabled DOFs are the generator, and the drive train fl exibilitiesDOFs.ThesearetheonlyDOFsthatcouldbeobservedof the measured generator speed. They are considered the dominantdynamics in our turbine [10]. As a result, the other enabled DOFs will be considered as unstructured model uncertainty.The state vector is  D  X   : D  X   ¼ ½ D  X  1 ; D  X  2 ; D  X  3  T  where:(1)  D  X  1 ¼ drivetrain rotational- fl exibility (perturbations in drive-train torsional displacement) (m);(2)  D  X  2 ¼ generator DOF (perturbations in rotor speed) (rad/s);and(3)  D  X  3 ¼ Drivetrain  fl exibility (perturbations in Drivetraintorsional velocity) (m/s). D u ¼ [ D b ],  D u  is the perturbation in the collective pitch (controlaction),  D u d ¼ [ D v w ] is the perturbation in the wind speed,  D  y ¼ [ D u gen ], D  y  is the perturbation in generator speed. The design model  p ( s ) is completely observable and completely controllable. It is Fig. 2.  Torque speed characteristics of the generator. Fig. 3.  The pitch controller synthesis. H.M. Hassan et al. / Renewable Energy 44 (2012) 63 e 71  65  a linearized model around the operating point:  ð u gen  ¼  u rated ¼  1173 : 6 rpm ; b  ¼  14 : 93 + ; v w  ¼  18 m = s Þ . The model could bewritten in the following form: P  ð S  Þ  : 8<: D _  X   ¼  A * D  X   þ  B 1 * D W   þ  B 2 * D u Z  N  ¼  C  1 * D  X   þ  D 11 * D W   þ  D 21 * D u Z  2  ¼  C  2 * D  X   þ  D 22 * D u (5) where (  Z  N ¼ D  y ) and (  Z  2 ¼ Q  * D  X  þ R * D u ) . The state feedbackcontroller takes the form: D u  ¼  b cpc  ¼  K  * D  X   (6) The LMI problem includes the optimization of a cost function  f  ( H  N / H  2  trade-off criteria)  f   ¼  a k T  N k 2 N þ b k T  2 k 22  (7) where  a  and  b  are some weighting scalars.In this design, the CPC is constructed to solve: minimize  a * g 2 þ b * Trace ð Q  Þ  Subject to:where  P   is a Lyapunov matrix that satis fi es all the previousconstraints, (*) denotes symmetrical element,  ð ; Þ  denotes thekroneker product,  A cl  is the state matrix of the closed loopsystem. ( G ,  p ) are the parameters ’  matrices of the desired poleclustering region. Further details and proofs are given in [11,12].Once a feasible solution for the LMI framework is reached, anoptimal value of the cost function in Eq. (7) is also reached. Thesolution yields (  p ,  Y  *,  g * ,  Q  *), where  g * is the optimal  H  N  perfor-mance, and  Q  * is the optimal  H  2  performance. This problem isconsidered a semi-de fi nite problem (SDP). LMI Lab solver in LMIcontrol toolbox [13] is used to solve this problem. The  fi nal statefeedback controller is calculated as: K   ¼  Y  * * ð P  Þ  1 (11) The values of   g 0  and  v 0  are chosen as: g 0  ¼  17 : 3 ; v 0  ¼  1 The criteria behind that choice were that these values should beassmallaspossibleforbetterperformancewhileafeasiblesolutionis obtained.a. Pole clustering regionsPole clustering regionsare chosen to guarantee improvementintransient response. This can be achieved by specifying regionsrepresenting limits on the system ’ s eigenvalues. The  fi rst region( R 1 ) guarantees an upper limit on settling time. The secondregion ( R 2 ) guarantees a lower limit on settling time (whichprevents excessive control action). Finally region three ( R 3 ) ischosen as an upper bound on damping ratio. The previousregions are convex subsets of the complex plane characterizedby [14]: D  ¼ n  z  ˛ c   :  p þ G  z   þ G T   z   <  0 o  (12) R 1  and  R 2  regions are de fi ned by a vertical strip between ( h 1 , h 2 ) with characteristic function ( F  D 1,2 ) as: F  D 1 ; 2 ð  Z  Þ ¼  2 h 1   ð  z     z  Þ  00  ð  z   þ  z  Þ  2 h 2   (13) Region ( R 3 ) is a conic sector centered at the srcin with innerangle (2 q ) as shown in Fig. 4. The characteristic function ( F  D 3 ) isgiven as: F  D 3 ð  Z  Þ ¼  sin ð q Þ * ð  z   þ  z  Þ  cos ð q Þ * ð  z     z  Þ cos ð q Þ * ð  z     z  Þ  sin ð q Þ * ð  z     z  Þ   (14) The region parameters are taken as  h 1 ¼ 1,  h 2 ¼ 4, q ¼ 80  (compatible with the system dynamics) [1].The resulting LMI region ( D ) is the intersection of the threeregions stated above as shown in Fig. 4.b.  H  N / H  2  trade-off criteriaThe relation between ( H  N  performance/ H  2  performance) iscalculated at different weights ( b ,  a ) of the trade-off criteriagiven in (Eq. (7)). Fig. 5 shows the relation between  H  N performance and  H  2  performance. Fig. 4.  Pole clustering regions. 8>>>>>>>>>>><>>>>>>>>>>>: H  N performance  : 8<:0@  AP   þ  PA T  þ  B 2 Y   þ  Y  T  B T  2  B 1  PC  T  1  þ  Y  T  D T  12 *   I D T  11 * *   g 2 I  1A < 0 H  2  performance  : 8>><>>: Q C  2 P   þ  D 22 Y  *  P   > 0Trace ð Q  Þ  <  v 20 g 2 <  g 20 Pole clustering  : n p ; P   þ G ; ð P  *  A cl Þ þ G T  ;   A T  cl * P   < 0(8)  e (10) H.M. Hassan et al. / Renewable Energy 44 (2012) 63 e 71 66  The optimal performance occurs at the minimumvalue of thecost function (Eq. (7)). This value exists as a Pareto-optimal-likepointwhichiscircledinFig.5.Theweightsforthecostfunction  f  in Eq. (7) that achieve the Pareto-optimal-like point are  b ¼ 47,and  a ¼ 0.1.Now, the LMI problem could be written in this form: minimize  f   :   f   ¼  0 : 1 * k T  N k 2 N þ 47 * k T  2 k 22  ; subject to 8<: k T  N k N <  17 : 3 k T  2 k 2  <  1closed loop poles ð  A cl Þ ˛ region ð D Þ (15) Solving for the controller ’ s gains yields: K  1  ¼ ½ 674 : 6 8 : 024 5 : 7  T  where  K  1  is the controller ’ s gain for the single operating model-based case. Fig. 6 depicts the resultant closed loop poles fordifferent operating points (different wind speeds) using theprevious controller  K  1 . Fig. 6 shows that the controller manages Fig. 6.  Closed loop poles in single design model case. Fig. 7.  Closed loop poles in polytopic model case. Fig. 5.  H  N  performance/ H  2  performance relation. H.M. Hassan et al. / Renewable Energy 44 (2012) 63 e 71  67
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