Mirzaei, Mahmood

Niemann, Hans Henrik

Poulsen, Niels Kjølstad

In this paper designing a controller for trailing edge flaps (TEF) as well as optimizing its position on the wind turbine blade will be considered. An integrated design approach will be used to optimize both TEF placement and controller simultaneously. Youla parameterization will be used to parameterize the controller and the plant. The goal is to maximize blade root bending moments while minimizing actuator activity. An optimization with linear matrix inequalities (LMI) constraints will be used to optimize the H1 norm of the system

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Downloaded from orbit.dtu.dk on: Dec 20, 2017Designing Trailing Edge Flaps of Wind Turbines using an Integrated Design ApproachMirzaei, Mahmood; Poulsen, Niels Kjølstad; Niemann, Hans HenrikPublication date:2012Link back to DTU OrbitCitation (APA):Mirzaei, M., Poulsen, N. K., & Niemann, H. H. (2012). Designing Trailing Edge Flaps of Wind Turbines using anIntegrated Design Approach. Paper presented at 10th European Workshop on Advanced Control and Diagnosis,Kgs. Lyngby, Denmark.Designing Trailing Edge Flaps of WindTurbines using an Integrated DesignApproachMahmood Mirzaei ∗ Niels Kjølstad Poulsen ∗Hans Henrik Niemann ∗∗∗Department of Informatics and Mathematical Modeling, TechnicalUniversity of Denmark, Denmark, (e-mail: mmir@imm.dtu.dk,nkp@imm.dtu.dk).∗∗Department of Electrical Engineering, Technical University ofDenmark, Denmark, (e-mail: hhn@elektro.dtu.dk)Abstract: In this paper designing a controller for trailing edge flaps (TEF) as well as optimizingits position on the wind turbine blade will be considered. An integrated design approach will beused to optimize both TEF placement and controller simultaneously. Youla parameterization willbe used to parameterize the controller and the plant. The goal is to maximize blade root bendingmoments while minimizing actuator activity. An optimization with linear matrix inequalities(LMI) constraints will be used to optimize the H∞ norm of the system.Keywords: Wind turbines, trailing edge flaps, integrated design, Youla parameterization1. INTRODUCTIONIn recent decades there has been increasing interest ingreen energies of which wind energy is one of the mostimportant. Wind turbines are the most common windenergy conversion systems (WECS) and are hoped to beable to compete with fossil fuel power plants on energyprice in the near future. However this demands bettertechnology to reduce the electricity production price. Oneway to decrease the electricity production price is to reducestructural fatigue and therefore increase the lifetime of thewind turbine. Recent studies have shown that employingtrailing edge flaps on wind turbine blades can decrease fa-tigue loads on the blades. There have been several researchin this area tackling the problem of designing trailing edgeflaps (Andersen, 2010), (Castaignet et al., 2011) and ().In this paper we try to design trailing edge flaps for awind turbine using an integrated design approach wherewe optimize the flap radial position on the blade andalso its length while optimizing the controller. In order toformulate the integrated design problem we parameterizethe plant and the controller using Youla parameterization(Tay et al., 1998). The paper is organized as follows: Insection 2.1 the problem will be stated and design objectiveswill be given. In section 2.2 controller parameterizationand optimization will be discussed and finally in section2.3 plant parameterization and optimization will be given.? This work is supported by the CASED Project funded by grantDSF-09- 063197 of the Danish Council for Strategic Research.Fig. 1. Plant and controller parameterized2. PROPOSED INTEGRATED DESIGN METHOD2.1 Problem formulation and design objectivesWe want to optimize plant and controller parameterstogether in the design process. One way to make theproblem solvable is to formulate two different optimizationproblems and solve them iteratively. We start be aninitial guess, then firstly we optimize controller parametersgiven a fixed plant model and then we optimize plantparameters, given a fixed controller. To do so, considerfigure 1. In this figure input output relationships could bewritten as: (szy)=(P11 P12 P13P21 P22 P23P31 P32 P33)(rwu)(1)r = S(R)s (2)u = K(Q)y (3)In which S(R) is used to denote parameters of the plantas the transfer function R and K(Q) is used to denoteparameters of the controller as the transfer function Q.Having this, the integrated design problem can be castFig. 2. Integrated design for a SISO systeminto the following optimization problem:minQ,R‖Hzw‖∞ (4)In which we try to minimize the H∞ norm of the transferfunction Hzw from disturbance w to exogenous output z.To do so, we can iterate between optimizing the plant andthe controller. which are:HPzw = Fu(Fl(P,K(Q)), S(R)) (5)HCzw = Fl(Fu(P, S(R)),K(Q)) (6)Fl and Fu denote lower and upper linear fractional trans-formations (LFT) respectively. Therefore the optimizationalgorithm becomes:• Step 1: Optimize controller parametersminQ‖Hzw‖∞ (7)• Step 2: Optimize plant parametersminR‖Hzw‖∞ (8)• Step 3: Check termination conditions, if satisfiedterminate else go to step 1.We start by defining a simple SISO example. The systeminterconnection is shown in figure 2.2.2 Controller OptimizationController Parameterization Having a plant model (ei-ther from a first guess or result of an optimization), we arenow in a position that we can optimize the controller. Todo so we start by writing down the transfer matrix for thesystem with disturbances and input/output.(zy)=(C11 C12C21 C22)(wu)(9)Therefore the transfer function from w to z becomes:HCzw = C11 + C12K(1−KC22)C21 (10)Now if we parameterize the controller K using Youlaparameterization as:K =Uq +NqQVq +MqQ(11)In which Uq, Vq, Nq and Mq are calculated by coprimefactorization of C22:C22 = G22 +G21(1− SG11)−1SG12 (12)C22 = NqM−1q (13)And solving the following Bezout equation:NqUq +MqVq = 1 (14)Using the parameterization given in (11), the transferfunction HCzw could be written as:HCzw = C11 + C12QC21 (15)For a SISO system given in figure 2, we know C11, C12, C21and C22 are:C11 = 1 (16)C12 = 1 (17)C21 = G22 +G21(1− SG11)−1SG12 (18)Coprime factorization of state space models After formu-lating the transfer functions from w to z for parameterizedcontroller as follows:HCzw = C11 + C12QC21Now it is time to find coprime factorization of C22. Oneway to do this, is to first find a state space realization ofit which we denote as:C22 =(Ac BcCc Dc)Next step is to calculate normalized coprime factorizationof this transfer functions which can be calculated by thefollowing formula (Vidyasagar, 1987): For a state spacerealization of G(s) normalized coprime factorization of Gis:(Nl(s) Ml(s))s=(A+HC B +HD HR−1/2C R−1/2D R−1/2)Where:H∆= −(BDT + ZCT ), H ∆= I +DDTIn which Z is the solution to the following algebraic Riccatiequation:(A−BS−1DTC)Z + Z(A−BS−1DTC)T− ZCTR−1CZ +BS−1BT = 0In which:S∆= I −DTDAnother way is to design a stabilizing controller, e.g.using LQG theory (Doyle et al., 1992), and then usethat controller and coprime factorization of the plant toparameterize all the stabilizing controllers. For the transferfunction C22 we have a state space realization as:C22 =(A BC D)And we need to find state space realizations for N,M,Xand Y such that we have:C22 =NM, NX +MY = 1We do not go into the details here and give the finaltransfer functions in terms of state space matrices of arealization of the transfer function C22 and matrices F andH are found such that A + BF and A + HC are stable,found for example using LQG method.M(s) =(A+BF BF 1)N(s) =(A+BF BC +DF D)X(s) =(A+HC HF 0)Y (s) =(A+HC −B −HDF 1)Optimization problem So far we have formulated theoptimization problem of the form:Controller optimization: minQ‖HCzw‖∞ (19)Knowing thatHCzw is an affine functions of Q (see equation(15)), now we proceed to formulate and solve an opti-mization problem of the form given below, which is theoptimization problem (19) after parameterization.minX(s)∈RH∞‖Y0(s) + Y1(s)X(s)Y2(s)‖∞ (20)X(s) is a transfer function which is in infinite dimensionalRH∞. However in order to be able to solve the optimiza-tion problem we approximate the solution with a finitedimensional parameterized transfer function of the form:X(s) =nq∑i=1αiXi(s) (21)In which for example we formulate Xi’s as:Xi =√2as+ 1(s− as+ a)i−1(22)we are optimizing only the parameters αi in the transferfunction X(s), therefore we can write:X(s) =(Ax BxCx(α) Dx(α))(23)In order to simplify, the transfer function given in (20) canbe written as (knowing it is a SISO system):Y (s) = Y0(s) + Y1(s)X(s)Y2(s) (24)= Y0(s) + Y1(s)X(s) (25)In which we have replaced Y1(s)Y2(s) with Y1(s).LMI formulation For minimizing H∞ norm of a transferfunction, having its state space realization we can solve thefollowing optimization problem:min γ (26)ATy P + PAy PBy CTy? −γI DTy? ? −γI ≺ 0 (27)P  0 (28)In which:Y (s) =(Ay ByCy Dy)(29)Now we need to find the state space realization of Y (s)parameterized in terms of parameters of the transferfunction X(s). And for that matter we use the followingequations:W1 +W2 =A1 0 B10 A2 B2C1 C2 D1 +D2 (30)W1W2 =A1 B1C2 B1D20 A2 B2C1 D1C2 D1D2 (31)Therefore we have:Y (s) = Y0(s) + Y1(s)X(s) (32)=(A0 B0C0 D0)+(A1 B1C1 D1)(Ax BxCx(α) Dx(α))(33)=(A0 B0C0 D0)+ Ax BxC1 BxD10 A1 B1Cx(α) Dx(α)C1 Dx(α)D1(34)=A0 0 0 B00 Ax BxC1 BxD10 0 A1 B1C0 Cx(α) Dx(α)C1 D0 +Dx(α)D1 (35)And then we get the following matrices for the LMIoptimization.Ay =(A0 0 00 Ax BxC10 0 A1)By =(B0BxD1B1)(36)Cy(α) = (C0 Cx(α) Dx(α)C1) Dy(α) = D0 +Dx(α)D1(37)2.3 Plant OptimizationA set of linearized transfer functions for a trailing edge flapcan be found using system identification. The set could bewritten as:H(s) =(s+ z1(r, d))(s+ z2(r, d))(s+ p1(r, d))(s+ p2(r, d))(38)In which r is the distance from the root of the blade anddetermines TEF placement and d determines the size ofthe TEF. The equation (38) shows that poles and zeros ofthe TEF are functions of r and d and we can determinethese functions by running a number of experiments withdifferent values of r and d, and using system identificationto find poles and zeros of the transfer function. The goalis to solve the following optimization problem:Plant optimization: minr,d‖HPz w‖∞REFERENCESAndersen, P.B. (2010). Advanced Load Alleviation forWind Turbines using Adaptive Trailing Edge Flaps: Sen-soring and Control. Ph.D. thesis, Technical Universityof Denmark, Denmark.Castaignet, D., Poulsen, N.K., Buhl, T., and Wedel-Heinen, J.J. (2011). Model predictive control of trailingedge flaps on a wind turbine blade. Proceedings of theAmerican Control Conference, 4398–4403.Doyle, J., Francis, B., and Tannenbaum, A. (1992). Feed-back Control Theory. Macmillan Publishing Company,New York,N.Y.Panofsky, H.A. and Dutton, J.A. (????). Atmosphericturbulence: Models and methods for engineering appli-cations.Tay, T.T., Mareels, I., and Moore, J. (1998). High perfor-mance control. Birkhauser, Boston.Vidyasagar, M. (1987). Control system synthesis / Afactorization approach. MIT Press, Cambridge,Mass.

Designing Trailing Edge Flaps of Wind Turbines using an Integrated Design Approach

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Designing Trailing Edge Flaps of Wind Turbines using an Integrated Design Approach

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