An optimal excitation controller design based on multirate-output controllers (MROCs) having a multirate sampling mechanismwith different sampling period in each measured output of the system is presented. The proposed H∞ -control techniqueis applied to the discrete linear open-loop system model which represents a wind turbine generator supplying an infinite busthrough a transmission line

A. K. Boglou

C. Potolias

D. I. Pappas

D. V. Bandekas

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43Research Article   Optimal Excitation Controller Design for Wind Turbine Generator Using H∞ Control Technique A. K. Boglou*,a, D. V. Bandekasb, D. I. Pappasa and C. Potoliasb  aDepartment of Sciences, Kavala Institute of Technology,St. Loukas, 654 04 Kavala Greece.bDepartment of Electrical Engineering, Kavala Institute of Technology, 654 04, Kavala Greece. Received 15 January 2011; Accepted 26 March 2011AbstractAn optimal excitation controller design based on multirate-output controllers (MROCs) having a multirate sampling mecha-nism with different sampling period in each measured output of the system is presented. The proposed H∞ -control technique is applied to the discrete linear open-loop system model which represents a wind turbine generator supplying an infinite bus through a transmission line. Keywords: Multirate – output controllers, H∞ -control, wind turbine generators, disturbance.Journal of Engineering Science and Technology Review 4 (1) (2011) 43-49JOURNAL OFEngineering Science and Technology Reviewwww.jestr.orgIn the recent years H∞-control problem has great interest [1-5, 7, 8, 12-17]. The H∞-control problem for discrete-time and sampled-data single rate and multirate systems has been treated success-fully [2-6]. In both the continuous and discrete-time cases, when the state vector is not available for feedback, the H∞-control prob-lem is usually solved using dynamic measurement feedback ap-proach. For the solution of the H∞-disturbance attenuation problem a new technique [6] is presented. In order to solve the sampled-data, based on multirate-output controllers (MROCs), H∞-disturbance attenuation  problem  relies  mainly  on  the  reduction,  under  ap-propriate conditions, of the original H∞-disturbance attenuation problem, to an associated discrete H∞-control problem for which a fictitious static state feedback controller is to be designed, even though same state variables are not available for feedback.In this paper discrete linear open-loop power system model was obtained through a systematic procedure, using a linearized continuous, with impulse disturbances, 8th-order SIMO open-loop model representing a practical power system. The sough digital controller, which will lead to the associ-ated designed discrete closed-loop power system model display-ing enhanced dynamic stability characteristics, is accomplished by applying properly the presented MROCs technique. Consider the controllable and observable continuous linear state-space system model of the general form   ) t ( x   =Ax(t)+Bu(t)+Dq(t), x(0) = 0        (1a)) t ( ) t ( ) t ( 1 m u J Cx y + =           (1b)Where:  are the state, input, external disturbance, measured output and controlled output vectors, respectively. In Eqn. 1 all matrices have real elements and appropriate dimensions. The definition below is useful in what will follow.Definition.  For  an  observable  matrix  pair  ( ) C A, ,  with []TpT2T1T1 c c c C L =  and ci with i=1,…,p1, the ith row of the matrix C, a collection of  1 p integers    { }1 p 2 1 n   ,   , n   , n L  is called an observability index vector of the pair ( ) C A, , if the fol-  * E-mail address: akbogl@teikav.edu.gr ISSN: 1791-2377 © 2011 Kavala Institute of Technology. All rights reserved. 1. Introduction 2. H∞-Control Technique Using MROCS [4, 6]) t ( ) t ( ) t ( 2 c u J Ex y + =  n R x ∈ ) t ( , ,m R u ∈ ) t (d2 ) t ( L ∈ q ,,1 p ) t ( R ym =∈2 p ) t ( R yc ∈44lowing relationships simultaneously holdNext the multirate sampling mechanism depicted in Figure 1. is applied to Eqn. 1. Assuming that all samplers start simultaneously at t=0, a sampler and a zero-order hold with period T0 is connected to each plant input u t i( ) , i=1,2,…,m, such that( 2)while the ith disturbance  ) t ( qi , i=1,…,d, and the ith controlled output  ) t ( y i , c , i=1,…, p2, are detected at time kT0, such that for The ith measured output  ) t ( y i , m , i=1,…, p1, is detected at every   Ti  period, such that for   (3)where, (J2)i is the ith row of the matrix J2. Here Ni∈Z+ are the out-put multiplicities of the sampling and Ti∈R+ are the output sam-pling periods having rational ratio, i.e. Ti =T0 / Ni with i=1,…, p1. The sampled values of the plant measured outputs obtained over [kT0, (k+1)T0), are stored in the N*-dimensional column vector given by (4)(where  ), that is used in the MROC of the form          (5)where  .  Assumptions: a) The matrix triplets (A,B,C) and (A,D,E) are stabilizable and detectable.b)    c)   d) There is a sampling period T0, such that the open-loop discrete-time system model in general form becomes      (6) where,   (7)is stabilizable and observable and does not have invariant zeros on the unit circle. From  the  above  it  fellows  that  the  procedure  for  H∞-disturbance attenuation using MROCs, essentially consists in find-ing for the control law a fictitious state matrix F, which equiva-lently solves the problem and then, either determining the MROC pair (Lγ, Lu) or choosing a desired Lu and determining the Lγ. As it has been shown in [3], matrix F takes the form          (8)where P is an appropriate solution of the Riccati equation               (9)It is to be noted that γ∈R+, such that ││   Tqyc(z)││   ≥  γ  where    is the H∞-norm of the proper stable discrete trans-fer function Tqyc(z), from sampled-data external disturbances to sampled-data controlled output yc(kT0). Once matrix F is obtained the MROC matrices Lγ and Lu (in the case where Lu is free), can be computed according to the A. K. Boglou, D. V. Bandekas, D. I. Pappas and C. Potolias / Journal of Engineering Science and Technology Review 4 (1) (2011) 43-49∑==1 p1 ii n, n() ()1 1111111 ... ... ...p nn TT TT TTpp rank n−−  =  cA cc AcFigure 1. Control of linear systems using MROCsu(t) = u () , 0 kT   [ ) 0 0 1)T + (k   , kT t ∈[ ) 0 0 1)T + (k   , kT t ∈q(t) = q() 0 kT ,() = 0 c kT y Ex () 0 kT +J2 () 0 kT0,...,1 i N μ =−() () ()() ,0 01 0 mi ii i i yk TT kT Tk T μ += ++ cx Ju μ() () 0, 10 ,1 01 1 ˆ () ...( 1) ... m m kT yk Ty kT NT γ  = + −   () ()1 1 11Tm,p0m,p0 pp ...yk T ...yk TN 1T    +−    ∑==1 p1 ii* N N() [] () () 0 0 0 kT ˆ kT T 1 k γ − = + γ L u L u u* mxN mxmu   , R L R L ∈ ∈ γd m n rank d, n rankxd p xm p xd p 1 1 1+ + = + = 0 0 CD B A0 CD A[] [] m m n m 2T2 × × = I 0 J E J() () () () 0 0 0 0 ˆ ˆ 1 kT kT kT kT += Φ+ +   x x Bu Dq() () () 0 0 20 c kT kT kT =+ yE xJ u() () () λ λ = = ∫ d ) ( exp ˆ ˆ   , T exp0 T00 B,D A D , B A Φ() Φ P B B P B I FT 1 T ˆ ˆ ˆ−+ =()1 ˆˆ ˆˆ...T T T T−=+ ΦΦ −Φ + Φ PE EP PB IB PB BP  () ˆˆ ˆˆ ...T T ++ PD ID PD DPD D ˆ ˆ 1 −γ γ =γ γ γγ  do q 2 ) kT ( l ∈││ Tqyc(z)│ │∞45following relations (10)   (11) Where,                            and                      is an arbitrary specified matrix. In the case where Lu = Lu,sp, we have      (12)where Ĥ [H  Θu  Θq]=I and Σ∈RmxN*  is arbitrary. The explicit expressions for  Θu and Θq are given in [4, 6]. The resulting closed-loop system matrix (Αcl/d) takes the fol-lowing general form(13)where cl = closed-loop, ol = open-loop and d = discrete. 3.   Open  and  Closed  Loop  Model  of  Power  System  Under StudyIn the present work the power system under study (shown in Fig-ure 2) consists of a 1 MVA wind turbine generator [9-10], which was extrapolated from that of the 100kW unit for the NASA-Lewis Research Center [11], supplying an infinite bus through a transmission line (i.e. it refers to the plant of Figure 1). The  detailed  equations  characterizing  the  mechanical  dy-namics, generator and excitor systems and blade pitch controls, leading to the state space formulation, are taken from [9]. Its con-tinuous description in the form of Eqn. 1 is as followsyc = x,      E = I8x8,       J1 = 02x2,       J2 = 08x8The computed discrete linear open-loop power system mod-el, based on the associated linearized continuous open-loop sys-tem model described in Appendix 2 of [9], is given below in terms of its matrices with sampling period T0 = 1.0 sec.  Based on Figure 1, the H∞-control using MROCs (given in section 2), and the computed discrete linear open-loop model of the power system under study, and the discrete closed-loop power system model were designed considering with γ = 0.08 the f feed-back gain computed as:The numerical values of the matrices referring to the dis-crete closed-loop power system model of the above case are not included here due to space limitations. The magnitude of the eigenvalues of the discrete original open-loop and of the designed closed-loop power system model are shown in Table 1. By comparing the eigenvalues of the de-signed closed-loop power system model to those of the original open-loop power system model the resulting enhancement in dy-namic system stability is judged as being remarkable. A. K. Boglou, D. V. Bandekas, D. I. Pappas and C. Potolias / Journal of Engineering Science and Technology Review 4 (1) (2011) 43-49[] () ** md NN γ × ×  =+ Λ− Θ  q LF 0H IH H %%[ ] ( ) { } * * NN md × ×   Lu u q = F OH H % H % + Θ Θ Λ Ι —  [] I H H q = Θ~ * mxN R Λ∈( ) * * NN u,sp × Lγ u q = F L Ĥ H Ĥ + Θ Θ Σ Ι —   mxd O        , , ˆ ,~, Λ H H HF B A A d ol d ol d cl / / / − =Figure 2. Schematic diagram of 1 MVA wind turbine generator , ] v v [T2 1 3 R fd q' θ θ ω δ Ε Ε = x  [] []Tref. m v, ,, uq uy ω == = δ − − − −− − − −− − − − −− − − −− − − −− − −− − − −− − − −=0106 . 0 0763 . 0 0800 . 0 0023 . 0 1098 . 0 1481 . 0 9751 . 21 0330 . 00011 . 0 0004 . 0 3505 . 0 0105 . 0 0044 . 0 2025 . 0 5445 . 4 0589 . 00005 . 0 0020 . 0 8825 . 0 0266 . 0 0481 . 0 0076 . 0 9717 . 0 0379 . 00380 . 0 2745 . 0 8617 . 8 2696 . 0 7826 . 2 2567 . 6 8777 . 121 2283 . 10143 . 0 1462 . 0 3517 . 5 1601 . 0 5589 . 0 1567 . 4 0600 . 107 0250 . 00020 . 0 0088 . 0 8903 . 2 0870 . 0 1748 . 0 0737 . 0 2387 . 2 1332 . 00 . 0 0001 . 0 0138 . 0 0004 . 0 0009 . 0 0069 . 0 0010 . 0 0004 . 00010 . 0 0019 . 0 5993 . 3 1084 . 0 2314 . 0 0905 . 0 1100 . 0 0010 . 0/ d ol A−−−=4304 . 04806 . 02659 . 06413 . 71723 . 119848 . 30234 . 07483 . 4/d ol B =1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1/d ol C] 10 = 4501  -1  -2600  9660  37687  729  -3457  967T D4 − [] 10 = 2184 -843087 -477 -1475  340 -11369  2418 336 f4 − [Table 1.   Magnitude of eigenvalues of discrete original open-loop and designed closed-loop power system models.Original open-loop power system model λ1.1294  1.1294  0.3450  0.1820  0.1820  0.0087  0.0087  0.0000Designed closed-loop power system model λ0.8314  0.5634  0.5634  0.3365  0.2069  0.0137  0.0119  0.000046The responses of the output variables (δ, ω, Efd, E′q, θ1, θ2, v3, and VR) of the original open-loop and designed closed-loop power system model for zero initial conditions and unit step input distur-bance are shown in Figure 3 and Figure 10, respectively. A. K. Boglou, D. V. Bandekas, D. I. Pappas and C. Potolias / Journal of Engineering Science and Technology Review 4 (1) (2011) 43-49Figure 3. (a) (b):   δ responses of original discrete open and close-loop power system model subject to unit step input disturbance Figure 4. (a) (b):   ω responses of original discrete open and close-loop power system model subject to unit step input disturbanceFigure 5. (a) (b):   E΄q responses of original discrete open and close-loop power system model subject to unit step input disturbance0     -0.5 -0.4 -0.3 -0.2 -0.1 0 0.10.2 0.3     open No. of samplesδ  (rad.)8 6 4 2 1 0 1 2 1 4 1 6 1 8 2 0(a)No. of samples          x 10-ω  (rad./sec.) open  2.521.510.50-0.5-1-1.5-20 2 4 6 8 10 12 14 16 18 203(a) -0.3-0.2-0.10.1 0.2 0.3 0.4  E´q (p.u.) open No. of samples4 6 8   10  12 14  16 18  20 2 00(a)5         closed  No. of samplesδ  (rad.)10 15 20 25 30 35-0.250-0.2-0.15-0.1-0.050(b) 0  5 10 15 20 25 30   -2 -1.5 -1 -0.5 0 0.5 x 10-4 ω  (rad./sec.) closed No. of samples-2.5(b)E´q (p.u.) No. of samples           closed  0 5 10 15 20 25 30 350.020.040.060.080.100.120.140.160.180.200(b)47A. K. Boglou, D. V. Bandekas, D. I. Pappas and C. Potolias / Journal of Engineering Science and Technology Review 4 (1) (2011) 43-49Figure 6. (a) (b): Efd responses of original discrete open and close-loop power system model subject to unit step input disturbanceFigure 8. (a) (b): θ2 responses of original discrete open and close-loop power system model subject to unit step input disturbanceFigure 7. (a) (b): θ1 responses of original discrete open and close-loop power system model subject to unit step input disturbanceNo. of samples        open θ1(p.u.)-0.05-0.04-0.03-0.02-0.010.010.020.0300 2 4 6 8 10 12 14 16 18 20(a)Efd (p.u.) No. of samples0 4  6 8   10 12 14   16 18  -1.5 -1.0 -0.5 0.00  0.5 1.0  1.5    open 2 2 0(a)Efd (p.u.) No. of samples0 4  6 8   10 12 14   16 18  -1.5  -1.0 -0.5 0.00 0.5 1.0 1.5   open 2 2 0 0 5   10  15  20   25  30 -0.10 -0.05 0.00.05 0.10 0.15  Efd  (p.u.) closed No. of samples(b)          closed x 10-3No. of samplesθ2    (p.u.)-10-8-6-4-2024680 2 4 6   8 1 0 1 2 1 4 1 6(b)       closed 0 5 1 0 1 5 2 0 2 5 3 0x 10-3 00.511.52 2.533.544.55(b)No. of samples  open θ2   (p.u.)-0.08-0.06-0.04-0.0200.020.040.060 2 4 6 8 1 0 1 2 14 16 18 20(a)48From Figure 3 to Figure 10 it is clear that the dynamic sta-bility characteristics of the designed discrete closed-loop system-model are far more superior than the corresponding ones of the original open-loop model, which attests in favour of the proposed H∞-control technique.4. ConclusionsAn efficient H∞-control technique based on multirate-output con-trollers has been presented in concise form for the purpose of attenuating  in  an  effective  manner  system  disturbances  which otherwise degrade the performance of the synchronous genera-tor. The method was applied successfully to a discrete open-loop power system model, which was computed from an original con-tinuous linearized open-loop model of the practical power sys-tem, resulting in the design of an associated discrete closed-loop power system model. The results of the simulations performed on the discrete open- and closed- loop power system models dem-onstrated clearly the significant enhancement of the dynamic sta-bility characteristics achieved by the designed closed-loop power system model. Thus the H∞-control technique used is an appropri-ate/reliable tool for the design of implementable multirate-output controllers.Acknowledgements The Project is co-funded by the European Social Fund and Na-tional Resources – (EPEAEK-II) ARCHIMIDIS ΙΙ.A. K. Boglou, D. V. Bandekas, D. I. Pappas and C. Potolias / Journal of Engineering Science and Technology Review 4 (1) (2011) 43-49Figure 9. 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Optimal Excitation Controller Design for Wind Turbine Generator

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