This paper deals with the concepts of functional
output-controllability character of a finite-dimensional
linear dynamical system. And a method for comput-
ing the functional outputcontrollability consisting on
the calculation of the rank of a certain constant ma-
trix related to the system dynamics is introduced. The
linear system under study is a fixed speed wind tur-
bine (FSWT) formed by a squirrel cage generator con-
nected directly to the grid. Due to the non-linear be-
haviour of such system, the linear system model is
calculated by means of a Taylor’s decomposition of
the non-linear equations of the squirrel cage induction
generator, being the system linearized around a steady
state operating point. Finally, the study of the func-
tional output-controllability of such system is done,
and some boundaries are given to ensure functional
output-controllability from some given operational val-
uesPostprint (published version

Domínguez García, José Luis

García Planas, María Isabel

English

UPCommons. Portal del coneixement obert de la UPC

PHYSCON 2013, San Luis Potosı´, Me´xico, 26–29th August, 2013FUNCTIONAL OUTPUT-CONTROLLABILITY ANALYSISOF FIXED SPEED WIND TURBINEJose´ L. Domı´nguez-Garcı´aCatalonia Institute for Energy Research, IRECSpainjldominguez@irec.catM. Isabel Garcı´a-PlanasDepartament de Matema`tica Aplicada IUniversitat Polite`cnica de CatalunyaSpainmaria.isabel.garcia@upc.eduAbstractThis paper deals with the concepts of functionaloutput-controllability character of a finite-dimensionallinear dynamical system. And a method for comput-ing the functional outputcontrollability consisting onthe calculation of the rank of a certain constant ma-trix related to the system dynamics is introduced. Thelinear system under study is a fixed speed wind tur-bine (FSWT) formed by a squirrel cage generator con-nected directly to the grid. Due to the non-linear be-haviour of such system, the linear system model iscalculated by means of a Taylor’s decomposition ofthe non-linear equations of the squirrel cage inductiongenerator, being the system linearized around a steadystate operating point. Finally, the study of the func-tional output-controllability of such system is done,and some boundaries are given to ensure functionaloutput-controllability from some given operational val-ues.Key wordsFunctional Output-Controllability, Squirrel Cage In-duction Generator, Linear System, Boundaries.1 IntroductionIn the control theory of continuous linear time-invariant dynamical systems the most frequently usedmathematical model is given by the following systemconsisting of a differential state equation and an alge-braic output equationx˙(t) = Ax(t) +Bu(t)y(t) = Cx(t) +Du(t)}(1)where x is the state vector, y is the output vector, uis the input (or control) vector, A ∈ Mn(R) is thestate matrix, B ∈ Mn×m(R) is the input matrix, C ∈Mp×n(R) is the output matrix, and D ∈ Mp×m(R) isthe feedthrough (or feedforward) matrix.For simplicity we will write the systems as a quadru-ples of matrices (A,B,C,D).For its analysis and solution, this system is usuallydescribed by the transfer function obtained by applyingLaplace transformation to equation (1). It is obtained inthe following formsX˙ = AX +BUY = CX +DU}C(sI −A)−1B +D. (2)A related problem to the control of the system iswhether it is possible to steer the output following apreviously assigned curve over any interval of time.The answer to this problem is given analyzing the func-tional output-controllability.On the other hand, the recent increasing of windpower in the electrical network, makes interesting thestudy and ensure the functional output-controllabilityof Fixed-Speed Wind Turbines (FSWT), which can af-fect directly the behavior of power systems.2 Functional output-controllabilityDefinition 2.1. A system is functional output-controllable if and only if its output can be steeredalong the arbitrary given curve over any interval oftime. It means that if it is given any output yd(t),t ≥ 0, there exists t1 and a control ut, t ≥ 0, such thatfor any t ≥ t1, y(t) = yd(t).Proposition 2.1 ([Chen, 1970]). A system is func-tional output-controllable if and onlyrankC(sI −A)−1B +D = pin the field of rational functionsA necessary and sufficient condition for functionaloutput-controllability isProposition 2.2 ([Chen, 1970]).rank(sI −A BC D)= n+ p,For systems in which the matrix D is the zero ma-trix in [M. Garcı´a-Planas, Domı´nguez-Garcı´a, 2013],a simple test to compute the functional output-controllability is obtained. In this section, a general-ization of this test is presented.For a linear continuous-time system, like (1), de-scribed by matrices A, B, C and D, the functionaloutput-controllability matrix can be defined as.Definition 2.2.oCf(A,B,C,D) =C DCA CB DCA2 CAB CB D.... . .. . .CAn CAn−1B . . . CAB CB D .and the following result is obtained.Theorem 2.1. The system (A,B,C,D) is functionaloutput-controllable if and only ifrank oCf(A,B,C,D) = (n+ 1)p.The null terms are not written in the matrix.In order to prove this theorem an equivalence relationpreserving the functional output-controllability is de-fined that permit to consider an equivalent simple re-duced form for the systemDefinition 2.3. Two systems (A,B,C,D) and(A1, B1, C1, D1) are equivalent if and only thereexist matrices P,∈ Gl(n;R), R,∈ Gl(m;R),S,∈ Gl(p;R), V ∈ Mm×n(R) and W ∈ Mn×p(R)such that(A1 B1C1 D1)=(P−1 W0 S)(A BC D)(P 0V R).This equivalence relation coincides with the strictequivalence relation defined over pencils in the formH(λ) =(λI −A BC D). So the Kronecker canonicalreduced form can be considered [Garcı´a-Planas, Ma-gret, 1999].Proposition 2.3. The functional output-controllabilitycharacter is invariant under equivalence relation.Proof.(sI −A1 B1C1 D1)=(P−1 −W0 S)(sI −A BC D)(P 0−V R)Proof of the Theorem. It suffices to consider the Kro-necker reduced form of the pencil associated to the sys-tem.Corollary 2.1. The system (A,B,C,D) is functionaloutput-controllable if and only ifrank(C D)= prank ( C DCA CB D ) = 2p...rankC DCA CB DCA2 CAB CB D.... . .CAi CAi−1B ... CAB CB D = (i+ 1)p...Analogously to the case of triples of matrices(A,B,C) [M. Garcı´a-Planas, Domı´nguez-Garcı´a,2013], this corollary provides an iterative method tocompute functional output-controllability. Calling oCfthe matrices in the corollary, it is shown an example ofa flowchart in Figure 1.(A, B, C,D) rank( C D) < p i= 0 Update oCf rank( oCf) <(i+1)p Yes No functional  output-controllable Yes i= n Yes Functional  output-controllable No No No i= i+1 Figure 1. Flowchart showing the required iteration for functionaloutput-controllability computation3 A lower bound on the distance to a non func-tional output-controllable systemThe goal is to obtain a bound for the value of the radiusof a ball which is neighborhood of a functional output-controllable element, containing only elements whichare also functional output-controllable.The distance considered is that deduced from theFrobenius norm. We recall that given a matrix A =(aij) =∈ Mn×m(C), its Frobenius norm is defined as‖A‖√∑ij a2ij .This norm leads to the natural definition of the normof quadruples inM and the corresponding definition ofthe distance in M .Definition 3.1. Given a quadruple (A,B,C,D) ∈ Mits norm is defined as‖(A,B,C,D)‖=√‖A‖2 + ‖B‖2 + ‖C‖2 + ‖D‖2and the distance between the quadruples (A,B,C,D),(A′, B′, C ′, D′) is defined asd((A,B,C,D), (A′, B′, C ′, D′)) =‖(A−A′, B −B′, C − C ′, D −D′)‖.Finally, the distance between a functional output-controllable quadruple and the nearest non-functionaloutput-controllable one is defined asinf‖(δA, δB, δC, δD)‖where (δA, δB, δC, δD) is a quadruple such that (A+δA,B + δB,C + δC,D + δC) does not satisfies thegiven property.The starting point to find a bound is the rela-tionship between the norm of the associated matrixoCf(A,B,C,D) to the quadruple (A,B,C,D) and thenorm of this quadruple.The difficulty to relate the norm of oCf(A,B,C,D)to the quadruple (A,B,C,D) induce to consider thefollowing matrixM(A,B,C,D) =A B −I 0 0 0 0 . . . 0 0C D 0 0 0 0 00 0 A B −I 0 00 0 C D 0 0 0.... . . A B −I 0. . . C D 0 0. . . 0 0 C D∈M((n2+(n+1)p)×(n(n+1+m(n+1))(C).The following theorem ensures that we consider thismatrix to study the functional output-controllability.Theorem 3.1.rankC DCA CB DCA2 CAB CB D.... . .CAn CAn−1B CB D+ (n− 1) =rankA B −I 0 0 0 0 . . . 0 0C D 0 0 0 0 00 0 A B −I 0 00 0 C D 0 0 0.... . . A B −I 0. . . C D 0 0. . . 0 0 C DProof. Making block elementary row and columnstransformations we haverankA B −I 0 0 0 0 . . . 0 0C D 0 0 0 0 00 0 A B −I 0 00 0 C D 0 0 0.... . . A B −I 0. . . C D 0 0. . . 0 0 C D=rankI. . .IC DCA CB DCA2 CAB CB D.... . .CAn CAn−1B CB D.Relating the norm of the associated matrixM(A,B,C,D) to the quadruple (A,B,C,D)and the norm of this quadruple.Theorem 3.2. Given a non-functional output-controllable quadruple (A,B,C,D) a lower boundfor the distance to the nearest non-functional output-controllable quadruple is given by‖(δA, δB, δC, δD)‖ ≥1√n+ 1σn2+(n+1)pM(A,B,C,D)where σn2+(n+1)pM(A,B,C,D) denotesthe smallest non-zero singular value ofM(A,B,C,D).Proof. The functional output controllability of(A,B,C,D) implies that rankM(A,B,C,D) = n2+(n+1)p and that if (A+δA,B+δB,C+δC,D+δD)is not functional output controllable, rankM(A +δA,B + δB,C + δC,D + δD) ≤ n2 + (n+ 1)p.The Eckart-Young and Minkowski theorem states thatthe smallest perturbation in the Frobenius norm that re-duces the rank of a matrix A with rankA = r from r tor − 1 is σr(A), the smallest non-zero singular value ofA.Noting thatM(A+ δA,B + δB,C + δC,D + δD) =M(A,B,C,D) +M(δA, δB, δC, δD)whereδA δB 0 0 0 0 0 . . . 0 0δC δD 0 0 0 0 00 0 δA δB 0 0 00 0 δC δD 0 0 0.... . . δA δB 0 0. . . δC δD 0 0. . . 0 0 δC δDand it is easy to prove that, for all quadruple(A,B,C,D),‖M(δA, δB, δC, δD)‖ ≤ √n+ 1‖(δA, δB, δC, δD)‖.It suffices to compute.‖M(δA, δB, δC, δD)‖2 =n(‖δA‖2 + ‖δB‖2) + (n+ 1)(‖δC‖2 + ‖δD‖2) ≤(n+ 1)‖(δA, δB, δC, δD)‖2.Therefore, the norm of the perturbation of thematrix M(δA, δB, δC, δD) must be at leastσn2+(n+1)(M(A,B,C,D)),Hence, a bound for the distance from (A,B,C,D) tothe nearest non-functional output-controllable quadru-ple, taking into account above proposition is‖(δA, δB, δC, δD))‖ ≥1√n+ 1‖M(δA, δB, δC, δD)‖ ≥1√n+ 1σn2+(n+1)p(M(A,B,C,D)).4 Modeling of FSWTThe global analyzed system is a wind power generatorconnected directly to the grid.The linear system is defined by means of the squir-rel cage induction generator differential equations anda first order mechanical system. The differentialequations considered within this system are time de-pendant [Domı´nguez-Garcı´a, M. Garcı´a-Planas, 2011][Ugalde-Loo, Ekanayake, Jenkins]. Its inputs are thevoltage of the grid. Supposing the system to be insteady state, the system can be described as:X˙ = AFSWTX +BFSWTUY = CciX +DciU(3)AFSWT =KG ·α11 α12 α13 α14 a15−α12 α11 −α14 α13 a25α31 α32 α33 α34 a35−α32 α31 −α34 α33 a45Kωiqr0 −Kωidr0 −Kωiqs0 Kωids0 0α11 = −RTXrr, α12 = α10ωs + XTXrr, α13 =−RrXm, α14 = −βr0ωs, α31 = RTXm, α32 =−βs0ωs −XTXm, α33 = −RrXss, α34 = −α20ωs.BFSWT =γXrr 0 00 γXrr 0γXm 0 00 γXm 00 0 12Hγ =−ωbXssXrrσ.Where,KG = ωb(XssXrrσ)−1,Kω = Xm(2HKG)−1 = −γ,RT = Rs +RE ,XT = XE +Xtr,Xss = X ls +Xm,Xrr = X lr +Xm,σ =1−X2mXrrXssα10 = XssXrr − s0X2m,βs0 = XmXss(1− s0),βr0 = XmXrr(1− s0),α20 = X2m − s0XssXrr,a15 = Xm(Xmiqs0 −Xrriqr0),a35 = Xss(Xmiqs0 −Xrriqr0),a45 = Xss(−Xmids0 +Xrridr0),a25 = Xm(−Xmids0 −Xrridr0).Subindex 0 in some terms of the matrix AFSIG in-dicates evaluation at the initial condition. Observethat the terms ai5 appear because the slip s is a func-tion of ωr. The effect of the transmission line hasbeen considered through the following relations vds =vd∞ −XT iqs +RT ids and vq∞ +XT ids +RT iqs.4.1 Output variables selection for C and D matri-ces definitionThe active and reactive power delivered by the induc-tion generator, and also the stator currents of the windturbine have been chosen for such analysis.4.1.1 Active and reactive power selection Theoutput system described as Y = Cc1X +Dc1U can bewritten as follows:(∆Qss∆Pss)=(vsd0 −vsq0 0 0 0vsq0 vsd0 0 0 0)︸ ︷︷ ︸Cc1isqisdirqirdωr+(−isd0 isq0 0isq0 isd0 0)︸ ︷︷ ︸Dc1 vsqvsdTm(4)where the subscripts 0 (as previously stated) representsthe operating point selected to linearize.4.1.2 Stator currents of the FSWT In this case,the state variables have been selected as outputs vari-ables to be analyzed. Then,Cc2 =(1 0 0 0 00 1 0 0 0)(5)and Dc2 = 02×35 Functional Output-controllability of FSWTIn the following subsections it is studied functionaloutput-controllability of FSWT.Applying the theorem 2.1 in the linearized system, itcan be computed rank oCf(A,B,C,D).5.1 Case FSWT with active and reactive power asmeasured outputsTaking into account that isd0 and iso0 can be not zerosimultaneously, the matrixD =(−isd0 isq0 0isq0 isd0 0)has full row rank, then applying the test 2.1 it canbe concluded that the system is functional output-controllable.Notice that if the matrix D has full row rank the testfinish in the first iteration, since the ranks of the itera-tive matrices differs exactly the rank of the matrix D.Taking particular values it is possible to compute abound ensuring functional output-controllability.5.2 Case FSWT with stator currents of the induc-tion generator as measured outputsIn this case the matrixD is the zero matrix, but matrixC has full row rank, and taking into account that γ 6= 0and Xrr 6= 0, the matrixCB =(γXrr 0 00 γXrr 0)has full row rank and applying the test 2.1, as beforeit is concluded that the system is functional output-controllable.Notice that if D = 0 but matrices C and CB have fullrow rank the test finish in the second iteration, since theranks of the iterative matrices differs exactly the rank ofthe matrix CB from second iteration.6 Computing bounds for particular casesDue to the fact that the linear system under study isderived from a non-linear system, it is important to de-termine the confidentiality of the linear system in com-parison with the real one.In order to be able to compute a boundary which en-sures functional output-controllability characteristic ofthe system, some values for the symbolic parameterspreviously presented are used. Such parameters are de-fined in Appendix A6.1 Case FSWT with active and reactive power asmeasured outputsIf matrixD has full row rank, the system can be deter-mined functional output-controllable. For that reason,a proper first boundary approximation can be obtainedby computing the singular values of matrix D.In order to calculate the boundary, the system is eval-uated under two different operational cases: a wind tur-bine connected into a weak network and a wind turbineconnected into a strong network. The initial conditionsderived from those cases are introduced in Appendix A6.1.1 Weak network (V ASC = 16MVA) In thisparticular case matrix D isD =(0.527 0.791 00.791 −0.527 0)and the smallest singular value of the matrix D is0.9488.It is worth to remark, that if the parameters selectedfor the analysis implies that the smallest singular valueof matrixD is 0, the boundary must be computed usingmatrix M .6.1.2 Strong network (V ASC = 40MVA) Inthis particular case matrix D isD =(0.458 0.789 00.789 −0.458 0)and the smallest singular value of the matrix D is0.9201.Analogously, it is worth to remark, that if the param-eters selected for the analysis implies that the smallestsingular value of matrix D is 0, the boundary must becomputed using matrix M .6.2 Case FSWT with stators currents of the induc-tion generator as measured outputsIn this case D = 0, but matrices C and CB has fullrow rank, taking into account C is a fixed matrix, thena proper first boundary approximation can be obtainedby computing the singular values of matrix CB.In our caseB =87.0454 0 00 87.0454 084.9079 0 00 84.9079 00 0 0.1429ThenCB =(87.0454 0 00 87.0454 0)and the smallest singular value of the matrix CBis 87.0454. Then, the nearest non-functional output-controllable system (A + δA,B + δB,C + δC) is insuch a way that ‖δ(C ·B)‖ > 87.0454.It is worth to remark, that if the parameters selectedfor the analysis implies that the smallest singular valueof matrix CB is 0, the boundary must be computedusing matrix M .6.2.1 Weak network (V ASC = 16MVA) In thisparticular case matrix A isA =( 1.5058 −369.1080 0.4661 345.9649 46.2303369.1080 1.5058 −345.9649 −0.4661 −282.1571−1.0525 360.1678 0.4769 337.5596 26.1976−360.1678 −10.5254 −337.5596 0.4769 −170.7528−0.4812 0.1740 −0.4511 −0.3007 0)Using Matlab it can be obtained the singular values ofthe matrix M , and the smallest one is 0.003.6.2.2 Strong network (V ASC = 40MVA) Inthis particular case matrix A isA =( 0.8614 −362.5157 0.4661 345.8275 20.5053362.5157 0.8614 −345.8275 0.4661 −143.3823−0.8399 345.2217 −0.4769 337.4178 20.9843−345.2217 −0.8399 −337.4178 −0.4769 −236.53900.4734 0.1272 −0.4511 −0.2578 0)Using Matlab it can be obtained the singular values ofthe matrix M , and the smallest one is 0.0002.7 ConclusionThis paper has presented the concept of functionaloutput-controllability and applied to a linearized sys-tem from the nonlinear equations representing thesquirrel cage induction generator. Functional output-controllability has been determined using the A, B, Cand D matrices. Moreover, the demonstration is madewith a generic system. Therefore, it can be ensurednot only for an example. Due to the functional output-controllability condition, it can be concluded that anyoutput can be reached regulating the voltage inputs.A System under study parametersThe parameters of the squirrel cage induction gener-ator used for the evaluation of the boundaries are thefollowing: Vb = 690V , Sb = 2MVA, fb = 50Hz,ωb = 2pifb, H = 3.5s, Xtr = 0.05, Rs = 0.00488,X ls = 0.09241, Rr = 0.00549, X lr = 0.09955,Rd = 0.2696, X ld = 0.0453, Xm = 3.95279 andXrm = 0.02.V ASC = 16MVA, X/R = 10, ZE = Sb/V ASC ,RE = ZE/(1 + (X/R)2)1/2, XE = RE(X/R).A.1 Weak network (V ASC = 16MVA)The operating point of a weak network used for sys-tem linearization is ids0 = −0.527, iqs0 = 0.791,idr0 = −0.306, iqr0 = 0.846 and s0 = −0.0055.A.2 Strong network (V ASC = 40MVA)The operating point of a strong network used for sys-tem linearization is ids0 = −0.458, iqs0 = 0.798,idr0 = −0.225, iqr0 = 0.838 and s0 = −0.0051.ReferencesChen, C., (1970). Introduction to Linear System The-ory, Holt, Rinehart and Winston, Inc., New York.Heniche, A., Kamwa, I., (2002). Using measures ofcontrollability and observability for input and outputselection, in: IEEE International Conference on Con-trol Applications, Vol. 2, pp. 1248–1251.P. Kundur, Power System Stability and Control,McGraw-Hill, New York, 1994.Domı´nguez-Garcı´a, J.L., Garcı´a-Planas, M.I. (2011).Output controllability analysis of fixed speed windturbine, in: Proceedings of 5th Physcon, 2011. IPACSLibrary.Garcı´a-Planas, M.I-, Domı´nguez-Garcı´a, J.L. (2013).Alternative tests for functional and pointwise output-controllability of linear time-invariant systems. Sys-tems & Control letters. 62, (5), pp. 382387.Germani, A., Monaco, S. (1983). Functional output-controlability for linear systems on hilbert spaces,Systems & Control Letters 2, (5) pp. 313–320.Gohberg, I., Lancaster, P., Rodman, L. (1982) “MatrixPolynomials”, Academic Press, New York.Garcı´a-Planas, Ma¯ I., Magret, D.M. (1999). An alter-native System of Structural Invariants for Quadruplesof Matrices, Linear Algebra and its Applications 291,(1-3), pp. 83–102.Ugalde-Loo, C.E, Ekanayake, J.B, Jenkins, N. (2013).State-Space Modelling of Wind Turbine Generatorsfor Power Systems Studies, IEEE Trans. on IndustryApplications 49, (1), pp. 223–232.

Functional output-controllability analysis of fixed speed wind turbine

This paper deals with the concepts of functional

output-controllability character of a finite-dimensional

linear dynamical system. And a method for comput-

ing the functional outputcontrollability consisting on

the calculation of the rank of a certain constant ma-

trix related to the system dynamics is introduced. The

linear system under study is a fixed speed wind tur-

bine (FSWT) formed by a squirrel cage generator con-

nected directly to the grid. Due to the non-linear be-

haviour of such system, the linear system model is

calculated by means of a Taylor’s decomposition of

the non-linear equations of the squirrel cage induction

generator, being the system linearized around a steady

state operating point. Finally, the study of the func-

tional output-controllability of such system is done,

and some boundaries are given to ensure functional

output-controllability from some given operational val-

ue

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Functional output-controllability analysis of fixed speed wind turbine

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