ASSESSING ROBUST STABILITY OF POWER SYSTEMS USING THE COMPLEX AND MIXED SSV

Transcription

1 Proeedings of the IASTE International Conferene Power and Energ (PE 23) November - 3, 23 Marina del Re, USA ASSESSING ROBUST STABILITY OF POWER SYSTEMS USING THE COMPLEX AN MIXE SSV Evaristo N. Rees, Maro A. Pérez, Artro R. Messina. Centro de Investigaión de Estdios Avanzados del I.P.N. Unidad Gadalajara Av. del Bosqe No. 45, Col El Bajío, Zapopan, Jal., 459, Méxio noereesp@gmail.om, [mperez, aroman]@gdl.investav.mx ABSTRACT The aim of this work is to evalate the robst stabilit in power sstems b omparing the se of the omplex against the mixed strtred singlar vale (SSV). This proposal inldes a perent of omplex pertrbations to avoid the semi-ontinit in the lower bond proded b the mixed SSV and onseqentl it allows to assess the impat of omplex nertaint in the sstem. Nmerial simlations are performed on a mlti-mahine power sstem to show the advantages of the method as well as the hallenges to overome. KEY WORS Eletromehanial osillations, Robst stabilit, Strtred singlar vale. Introdtion Man linear methods have been proposed to evalate the nominal stabilit in power sstems based on the se of the small-signal stabilit [-4]. These linear tehniqes have demonstrated their effetiveness in several researhes to std poor damping of the osillations of snhronos mahine rotors, instabilit, ndesired stead-state network performane, et. Frthermore these methodologies have the abilit of the power sstem to maintain snhronism when sbjeted to small distrbanes [22-23]. However, these linear methodologies do not model the nertaint in the nominal sstem and the do not provide robst properties for a wide range of operating onditions. In several stdies, the nertaint arises as a natral expression of nmodelled dnamis, parameter variations, nonlinear variations, et. that an lead to instabilit. The investigation of losed-loop sbjet to modelled nertaint is an important isse to assre robstness in stabilit and performane. The strtred and nstrtred nertaint in power sstems emerges de to load flow variations, transmission line parameters, hanges in the network topolog, dereglation in the setor, negleted dnamis, et. all these fators have ased the sstem to be loser to its operational limits. The strtred singlar vale (SSV) theor was developed as a mean to determine the robst stabilit and performane of a linear sstem with parameter nertaint. Compting the exat vale of the SSV has shown to be NPhard (Non-deterministi polnomial-time hard), ths in pratie is not possible to ompte this vale exept for ver low order sstems. However, sing saling properties an pper and lower bond an be defined to approximate the SSV. Algorithms for ompting these bonds have been domented in several researhes [5-8]. An essential and sometimes nderrated onept regarding to the SSV is the differene between the omplex and the mixed SSV. The omplex ase was derived sing pper and lower bonds ot of real parametri nertainties [9-]. Unfortnatel, the gap between these bonds an be onservative. In fat, the mathematial formlation of the omplex SSV shows that it ma be onsidered as a first approximation with respet to the mixed SSV. The mixed SSV has been the fos of researhes de to the fat that it inorporates real parametri nertainties [-4]. In the ase where have real parametri nertainties, a soltion is to reformlate the pper and lower bonds with new saling matries. However, the omptational brden of the proedre grows and the mixed SSV is disontinos. This means that the possibilit of missing a point does exist. A partial soltion for the disontinit behavior of the mixed SSV is to inorporate a perent of omplex nertaint, bt this will reslt in an inrease of the nertaint [5]. The introdtion of tehniqes to power sstems has been applied in different analses [6-8]. In short, two methodologies have been developed b researhers to represent the nertaint in power sstems. A perent representation of the strtred nertaint from the nominal linear sstem added to the nstrtred OI:.236/P

2 nertaint. The other methodolog is the polnomial representation sing least sqare minimization derived from different operating onditions. However, the first nertaint representation is more onservative and is limited to eah pertrbation in a given parameter. In addition, the proess of designing ontrollers sing has been implemented in different works sing a snthesis proess [9-2]. The main ontribtions of this paper are: The omparison between the omplex and mixed SSV to evalate the robst stabilit in power sstems and the inorporation of a perent of omplex pertrbations to real parameters to avoid the semiontinit of the mixed SSV. The rest of the paper is organized as follows: Setion 2 introdes the nominal sstem model. Then the theor of the SSV for linear sstems and the differenes between omplex and mixed SSV are presented in setion 3. Setion 4 desribes the ase std for a mltimahine power sstem. Finall, the onlsion of this paper is given in setion Sstem Modelling Let a finite dimensional LTI dnamial sstem be desribed b the following linear onstant oeffiient differential eqations: A B G( s ) : C( I A) B. C s (3) 3. Complex and Mixed SSV The nominal sstem desribed in () an be extended inlding nertaint. The representation of a pertrbed state spae sstem with nertaint is given b the next eqation: x t t t A C C B B x t t t The above sstem an be desribed in Fig. with an pper Linear Frational Transformation (LFT) to obtain the representation for robst performane where v t, t : : and is de to sores of nertaint. It is assmed that the matrix A is stable, and the inpt t is desribed as: t t. (4) (5) x t Axt Bt xt t Cxt t x, () n where x t : is the sstem state, initial ondition of the sstem, p sstem inpt, q x is the t t : is the t : is the sstem otpt, and A nn, B n p, C qn, q p are state, inpt, otpt matries, respetivel. The orresponding transfer matrix from (s) to (s) defined as: is Figre. Upper LFT for robst performane analsis. The nertain losed-loop transfer fntion from (s) to (s) is related b an pper LFT, s ( s) [ C( si A) B ] ( ) ( s) G s (2) N, N 22N 2I N N2 F F (6) where (s) and (s) are in the freqen domain. The sstem an be desribed with the following notation alled a realization of G (s). To analze the robst stabilit of (6), the sstem an be redraw into the M -strtre of Fig. 2. where M N is the transfer fntion from the otpt to the inpt of pertrbations. The matrix M an be obtained as: 48

3 M C II A s s B (7) efinition [4]: The mixed strtred singlar vale, M of a From nn C matrix M with respet to a blok strtre is defined as A B Γ (8) C M (2) min BX : deti M With M if no BX solves det M I. For large sstems the omptation of the exat vale of the SSV has shown to be NPhard. However; it is possible to ompte pper and lower bonds to approximate this analtial problem. In [] pper and lower bonds were expressed for the mixed ase as: Figre 2. M -strtre for robst stabilit analsis. To std the robst stabilit, the mixed SSV onsiders three tpes of bloks: repeated real salar, repeated omplex salar and fll bloks. In eah tpe of blok are three nonnegative integers, S, S and F. So that the total nmber of bloks in the sstem is defined b: S r S k i r j i j l r F The blok strtre an be defined as: m l n (9) k,..., ks, r,...,rs,m,...,mf () r Hene the normalized nertaint an be desribed BX r r Ik,..., S I r diag C C m,..., mf :, k I S r r S,..., I r S, () The prel real ase orresponds to S F and the prel omplex ase orresponds to S r F. max QQ k inf QM M min, GG k : M M j GM M G 2 (3) Similarl, for the omplex SSV,, the pper and lower bonds an be written as: max UU C UM M inf M L, R L,R L R (4) Let BX denote the set of all real-rational, proper, stable, blok diagonal transfer matries, with blok strtre like BX : where BX RH : j BX for all (5) RH represents the spae of real-rational proper transfer matries whit no poles in Res following theorem addresses robst stabilit. Theorem (Robst stabilit with ) [4]:. The Sppose that M s RH. Then for all BX with, the pertrbed losed-loop sstem is wellposed and stable if and onl if 49

4 s : sp M j M (6) In the ase of prel real nertainties, is not neessaril a ontinos fntion [-2]. These disontinities an ase problems in the onvergene of the lower bond mixed- algorithm. A method to avoid the semiontinit of M is to introde small levels of omplex nertaint into the nertaint model, in order, to improve the onvergene properties of the lower bond of. This proedre an be shown in Fig. 3 where represents a saling fator. The sstem data have been slightl modified to introde more pertrbation and nertaint to the nominal sstem. In addition, the tie line between bses and 27 is open. All generators are represented b a two-axis model with a slow exitation sstem (IEEE-CA). The strtred nertaint was ompted sing the method desribed in [6-7]. The nominal modal analsis with different tpes of nertaint is performed and the eletromehanial osillations identified. The reslts are smmarized in Table where the inter-area eletromehanial osillations with damping less than.5 are identified. Figre 3. Replaing real nertaint with real+omplex nertaint. 4. Nmerial Simlations The sstem shown in Fig. 4 is a sixteen mahine five area with 68 bses. This sstem is a reded-order model of the New England/New York interonnetion. The detailed desription of the sstem and network parameters an be fond in []. AREA #3 AREA # AREA # AREA # AREA # Figre 4. Sixteen mahine five area std sstem Table Inter-area mode and damping ratio of the nominal sstem. Linear nertaint Nonlinear nertaint Inter-area mode Inter-area mode i i i i i i i i.482 The nonlinear and linear strtred nertaint was ompted sing two parameter variations. The variation range in the reatane line onneting bses and 2 is [ p p..] and the ative load range is [ p.. 4 p..]. In Fig. 5 and Fig. 6 the reslts evidene that the sstem does not have robst stabilit with omplex SSV (dash line). On the other hand sing the mixed SSV the sstem has robst stabilit (solid line). It mst be noted that lower bonds are not depited in both figres. The gap between the lower and pper bond for the omplex SSV is pratiall zero. The lower bond of the mixed SSV is zero. The dimensions of the nonlinear and linear nertaint matries are smmarized in Table 2. The omptational brden is notorios as the size of the nertainties 4545 and 6262 is revealed for these ases. Table 2 Size of nertaint for robst stabilit allation. esription of Parameter size nertaint Nonlinear , Linear 8 8, Unertaint

5 M pper bonds M pper bonds M pper bonds M pper bonds shows that as the amont of omplex nertaint inreases the sstem approahes to its robst stabilit limit whih is srpassed when this qantit reahes approximatel 3% % % 3 % Freqen (rad/s) Figre 5. bonds with nonlinear nertaint Freqen (rad/s) Figre 6. pper and lower bonds with linear nertaint. The omparison of the robst stabilit is given in Table 3, where the SSV peak vales are shown for the different tpes of modeled pertrbations. Frthermore from Table, Fig. 5 and Fig. 6 is evident that the peaks of the omplex SSV identif inter-area and loal osillations modes, a featre that the mixed SSV laks. Table 3 Robst stabilit allation with nonlinear and linear nertaint representation. esription of nertaint Freqen Nonlinear CSSV Linear CSSV Nonlinear MSSV Linear MSSV Fig. 7 and Fig 8 show the pper and lower bonds of the mixed SSV for different levels of nertaint inlding omplex pertrbation. For the non-linear nertaint Fig Freqen (rad/s) Figre 7. pper and lower bonds with omplex pertrbation and non-linear nertaint. On the ontrar Fig. 8 evidenes the disadvantages of sing the linear pertrbation modelling, sine the mixed SSV does not reah the robst stabilit limit whih makes this measre within this modelling not be the best indiator to identif this senario Freqen (rad/s) Figre 8. pper and lower bonds with omplex 5. Conlsion pertrbation and linear nertaint. In this paper robst stabilit assessment reslts between omplex and mixed SSV have been stdied to analze and qantif the pper bonds of the SSV. These reslts show % % 3 % 5

6 that for large power sstems the modelling of the strtred nertaint an have a great impat in measring the limits of the robst stabilit. Frthermore, b inlding omplex pertrbations the lower bond of the mixed SSV an hange sbstantiall. Ftre work shold be fosed to obtain reded order models of the strtred nertaint to design robst deentralized ontrollers with real parameters. Finall, the proposed methodolog provides a nified framework for the analsis of the omplex and real parametri nertaint ases in power sstems. Referenes [] G. Rogers, Power Sstem Osillations (Klwer Aademi Pblishers, 2). [2] F. L. Pagola, I. J. Pérez-Arriaga, & G. C. Verghese, On sensitivities, resides, and partiipations, IEEE Trans. Power Sst., 4(), 989, [3] A. Henihe, I. Kamwa, Control loops seletion to damp inter-area osillations of eletrial networks, IEEE Trans. Power Sst., 7(2), 22, [4] M. Klein, G. J. Rogers, P. Kndr, A fndamental std of inter-area osillations in power sstems, IEEE Trans. Power Sst., 6(3), 99, [5] M. K. H. Fan, A. L. Tits, Charaterization and effiient omptation of the strtred singlar vale, IEEE Trans. Atomat. Contr., 3(8), 986, [6]. W. G, P. Hr Petkov, M. M. Konstantinov, Robst ontrol esign with Matlab (Springer, 25). [7] R. R. E. e Gaston, M. G. Safonov, Exat allation of the mltiloop stabilit margin, strtred singlar vale, IEEE Trans. Atomat. Contr., 33(2), 988, [8] A. Pakard, J. C. ole, The omplex strtred singlar vale, Atomatia, 29(), 993, 7-9. [9] S. Skogestad, I. Postlethwaite, Mltivariable Feedbak Control Analsis and esign, (John Wile and Sons, 996). [] G. J. Balas, J. C. ole, K. Glover, A, Pakard, R. Smith, µ-analsis and snthesis toolbox (Msn In. and The Mathworks In. version online, 998). [2] M. K. H. Fan, A. L. Tits, J. C: ole, Robstness in the presene of mixed parametri nertaint and nmodeled dnamis, IEEE Trans. Atomat. Contr., 36(), 99, [3] P. M. Yong, J. C. ole, Properties of the mixed m problem and its bonds, IEEE Trans. Atomat. Contr., 4(), 996, [4] P. M. Yong, Robstness with parametri dnami nertaint (Ph.. Thesis, California Institte of Tehnolog, 993). [5] S. L. Gatle,.G. Bates, M. J. Haes, I. Postlethwaite, Robstness analsis of an integrated flight and proplsion ontrol sstem sing µ and the υ-gap metri, (3), 22, [6] M. jkanovi, M. H. Khammash, V. Vittal, Appliation of strtred singlar vale theor for robst stabilit and ontrol analsis in mltimahine power sstems part-i: Framework development, IEEE Trans. Power Sstems, 3(4), 998, [7] M. jkanovi, M. H. Khammash, V. Vittal, Appliation of strtred singlar vale theor for robst stabilit and ontrol analsis in mltimahine power sstems part-ii: Nmerial simlations and reslts, IEEE Trans. Power Sstems, 3(4), 998, [8] R. Castellanos, A. R. Messina, H. Sarmiento, Robst stabilit analsis of large power sstems sing the strtred singlar vale theor, Eletri Power and Energ Sstems, 27(5-6), 25, [9] M. jkanovi, M. Khammash, V. Vittal, Seqential snthesis of strtred singlar vale based deentralized ontrollers in power sstems, IEEE Trans. Power Sstems, 4(2), 999, [2] X. Y, M. Khammash, V. Vittal, Robst design of a damping ontroller for stati var ompensators in power sstems, IEEE Trans. Power Sstems, 6(3), 2, [2] M. Ye, R. A. Shleter, µ-snthesis power sstem stabilizer design sing a bifration sbsstem based methodolog, IEEE Trans. Power Sstems, 8(4), 23, [22] P. Kndr, Power Sstem Stabilit and Control (MGraw-Hill, 994). [23] P. W. Saer, M. A. Pai, Power Sstem namis and Stabilit (Prentie Hall, 998). [] K. Zho, J. C. ole, Essentials of Robst Control, (Prentie Hall, 998). 52