A New Nonlinear Excitation Controller for Synchronous Power Generator

Transcription

1 Proc. of he 5h WSEAS/IASME In. Conf. on Elecrc Power Sysems, Hgh Volages, Elecrc Machnes, Tenerfe, Span, December 16-18, 2005 (pp98-103) A New Nonlnear Excaon Conroller for Synchronous Power Generaor M. OUASSAID*, A. NEJEMI**, M. CHERKAOUI* * Deparmen of Elecrcal Engneerng, Ecole Mohammaa ngéneur, Mohamme V Unersy, B.P. 765, Agal, Raba, Morocco **Faculé es scences e echnques Benmellal, Ka Ayya unersy, Marrakech, Morocco Absrac: - Ths paper presens a new an smple conroller for ransen sablzaon wh olage regulaon of a synchronous power generaor connece o an nfne bus. The oerall sably of he sysem s shown usng Lyapuno echnque. The esgn of he propose conroller akes no accoun he mporan non lneares of he power sysem moel an s nepenen of he equlbrum pon. Smulaon resuls hae been shown o emonsrae he effeceness of he propose conroller for he enhancemen of he ransen sably an olage regulaon of he power sysem uner a large suen faul an a we range of operang conons. Key-Wors: - Excaon conrol, Lyapuno echnque, non lnear conrol, synchronous generaor, ransen sably, olage regulaon 1 Inroucon The hgh complexy an nonlneary of power sysems, ogeher wh her almos connuously me aryng naure, hae eal of challenge of power sysem conrol engneers for ecaes. A parcular ssue encounere a he generang plan leel s o manan sably uner arous operang conons. In orer o oban hgh qualy for synchronous generaor conrollers, many researches has been esablshe an numerous paper are publshe. Conenonal excaon conrollers such as he auomac olage regulaor (AVR) an power sysem sablzer (PSS) are manly esgne by usng lnear conrol heory [1], [2]. These excaon conrollers can be use effecely o amp oscllaon an nsure asympoc sably of he equlbrum followng small perurbaon. In case of a large faul, he operang pon of he sysem may ary conserably; non lneares begn o hae hen sgnfcan effecs an a lnear conrollers may no be able o manan asympoc sably [3]. Recenly, aance nonlnear conroller echnque, whch are nepenen of he equlbrum pon an ake no accoun he mporan non-lneares of he power sysem moel hae been use n he excaon conrol of power sysems [4], [5], [6]. Mos of hese conrollers are base on feeback lnearzaon echnque [3], [7], [8]. I was shown n he leraures ha he ynamcs of he power sysem coul be exacly lnearze by employng nonlnear sae feeback. The essence of hs echnque s o frs ransform a nonlnear sysem no a lnear on by a nonlnear feeback, an hen uses he well-known lnear esgn echnques o complee he conroller esgn. Consequenly one can use conenonal lnear conrol o ge accepable performance [9], [10] an [11]. Neerheless n many cases he feeback lnearzaon meho requres precse parameers plan an ofen cancels some useful non-lneares. I s well known ha power sysems conan some paramerc unceranes n pracce. In hs case, s ffcul o exacly lnearze he sysem wh nomnal parameers. Aape ersons of he feeback lnearzng conrols are hen eelope n [12], [13]. Feeback lnearzaon s recenly enhance by usng robus conrol esgns such as H conrol an L2 surbance aenuaon [14], [15]. Lyapuno heory has for a long me been an mporan ool n lnear as well as nonlnear conrol [16]. Howeer, s use whn nonlnear conrol has been hampere by he ffcules o fn a Lyapuno funcon for a gen sysem. If one can be foun, he sysem s known o be sable, bu he ask of fnng such a funcon has ofen been lef o he magnaon an experence of he esgner.the am of hs paper s he esgn of a conrol law for a nonlnear excaon conroller o enhance he ransen sably an o ensure goo pos-faul olage regulaon for synchronous generaor connece o an nfne bus hrough a ransmsson lne, as shown n Fg. 1. The moel of he synchronous machne use s a 7 h orer moel, 5 for he elecrcal ynamcs an 2 for he mechancal ynamcs, whch akes no accoun he saor ynamcs as well as he amper wnng effecs an praccal lmaon on conrol. The feeback sysem s globally asympocally sable n he sense of he Lyapuno sably heory. The res of hs paper s organze as follows. In secon 2, we escrbe he sngle-machne-nfne-bus power sysem moel n a sae space form suable for

2 Proc. of he 5h WSEAS/IASME In. Conf. on Elecrc Power Sysems, Hgh Volages, Elecrc Machnes, Tenerfe, Span, December 16-18, 2005 (pp98-103) Lyapuno-base conrol esgn. In secon 3, he nonlnear excaon conroller s ere. The sably of hs conroller s proen. Some llusrae smulaon resuls are presene an compare o he performance of a sanar regulaor olage AVR an PSS n secon 4 o alae he propose conroller an some conclung remarks are menone n he fnal secon. 2 Mahemacal moel of power sysem The generaor o be conrolle, sue n hs work, s shown n Fg. 1. I consss of synchronous generaor connece o an nfne bus a a ransmsson lne (SMIB). The synchronous machne equaons n erms of Park s -q axs are expresse as follows [17], [18]: Armaure wnngs where Fel wnng Damper wnngs Rs ωλq λ = + (1) q Rsq ωλ λq = + + (2) λ = L + L ( + ) (3) m k λ q = L q q + Lmq kq (4) = R L + L + L (5) s m m k 0 = Rkk Lm + Lm + Lk k 0 = Rkqkq Lmq + Lkq kq (6) (7) where, q are rec an quaraure axs saor ermnal olage componens, respecely; s he excaon conrol npu;, q rec an quaraure axs saor curren componens, respecely; he fel wnng curren; k, kq rec an quaraure axs amper wnng curren componens, respecely; λ,λ q rec an quaraure axs flux lnkages, respecely; R s he saor ressance; R he fel ressance; R k, R kq he amper wnng ressances; L, L q he rec an quaraure self nucances, respecely; L he roor self nucance; L k, L kq rec an quaraure amper wnng self nucances, respecely an L m, L mq rec an quaraure magnezng nucances, respecely. Mechancal equaon δ = ω 1 2 ω m e H = T T Dω (8) (9) where ω s he angular spee of he generaor; δ he roor angle of he generaor; T m he mechancal orque; T e he elecromagnec orque; D he ampng consan an H he nera consan. Generaor G Lne (R e, L e) Fg. 1. Sngle machne nfne bus. Infne Bus The equaon for ransmsson nework wh exernal ressance Re an nucance Le, n he Park ransforme coornaes are = R + L ωl + V cos( δ a) e e e q = R + L + ωl V sn( δ a) q e q e q e I.B (10) (11) where V s he nfne bus olage an a s s phase angle. By combnng equaons (1) o (11), he resulan generaor sysem moel, n per un, has he followng form = a + a + a ω + a + a ω q 14 k 15 kq + a cos( ) + b 16 1 = a + a + a ω + a + a ω q 24 k 25 kq + a cos( ) + b 26 2 (12) (13) q = a31ω+ a32ω+ a33q + a34kω+ a35kq (14) + a sn( ) 36 = a + a + a ω + a + a ω k q 44 k 45 kq + a cos( ) + b 46 3 = a ω+ a ω+ a + a ω+ a kq q 54 k 55 kq + a sn( ) 56 ω = a + a + a + a 61 q 62 q 63 q k 64 kq + a ω + a T m (15) (16) (17) δ = ω ( 1 R ω ) (18) where ω R s he elecrcal frequency; a j an b are consans whch epen on he generaor an on he loa parameers [3]. 3 Desgn of a nonlnear excaon conroller Lyapuno s secon or rec meho s a ery powerful ool of assessng sably of a nonlnear sysem [16], [19]. In hs paper, he concep of Lyapuno s sably creron s use o selec he conrol sraegy of he Sngle Machne Infne Bus (SMIB), n orer o ensure

3 Proc. of he 5h WSEAS/IASME In. Conf. on Elecrc Power Sysems, Hgh Volages, Elecrc Machnes, Tenerfe, Span, December 16-18, 2005 (pp98-103) goo seay an ransen sably. To reach hs objece, we efne he ermnal olage error as e= (19) where = 1 s he esre rajecory an V ref Nonlnear Conroller P m Synchronous Generaor V = + q (20) 2 2 Sysem sae arables The expressons of an as a funcon of he sae arables can be expresse as follow = c + c + c ω + c + c ω q 14 k 15 kq + c cos( ) + c q (21) q = c21ω + c22ω+ c23q + c24kω+ c25kq c sn( ) (22) + 26 where c j are consans whch epen on he consans a j an on he loa parameers [3]. A pose efne Lyapuno funcon of he SMIB can be consere as 1 2 V = e (23) 2 The me erae of he V() ecan be wren as V = ee (24) From he erae of he ermnal olage error an by usng (12)-(16) an (20)-(22), we obans he followng expresson 1 e= = + qq (25) q = c17 + bc g1 + c14 g2 + q where g1 = c11+ c12+ c13 ωq+ qω (26) + c sn( ) 15 ωkq + kq ω + c16 an g2 = a41 + a42 + a43qω + a44k + a45k qω a cos( ) (27) + 46 Then he erae of he Lyapuno funcon s compue as V = c e+ bc e+ g e q gc 2 14 e q e (28) Thus, he Lyapuno s sably creron can be sasfe by makng erm on he rgh han se of (28) negae sem efne n orer o guaranee he global asympoc sably of he sysem. The canaes of ha Fg. 2 Bloc agram of he propose nonlnear scheme guaranees he sem negae efneness creron of equaon (28) can be consere as Ke + b3c14 + g1 = c 17 q (29) + c14 g2 + q where K s a pose consan feeback gan. Subsung (29) no (28) he erae of he Lyapuno funcon becomes 2 V = Ke (30) Defne he followng equaon 2 W () = Ke 0 (31) Furhermore, by usng LaSalle Yoshzawa s heorem [16], s can be shown ha W() en o zero as. Therefore, e wll conerge o zero as. 4 Smulaon resuls an scusson In orer o alae he mahemacal analyss an, hence, o esablsh he effeceness of he propose nonlnear conrol scheme, Smulaons works are carre ou for he Sngle Machne Infne Bus Sysem. The sysem confguraon s presene as shown n Fg.2.The performance of he nonlnear conroller was ese on he complee 7 h orer moel of he generaor sysem wh he physcal lm of he excaon olage of he generaor. The parameer alues use n he ensung smulaon are gen n he appenx. The faul consere n hs paper s a symmercal hree-phase shor crcu. The faul locaon s nexe by a consan λ whch s he fracon of he lne o he rgh of he faul. If λ = 1 he faul s a he nfne bus bar. The smulae resuls are gen n Fg. 3. I s shown ermnal olage, roor spee, roor angle an excaon olage, respecely. The operang pon consere s P mo = 0.6 p.u. The faul occurs closer o he nfne bus bar a = 5s an remoe by openng he barkers of he faule lne a = 5.1s. As can be seen he pos faul ermnal olage s regulae o s prefaul alue ery quckly. I s que een ha he propose

4 Proc. of he 5h WSEAS/IASME In. Conf. on Elecrc Power Sysems, Hgh Volages, Elecrc Machnes, Tenerfe, Span, December 16-18, 2005 (pp98-103) Termnal olage (p.u.) Roor spee (p.u.) Tme (s) Tme (s) Excaon olage (p.u.) Roor angle (egree) Tme (s) Tme (s) Fg. 3 Smulae resul of he propose nonlnear excaon conroller. conroller achees goo ransen sably, an ampens ou he power angle oscllaons. In orer o proe he robusness of he propose conroller, he resuls are compare wh hose of he lnear conrollers (conenonal IEEE ype 1 AVR an PSS). Fg. 4 an Fg. 5 show he responses of he ermnal olage uner P mo = 0.9 p.u, λ = 0.01 an P mo = 0.3 p.u, λ = 1, respecely. I s seen how ynamcs of he ermnal olage exhb large oershoos urng pos faul sae before he sele o s seay sae alue wh he sanar lnear scheme han wh he nonlnear conroller. The propose conroller can quckly an accuraely rack he esre ermnal olage espe he fferen faul locaons an operang pons. Anoher smulae ermnal olage responses for a suen ncrease n he mechancal power s shown n Fg.6. The power sysem s sare a mechancal power of P mo = 0.3 p.u. Aroun = 6s, he mechancal power s se a P mo = 0.75 p.u. I s shown ha he ermnal olage of he lnear conroller shows remarkable ransen before o seles o esre alue, whle he ermnal olage of he propose conroller s unaffece by hs araon. From he resuls presene earler, agan he superory of he nonlnear conroller s obsere. 5 Concluson Ths paper has successfully emonsrae he esgn, an sably analyss of Lyapuno echnque approach for he ransen sably an olage regulaon of a SMIB power sysem base on he complee 7 h orer moel of he generaor sysem. Frs, he ynamc moel of a SMIB was nrouce. Then, a nonlnear conrol sysem was esgne n he sense of Lyapuno conrol echnque. The feeback sysem s globally asympocally sable. The esgn of he conroller s nepenen of he operang pon. Smulaon resuls emonsrae ha wh he propose conroller, he generaor excaon conroller can effecely mproe he olage sably amp oscllaon an enhance he ransen sably of power sysem uner a large suen faul. Wh he ere conrol hgh an accuracy sably can be achee compare o he conenonal AVR an PSS conrollers.

5 Proc. of he 5h WSEAS/IASME In. Conf. on Elecrc Power Sysems, Hgh Volages, Elecrc Machnes, Tenerfe, Span, December 16-18, 2005 (pp98-103) Termnal olage (p.u.) Tme (s) Fg. 4 Responses of he ermnal olage uner P mo = 0.9 p.u an λ = 0.01; (sol) propose conroller; (o) lnear conrollers. Termnal olage (p.u.) Tme (s) Fg. 5 Responses of he ermnal olage uner P mo = 0.3 p.u an λ = 1; (sol) propose conroller; (o) lnear conrollers. Termnal olage (p.u.) Tme (s) Fg. 6 Responses of he ermnal olage for a suen ncrease n mechancal power, P mo =0.45 a = 6 s; (sol) propose conroller; (o) lnear conrollers.

6 Proc. of he 5h WSEAS/IASME In. Conf. on Elecrc Power Sysems, Hgh Volages, Elecrc Machnes, Tenerfe, Span, December 16-18, 2005 (pp98-103) Appenx Table 1 Parameers of he Transmsson Lne n p.u. Parameer L e, nucance of he ransmsson lne. R e, ressance of he ransmsson lne. Value Table 2 Parameers of he Power Synchronous Generaor n p.u. Parameer R s, saor ressance. R, fel ressance. R k, rec amper wnng ressance. R kq, quaraure amper wnng ressance. L, rec self-nucance. L q quaraure self-nucances. L, roor self nucance. L k, rec amper wnng self nucance. L kq, quaraure amper wnng self nucance. L m, rec magnezng nucance. L mq, quaraure magnezng nucance. V, nfne bus olage D, ampng consan. H, nera consan. Value s References: [1] M.S. Ghazzaeh, F. M. Hughs, A Generaor Transfer Funcon Regulaor for Improe Excaon Conrol, IEEE Trans on. Power Sysems, Vol. 13, N 2, 1998, pp [2] Al. A Ghanakly, A. M. Farhou, A Paramercally Opmze Self Tunng Regulaor for Power Sysem Ssablzers, IEEE Trans on. Power Sysems, Vol. 7, N 3, 1992, pp [3] O. Akhrf, F. A. Okou, L. A. Dessan, R. Champagne, Applcaon of Mularable Feeback Lnearzaon Scheme for Roor Angle Sably an Volage Regulaon of Power Sysems, IEEE Trans on. Power Sysems, Vol. 14, N 2, 1999, pp [4] Q. Zhao, J.Jang, Robus Conroller Desgn for Generaor Excaon Sysems, IEEE Trans on. Energy Conerson, Vol. 10, N 2, 1995, pp [5] Z. X, G. Feng, D. Cheng, an Q. Lu, Nonlnear ecenralze saurae conroller esgn for power sysems, IEEE Trans. on Conrol Sysems Technology, Vol. 11, N 4, 2003, pp [6] Y. Wang, G. Guo, an D. J. Hll, Robus ecenralze nonlnear conroller esgn for mulmachne power sysems, Auomaca, Vol. 33, N 9, 1997, pp [7] W. Ln, T. Shen, Robus passy an feeback esgn for mnmum-phase nonlnear sysems wh srucural uncerany. Auomaca, ol. 35, 1999, pp [8] J. Chapman, M. Ilc, C. Kng, L. Eng, an H. Kaufman, Sablzng a mulmachne power sysem a ecenralze feeback lnearzng excaon conrol, IEEE Trans. on Power Sysems, ol. 8, 1993, pp [9] L. Gao, L. Chen, Y. Fan an H. Ma, A nonlnear conrol esgn for power sysems, Auomaca, Vol. 28, N 5, 1992, pp [10] C. A. Kng, J. W. Chapman, an M. D. Ilc, Feeback lnearzng excaon conrol on a fullscale power sysem moel. IEEE Transacons on Power Sysems, Vol. 2, 1994, pp [11] S. Jan, F. Khorram an B. Faranesh, Aape nonlnear excaon conrol of power sysem wh unknown nerconnecons, IEEE Transacons on Conrol Sysems Technology, Vol. 2, 1994, pp [12] Y. Tan an Y. Wang, Augmenaon of ransen sably usng a supperconucon col an aape nonlnear conrol, IEEE Trans. On Power Sysems, ol. 13, N 2, 1998, pp [13] Y. Wang, D. J. Hll, R. H. Mleon, an L. Gao, Transen sablzaon of power sysems wh an aape conrol law, Auomaca, Vol. 30, 1994, pp [14]T. Shen, S. Me, Q. Lu, W. Hu, an K. Tamura, Aape nonlnear excaon conrol wh L2 surbance aenuaon for power sysems, Auomaca, Vol. 39, N 1, 2003, pp [15]Y. Wang, G. Guo, an D. Hll, Robus ecenralze nonlnear conroller esgn for mulmachne power sysems, Auomaca, Vol. 33, N 9, 1997, pp [16] M. Krsc, I. Kanellakopoulos, an P.V. Kokooc, Nonlnear an Aape Conrol Desgn. New York: Wley Inerscence, [17] P. M. Anerson, A. A. Foua, Power sysem conrol an sably, IEEE Perss, [18] C. H. Cheng, Y. Y. Hsu, Dampng of Generaor Oscllaon Usng an Aape Sac Var Compensaeur, IEEE Trans on. Power Sysems, Vol. 7, N 2, 1992, pp [19] M. Vysagar, Nonlnear sysem Analyss, SIAM, Phlaelpha 2002.