Non-linear Analytical Extended Poincare's Model of Phase Saturated Self Excited Series Connected Synchronous Generators

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1 Seye Mhaa SHARIAMADAR 1, Jalal NAZARZADEH 2, Meha ABEDI 3 Science an Reeach Banch, Ilaic Aza Univeity (1), Shahe Univeity (2), Ai-Kabi Univeity f echnlgy (3) Nn-linea Analytical Extene Pincae' Mel f Phae Satuate Self Excite Seie Cnnecte Synchnu Geneat Abtact. In the peent pape, fit a agnetizatin cuent bae el i intuce f the electical achine analyzing that atuatin i ccue f the phae that it cuent entee t atuatin egin. Othe pat f achine ay be atuate in the linea cnitin in e f thei cuent. hen a new analytical extene Pincae' ap i intuce f elling f elf excite eie cnnecte ynchnu geneat in atuate cnitin. Uing the nn-linea cntl they, an analytical fit e Pincae' ap f the achine i cpute. hen extene Pincae' ap f the achine i intuce. Afte that chaacteitic ultiplie f the elf excite eie cnnecte ynchnu geneat ae eteine. Nnlineaity f the achine can be elle with the new ap an tability analyi f the yte i invetigate. he new ap i capable f elling, cntl, bifucatin an cha analyi in the nn-linea (atuatin) cnitin. Finally the iulatin eult f the new extene Pincae' ap with expeiental labaty et-up eult ae cpae. he eult hw that the new Pincae' ap i an effective eth f elling an analyi f any ac electical achine. Stezczenie: Wpwazn el pąu agneująceg azyny eletycznej w ejnie naycenia analizy tanu naycenia awzbunych geneatów ynchnicznych. Wyzytan el Pincae. Zaelwan nieliniwść azyny i bliczn wauni tabilnści. Rezultaty yulacji pazują że zappnwany el że być wyzytany analizy óżnych azyn eletycznych AC. (Nieliniwy analityczny el awzbunych płącznych zeegw geneatów ynchnicznych) Keyw: Cha, Pincae' Map, Self Excite Seie Cnnecte Synchnu Geneat, Satuatin. Słwa luczwe: awzbuny geneat ynchniczny, apa Pincae, naycenie. Intuctin Self excite ac geneat ae uitable achine f win enegy cnvein. Inuctin, eluctance an peanent agnetic geneat ae epeentative f uch applicatin. Self Excite Inuctin Geneat (SEIG) wa ne f ealiet type f elf excite ac geneat. It ha the avantage f buh-le cntuctin with quiel cage t, euce ize, abence f c pwe upply f excitatin, euce aintenance ct an bette tanient chaacteitic [1, 2, 3]. SEIG will have a eiu pble in vltage egulatin when la pee change. Bth the agnitue an fequency f puce vltage f a SEIG ae affecte by the ynaic f la. Cnnecte Synchnu Geneat (SESCSG) ha e avantage ve SEIG an act a a hypthetical alient ple achine. It peate at half the t angula fequency which i inepenent f laing cnitin [4]. Mutafa et al. [5, 6] invetigate the teay-tate pefance f thee-phae SESCSG by cnieing atuatin effect uing the q el. anient an teay tate pefance analyi f SESCSG can be applie with the genealize they an the q efeence fae [7]. Effect f uen icnnectin f ne excitatin capacit n the utput vltage in the abc fae wa invetigate [4]. Wang et. al. cntinue the eeach abut vaiu in f inuctin geneat f e yea until [8]. Self excitatin phenena an la pefance ae invetigate with tw-ieninal finite eleent analyi in [9]. hei eth, hweve, i nt uitable f nn-linea pble uch a cha ientificatin. Mt f the pape have tuie the SESCSG in the teaytate cnitin an cniee ipving the achine pefance. Dynaic analyi an vltage builing pce in the q fae have been peente in the few pape but geneally peaing, the q el i nt able t analyze SESCSG in the nn-linea an unbalance cnitin. hu, uen icnnectin vaiatin excitatin capacit f ne phae can be analyze with abc fae elling eth. Output ignal f the achine ae peiic an tie vaiant. Cnequently, which accingly lineaizatin f the yte cul nt be caie ut in the whle pei f the yte. Jacbian atix f the yte cul nly be eive in the ingle peating pint f the utput an all f the yte pefance in the whle pei cul nt be hwn. Al, the Pincae ap f analyi f the peiic yte i ecene [1]. he Pincae ap el f tie vaying yte uch a buc an bt c-c cnvete wa btaine [11, 12]. Stability an ynaic analyi f the ppe yte ae invetigate with the cntl cnieatin. Nn-linea phenena uch a inteittent bifucatin an cha in bt cnvete une pea-cuent cntl e wee elle by the Pincae ap [13]. w nn-linea el in the f f icete ap wee eive t ecibe peciely the nn-linea ynaic f bt cnvete f the tw pepective: lw an high fequency egie. Althugh the Pincae ap i a pweful eth in the nnlinea cntl they, it i nt uitable f the yte with tie vaiant utput ignal (inuial utput). Mifie Pincae ap (highe e Pincae ap) appea t be a caniate in uch cae [11]. hi ap i the eult f n equential cnventinal Pincae ap in which the chaacte f each peating pint f the yte in the whle pei the yte, i canne an gathee in the final ifie Pincae ap. In alt all f peviu w, c t c cnvete wee invetigate by the Pincae ap. F a yte with ac utput, uch a ac vaiable active paive eactance yte, the ifie Pincae ap wa al ue in e t ipving f cha phenena by ptial cntl appach [14, 15]. In thi pape, a nvel el bae n agnetizing cuent i initially intuce which i capable f invetigating atuatin in each phae f the achine. hugh the ue f thi elling a pat f achine atuate whe cuent entee the atuatin egin an the phae ae nt pactically atuate. A new extene Pincae ap el f electical achine i then evie by uing the nn-linea cntl they. he ynaical peiic yte can be lineaize aun the ap n the electe plane in the bit f the yte by eiving the extene Pincae ap. hi ap i an effective eth f elling an analyi f the achine in atuatin an unbalance cnitin. hugh the chaacteitic ultiplie f the 126 PRZEGLĄD ELEKROECHNICZNY (Electical Review), ISSN , R. 88 NR 3b/212

2 lineaize extene Pincae ap, tability, cha an bifucatin f the yte can be analyze. In e t evaluate the eth, a cuple f labaty tet eult in which the efficiency f the new el i entate. Matheatical el f the yte Dynaical el f SESCSG Stat an t vltage equatin f a atuate inuctin achine ae: V () t R i () t L i () t Λ () t l (1) t t V () t R i () t L i () t Λ () t l t t whee R, R, L an L ae eitance an leaage l l inuctance iagnal atice f the t an tat wining which have,, l l an l l in thei iagnal eleent, epectively. he vect V () t, V () t, i () t an i () t epeent the vltage an cuent f tat an t wining, epectively. Al, Λ () t an Λ () t ae flux linage vect f tat an t wining, which can be futhe expee by atix f a the fllwing: Λ() t a() t b() t c () t (2) Λ () t a () t b () t c () t Uing the chain ule f eivatin in Eq. (2), we can wite: a () t ia () t ia () t t b () t ib () t Λ() t t ib () t t c () t ic () t i () (3) c t t a () t ia () t ia () t t b () t ib () t Λ () t t ib () t t c () t ic () t ic () t t i t t i () t ae the agnetizatin cuent f whee () a c tat an t wining, epectively. Accing t the electical cicuit appach, agnetizatin eactance f each phae i the lpe f it agnetizatin flux t the cepning agnetizatin cuent. Obviuly, in the linea yte thi lpe i cntant. hu we have: (4) () t j L ( i ) j a,, c j i () t j whee L ( i ) i agnetizatin eactance i.e. the lpe f i in the atuatin cuve. Subtituting Eq. (4) int Eq. (3), we have: Λ() t L ( i ) i () t t t (5) Λ() t L ( i ) i () t t t whee (6) L L ( i ) iag L ( i ) L ( i ) L ( i ) ( i ) iag L ( i ) L ( i ) L ( i ) a b c a b c an i () t ( i () t i () t i ()) t a b c (7) i () t ( i () t i () t i ()) t a b c Eq. (1) an (5) can ecibe atheatical el f an inuctin achine, if thee ae elatinhip between teinal an agnetizatin cuent f the t an tat wining. In the next ectin, thee elatin will be btaine in a SESCSG. Vltage equatin in SESCSG Fig. 1 hw a thee-phae cnnectin iaga f a SESCSG with an excitatin capacit ban c an a la ban. If the t an tat eie wining f the inuctin achine ae aange in Y cnnectin with fee neutal pint, the thi line cuent i 3 (t) can be expee by: (8) i t = - 3 i1 t + i2 t hu, we can wite: (9) i () t i () t i () t ( i () t i ()) t Al, t an tat phae equence ae ppite, thu we can wite: i () t (1) i () t ( i () t i ()) t i () t Futhe, accing t Fig. 1, the utput teinal vltage f the SESCSG can be btaine a: v () t v () t v () t a a a (11) v () t v () t v () t b b c v () t v () t v () t c c b whee v () t, v () t an v () t ae thee phae utput a b c teinal vltage t neutal pint in the achine ie. In vect f, Eq. (11) can be witten a: V() t V () t V () t t (12) whee V t (13) t v t v t v t a b c () () () () 1 (14) 1 1 ubtituting Eq. (1) int Eq. (11), we have: (15) V() t R i() t L i() t ( Λ () t Λ ()) t t c l t t whee (16) R c l l l l (17) L l l l l l l l l l l l l l Al, the KCL cntaint in the la neutal pint f the yte can be expee a: PRZEGLĄD ELEKROECHNICZNY (Electical Review), ISSN , R. 88 NR 3b/

3 Fig.1. SESCSG Cnfiguatin. (18) ( v ( ) ( ) ( )) 1 t v 2 t v 3 t t 1 ( v1 ( t ) v2 ( t ) v3 ( t )) c whee v () t, v () t an v () t ae phae vltage in the la ie. Eq. (18) i a linea autnu iffeential equatin which will have ze lutin, when the initial cnitin f the iffeential ae ze. hu, if u f the phae vltage at t i ze, we can wite: (19) v () t v () t v () t ft On the the han, elatin between la phae vltage an neutal ptential can be witten a: v () t v () t v () t 1 a N (2) v () t v () t v () t 2 b N v () t v () t v () t 3 c N whee v () t i a vltage f the neutal pint t gun. By N ubtituting Eq. (19)int Eq. (2), we have: (21) 1 v () t ( v () t v () t v ()) t N a b c 3 Meve, Eq. (2) can be witten a: (22) V () t V() t Q v () t whee t N (23) Q (24) V () t v () t v () t v () t Cnequently, by applying KCL law t the la teinal, we can wite: 1 1 (25) V t C R V() t i () t t t t whee C an R ae capacitance an eitance iagnal atice f the la which have c an in thei iagnal eleent, epectively. Relatin (1) an (25) illutate a nn-linea ynaical el f the yte which i hwn in Fig. 1 Al, Eq.(1) an (25) hw that the inepenent egee f the nn-linea iffeential equatin i 4 an ean that the yte inepenent vaiable ae i () t, i () t, 1 2 v () 1 t an v () t. Eventually the tate 2 vaiable atice ae: (26) V() t ( v () t 1 v ()) t 2 i() t ( i () t i ()) t 1 2 Satuatin an agnetizatin cuent buil up utput vltage ac the geneat teinal, excitatin ut be pvie by a uitable capacit ban cnnecte ac the geneat teinal. When an inuctin geneat fit tat t un, the agnetic eiual in the t cicuit puce a all vltage. hi all vltage puce a capacit cuent flw, which inceae the vltage an fth until the vltage i fully built up. Al, the elatin between the agnetizatin an wining cuent can be fun by q efeence el f an electical achine. Fig. 1 hw iect () an quaatue (q) axe ue f eteining the iect an quaatue cpnent f the agnetizatin cuent in the tatinay efeence fae. Since the thee phae cuent f the achine cul nt be axiize iultaneuly. Cnequently, the pat f ai gap an wining can be atuate that i lcate une axiu cuent phae. he achine elling ut theeby be pefe by the agnetizing cuent f each wining, epaately. hu, accing t Fig. 1 an uing Eq. (9) an (1), the agnetizatin linage cuent f t efe t tat ( i ) an tat ( i ) in a axi can a a be witten by: (27) 3 i () t i () t a 1 2 i () t 3 ( i ()c( t ) i ()in( t )) a whee i the t wining tun t tat wining tun ati. tal agnetizatin linage cuent in the a (the phae a f tat wining) can be btaine by: i () t i () t i () t a a a (28) 3 i ( t)( 3c( )) 3 i ( t)in( ) Liewie, we can fin agnetizatin linage cuent f b an c tat wining. In vect f, the thee-phae agnetizatin cuent can be peente a: (29) i () t G ( )() i t whee 128 PRZEGLĄD ELEKROECHNICZNY (Electical Review), ISSN , R. 88 NR 3b/212

4 3 3c( ) 3in( ) (3) ( ) G 3c( ) 3 c( ) in( ) 3 c( ) Siilaly, vect f f t agnetizatin cuent bae n teinal cuent f achine i 1 () t an i () t can be 2 expee a: (31) i () t G ( )() i t whee c( ) in( ) (32) G ( ) in( ) c( ) c( ) c( ) Accing t Eq. (5), the patial eivatin f the phae agnetizatin cuent ut be cntucte. Al, egaing Eq. (29) an (31), the t an tat agnetizatin cuent vect ae epene n the i 1 () t, i 2 () t an () t.heefe, patial eivatin f the phae agnetizatin cuent f the t an tat wining i: i i i i i 1 2 i () t t i t i t t 1 2 (33) i i i i i 1 2 i () t t i t i t t 1 2 Eq. (33) epeent the patial eivatin f tw agnetizatin cuent vect i () t an i () t ubject t a cala functin i 1 () t, i 2 () t an () t.hu, by uing Eq. (31) an (29), we can eaange Eq. (33) t atix f a: (34) i () t G ( ) i() t ( G ( ))() i t t t t i () t G ( ) i() t ( G ( ))() i t t t t Uing patial eivatin f G ( ) an G ( ) in Eq. (3) an (32), we can btain atix cefficient in Eq. (34). Subtituting Eq. (34), (5) int (15), the vltage equatin f a SESCSG in tate pace f can be cntucte a: (35) V() t R (, i t)() i t L (, i t) i () t t t t t whee (36) 1 R (,) i t R ( I Q Q) [ L ( i ) t c 3 t G ( ) L ( i ) G ( )] Fig.2 Magnetizing Cuve (37) 1 L (,) i t L ( I Q Q)[ L ( i ) G ( ) t l 3 L ( i ) G ( )] If the neutal pint f achine i cnnecte t la, the neutal vltage will be equal t ze ( v ( t) ) in Eq. (21). N In thi cae, the vect f Q equal t ze. It can be beve that the cuent f the achine ae epenent n eitance, inuctance, t pitin an pee f the achine, althugh agnetizing inuctance i epenent n the agnetizing cuent, the yte equatin ae nnlinea yte an inteelate. he agnetizatin cuve ( L veu i ) ay be btaine by uing ynchnu pee tet. N-la teinal vltage f the inuctin geneat i the inteectin f the geneat' nn-linea agnetizatin cuve with capacit la line [5]. Cnequently, atuatin f the t ce i the eential nee f apppiating peatin f the achine. Othe paaete f the achine can be eteine f lce t an tat-eitance tet. A typical nn-linea elatin between agnetizing eactance L an agnetizing cuent i i hwn in Fig. 2, he Acengentian cuve i fitte n the expeiental eult with the leat quae appach. In thi cae, agnetizing eactance can be peente by: (38) L ( i ) L L ( i ) in line whee 1 i i i at at (39) ( i ) tan the 1 ( i i ) at he paaete in Eq. (38) an (39) f Fig. 2 ae: L 25 L i.18 in line at Nn-linea Augente Dynaical Mel f SESCSG A it wa icue ealie, the final elatinhip i Eq. 35) an hee, i () t an i () t t be eplye a 1 2 inepenent vaiable in the Eq. (25) an (35) ( itting v () t an i 3 3 () t ). Meve, ince the cefficient atice in the utput vltage equatin (Eq. (25)) ae iagnal, thu thi equatin can be euce t a 2 2atix by itting v () t an i () t.finally, by uing Eq. (25) an (35), 3 3 augente ynaical nn-linea tate pace equatin f a SESSGS can be expee a: PRZEGLĄD ELEKROECHNICZNY (Electical Review), ISSN , R. 88 NR 3b/

5 (4) X() t A( X(),) t t X () t t whee () X t i () t V () t (41) 1 1 L (,) i t R(,) i t L (,) i t (42) AX ( ( t), t) C C R Ri (,) t an Li (,) t ae 2 2euce atice in Eq. (36) an (37). Finally, cplexity f equatin an the wie ange vaiatin f utput vaiable caue the yte nt t have an peating pint, theeby the yte utput ae lage inuial peiic an cannt be lineaize by cnventinal eth uch a all ignal analyi appach. In the fllwing a Pincae' ap which i an effective technique f peiic yte, will be intuce. Pincae' ap in peiic yte he Pincae' ap i a tana eth f ynaical yte they t tuy the ynaic f peiic yte [16]. he ain iea f thi appach i t beve the yte tate nce pe cycle. Cnie a nn-linea ynaical peiic yte a: (43) X() t AX (), t t t whee (), t t pei (i.e. ( t), t ( t), t A X i an n yte vect with a fixe A X A X ). By applying a uy vaiable () t t Eq.(43), we have: X() t AX(), t () t (44) t 2 () t t Eq.(44) peent a ( n1) ( n1) autnu ynaical yte with pei. In thi yte, we can fin a n n pen pace V with a cle ub-pace S in it (V S ) that any yte tate X() t an () t in V tanvee t S in tie cle t. he ub-vect pace S ay be electe a: X (45) S t t t t n1 11 (), () X(), (), () t, 2 A Pincae' ap P f the ynaical yte (Eq. (43)) can be efine a: (46) P : V S X(), t (), t a X whee (), t a X i a tanitin atix f the yte which can be eteine by nueical eth in nnlinea cae. Eq. (46) can be etate a: X( t) X( t), P X ( t) (47) X(), t i a ub-et f X (), t a. If we call the yte tate clliin t V at tie t a X, the Pincae' ap f the yte (tbcpic ap) in Eq.(47) can be witten a: (48) X P ( X ) 1 Fig.3. Obit f a Syte an a Selecte Sectin. Al, if X i a teay tate value f the tate X, then we have: (49) X P( X ) A it i hwn in Fig. 3, the aple equence ae X, X, X an n. Meve, the aple-ata 1 2 appach ay be ue f tability analyi f the yte [14,15]. In Fig. 3, S i a Pincae' ectin that i cutting bit f the yte tajecty at X, PX ( ), PX ( ),... 1 cnequently eult f the Pincae' ap appea a clliin pint n S plane. hee pint will cnvege t a fixe pint X if the yte bece table in neighbuh f a fixe pint in S. In thee cae, we can lineaize the yte at the fixe pint by applying the petubatin technique. Al, the tability analyi f a peiic lutin i eteine by it chaacteitic ultiplie naely Flquet' ultiplie. Chaacteitic ultiplie ae a genealizatin f the eigenvalue at an equilibiu pint i petube f it X. If the tate X() t teay tate value by X, a ynaical tanient will appea in the yte which can be btaine by applying the Pincae' ap ucceively t X. In paticula, the linea ap can be peente a: (5) X P ( X ) X 1 whee PX ( ) i the Jacbian' f the ap at X which gven the evlutin f a petubatin X in a neighbuh f the fixe pint. Suppe that q i the ienin f the Pincae' ectin S an the eigenvalue f P( X ) i C, with cepning eigenvect i n C f i 1,2,..., i q. Auing that the eigenvect ae itinct, the bit f P with initial cnitin X fit e. X X X (51) X ( P( X )) X X C... C q q q X i, whee C C ae cntant btaine f the initial i cnitin. Afte uing the peviu equatin f the ifie Pincae' ap, the yte petubatin ae then 13 PRZEGLĄD ELEKROECHNICZNY (Electical Review), ISSN , R. 88 NR 3b/212

6 invetigate f the whle pei an the Jacbian atix can be fun. A a eult the ultiplie ecibe the cnitin f the yte. Finally it ut be cniee that the Pincae' ap w in the icete pace an the yte have t be cnvete t icete el [1]. Nnlineaity te f the yte nt peit u t ue cnventinal eth accingly, a eth will be intuce in the next ectin. Pincae' ap f SESCSG A entine in the peviu ectin ectin, in e t eive a Pincae' ap f the yte, tajecty f the yte ut be fun. A cnventinal eth i integatin f tate pace equatin f the achine abve the whle pei f the yte [1]. he cnventinal integatin eth pve t be incapable an the tapezial integatin eth i thu ppe. apezial eth in the electical achine analyi i a well nwn iea. It i the bae f the phae ain eth [17]. F the pupe f lving the achine' electical quantitie in the tie ain, the tate pace equatin ae change int icete f with the tapezial ule f integatin. Dicete fat f equatin i cn in the phae ain eth an the Pincae' ap eth. By applying tapezial integatin ule t Eq. (4), we can fin: t X( t t) I A( X( tt), tt). 2 (52) t I AX(), t tx() t 2 whee I i a 4 4 ientity atix. Al, accing t the atuatin functin in Eq. (38), the inuctance f the achine ae epenent n wining cuent an vice vea. Cnequently, the ight han ie f Eq. (52) ha eleent which epen n the yte tate at ttan ae nt cpatible with the Pincae' ap efine in Eq. (48). In e t vecing thi pble an t fin ecuive Pincae' ap f the achine, we can peict the tate f the yte by the Eule eth [18] a: X ˆ ( t t ) I ta ( X ( t ), t ) X ( t ) If in the ight han ie f the Eq. (52), X ( tt) i eplace by X ˆ ( tt) in Eq. (53), the ecuive elatin f eteining the tate vaiable f the yte will be bece inepenent yte tate at t t. hu, we have: (54) X( tt) X( t), tx ( t) whee 1 t (), ( ˆ X t t I A X( t t), t t). 2 (55) t I AX(), t t 2 he ecuive ap f Eq. (54) will have a ppe eult, when the tep f integatin ( t ) i electe vey alle than pei f yte. hu iila t Eq. (54) an with t= an t, the elatinhip between X( tnt) an X( t( n1) t) can be peente: (53) 1 n n1 n1 X(( ) ) X(( ) ),( ). (56) n1 X(( ) ) Accing t Eq. (56), we have t btain a tie invaiant ecuive ap f tate vaiable. F thi pupe, if the SESCSG peate in a cntant pee, we can ientify a elatin between an t. In thi cae, we can wite t an tie equence tt, tt, 2 t, an tntcan eplace by, 2,, 2n. hu, by ubtituting with t in Eq. (56), a ecuive ap f tate vaiable f t( n1) t t tnt can be expee a: (57) 2( n 1) X, n X n1 X n1 whee X i tate vect f yte at tn t. n Cpaing Eq. (57) with (48), the Pincae' ap f the yte can be expee a: 2( n 1) X, 1 n 1 (58) X X n1 PXX hu, the final icete ap change the pble f analyzing the tability f the bit int ne f the tability f a fixe pint. Afte each t, all tate vaiable f the atuate achine ae calculate accing t the peviu tate vaiable. Unbalance an unyetical yte (with initial cnitin auptin) with eep nn-lineaity can be elle by the ue f thi eth. Cha an bifucatin f the yte can be ecgnize an cntlle. A entine chaacteitic ultiplie pitin in the cplex plane eteine the tability f the yte. Ablute value f the chaacteitic ultiplie hw the yte tability cnitin lie tability, bifucatin, cha an intability. Cae Stuy An expeiental et-up ha been pepae f iulatin eult valiatin. he tet yte cnit f a wune t inuctin achine whe paaete ae given in the table I. In Fig. 4-a, a ynchnu t i hwn which i cuple with the inuctin achine in which an avance invete, i ue t upply the ynchnu t, agnitue an fequency f the ynchnu t ae ajute by the invete. he invete utput vltage an fequency ati ut be ept cntant thughut the tet. t i electe equal t.6283 ec all enugh t cve all ignal phenena. In e t iplify the vltage builing at ninal netw fequency with all excitatin capacit the geneat-t et i pvie which peate a a t with SW1 witch by pening SW1, it act a a geneat. F fit expeiental tet cnitin, the achine i cnnecte t 16 with excitatin capacit C = 1 F. PRZEGLĄD ELEKROECHNICZNY (Electical Review), ISSN , R. 88 NR 3b/

7 (a )Expeiental etup. (b) Siulatin eult f 1 () Fig. 4. Cae tuy etup an eult with c 1 F. i t an v () t. (c )Expeiental eult fi 1 1 () t. Siulatin hw that, achine w in the table cae but pea f ignal change ue t atuatin. Fig. 4-b an c hw the fit e Pincae' ap eult f an utput cuent, vltage an expeiental eult f utput tat cuent change in fequency in the tw e can be een. It can be een that the tie vaiable yte cannt be pecifie with the fit e Pincae' ap. Accing t Eq. (52) aue that the yte peating pint i: (59) Xt ( ) (.11,.6, 3.848,6.77) he fit e Pincae ap that tanfe the initial pint X() t t next pint with tie tep t i: (6) ( X( t), t) In the ecn tep with the ae tie tep an the cepning ap the yte peating pint ve t the next pint. Othe pint hul be calculate an afte cacaing the ap the highe e ap i appeae that tanfe fit pint t final pint. 2( n 1) PX ( ), X n1 n (61) he teay tate yte peating pint with the final ap can be btaine by: (62) X ( t) Chaacteitic ultiplie f the ap ae: (63) 6 6 ( P).11,.12, 7.81, j4.451 whee j inicate iaginay pat f the chaacteitic ultiplie. It can be beve that the yte i in the table egin but in thi cnitin, the excitatin capacit value i nt a pactical value an i vey all t buil the ninal vltage. he eult f inceaing the excitatin capacit t c 15F ae hwn in the Fig. 5. he vltage an cuent f phae a ae hwn in Fig. 5-a. In thi peatin pint, the Jacbian atix an it chaacteitic ultiplie ae: (64) P (65) ( P ).564,.1 j.2,.2 Liewie, f the excitatin capacitance c 35 F, the Jacbian f the ap an it chaacteitic ultiplie ae: P PRZEGLĄD ELEKROECHNICZNY (Electical Review), ISSN , R. 88 NR 3b/212

8 a) vey enitive t initial cnitin an thei value. Al, yte tajecty i hwn in Fig. 7. he tat pint f bifucatin i tate f c 2 F. he yte with thi value f excitatin capacit ha the chaacteitic ultiplie equal t 1. Fig. 8 hw bifucatin iaga f the yte with ifie Pincae ap. It can be een that when value f excitatin capacit ae inceae the yte ve in chatic egin. b) a) b) Fig. 6. Siulatin an expeiental eult f C = 15 F : a) Cuent an vltage f Phae, b) Expeiental eult i 1 (t). a) Fig. 7. ajecty f the yte: i 2 (t)-v 2 (t) tajecty f C = 1 F, b) i 2 (t)-i 1 (t) tajecty f C = 35 F b) Fig. 8. he SESCSG Bifucatin Diaga. Fig. 6. Siulatin an expeiental eult f C = 35 F : a) Cuent an vltage f Phae, b) Expeiental eult i 1 (t). (67) ( P ) 1.176,.1 j.28,.6 It ean that, the yte i bifucate an the type f it bifucatin i Pitchf bifucatin [1]. Fig. 6-a an 6-b hw the cuent an vltage f phae a with the fit e Pincae ap an expeiental eult. Fig. 6 inicate that with thi capacitance value, the yte behaviu ha bifucate an ha chatic anne. he yte i in the chatic egin an nn-linea e. hi in f yte i able 1. Machine paaete an value Deciptin Paaete Value Relate Vltage v n 38 V Rate Pwe p n 175 W Magnetizatin eactance X 472 Ω Ce l eitance R 16 Ω Stat eitance 42 Ω Rt eitance 75 Ω Stat leaage eactance X l 11.2 Ω Rt leaage eactance X l 11.2 Ω Rt/tat cil tun ati K.5873 Suce intenal eitance R uce 15 Ω Suce intenal inuctance L uce.5 H PRZEGLĄD ELEKROECHNICZNY (Electical Review), ISSN , R. 88 NR 3b/

9 Cncluin In thi pape, the analytical fit e an ifie Pincae' ap f the elf excite eie cnnecte ynchnu geneat wa eive f the fit tie bae n the nvel agnetizatin cuent el. SESCSG wa electe becaue it w in the nn-linea egin (atuate ce). Patial atuatin can be invetigate in the abc quantitative eth with the pape el. Chaacteitic ultiplie f the lineaze ap aun the peatin ute ecibe cae by changing in the capacit value, untable, bifucatin an cha cae. he el f SESCSG can be efine in the atuate cae with capacit an chaacteitic ultiplie f the yte ae eive. he in f bifucatin can be ientifie by the ue f elling eth. Expeiental eult in the electical achine labaty ientifie by the ue f the new el' eult. REFERENCES [1] Muthy, S. S., Singh, B., Gupta, S., an Gulati, B. M., Geneal Steay-State Analyi f hee-phae Self-Excite Inuctin Geneat Feeing hee-phae Unbalance La/Single- Phae La f Stan-Alne Applicatin, IEE Pc. Geneatin, aniin an Ditibutin, 15(1), pp , Ap. 23. [2] Faet, F. A., Palle, B., an Sie, M. G., Full Expanable Mel f Paallel Self-Excite Inuctin Geneat, IEE Pceeing Electic Pwe Applicatin, 152(1), pp , Jan. 25. [3] Jain, S. K., Shaa, J.D., an Singh, S. P., anient Pefance f hee-phae Self-Excite Inuctin Geneat Duing Balance an Unbalance Fault, IEE Pc. Geneatin, aniin an Ditibutin, 149(1), pp. 5-57, Jan 22. [4] Huang, M. Y., anwang, L., Suen Dicnnectin f an Excitatin Capacit n anient Synchnu Geneat Pefance f a Self-Excite Seie Cnnecte Synchnu Geneat, Pwe Engineeing Sciety Winte Meeting IEEE, 1(1), pp , Jan. 2. [5] Mutafa, A. S., Mhaaein, A. L., an Raha, E. M.,Applicatin f Flquet hey t the Analyi f Seie- Cnnecte Wun-Rt Self-Excite Synchnu Geneat, IEEE an. Enegy Cnvein, 8(3), pp , Septebe [6] Mutafa, A. S., Mhaaein, A. L., an Raha, E. M., Analyi f Seie-Cnnecte Wun-Rt Self-Excite Inuctin Geneat, Electic Pwe Applicatin, IEE Pc., 14(5), pp , Sep [7] Mhaaein, A. L., Yuef, H. A., an Deuy, Y.G., Seie- Cnnecte Self-Excite Synchnu Geneat: Steay State an anient Behavi, IEEE an. Enegy Cnvein, 14(4), pp , Dec [8] Wang, Y. J., an Huang, Y. S., Analyi f a Stan-Alne hee-phae Self excite Inuctin Geneat with Unbalance La Uing a w-pt Netw Mel, Electic Pwe Applicatin, IE, 3(5), pp , Septebe 29. [9] Chan. F., Wang W., an Lai L. L., Fiel Cputatin an Pefance f a Seie-Cnnecte Self-Excite Synchnu Geneat, IEEE an. MAGNEICS, 46(8), pp ,Aug. 21. [1] Banejee, S., an Veghee, G. C. Nn-linea Phenena inpwe Electnic. Attact, Bifucatin, Cha, an Nnlinea Cntl, New Y, Jhn Wiley-IEEE Pe, 21. [11] Mazue, S. K., Nayfeh, A. H., an Byevich, D., An Invetigatin int the Fat an Slw-Scale Intabilitie f a Single-phae Biiectinal Bt Cnvete, IEEE an. Pwe Electnic, 18(4), pp , July 23. [12] Mazue, S. K., Nayfeh, A. H., an Byevich, D., heetical an Expeiental Invetigatin f the Fat- an Slw-Scale Intabilitie f a c-c Cnvete, IEEE an. Pwe Electnic, 16(2), pp , Ma 21. [13] Zhang, H., Ma, X., Xue, B., an Liu, W., Stuy f Inteit-tent Bifucatin an Cha in Bt PFC Cnvete by Nn-linea Dicete Mel, Cha, Slutin an Factal, 23(2),pp , Jan. 25. [14] Shaiataa, S. M., an Nazazaeh, J., Mifie Pincae Map f Vaiable Active Paive Reactance f Stability Evaluatin With Cnieatin f Capacit Me, 7th IEEE cnfeence n pwe electnic an ive yte, pp ,hailan, May 27. [15] Shaiataa, S. M., an Nazazaeh, J., Optial Output Feebac f Cha Ipveent in AC Vaiable Active Paive Reactance, IEEE Intenatinal Cnfeence n Inutial echnlgy ICI, pp. 1-6, China, Ap. 28. [16] Wiggin, S., Intuctin t Applie Nn-linea Dynaical Syte an Cha, Spinge-Velag, New Y, Inc., 199. [17] Riguez, O., an Meina, A., Efficient Methlgy f the anient an Peiic Steay-State Analyi f the Synchnu Machine Uing a Phae Cinate Mel, IEEE an. Enegy Cnvein, 19(2), pp , June 24. [18] Naaua, J., Applie Nueical Meth with Sftwae, 1 t eitin, Pentice Hall, Auth: Seye Mhaa Shaiataa, PhD Stuent f Science an Reeach Banch, Ilaic Aza Univeity, Ahfi Efahani Highway, Pna, ehan, Ian, Shaiataa@IEEE.g. Jalal Nazazaeh, Aciate pfe f Shahe Univeity, Depatent f Electical Engineeing, Shahe Univeity, P. O. Bx , ehan, Ian, E-ail: Nazazaeh@hahe.ac.i. Meha Abei, Pfe f Ai Kabi Univeity f echnlgy, P.O. Bx 15914, ehan, Ian, Eail: Abei@aut.ac.i. 134 PRZEGLĄD ELEKROECHNICZNY (Electical Review), ISSN , R. 88 NR 3b/212