SIMPLIFICATIONS OF SYNCHRONOUS MACHINE PARAMETERS IN STABILITY STUDIES AND REACTIVE CAPABILITY LIMITS

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1 Rgua pap SIMPIFICATIONS OF SYNCHRONOUS MACHINE PARAMETERS IN STABIITY STUDIES AND REACTIVE CAPABIITY IMITS Sjan MAZAICA, Mćo GAĆANOVIĆ 2 Abstact: In ths pap, bfy, th pobm of pow sytm stabty s cons. Aft ths ntoucton, som smpfcatons u fo th psntaton of synchonous machns n stabty stus a scuss. Aso cons a vaous gs of appoxmatons that can b ma to smpfy th machn mo, mnmzng ata umnts an computatona ffot. Kywos: Synchonous machn paamts, Stabty, Ractv capabty mts, akag fux. INTRODUCTION Pow systm stabty s a sng pobm; howv, t s mpactca to stuy t as such. Instabty of a pow systm can tak ffnt foms an can b nfunc by a w ang of factos. Anayss of stabty pobms, ntfcaton of ssnta factos that contbut to nstabty, an fomaton of mthos of mpovng stab opaton a gaty factat by cassfcaton of stabty nto appopat catgos. Ths a bas on th foowng consatons: - th physca natu of th sutng nstabty; - th sz of th stubanc cons; - th vcs, pocsss, an tm span that must b takn nto consaton n o to tmn stabty; an - th most appopat mtho of cacuaton an pcton of stabty. Pow systm stabty can b fn as abty to man n opatng ubum (ubum btwn opposng focs), an b cassf nto: - ang stabty (abty to mantan synchonsm, tou baanc of synchonous machns) - votag stabty (abty to mantan stay accptab votag, actv pow baanc) Wh cassfcaton of pow systm stabty s an ffctv an convnnt mans to a wth th compxts of th pobm, th ova stabty of th systm shou aways b kpt n mn. Soutons to stabty pobms of on catgoy shou not b at th xpns of anoth. It s ssnta to ook at a aspcts of th stabty phnomna an at ach aspct fom mo than on vw pont. SIMPIFICATIONS ESSENTIA FOR ARGE- SCAE STUDIES Concnng p unt stato votag uatons []: = p ω R () a = p ω R (2) a t s ncssay to ngct th foowng fom abov uatons, fo asons of stabty anayss of ag systms: - th tansfom votag tms, p an p ; - th ffcts of sp vaatons. Th asons fo an th ffcts of ths smpfcatons a scuss bow. Ngct of stato p tms Th p an p tms psnt th stato tansnts. Wth ths tms ngct, th stato uantts contan ony funamnta funcy componnts an th stato votag uatons appa as agbac uatons. Ths aows th us of stay-stat atonshps fo psntng th ntconnctng tansmsson ntwok. Fgu shows th systm stu. It conssts of a sant-po gnato connct to an nfnt bus though two tansmsson ns. Th stubanc app s a th-phas shot-ccut at th snng n of on of th ns, ca n.9s by soatng th faut ccut. Th sponss of gnato vaabs comput wth an wthout ncuson of th stato p tms a compa n Fgus 2, 3 an 4 []. Fgu Systm confguaton an paamts Whn th stato p tms a omtt, w s fom Fgu 2 that an hav ony unctona componnts; ths cospon to th funamnta funcy componnt of phas cunts. Th sutng a-gap tou s unctona an sma n magntu; t s u to th stato sstanc osss. On th oth han, whn th p tms a ncu, an contan funamnta funcy (5 Hz) componnts, whch cospon to th c offst n th phas cunts. Ths n tun sut n th foowng componnts of a-gap tou: - a funamnta funcy oscatoy componnt, u to ntacton wth th oto f; - a unctona componnt, u to oto sstanc osss caus by th funamnta funcy cunts nuc n th oto. Unvsty of Banjauka, Facuty of Ectca Engnng, Pat 5, 78 Banjauka, Bosna an Hzgovna, E-ma: bagus@bc.nt 2 Unvsty of Banjauka, Facuty of Ectca Engnng, Pat 5, 78 Banjauka, Bosna an Hzgovna, E-ma: bchy@bc.nt

2 Snc, fo ag-sca stabty stus, t s ncssay to ngct th p tms, th ffcts of th unctona (c) bakng tou an th oscatoy tou may b stmat an ncu n th cacuatons. Wth th stato tansnts ngct, th p unt stato votag Euatons an 2 appa as agbac uatons: = ω R (3) a = ω R (4) a Fgu 2 Effct of ngctng stato tansnts on a-gap tou an - componnts of stato cunts Th unctona componnt of tou, u to oto sstv osss, can b ut hgh an has a bakng ffct. Thfo, t s f to as th c bakng tou; ts ffct of th oscatoy componnt s to cat th oto ung th fst haf cyc an to accat t to na ts nta sp ung th scon haf cyc, an so on ung subsunt cycs. Th nt ffct of th oscatoy tou s thfo a ucton of th man sp of th oto [2]. Th ova ffct of ths two componnts fo a cos-up smutanous th-fas faut cou b ag nough to ntay caus taaton of th oto o a back swng. Ths cou hav a sgnfcant bnfca ffct on systm stabty as sn fom pots of sp vaton an oto ang n Fgus 3 an 4. Fgu 3 Effct of ngctng stato tansnts on sp vatons Fgu 4 Effct of ngctng stato tansnts on oto ang swngs Ngctng th ffct of sp vaatons on stato votags Anoth smpfyng assumpton nomay ma s that th p unt vau of ω s ua to. n th stato votag uatons. Ths s not th sam as sayng that sp s constant; t assums that sp changs a sma an o not hav a sgnfcant ffct on th votag. Th assumpton of p unt ω =. (.., ω =ω o a/s) n th stato votag uatons os not contbut to computatona smpcty n tsf. Th pmay ason fo makng ths assumpton s that t countbaancs th ffct of ngctng p, p tms so fa as th owfuncy oto oscatons a concn. Wth p unt ω =., th stato votag uatons ( an 2) uc to = R (5) a a = R (6) Thus, p unt oto votag, stato fux nkag, an oto fux nkag uatons bow [] man th sam, as sam as p unt a-gap tou xpsson: P unt oto votag uatons: = p R (7) f f f f p R p R p 2 R22 = (8) = (9) = () P unt stato fux nkag uatons: = () ( a ) a f a ( a ) a a2 = (2) = (3) P unt oto fux nkag uatons: (4) f = ff f f = f f = a2 2 = a 222 a (5) a (6) a (7) a P unt a-gap tou: T = (8) In wtng Euatons 6 an 7, w hav assum that th p unt mutua nuctanc 2 s ua to a. Ths mps that th stato an oto ccuts n th -axs a nk a sng mutua fux psnt by a. Ths s

3 accptab bcaus th oto ccuts psnt th ova oto boy ffcts, an actua wnngs wth physcay masuab votags an cunts o not xst. SIMPIFIED MODE WITH AMORTISSEURS NEGECTED Th fst o of smpfcaton to th synchonous machn mo s to ngct th amotssu ffcts. Ths mnmzs ata umnts snc th machn paamts at to th amotssus a oftn not ay avaab. In aton, t may contbut to ucton n computatona ffot by ucng th o of th mo an aowng ag ntgaton stps n tm-oman smuatons. Wth th amotssus ngct, th stato votag uatons 5 an 6 a unchang. Th manng uatons (7 to 7) smpfy as foows. Fux nkags: = (9) a f = (2) = (2) f a ff f Roto votag: = p R (22) f f f f o p = R (23) f f f f Euaton 23 s now th ony ffnta uaton assocat wth th ctca chaactstcs of th machn. In th abov uatons a uantts, ncung tm, a n p unt. CONSTANT FUX INKAGE MODE T Cassca mo Fo stus n whch th po of anayss s sma n compason to (opn ccut tm-constant), th machn mo of pvous Scton s oftn smpf by assumng E (o f) constant thoughout th stuy p- o. Ths assumpton mnats th ony ffnta uaton assocat wth th ctca chaactstcs of th machn. A futh appoxmaton, whch smpfs th machn mo sgnfcanty, s to gno tansnt sancy by assumng X =, an to assum that th fux nk- X ag (assocat wth th -axs oto ccut cosponng to ) aso mans constant. Wth ths as- X sumptons, as shown bow, th votag bhn th tansnt mpanc R has a constant magntu. a jx Th - an -axs uvant ccuts wth ony on ccut n ach axs a shown n Fgu 5 []. Th p unt fux nkags ntf n th -axs a gvn by = (24) a a a f = (25) f a a = (26) f Fom uaton 26 f f a f = (27) f Fgu 5 Th - an -axs uvant ccuts wth on oto ccut n ach axs Substtutng n Euaton 24 gvs ( ) a a = a f a (28) f Raangng to xpss a n tms of f, w fn f = a a f a = a f = (29) (3) Smay, fo th -axs = a a (3) a = Fom Euaton 3, th -axs stato votag s gvn by = R ω = R ω a a ( ) ω=ω =ω =. pu. Substtutng fo a fom Euaton 3 gvs = R = R a = R a a ω X ω ( ) E ω a a ω a a (32)

4 E = ω a (33) Smay, th -axs stato votag s gvn by = R X E (34) a f E = ω a (35) f Wth tansnt sancy ngct ( X = tmna votag s = X j = ( E je ) Ra ( j ) X ( ( E je ) R ( j ) jx ( j ) a ), th stato j ) Usng phaso notaton, w hav ~ ~ E = E (36) t ~ E = E E ~ ( Ra jx ) It je = a j Th cosponng uvant s shown n Fgu 6. Wth oto fux nkags ( f an ) constant, an a constant. Thfo, th magntu of E s E constant. As th oto sp changs, th - an -axs mov wth spct to any gna fnc coonat systm whos R-I axs otat at synchonous sp, as shown n Fgu 7. Hnc, th componnts chang. f f E R Fgu 6 Smpf tansnt mo an Fgu 7 Th R-I an - coonat systms E I Th magntu of E can b tmn by computng ts p-stubanc vau. ~ E ( Ra jx ) It ~ ~ Et = (37) E Its magntu s thn assum to man constant thoughout th stuy po. Snc R a s sma, t s usua to ngct t. Wth th componnts an ach havng a constant magntu, E w hav constant ontaton wth spct to - an -axs, as th oto sp changs. Thfo, th ang of E wth spct to synchonousy otatng fnc axs (R-I) can b us as a masu of th oto ang. Ths mo offs consab computatona smpcty; t aows th tansnt ctca pfomanc of th machn to b psnt by a smp votag souc of fx magntu bhn an ffctv actanc. It s commony ff to as th cassca mo, snc t was us xtnsvy n ay stabty stus. REACTIVE CAPABIITY IMITS In votag stabty an ong-tm stabty stus, t s mpotant to cons th actv capabty mts of synchonous machns. Synchonous gnatos a at n tms of th maxmum MVA output at a spcf votag an pow facto (usuay.85 o.9 aggng) whch thy can cay contnuousy wthout ovhatng. Th actv pow output s mt by th pm mov capabty to a vau wthn th MVA atng. Th contnous actv pow output capabty s mt by th consatons: amatu cunt mt, f cunt mt, an n gon hatng mt. Amatu cunt mt Th amatu cunt suts n an RI 2 pow oss, an th ngy assocat wth ths oss must b mov so as to mt th ncas n tmpatu of th conucto an ts mmat nvonmnt. Thfo, on of th mtatons on gnato atng s th maxmum cunt that can b ca by th amatu wthout xcng th hatng mtatons. F cunt mt Bcaus of th hat sutng fom th R 2 f f pow oss, th f cunt mposs a scon mt on th opaton of th gnato. En gon hatng mt Th ocaz hatng n th n gon of th amatu mposs a th mt on th opaton of a synchonous machn. As xpan bow, ths mt affcts th capabty of th machn n th unxct conton. Fgu 8 s a schmatc of th n-tun gon of a gnato. Th n-tun akag fux, as shown n th fgu, nts an avs n a cton ppncua (axa) to th stato amnatons. Ths causs y cunts n th amnatons, sutng n ocaz hatng n th n gon. Th hgh f cunts cosponng to th ovxct conton kp th tanng ng satuat, so that n akag fux s sma. Howv, n th unxct gon th f cunt s ow an th tanng ng s not E

5 satuat; ths pmts an ncas n amatu an akag fux [3]. Aso, n th unxct conton, th fux pouc by th amatu cunts as to th fux pouc by th f cunt; thfo, th n-tun fux nhancs th axa fux n th n gon an th sutng hatng ffct may svy mt th gnato output, patcuay n th cas of a oun oto machn. a sv stubanc such as a faut on tansmsson facts, oss of gnaton, o oss of a ag oa. Th systm spons to such stubancs nvovs ag xcusons of gnato oto angs, pow fows, bus votags, an oth systm vaabs. Stabty s nfunc by th nonna chaactstcs of th pow systm. If th sutng angua spaaton btwn th machns n th systm mans wthn ctan bouns, th systm mantan synchonsm. oss of synchonsm bcaus of tansnt nstabty, f t occus, w usuay b vnt wthn 2 to 3 scons of th nta stubanc. Cons th systm shown n Fgu, consstng of a gnato vng pow to a ag systm psnt by an nfnt bus though two tansmsson ccuts. An nfnt bus psnts a votag souc of constant votag magntu an constant funcy. Fgu 8 Sctona vw of n gon of a gnato V cuvs an compounng cuvs Th cuv showng th aton btwn amatu an f cunt at a constant tmna votag an wth constant actv pow output s known as a V cuv. Fgu 9 shows so cuvs shown fo th vaus of P (.5,.7,.85 pu). Th ash ns a oc of constant pow facto an a known as compounng cuvs. Each of ths cuvs shows how f cunt has to vay n o to mantan a constant pow facto. Aso shown n Fgu 9 a th actv capabty mts fo on vau of hyogn pssu (45 PSIG). Th th sgmnts AB, BC an CD cospon to th f cunt mt, amatu cunt mt, an n gon hatng mt, spctvy. Snc th chaactstcs shown n Fgu 9 appy at at stato tmna votag, th p unt vaus of amatu cunt an appant pow output a ua, an hnc both a shown aong th onat. Th f cunt pott aong th abscssa s th nomaz vau, wth. pu psntng th f cunt cosponng to at MVA output an pow facto []. Fgu Sng-machn nfnt bus systm W w psnt funamnta concpts an pncps of tansnt stabty by anayzng th systm spons to ag stubancs, usng vy smp mos, gvn by som of pvous smpfcatons. A sstancs a ngct. Th gnato s psnt by th cassca mo an th sp govno ffcts a ngct. Th cosponng systm psntaton s shown n Fgu (a). Th votag bhn th tansnt actans ( X ) s not by E. Th oto ang δ psnts th ang by whch E as E B. Whn th systm s ptub, th magntu of E mans constant at ts pstubanc vau an δ changs as th gnato oto sp vats fom synchonous sp ω. Fgu 9 Th V cuvs an compounng cuvs fo a gnato at at amatu votag AN EEMENTARY VIEW OF TRANSIENT STABIITY Tansnt stabty s th abty of th pow systm to mantan synchonsm (ang stabty) whn subjct to Fgu Systm psntaton wth gnato psnt by cassca mo Th systm mo can b uc to th fom shown n Fgu (b). It can b anayz by usng smp anaytca mthos an s hpfu n acung a basc un-

stanng of th tansnt stabty phnomnon. Th gnato`s ctca pow output s E EB P = snδ = Pmax snδ (38) X P T E E X T B max = (39) Snc w hav ngct th stato sstanc, P psnts th a-gap pow as w as th tmna pow.

Wth a mchanca pow nput of P m, th staystat ctca pow output P s ua to P m, an th opatng conton s psnt by pont a on th cuv. Th cosponng oto ang s δ a.

6 stanng of th tansnt stabty phnomnon. Th gnato`s ctca pow output s E EB P = snδ = Pmax snδ (38) X P T E E X T B max = (39) Snc w hav ngct th stato sstanc, P psnts th a-gap pow as w as th tmna pow. Th pow-ang atonshp wth both tansmsson ccuts n svc (I/S) s shown gaphcay n Fgu 2 as cuv. Wth a mchanca pow nput of P m, th staystat ctca pow output P s ua to P m, an th opatng conton s psnt by pont a on th cuv. Th cosponng oto ang s δ a. Fgu 2 Pow-ang atonshp If on of th ccuts s out of svc (O/S), th ffctv actanc X T s hgh. Th pow-ang atonshp wth ccut 2 out of svc s shown n Fgu 2 as cuv 2. Th maxmum pow s now ow. Wth a mchanca pow nput of P m, th oto ang s now δ b cosponng to th opatng pont b on cuv 2; wth a hgh actanc, th oto ang s hgh n o to tansmt th sam stay-stat pow. Dung a stubanc, th oscaton of δ s supmpos on th synchonous sp ω, but th sp vaton ( ω =δ/t) s vy much sma than ω. Thfo, th gnato sp s pactcay ua to ω, an th p unt (pu) a-gap tou may b cons to b ua to th pu a-gap pow. W w thfo us tou an pow ntchangaby whn fng to th swng uaton. Th uaton of moton o th swng uaton may b wttn as 2 2H δ = P P snδ (4) ω t 2 m max CONCUSION Wth ths smpfcatons, as t has bn shown n th vw of tansnt stabty, t s possb to mo asy unstan a pobm of pow systm stabty. Gnay, ths smpfcatons an ngctons contbut to gath wth ths pobms mo asy, mnmzng aso computatona ffot. REFERENCES [] P. Kunu: Pow Systm Stabty an Conto, McGaw-H, Inc., USA, 994. [2] G. Shackshaft: Effct of Oscatoy Tou on th Movmnt of Gnato Rotos, Poc. IEE, Vo. 7, No., pp , 97. [3] S. B. Fanham, R. W. Swatnout: F xctaton n Raton to Machn an Systm Opaton, AIEE Tans., pp , Dcmb 953. Sjan Mazaca was bon n Banjauka on Dcmb 26, 978. In 997 h no n th Facuty fo Ectca Engnng, Unvsty of Banjauka, Rpubc of Spska, Bosna&Hzgovna. H gauat n 25 wth th fna thss n Pow Systm Anayss ntt Pvnton of Backouts kpng Pow Systm Stabty. Snc Jun 25 h has wok fo Tansmsson Company of Rpubc of Spska as pow systm opato. Snc Juy 26 h has stat to wok n potctv ayng scto. H spaks funty Engsh, Gman an Itaan. D. Mćo Gaćanovć was bon n 952. H s cognz an known ntnatonay as a scntst n th f of app ctostatcs, h has gvn hs contbuton though ogna soutons, whch a patnt n 36 counts thoughout th wo an app n poucton. H cv many pstgous wo-known awas an ctfcats fo hs catv wok. Hnc, h s ncu n th wok of wo goups of catvty, sach an nw tchnoogy n Busss, Moscow, Pttsbugh an oth wo cts. H s aso nvov n sach pojcts fom th f of thotca ctca ngnng n Gmany, Bgum an Russa. P m = mchanca pow nput, n pu P max = maxmum ctca pow output, n pu H = nta constant, n MWs/MVA δ = oto ang, n c. a t = tm, n s

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