A. P. Sakis Meliopoulos Power System Modeling, Analysis and Control. Chapter 5 2 Modeling - Synchronous Machines 2

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1 AF n NCOMPEE bl o Conns om A. P. Sks Mlopoulos Pow Sysm Molng, Anlyss n Conol Chp 5 Molng - Synchonous Mchns 5. noucon 5. h Consucon o Synchonous Mchn 5. h ynmcl Euons o Mchn 5.4 Pks nsomon h P Un Sysm h P Un Sysm o Synchonous Mchn Slcon o s uns Euln Ccus o Synchonous Mchn 5.7 Synchonous Mchn ou Euon S Spc Mol o Synchonous Mchn Sy S Anlyss 4 5. Synchonous Mchn Pomnc Un Fuls Subnsn n nsn nucncs m Consns o Synchonous Mchn Summy o Synchonous Mchn Pms Synchonous Mchn Smpl Mols h Sy S Mol Consn Mn Fl Wnng Flux Mol Consn Mn Fl n mp Wnngs Flux Mol Summy o Smpl Mols On Axs Synchonous Mchn Mol 7 5. Exc Mol n Volg Conol Synchonous Gno Cpbly Cus Summy n scusson 74

2 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos 5.4 Poblms 75 Chp 5 Molng - Synchonous Mchns 5. noucon h synchonous mchn s lmos xclusly us o bulk gnon o lcc pow. h synchonous mchn s on o h mos complx componns o n lcc pow sysm s wll s n mpon componn o n lcc pow sysm. n hs chp w xmn h synchonous mchn n lop ppop mols o ous pplcons. Synchonous mchns mnucu om smll szs o y lg szs up o,5 MVA n o o cplz on conoms o scl nhn n lg pow gnon plns. h gowh n sz bough bou gowh n sgn sophscon. n on, h concn bou h synchonous gno pomnc ung sy s opon n nsns po h mpus o pplcon o sophsc conol schms o h synchonous mchn. Fo xmpl, ung sy s opon, conol loop s c o mnn mnl olg spc lus, whl ung nsns noh conol loop pow sysm sblz s c o sblz h mchn. h suy o hs phnomn n h conol umns c h n o ppop mhmcl mols o h synchonous mchn. h objc o hs chp s o ll n hs n. A b ouln o h chp s s ollows: Fs, h consucon o h synchonous mchn s xmn. h mchnsm o ngy conson s xpln n ms o ncng mgnc ls pouc by cols. hs nlyss ls o h gnl ynmcl uons. hn s shown h n ppop slcon o p un sysm wll l o uln ccu psnon o synchonous mchn. h uln ccus ogh wh h ou uon l o nonln s spc mol o synchonous mchn. h pcul s spc mol mploy h slcs s ss h lcc cun lowng n h mchn. hs mol s hn ulz o suy h sy s pomnc o h mchn, b h pomnc un uls, n c h pomnc ung nsns. n h nl scon o h chp, w nsg h lonshp bwn h s spc mol o h synchonous mchn n ous smpl mols whch my b us o spcc sus,.. h clsscl mol, c. h smpl mols psn n ms o h ssumpons whch mus b ppl Pg Copygh A. P. Sks Mlopoulos 99-6

3 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos o h gnl mol o h on. hs nlyss po nsgh no h pplcbly o h smpl mols. Fgu 5. sgn ls o Sln Pol Synchonous Mchn 5. h Consucon o Synchonous Mchn h wo bsc sgns o synchonous mchn: h sln pol mchn n b h cylncl oo mchn. hs wo sgns llus n Fgus 5. n 5. spcly. n gnl sln pol mchns chcz wh lg numb o pols. h sp o h mchn s pnn upon h numb o pols, ω m P ω n ho h lg h numb o pols h low h sp wll b. hs uns subl wh slow sp pm mos, o xmpl, hyoubn. Cylncl oo mchns, s Fgu 5., chcz wh smll numb o pols o 4 n us wh hgh sp pm mos, o xmpl, sm ubns. h oo s cyln on whch slos h bn ng o ccp h ss o h l col. Fo h llus wo pol mchn wo ss o slos u. n gnl o P-pol mchn P-ss o slos wll b u. h l col s sbu n h oo slos n such wy h c cun hough h col wll pouc mgnc lux nsy n h gp whch hs ppoxmly snusol sbuon. h mxmum o h mgnc lux occus long h -xs. n noml opng conons h oo, h l Copygh A. P. Sks Mlopoulos 99-6 Pg

4 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos col, n h -xs ong wh h synchonous mchncl sp. h oo n h l wnng schmclly psn wh h nucnc. Fgu 5. sgn ls o h-phs Synchonous ubogno n lg synchonous mchns h oo bs noh s o wnngs, h mp wnngs. hs sho ccu wnngs whch plc n slos cu on h suc o h oo. ung noml opng conons hs wnngs nc bcus, s wll b shown, hy lnk consn mgnc lux n hus no olg wll b nuc n hs cols. How ung nsn conons, lcc olg n cun s nuc n hs wnngs. A ou s gn n con such s o bng h sysm bck o synchonous sp. ho hy nouc mpng o h sysm, whch s xmly bncl o h pomnc o h mchn ung nsns. Slos sml o hos o h oo ng n h so s llus n Fgu 5.. h ss o wnngs plc n hs slos: phs A, phs n phs C wnngs. h gu lluss h ss o slos whch cy h phs A wnngs. No h ch s occups 6 o h so. n ly, h sgn o h so s mo complc hn h on llus n Fgu 5.. Fo xmpl h s o slos o phs A olps wh h ss o slos o phss n C. hs s on n such wy s o ch snusol sbuon o h mgnc lux ns h gp u o h cun lowng n ny on o h h phss. Fo smplcy, how, hs spc o h sgn s no shown. Pg 4 Copygh A. P. Sks Mlopoulos 99-6

5 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos oo mgnc xs o -xs sony nc Fgu 5. Schmc psnon o h Mn Wnngs o A Synchonous Mchn Fgu 5. summzs h sgn o synchonous mchn n schmc shon. All h phs wnngs llus. Also h l wnng s llus. No lso h mgnc xs o h mgnc l gn by h ous wnngs. s xpn o xmn h opon o h mchn ung sy s conons. Fo hs pupos h poson o h oo wll b msu om h mgnc xs o phs A whch s sony. Spcclly, θ m wll n h ngl bwn h phs A mgnc xs n h oo mgnc xs o -xs. Snc h oo os, h ngl θ m s uncon o m. W posul h h oo poson s gn wh: π θ m ωms δ m 5. P Wh ω ms s h synchonous ngul sp mchncl o h gnon oo n P s h numb o mgnc pols. No h sy s h oo os wh consn sp ω ms. n hs cs h ngl s consn n npnn o m,.. δ δ. n hs cs, h bl m m δ m s h ngl bwn phs A mgnc xs n oo uu xs m zo, snc π π θ m δ m δ m θ. P P Copygh A. P. Sks Mlopoulos 99-6 Pg 5

6 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos Un sy s conons, h ollowng lonshps hol ω ms s P ω, ωs π wh ω s s h synchonous lccl ngul sp, s h uncy o h gn olg, n P s h numb o mgnc pols o h synchonous mchn. Upon mulplcon o uon 5. by P/, uon 5. s obn: π θ ωs δ 5. P wh δ δ m P θ θ m n subsun scussons, w us bo nsom uon. Fo smplcy, w op h subscp. n summy, n Fgu 5. wo ncs h bn n. h spl nc s n o b h sony mgnc xs o phs A. Spl ngls msu om hs nc n h con o oon. Also m nc hs bn slc wh uon 5.. Fgu 5.4 lluss h mgnc lux sbuon n h gp o synchonous mchn u o phs A lcc cun p, phs lcc cun p b, phs C lcc cun p c, suln mgnc lux u o phss A,, n C cuns p, n mn l wnng cun p. n h Fgu h m o h so hs bn skch n sgh ln. hus pon A s ncl o pon. Cons Fgu 5.4. Whn lcc cun lows hough phs A, mgnc lux s sblsh n h gp whch hs h nc spl sbuon. Anlyclly, h mgnc lux nsy s xpss wh wh s α, k cosα 5. bs cs π 4π cos α α, k cos α b α, k c Pg 6 Copygh A. P. Sks Mlopoulos 99-6

7 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos s α, s h mgnc lux nsy n poson α o h gp n m, u o lcc cun,, m n phs A bs α, s h mgnc lux nsy n poson α o h gp n m, u o lcc cun, b, m n phs cs α, s h mgnc lux nsy n poson α o h gp n m, u o lcc cun, c, m n phs C k s popoonly consn pnng on h sgn o h synchonous mchn n spclly h lngh o h gp. No h α, h spl ngl s msu om h phs A mgnc xs. h cuns, b, n snusol cuns, h mgnc lux nsy s α,, c bs α,, n cs α, wll b lso uncon o m. Un snusol sy s n blnc conons, h lcc cuns wll b: m θ ϕ cos π b m cos θ ϕ 5.4 4π c m cos θ ϕ wh ϕ s h phs o h phs A lcc cun phso. Now cons h mgnc lux u o cun,, o phs A. Fom uon 5. s π obous h h pons o zo mgnc lux sony n occu α ±. hus h mgnc lux nsy s α, s pulsng w. h gu lluss snpsho kn m whn h lcc cun s pos mxmum,.. θ ϕ. Smlly h pulsng mgnc lux nsy woms o phss n C n shown n Fgus 5.4b n 5.4c. Copygh A. P. Sks Mlopoulos 99-6 Pg 7

8 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos b c Fgu 5.4 lluson o Synchonous Mchn Mgnc Fluxs shoul b mnon h wh s shown n Fgu 5.4 s h unmnl componn o h mgnc lux nsy. n ly h wom os h hmoncs bcus s pccl mpossbly o sgn wnng whch wll pouc pcly snusol wom. h suln mgnc lux nsy n h gp s compu by supposon o h nul conbuons: α, α, s α, s bs cs α, 5.5 c subsuon n som gonomc mnpulons: Pg 8 Copygh A. P. Sks Mlopoulos 99-6

9 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos s α, cos θ ϕ α π 5.6 h ls uon psns snusol wom whch os wh sp α θ ω s 5.7 n concluson, h h pulsng mgnc lux woms gn ong snusol mgnc lux wom. h sp o oon s h synchonous sp ω. Now cons h l wnng. A c lcc cun lows hough un sy s conons. hs cun gns mgnc lux nsy wom o snusol shp. h nlyc xpsson o hs lux s α, cos θ α π. hs wom s sony n consn wh spc o h oo. u snc h oo os wh synchonous sp, so os h l mgnc lux wom. s A mu wnng pouc mgnc l ll phss α gp oo wnng pouc mgnc l Fgu 5.5 lluson o h So n oo Pouc Mgnc Fluxs Abo suls summz n Fgu 5.5. n h gu, α msus h ngl o pon on h so om h spl nc n θ msus h poson o h -xs om h nc. wo mgnc lux woms gn, on u o so cuns n h oh u o h l wnng cun. oh woms o wh synchonous sp. hs woms y o lgn wh ch oh hnk o hm s wo lcomgns. hs con gns h lcomgnc ou. n h nx scon wll nlyz h sm phnomn n un wy lng o s o ynmcl uons scbng h mchn. Copygh A. P. Sks Mlopoulos 99-6 Pg 9

10 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos 5. h ynmcl Euons o Mchn ul nlyss o synchonous mchn, sml o h on n h pous scon, pos nsgh n h opon o such mchns bu os no nbl ngnng nlyss n sgn. Fo hs pupos, nlycl mols o synchonous mchns n o pcs un nlyss o h pomnc hs mchns. Fo mol lopmn, wo ppochs cn b kn: Fl Appoch: Gn h sgn o h synchonous mchn, h mgnc lux nss cn b compu s wll s h ncon lng o h compuon o h lcomgnc ou. Also n ppop mhmcl mol cn b lop o h lccl sysm o h mchn. b Ccu Appoch: h synchonous mchn cn b w s s o muully coupl nucos, whch nc mong hmsls o gn h lcomgnc ou. Sghow ccu nlyss ls o h on o n ppop mhmcl mol. oh ppochs uln, lng o uln mhmcl scpons o synchonous mchn. h ccu ppoch how s smpl n wll b ollow h. Fgu 5.6 lluss h wnngs o synchonous mchn: h phs wnngs, A,, n C, l wnng,, n wo mp wnngs, cng long h - n - xs spcly. h wnng o phs A s schmclly psn wh h nucnc. Smlly phss n C schmclly psn wh h nucos b n c. n gnl h h phs wnng my b l o wy connc. Fgu 5.6 lluss wy conncon. Fo gnly wll b ssum h h nul o h synchonous mchn s goun hough n mpnc conssng o nucnc n n ssnc n. No h wh h xcpon o h nuco n, ll oh nucos moun on h sm mgnc ccu n hus hy ll mgnclly coupl. Pg Copygh A. P. Sks Mlopoulos 99-6

11 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos θ phs mgnc xs nc -xs -xs ω _ n n c b b b c c Fgu 5.6 psnon o Synchonous Mchn s S o Muully Coupl Wnngs Applcon o Kchos olg lw o h ccu o Fgu 5.6 yls b b b b c c c c Copygh A. P. Sks Mlopoulos 99-6 Pg

12 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos n bo uons s h mgnc lux lnkg o phs A b s h mgnc lux lnkg o phs c s h mgnc lux lnkg o phs C s h mgnc lux lnkg o h l wnng, s h mgnc lux lnkg o h -mp wnng s h mgnc lux lnkg o h -mp wnng. All oh bls n n Fgu 5.6. s xpn o w bo uons n compc mx noon. o hs pupos n: c b bc c b bc bc, b c, h s h x ny mx. Pg Copygh A. P. Sks Mlopoulos 99-6

13 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos c b bc c b bc h olg uons now cn b wn n h ollowng compc om: bc bc bc n bc n bo uons h mgnc lux lnkgs complx uncons o h oo poson n lcc cuns lowng n h ous wnng o h mchn. h gnl xpssons : So Wnngs c c b b b b b c bc b bb b b c c c c cc b cb c c oo Wnngs c c b b Copygh A. P. Sks Mlopoulos 99-6 Pg

14 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos b b c c b b c c h noon n bo uons s obous. s h sl nucnc o wnng, whl j s h muul nucnc bwn wnngs n j. Mny o h nucncs n bo uons pnn on h poson o h oo whch s m yng. hus hs nucncs m pnn. n mx noon bo uons wh bc ss b c ss s b bb bc s c bc cc bc s b c s s b c b c h pnnc o h nucncs on oo poson s xpln nx. So Sl nucncs, bb, n cc n gnl pn on oo poson. An ppoxm xpsson o hs pnnc s: s m cos θ π bb s m cos θ 5.8 4π cc s m cos θ Pg 4 Copygh A. P. Sks Mlopoulos 99-6

15 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos wh θ s h c xs poson l o h phs A mgnc xs. No h s π h oo os, s poson θ s scb wh uon θ ωs δ. A ypcl on o s shown n Fgu 5.7 s m π θ Fgu 5.7 ypcl Von o Phs A Sl nucnc s Funcon o θ oo Sl nucncs,, n ppoxmly consns n cn b symbolz wh:,, n 5.9 So Muul nucncs b, c, n bc ng. hy uncons o h oo poson θ. Appoxm xpssons o hs uncons : π b b M s m cos θ 6 π π bc cb M s m cos θ 5. 6 π 4π c c M s m cos θ 6 A ypcl on o j s shown n Fgu 5.8 Copygh A. P. Sks Mlopoulos 99-6 Pg 5

16 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos b π θ -M s m Fgu 5.8 ypcl Von o h Muul nucnc wn wo Phs Wnngs oo Muul nucncs,, n consns, npnn o h oo wnngs sony wh ch oh. θ, bcus M 5. So o oo Muul nucncs, b, n c pnn upon h oo poson θ s ollows. M F cos θ π b b M F cos θ 5. 4π c c M F cos θ Smlly M cos θ π b b M cos θ 5. 4π c c M cos θ Pg 6 Copygh A. P. Sks Mlopoulos 99-6

17 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos h mp wnng s ohogonl o h wnng. ho π M cos θ π π b b M cos θ 5.4 π 4π c c M cos θ Acully h nucncs pub om snusol on wh hmoncs. Gnlly spkng hs hmoncs kp low wh h us o sbu cols, oubl lys n conl pch. n hs xbook, hs hmoncs wll b nglc. π Snc θ ωs δ, s obous h h nucnc mx s m pnn bc ss s s bc o n summy h olg uons : bc n bc bc bc n bo uons, h mcs ss, s, s n shoul b subsu wh h xpssons 5.8 hough 5.. h moon o h gno oo s mn by h moon uon whch n h gnl om s J θ m m wh J s h oo momn o n, Copygh A. P. Sks Mlopoulos 99-6 Pg 7

18 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos h oo ylng m s h mchncl ou ppl o h oo, s h lcomgnc ou lop by h gno, n θ m s h oo poson. poson cn b subsu wh h lccl oo poson s uon 5. J θ P m h mchncl ou m s mn by h pm mo. h lccl ou s mn by h moun o pow con om mchncl no lccl. cn b c ompu by usng h pncpl o ul wok splcmn: W θ l m wh W l s h ol ngy so n h mgnc l o h synchonous mchn, n θ m s h poson o h oo. h ol mgnc ngy so n h wnngs o h synchonous mchn s gn by W l bc ss s bc [ ] [ ] bc bc s An ln wy s o compu h ol pow con om mchncl no lccl n o hs pow by h sp o h oo. oh ppochs g h sm nsw. W us h h l. h ol pow con om mchncl no lccl, P m, s: θ W m l θ m Pm θ m compu wh: n P P m bc bc, o m bc bc Pm ω m P Pm ω Pg 8 Copygh A. P. Sks Mlopoulos 99-6

19 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos Obously, h uons bcom u complx. n h nx scon wll nouc nsomon whch smpls bo uons., h mol wll b uh smpl wh h pplcon o h pun sysm lng o uln ccus. 5.4 Pks nsomon Pk s s bs on h obson h h synchonous mchn ccu uons mnously smpl nsomon s ppl o h mol h wll mk h mx consn mx npnn o m. hs nsomon b, n wh: o o - - Upon subsuon o n n h gno uons: - Pmulplcon o h uon wh yls o wh: Pk s nsomon s slc n such wy s o mk h mx consn mx. hs nsomon s n blow P wh Copygh A. P. Sks Mlopoulos 99-6 Pg 9

20 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos α θ α θ θ α θ α θ θ sn sn sn cos cos cos P π α h x ny mx x zo mx π δ ω θ s poson o -xs h bo slc mx P hs h ollowng popy whch cn b sly chck: P P h poo h h mx s consn s obn by c subsuon n subsun mnpulons. Fs obs h h ns o Pk s nsomon s P P cll h h mx s: s s ss Subsuon no h xpsson o h nsom nucnc mx yls s s ss s s ss P P P P P P y c subsuon, h ollowng s obn nn n m cons P P o ss Pg Copygh A. P. Sks Mlopoulos 99-6

21 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos nn n m cons P F s nn n m cons P F s nn n m cons M M F wh: s M s m s s M m s s M k o compl h mol, cons h gno olg uons: n Upon subsuon o h cuns n lux lnkg wh n pmulplcon o h sulng uon wh, h ollowng s obn n } { Now n: Copygh A. P. Sks Mlopoulos 99-6 Pg

22 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos P c b o } { o n h olg uons o h gno cn b xpss n ms o h lcc cuns, ylng h so cll cun mol o hy cn b xpss n ms o h mgnc lux lnkgs, ylng h so cll lux mol. oh mol gn blow. Elcc Cun Mol Mgnc Flux Mol oh mols cn b uh smpl by h ollowng suls whch obn by sghow mnpulons: y c compuon Pg Copygh A. P. Sks Mlopoulos 99-6

23 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos ω ω, θ ω hus o ω ω Upon c subsuon o bo suls n ngng h uons h cun mol s: o F o ω ω ω ω ω o M M wh k n h lux mol s: Copygh A. P. Sks Mlopoulos 99-6 Pg

24 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos o F o ω ω ω ω ω Anoh usul om o h gnon mol uons s n h s spc om x x u, whch s obn s ollows. Elcc Cun Mol. n h ollowng [ ] G o [ ] G o [ ] G o F Pg 4 Copygh A. P. Sks Mlopoulos 99-6

25 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos o M M Now h cun mol s wn s ollows: G G G ω G G o G G G ω G G h lux mol s obn by lmnng h lcc cun om bo mol,.. G G G ω G G Abo mols o synchonous mchn llus n Fgu 5.9. Copygh A. P. Sks Mlopoulos 99-6 Pg 5

26 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos oo So o o V o - V - M V - - V - - Fgu 5.9. Euln Synchonous Mchn Ccus ω ω ω ω ω h uln synchonous mchn ccus o Fgu 5.9 cn b uh smpl wh h noucon o h p un sysm. hs s scuss nx. 5.5 h P Un Sysm Mny ms s xpn o wok wh nomlz p unz uns ns o physcl uns. Fo hs pupos, n ppop sysm o uns s nouc n ll uns xpss s mulpls o h cosponng un. Jucous slcon h uns cn sul n: h uons xpssng physcl lws, such s ohm s lw, Kcho s lws, c. mn o h sm om n h p un sysm, b som mols Pg 6 Copygh A. P. Sks Mlopoulos 99-6

27 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos smpl, n c h sulng numbs o h sm o o mgnu n ho s o compu. n compu bs nlyss pocus, numcl oun o os much low. bl 5. Physcl uns, Symbolsm n Uns. Symbol Uns Volg V Cun A Pow p, o S W, VA o VA Flux nkg Wb ssnc Ω nucnc H m sc Angul Vlocy ω sc - Angl δ h nomlzon pocu s u smpl n mmcs h bsc pocu by whch h ous un sysms h bn lop o xmpl h mc sysm. Cons o xmpl h physcl uns o bl 5.. s uns cn b bly slc o ch on o h ls physcl uns. Fo xmpl Vols o h olg bs uny, 5 Amps o h cun bs uny, c. hn h p unz lu o uny wll ul h u lu o hs uny by h bs lu. o connu wh h xmpl 5 ol olg wll ul.5 5/ n p un c. hus h p unzon pocu s y smpl. How ll bs uns bly slc, uons scbng physcl lws wll h o b mo n h nw bs sysm. n gnl uons scbng physcl lws n n bly slc p un sysm complx. o o hs complcon s ncssy o slc h bs uns n such wy h physcl lws xpss wh h sm uons n ms o p unz uns n physcl sysm o uns such s h mc MKSA sysm o xmpl. n o o cly h pon ssum h w slc om h uns o bl 5. h bs o pow, olg n sp bly s S, V, n ω whl h mnng slc wh: scons ω S Amps V V Wbs Hnys Copygh A. P. Sks Mlopoulos 99-6 Pg 7

28 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos V Ohms hn goo numb o physcl lws wll b xpss wh h sm uons n boh h pun sysm n mc sysm o uns. h ollowng xmpl lluss hs smpl concp Exmpl E5.: Cons h uon whch s xpss n h mc sysm. h s s msu n ols, n ohms, n Amps, n hns, n scons. lop h p unz son o hs uon. Soluon: ch m o h uon by V o ny o s uln xpssons: V h sul s: V o u u u u u u wh h subscp u nos p unz uns. Obously h p unz uon s o ncl om s h ognl uon. n mulccu nwoks whch no nconnc, bu mgnclly coupl, h s cn g o om n slcng h bs uns o h p un sysm. On shoul xcs hs om ows smplcon o h uons n possbl physcl npon o mhmcl mols. hs s objc cn b n s ollows: h om o h sysm olg uons shoul b xcly h sm whh h uons xpss n pu o mc sysm uns. b h om o h sysm pow uons mus b nn sm n pu o cul mc uns. c h p un sysm shoul b slc n such wy h muul nucncs cn b psn wh -uln ccus, p unzon. Pg 8 Copygh A. P. Sks Mlopoulos 99-6

29 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos n chp ou h p un mol o pow nsom ws lop. H w pply h sm pocu o lop h p un mol o gno h P Un Sysm o Synchonous Mchn h concps lop n Chp 4 ly pplcbl o h nomlzon o h synchonous mchn ccus. h ou mgnclly coupl ccus. Nomlzon o h synchonous mchn uons us h ollowng sps: Sp : Slc bs o h mu wnngs S usully h on phs pow o h mchn V usully h phs o nul olg o h mchn ω usully h synchonous ngul uncy Sp : Slc h bs o h mnng wnngs mn l, -mp, -mp s ollows: sm S n ω V on h ssumpon o ul muul lux Abo pocu guns h h om o h uons scbng h synchonous mchn mns h sm n h p un sysm. hs ls o h smpl pocu o smply plcng h cul uns wh h p un uns. Subsunly h bo sps scb n mo l. Enully h suls o sps n wll b summz n bl 5.. h pocu ls o h gno uln ccu o Fgu Slcon o s uns Assum h bs uns h bn slc o h so wnng s scb n sp. hn cons h mgnc lux lnkg uons: n gnl G m m G l l m l m l l m Noc h lwys Copygh A. P. Sks Mlopoulos 99-6 Pg 9

30 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos m m m m mm M mm h slcon o h bs uns s on on h bss o ul muul lux lnkgs s ollows: n: hn k k k M m m M m m k k k k k k c. m m Abo uons compl h slcon o h bs uns. h suls summz n bl 5.. bl 5.. Summy o Synchonous Mchn P un Sysm. Slc So Ccu s uns S ω - p phs pow bs pow/ - ngul sp bs, lccl V - phs o ln olg bs - olg/ s hn h bs uns : / ω Pg Copygh A. P. Sks Mlopoulos 99-6

31 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos S / V s s s V s s s / V / s s s s. n h ollowng consns: M m k M m k m k m m m Abo suls summz n bl 5.. bl 5.. Summy o Synchonous Mchn P un Sysm Amu Fl Ccu -Axs mp Ccu -Axs mp Ccu S, ω, Vs S, ω, k Vs S, ω, k Vs S, ω, kvs / ω S / V s s s V s s / V / s s s s s / ω / k s k k k s s s / ω / k s k k k s s s / ω / k s k k k s s s. Muul nucnc s M k s s s M k s M k s M s s s s 5.6 Euln Ccus o Synchonous Mchn Copygh A. P. Sks Mlopoulos 99-6 Pg

32 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos Wh h n p un sysm, uln ccus o h psnon o synchonous mchn cn b lop. h pocu wll b mons o h lopmn o h -xs uln ccu whch hs bn slc o s smplcy. h suls wll b xn o h lopmn o h -xs uln ccu. h - xs uln ccu s u smpl. -Axs Euln Ccu: Cons h olg uons o h -xs olg n cuns: ω ω ω h s uon by V s, mmbng h V s ω ω M ω M s s s s s s s s / M s / h sul wll b V o s ω ω s s ω ω M s ω ω M s u u u u u u mu l u s u s u u s s M s Smlly h scon uon by V s, mmbng h V M / / s s s h sul wll b M s s o Pg Copygh A. P. Sks Mlopoulos 99-6

33 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos u u uu u mu l u u u A summy o bo suls s gn blow. u V u u u u mu l u u u u uu u mu l u u ω ω ω u u u u u u u u u u u u u u Now s obous h h ccu o Fgu 5. cospons xcly o bo wo u l u u u l u Au V u - u - Fgu 5. -Axs uln Ccu n p.u. Now cons h -xs uons: A sml pocu on hs uons yls h punz uons: u u u u mu u u bu l u u u mu u u u l u uu mu u u u l u u u u u u n u u u u Wh: u ω ω u u u u u u Abo uons psn h uln ccu o Fgu 5. Copygh A. P. Sks Mlopoulos 99-6 Pg

34 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos u lu u u u - V u l u l u mu - u V u - Fgu 5. -Axs Euln Ccu n p.u. h suls o h -xs n -xs uln ccu summz n Fgu 5. whch lso nclus h -xs uln ccu. Pg 4 Copygh A. P. Sks Mlopoulos 99-6

35 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos o V o - l - V l l b A - V - l l A V - - c Fgu. 5. Euln Ccus o Synchonous Mchn wh wo mp Wnngs n p.u. -xs Euln Ccu b -xs Euln Ccu c -xs Euln Ccu h p unz mol o synchonous mchn wll b llus wh n xmpl. Exmpl E5.: A MVA, 6 Hz, 4 pol, 5 kv gno hs h ollowng pms: Copygh A. P. Sks Mlopoulos 99-6 Pg 5

36 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos H.8 sc. ohms.6 ohms.77 ohms.58 ohms.75 mh.474 mh l l. mh mh l 4. 9 mh M mh.486 mh o.7 mh l. mh l mh Us h ollowng bs sysm o h so ccu S MVA ω π 6sc 5 V Vols n compu h p un lus o h ollowng pms:,,,, A, A, l, l, l, l, l,o Soluon: h bss compu usng h uons n bl 5.. Fs w compu som o h pms. m l. 47mH m l. 85mH m m mH k m.6975 M. 6899mH k m l. 54mH Now k. 9 m k.977mh, m l m. 96mH An k. 69 m k.54mh, m l m. 59mH Fom bo uns, h bs uns o ll ccus compu s ollows: A. So: Pg 6 Copygh A. P. Sks Mlopoulos 99-6

37 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos S 4. MVA ω 77 s c V Vols S.6558 S.8 scons mps S Wb S.758 H S ohms. Mn Fl k.6975 S 4. MVA ω 77 sc - V V S.6558 sc mps Wb.689 H 6.48 ohms C. - Ccu k.9 S 4. MVA ω 77 sc - V 4.7 V.6558 sc mps Wb.5659 H.985 ohms. - Ccu k.69 S 4. MVA ω 77 sc - V V.6558 sc mps Wb.676 H ohms Copygh A. P. Sks Mlopoulos 99-6 Pg 7

38 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos E. Muul nucnc F. ou s M S S H M S S. 65H M S S H M. 44H S ω,49,46.446n m S. S hn h p un lus o h mchn pms : A.869 A.689 l l.59 l.44 l.856 l o 5.7 Synchonous Mchn ou Euon o compu h ou n pow on hs o nlyz h lcomgnc l n hus compu h ngy ns hough h gp. hs ngy s known on cn compu h ou. h sm sul cn b obn s ollows: h oupu lccl pow s: p b b c c o o p- -o -o o o ω - ω o ] wh p P ohmc P s. mgn. P n. Pg 8 Copygh A. P. Sks Mlopoulos 99-6

39 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos P ohmc : ohmc losss on ssos P s. Mgn. : o chng o so mgnc l ngy : pow ns coss gp P n. U sng h pncpl o ul wok splcmn w obn Wl Pl m θ ω m - Abo uon yls h ol lcomgnc l ou. n co noon m [ ] o h pow ns hough h gp s h on po wh h olg souc, n. P m - An h lcomchncl ou s m Pm ω m P ω h gno oo moon uon s θ m J m m o J P ω m m No h bo uon s h so-cll swng uon. hs uon cn b wn n p.u. Fo hs pupos l S h p phs bs pow. h -Phs pow wll b S. An h -phs bs ou Copygh A. P. Sks Mlopoulos 99-6 Pg 9

40 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos P S ω Upon son o h swng uon wh h bo bs ou, on obns: Jω ω m P S m wh m s nomlz mchncl ou m s nomlz lcomgnc ou No h Jω ω ω / ω ω ω ωu J Hω P S / P S u ω u u τ j Hω. hus h nomlz swng uon bcoms τ j ω m mu π Now cons h uon θ ω s δ. h o hs uon s θ δ ω ω s Nomlz uon by ng by ω no ω s ω δ δ ω u, o ω u δ ω n summy, h s spc swng uon n p.u. o gno s gn by Pg 4 Copygh A. P. Sks Mlopoulos 99-6

41 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos [ ] o j j m τ τ ω ω δ Abo wo uons known s h swng uon o synchonous gno. 5.8 S Spc Mol o Synchonous Mchn n hs scon w summz h mol o synchonous mchn. cll h wo mols w lop: h cun mol n b h lux mol. A summy o h sx olg-cun uons n h swng uons s gn blow cun mol. Synchonous Mchn Elcc Cun Mol Summy G G G ω m G G j ω τ ω ω δ wh o G,, o G o M M [ ] C Copygh A. P. Sks Mlopoulos 99-6 Pg 4

42 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos wh j Hω τ h cul olg o cun phs uns my b obn om P - o c b P - o c b Pg 4 Copygh A. P. Sks Mlopoulos 99-6

43 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos 5.9 Sy S Anlyss n hs scon w xmn h opon o synchonous mchn n sy s. hs nlyss pos nsgh no h opon o synchonous mchn n h uscn opon pon o h mchn. A sy s conons, h cun n olg wll b ssumng blnc opon conons cos π ϕ δ ω s cos π π ϕ δ ω s b 4 cos π π ϕ δ ω s c cos π ϕ δ ω V s V cos π π ϕ δ ω V s b V 4 cos π π ϕ δ ω V s c V c V c n π δ ω θ s cos sn V V c b o V V P ϕ ϕ cos sn c b o P ϕ ϕ Obs h h o-- xs cuns n olgs consn sy s, blnc conons hus o Copygh A. P. Sks Mlopoulos 99-6 Pg 4

44 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos o bo lonshps subsu no h synchonous mchn mol, w obn: o - - ω - ω - ω ω ω Fom h ls wo uons, s ppn h: hus o - - ω - ω - ω - - No h:,,,,, psn c-uns. Fom hos, cully cuns whl h ohs pojc nsom uns. o obn cul phs uns Pk s nsomon s ppl. Spcclly, h phs A olg n cun s obn n ms o h o-- uns. cosθ snθ cosθ snθ π θ ωs δ All oh phss splc by. A gomc npon cn b gn o h pous uons. Cons h c h sy s h oo os wh synchonous sp n s poson s n wh π θ ωs δ. h uns, psn cun n olg on h -xs o h oo, n h uns, psn cun n olg h -xs o h oo. h ngl θ ncs h poson o h -xs. hs s shown n Fgu 5.. Pg 44 Copygh A. P. Sks Mlopoulos 99-6

45 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos -xs π θ ωs δ -xs ωs δ nc Phs A Mgnc xs Fgu 5.. Schmc psnon o h - Axs o Synchonous Mchn oo n Elcc Cun s cusomy o w h gm o Fgu 5. n o cons h wo mnsonl gm s h complx pln wh h nc s h l xs. hn cuns n olgs bcoms complx uns whch xpss s : π j δ jδ π j δ jδ Abo conclusons cn b jus nlyclly by usng h usul non o phsos. Now, cons h phs A olg: cosθ snθ Upon subsuon o h olgs n conng h sns no cosns, w h: π [ ω cos ω δ ω ω cos ω δ ] No h h ms phso psnon o s: Copygh A. P. Sks Mlopoulos 99-6 Pg 45

46 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos δ δ π ω ω ω j j V hs uon cn b wn n h om E jx jx V wh x ω -xs synchonous cnc x ω -xs synchonous cnc E M ω gn olg δ π j jδ Sm nlyss cn b p o phss n C. h nl suls s ollows: E jx jx V π π π π j j j j b E jx jx V π π π π j j j j c E jx jx V h pous nlyss o h olg V cn b p o h cun. Spcclly, h m uncon o h cun s: ] cos cos [ sn cos δ ω δ π ω θ θ h ms phso psnon o h cun s : j j δ δ π h uons summz n Fgu 5.4. Pg 46 Copygh A. P. Sks Mlopoulos 99-6

47 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos -xs δ E -xs V V δ jx jx nc V Fgu 5.4. Phso gm o Phs A o Synchonous Mchn Sy S Usng h sy s mol o synchonous mchn w ss h ollowng unmnl poblm. Gn h gno mnl uns V,, P, c. mn compu h nsom uns, n h poson o h xs δ. Fom h gn mnl conons s lwys possbl o compu h phsos V n. hn, h phsos n cn b obn om h soluon o h ollowng uons. V jx jx E 5.5 An ln wy s o sol o, soluon s bs on wng h uon 5.5 n h om V jx j x x E hough gphcl soluon. h gphcl Obs h h phso j x x E s long h -xs n uls h phso: A V jx whch s compubl. hus, on cn s compu h phso o h -xs om h phs o h phso A A n n h locon Copygh A. P. Sks Mlopoulos 99-6 Pg 47

48 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos h gphcl soluon s shown n Fgu 5.5. h sps o h gphcl consucon s ollows. Sp : Fom h p o h phso V w h phso. Sp : w h phso jx. h ln OA ns h -xs,.. h ngl δ. Sp : Compu, s pojcons o on h n -xs n compl h gm. h pocu wll b llus wh n xmpl. -xs O V jx A jx E jx -xs δ nc Fgu 5.5 Gphcl Soluon o mnng h Poson o h -Axs Exmpl E5.: A 5 MVA gno, 5 kv, wy-connc ls A.85 pow co lggng n olg. mn h sy s opng conons. h un s connc wh nsmsson ln o.5 j.4 ohms o n nn bus. h ollowng gn: 96 A 6.4x - H 6.8x - H 5 C.54x - ohms Whn h mchn ops unlo wh olg h mnls, h l cun s 65 A opn ccu s. Soluon: Compu h ollowng x ω 9. Ω x ω. Ω Pg 48 Copygh A. P. Sks Mlopoulos 99-6

49 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos co s φ 85. φ V x 464. V Fom bo uns, plo s wn. -xs E -xs 6.75 o o.78 V jx jx jx nc Fom h plo w msu: δ 6.75 V V Also An cos A - sn A A 94 A jx j,697. V jx j5,84. V y complng h gph n msung E w obn E,7 V h poblm cn b lso sol nlyclly. Spcclly, s w compu h co A: Copygh A. P. Sks Mlopoulos 99-6 Pg 49

50 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos A V jx 6,8.4 j,5.76,4. j6.7 ols Abo compuon pos h ngl o h -xs. hn h - n -xs cuns compu s pojcons o h phs cun on h - n -xs. Knowng hs cuns, h gn olg E s compu s h co sum o h componns nc n h Fgu. h sul s: E,7 j6.7 V Exmpl E5.4: h gno o xmpl E5. s connc o n nn bus hough ss compns nsmsson ln. h pms o h ln :. ohms,. hns, n C 65 µf h gno oupus MVA mnl olg n.9 pow co lggng, compu h sy s olgs,, h cun, n h gn olg E. Also compu h olgs, n h nn bus. Soluon: Compu co A phso. A V jx A jδ wh j j V V. j 9. 98A. 769 x ω j. 45. A 98. j cos sn cos sn E.98 x -x.546 h nsmsson ln sy s uons nsom usng Pk s nsomon: -. ω ω. Pg 5 Copygh A. P. Sks Mlopoulos 99-6

51 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos o nomlzon ω ω h ss cpco sy s uons, nomlzon, : Z c.786 pu.786ω -.786ω ω.786ω E Synchonous Mchn Pomnc Un Fuls A ypcl synchonous gno, whn ul, xhbs h snc phnomn. mmly upon h pplcon o h ul, h ul cuns lly hgh n cy s ows sy s lu. On cn ny ls wo snc s o cy. hs s shown n Fgu 5.6 o h phs ul n h phs A ms lu o cun s shown n Fgu 5.7. s ppn om hs Fgu h h synchonous mchn mpnc chngs wh m. h h snc m pos nm subnsn b nsn n c synchonous s llus n Fgu 5.7. n hs scon w xmn h pm n m consns o hs h m gons. Copygh A. P. Sks Mlopoulos 99-6 Pg 5

52 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos Fgu 5.6 Synchonous Mchn Sho Ccu Cuns Followng h Phs Ful Subnsn nsn Synchonous " o o s mscons Fgu 5.7 lluson o h cyng ms Vlu o h Phs A Cun 5.. Subnsn n nsn nucncs Pg 5 Copygh A. P. Sks Mlopoulos 99-6

53 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos h synchonous mchn compss numb o coupl mgnc ccus. h mgnc lux n hs ccus hs cn n. Whn sun chngs occu n h mchn, h mgnc luxs s chngng. ypcl consucon chcscs o synchonous mchns such h h m consns ssoc wh h mgnc lux chngs u o sun chng on h o o w cycls o h mp wnng mgnc lux n on h o o scons o h mn l wnng lux. hus, n mnng h nl hou o synchonous mchn, w cn mk h ollowng ssumpons: nlly, ll mgnc luxs mn consn o sun chng o h xnl conons. hs ssumpon pos h subnsn pms o h synchonous mchn. b Fw cycls subnc, h mp wnng mgnc lux ssums sy s lus, whl h mn l wnng lux mns consn. hs ssumpon pos h nsn pms o h synchonous mchn. W wll us hs ppoch o lu h subnsn n nsn pms o synchonous mchn. Fo smplcy, w nglc h ssncs. -xs subnsn & nsn nucncs. hs nucncs wll b n hs pgph. h sul s gn by: Wh: " " A A A A A s h -xs subnsn nucnc, n s h -xs nsn nucnc. l l l l A l l h -xs subnsn n nsn nucncs n un h ollowng conons: Assum mchn os wh synchonous sp, Assum ll l ccus sho,, n Assum olg n sunly ppl o h mnls n such h o Vu wh u s h sp uncon. n hs cs h mchn wll bh nlly s n nucnc, o lu - subnsn nucnc -xs, n w cycl wll bh s n nucnc o n lu, - nsn nucnc -xs. Spcclly, ssumng h h mn l wnng lux n h mp wnng lux wll mn consn, wll yl h subnsn -xs nucnc. gnong h mp wnng lux n ssumng h h mn l wnng lux mns consn wll yl h nsn nucnc. h pocu o obnng h subnsn n nsn -xs nucncs s gn blow. Copygh A. P. Sks Mlopoulos 99-6 Pg 5

54 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos h on o bo uons s s ollows. Snc h gno l wnng s sho, hn, h mgnc lux n h mn l wnng n h -xs mpo wnng wll b: M M Soluon o bo uons o h cuns, n ms o yls: M M M M h mgnc lux n h -xs,, s compu om Subsung h l cun n h -xs mp wnng cun, yls: " M M cll h whn h synchonuous mchn pms xpss n pu, hn: M A hus: " A A A h mp wnng s nglc M, hus: A Pg 54 Copygh A. P. Sks Mlopoulos 99-6

55 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos -xs subnsn n nsn nucncs. hs nucncs compu n sml wy s h -xs nucncs. h nl sul o h -xs subnsn n nsn nucncs s: " A Wh: " s h -xs subnsn nucnc, n s h -xs nsn nucnc. h -xs subnsn n nsn nucncs n un h ollowng conons: Assum mchn os wh synchonous sp, Assum ll l ccus sho,, n Assum olg n sunly ppl o h mnls n such h o Vu wh u s h sp uncon. n hs cs h mchn wll bh s n nucnc o lu - s yng cuns subnsn nucnc -xs n s n nucnc o n lu - moly yng cuns nsn nucnc -xs. h on o h bo uons s s ollows. Snc h gno l wnng s sho, hn Soluon o bo uons o h cuns, yls: h -xs mgnc lux o hs conon wll b: "" hus Copygh A. P. Sks Mlopoulos 99-6 Pg 55

56 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos "" A h o -xs nsn cnc s by nglcng h -xs mp wnng. hn: 5.. m Consns o Synchonous Mchn h m consns compu by consng h conons subnsn o nsn n mnng h m consns nol. As n xmpl cons h -xs uln ccu o gno. h -xs ccu s opn, hn h only loop n h ccu s h on conssng o, n A. h m consn o hs ccu s l l " A τ o whch s h opn ccu -xs subnsn m consn A mo sysmc ppoch o mn h m consns o gno s o compu h plc nsom o h uln ccu. h oos o h plc uncon po h m consns. hs pocu s ous n s om. ns, ppoxm lu o h ous m consns h bn compu n bul n bl 5.. An xmpl pos ypcl lu o m consns. bl 5. Appoxm m Consn o Synchonous Gno -xs " M / τ o subnsn m consn opn ccu τ τo subnsn m consn sho ccu " " " τ o τ nsn m consn opn ccu τo nsn m consn sho ccu -xs Pg 56 Copygh A. P. Sks Mlopoulos 99-6

57 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos " τ o τ τ subnsn m consn opn ccu τo subnsn m consn sho ccu " " τ o " Amu m Consn: scbs h cy chcscs o h - phs sho ccu ms cuns C o so o ms o l cun τ Exmpl E5.5: h ollowng pms h bn compu o h gno o Exmpl E A l l 59.. l l 865. A.869 l. 8 o.5669 Compu h p un lus o h - n - xs subnsn n nsn cncs,,, Also compu h ollowng m consns Soluon: " " " " τo, τ, τo, τ, τo, τ, τo, τ n τ A " - " l A A A.4669 A A A l l ll l l A 57. Copygh A. P. Sks Mlopoulos 99-6 Pg 57

58 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos.999 " / τ o τ M 566. o.686 sc τo 9. 7 o.5 sc " " τ o τ " τ o τ τ " 7.99 o 6. sc τo 58. o.48 sc o.5757 sc τo 74. o.64 sc " " o ", τ mnnglss o hs m ol. τ o.767 sc 5.. Summy o Synchonous Mchn Pms A comphns ls o synchonous mchn pms s gn n bl 5.4. h sho ccu o SC s n n Fgu 5.. All hs pms lbl om h mnucu. cnly, chnus h bn lop by whch hs pms cn b sm wh l m monong sysms. bl 5.4 s o Gno Pms Mnucu Sho Ccu osc gulo Un MVA Un KV Un PF x, x,, x x, x, x l, xo Pg 58 Copygh A. P. Sks Mlopoulos 99-6

59 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos x, x, " " τ, τ, τ τ o, τ, τ " o o " τ, τ, τ H o V ull lo c -gp ln occ Volg scc h o b g o SC Fl Cun Ob gp lngh Og Fgu 5.8 non o h SC o Synchonous Mchn Copygh A. P. Sks Mlopoulos 99-6 Pg 59

60 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos 5. Synchonous Mchn Smpl Mols Mny ms, smpl mol wll suc o sbly sus un cn conons. Fo xmpl h s swng o h sysm s o b su o h pomnc o synchonous mchn sblz. onlly, smpl mols h bn mploy o hs sus. hs scon psns such mols. n on s o pspc o hs mols ows ncsng h nuon o h. Fo xmpl h usbly n lmons o hs mols xplcly xpln. h ppoch kn h s h o mnng h lcc pow oupu/npu o h mchn sus slow yng bls such s gn olg, nn bus olg, c. hus smpl xpsson o h lcc pow o h swng uon suls. s how mpon o lz h lmons o hs mols. h smpl mols nouc on h bss o h mn ssumpons lng o h smpl mol. As such, h smpl mols wll b nouc. Sy S Mol. b Consn Mn Fl Wnng Flux Mol. c Consn mn Fl n mp Wnngs Flux Mol. h lng ssumpons hmsls cn po nomon gng h pplcbly o h sulng smpl mols. Fo xmpl h sy s mol cn b mploy whn h nsons om on opng pon o noh smooh. Smlly h consn mn l wnng lux mol shoul b mploy o suy phnomn whch lss lss hn h m consn o h mn l wnng, c. Elboon on h usbly n pplcbly o h mols s uh po n h nul scon scbng h mol. 5.. h Sy S Mol h on o hs mol s bs upon h ssumpon o sy s conons o h synchonous mchn. hus h pplcbly o hs mol xns o css n whch h synchonous mchn ops n sy s o slowly chngs s opng pon. On such cs s gno plc un gono conol wh slow mp. Fo hs mol s uh ssum h plng conons blnc. h uons scbng hs mo : - - ω ω ω P 5. Pg 6 Copygh A. P. Sks Mlopoulos 99-6

61 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos n bo uons h o -xs uon n h mp wnng uons nglc o obous sons. Fgu 5.9 lluss h ssoc phso gm o h synchonous mchn. n hs gm h gn olg phso E hs mgnu E ωm 5. -xs E -xs V V V δ jx jx nc Fgu 5.9 Phso gm o Synchonous Mchn Sy S Mol No h suon cn b ccoun o n h usul wy by obsng h h nucncs n n uncons o h suon ll, h s Fom h uons scbng hs mol h lcc pow h mnls o h mchn cn b xpss n ms o h mnl olg V n h gn olg E. Fo hs pupos cll h V snδ 5.6 V cosδ 5.7 Copygh A. P. Sks Mlopoulos 99-6 Pg 6

62 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos hn sol uons 5.8 n 5.9 o n o obn x x x x x x E 5.8 x x x x x x x E 5.9 x x Upon subsuon no uon 5. P x x x V V sn δ V E cosδ V E snδ 5. x x x x x x x x Usully h ssnc o synchonous mchn s y smll s comp o x o x. hus cn b nglc. n hs cs uon 5.9 s smpl o : P V E V snδ sn δ 5. x x x Also h c pow s compu o b - 5. Upon subsuon w obn: V V E x sn δ cos δ snδ cosδ x x 5. x x x x h ssnc s nglc V VE x x sn δ cos δ cosδ 5.4 x x x Exmpl E5.6: A 65 MVA, 6 Hz, 8 kv synchonous gno hs h ollowng pms:.66 pu, x.5645 pu, x.4 pu, E.87 pu, V. 5 pu Compu h smpl sy s mol o hs un un h ssumpon o consn gn olg. Soluon: y c compuon h mu ssnc s nglc Pg 6 Copygh A. P. Sks Mlopoulos 99-6

63 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos P 4.95 snδ.7 sn δ sn δ -.4 cos δ 4.95 cosδ b h mu ssnc s no nglc. P sn δ.54 cosδ 4.95sn δ sn δ 4.48sn δ sn δ -.47cos δ 4.954cosδ -.sn δ 5.. Consn Mn Fl Wnng Flux Mol h lng ssumpon n h on o hs mol s h h mgnc lux lnkg o h mn l wnng mns consn,.. cons n 5.5 Snc h m consn o h mn l wnng s n h o o scons, s obous h hs mol s ppop whn phnomn whch ls lss hn scon o b su. Such n xmpl s h clsscl nsn sbly poblm. ws posul l h h sbly o h pow sysm s mn om h s swng o h sysm ollowng h subnc. Snc h s swng lss lss hn scon, h clsscl mol ulzs n ssumpon whch s uln o consn mn l wnng lux. As w shll s hs ssumpon s uln o h ssumpon o h clsscl mol consn olg bhn nsn cnc. Fo hs son hs mol wll b lso o s h clsscl nsn mol. An onl ssumpon o hs mol s h h mp wnng nglc. h uons scbng hs mol om h gnl uon by ulzng h s ssumpons: - - ω - - ω ω - P - Euon 5.8 yls Copygh A. P. Sks Mlopoulos 99-6 Pg 6

64 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos 5. n snc s ssum consn: 5. Upon subsuon o, n uons 5.6 n ω - - ω 5. ω / n bo uons h m nucnc, : cn b cognz s h -xs nsn 5.5 h nsn cnc wll b x ω 5.6 A uh ppoxmon cn b nouc no h mol by obsng h o h ng o pplcbly o hs mol h -xs n -xs cun, n y slowly. n hs cs h ms n y smll comp o oh ms o h uon. hus, hy my b om. Fnlly, snc s ssum o b consn h m ω / s lso consn. n ωm E 5.7 whch shll b cll h gn olg bhn nsn cnc. Pg 64 Copygh A. P. Sks Mlopoulos 99-6

65 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos -xs -xs nc E V jx jx Fgu 5. Phso gm o h nsn Mol Now uons 5. n 5.4 bcom - - x x E 5.9 n phso psnon bo uons E jx jx V 5.4 h phso gm s llus n Fgu 5.. h pow uons n sghow mnn s n h pous mol. h sul s δ δ δ sn cos sn x x E V x x x E V x x x x V x x V P 5.4 cos sn cos sn δ δ δ δ x x x E V x x x x V 5.4 Agn h ssnc s nglc h pow uons smpl o: δ δ sn sn x x V x E V P 5.4 Copygh A. P. Sks Mlopoulos 99-6 Pg 65

66 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos V V E x sn δ x cos δ cosδ 5.44 x x x Exmpl E5.7: Cons 65 MVA, 6 Hz, 8 kv synchonous mchn wh h ollowng pms.66 pu, x.5645 pu, x.4 pu, E.87 pu, V. 5 pu An x.776 pu, E. 7 pu Compu h smpl mol un h ssumpon o consn l wnng lux. Soluon: y c subsuon: V.5 pu E 98 V o E.7 pu x.94 ohms o x.776 pu x.4 pu P. 8sn δ 5. snδ o P sn δ. 959cos δ snδ 5.. Consn Mn Fl n mp Wnngs Flux Mol h lng ssumpon n h on o hs mol s h h mgnc lux lnkg o h mn l wnng n h mp wnngs mn consn,.. consn consn consn Fom h ssumpon, h ng o pplcon o hs mol s obous: phnomn whch ls lss hn h smlls m consn o h h wnngs: mn l, n mp wnngs. hs m consn s n h o o w 6 Hz cycls. hus hs mol s subl o h mnon o h lcc cuns mmly ollowng ul. hs s mpon n h pplcon o mon swchg whch op n o cycls. h uons scbng hs mol om h gnl uons: - - ω - ω Pg 66 Copygh A. P. Sks Mlopoulos 99-6

67 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos - ω ω ω M M 5.5 P Euons 5.5, 5.5 n 5.5, sol o h lcc cuns,, yl h ollowng M M M M 5.55 n upon non ssumng consn, n : M M 5.56 Subsung o h cuns, n n uons 5.48 n 5.49 on obns: - - ω - - ω M A 5.57 M - ω A - M ω A ω M A 5.58 wh A - M Copygh A. P. Sks Mlopoulos 99-6 Pg 67

68 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos n bo uons h subnsn nucncs n cncs cn b n: M A x ω 5.6 x ω 5.6 Snc, by ssumpon, n mn consn, uon 5.57 n 5.58 conn h consns. h consn ω M whl h consns ω A olgs gn long h -xs. psns olg gn long h -xs, n ω M A psn n E E ω M 5.6 E ω M M M A 5.64 E ω M M M A 5.65 E E E 5.66 Subsung bo nons no uons 5.57 n 5.58 o obn - - ω " - E" - " ω " - " E E 5.68 On nl ppoxmon s nouc by obsng h h ms " n " smll comp o oh ms, ylng - - ω " - E" ω " E 5.7 h phso psnon o bo uons s: Pg 68 Copygh A. P. Sks Mlopoulos 99-6

69 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos " " " V jx jx E E " 5.7 h phso gm o hs mol s llus n Fgu 5.. -xs -xs E" E" jx" E" jx" nc Fgu 5. Phso gm o h Subnsn Mol h pow uons n sghow mnn s o h pous mol. cus o h complxy o h suls only h cs o nglgbl mu ssnc s psn. P VE " V VE " sn δ sn δ cosδ 5.7 x" x" x" x" V x" x" V E " E" sn δ cos δ cos δ sn δ 5.7 x" x" x" x" Exmpl E5.8: A synchonous gno ls pow, mnl olg n pow co ul.. Sunly h phs ul s plc h mnls o h gno. Compu h ms lu o h ul cun mmly h ul. Nglc h c componn. h pms o h gno :.7.57 A A Copygh A. P. Sks Mlopoulos 99-6 Pg 69

70 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos Soluon: h subnsn mol wll b mploy. h subnsn nucnc compu o b: hn h nl conons compu usng h sy s mol. j. A V jx j. E A j x x δ. 655 h gn olg bhn h subnsn cnc s: " " " j. E V jx jx mmly ul non, E mn consn, n V hus E" jx " jx " A jδ j δ π A Subsu no uon o obn j A j δ j.5575 A j δ No h mmly ul non, δ Upon soluon o bo uon o A n A : A A A 448. pu A pu.566 pu 5..4 Summy o Smpl Mols Pg 7 Copygh A. P. Sks Mlopoulos 99-6

71 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos h h smpl mols psn n h pous pgphs summz wh h ollowng uons E V jx jx sy s mol E V jx jx E " " V jx jx " nsn mol subnsn mol P P P V x E V snδ sn δ x x V E V snδ sn δ x x x V E" V x" snδ x" x" V E" sn δ x" cosδ wh n ng h P uons, h ssnc hs bn nglc. h h mols known s: Sy s mol b nsn mol n c Subnsn mol. Fgu 5. summzs h phso gms o ll mols. -xs π θ ωs δ -xs E" V V V E E" E" jx" jx" E jx jx jx jx nc Fgu 5. Summy o Smpl Mols 5..5 On Axs Synchonous Mchn Mol Copygh A. P. Sks Mlopoulos 99-6 Pg 7

72 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos A uh smplcon o h psn mols s h so cll on xs mols. h lng ssumpon o hs mols s h h cncs o synchonous mchn ul long h n -xs. h s Sy s mol x x x nsn mol x x x Subnsn mol x x x Ulzng hs ssumpons, h mol uons o ch on o h h mols : Sy s mol nsn mol V jx E VE P snδ x V jx E P Subnsn mol VE sn δ x " V jx E " P VE " sn δ x" Fom bo uons s obous h n uln ccu cn b o ch mol. hs ccus llus n Fgu 5.. shoul b mnon h mny ms w o h on xs nsn mol s h clsscl mol. Pg 7 Copygh A. P. Sks Mlopoulos 99-6

73 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos x E x E b x" E" Fgu 5. On Axs Synchonous Mchn Smpl Mols Sy S Mol b nsn Mol c Subnsn Mol h ssumpons o h on xs mols po nsgh no h g o ppoxmon pom. Fo xmpl snc h - n - xs synchonous cncs o cylncl oo mchn h numcl lus ppoxmly ul whn % o xmpl, s conclu h h on xs sy s mol o cylncl oo mchn s sonbl ppoxmon. hs cn no b clm o sln pol synchonous mchn. gng h on xs subnsn mol on cn obs h mos synchonous mchns h - n - xs subnsn cncs ppoxmly ul. hus h on c Copygh A. P. Sks Mlopoulos 99-6 Pg 7

74 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos xs subnsn mol cn b cons o b sonbl ppoxmon. hs clm, how, cn no b xn o h on xs nsn mol. n gnl h xs nsn cnc s much smll hn h -xs nsn cnc. 5. Exc Mol n Volg Conol o b 5. Synchonous Gno Cpbly Cus o b 5.4 Summy n scusson hs chp psn h gnl synchonous mchn mol. h physcl consucon o h synchonous gno ws scuss n h mol scbng h nnl consucon o h mchn ws nouc. h mol ws mnpul o yl sl mol un spcc smpl ssumpons. hs mol l o suyng h pomnc o h mchn o cn phnomn. h on lso pos nsgh no h oll bho o synchonous gno. Pg 74 Copygh A. P. Sks Mlopoulos 99-6

75 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos 5.4 Poblms Poblm 5. A -phs, 6 Hz, 5 MVA, 5 kv n o n gno hs h ollowng pms: x. 75 ohms x. 7 ohms. Hns. 5 ohms. 65 Hns. Hns l. 976 Hns l. Hns Expss n p un h ollowng uon o h mhmcl mol o bo gno V ω ω ω Poblm 5. On o h ssumpons n h clsscl mol o synchonous gnos o nsn nlyss consn olg souc bhn nsn cnc s h ollowng: h mchncl ngl o h synchonous mchn oo -xs concs wh h lccl phs ngl o h olg bhn nsn cnc. Show h hs ssumpon s uln o h ollowng conon: h -xs nsn cnc s ul o h -xs synchonous cnc. Poblm 5. A h phs, 6Hz, 9MVA, 5kV ln o ln, ms synchonous gno hs h ollowng pms n p un sysm wh S MVA, ω π 6sc -, n V 5/ Vols o h so.7 pu,.49 pu,.57 pu,.5 pu,.8 pu, A.77 pu, n A.79 pu, Compu h ollowng: nsn cnc, -xs Subnsn cnc, -xs nsn cnc, -xs Subnsn cnc, -xs b h gno ls pow, mnl olg n pow co ul o., compu: h gn olg E h olg bhn nsn cnc, E h olg bhn subnsn cnc, E Assum h h phs ul s sunly ppl o h mnls o h gno. Pul conons sm s s bo. Compu h ms lu o h ul cun mmly ollowng h ul. Nglc h so-cll c-componn. Copygh A. P. Sks Mlopoulos 99-6 Pg 75

76 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos Poblm 5.4 h uln o-- xs nwok psnon o synchonous 6 Hz, 5kV ln o ln, ms, 8 MVA gno s gn n Fgu P5.4. All uns xpss n p un sysm wh h ollowng bs uns o h so: S 7 MVA p phs, b ω π 6 sc -, V 5,/ Vols. Compu h p un lus o h ollowng uns: Synchonous nucnc, -xs Synchonous nucnc, -xs nsn nucnc, -xs nsn nucnc, -xs Muul nucncs M,,, Mn Fl Wnng sl nucnc, -wnng sl nucnc, -Wnng sl nucnc, b n h mnmum moun o nomon u o compu bo uns n cul MKSA uns mc sysm.. j. o-xs. j V j.5 j.8 j.48 - ω -xs ccu. j..4 j.45 j.7 -xs ccu - ω Fgu P5.4 Pg 76 Copygh A. P. Sks Mlopoulos 99-6

77 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos Poblm 5.5 A -phs, 6 MVA, 6 Hz, synchonous gno ls 5 MW o l pow o n nn bus hough -phs nsom n nsmsson ln, s n Fgu P5.5. h pow co h mnls o h gno s.95, cun lggng. h nn bus s mnn o olg o 5 kv ln o ln. mn h sy s conons o h sysm. G -Phs n nn us Aonl nomon Fgu P5.5 h pos sunc ss mpnc o h nsmsson ln s: z 5. j5. ohms. b h nsom s 6 MVA, 5kV: kv, 6 Hz, -phs wh h ollowng mpnc: z.4 j. 78 pu on h nsom ngs. c h pms o h synchonous gno :.5 pu, x. pu, x. pu,. 8 H Poblm 5.6 Compu h p-unz olg uons o h gno o poblm 5.5. h gno s 6 MVA, 5 kv ln o ln un. Aonl nomon s s ollows: x. ohms x. ohms. 8 Hns. 5 ohms. Hns. 4 Hns l. 5 Hns l. 5 Hns M. 9 Hns 5 C. 7 ohms. 75 ohms. 95 ohms. Hns. 55 Hns. 45 Hns. 856 Hns Poblm 5.7 h uln ccu psnon o synchonous mchn s gn n Fgu P5.7 n p un. No h h ssncs nglc. h mchn s connc o n nn bus hough nsmsson ln whch hs j.5 pu cnc n nglgbl ssnc. ls pow mnl olg n pow co... Compu h sy s opng conons. Spcclly compu h uns α δ, E, E n E j oo ngl, gn olg, olg bhn nsn cnc n olg h nn bus.. Compu n xpsson o h nsn pow sus ngl δ un h ssumpon o consn olg E. Copygh A. P. Sks Mlopoulos 99-6 Pg 77

78 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos j.5669 o-xs j. 5 j. 4 j. 9 j.8 7 V -xs - j. 5 j. j.6 9 -xs - Fgu P5.7 Poblm 5.8 A h phs, 6 Hz, 5 MVA, 5 kv ln o ln, wo pol gno hs h ollowng pms: x. 75 ohms x. 7 ohms. Hns. 5 ohms. 65 Hns. Hns l. Hns l. Hns Cons h -xs olg uon ω ω ω W h p unz son o bo uon ll pms shoul b subsu wh h numcl lus. Pg 78 Copygh A. P. Sks Mlopoulos 99-6

79 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos Poblm 5.9 Cons h lcc pow sysm o Fgu P5.9. h sy s opng conons o h sysm n wh h pow low soluon o bl P5.9. o b mn s h sy s conons o gno G. h - n -xs uln ccus o hs MVA, 6 Hz, pols gno gn n Fgu P5.9b. Compu h nsn n subnsn nucncs o h gno G. b Compu h sy s conons o gno G,..,,,, E, E, E, n δ. V5kV V5kV G 4 G x6% x6% P G 4MW 5 MW j MVA Fgu P5.9 Elcc Pow Sysm bl P5.9 Pow low Soluon V V V V 4 V 5 Copygh A. P. Sks Mlopoulos 99-6 Pg 79

80 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos. j... - V j. j. 5 j.5 - ω -xs ccu. j..4 j. 5 j.4 - ω -xs ccu Fgu P5.9b - n -xs Ccus o Gno G Poblm 5. h p un pms o 45 MVA, 6 Hz, 4-pol synchonous gno gn n bl P5.. s sbl o compu h ohmc losss o h synchonous mchn o y o opng conons. Fo hs pupos compl h bl P5.b. Fo ch ny o bl P5.b, compu h sy s opng conon o h synchonous mchn n h ohmc losss n h so n oo. Exmn you soluon n s you obsons. bl P5.. Synchonous Mchn Pms n pu. 46 pu. 565 pu. pu l. 59 pu l. 59 pu. 956 pu l. 44 pu l. 856 pu l. 8 pu pu pu A. 869 A. 689 pu Pg 8 Copygh A. P. Sks Mlopoulos 99-6

81 Pow Sysm Molng, Anlyss n Conol: Chp 5, Mlopoulos bl P5.b Opng Conon S45 MVA Volg PowFco. S5 MVA Volg PowFco. S45 MVA Volg PowFco.9 lggng S45 MVA Volg PowFco.9 lng oo Poson Angl δ Gn Volg pu So osss W oo osss W Copygh A. P. Sks Mlopoulos 99-6 Pg 8