A. P. Sakis Meliopoulos Power System Modeling, Analysis and Control. Chapter 4 2 Modeling - Power Transformers and the Per Unit System 2

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1 DRAFT nd COMPETE Tle f Cnens frm A. P. Skis Melipls Pwer Sysem Mdeling, Anlysis nd Cnrl Chper 4 Mdeling - Pwer Trnsfrmers nd he Per Uni Sysem 4. nrdcin 4. Single Phse Trnsfrmers 4.. The del Single Phse Trnsfrmer 4.. mpednce Trnsfrmin Thery f Mlly Cpled ndcnces 7 4. The Per Uni Sysem Aximic Definiin f he Per-Uni Scling Sysem Per-Uni Scling Prcedre Mlly Cpled Circis 4..4 The Per Uni Sysem nd mericl Accrcy Discssin Three Phse Trnsfrmers The del Three Phse Trnsfrmer 4.4. n-del Three Phse Trnsfrmer Mdel 4.4. Seqence Circis f Three Phse Trnsfrmers Arnsfrmers Single-Phse Arnsfrmers Three Phse Arnsfrmers The Regling Trnsfrmer Trnsfrmer Pwer Flw Eqins Trnsfrmer Prmeer denificin Frm Tess The Open Circi Tes The Shr Circi Tes rmlized Pwer Sysem Mdel 5 4. Smmry nd Discssin Prlems 56

2 Chper 4 Mdeling - Pwer Trnsfrmers nd he Per Uni Sysem 4. nrdcin n his chper we exmine he pwer rnsfrmer. Single phse, hree phse s well s regling rnsfrmers re cnsidered. We sr frm he physicl cnsrcin f rnsfrmer, we exmine he perin f he rnsfrmer nd develp mhemicl mdel which nrlly leds n eqivlen circi. This pprch is fllwed fr ll ypes f rnsfrmers, single phse, hree phse r regling rnsfrmers. Trnsfrmer mdels cn e frher simplified wih he se f specific scling f elecricl qniies, herwise knwn s he per ni sysem. The per ni sysem hs een exensively sed in mdeling nd nlysis f pwer sysems wih mliple rnsfrmin levels f vlge rnsfrmers. is nrl inrdce he per ni sysem in his chper. 4. Single Phse Trnsfrmers A single phse pwer rnsfrmer cnsiss f mgneic cre nd w windings s i is illsred in Figre 4.. This is rher simple cnsrcin f mgneic circi. The exciin f he mgneic circi is resl f elecric crren flw in he w windings. Specificlly when elecric crren flws in ne r h windings, mgneic flx will e genered inside he irn cre f he rnsfrmer. The lerning mgneic flx indces vlges n he w windings nd i is respnsile fr he pwer rnsfer frm ne winding nher. The w windings mny imes re referred s primry nd secndry, which ypiclly indice he srce side nd he ld side, respecively. Anher descripin f he w windings is high vlge side r lw vlge side depending n he vlge level he w windings. When he elecric crren flws in ne winding f he rnsfrmer f Figre 4., mgneic field will e genered in he cre f he rnsfrmer. f he elecric crren is lerning, he mgneic field will e ls lerning. Cnsider he her cil. The lerning mgneic flx links his cil. As resl, vlge will e indced in his cil. The plriy f he indced vlge wih respec he crren i will depend n hw he cil is wnd. We cn define he plriy f he rnsfrmer, nd we symlize he plriy wih w ds s i is illsred in Figre 4., s fllws. f he direcin f he elecric crren flw in cil is in he d, hen he indced vlge n he her cil will hve psiive plriy n he d side f his cil. Cnsidering he cnsrcin Pge Cpyrigh A. P. Skis Melipls 99-6

3 f he rnsfrmer f Figre 4., n elecric crren i flwing in he d, will genere mgneic flx in he indiced direcin se he righ hnd rle. This mgneic flx will indce vlge n cil which will hve psiive plriy n he p erminl. This discssin shld explin hw he ds hve een plced. w, sing he ds we cn frge he cnsrcin f he rnsfrmer nd insed se he symlic represenin f Figre 4.. Φ m i i' Φ l Φ l i i' Figre 4. Single Phse Trnsfrmer Cnsrcin Symlic Represenin 4.. The del Single Phse Trnsfrmer Pwer rnsfrmers re ypiclly cnsrced wih high cndciviy wires cpper, nd herefre lw resisnce. The mgneic cre is ypiclly mde frm crn seel which hs very high permeiliy nd i is lmined minimize lsses. is expedien ke dvnge f hese servins, fr he prpse f exmining he sic prperies f rnsfrmers. Specificlly, we cn idelize he prmeers f he rnsfrmer, yielding he cncep f he idel rnsfrmer. The specific ssmpins f he idel rnsfrmer re lsses in he windings r in he cre Cre meril hs infinie permeiliy, i.e. µ which implies h here is n lekge flxes The mgneic resisnce f he mgneic circi f Figre 4. is: Cpyrigh A. P. Skis Melipls 99-6 Pge

4 A l µ R, where is he lengh f he mgneic circi, A is he crss secin f he cre, nd µ is he permeiliy f he cre meril. Fr he idel rnsfrmer, l R, nd he l mgnemive frce fr he rnsfrmer f Figre 4. is P R i i ϕ Therefre: i i e h he mgneic flx f is linked y h windings. The mgneic flx linkge f he w windings re ϕ λ nd ϕ λ respecively. Therefre he fllwing indced vlges will pper n he w windings: d d d d e ϕ λ, d d d d e ϕ λ Therefre pn eliminin f he flx vrile: e e Ave relinship indices h he vlges he erminls f n idel rnsfrmer re inversely prprinl he nmer f rns f he w windings. The l pwer sred y he idel rnsfrmer is i e i e i e i e p Ave relinships re vlid fr he insnnes vles f vlge, crren nd pwer. Under sinsidl sedy se cndiins, hese relinships re ls vlid fr he phsrs f he vlge nd crren nd he cmplex pwer, i.e. E E Pge 4 Cpyrigh A. P. Skis Melipls 99-6

5 * * E E S The ve relinships chrcerize n idel rnsfrmer. Prcicl rnsfrmers re designed s h heir perin nder nrml pering cndiins is clse he idel cse. 4.. mpednce Trnsfrmin A rnsfrmer insered eween srce nd n impednce ld cn rnsfrm he ppren impednce seen y he srce s i is illsred in Figre 4.. f he rnsfrmer is ner idel, his rnsfrmin f he impednce cn e chieved wih ny ssnil pwer lsses in he rnsfrmer. Cnsidering Figre 4., he impednce he lef-hnd side f he rnsfrmer is given wih: z : z z Srce Srce Figre 4. mpednce Trnsfrmin y Trnsfrmer e h he inserin f rnsfrmer wih rnsfrmin ri / eween srce nd ld f impednce, resled in n ppren impednce /. This prpery is mny imes sed fr mching ld specific srce. Cnsider he prlem f cnnecing srce ld nd h he srce hs n inernl impednce f S while he ld hs n impednce. Fr wh vle f he ld impednce, he srce will rnsfer mximm pwer he ld? The pwer sred y will e * Re Re E X X R R R E P S S s Cpyrigh A. P. Skis Melipls 99-6 Pge 5

6 where is he vlge he erminl f he ld, E is he inernl vlge srce. R jx s s R jx s w we wld like find he mximm f P y llwing he ld impednce chnge. The mximm cn e cmped y slving he eqins. P R P X The slin f ve eqins yields: r R R S X X S * S Mny imes i is pssile chieve he ve cndiins r cme very clse i y insering rnsfrmer eween he srce nd he ld. An exmple will illsre his pplicin. Exmple E4.: Cnsider srce wih 5 hm inernl impednce. The srce is e cnneced n 8 hm ld s i is illsred in Figre E4.. Hw cn we rnsfer he mximm pssile pwer he ld. 5Ω rms 8Ω Amplifier Speker Figre E4. A Srce nd d Slin: f he ld is direcly cnneced he srce, he pwer rnsferred he ld will e cmped s fllws. Firs he elecric crren is Pge 6 Cpyrigh A. P. Skis Melipls 99-6

7 /58.74 A rms The pwer sred y he ld is P Re * Re[8.74 ].78 Ws Fr mximm pwer rnsfer, he ld ms e cnneced he srce vi rnsfrmer in sch wy h he impednce f he rnsfrmer/speker sysem, s indiced in Figre E4., e: * S This mens h / 8Ω 5 Ω nd herefre /.5. n his cse ' /Ω A, nd.5.5a. The mximm pwer sred he ld is P 8Ω. 5A 5Wls. 5Ω ' ' rms 8Ω : Figre E Thery f Mlly Cpled ndcnces n his secin we cnsider single phse rnsfrmer wih he idel ssmpin, i.e. he windings f he rnsfrmer hve finie resisnce nd he mgneic cre hs finie permeiliy. n his cse, he mgneic flx genered y he flw f crrens in he windings is n cnfined in he mgneic cre sme mgneic flx leks in he ir s i is illsred in Figre 4.. Specificlly Figre 4. illsres he mgneic flx de elecric crren in winding f he rnsfrmer nd Figre 4. illsres he mgneic flx de elecric crren in winding f he rnsfrmer. The nin in Figre 4. is s fllws: ϕ is he l mgneic flx de crren i. ϕ is he mgneic flx de crren i nd which links he winding. ϕ l is he mgneic flx de elecric crren i nd which leks in ir nd herefre i des n link winding. The crrespnding mgneic flxes in Figre 4. de crren i re ϕ, ϕ nd ϕ l. w if he w windings crry crrens i nd i simlnesly, he mgneic flx linkge fr windings nd cn e fnd y pplying sperpsiin ssming h he rnsfrmer peres in he liner regin: λ ϕ ϕ ϕ l ϕ ϕ λ ϕ ϕ ϕ ϕ ϕ l Cpyrigh A. P. Skis Melipls 99-6 Pge 7

8 The vlge f he w cils will e: dλ dϕl d ϕ ϕ v ri ri d d d dλ dϕ l d ϕ ϕ v ri ri d d d e h he flx lekge ϕ l is genered y he elecric crren i nd herefre i will e prprinl his crren. Similrly, he flx lekge ϕ l is prprinl he elecric crren i, i.e. ϕ l i ϕ l i i ϕ ϕ i v ϕ l v i ϕ ϕ i v ϕ l v Figre 4. Mgneic Flx in Single Phse Trnsfrmer Primry Cil Energizin Secndry Cil Energizin Pge 8 Cpyrigh A. P. Skis Melipls 99-6

9 y defining in: φ ϕ ϕ, nd ssiing in he ve eqins, we di dφ v ri l d d di dφ v ri l d d Where l nd l w le R e he relcnce f he mgneic cre. Then ϕ ϕ i / R i / R nd ϕ i i / R Fr mgneic cre rnsfrmers, he mgneic relncnce R is very smll nd herefre i i is ls very smll. Define n eqivlen mgnemive frce in erm f he mgneizing crren i m: i i im Ave eqin cn e rewrien: m i i i Define where: dφ e d R d e φ d m R dim dim m d d Upn ssiin, he eqins f he rnsfrmer cn e rewrien in he fllwing frm: di v ri l e d di v ri l e d Cpyrigh A. P. Skis Melipls 99-6 Pge 9

10 dim e m d i im i e e Ave eqins re represened wih he eqivlen circi f Figre 4.4. e h he ls w eqins represen n idel rnsfrmer which is illsred in Figre 4.4. i r l i -i m l r i i m v m e e v del Trnsfrmer Figre 4.4 Eqivlen Circi f Single Phse Trnsfrmer The indcnce m is referred s he mgneizing indcnce nd he crren i m s he mgneizing crren. Fr pwer rnsfrmers nder nrml pering cndiins, he mgneizing crren is very smll nd mny imes i is negleced, i.e. he indcnce m is remved frm he eqivlen circi. Thrgh simple rnsfrmin, he impednces cn e mved he her side f he idel rnsfrmer. Under sedy se cndiins, he vlges nd crrens cn e expressed in erms f heir phsrs see Chper. n his cse, he rnsfrmer eqins re rnsfrmed in: r jω l E r jω l E E jω m m m E E is imprn ne h he resisnce nd he lekge indcnces f ypicl rnsfrmer re very smll while he mgneizing indcnce is very lrge. Under hese cndiins, he lekge impednce f ne side cn e rnsferred he her side. Recll h y rnsferring n impednce frm ne side f he rnsfrmer he her, he Pge Cpyrigh A. P. Skis Melipls 99-6

11 impednce is mliplied y he sqre f he rnsfrmin ri. ecse he mgneizing impednce is very lrge, he w lekge impednces cn e mved he sme lcin prviding n eqivlen lekge impednce. The resling eqivlen circi is shwn in Figre 4.5. e h his is n pprximin. The errr cmmied y his pprximin is relively smll. e h he simplified rnsfrmer mdel eqivlen is descried wih he eqivlen series impednce eq, he mgneizing impednce m nd he rnsfrmin ri. r jω l z eq r jω l del Trnsfrmer jω m : Figre 4.5 Simplified Eqivlen Circi f Single Phse Trnsfrmer is imprn ne h fr he single phse rnsfrmer, i is lwys pssile develp n eqivlen circi wih he presence f he idel rnsfrmer. The prcedre will e illsred n he simplified eqivlen circi f Figre 4.5. e h /. Fr his prpse, he erminl elecric crrens re expressed s fncin f he erminl vlges yielding deils re mied: where: Y eq Y eq eq Y eq, nd Y Y eq eq Y m Y m m The eqivlen circi f Figre 4.6 represens excly he ve eqins. The reder is encrged verify h he circi f Figre 4.6 exhiis he sme crren/vlge relinships s he ve eqins. This cn e dne y simply expressing he erminl crrens f he circi f Figre 4.6 s fncins f he erminl vlges nd cmpring he resling eqins he ve eqins. Cpyrigh A. P. Skis Melipls 99-6 Pge

12 : eq Y eq - del Trnsfrmer Y eq -Y eq -Y eq Figre 4.6 Simplified Single Phse Trnsfrmer Mdels The perfrmnce f rnsfrmers is chrcerized wih nmer f indices. The fllwing ms cmmn indices re inrdced: Trnsfrmer efficiency is defined s he ri f he nminl rel pwer p ver he reqired rel pwer inp, i.e. η P /P in. lge Reglin is defined s he rnge f he ri f he nrmlized p vlge mgnide ver he nrmlized inp vlge mgnide fr lding cndiins rnging frm n ld fll ld. Fr ypicl rnsfrmers, he ri f he nrmlized vlges he w ends nder n ld cndiins is pprximely.. Therefre he vlge reglin will e: R. fll ld, where:, nd. The sscrip n mens nminl vlge. Exmple E4.: The eqivlen circi f 4.4 k/, 6 Hz, 5 ka rnsfrmer is illsred in Figre E4.. The rnsfrmer spplies he secndry red pwer nder red vlge wih pwer fcr.. Cmpe: n Cmpe he vlge he high side n Pge Cpyrigh A. P. Skis Melipls 99-6

13 Cmpe he rnsfrmer efficiency, nd c Wh is he rnsfrmer vlge reglin j MΩ g j.4mω E j e Red d Pwer Fcr High lge Side w lge Side Figre E4.. Prmeers f he Trnsfrmer f Exmple E4. Slin: The d re mrked n Figre E4.. e h j j E 44 e 44e vls A. e j 46 A 44 The vlge he rnsfrmer high side is cmped s fllws: 4, 4 g. M e j e j 48 A Ω 4, 4. j.4m e j e j9 8 A Ω j j j g. 46e. 48e. 8e. 465e j. j. 4, 4 j. 465 e 4, e 4, 47. 4e j5. The rnsfrmer lsses re: 9 j P lss Ws P 5, Ws P in 5,9. Ws η.999 c The rnsfrmer vlge reglin is Cpyrigh A. P. Skis Melipls 99-6 Pge

14 / R..87 4,47 /4,4 4. The Per Uni Sysem The per ni sysem is scling prcedre y which physicl qniies re expressed in ser seleced se f nis, clled he ses. This rnsfrmin f physicl qniies resls in rnsfrmin f he mhemicl mdels fr pwer pprs. The sic ide f he prcedre will e descried firs fllwed wih exmples. Then, he per ni sysem will e inrdced ximiclly. The sic ide f he per ni sysem is express physicl qniies in new sysem f nis. Cnsider fr exmple he physicl qniy f cmplex pwer, S. S is cmplex qniy nd he ni f his qniy is A vl.mpere. w ssme h we wn eslish new ni fr S which is S A, fr exmple S A. e S is sclr. w he physicl qniy S expressed in he new ni is S, where S S/S. e h S is nw dimensinless. S is referred s he se fr S nd S is he per ni vle f S. Cnceplly, his simple scling prcedre cn e pplied ll physicl qniies. The ses fr ll physicl qniies cn e rirrily seleced. n prcice, hwever, we plce cnsrins in he selecin f he ses. Fr exmple, we reqire h physicl lws sch s Ohm s lw nd he pwer eqin re sisfied y he ses, i.e. S Where,,, S re he ses fr vlge, impednce, crren nd pwer respecively. Ave reqiremen limis he nmer f ses which cn e seleced rirrily. Fr he menined fr qniies,, nd S, since hey ms sisfy w eqins ve, hen nly w qniies cn e rirrily seleced. Uslly, we rirrily selec S,. Then, he ses nd re deermined frm he pwer eqin nd Ohm s lw: S S The ve descried sic prcedre will e illsred wih n exmple. Pge 4 Cpyrigh A. P. Skis Melipls 99-6

15 Exmple E4.: Cnsider he simplified circi f Figre E4.. Selec s ses fr pwer nd vlge S W nd, respecively. Then express he circi eqins in he new ses. r Ω rms jx j Ω Figre E4.. A Simple Elecric Circi Slin: Since S W, Then 4.4 Ω, nd 8. Amps The eqin fr he circi f Figre E4. is r jx Dividing ve eqin y he vlge se,, nd recgnizing h, r jx r r jx where., r.89, nd x.89. Ths..89 j.89 The ls eqin is represened wih he circi f Figre E4.. e h he circi f Figre E4. is dimensinless. Cpyrigh A. P. Skis Melipls 99-6 Pge 5

16 .89. j.89 Figre E4.. Scled ersin f he Simple Circi f Figre E Aximic Definiin f he Per-Uni Scling Sysem The per ni sysem ws inrdced simplify he mdeling prcedre f pwer sysems. n his respec, here re sme desirle chrcerisics h he per-ni scling prcedre shld pssess. These re: The per ni sysem shld e cmpile wih her sndrd sysems, sch s he meric sysem. Cmpiiliy mens h he physicl lws shld e descried wih he sme fmilir mhemicl expressins, nd ses shld e seleced s h scled vles re nrmlly ner niy. These reqiremen re me wih per ni sysem which sisfies he fllwing xims [???]: Axim : Unscled eqins which descrie physicl lws shll e idenicl in frm heir scled versins. Fr exmple, scling prcedre which cnvers Ohm's lw he frm " k," where k is ncceple. Axim : A given lcin in pwer sysem, he se vle fr ech qniy e scled vlge, crren, impednce, pwer, ec. ms e niqe. Axim : Trnsfrmer mdels ms e simplified y he scling prcedre. n he nex secin, we descrie he prcedre fr selecing per-ni sysem h mees ve xims. 4.. Per-Uni Scling Prcedre The fndmenl scling eqin is: cl vle per ni vle 4. se vle where cl vle he nscled qniy in pprprie S ni; phsr r cmplex nmer r fncin f ime. se vle rel nmer seleced cnfrm he scling xims. per-ni vle he scled dimensinless qniy; phsr r cmplex nmer, r fncin f ime. The erm "qniy" here refers ny physicl r mhemicl eniy reling pwer sysem nlysis, inclding, n limied, vlge, crren, pwer, impednce, Pge 6 Cpyrigh A. P. Skis Melipls 99-6

17 resisnce, indcnce, cpcince, chrge, flx linkge, ime, rqe, psiin liner nd nglr, velciy, nd ccelerin. The prcedre fr selecing he ses fr he per-ni sysem is similr he prcess f selecing he S sysem f nis. Recll h he S sysem f nis [???] is sed n nmer f rirrily seleced se nis. All her nis re expressed s fncins f he se nis nd re clled derived nis. Cnseqenly, i is necessry fllw he sme prcedre: rirrily selec minimm nmer f ses. These ses shll e clled ssigned ses. The remining ses cn e derived nd shll e clled derived ses. The derived ses ms e seleced sch s xim is sisfied. Fr pwer engineering pplicins, is n necessry selec s ssigned ses he fll cmplimen f se S nis. Fr ll prcicl pwer engineering pplicins, i sffices selec nly he fllwing hree ssigned ses: pwer, vlge, nd c freqency. The derived ses f ineres will e ime, crren, c impednce, d indcnce, e cpcince, f elecric chrge, nd g mgneic flx. The prcedre is lined elw. Sep : Selec n rirry lcin in pwer sysem s he reference s. A his lcin, define: Assigned ses f freqency se vlge se S pwer se nrml pering sysem freqency, in Hz nminl rms line--nerl vlge, in vls nminl per phse pwer in A ses fr ll her physicl qniies his lcin cn e derived frm relins idenicl hse physicl lws which inerrele he nscled vriles. Derived ses ime se crren se impednce se indcnce se f S S C cpcince se S πf πf Q elecric chrge se S C πf 4. πf S πf Cpyrigh A. P. Skis Melipls 99-6 Pge 7

18 λ mgneic flx se πf Sep : Selec he ssigned se vles he nex lcin s f he sysem. Tw cses re recgnized: Cse : There is n rnsfrmer eween he presen lcin nd nher lcin fr exmple s k where he ses hve lredy een ssigned. n his cse he ssigned ses shld e he sme s he ssigned ses s k. e h he derived ses will e ls he sme. Cse : There is rnsfrmer eween he presen lcin nd nher lcin fr exmple s k where he ses hve lredy een ssigned. n his cse we pply xim which reqires h he rnsfrmer mdel e simplified y he scling prcedre. e s exmine he rnsfrmin f rnsfrmer mdels y he scling prcedre nd hen decide hw chse he ses. Cnsider single phse rnsfrmer f rnsfrmin ri /. Assme h he ssigned ses hve een deermined ne side f he rnsfrmer, f, S, nd. is self-ndersd h he derived ses hve een ls deermined vi eqins 4.. We cnsider he prlem f selecing he ssigned ses he her side side f he rnsfrmer, i.e., f, S, nd. The eqins descriing he rnsfrmer re fr simpliciy he mgneizing crren is negleced: di dφ v ri l d d di dφ v ri l d d i i The per-niized eqins re ined y dividing he firs eqin y, nd he secnd eqin y. n he prcess, we cn ssie wih is eql, ec., s pprprie yielding: v r i l i d d dφ d 4. Pge 8 Cpyrigh A. P. Skis Melipls 99-6

19 d d d i d i r v φ l 4.4 i i r d d d di i r v φ l d d d di i r v φ l i i mpsing h he pwer flw eqins will remin invrin xim resls in he ddiinl reqiremens: S S, n his cse, S /, S / nd herefre / /. e nw h he ls eqin ecmes i i i / /. Ave eqins represen rnsfrmer wih rnsfrmin ri / / which is illsred wih he circi f Figre 4.7. The circi f Figre 4.7 is simplified if he ses re seleced s h he rnsfrmin ri f he idel rnsfrmer is :: r 4.5 n his cse, per niized eqins f he rnsfrmer ecme: e d di i r v l Cpyrigh A. P. Skis Melipls 99-6 Pge 9

20 where: di v r i l e d i i e dφ d d d φ Ave eqins re represened wih he circi f Figre 4.7. e h in his cse xim is sisfied, h is he eqivlen circi f he rnsfrmer hs een simplified, he idel rnsfrmer hs een remved. Hwever, mny imes i is n pssile selec he ses ccrding Eqin 4.5. An exmple will e given ler. n his cse, he per niized rnsfrmer mdel incldes n idel rnsfrmer s illsred in Figre 4.6. i r l del Trnsfrmer l r i v v i r l l r i v Figre 4.7. Perniizin f Trnsfrmer Circi Relizin f Eqins 4. nd 4.4 Eqivlen Circi in Cse / / Ave cmplees he prcedres fr he selecin f he ses. Pge Cpyrigh A. P. Skis Melipls 99-6

21 n smmry, prcicl wy pply ve prcedre pwer sysem is s fllws: Clssify ll pwer sysem cmpnens s shn r series cmpnens. Shn cmpnens re hse which re cnneced ne s nly, i.e. generrs, lds, cpcir nks, ec. Series cmpnens re hse which inercnnec w ses. Exmples inclde pwer rnsmissin lines, rnsfrmers, slid se cnverers, ec. Selec s f he pwer sysem, ny s. A his s le i e s i, selec he "ssigned ses" s fllws: The vlge se is seleced e he "nminl" line grnd rms vlge f he s. The pwer se s i is rirrily seleced e S The freqency se s i is seleced e f nrml pering sysem freqency s i. 4 All her ses derived s i re cmped frm Eqins 4.. Selec he ssigned ses fr ll her ses prceeding ne s ime. Specificlly: Cnsider series cmpnen eween ses j nd k, fr which he ssigned ses hve een deermined s j n s k. f he series cmpnen is: her hn rnsfrmer: ssign ses s k sme s s j, rnsfrmer wih nminl rnsfrmin ri : ssign ses his s s fllws: S k S j k j f k f j where is he rnsfrmin ri f he rnsfrmer. Cmpe he derived ses s k wih Eqins 4.. The per-niizin prcess is cmpleed when ssigned nd derived ses hve een defined fr ll ses in he sysem. 4.. Mlly Cpled Circis Cnsider w mlly cpled circis nd. Assme h he w circis re mlly cpled nd h he ssigned ses hve een deermined fr he w circis, i.e. f, S, nd f f, S S, nd. This sysem is illsred in Figre 4.7. The eqins descriing his sysem re: j k m Cpyrigh A. P. Skis Melipls 99-6 Pge

22 m k j Ave eqin cn e scled y dividing he firs eqin y nd he secnd eqin y, yielding m k j m k j j m j k k Figre 4.8 Tw Mlly Cpled Circis n ve eqins we sed he fc h nd. w serve h The prf f ve relinship cn e ffeced y direc ssiin f he eqins f derived se qniies in erms f he ssigned se qniies. Upn ssiin in he eqin fr he cpled circis: m k j m k j Ave eqins sgges h he se fr he ml impednce shld e m Pge Cpyrigh A. P. Skis Melipls 99-6

23 m m m w he eqins f he mlly cpled circis re j k m j k m The per-niizin prcess f sysem wih mliple mlly cpled circis, sch s rnsfrmer wih mliple windings r synchrns generr, is rher cmplex. The per-niizin f synchrns generr mdel will e ddressed in Chper 5. The ineresed reder shld ls cnsl Reference [???] fr cmprehensive per-niizin prcedre f he synchrns generr mdel. The per niizin prcedre will e illsred wih w exmples. Exmple E4.4: Cnsider he eqivlen circi f single phse rnsfrmer nd ld s i is illsred in Figre E4.4. Selec per ni sysem, nd per-niize he eqivlen circi f Figre E4.4. The rnsfrmer is 4/4, 5 ka rnsfrmer. Ω j Ω jω.ω x Ω -j Ω Ω jω : Figre E4.4 Eqivlen Circi f Trnsfrmer nd Elecric d Slin: The ssigned nd derived ses re: Primry Side: 4 S 5 A A 4 5 Ω 5 Y S Cpyrigh A. P. Skis Melipls 99-6 Pge

24 Secndry Side: 4 S 5 A.8 A.5 Ω Y S Upn cmpin f rnsfrmer prmeers n he new se, he eqivlen circi f Figre E4.4 is ined..868 j.868 j j.5 j.64 Figre E4.4 Per-Uniized Circi f he Sysem f Figre E4.4 Exmple E4.5: Cnsider n exmple pwer sysem. The psiive seqence mdel f his sysem is illsred in Figre E4.5. Assme h he ssigned ses s re f 6 Hz, S. MA, nd 7. k. Selec he ssigned ses ll ses f he sysem. Cmpe he per ni psiive seqence newrk f he sysem. j. hms j. hms 4j5 hms j hms 4 7. k j.5 hms.4 k 7.: :.4.5j8 hms.5j5 hms 5 j.5 hms 6 :5 Figre E4.5 Psiive Seqence ewrk f n Exmple Pwer Sysem Slin: The ssigned ses fr ll ses re seleced nd lised in Tle E4.5. Pge 4 Cpyrigh A. P. Skis Melipls 99-6

25 Tle E4.5 Assigned ses fr he Exmple Sysem f Figre E4.5 s k S MA f Hz e h he derived ses re clcled direcly frm Eqins 4.. Fr exmple, s 6: S /.596 ka /.8 Ohms /πf.589 H /f ms, ec. The se impednces re cmped sing eqins 4. nd lised in Tle E4.5. Using he ses f Tles E4.5 nd E4.5, he per ni psiive seqence newrk is cmped nd shwn in Figre E4.5. Tle E4.5 Derived mpednce ses fr he Exmple Sysem f Figre E4.4 s hms Cpyrigh A. P. Skis Melipls 99-6 Pge 5

26 j.86 j.77.j.78 j.98 4 j j.7.j j :. Figre E4.5 Per Uni Psiive Seqence ewrk f he Exmple Sysem f Figre E The Per Uni Sysem nd mericl Accrcy The dvnges f he per ni sysem re w: he nmericl vles f he per ni vlges indice hw clse he vlges re ner nminl vles, i.e. per ni vle f.98 indices h he vlge is 98% f he nminl vle, nd he per ni vles re dimensinless. Her we will discss nher less vis dvnge. Specificlly, he per ni sysem cndiins he pwer sysem eqins in sch wy h he nmericl precisin f cmpins n finie precisin cmper will e incresed. Recll recen resl [???] which ses h he nmericl errr f sysem f eqins is prprinl he mrix cndiin nmer k which is defined s he ri f he lrges he smlles singlr vle f he mrix, i.e. k σ mx /σ min. The per ni sysem resls in sysem eqins wih smller cndiin nmer nd hs incresed nmericl ccrcy. This prpery will e illsred wih n exmple. Exmple E4.6: Cnsider 5 ka, 4.4 k/, 6 Hz single phse disriin rnsfrmer. The prmeers f he rnsfrmer re illsred in Figre E4.6. Fr simpliciy, he cil resisnce will e negleced. The mdel descriing he rnsfrmer f Figre E4.6 see secin??? is: di dim v l m d d di dim v l d d m Pge 6 Cpyrigh A. P. Skis Melipls 99-6

27 i m i i Upn eliminin f he mgneizing crren, i m : v di m d di d di l v d di d m di d di m m l d i l l i v m v 44: Figre E4.6 Single Phse Disriin Trnsfrmer Assme his mdel is e sed fr rnsien nlysis. negring ve eqins sing he rpezidl rle, wih n inegrin sep f h, we in: v v l h m h m l m h h m i i i i h v h v h h Wh we wld like d is sdy he nmericl prperies f he mrix efre nd fer per-niizin. Fr his prpse, selec: Assigned ses: High vlge side: Pwer : 5 ka lge: 4.4 k Freqency: 6 Hz w vlge side: Pwer : lge: 5 ka Cpyrigh A. P. Skis Melipls 99-6 Pge 7

28 Derived ses: Freqency: 6 Hz High vlge side: w vlge side: Ml indcnce se: H.856 H m.566 H Using hese ses nd n inegrin ime sep f h micrsecnds, he mrices re: Acl Unis: Per-Uniized Mrix: Upn cmpin f he singlr vles f ve mrix sing mhemicl pckge, we in: Acl Unis Mrix: rges singlr vle: Smlles singlr vle:.694 x -5 Mrix cndiin nmer:. x 7 Per-Uniized Mrix: rges singlr vle: Smlles singlr vle:.597 x - Mrix cndiin nmer:.877 x 4 e h he per-niizin redced he cndiin nmer f he mrix y fcr f 7.5! redcin f hree rders f mgnide. This mens he nmericl rncin errr will e redced y fcr f 7.5 sing he per-niized eqins [???] Discssin The per ni sysem hs hree dvnges: pwer sysem cmpnen mdels re simplified, fr exmple recll he cse f rnsfrmer, he per ni vles f vlge r pwer my e meningfl, fr exmple vle f. p fr vlge my men h Pge 8 Cpyrigh A. P. Skis Melipls 99-6

29 he vlge is eql he nminl vlge, nd he mdel eqins re eer cndiined frm he nmericl pin f view. Advnge is very imprn frm he prcicl pin f view nd pplies her qniies s well. Cnsider fr exmple he lekge impednce f rnsfrmer. When i is expressed in p.. sing rnsfrmer rings s he ses, he p lekge impednce is ypiclly cnsn fr specific designs. Specificlly, lrge pwer rnsfrmers hve lekge impednce f.7. p. Smll disriin clss rnsfrmers hve lekge impednce f.. p., ec. n her wrds while he prmeers f rnsfrmer my spn lrge rnge, when hey re perniized sing he red vles s ses, he prmeer vles re lced in relively nrrw rnge. 4.4 Three Phse Trnsfrmers Three phse rnsfrmers cn e cnsrced in nmer f wys. Three f he ms sl cnsrcins re illsred in Figre 4.9. Figre 4.9 illsres hree phse cre ype rnsfrmer. The cre hs hree legs, n ech leg here re w windings fr l f six windings. Similrly, Figre 4.9 illsres shell ype rnsfrmer which ls hs six windings. Figre 4.9c illsres nk f hree single phse rnsfrmers. This rrngemen ls hs six windings. The six windings f ny cnfigrin,, r c re grped in w grps f hree, he primry nd he secndry. Fr exmple, in Figre 4.9 he primry my e he hree windings lced n he pper pr f ech leg nd he secndry my e he her hree winding. h he primry nd secndry windings my e cnneced in del r wye cnfigrins leding fr pssile rrngemens f hree phse rnsfrmer: del-del, wye-del, c del-wye nd d wye-wye. These rrngemens re schemiclly represened in Figre 4.. e h frm he circi pin f view, ll hree phse rnsfrmer cnsrcins re similr, i.e. ll hve six winding grped in hree phses. Hwever he mgneic circi f ech ne f hese cnsrcins is differen. Fr exmple, he hree phse rnsfrmer nk cnsiss f hree independen mgneic circis. The shell nd cre ype hree phse rnsfrmers re chrcerized wih cpled mgneic circis f he hree phses. Cpyrigh A. P. Skis Melipls 99-6 Pge 9

30 c Figre 4.9 Three Phse Trnsfrmers Cre Type Shell Type c Three Single Phse Trnsfrmer nk The mdel f hree phse rnsfrmer nk is he simples since i cnsiss f he inercnnecin f hree single phse rnsfrmers. Replcing ech ne f he single phse rnsfrmers wih is eqivlen circi, he eqivlen circi f he hree phse rnsfrmer is ined. This hs een dne in Figre 4. where he simplified eqivlen circi f single phse rnsfrmer hs een sed. The Figre illsres del-wye cnnecin. n sseqen prgrphs we will cnsider firs he idel hree phse rnsfrmer mdel fr he prpse f exmining is sic chrcerisics. Then he nn-idel rnsfrmer mdel will e sdied. The se f he symmericl rnsfrmin he hree phse rnsfrmer mdel will resl in he seqence mdels. Pge Cpyrigh A. P. Skis Melipls 99-6

31 Y Y Y Y Figre 4. Schemic Represenin f Three Phse Trnsfrmers A ' A ' C C ' c c c α: Figre 4. Del-Wye Cnneced Three Phse Trnsfrmer Mdel 4.4. The del Three Phse Trnsfrmer Cpyrigh A. P. Skis Melipls 99-6 Pge

32 An idel hree phse rnsfrmer cnsiss f hree idel single phse rnsfrmers. The rnsfrmer f Figre 4. will e idel if Y shr circi. The vlge relinships f n idel hree phse rnsfrmer re: A n C n CA cn Under lnced pering cndiins, he vlges will e: j n Ane j4 Cn Ane j n ne j4 cn ne e h: e j j A An n An An An j j j9 C n Cn Ane Ane Ane j j j CA Cn An Ane Ane Ane w he relinship eween he primry nd secndry vlges cn e fnd. e j An ne j n ne j Cn cne An n n n Cn cn e e e j j j Ave eqins indice h he per phse psiive seqence eqivlen mdel f del-wye cnneced idel hree phse rnsfrmer is single phse idel rnsfrmer wih rnsfrmin ri: An j j n e e n e h in his cse he rnsfrmin ri is cmplex nmer. Pge Cpyrigh A. P. Skis Melipls 99-6

33 4.4. n-del Three Phse Trnsfrmer Mdel The nn-idel hree phse rnsfrmer mdel cn e derived frm he prper inercnnecin f he nn-idel single phse rnsfrmers. Fr simpliciy we ssme h ech single phse rnsfrmer is represened wih is simplified nn-idel mdel. Fr he cse f del-wye cnneced rnsfrmer, he resl is illsred in Figre 4.. Fr he circi f Figre 4., he fllwing relinships re vlid: ' Y n A ' Y n C ' ' Y cn A C c ' ' c c ' ' c A ' ' ' ' c C Upn eliminin f he vriles E E E,, c nd, ' ' ' nd expressing he remining crrens s fncin f he vlges we in se f six eqins which, wrien in mrix nin, re: c, C A c C A c Y e h ve eqin expresses he inp/p relinship f he hree phse rnsfrmer. n cmpc mrix frm, ve eqin cn e wrien s AC c T AC c F E E Y where is x idenify mrix, nd he mrices E nd F re: E Cpyrigh A. P. Skis Melipls 99-6 Pge

34 F Ave eqins represen he simplified mdel f del-wye cnneced hree phse rnsfrmer. The sme prcedre cn prvide he mdels fr her cnnecins, i.e. deldel, wye-wye nd wye-del cnnecins Seqence Circis f Three Phse Trnsfrmers Three phse rnsfrmers re inherenly symmeric hree phse elemens. This mens h y pplying he symmericl rnsfrmin, heir mdel cn e rnsfrmed hree eqivlen circis, nmely he psiive, negive nd zer seqence eqivlen circis. The prcedre will e illsred n del-wye cnneced rnsfrmer mdel develped in he previs prgrph. shld e ndersd h he prcedre eqlly pplies ny her cnfigrin. The phse vlges nd crrens re ssied wih heir crrespnding symmericl cmpnens s fllws T c R T AC c T R AC T where R nd indice righ nd lef side respecively. Replcing he phse qniies wih he symmericl cmpnens, he eqin fr he hree phse rnsfrmer ecmes: R TT Y T TE T TET TFT R e h y direc evlin, he fllwing pply: TT Pge 4 Cpyrigh A. P. Skis Melipls 99-6

35 j j e e TET j j T e e T TE TFT Upn ssiin nd grping he six eqins in hree grps f w we in: j R j R R Y e Y Ye Y R j R j Y Y e Y Ye j R j R R Y e Y Ye Y R j R j Y Y e Y Ye R R R Y Y e h ve relins represen hree independen se f eqins crrespnding hree eqivlen circis which re illsred in Figre 4.. The vriles ppering in Figre 4. re: j j e e, Therefre: The reder is encrged verify h he circis f Figre 4. crrespnd he ve develped eqins. Fr his prpse, he erminl crrens in Figre 4. shld e expressed s fncins f he erminl vlges nd he resling eqins shld e cmpred he ve eqins. Cpyrigh A. P. Skis Melipls 99-6 Pge 5

36 An exmple will illsre he prcedre. Y Y Y del Trnsfrmer j e : del Trnsfrmer j e : del Trnsfrmer : R R R R R R Figre 4. Seqence Eqivlen f Del - Wye Cnneced Three Phse Trnsfrmer Exmple E4.7: A hree phse rnsfrmer nk is mde frm hree phse rnsfrmers. Ech single phse rnsfrmer hs he eqivlen circi f Figre E4.7. The hree phse cnnecins re illsred in Figre E4.7. Drw he psiive seqence eqivlen circi f he hree phse rnsfrmer nk wih ll impednces shwn n he lef hnd side. The rnsfrmer ri nd he impednce vles shld e clerly mrked in cl qniies, i.e. vls nd hms. Drw he psiive seqence eqivlen circi f he hree phse rnsfrmer nk wih ll impednces shwn n he righ hnd side. The rnsfrmer ri nd he impednce vles shld e clerly mrked in cl qniies, i.e. vls nd hms. c Drw he per phse eqivlen circi in per ni f he hree phse rnsfrmer nk sing he fllwing ses. ef Hnd Side: S MA ne phse Pge 6 Cpyrigh A. P. Skis Melipls 99-6

37 66.95 k line nerl Righ Hnd Side: S MA ne phse 6.98 k line nerl j6.65ω j.4ω A C 5k: k SMA c Figre E4.7 Cnsrcin f Three Phse Trnsfrmer nk Circi Mdel f Single Phse Trnsfrmer Three Phse Cnnecins. Ech lck Represens he Single Phse Trnsfrmer f Slin: The eqivlen circi f Figre E4.7 cn e mdified y referring he j.4 hm lekge impednce n he lef hnd side. y ding s, he hree phse rnsfrmer mdel ecmes idenicl he circi f Figre 4. wih Y S j. S j nd y ilizing he resls f he previs secin, he psiive seqence eqivlen circi f he rnsfrmer is shwn in Figre E4.7. Cpyrigh A. P. Skis Melipls 99-6 Pge 7

38 -j.68 S del Trnsfrmer Figre E4.7 j e y referring he dmince n he lef hnd side, he psiive seqence eqivlen circi f Figre E4.7 is ined. del Trnsfrmer : -j6.496 S j e : Figre E4.7 The se dmince he lef hnd side is: S Y. 68 S. The per ni vle f he dmince is: j Y j The per ni rnsfrmin ri is: j j e : e :. 5 Therefre he per ni eqivlen circi is shwn in Figre E4.7c. Similrly he circi f Figre E4.7d is develped. Pge 8 Cpyrigh A. P. Skis Melipls 99-6

39 -j. del Trnsfrmer del Trnsfrmer Figre E4.7c j e : -j. j e : Figre E4.7d 4.5 Arnsfrmers Arnsfrmers re exensively sed in pwer sysems fr inercnnecing w prs f sysem pering differen vlges. Arnsfrmers re ls sed in mny her pplicins. Fr exmple, vric is n rnsfrmer. Arnfrmers cn e single phse devices r hree phse devices. h f hese devices will e exmined nex Single-Phse Arnsfrmers The cnsrcin f single phse rnsfrmer is simpler hn he sl rnsfrmer, ecse i reqires nly ne winding. The cnsrcin f single phse rnsfrmer is illsred in Figre 4.. Cpyrigh A. P. Skis Melipls 99-6 Pge 9

40 i φ _ A i _ Figre 4. Single Phse Arnsfrmer Cnsrcin Symlic Represenin The sic relins f n rnsfrmer cn e derived y ssming h he rnsfrmer is idel. e ϕ e he mgneic flx inside he cre f he rnsfrmer f Figre 4.. e h d v ϕ d dϕ v d Ρ i i i Rϕ Frming he ri f he vlges nd crrens: v v i i The pwer relinships re p v i v i Pge 4 Cpyrigh A. P. Skis Melipls 99-6

41 Ave relinships f he idel rnsfrmer re ls vlid fr he phsrs f he vlges nd crrens, i.e. * S * e h ve eqins re idenicl he eqins f he idel w winding rnsfrmer Three Phse Arnsfrmers Three phse rnsfrmers hve mgneic circis similr hse f he sl rnsfrmers, excep h here is nly ne winding n ech mgneic leg f he rnsfrmer wih secndry p. This winding frms ne phse f he hree phse rnsfrmer. ecse f his rrngemen nd ecse f he need grnd he rnsfrmer, hree phse rnsfrmers re lwys wye-wye cnneced. The nlysis f hese rnsfrmers fllws he sme prcedre s he sl hree phse rnsfrmers. 4.6 The Regling Trnsfrmer The regling rnsfrmer is rnsfrmer rrngemen which llws vrile rnsfrmin ri. This ri my e rel r cmplex nmer. Typicl cnsrcins f regling rnsfrmers re shwn in Figre 4.4. Clse exminin f he plgy f hese rnsfrmers will revel h h rnsfrmers 4.4 nd 4.4 simply regle vlge mgnide while rnsfrmer 4.4c regle h vlge mgnide nd phse. Trnsfrmers h regle h mgnide nd phse f he vlge re referred s phse shifers. Cpyrigh A. P. Skis Melipls 99-6 Pge 4

42 A c C c A C A c C c Figre 4.4 Exmple Cnsrcin f Regling Trnsfrmers rile Tp Underd Trnsfrmer lge Mgnide Regling Trnsfrmer c Phse Shifer Trnsfrmer Pge 4 Cpyrigh A. P. Skis Melipls 99-6

43 Cnsider regling rnsfrmer which regles h vlge nd rel pwer phse shif. This rnsfrmer is srcrlly symmeric hree phse elemen. y mdeling ech cmpnen f his rnsfrmer nd sseqen pplicin f he symmericl rnsfrmin, he psiive, negive nd zer seqence eqivlen circis cn e develped. The deils re mied. The resl is illsred in Figre 4.5. The regling prperies f he rnsfrmer re expressed wih he cmplex rnsfrmin ri n e j α. n Figre 4.5, y is he series dmince f he rnsfrmer while ys is he shn dmince mgneizing dmince. shld e cler h he per ni sysem cn e pplied he mdels f Figre 4.5 yielding he perniized psiive, negive, nd zer seqence mdels f he rnsfrmer. The mdels will e idenicl hse f Figre 4.5 wih he excepin h he prmeers y, y s, nd will ssme heir per ni vles. Ths Figre 4.5 illsres eiher he cl seqence mdel r he perni seqence mdel. y y y y S y S del Trnsfrmer j e : del Trnsfrmer j e : R R R R R y S del Trnsfrmer R : Figre 4.5 Seqence Mdels f he Phse Shifer Psiive Seqence Mdel egive Seqence Mdel c er seqence Mdel Cpyrigh A. P. Skis Melipls 99-6 Pge 4

44 The vlge/crren relinship fr he regling rnsfrmer cn e develped direcly frm he eqivlen circis. As n exmple cnsider he psiive seqence mdel Figre 4.5. e h he vlge nd crren in per ni he righ hnd side f he idel rnsfrmer re R, nd R. The vlges nd crrens he her side f he idel rnsfrmer will e: j e α R nd j R respecively. e α These expressins re derived n he sis f zer pwer cnsmpin inside he idel rnsfrmer. ex, nd wih he help f Figre 4.5, he crrens nd R re expressed in erms f, R : R y jα e y jα e y y ys R 4.6 Fr n eqivlen circi exis, he dmince mrix ms e symmeric. Ths nly when α is rel, n eqivlen circi exiss nd i is illsred in Figre 4.6. The dmince eween nde nd eqls he negive f he enry, f he dmince mrix, i.e., he shn dmince nde will e eql he sm f enries, nd, f he dmince mrix i.e., he shn dmince nde will e eql he sm f he enries, nd, f he dmince mrix. The mdeling f regling rnsfrmers is illsred wih n exmple. y R -y -y y s R Figre 4.6 pi-eqivlen Circi f he Regling Trnsfrmer f Figre 4.4, fr α Exmple E4.8: Cnsider he fr s sysem f Figre E4.8. The elecric ld s 4 is. p. The genering ni generes.8 p rel pwer. The rnsfrmer is n ff nminl p rnsfrmer, is recnce is j.8 p nd he rnsfrmer p is se.5 p s 4 is he high side. The line d re given in Tle E4.8. Derive he psiive seqence eqivlen circi f he sysem in per ni. Pge 44 Cpyrigh A. P. Skis Melipls 99-6

45 Tle E4.8 Sysem D Circi Frm s T s Series Admince Shn Admince -j. J. -j9. J. 4 -j.5 J.5 Slin: The psiive seqence pi-eqivlen prmeers f he ff-nminl p rnsfrmer re : y s j.5 -j.6565 y s j.5 j.65 y 4.5 -j.5 -j.5 The verll psiive seqence eqivlen circi is illsred in Figre E4.8. G Slck s C T 4 S D4 Figre E4.8 A Fr s Exmple Pwer Sysem Cpyrigh A. P. Skis Melipls 99-6 Pge 45

46 G G j.5 -j j. j. -j9 j. j. j. -j.5 j.5 -j j.5 j.65 4 S D4 Figre E4.8 Psiive Seqence Eqivlen Circi in per ni f he Sysem f Figre E4.8 Exmple E4.9: Cnsider he simplified pwer sysem f Figre E4.9. Selec s e he reference s nd ssme: S MA /, 5 k /, f 6 Hz All he rnsmissin lines hve he sme series impednce f.j.5 hms nd negligile shn impednce. is desirle develp he per-ni mdel f his sysem. Fr his prpse selec he ssigned ses fr he remining ses,, 4, 5, nd 6, cmpe he psiive seqence per-ni mdel f his sysem. Pge 46 Cpyrigh A. P. Skis Melipls 99-6

47 5MA % Pwer Sysem Mdeling, Anlysis nd Cnrl: Chper 4, Melipls 5MA 5k 6 Hz 45MA 5k 6 Hz Y Y 4 5/k 5 MA 7.5%.j.5 Ohms /5k 45 MA 8% 6 5 Y Y 5/ Figre E4.9 A Simplified Pwer Sysem Slin: The seleced ses pper in Tle E4.9. The derived ses cn e cmped wih he eqins h hve een presened in secin???. Tle E4.9 Assigned ses fr Simplified Pwer Sysem f Figre E4.9 s k S MA f Hz e dded. ineninlly mied 4.7 Trnsfrmer Pwer Flw Eqins n his secin we derive he pwer flw eqins fr vris rnsfrmer mdels discssed in his chper. We fcs n pwer flw nder sedy se lnce cndiins nd herefre we need e cncerned nly wih he psiive seqence mdel. n prcice, rnsfrmer mdels f vrying ccrcy re sed sed n he desired precisin. Mdels f vrying precisin re shwn in Figre 4.7. Cpyrigh A. P. Skis Melipls 99-6 Pge 47

48 y y sh y e jα : y sh y y e jα : y sh y c y e jα : y sh d Figre 4.7 Psiive Seqence Trnsfrmer Mdel n-regling, nminl rnsfrmin ri Simplified Mdel f Regling Trnsfrmer Pge 48 Cpyrigh A. P. Skis Melipls 99-6

49 c nermedie Mdel f Regling Trnsfrmer d Relisic Mdel f Regling Trnsfrmer The nminl prmeers f he rnsfrmer in per ni re: y s g j, nd y sh js. e h mdel is he per ni mdel f rnsfrmer wih fixed p nminl p. Mdel is simplified mdel f n ff-nminl p rnsfrmer. The simplificin cnsiss f ssming h he lekge impednce f he rnsfrmer is independen f he p seing nd h i is ll n he lef side f he rnsfrmer. Mdel c is simplified mdel f n ff-nminl p rnsfrmer. The simplificin cnsiss f ssming h he lekge impednce f he rnsfrmer is independen f he p seing. Mdel d is he ms ccre rnsfrmer mdel. The lekge impednce n he nn-pped side is cnsn while he lekge impednce n he pped side is prprinl he qniy s- where is he p seing. This mdel resls frm cnsidering he cnsrcin chrcerisics f he vrile p rnsfrmers. Fr ccre pwer flw cmpins, mdel d shld e sed. The pwer flw eqins fr he fr mdels re: Pwer flw eqins fr mdel. THESE EQUATOS ARE WROG- Upde P g [ g cs δ δ sin δ ] Q [ cs δ δ g sin δ δ s δ g [ g cs δ δ sin δ δ ] k [ cs δ δ g sin δ δ ] P Q Pwer flw eqins fr Mdel. P g [ g cs δ δ α sin δ δ α] Q [ cs δ δ α g sin δ δ P Q S α g [ g cs δ δ α sin δ δ α] [ cs δ δ α g sin δ δ α] Pwer flw eqins fr Mdel. P s ga α A β Q s g A α β A P s ga α A β ] ] Cpyrigh A. P. Skis Melipls 99-6 Pge 49

50 Q s g A α β A where: α g cs δ δ α sin δ δ α β g sin δ δ α cs δ δ α α g cs δ δ α sin δ δ α β g sin δ δ α cs δ δ α A 4 g 4 g 4 g s 4 gs s s s 4 s Pwer flw eqins fr Mdel 4. P s ga s α A β Q s g s A α β A P s ga α A β Q s g A α β A where: α g cs δ δ α sin δ δ α β g sin δ δ α cs δ δ α α g cs δ δ α sin δ δ α β g sin δ δ α cs δ δ α A 4 g 4 g s ss s s 4s s s s Pge 5 Cpyrigh A. P. Skis Melipls 99-6

51 4 g s sgs s s 4s s s, if. s, if. 4.8 Trnsfrmer Prmeer denificin Frm Tess Acl mdeling f rnsfrmers reqires h heir prmeers e ccrely deermined. Fr his prpse, ess cn e perfrmed mesre he prmeers f he rnsfrmer. Tw simple ess re qie sl fr his prpse: he pen circi es nd he shr circi es The Open Circi Tes The pen circi es is shwn in Figre 4.8. Specificlly Figre 4.8 shws cnnecins nd mesremens fr n pen circi es f single phse rnsfrmer. e h he secndry f he rnsfrmer is pen, while he primry is cnneced srce. The vlge he erminls f he rnsfrmer is ypiclly ner he nminl vle. The vlge he primry side is mesred, he elecric crren f he primry is ls mesred A s well s he rel pwer flw in he rnsfrmer W. Trnsfrmer Under Tes W A Open W A Trnsfrmer Under Tes r jxl r jxl g -j Open Figre 4.8 Open Circi Tes Sep fr he Open Circi es Eqivlen Circi f he Open Circi Tes Cpyrigh A. P. Skis Melipls 99-6 Pge 5

52 Cnsider he mesremens f vlge, crren nd pwer, c, c nd P c respecively. y replcing he rnsfrmer nder es wih is eqivlen circi, s i is shwn in Figre 4.8, he fllwing relinship cn e derived: g P c c g c c Pc j 4 c c c Therefre his es prvides he mgneizing recnce nd he cre lss f he rnsfrmer The Shr Circi Tes The shr circi es is shwn in Figre 4.9. Specificlly, Figre 4.9 shws cnnecins nd mesremens fr shr circi es f single phse rnsfrmer. e h he secndry f he rnsfrmer is shred, while he primry is cnneced srce. The pplied vlge he erminls f he rnsfrmer is djsed s h he crren flwing in he rnsfrmer is ypiclly ner he nminl vle. The vlge he primry side is mesred, he elecric crren f he primry is ls mesred A s well s he rel pwer flw in he rnsfrmer W. Red mps Trnsfrmer Under Tes W A Trnsfrmer Under Tes W A r jxl r jxl g -j Figre 4.9 Shr Circi Tes Sep fr he Shr Circi es Eqivlen Circi f he Shr Circi es Pge 5 Cpyrigh A. P. Skis Melipls 99-6

53 Cnsider he mesremens f vlge, crren nd pwer, sc, sc nd P sc respecively. y replcing he rnsfrmer nder es wih is eqivlen circi s i is shwn in Figre 4.9, he fllwing relinships cn e derived: P r r sc sc x x r r l l x x l l sc sc P sc sc sc 4 sc Therefre his es prvides he lekge impednce nd eqivlen resisnce f he rnsfrmer. e h his es des n prvide he impednces f he individl cils high nd lw side rher he verll eqivlen. 4.9 rmlized Pwer Sysem Mdel Del-wye cnneced hree phse rnsfrmers re exensively sed in hree phse pwer sysems fr vriey resns. These rnsfrmers genere phse shif f he vlge phse ngles. Sysems re designed in sch wy h hese phse shifs re cnsisen in he sense h fr ny lp in he newrk he ne phse shif will e zer. n his cse, we cll he sysem nrml. n erms f mdeling he sysem, he phse shif f he del-wye cnneced rnsfrmers cn e ignred if he sysem is nrml. Hwever, if i is necessry cmpe he cl sysem vlges, hese phse shifs ms e mdeled. Cnsider fr exmple he sysem f Figre 4.. The Figre illsres ypicl sysem. e h fr his sysem, here is n newrk lp in which he ne phse shif y delwye r wye-del cnneced rnsfrmers is differen hn zer. There re w delwye cnneced rnsfrmers h re cnneced rdilly. Ths he sysem is nrml. n mdeling his sysem, he phse shif f he del-wye cnneced rnsfrmers re mied fr simpliciy. Cpyrigh A. P. Skis Melipls 99-6 Pge 5

54 8k:k k k:5k Y Y Y 5k:.8k Y Y A 5k:k k:5k Figre 4. Exmple rml Pwer Sysem Y 5k:.8k Pge 54 Cpyrigh A. P. Skis Melipls 99-6

55 4. Smmry nd Discssin n his chper, he mdeling f pwer rnsfrmers hs een ddressed. The pwer rnsfrmer nder nrml pering cndiins ehves s n pprximely liner device. Three phse rnsfrmers re symmeric nd herefre cn e represened wih heir psiive, negive nd zer seqence newrks. We develped hese mdels fr hree phse rnsfrmers. These mdels re sefl fr pwer flw nlysis, shr circi nlysis nd similr nlysis prlems. Regling rnsfrmers were ls ddressed nd heir mdels were discssed. n his chper we ls inrdced he per ni sysem. The per ni sysem is sefl fr simplifying rnsfrmer mdels, expressing pwer sysem qniies in meningfl wy nd cndiining he pwer sysem mdel fr eer nmericl siliy. Cpyrigh A. P. Skis Melipls 99-6 Pge 55

56 4. Prlems Prlem P4.: A single phse rnsfrmer hs he fllwing rings: -ka, 6 Hz, /. The impednces f he rnsfrmer re: r 4. hms, r. 4 hms x 5.5 hms, x.55 hms, l l m 7. Henries n side Sscrip denes he winding, sscrip he winding. The rnsfrmer delivers 9% f is red vl-mperes ppren pwer.85 pwer fcr crren lgging ld n he lw vlge side wih crss he ld. The cre lsses re 7 Ws. Cmpe he l lekge impednce referred he high vlge side, nd he lw vlge side. Cmpe he high side erminl vlge, Cmpe he pwer fcr he high side erminls, Cmpe he rnsfrmer efficiency. Slin: s sep: Cmpe.9, A 9. A. ϕ j e csϕ - ϕ.85 ϕ e j j Ω e R jx Ω j.96 Ω [ j ] e 57.4 j.6.9 e e j.875 j.5548 j. 5548,48.64 Pwer fcr pf cs φ, ' cs Pge 56 Cpyrigh A. P. Skis Melipls 99-6

57 pf.8456 c Efficiency - sses P P Ws sses 7 W Ws Efficiency.974 Prlem P4.: Cnsider he hree winding rnsfrmer f Figre P4.. An elecric crren i A cs ω, ω 77 sec - is injeced in winding. Winding is lded wih series, C circi while winding is pen, i.e., i. The vles f nd C re:.5h, CµF. The nmer f rns re:, 4,. Assme idel mgneic meril µ nd idel windings zer resisnce. Cmpe he vlge v nd v crss he windings nd. i i i C Slin: Figre P4. i i i i 5Acsω jω j8. 76 jωc Recll jω Φ Φ.545T Φ.545T cs ω 8 Cpyrigh A. P. Skis Melipls 99-6 Pge 57

58 dφ 9.89 cs ω 9 d dφ cs ω 9 d Prlem P4.: A single phse disriin rnsfrmer hs he fllwing prmeers: red vlges 4.4 k/, red pwer 7.5 ka, red freqency 6 Hz, lekge impednce.7%, l winding resisnce.5%. c Cmpe nd drw he eqivlen circi f he rnsfrmer inclding n idel rnsfrmer. Refer ll impednces he high vlge side. Cmpe nd drw he pi eqivlen circi f he rnsfrmer in cl nis. Cmpe nd drw he per ni eqivlen circi f he rnsfrmer ssming he fllwing ses n he high vlge side f he rnsfrmer: 4.4 k, S 7.5 ka. Selec he lw side ses s y like. Prlem P4.4: Given 5 MA, k/5k hree phse, wye/wye cnneced rnsfrmer wih % lekge impednce nd negligile resisnce. The rnsfrmer hs severl ps n he 5 k side. Assme h he p f he rnsfrmer is se fr k. Cmpe he per ni psiive seqence eqivlen circi f he rnsfrmer. Use s ses he fllwing nminl vlges: High vlge side: / k, S 5/ MA, lw vlge side: 5 / k, S 5/ MA. Slin: The per phse eqivlen circi in perni is shwn in Figre. Eliminin f he idel rnsfrmer yields he circi f Figre. Prlem P4.5: A k rnsmissin line nd 5 k line prllel ech her fr disnce f miles. The zer seqence eqivlen mdel f he sysem is illsred in Figre P4.5. The ssigned ses fr he k line re: f, 6 Hz, S, MA/,, k / Pge 58 Cpyrigh A. P. Skis Melipls 99-6

59 The ssigned ses fr he 5 k line re: f, 5 6 Hz, S,5 MA /,, 5 5 k / Cmpe he per ni zer seqence eqivlen mdel f his sysem. Wh is he se impednce fr he ml impednce? hms.5 hms j hms j8 hms j hms k ine 5 k ine Figre P4.5 Prlem P4.6: Cnsider he elecric pwer sysem f Figre P4.6. The rnsfrmer recnces re given in per ni n ech rnsfrmer rings. The generr cnrls he vlge s nminl vle. Assme h he indiced elecric ld is cnsn impednce ld nd i srs he indiced mn f pwer when he vlge is nminl. Cmpe nd drw he psiive seqence eqivlen circi f he sysem in per ni sing / MA pwer sis ne phse nd nminl vlges. G 4 Y 45MA 5k/5k X8% 6j6Ω Y MA 5k/5k X7.5% S d 4MW jmar Figre P4.6 Slin: The se impednce he hree differen k levels f his sysem re: 5, 5k. 5 hms 5, 5k. 5 hms 5, 5k 6. 5 hms The rnsfrmer impednces referred MA sis re: z T j.8 45 j.778 Cpyrigh A. P. Skis Melipls 99-6 Pge 59

60 z T j.75 j.5 The psiive seqence eqivlen circi is: j j.469 j.5 4.p j :e j :e -j Prlem P4.7. Cnsider he phse-shifer rnsfrmer f Figre P4.7. e h he phse shifer rnsfrmer is mde f six idenicl single-phse rnsfrmers. Assme h ech rnsfrmer is idel nd he primry secndry rnsfrmin ri is.5. Wh is he psiive seqence eqivlen f his phse-shifer rnsfrmer? Wh is he negive seqence eqivlen f his phse-shifer rnsfrmer? Wh is he zer seqence eqivlen f his phse-shifer rnsfrmer? Hin: Apply he pprprie vlge psiive, negive, r zer seqence ne side f he phse-shifer rnsfrmer nd cmpe he vlges he her side. A S P S P P S C P S P S S c P Phse Shifer Figre P4.7 Prlem P4.8: An elecric pwer sysem cnsiss f rnsmissin line nd rnsfrmer s i is illsred in Figre P4.8. Cmpe he dmince mrix f he sysem cmprising he line nd he rnsfrmer, i.e. neglec he presence f he generr nd elecric ld. All relevn vles re given in he Figre. Pge 6 Cpyrigh A. P. Skis Melipls 99-6

61 G j. p Figre P4.8 : j. p.5 Prlem P4.9. Figre P4.9 illsres simplified pwer sysem cnsising f w synchrns mchines, phse shifing rnsfrmer nd line. Assme h he synchrns mchines re idel vlge srces nd h he phse shifer cnsiss f nmer f idel rnsfrmers. Specificlly, ll illsred rnsfrmers re idel wih rnsfrmin ri f secndry primry vlge eql.5 Secndry lge/primry vlge.5 nd hey re cnneced s illsred in he figre. e h he primry is designed s P nd he secndry s S. Cmpe he phse grnd vlge h mgnide nd phse he w sides f he phse shifer. Cmpe he l elecric pwer sred r genered y he synchrns mchine. The fllwing ddiinl infrmin is given. j E 8. 66e k, j E E e, j E 9. e k, j E E e, j4 E c E e j4 Ec Ee S y. - j. mhs E P E E c E S P P S P y. - j. mhs E E c S P S S y. - j. mhs Synchrns Mchine P Phse Shifer Trnsmissin ine Synchrns Mchine Slin: Figre P4.9 The vlge phse A lef hnd side f he line is: Cpyrigh A. P. Skis Melipls 99-6 Pge 6