On parameters determination of multi-port equivalent scheme for multi-winding traction transformers

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1 ARCHIVES OF EECRICA ENGINEERING VO. 6(), pp. 7-7 (5) DOI.55/a-5- On paramtrs dtrmination of multi-port quivalnt schm for multi-winding traction transformrs ADEUSZ J. SOBCZYK, JOSEPH E HAYEK Cracow Univrsity of chnology Institut on Elctromchanical Enrgy Convrsion Warszawska, -55 Kraków, Poland tl./fax: mail: psobczy@cyf-kr.du.pl Univrsity of Applid Scincs and Arts Wstrn Switzrland School of Enginring Rout du Rawyl 7, CP, 95 Sion, Switzrland -mail: josph.lhayk@hvs.ch (Rcivd: 7.., rvisd: 8..) Abstract: his papr aims to prsnt a nw quivalnt schm of multi-windings traction transformrs, basd on multiport purly inductiv circuit. h mathmatical background of this quivalnt schm is dscribd. h dtrmination of th diffrnt schm lmnts is mad through a finit-lmnts calculation of both main and lakag inductancs, for th cas of a four-winding transformr. A procdur is dfind, which allows to stimat th valus of ths lmnts from som masurmnts on th transformr at no-load and short-circuit oprations. A spcific stratgy of short-circuit tsts is dscribd, allowing to dtrmin all paramtrs in a rathr simpl way. Ky words: multi-winding transformr, quivalnt schm, multi-port circuit, traction transformrs. Introduction h quivalnt schm of a transformr is a basic tool in lctrical nginring of altrnating currnts. Its classical form is usd vrywhr whn magntic coupling xists. Commonly an quivalnt schm of -typ with on vrtical magntizing branch is usd. It is wll known that a transformr having mor than thr magntically coupld windings cannot b rprsntd uniquly by a -typ quivalnt schm. In ordr to b dscribd corrctly, thr coupld windings nd thr slf- and thr mutual inductancs. An quivalnt schm has to includ th sam numbr of indpndnt paramtrs. h -typ quivalnt schm for thr windings nds accordingly six paramtrs: thr lakag inductancs, on common magntizing inductanc and two winding ratios, rcalculating th diffrnt paramtrs to a rfrnc winding. So, six indpndnt inductancs can b uniquly rprsntd by six paramtrs of th quivalnt schm. In cas of four coupld windings thr ar tn indpndnt quantitis: four Brought to you by Bibliotka Glówna Zachodniopomorskigo Uniwrsyttu chnologiczngo w Szczcini Authnticatd Download Dat 6/5/5 :5 PM

2 8.J. Sobczyk, J. El Hayk Arch. Elct. Eng. slf-inductancs and six mutual ons, but th -typ quivalnt schm with on magntizing branch prsnts only ight paramtrs: four lakag inductancs, th common magntizing inductanc and thr winding ratios. It is thrfor impossibl to rprsnt in a uniqu way mor than thr magntically coupld windings by th quivalnt schm of -typ []. In [] th multi-port circuit has bn proposd as an quivalnt circuit of st with an arbitrary numbr (N) of coils. In that rprsntation th numbr of inductiv lmnts is always qual to th numbr of indpndnt slf and mutual inductancs. In [,, 7] this approach has bn applid as an quivalnt schm of a multi-winding traction transformr. raction transformrs hav many windings with vry diffrnt tasks within th locomotiv. hr ar windings conncting th lin catnary to th traction supply systm, som ons supplying powr lctronics drivs and othrs fding th auxiliary systms of th train. A propr rprsntation of traction transformrs is of a grat intrst for dsignrs as wll as for usrs. In [6] an quivalnt schm of a laboratory modl traction transformr with four windings has bn dvlopd and a procdur of dtrmining its paramtrs by fild computation has bn dscribd, including magntic non-linarity of th transformr cor. his papr aims to prsnt a procdur, which allows stimating ths paramtrs from som masurmnts on th transformr by no-load and short-circuit oprations.. Background for th multiport quivalnt schm A st of magntically coupld coils is modlld by rlations btwn coils flux linkags and coils currnts. Assuming magntic linarity, this is dscribd by rlation () whr: Ψ is th vctor of flux linkags = [ ψ ψ ψ ], i is th vctor of winding currnts Ψ = i, () Ψ N i = [ i i ], in is th inductanc matrix = ( sym) O M In ordr to obtain th multi-port quivalnt schm, on has to xprss th coil currnts in function of th flux linkags []. his mans that, instad of th classical rlation, N N NN. ψ = i + i +, () n n, n, + n,n in Brought to you by Bibliotka Glówna Zachodniopomorskigo Uniwrsyttu chnologiczngo w Szczcini Authnticatd Download Dat 6/5/5 :5 PM

3 Vol. 6 (5) On paramtrs dtrmination of multi-port quivalnt schm 9 w should apply th xprssion i = W ψ + W ψ + + W ψ () n n, n, for n =,,..., N, in which th currnt of th winding n dpnds on ach flux linkd to ach winding. Bcaus th windings magntic coupling is not idal, th matrix of inductancs is not singular and th rlations () xist. Using an analogy to a dscription of purly rsistiv circuit by th nod potntial mthod, rlations () can b writtn as () in = ( ψ - ) ( - ). n ψ + + ψ n + + ψ n ψ N () n, n In this xprssion th flux linkags rplac th nod potntials and th inductancs n, k and n rplac th rspctiv rsistancs. Actually, this substitution is possibl, sinc th rlations btwn flux linkags and winding currnts ar algbraic, lik th rlations btwn potntials and currnts in a rsistiv ntwork. hrfor on can writ th rlation whr n,n n,n N i = W Ψ, (5) W, W, W,N W W,,N W = =. O M ( sym) WN,N akn into account (), th matrix W can b writtn in th following form N + =,, N, k k k N + W = =, N, k k (6) k O M N (sym) + N k = N, k k N h inductancs in this matrix with th suprscript '' ar rlatd to th lmnts of th matrix W according to th following xprssions: n, k = W, for k n, (7) n, k Brought to you by Bibliotka Glówna Zachodniopomorskigo Uniwrsyttu chnologiczngo w Szczcini Authnticatd Download Dat 6/5/5 :5 PM

4 .J. Sobczyk, J. El Hayk Arch. Elct. Eng. n = N k = W n, k. (8) rating th matrix W in th form (6) as a conductanc matrix G of purly rsistiv circuit dscribd by th rlations i = Gv (whr v is a vctor of nod potntials), th rlations i = WQ can b intrprtd as an N-port, purly inductiv circuit with N nods, providing th flux linkags ψ, ψ,..., ψ N. h nods ar connctd to ach othr by th inductancs n, k and to a rfrnc nod by th inductancs n. h winding currnts i, i,..., in supply rspctiv nods. o fulfil th voltag quations dψ n un = R n in +, for n =,,..., N, (9) dt th winding rsistanc R n has to b addd to ach port. A typical multi-port quivalnt schm for a four-winding transformr is shown in Figur, in which th inductancs n lay in vrtical branchs conncting all nods to th rf- rnc on and th inductancs n k connct all nods constituting th N-polygon., R u R i,,,,, i ψ i, i ψ R R u u ψ ψ u Fig.. Equivalnt schm of a four-winding transformr In th nw multi-port quivalnt schm thr ar many vrtical inductancs instad of a singl common magntizing inductanc in th classical -typ on. h total numbr of quivalnt inductancs in that multi-port quivalnt schm is xactly qual to th numbr of indpndnt lmnts of th inductancs matrix. So, th multi-port quivalnt schm can rprsnt prcisly any multi-winding transformr, vn without rcalculating th paramtrs to on rfrnc sid. Inductancs apparing in th multiport quivalnt schm can b collctd into th matrix ().,, N, N =. () O M ( sym) N h lmnts of that matrix charactriz all inductancs of th multi-port quivalnt schm. Brought to you by Bibliotka Glówna Zachodniopomorskigo Uniwrsyttu chnologiczngo w Szczcini Authnticatd Download Dat 6/5/5 :5 PM

5 Vol. 6 (5) On paramtrs dtrmination of multi-port quivalnt schm. Equivalnt schm of traction transformrs A traction transformr is locatd in th train. In this sns, it dos not blong to a static supply station. Its main us is to bring th singl-phas catnary s voltag lvl to valus, which ar appropriat to th powr lctronics and th traction motors. Static convrtrs ar fd through low voltag sourcs, which ar th scondary windings of th transformr. Whnvr mor than two singl-phas powr sourcs ar ndd, w hav rcours to a transformr with svral scondary coils in ordr to sav plac and wight. Gnrally such transformrs hav also othr low voltag windings usd for supplying diffrnt auxiliary dvics within th train, such as lights, hatrs or air conditionrs for xampl. Morovr, spcially in Europ whr th railways supply ntworks oprat at diffrnt voltags and frquncis, transformrs should b abl to work undr divrs systms; this lads to us vn mor coils in ordr to achiv a wid rang of oprations. ypically, a transformr can involv up to diffrnt winding parts. In [] th multiport quivalnt schm for a modl transformr has bn cratd and invstigatd. his transformr oprats at two ntwork s frquncis: 6.7 Hz and 5 Hz. It consists of a two-limb magntic cor, and svral windings dispatchd as in Figur : on th primary sid: H high-voltag windings (H) connctd in paralll in normal configuration, on th scondary sid: H traction windings (r) which supply th locomotiv motors through static convrtrs, H filtr windings (Fi) dsignd for filtring harmonic currnts on th transformr primary sid and furthr to th supply ntwork. In ordr to insur th dual frquncy opration, on winding pr block had to b dividd into two parts. his lads to th rprsntation of ach windings group by four windings, instad of thr, lik in Figur. Fig.. ayout of a traction transformr Brought to you by Bibliotka Glówna Zachodniopomorskigo Uniwrsyttu chnologiczngo w Szczcini Authnticatd Download Dat 6/5/5 :5 PM

6 .J. Sobczyk, J. El Hayk Arch. Elct. Eng. In [] a group of four windings has bn modlld and th quivalnt schm has bn dtrmind. Such windings ar magntically coupld through a common flux in th iron cor. Howvr, windings ar magntically coupld in th air also, which diffrs for ach pair of windings. So, th inductanc matrix can b dividd into two parts: th matrix of main magntizing inductancs, du to th coupling through th iron, and th matrix of lakag inductancs rprsnting th coupling in th air. Aftr rcalculating all inductancs to a rfrnc numbr of turns, for xampl th winding dnotd '', th inductanc matrix taks th form σ σ σ ', ',N σ σ = μ ' ',N +. M M O O M σ (sym) ' N, N () In [7] calculation rsults of inductancs in thos matrics ar prsntd, using FUXD as finit lmnts softwar. It has bn shown that whras th : valu changd with th saturation of th iron cor, th lakag inductancs in th scond matrix rmain constant. For unsaturatd stat of th transformr th following data hav bn obtaind: th matrix of main inductancs th matrix of lakag inductancs μ = 6 [H]. () σ = [H]. () h quivalnt schm of th modlld group of windings is xactly th sam as in Figur. h inductancs apparing in this quivalnt schm, obtaind by th procdur dscribd in th prvious sctions and arrangd as in (), ar summarizd in (): = [H]. () Brought to you by Bibliotka Glówna Zachodniopomorskigo Uniwrsyttu chnologiczngo w Szczcini Authnticatd Download Dat 6/5/5 :5 PM

7 Vol. 6 (5) On paramtrs dtrmination of multi-port quivalnt schm hs rsults show consquntly, that th procdur basd on fild computation, allow th dtrmination of th multi-port quivalnt schm.. Dtrmination of th multiport quivalnt schm paramtrs by masurmnts h multiport quivalnt circuit of N-winding transformr has N(N+)/ indpndnt paramtrs. In gnral th dtrmination of thos paramtrs is much mor complicatd as for -typ quivalnt schm. In th considrd cas of four windings traction transformr, quivalnt inductancs should b found. It mans that at last quations hav to b statd to calculat thm. Howvr, th stimation of paramtrs valus can b rlativly asily don as it will b prsntd in this chaptr. Usually th schm paramtrs of a transformr ar dtrmind from no-load and shortcircuit tsts undr sinusoidal voltag supply. h lmnts of th multi-port quivalnt schm can b also stablishd from such tsts, but carrid out in a slightly diffrnt mannr []. For th no-load tsts th individual windings should b supplid succssivly, kping th othr windings opn. For th short-circuit tst th sam is don, but all non-supplid windings ar short-circuitd... h no-load tsts For th no-load opration, whn only th winding '' is supplid by a sinusoidal voltag with a pulsation S, and all th othr windings ar opn, th transformr is dscribd by th Equations (5), o ( R + Ω( μ σ ) I, o in which: I [ ], = [ U ], o = I, o U = j + (5) U, o, whr, o I is a masurd currnt, U is th supplid voltag and,, ar masurd voltags on th opn windings, rcalculating to th winding. In ordr to link th masurd valus to th paramtrs of th multiport quivalnt schm, Equation (5) should b rwrittn as whr I = Y ( U R ), (6), o,o I,o Y = W. j Ω h sum of all Equations (6) givs th quation I, o + = Y + Y + Y Y, (7) whr = U RI, o and Y = j Ω. Rpating tsts for th windings, and th quation st could b writtn Brought to you by Bibliotka Glówna Zachodniopomorskigo Uniwrsyttu chnologiczngo w Szczcini Authnticatd Download Dat 6/5/5 :5 PM

8 .J. Sobczyk, J. El Hayk Arch. Elct. Eng. I I I I,o,o,o,o =,,,,,,,,,,,,,,,, Y Y Y Y, (8) in which th scond subscript in,n dnots th numbr of th tst. It mans that valus of vrtical branchs in multi-port schm can b dtrmind sparatly from th valus of ractancs in th uppr polygonal. raction transformr is a singl-phas multi-windings on, and th main magntic flux is common for all windings, which ld to th matrix (). h voltags in (8) hav vry clos valus, and thy should thrfor b masurd prcisly. Assuming that thy ar qual, th quations (8) rducs to only on quation I = + (9), o (Y + Y + Y Y ) U from which th valus of individual paramtrs cannot b found. h multi-port quivalnt schm undr such assumption bcoms as in Figur [8]. = = I jx U jx jx jx = Fig.. Equivalnt schm of a four-winding on-phas transformr for th no-load condition Calculations of th multiport schm paramtrs from dsign data in prvious chaptr showd that vrtical lmnts hav almost th sam valus = = =. On th othr hand, th first stimation from (5), if rsistancs and lakag inductancs ar nglctd, lads to U = j Xμ I. So, in gnral cas th vrtical inductancs in th multiport quivalnt schm of singl-phas transformrs can b stimatd as = = = N = N. () Howvr, for thr-phas powr transformrs such stimation is not valid [6]... h short-circuit tsts h classical short-circuit tst for th -typ quivalnt circuit is not usful for th multiport on. In [] it has bn shown that th following stratgy is vry ffctiv: on winding is μ Brought to you by Bibliotka Glówna Zachodniopomorskigo Uniwrsyttu chnologiczngo w Szczcini Authnticatd Download Dat 6/5/5 :5 PM

9 Vol. 6 (5) On paramtrs dtrmination of multi-port quivalnt schm 5 supplid by a limitd voltag having all othr windings short-circuitd; and this is succssivly rpatd for all windings. h quations, whn th winding '' is supplid, tak th form I = Y ( U R ), (), sc,sc I,sc whr: sc = [ I,sc,sc,sc, sc ], U sc = U, sc. All lmnts of ths vctors ar masurabl from th short-circuit tst. As it has bn statd, th matrix Y has indpndnt lmnts. Howvr, of thm th vrtical ons can b dtrmind from th no-load tsts by th Equations (8). Valus of th lmnts Yn,k = j Ω n,k in th uppr polygonal fulfil quations I [ ] I,sc,sc,sc,sc Y Y = Y Y Y Y Y Y Y Y,,,,,, () or shortly, I = U Y. Voltags in ths quations ar: = U,sc R I, sc, = R', sc, = R', sc, = R', sc. So, it is ncssary to know winding rsistancs to writ it. h short-circuit tst for th winding givs quations but thr ar 6 unknown valus. So, on short-circuit tst is no nough to dtrmin th uppr paramtrs of th multiport schm. Rpating th short-circuit tst for th windings, and th st of 6 quations with 6 unknown valus can b writtn I U I U = Y. I U I U () Writing () in th form I = U Y, th vctor Y can b found, applying a linar rgrssion mthod, from formula [5] Y = (U U)! (U I) () Such approach is rathr complicatd and not suitabl for nginring application. Assuming that all windings rsistancs can b omittd, voltags in () ar: U, =, = and =. Additionally, for traction transformrs vrtical admittancs ar vry high, which lads to vry simpl formula for th currnts in th short-circuitd windings =, sc Brought to you by Bibliotka Glówna Zachodniopomorskigo Uniwrsyttu chnologiczngo w Szczcini Authnticatd Download Dat 6/5/5 :5 PM

10 6.J. Sobczyk, J. El Hayk Arch. Elct. Eng.,sc U,sc U,sc =,,,sc j X j X and th currnt in th supplid winding is, I,sc =,sc, j X,, sc,sc,sc,sc U = (5) = (6) i.. it is just th sum of th currnts circulating in th short-circuitd windings. hs rsults ar vidnt from th quivalnt schm at a considrd short-circuit condition, shown in Figur [8]. his tst allows to dtrmin th inductancs,, and,. Whn th winding '' is supplid and th othr ons ar short-circuitd, th inductancs, and, can b additionally found. From th tst for winding th last inductanc, can b found. hn, from only thr short-circuit tsts all th inductancs constituting th uppr polygonal of th quivalnt circuit can b dtrmind vry asily. From th som tst for winding th last inductanc,sc U,sc U,sc =,.,sc = (7) j X j X,,sc,, can b found U,sc =. (8) j X hn, all inductancs constituting th uppr polygonal of th quivalnt circuit can b vry asy dtrmind from only thr short-circuit tsts., jx, jx,, jx, jx = = jx, I jx, U = Fig.. Equivalnt schm of a four-winding transformr for th short-circuit condition whn th winding '' is supplid Howvr, it should b noticd that som valus of som inductancs n, k for th analysd traction transformr could b ngativ, as it can b sn in th matrix (). So, whn masuring th short circuit currnts, on has to tak, not only thir rms valus, but also thir phass with rspct to th phas of th supplid winding currnt. Brought to you by Bibliotka Glówna Zachodniopomorskigo Uniwrsyttu chnologiczngo w Szczcini Authnticatd Download Dat 6/5/5 :5 PM

11 Vol. 6 (5) On paramtrs dtrmination of multi-port quivalnt schm 7 5. Conclusions In this papr, a nw multi-port quivalnt schm of th multi-winding traction transformrs is prsntd. It is univrsal bcaus th numbr of its inductiv lmnts is xactly qual to th numbr of slf and mutual inductancs of transformr s windings. hanks to that, any traction transformr with an arbitrary high numbr of windings can b corrctly rprsntd by such an quivalnt schm. h cntrally locatd magntizing inductanc in th -typ quivalnt schm is dcntralizd in th multi-port quivalnt schm to many magntizing inductancs distributd at ach port rspctivly. In this papr th procdurs of paramtrs dtrmination of th multi-port quivalnt schm from calculations and masurmnts ar prsntd, basing on fild calculations as wll as no-load and short-circuit tsts. Dtrmining th paramtrs of th multiport quivalnt schm rquirs simulation of magntic fild distribution by th finit lmnts mthod, from which th slf and mutual inductancs of transformr s windings hav to b calculatd prcisly, spcially thir coupling by lakag fluxs in th air. Accurat dtrmination of th multiport quivalnt schm paramtrs from masurmnts is rathr complicatd. Howvr, thir stimation from no-load and short-circuit tsts is rathr simpl, similar as for th classical -typ schm, which should b wll accptd by a majority of nginrs. Rfrncs [] Erickson R.W., Maksimovič D., A multipl-winding magntic modl having dirctly masurabl paramtrs. IEEE PESC : 7-78 (998). [] Sobczyk.J., On a circuital rprsntation of magntically coupld coils. Proc. of Int. Conf. on Fundamntals of Elctrical Enginring and Circuit hory (IC-SPEO), Poland, : 9-96 () (in Polish). [] Sobczyk.J., Equivalnt schms of multi-winding on-phas transformrs. Proc. of Int. Symp. on Elctrical Machins (IS-SME), Poland, pp () (in Polish). [] El Hayk J., Sobczyk.J., Equivalnt circuit of multi-windings traction transformrs including magntizing currnts. Proc. of Int. Conf. on Elctrical Machins & Systms, Nanjing, China, : 7-75 (5). [5] Bishop Ch.M., Pattrn Rcognition and Machin arning. Springr Scinc + Businss Mdia, (6). [6] Sobczyk.J., Multi-port quivalnt schm of thr-phas powr transformr. Proc. of Int. Conf. on Fundamntals of Elctrical Enginring and Circuit hory (IC-SPEO), Poland, pp.9- (7). [7] El Hayk J., Sobczyk.J., Multi-port quivalnt schm for multi-winding traction transformrs. COMPE (): (). [8] El Hayk J., Sobczyk.J., Estimation of paramtrs of multi-port quivalnt schm for multi-winding traction transformrs. Maszyny Elktryczn Zszyty Problmow (): 9- (). Brought to you by Bibliotka Glówna Zachodniopomorskigo Uniwrsyttu chnologiczngo w Szczcini Authnticatd Download Dat 6/5/5 :5 PM