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IEEE TRANSACTIONSONMAG ICS, VOL. 8, N0.2 ,MARCH 1992 1363FORCE CALCULATION IN TRANSFORMER WINDINGS

UNDER UNBALANCED MMFsBY A NON LINEAR FINITE ELEMENT CODEC. M Ar t uri , I EEE Member

Di part i ment o di El ett r otecni caPol i tecni co di M l anoPi azza Leonard0 da Vi nci 3 2 - 2 0 1 3 3 M l ano - I TALYbstrac t - Thi s paper deal s w t h thecomput ati on of t he el ectr omagneti c axi alf or ces on the w ndi ngs of a st ep- up- gener at ort r ansf ormer under unbal anced MMF of t he samephase, by means of a f i ni t e el ement code.The unbal ance bet ween t he MMF of t he samephase, whi ch mght be of t he order of t henomnal MMP, i s caused by t he hi gh saturati onof t he i r on cor e, whi ch occur s dur i ng t het r ansi ent due t o a wr ong par al l el of thet r ansf ormer w t h a net work havi ng a ver y l owi mpedance. The eval uat i on of t hese MMF i smade by a non l i near ci r cui t al model of t het hr ee- phase f i ve- l i mb tr ansformer[ l ].W t h the peak val ues of MMFs , t he magnet i cf i el d i n the t r ansf ormer w ndow i s computedby a t wo-di mensi onal nonl i near code, andt hen, t he Lor ent z f orces on t he w ndi ngs areeval uat ed. The r esul t of t he comput ati on i s

expr essed as rati o bet ween axi al f or ces w t hand w t hout MMP unbal ance, as a f uncti on oft he i mpedance of t he net work w t h whi ch thewrong par al l el occur s.I . I NTRODUCTI ON

power t r ansf ormer shoul d be desi gned t ow t hst and the dynamc f orces caused by f aul tcurr ent s.I n t ransf ormers w t h concent ri c w ndi ngs,t he el ect r omagnet i c f orces gener ated by t hei nt er act i on of cur r ent s and magnet i c f i el d,have radi al components, due to t he axi alf i el d, and axi al components , due t o t heradi al f i el d. The radi al f or ces act i nwardson t he i nternal w ndi ng and outwards on theext ernal w ndi ng. The axi al f orce i sf r equent l y a compr essi on f orce, i . e. di r ect edt owards the m ddl e, but t here are w ndi ngzones i n whi ch t he axi al f orce mght bedi rect ed t owards t he end. The axi al f orces,i f t he two concent r i c w ndi ngs are w t houtaxi al di spl acement and asymmetr y r espect t ot he yokes and have uni f orm y di st r i but edt urns, have zer o summed axi al f orce. I n t hi scondi t i on, t he upper hal f w ndi ng exer t s af orce on the m ddl e whi ch i s exact l y opposi t et o that exer t ed by t he l ower hal f w ndi ng.However t he equi l i bri um i s unstabl e. I nf act , i f a smal l axi al di spl acement i sproduced bet ween t he magnet i c cent r es ofgravi t y of t he t wo w ndi ngs, a t ot al axi alf orce i s generated on each w ndi ng whi cht ends t o i ncr ease the di spl acement .When the axi al f orce on a di sc w ndi ngexceeds t he st r ength, f ai l ur e usual l y occur sby a mechani sm charact er i zed by t i l t i ng oft he conduct or t ur ns i nt o a coni cal shape [ 2 -

The dynam c anal ysi s of t he axi al mechani -3 1cal behavi our , f or t he comput ati on of bot h

Munuscri pt r ecei ved J ul y 7 , 1991.

t he dynam c f orce and t he di spl acement i n anypoi nt of w ndi ngs and cl amps, i s ver ycompl ex. The dynam c f orce can be compl etel ydi f ferent f rom the appl i ed (el ect r o-magnet i cal l y generated) f orces. Possi bi l i t i esal so exi st i n both t he w ndi ng and t he cl ampf or a f orce ampl i f i cati on due to r esonanceef f ects or f or a f orce reduct i on due toi nert i al ef f ects [ 4 ] .However, every accur ate anal ysi s of t hemechani cal behavi our r equi r es t hat t heel ect r omagneti c f orce appl i ed t o any poi nt ofw ndi ngs under t he assumed f aul t condi t i on beknown.

11 WRONG- PARALLEL MMFsOF A STEP- UP-TRANSFORMER

For a step- up t r ansf ormer, a short ci r cui tbet ween the LV t erm nal s i s ext r emel yunl i kel y, si nce t he connecti ons w t h thesynchr onous generat or i s made by segregat edbar s, and a shor t ci r cui t on t he HV t er m nal soccur s w t h moderate cur r ent , si nce i t i ssuppl i ed t hroughout t he subt r ansi enti mpedance of t he synchronous gener at or , whi chhas a r el at i vel y hi gh val ue.Experi ence shows, however, t hat f or st ep-UP t r ansf ormer an out - of - phasesynchr oni zat i on oper at i on i s mor e l i kel y thesour ce of very hi gh cur r ent s and possi bl emechani cal f ai l ur e. A compl ete t heor eti caland experi ment al anal ysi s of t he t r ansi entf ol l ow ng a wr ong par al l el has been made i nr efer ence [ l ] The essent i al r esul t of t hi sanal ysi s i s t hat , because of t he hi ghsaturat i on of t he i r on br anch of t he magneti cci r cui t , t he w ndi ng MMFs of t he same phaseare not bal anced.The unbal ance of t he MMF, whi ch i s of t heorder of t he nom nal MMF, can be est i mated bya non- l i near ci rcui t al model of t he three-phase f i ve- l i mb t r ansf ormer [ l ] .I f saturati on i s negl ect ed, t he t ransi entof an out- of- phase synchr oni zat i on i sequi val ent to a si mpl e l i near RL t ransi entsuppl i ed by an ac vol t age source the val ue ofwhi ch i s doubl e t he nom nal one. Thet r ansi ent MMF i s expressed by:

F(t )=Fmexp(- t/ r) cos(wt) l ( 1)wher e Fm i s the symmet r i cal peak val ue and Ti s the t i me const ant of t he ci r cui t . Theasymmet r i cal peak val ue comput ed i n t hi s way( i n P. u. ) i s l ess than t he actual MMF of t heext ernal w ndi ng and great er t han t hat of t hei nternal w ndi ng.

Conventi onal l y, the computat i on of t heel ect r omagnet i c f orce i s made assumng t hatt he t wo concentr i c w ndi ngs of each phase ofa t hree- phase t r ansf ormer are i nt erest ed,0018-9464/92$03,00 1992 IEEE


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1364dur i ng t he f aul t , by bal anced MMF whi chcorr espond to t he maxi mum peak val ue of af ul l y asymmet r i c cur rent . Act ual l y, as hasbeen previ ousl y out l i ned, t he MMF whi chshoul d be consi der ed are unbal anced.

For a step- up t r ansf ormer, t hechar act er i st i cs of whi ch are repor t ed i nApp. A together w t h t hose of bot h thesynchronous generator and t he power net work,Fi g. 1 shows t he p. u. val ues of t he f i r st peakof t he w ndi ng MMFs of a l ateral phase,dur i ng a wrong paral l el w t h a phase err or of180' , as a f unct i on of t he percent agei mpedance of t he power net work.I t can be observed t hat t he MMF unbal ancei ncreases as t he power network i mpedancedecr eases. W t h a zer o i mpedance of t he powernet work, t he unbal ance r eaches, f or t hi st r ansf ormer, t he val ue of about 2. 08 p. u.Thi s MMF i s a magnet i zi ng MMF of hi ghsaturated branches of t he i r on core. I t sval ue gi ves an i dea of t he saturati on st ateo t he magneti c ci r cui t , whi ch r eaches val uesbetween 2. 1 and 2. 4T [l].

111. COMPUTATI ON OF THE ELECTROMAGNETI C AXI ALFORCEThe el ectr omagnet i c axi al f orce on a smal lvol ume dV of w ndi ng, havi ng a curr ent

densi t y J and pl aced i n a poi nt wher e theradi al component of t he f l ux densi t y i s B ( r )i s expr essed by:df = B(r ) J dV . ( 2)

For t he f orce comput at i on one has t o comput et he f l ux densi t y i n every poi nt of t he vol umeoccupi ed by w ndi ngs. A two-di mensi onalf i ni t e el ement code comput es t he vect ormagnet i c potent i al A i n the nodes of t he meshand then eval uates t he f l ux densi t y by ther el at i onshi p B=r otA.The t wo-di mensi onal code avai l abl e atpresent onl y al l ows a rotati onal or at r ansl ati onal symmetr y to be sol ved.Consequent l y we are f orced to repl ace t het hree- di mensi onal model of t he f i ve- l i mb coret r ansf or mer w t h an i deal i zed si ngl e-phaseshel l - t ype t r ansf ormer havi ng rotati onalsymmet r y, whi ch approxi mates t he behavi our of

Fi g. 1. Per uni t val ue of t he f i rst peak ofw ndi ng MMFs of a st ep- up gener atort r ansf ormer r ated 370 MVA, f or a wrongparal l el w t h a phase err or of 180' . Fa1 andFa2 are eval uat ed by t he nonl i near model [ l ]whi l e Fa i s eval uat ed i n a convent i onal way,negl ecti ng sat urat i on.

the w ndi ngs i n a l ateral w ndow of thet hr ee- phase uni t ( Fi g. 2) .The yokes and t he l ater al l i mbs have acr oss secti on of about 5 7 of t he woundl i mbs. The shape of t he yoke i s such t hat i tgi ves a const ant cr oss sect i on as t hedi stance f rom t he axi s i ncreases. I n t headopt ed model , t he w ndow covers al l t heci r cum er ence of w ndi ngs.The adopt ed model can be cri t i ci sed asf ol l ows: i t does not t ake i nt o account t hei nf l uence of ( 1 ) MMFs of t he adj acent phaseand ( 2) t he r eal shape of bot h t he yoke andt he l at er al l i mb, whi ch cover t he w ndi ngsonl y part i al l y, i nf l uences t he permeancedi st ri but i on part i cul arl y ar ound t he ext ernalw ndi ng.An approxi mate eval uat i on of t he wei ght oft he above si mpl i f i cat i ons can be made, by at wo-di mensi onal code, compari ng t he axi alf orces on two separate model s: ( l ) underw ndow, on a tr ansl ati onal symmetr y, w t h orw t hout t he MMF of t he adj acent phase, and( 2) i n a model w t h r ot at i onal symmetr y, w t hyoke and l ateral l i mb of whi ch t he di st ancef rom t he w ndi ngs i s equal t o the real one ormuch mor e.Of t hese t wo ef f ects , t hose r el ated t o t heMMF of t he adj acent phase i s heavi er t hant he other r el ated t o per meance di st r i buti onal ong the ci rcumerence of t he w ndi ng. I nf act, t he val ue of t he MMF of t he cent r alphase i s al most hal f t hat of t he l at eralphase but w t h a changed si gn. I t f ol l owst hat t he ext ernal w ndi ngs have concordantMMFs i n the common w ndow Thi s gi ves asubst ant i al i ncr ease of t he r adi al componentof t he f l ux densi t y i n t he external w ndi ngunder t he w ndow

The avai l abi l i ty o a three- di mensi onalmagnetost ati c non- l i near code woul d perm t t ot ake i nto account si mul t aneousl y bot h t he MMFof t he adj acent phase and the permeanceeff ect of t he f i ve- l i mb core whi ch can behi ghl y sat urated i n some br anches.The equi val ent si ngl e-phase tr ansf ormerhas been st udi ed w t h bal anced and unbal ancedMMF, and network i mpedance varyi ng f r om zer ot o 10%

Fi g. 2 - Geometr i c model of an i deal i zedsi ngl e- phase shel l - t ype t r ansf ormer, adopt edf or t he f i ni t e- el ement anal ysi s, whi chapproxi mate a l ateral phase of a f ve- l i mbcore three- phase t r ansf ormer.


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1365symmet r y); on the i nt er nal w ndi ng i t i scompressi ve anywhere.The axi al f orce on the i nt ernal w ndi ng,w t h unbal anced MMF, i s compress i ve aroundt he end and t ensi l e ar ound t he m ddl e ( onabout t he 2/ 3 of t he hei ght near t he pl ane ofsymmet r y) and i t i s compressi ve anywhere ont he external w ndi ng.I n ot her words, t he axi al f or cedi st r i but i on i s r eversed i n i nt ernal andext er nal w ndi ngs, passi ng f r om bal anced t ounbal anced MMFs.The axi al f orce di st r i but i on here observedi s obvi ousl y not val i d f or any t r ansf or merbut i s rel at ed to t he speci f i c hei ght rati oof t he exam ned concent r i c w ndi ngs.The summed axi al f orce, st art i ng f r om t heend of t he w ndi ngs, can be obt ai ned by i nt e-grat i ng t he axi al f or ce - per uni t of l engt h- shown i n Fi g. 4a and 4b. The r esul t i srepresent ed i n Fi g. 4~and 4d, r especti vel y.The summed f orces on t he m ddl e of t hew ndi ngs, f or Z r = l , have t he fol l ow ngval uesW ndi ng unbal anced( U) bal anced( B) Rat i o U/ Bi nternal - 4. 003*10 5 N - 1. 993. 10- 6 N 0. 201external - 2. 425. 10- 6 N - 2.314.10*5 N 10. 48total - 2. 825- 10- 6N - 2. 224*10 6N 1. 27

The i r on core has been model l ed by i t snonl i near B-8 char acter i st i c when unbal ancedMMF case i s si mul ated. Ot herw se, when MMFsar e bal anced, t he i r on has i nf i ni t epermeabi l i t y, i n or der t o repr oduce t heconventi onal procedur e.W t h a 1% network i mpedance, t he f orcel i nes of t he magneti c f i el d are shown i nFi g. 3, w t h and w t hout MMF unbal ance.I t i s i nt eresti ng to compare t he di st r i but i onof t he axi al f or ce (per uni t l ength) on bot ht he i nt ernal and t he ext ernal w ndi ng. Eachw ndi ng has pur posel y been di vi ded i nto anumber of r adi al l y adj acent sl i ces havi ngradi al t hi ckness Dr.The axi al f or ce - per uni t l engt h - hasbeen comput ed f or each sl i ce and then thevar i ous r adi al cont r i but i ons are summed overt he total radi al t hi ckness of t he w ndi ng.The axi al f or ce - per uni t of l engt h - i sgi ven by t he f ol l ow ng r el at i onshi p:f = Z B( r) 2nr Dr [N/ m ( 3)

and i s shown i n Fi g. 4a and 4b f or Zr %= % andMMF bal anced r unbal anced.The axi al f or ce on t he external w ndi ng,w t h bal anced MMF, i s compress i ve ar ound theend and t ensi l e ar ound the m ddl e ( on aboutt he 2/ 3 of t he hei ght near t he pl ane of

Fi g. 3 - Magnet i c f i el d f orce l i nes w t h andw t hout MMF unbal ance; a) w t h MMF unbal ance:Fa1/ 2=- 1. 842- 10A6 A ( 9. 81 P. u. ) ; Fa2/ 2=+2. 1554. 10 6 A ( 11. 48 P. u. ) ; b) w t hout MMFunbal ance: Fa/ 2= f 1. 95985*10 6 ( 10. 44 P. u. ) .

Compar i ng the summed axi al f orces on hal fw ndi ng w t h and w t hout MMF unbal ance, t he2F -- unbal.

-00 1-IOo a)-0.5

1

/

-2.5 500 11O c)

x106bolon.

- 1 4 11 0 a 5 1x1060

External i n d i n g

O 4 500 1000Fi g. 4 - Axi al f or ce per uni t of l engt h - ( aand b) and summed axi al f orce ( c and d) .Net work i mpedance 1


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1366f ol l ow ng can be obser ved: ( l ) an i ncrease ofabout 10 ti mes on the external w ndi ng; ( 2) adecr ease of about 5 t i mes on t he i nter nalw ndi ng; ( 3) a 27% i ncrease of the totalaxi al f orce for bot h w ndi ngs.

Fi nal l y, Fi g. 5 shows the val ues of t herat i o K between the summed axi al f orce onhal f exter nal w ndi ng w t h and w t hout MMFunbal ance, as a f unct i on of t he networki mpedance, f or a power t r ansf ormer r ated 370MVA ( Fi g. 5a) and f or a second powert r ansf or mer w t h si m l ar r at i ng (Fi g. 5b). Thesecond power t r ansf ormer , corr espondi ng toFi g. 5b, has di f f erent geometr i cal parametersof t he core, al t hough i t t oo i s a f i ve- l i mbone, and a gr eat er per cent age l eakagei nductance i n compari son w t h t he 370 MVAt r ansf ormer. I t can be observed t hat t het r ansf or mer of Fi g. 5b has a r at i o K whi ch i sabout 1/ 5 t han t he ot her . Thi s r educt i on canbe j ust i f i ed by a smal l er per uni t unbal ancebetween the MMF, because of a l owersat ur ati on of t he core, and a gr eaterper cent age l eakage i nductance.

I V. CONCLUSI ONSThe paper has present ed t he compar i sonbetween the axi al f orce on t he concent r i cw ndi ngs of a st ep- up generat or t r ansf ormerw t h and w t hout MMF bal ance, duri ng a wrongparal l el w t h phase er ror of 180 .The anal ysi s has shown t hat t he axi alf orce on t he ext ernal w ndi ng w t h unbal ancedMMFs can be consi der abl y greater - bet ween 2and 10 t i mes - f or t he two power t r ansf ormerconsi der ed - t han t hose eval uated w t houtunbal ance of t he MMF of t he same phase, as i ti s convent i onal l y made t o ver i f y t hemechani cal st r ength of t he t r ansf ormers i nt he desi gn st age.

14K

2.2

1.5- - -

network imped.7.1.3 5 10b)

Fi g. 5. Rat i o K between t he summed axi alf orces on hal f ext ernal w ndi ng, w t h andw t hout MMF unbal ance, as a f unct i on of t hei mpedance of t he power networ k, f or a powert r ansf ormer r ated 370 MVA ( a) and a secondpower t r ansf ormer havi ng si m l ar r ati ng butdi f f erent geometr i c par ameter of t he cor eb).

Theref ore, t he desi gners shoul d be aware oft he i mpor t ance of t he sat ur ati on of t hemagnet i c ci r cui t duri ng some faul tt r ansi ents, whi ch gi ves MMF unbal ance betweenw ndi ngs of t he same phase.The t wo- di mensi onal code used has onl yper m t t ed an approxi mate anal ysi s. Moreconsi der at i on shoul d be devoted t o t hei nf l uence of t he MMF of t he adj acent phase ont he val ue of t he maxi mum axi al f orce on theext ernal w ndi ng i n t he w ndow Thi s coul deasi l y be accompl i shed by a thr ee- di mensi onalcode.

The novel t y pr esented i n t he paper shoul dbe s t r essed agai n agai nst t he conventi onalapproach, havi ng out l i ned t he i mport ance oft he sat ur ati on of t he i r on core i n some t ypesof t ransf ormer f aul ts.

ACKNOWLEDGMENTThe aut hor w shes t o t hank Mr . M. Borsaniof t he Soci et h Nazi onal e del l e Of f i ci ne diSavi gl i ano of Turi n ( I ta l y) f or hi sst i mul at i ng di scussi ons and t he usef ulsuggest i ons made duri ng the devel opment oft hi s r esear ch.

APPENDI X ASt ep- up- generator t r ansf ormerNom nal power : 370MVANomnal vol t age: 400kV/ 20kVConnect i on: st ar w t h gr ounded neut r al / del t aLeakage i nductance: 12. 8% Resi st ance: 0. 2%Tur ns: 397/43Synchr onous gener atorDi r ect sub-t r ansi ent i nduct ance: 22. 69%Resi st ance: 1%Power network: r / x = 0. 1.REFERENCES

l ] C. M Ar t ur i : Tr ansi ent Si mul at i on andAnal ysi s of a Thr ee- phase Fi ve- l i mb Step-up Tr ansf ormer f ol l ow ng an Out - of - PhaseSynchr oni zati on , I EEE Tr ansacti ons onPower Del i ver y, Vo1. 6, No. 1, J an. 1991,pp. 196- 207.[ 21 M Wat ers: The Shor t - ci r cui t St r engt h ofPower Tr ansf ormer , McDonal d Co. Ltd. -London 1966.[ 3] W J . McNut t , W M J ohnson, R. A. Nel son,R. E. Ayer s : Power Transf ormer Shor t -Ci rcui t St r engt h - Requi r ement s, Desi gn,and Demonst r at i on , I EEE Trans. on PAS,Vol . 89, No. 8, Nov/ Dec 1970, pp. 1955- 1969.[ 41 M R. Pat el : Dynamc Responce of PowerTransf ormers under axi al shor t ci r cui tForces; Par t I W ndi ng and Cl amp asi ndi vi dual component s; Par t I 1 - W ndi ngsand Cl amps as a combi ned syst em ; I EEETrans. on PAS, Vol . 92, Sept / Oct . 1973,pp. 1558- 1576.

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