omprison of Geometry-Bsed Trnsformer Iron- ore Models for Inrush-urrent nd Residul-Flux lultions R. Yonezw, T. Nod Astrt--When trnsformer is energized, oltge drop is osered due to the inrush urrents. An urte trnsformer model tht n reprodue the inrush urrents is required for preisely ssessing the oltge drop. Sine the mgnitude of the inrush urrent is ffeted y the iron-ore mgnetizing hrteristis nd iron-ore geometry, the modeling of the ironore mgnetizing hrteristis nd the representtion of the iron-ore geometry re the two key ftors in deeloping n urte trnsformer model. In this pper, the inrush urrents nd residul flux re lulted using newly proposed iron-ore mgnetizing model nd geometry representtion, nd the results re lidted y omprison with lortory test results. Keywords: Trnsformers, Inrush urrents, Residul flux, Mgnetizing iruit, Iron-ore geometry, nd Eletromgneti trnsient (EMT) simultions. I. INTRODUTION hen trnsformer is energized, oltge drop is W osered due to the inrush urrents []. From the iewpoint of power qulity, n urte trnsformer model tht n reprodue the inrush urrents is required for preisely ssessing the oltge drop. Now then, residul flux is usully set 8 % of the rted flux for lulting the worst se in the nlysis of the inrush urrents []. Howeer, if trnsformer model is le to reprodue urte residul flux, it eomes possile to set prtil residul flux nd to lulte pproprite inrush urrents. Thus, trnsformer model is required to reprodue not only inrush urrents ut lso residul flux. Sine the mgnitude of the inrush urrent nd residul flux re ffeted y the iron-ore mgnetizing hrteristis nd iron-ore geometry, the modeling of the iron-ore mgnetizing hrteristis nd the representtion of the ironore geometry re the two key ftors in deeloping n urte trnsformer model. An Iron-ore mgnetizing hrteristi n e represented y the mgnetizing iruit in Steinmetz s trnsformer equilent iruit. Existing mgnetizing iruit models n e diided into two lsses: ehior-sed models nd physissed models. The prmeter determintion proess for ehior-sed models is reltiely simple, ut these models R. Yonezw nd T. Nod re with Eletri Power Engineering Reserh Lortory, RIEPI (entrl Reserh Institute of Eletri Power Industry), -6-, Ngsk, Yokosuk, Kngw 4-96, Jpn (e-mil: {yonezw, tkunod}@riepi.denken.or.jp). Pper sumitted to the Interntionl onferene on Power Systems Trnsients (IPST5) in tt, roti June 5-8, 5 nnot urtely reprodue the residul flux, whih my he signifint effet on the mgnitude of the inrush urrent. The hu model [3], whih onsists of nonliner resistne nd nonliner indutne onneted in prllel, is one of the most well-known ehior-sed models. On the other hnd, the physis-sed models n reprodue residul flux fter deenergiztion, ut their prmeter determintion proess is quite omplited [4]. The Jiles-Atherton model [5], [6] nd Preish model [7], [8] re exmples of physis-sed models. To oerome the limittions of two types of model, ehior-sed model tht is ple of reproduing the residul flux hs reently een proposed [9]. This model onsists only of liner nd nonliner RL elements, nd their prmeters n e determined in simple wy using informtion otined from the nmeplte, test report, nd urrent-flux ure. It hs een shown tht the model preisely reprodues lortory test results in terms of the inrush urrents nd residul flux [9]. The other key to deeloping n urte trnsformer model is to represent of the iron-ore geometry. Mny power trnsformers he three-legged iron ore or fie-legged iron ore; these iron-ore geometries re not three-phse symmetril. To urtely reprodue the mgneti flux pths through the iron ore, it is neessry to model the mgneti iruit representing the iron-ore geometry. The following three pprohes n e used to represent the mgneti iruit: ) the trditionl pproh [], ) the dulity-sed pproh [], [], nd 3) the gyrtor-pitor-sed pproh [3]- [5]. Although the equtions desriing these three representtions re identil, their implementtion is different in generl-purpose eletromgneti trnsient progrms nd their ese of understnding y engineers is different. In this pper, we ompre the oe three pprohes to representing the iron-ore geometry nd propose to the use of the gyrtor-pitor-sed pproh owing to its ese of implementtion nd understnding. In ddition, the inrush urrents nd residul flux re lulted using the proposed iron-ore mgnetizing model nd the gyrtor-pitor-sed pproh, nd the results re lidted y omprison with lortory test results. II. MAGNETIZING IRUIT MODEL This setion riefly desries the mgnetizing iruit model proposed in Ref. [9]. Figure shows the mgnetizing iruit model, whih is omposed of non-liner indutne L mg, non-liner
L mg i L i m R L R mg i R Fig.. The mgnetizing iruit model hs een proposed in Ref. [9]. resistne R mg, liner resistne R, nd liner indutne L. L mg represents the mgneti sturtion hrteristi. R mg represents the iron-ore loss nd is hrterized y low resistne (the slope of the oltge-urrent hrteristi, ) when oltge ross the model is smll; otherwise it is hrterized y the onstnt urrent h s shown in Fig.. R nd L re elements tht re inluded to perform smooth swithing of the opertion mode to e desried lter. R is set to lue orresponding to the iron loss nd L is set to the time onstnt of L nd is out ms. The set lue of h is otined from the rted oltge nd R using the following eqution: h = () R The proposed model hs four operting modes s shown in Fig.. is the pitne etween the trnsformer winding nd the ground. A. Stedy Stte When the oltge ross the model is suffiiently lrge, R mg enters the onstnt-urrent (= h) mode. i m is the sum of i L nd i R (mode ), where i m is the mgnetizing urrent, i L is the urrent in L mg, i R is the urrent in R mg. At this time, R mg represents the iron loss euse i R flows through L insted of R. When the oltge ross the model is smll, R mg enters the low-resistne mode nd i R suddenly hnges. At this time, i R flows through R euse the impedne of L is higher thn tht of R (mode ). R represents the iron loss euse R mg is in the low-resistne mode. Therefore, the model repetedly opertes in mode nd mode in the stedy stte, R mg nd R represent the iron loss in modes nd, respetiely. B. After De-Energiztion If high oltge ppers fter de-energiztion, the impedne R mg will e lrger thn the impedne nd L mg sine R mg enters the onstnt-urrent mode. Thus, most of the urrent flowing through the iruit flow only L mg nd, nd nturl L response ppers (mode 3). After the ringdown trnsient hs onerged, R mg enters the low-resistne mode. Sine i R hnges slowly, i R flows through L insted of R. onsequently, the irulting urrent ontinues flowing etween R mg nd L mg for long durtion euse R mg is in the low-resistne mode (mode 4). In ft, the residul flux is stti mgneti phenomenon in the iron ore; this mgnetizing iruit model represents it φ i h i using the ontinuous urrent. As noted oe, owing to the two opertion modes of R mg, this model is ple of the representing two different onditions: the stedy stte nd the ondition fter de-energiztion. The prmeters for the model re determined in simple mnner using informtion otined from the nmeplte, test report, nd urrent-flux ure. The detiled prmeters of trnsformers, suh s the numer of turns of the winding nd iron-ore size, re usully not ille. But the informtion of nme plte, test report nd urrent-flux ure re ille in mny ses. III. REPRESENTATION OF IRON-ORE GEOMETRY In this setion, the three pprohes to modeling the mgneti iruit, the trditionl pproh, the dulity-sed pproh, nd the gyrtor-pitor-sed pproh, re ompred. A. The Trditionl Approh The trditionl pproh to representing mgneti iruit is to use relutne-resistne nlogy []. The mgnetomotie fore (MMF) F is regrded s eing nlogous to the oltge nd the mgneti flux Φ s eing nlogous to the urrent. Figure 3 shows single-phse two-winding trnsformer iruit representing the eletril iruit nd the Primry winding Stedy stte After de-energiztion L i i L mg When the oltge ross the proposed model is lrge R mg Mode Mode 3 Mode 4 Seondry winding L Φ Φ A) The eletril iruit B) The mgneti iruit Fig. 3. A single-phse two-winding trnsformer iruit representing the eletril iruit nd the mgneti iruit using the trditionl pproh. Lrge Lrge When the oltge ross the proposed model is smll Mode Fig.. The proposed mgnetizing iruit model hs four operting modes. R Φ Lmg R mg Smll Smll
mgneti iruit using the trditionl pproh ssuming tht it is not onsidered the winding resistne nd the mgnetizing iruit is represented y liner resistne nd liner indutne. The prts indited y the dotted line in Fig. 3 re the mgnetizing iruit. The mgneti iruit n represent the lekge flux Φ s lekge mgneti pths, ut n eletri iruit nnot represent it diretly. The lekge indutne is distriuted to the primry side L nd seondry side L. The indutne L in the eletri iruit is onerted into relutne R in the mgneti iruit, nd the resistne R in the eletri iruit is onerted to the mgneti indutne L in the mgneti iruit. The lue of eh element is trnsformed using the following equtions: N N R =, L = () L R where N is the numer of turns of the winding. The onnetions in series in the eletri iruit re onerted into prllel in the mgneti iruit, nd ie ers. An exmple of the mgneti iruit of two-winding, three-phse, three-leg trnsformer is shown in Fig. 4 []. nd re the nonliner relutnes of the trnsformer lim nd yoke, respetiely. R is the liner relutne of the lekge pths etween the primry nd seondry windings, R is the liner relutne of the lekge pths etween the seondry (inner) winding nd the iron ore, R is the liner relutne of the zero-sequene lekge, nd nd re the MMF for indued y the primry nd seondry windings, respetiely. The mgneti iruit represented using the trditionl pproh is ple of presering physil topologies, mking it esy to intuitiely understnd. Howeer, it should e noted tht the losses re represented y indutors, unlike in n eletri iruit. In the implementtion of mgneti iruit using the trditionl pproh in generl-purpose eletromgneti trnsient progrms, it is possile to represent the oupling etween the eletri iruit nd the mgneti iruit using the following three omponents: ) urrentontrolled oltge soure relting the MMF in the mgneti iruit nd the winding urrent; ) differentil iruit used to lulte the rte of hnge of the mgneti flux flowing in the mgneti iruit; 3) oltge-ontrolled oltge soure relting the winding oltge nd the rte of hnge of the mgneti flux. R R R B. The Dulity-Bsed Approh The dulity-sed pproh to representing the mgneti iruit is to use n eletri iruit tht hs een onerted from mgneti iruit using dulity theory [], []. The equilent eletri iruit in Fig. 5 is otined y the dulity trnsformtion of the mgneti iruit in Fig. 3. The lue of eh element is trnsformed using the following equtions: N N L = = L, R = = R (3) R L This pproh n e diretly used for eh element in the eletril iruit. The equilent eletri iruit in Fig. 6 is otined y the dulity trnsformtion of the mgneti iruit in Fig. 4. Unlike the trditionl pproh, mgneti pths eome nodes, nd ie ers, sine the physil topologies re not presered. Therefore, some effort is required to trnsform the iruit nd to the otined result. In the implementtion of the mgneti iruit using the dulity-sed pproh in generl-purpose eletromgneti trnsient progrms, it is possile to represent the oupling etween the eletri iruit nd the mgneti iruit using the following two omponents: ) urrent-ontrolled urrent soure relting the urrent in the mgneti iruit nd the winding urrent; ) oltge-ontrolled oltge soure relting the winding oltge nd the oltge in the mgneti iruit. In prtie, the pir of ontrol soures is the sme s tht in n idel trnsformer itself. Therefore, it n e implemented using idel trnsformer omponents.. The Gyrtor-pitor-Bsed Approh The gyrtor-pitor-sed pproh to representing the mgneti iruit is to use relutne-pitne nlogy [3]-[5]. The MMF is regrded s eing nlogous to the oltge nd the rte of hnge of the mgneti flux Φ s eing nlogous to the urrent. The iruit representing the one of Fig. 3 in the gyrtor-pitor-sed pproh is shown in Fig. 7. R in the eletri iruit is onerted into ondutne L R mg L mg Fig. 5. The equilent eletri iruit otined y the dulity trnsformtion of the mgneti iruit in Fig. 3. i i R R R R R R R R R R R R R R R Fig. 4. An exmple of the mgneti iruit of two-winding, three-phse, three-leg trnsformer using the trditionl pproh. Fig. 6. The equilent eletri iruit otined y the dulity trnsformtion of the mgneti iruit in Fig. 4.
Φ Φ Φ Fig. 7. The equilent eletri iruit otined y the gyrtor-pitnesed pproh of the eletri iruit in Fig. 3. G in the mgneti iruit, nd L in the eletri iruit is onerted to the mgneti pitne in the mgneti iruit. The lue of eh element is trnsformed using the following equtions: N N =, G = (4) L R In this pproh, it is diffiult to intuitiely understnd tht the indutne is onerted to the pitne. Howeer, the physil topologies re presered, nd the losses re represented s resistne. Furthermore, the produt of the ross rile nd the through rile is the power, onsistent with the eletri iruit. (The produt of the ross rile nd the orresponding through rile is the energy in the trditionl pproh.) In the implementtion of mgneti iruit using the gyrtor-pitor-sed pproh in generl-purpose eletromgneti trnsient progrms, it is possile to represent the oupling etween the eletri iruit nd the mgneti iruit using the following two omponents: ) urrent-ontrolled oltge soure relting the winding urrent nd the MMF in the mgneti iruit; ) urrentontrolled oltge soure relting the rte of hnge of the mgneti flux flowing in the mgneti iruit nd the winding oltge. The omponent with the pir of ontrol soures tht n inert the reltionship etween the oltge nd urrent of the two iruits is lled the gyrtor. Thus, this pproh is lled, gyrtor-pitor-sed pproh. A summry of the fetures of three methods of representing the mgneti iruit is gien in Tle I. Although the mg equtions desriing these three representtions re identil euse they only onert the riles, there re some differenes s desried oe. We propose the use of the gyrtor-pitor-sed model to represent the mgneti iruit euse the physil topologies re presered, the losses re represented s resistnes, nd differentil iruits re not required. I. ERIFIATION In this setion, the inrush urrents nd residul flux re lulted using the proposed iron-ore mgnetizing model nd the gyrtor-pitor-sed pproh, nd the results re ompred with those of lortory tests for lidtion. The speifitions of the test trnsformer used in the erifition re shown in Tle II. The experimentl setup is shown in Fig. 8. The test trnsformer is dry-type trnsformer, nd the pitne etween the windings nd the housing is smller thn tht of typil oil-insulted power trnsformer. To simulte typil power trnsformer, pitnes of, pf re onneted etween the primry winding terminl nd the ground of eh phse. The neutrl point on the primry side is grounded using 47 Ω resistor. Figure 9 shows the iruit of the test trnsformer using the gyrtor-pitorsed pproh. The eletri iruit is onneted to the winding resistnes R w, nd R w of eh winding. The mgneti iruit is onneted the proposed mgnetizing iruit, nd representing the lekge pths orresponding to eh lekge indutne L. R nd R in Fig. 4 re ssumed to e suffiiently lrger thn the mgneti relutnes of the iron ore nd re remoed from the model in Fig. 9. Beuse the numer of turns of the primry winding N nd the numer of turns of the seondry winding N re unknown, N is ssumed e 3 nd N is ssumed euse the rtio etween the numer of turns of the windings nd the rtio etween the oltges re suequl. The hrteristis of mg nd re shown in Fig. nd the lue of G is 4.57 [S], nd is 6.4 [μh]. Owing to the effet of the yoke, the phse orresponds to the enter-leg winding nd the other phses he different urrent-flux hrteristis [6]. Howeer, in this simultion, ll three phses re pplied to the i Φ hrteristi of the phse winding. is pproximted y TABLE I A SUMMARY OF THE FEATURES OF THREE METHODS OF REPRESENTING THE MAGNETI IRUIT. Eletril domin The trditionl pproh Mgneti domin The dulity-sed pproh The gyrtor-pitor-sed pproh The ross rile oltge [] MMF [A] oltge [] MMF [A] The through rile urrent [A] Mgneti flux [W] urrent [A] Mgneti flux rte [W/s] The produt of the ross rile nd the through rile Power [W] Energy [J] Power [W] Power [W] The element of representing the iron loss Resistne Indutne Resistne Resistne The physil topologies - Presere Not presere Presere The oupling of the eletri iruit nd mgneti iruit - A differentil iruit nd two ontrolled sorues An idel trnsformer (two ontrolled soures) A gyrtor (two ontrolled soures)
Fig. 8. 3φ TABLE II THE SPEIFIATIONS OF THE TEST TRANSFORMER. Winding onnetion Rted power Rted oltge (H) Rted oltge (L) Frequeny No-lod loss Lod loss Impedne oltge Lekge indutne Winding resistne (H) Winding resistne (L) iruit rekers A A A The experimentl setup. S f W o W s s L R w R w unit A Hz W W mh Ω Ω Tr Y Δ 5 6 5 3.7.7 8.3.4.8.434 Trnsformer 47Ω N A. Residul Flux A typil iruit reker n generlly turn off when the urrent eomes zero. Howeer, it might turn off t nonzero urrent sine the mgnetizing urrent is quite smll s the trnsformer is not sujeted to lod. The oltge, urrent, nd flux weforms of the primry winding when the iruit rekers of eh phse re simultneously turned off from the stedy stte re shown in Fig.. The flux weforms re lulted from the mesured oltge weforms y numeril integrtion. The phenomenon tht the residul flux is retined fter the iruit rekers re turned off is represented, nd the lues of residul flux of eh phse he similr lues to the mesured lues. Figure shows n enlrged iew of Fig. round the region where the iruit rekers re turned off. Oeroltges pper fter the iruit rekers re turned off euse the rekers ut smll urrents. These phenomen re reprodued in the simultion results, ut the mgnitudes of the oeroltge re smller thn the mesured lues. This simultion tkes into ount the iron-ore geometry ut not the different of the i Φ hrteristis etween the enter leg nd the other leg. This is onsidered to e one of the uses of the differenes etween the mesured nd lulted results. In ddition, the results when the iruit rekers of eh phse re turned off t urrent zeros re shown in Fig. 3. The oeroltges do not pper in this ondition, nd the lues of residul flux of eh phse he similr lues to the mesured lues. R w R w i R w i R w i mg G mg G mg G i R w i R w i Eletril domin Mgneti domin Eletril domin Fig. 9. The iruit of the test trnsformer using the gyrtor-pitor-sed pproh. oltge [] 5 5-5 -5 5 5 5 3 35.. -. -. 5 5 5 3 35.5 -.5-5 5 5 3 35 Flux [W]. 6.. -6. mg -. -3-3 MMF [A] Fig.. The hrteristis of mg nd. -8-4 - 4 nonliner resistne, nd suffiiently lrge lue ompred with G is set s the slope of the onstnt-urrent portion. The EMT nlysis progrm XTAP (expndle Trnsient Anlysis Progrm) [7] is used in these simultions. MMF [A] 8 4-4 Mgneti flux rte [W/s] Fig.. The oltge, urrent, nd flux weforms of the primry winding when the iruit rekers of eh phse re simultneously turned off from the stedy stte (Solid line: mesured, dshed line: lulted). oltge [] 5 5-5 -5.6.3 -.3 -.6 7. 7.4 7.6 7.8 8 8. 8.4 8.6 8.8 9 7. 7.4 7.6 7.8 8 8. 8.4 8.6 8.8 9 Fig.. An enlrged iew of Fig. round the region where the iruit rekers re turned off (Solid line: mesured, dshed line: lulted).
oltge [] - - 5 5 5 3 -. 5 5 5 3.. -. - - 5 5 5 3 oltge [] - - - 3 4 5-5 - 3 4 5 5 5-5 - - - 3 4 5 Fig. 3. The oltge, urrent, nd flux weforms of the primry winding when the iruit rekers of eh phse re turned off t urrent zeros (Solid line: mesured, dshed line: lulted). Fig. 4. The oltge, urrent, nd flux weforms when the energiztion ours t 54 (Solid line: mesured, dshed line: lulted). B. Inrush urrents The inrush urrents re mesured under the ondition fter the iruit rekers of eh phse re turned off t urrent zeros. The timing of the energiztion is ried from to 36 with inrements of 8. In simultions, the sme simultion se in the preious setion is used. The omputtion is proeeded for ms from the de-energiztion timing, nd the next timing tht the oltge of phse eomes is regrded s in this simultions. The residul flux of phse (solute lue) is the lrgest mong ll phses in these experiments. In this se, it is expeted tht the inrush urrent will e lrgest when the oltge of phse is energized t 8 (6 for phse ). Figure 4 shows the oltge, urrent, nd flux weforms when the energiztion ours t 54. The mesurement results nd lulted results re in good greement. The pek lues (solute lue) of the inrush urrent of eh phse t eh timing re shown in Fig. 5. As predited oe, the pek inrush urrent ours when the energiztion ours t pproximtely 6, nd it is possile to onfirm the sme tendeny in the lultion.. ONLUSIONS In this pper, trnsformer models tht tke into ount the iron-ore geometry nd n e used for inrush urrent nd residul flux lultions re desried. The three pprohes to modeling the iron-ore geometry re ompred nd the use of the gyrtor-pitor-sed is proposed owing to its ese of implementtion nd understnding. It ws shown tht the proposed mgnetizing model with the mgneti iruit model using the gyrtor-pitor-sed pproh preisely reprodues lortory test results for the inrush urrents nd residul flux. I. FURTHER WORK In this pper, we he only lidted the proposed model using the smll trnsformer not n tul trnsformer. Thus, we will rry out the proposed model lidtion using tul power trnsformers. 3 5 5 5 35 3 5 5 5 5 45 4 35 3 5 5 5 8 36 54 7 9 8 6 44 6 8 98 6 34 5 7 88 36 34 34 36 energiztion timing [deg] A) Phse Fig. 5. The pek lues (solute lue) of the inrush urrent of eh phse t eh timing. II. REFERENES mesured lulted 8 36 54 7 9 8 6 44 6 8 98 6 34 5 7 88 36 34 34 36 energiztion timing [deg] B) Phse mesured lulted 8 36 54 7 9 8 6 44 6 8 98 6 34 5 7 88 36 34 34 36 energiztion timing [deg] ) Phse mesured lulted [] L. F. Blume, G. milli, S. B. Frnhm, nd H. A. Peterson, Trnsformer mgnetizing inrush urrents nd influene in system opertion, AIEE Trns., ol. 63, no. 6, pp. 366-375, 944. [] IGRE WG 4.37, Trnsformer energiztion in power systems: study guide, IGRE Tehnil Brohure 568, Fe. 4. [3] L. O. hu nd K. A. Stromsmoe, Lumped-iruit models for nonliner indutors exhiiting hysteresis loops, IEEE Trns. on iruit Theory, ol. T-7, no. 4, pp. 564-574, No. 97.
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