ekge nductnce Model for Autotrnsformer rnsient imultion B. A. Mork, Member, EEE, F. Gonzlez, Member, EEE, D. shchenko, Member, EEE Abstrct his pper provides thorough nlysis of the lekge inductnce effects of n utotrnsformer, reconciling the differences between the 3-winding blck box ssumption mde in fctory short-circuit tests nd the ctul series, common, nd delt coils. An importnt new contribution is inclusion of the lekge effects between coils nd core nd creting topologiclly correct point of connection for the core equivlent. Keywords: Autotrnsformers, EMP, nductnce, rnsformer Models, rnsient imultions.. NRODUON methodology is developed to clculte utotrnsformer coil rectnces nd formulte the inverse lekge inductnce mtrix. he formultion is bsed on shortcircuit impednce dt typiclly provided in stndrd fctory test reports. Ultimtely this lekge representtion is being incorported into new hybrid trnsformer model for simultion of low- nd mid-frequency trnsient behviors [4]. he key dvncement is to estblish topologiclly correct point of ttchement for the core. he resulting N+ winding lekge representtion cnnot be directly produced by the commonly used BRAN supporting routine of EMP. herefore, it is useful to document the development of this N+ lekge inductnce representtion, which lso is in the form of the [A] mtrix (inverse inductnce mtrix [] - ) []. he elements tht form this mtrix include the effect of the respective turns rtios between coils. he representtion is of the ctul series, common, nd delt coils, nd not of the threewinding blck box equivlent typiclly ssumed. A. hort-circuit est Dt. MODE DEVEOPMEN n generl, for n N-winding trnsformer, the per-phse representtion of the lekge rectnces of the windings is fully-coupled N-node inductnce network, s shown in Fig.. [A] cn be topologiclly constructed just s ny nodl dmittnce mtrix. he individul - lekge vlues cn be determined from binry short-circuit tests. n the specil cse upport for this work is provided by Bonneville Power Administrtion, prt of the U Deprtment of Energy, nd by the pnish ecretry of tte of Eduction nd Universities nd co-finnced by the Europen ocil Fund. B. A. Mork, F. Gonzlez nd D. shchenko re with the Dept. of Electricl Engineering, Michign echnologicl University, oughton, M 4993, UA. Presented t the nterntionl onference on Power ystems rnsients (P 05) in Montrel, nd on June 9-3, 005 Pper No. P05-48 of n utotrnsformer, or for three-winding trnsformer in generl, [A] reduces to simple three-node delt-connected circuit. ommon prctice hs been to convert this delt to wye or str equivlent for stedy-stte short-circuit or lodflow clcultions. his str equivlent often contins negtive inductnce t the medium voltge terminl, which cn be of concern for some trnsient simultions [], [3]. Fig.. hort circuit representtion for N-winding trnsformer. Dt typiclly vilble from fctory short-circuit tests re: hort-circuit impednces in %, MVA bse of ech winding, nd short-circuit losses in kw. n the lekge representtion developed here s prt of the hybrid model [4], [5], short-circuit rectnces nd coil resistnces re seprtely represented, mking it convenient to work directly with [A] nd its purely inductive effects. he turns rtios of the coils cn lso be directly incorported into [A]. ekge effects lso exist between the core nd the coils. his is conceptully delt with by ssuming fictitious infinitely-thin N+ th coil t the surfce of the core. Fig. shows the conceptul implementtion of the N+ winding flux lekge model for the reduced cse of two winding trnsformer. he reder is directed to Appendix B for definitions of terminl lbels nd subscript nottions. ylindricl coils re ssumed, but this pproch is generlly pplicble for other coil configurtions [9]. hese core-to-coil lekge effects re importnt for detiled models but re not considered (or mesured) in fctory tests. he N+ th winding serves s n ttchment point for the core equivlent [4].
ORE ow voltge igh voltge oil-to-coil flux linkge ( K + ) + () ORE ORE D ORE D D ( K + ) () K (3) D del N+th coil of zero thickness, resting on surfce of core oil-to-core flux linkges λ - λ -ORE λ -ORE Fig.. onceptul implementtion of N+ th winding flux lekge model. he generic blck box equivlent does not represent the ctul series, common nd delt coils of the utotrnsformer, but rther ssumes tht the three windings re rted ccording to respective terminl voltges nd currents. owever, for detiled model, the ctul coil topology must be represented. his is illustrted on per-phse bsis in Fig. 3. Z usully is quite smll since the series nd common coils re relly one coil with tp point. D is the lrgest since the series coil will hve the highest voltge nd insultion buildup. D is not quite s lrge, but significnt, since there cn be extr insultion nd brriers nd oil duct spce between the delt nd the medium-voltge winding. ekges between coils depend on the type of core (shell or core) nd the coil configurtion (cylindricl or pncke). As first pproximtion, K might be estimted s 0.5, if one ssumes tht the innermost coil is lso the lowest voltge coil, with modest coil-to-core insultion requirements. his rtionle is bsed on [9] nd the flux linkge distributions of Fig.. B. oil rectnces he methodology to clculte the utotrnsformer coil rectnces is presented here. his formultion is bsed on the short-circuit rective power tht cn be obtined from the short-circuit losses. Results obtined re equl to nother pproch by Dommel [], but the derivtion dels directly with ctul impednce vlues thus voiding the complexities of trnsforming per unit vlues ccording to the voltge nd MVA bses of the series, common nd delt coils. hree binry short-circuit tests re usully performed for the utotrnsformer (Figs. 4-6). Z Z Z 0 Fig. 3.Autotrnsformer configurtion nd impednces of ech coil. Z Z opened he lekge rectnces between the coils cn be clculted ccording to the flux linkges of Fig. [9]. ince the fictitious N+ th coil is interior to ll other coils, lekge flux between the core nd coils will typiclly be more thn tht between the innermost coil nd other coils. ence, ()-(3) cn be used to describe the lekge rectnces between core nd the series, common nd delt coils respectively, where K represents the dditionl effect of lekge between innermost coil nd core. Note tht the delt coil is ssumed to be innermost, i.e. hving the lest coil-to-core lekge of ny coil. n generl, (3) is ssocited with the coil hving minimum coil-to-core lekge. Fig. 4. Per-phse short-circuit test -. hort-circuit - (Fig. 4): /3 V / 3 3 V (4) Q V /,, /3 / 3, 3 V, 3 Q + Q + (5) (6)
with Q being the rective power in this test, Q the rective power corresponding to the series coil, nd Q the rective power corresponding to the common coil. ( Q / 3 Q / 3) Q / 3 (3) Q / 3 (4) Z Q / 3 (5) opened 0 Z Z. hort-circuit winding rectnces he str-delt trnsformtion [] is then pplied to obtin the delt equivlent for the three coils. Fig. 7 illustrtes the reltionship between binry short-circuit test mesurements nd the individul elements of the delt equivlent. Fig. 5. Per-phse short-circuit test -. opened 0 Fig. 6. Per-phse short-circuit test -. Z Z Z hort-circuit - (Fig. 5): /3 V / 3 3 V (7), V, /3 3 V,, ( + ) + Q / 3 Q + Q + Q (9) hort-circuit - (Fig. 6): /3 V / 3 3 V (0), V /3 3 V,,, (8) () Q / 3 Q + Q + () he three equtions with three unknowns (6), (9), nd (), cn be solved to find coil rectnces s function of rective powers nd rted currents. Results re shown s follows: Dpu D Dpu pu -pu Fig. 7. lcultion of short-circuit winding rectnces. Mesurement of short-circuit rectnce between series nd common coils hese reltionships for ll three binry short-circuit tests re given in (6)-(8). Rectnces re in per unit using common MVA bse nd the voltge bse of ech coil. pu + pu D pu + (6) D pu D pu + D pu pu + (7) D pu D pu + D pu pu + (8) D pu D. Admittnce formultion An dmittnce-type formultion is used to obtin [A] directly from individul inverse inductnces [6]. hese inverse inductnces re nlogous to trnsfer dmittnces. he circuit in Fig. 8 represents the overll lekge inductnce effects, including the N+ th coil. del trnsformers re used to represent the turns rtios of the coils. nverse inductnce vlues re referred to the lower voltge side in ech cse. he methodologies of [], [8], nd [0] cn be dpted to obtin the [A] mtrix, using the rectnces, D, D solved for from (6)-(8) nd converted from per unit to ctul vlues. nverse inductnce vlues re simply ω /. Fig. 9 illustrtes the contribution of turns rtios nd individul inverse inductnces to [A]. onceptully this is submtrix whose elements re dded into the pproprite row-
column positions of [A]. Prmeters re clculted ccording to (9), which cn be derived vi the sme short-circuit method used to obtin dmittnce mtrix vlues. his submtrix cn lso be visulized s Pi-equivlent. he resulting [A] is symmetric. his mtrix is symmetric nd its elements cn be obtined directly from those clculted by mens of (6)-(8). o include ll coils, the dmittnce mtrix formultion will be used. [ red pu ] is first inverted [ A ] [ ] red red pu pu () N : N - N : N D N : N D N : N D - D - D ORE D - - D ORE hen, the M th row nd column re dded to obtin the full [A] mtrix M k (3) im pu M i ik pu (4) MM pu im pu o convert from per unit to the ctul vlues required for EMP simultion, ll elements of [A] re multiplied by the common VA bse, nd ech row nd column i is multiplied by /V i. N: N D - ORE ore reference Fig. 8. Admittnce formultion for n utotrnsformer (fictitious winding is included). c N : N reference -. AP MODE he following utotrnsformer is implemented here s n exmple: 40/40/63 MVA utotrnsformer Wye-wye-delt utotrnsformer configurtion 345GRY/99.:8GRY/68.:3.8 kv. ble summrizes the intermedite steps in clculting [A] for this trnsformer. he full three-phse [A] is given in AP formt in Appendix A. Figs. 0 nd show how the individul series, common nd delt coils re connected. EMP simultions of the binry short-circuit tests mtch with vlues reported in the fctory test report. BUA BUA R NODEA NODAA R [A] R D BUA delt connection Fig. 9. ncorportion of turns rtio, resulting Pi-equivlent, nd contribution ech inverse inductnce to [A]. c c c (9) Algorithmiclly, [A] cn be constructed using the methods of []. For generl M M cse, the first step is to clculte the reduced M- M- mtrix [ red ], whose digonl elements cn be obtined s red ii pu pu (0),iM pu nd whose off-digonl elements re red + - ik pu [ ],im pu,km pu,ik pu. () NODEA NODAA NODEA NODAA N+th winding Fig. 0. Autotrnsformer. ingle-phse representtion. V. ONUON [] he N+ lekge representtion developed here includes the lekge inductnces between core nd coils, which re not considered in typicl EMP implementtions, such s BRAN. he ctul coils of the trnsformer re represented, s opposed to blck box N-winding equivlent. Prmeters cn be obtined from design informtion or from fctory
Dt (rtings) V -, 345 kv V -, 8 kv V -, 3.8 kv bse 00 MVA Dt (short-circuit) pu 0.0584 pu 0.089 pu 0.0878 oil rectnces [from (3)-(5)] 3.5475 Ω -0.9700 Ω D 0.8359 Ω ABE [A] MAR AUAON hort-circuit rectnces [from (6)- (8)] -pu 0.056 -Dpu 0.095 -Dpu 0.393 Assumptions V ORE V D ORE- (K+) D + ORE- (K+) D ORE-D K D K 0.5 [A] mtrix [from (3)-(4)] 3.3 (/) 73.6 (/) 33 448.3 (/) 44 878.4 (/) -6.8 (/) 3 8.9 (/) 4 -.9 (/) 3-34.3 (/) 4 44.3 (/) 34-67.3 (/) BUA NODEA eg NODAA BUA BUA NODBB NODEA NODEA eg NODEB eg NODE NODAA NODBB NOD NODEB NODEA eg eg 3 NOD NOD NODAA NODBB NODEB NOD BU NODBB BUB NODE eg 3 eg BU NODE eg 3 eg NODEB BU NODAA NODE BUB BUB G/OW VOAGE WNDNG ERARY WNDNG N+th WNDNG Fig.. Autotrnsformer. onnections. mesurements. mplementtion is intuitive nd bsed on the [A] mtrix tht is commonly used in EMP progrms. Results of the model constructed nd tested here mtch with fctory test reports. his lekge model cn be used s the bsis of low- nd mid-frequency topologiclly-correct trnsformer models, with representtions for the core, cpcitive effects, nd coil resistnces built round it [4]. V. APPEND A. [A] mtrix prmeters ------------------------------------------------- [A] MAR [A] [R] ------------------------------------------------- UE AR $VNAGE,, NODEANODAA 3.67 0.0 NODEANODAA -6.805 0.0 73.598 0.0 3NODEANODAA 8.9946 0.0-34.33 0.0 448.30 0.0 4NODEANODAA -.933 0.0 44.335 0.0-67.3 0.0 878.40 0.0 5NODEBNODBB 3.67 0.0 6NODEBNODBB -6.804 0.0 73.598 0.0 7NODEBNODBB 8.9946 0.0-34.33 0.0 448.30 0.0 8NODEBNODBB -.933 0.0 44.335 0.0-67.3 0.0
878.40 0.0 9NODENOD 3.67 0.0 0NODENOD -6.805 0.0 73.598 0.0 NODENOD 8.9946 0.0-34.33 0.0 448.30 0.0 NODENOD -.933 0.0 44.335 0.0-67.3 0.0 878.40 0.0 UE R $VNAGE, 0, B. erminl nd subscript definitions Nottion D or ORE - Definition igh voltge terminl ow voltge terminl ertiry voltge terminl eries coil ommon coil Delt (tertiry) coil Fictitious coil on surfce of core ine-to-line voltges V. REFERENE [].W. Dommel with. Bhttchry, V. Brndwjn,.K. uw nd. Mrtí, Electromgnetic rnsients Progrm Reference Mnul (EMP heory Book), Bonneville Power Administrtion, Portlnd, UA, 99 nd Edition. []. enriksen, rnsformer ekge Flux Modeling, Proc. nterntionl Power ystems rnsients onference P 00, Rio de Jneiro, Brzil, June 00. [3] P. olenrsipur, N. Mohn, V.D. Albertson, nd J. hristofersen, Avoiding the Use of Negtive nductnces nd Resistnces in Modeling hree-winding rnsformers for omputer imultions, Proc. EEE Power Engineering ociety 999 Winter Meeting, Vol., pp. 05-030, Jnury 3 Februry 4, 999. [4] B.A. Mork, F. Gonzlez, D. shchenko, D.. tuehm, J. Mitr, "ybrid rnsformer Model for rnsient imultion: Prt - Development nd Prmeters", EEE rns. on Power Delivery, to be published, PWRD-0005-004. [5] B.A. Mork, F. Gonzlez-Molin, nd J. Mitr, Prmeter Estimtion nd Advncements n rnsformer Models For EMP imultions. sk/activity MU-4/NDU-: ibrry of Models opologies, report submitted to Bonneville Power Administrtion, Portlnd, UA, June 5, 003. [6] B.A. Mork, F. Gonzlez-Molin, nd D. shchenko, Prmeter Estimtion nd Advncements n rnsformer Models For EMP imultions. sk/activity MU-6: Prmeter Estimtion, report submitted to Bonneville Power Administrtion, Portlnd, UA, December 3, 003. [7] J.W. Nilsson, Electric ircuits nd Edition, Addison Wesley, 987. [8] R.B. hipley nd D. olemn, A New Direct Mtrix nversion Method, AEE rnsctions, Februry 959. [9] A.K. whney, A ourse in Electricl Mchine Design, Dhnpt Ri & ons, 994. [0] R.B. hipley, D. olemn,.f. Wtts, rnsformer ircuits for Digitl tudies, AEE rnsctions, Prt, vol. 8, pp. 08-03, Februry 963. V. BOGRAPE Bruce A. Mork (M'8) ws born in Bismrck, ND, on June 4, 957. e received the BME, MEE, nd Ph.D. (Electricl Engineering) from North Dkot tte University in 979, 98 nd 99 respectively. From 98 through 986 he worked s design engineer for Burns nd McDonnell Engineering in Knss ity, MO, in the res of substtion design, protective relying, nd communictions. e hs spent 3 yers in Norwy: 989-90 s reserch engineer for the Norwegin tte Power Bord in Oslo; 990-9 s visiting resercher t the Norwegin nstitute of echnology in rondheim; 00-0 s visiting enior cientist t NEF Energy Reserch, rondheim. e joined the fculty of Michign echnologicl University in 99, where he is now Associte Professor of Electricl Engineering, nd Director of the Power & Energy Reserch enter. Dr. Mork is member of EEE, AEE, NPE, nd igm i. e is registered Professionl Engineer in the sttes of Missouri nd North Dkot. Frncisco Gonzlez ws born in Brcelon, pin. e received the M.. nd Ph.D. from Universitt Politècnic de tluny (pin) in 996 nd 00 respectively. As visiting resercher, he hs been working t Michign echnologicl University, t ennessee echnologicl University, t North Dkot tte University, nd t the Norwegin nstitute of cience nd echnology (Norwy). is experience includes eight yers s resercher involved in projects relted to Power. n October 00, Dr. Gonzlez ws wrded Postdoctorl Fellowship from the pnish Government. ince then, he hs been working s Postdoc t Michign echnologicl University. is reserch interests include trnsient nlysis of power systems, lightning performnce of trnsmission nd distribution lines, FA, power qulity, nd renewble energy. Dr. Gonzlez is member of the EEE Power Engineering ociety. Dmitry shchenko ws born in Krsnodr, Russi. e received his M.. nd Ph.D. degrees in Electricl Engineering from Kubn tte echnologicl University, Russi in 997 nd 00 respectively. n eptember 000 he ws wrded with the Norwegin Government Reserch cholrship nd worked s visiting resercher t the Norwegin nstitute of cience nd echnology (Norwy). is experience includes 5 yers s Power ystems Engineer t the outhern Division of the Unified Energy ystem of Russi. n Februry 003 he joined the Electricl nd omputer Engineering Deprtment of Michign echnologicl University s postdoctorl resercher. is reserch interests include computer modeling of power systems, power electronics, nd power system protection. Dr. shchenko is member of the EEE Power Engineering ociety.