Harmonic Analysis of the DC Biased Epstein Frame-Like Core Model by the Harmonic Balance Finite Element Method

Transcription

1 Harmonic Analysis of th DC Biasd Epstin Fram-Li Cor Modl by th Harmonic Balanc Finit Elmnt Mthod Author Zhao, Xiaojun, Lu, Junwi, Li, Lin, Chng, Zhiguang Publishd 2 Confrnc Titl Procdings of IEEE APEMC2 DOI Copyright Statmnt 2 IEEE. Prsonal us of this matrial is prmittd. Howvr, prmission to rprint/ rpublish this matrial for advrtising or promotional purposs or for crating nw collctiv wors for rsal or rdistribution to srvrs or lists, or to rus any copyrightd componnt of this wor in othr wors must b obtaind from th IEEE. Downloadd from Griffith Rsarch Onlin

2 2 Asia-Pacific Intrnational Symposium on Elctromagntic Compatibility, April 2-6, 2, Bijing, China Harmonic Analysis of th DC Biasd Epstin Fram-Li Cor Modl by th Harmonic Balanc Finit Elmnt Mthod Xiaojun Zhao #, Junwi Lu *2, Lin Li #, Zhiguang Chng 4 # School of Elctrical & Elctronic Enginring, North China Elctric Powr Univrsity PO Box, 69 Yonghua North Strt, Baoding, 7, China @6.com lilin@ncpu.du.cn * Griffith School of Enginring, Griffith Univrsity Brisban, 4, Australia 2 j.lu@griffith.du.au R&D Cntr of Baoding Tianwi Group Baoding, 756, China 4 mlab@btwtc.com Abstract Th DC biasd problm of th Epstin Fram-li cor modl is invstigatd by using th Harmonic Balanc Finit Elmnt Mthod (HBFEM). It considrs coupling btwn th lctric circuit and th magntic fild. Th magntizing currnt and its harmonic componnts can b calculatd dirctly in th harmonic domain. Comparing th calculatd rsults with th xprimntal data shows consistncy. Th DC bias ffct on th harmonic componnts of th xciting currnt is analyzd dirctly with th distribution of th magntic flux visually obsrvd through th harmonic solutions. I. INTRODUCTION With th dvlopmnt of HVDC transmission systms, rsarchrs and nginrs hav paid mor attntion to th DC biasd problm in HVDC transformrs. Th HVDC systm oprats in a monopolar mod and uss th arth as a rturn path. This will lad to a potntial diffrnc btwn th two convrting plants. Th rsulting lctric potntial diffrnc gnrats a dirct currnt that flows into th windings of HVDC transformrs through arthd nutrals in th AC ntwor. Anothr form of DC bias, Gomagntically Inducd Currnt (GIC) [], is causd by solar magntic disturbancs. Th apparanc of th DC in windings may damag th transformrs and lctric ntwors sriously. Various mthods hav bn usd to invstigat th DC biasd phnomnon in HVDC transformrs. Th lctric circuit modl [2] and magntic circuit modl [] wr proposd by som rsachrs to calculat th xcitation currnt in winding, whil th tim-stpping finit lmnt mthod [4] was usd by many rsarchrs to comput th magntizing currnt and magntic fild. Howvr, it is difficult to obtain accurat rsults from th lctric and magntic modls (spcially whn magntic fild analysis is rquird to invstigat th mchanism of th DC biasd problm). Th tim-stpping finit lmnt mthod is an ffctiv mthod to calculat th transint magntic fild. Howvr, accurat solutions of high ordr harmonics in th xcitation currnt and flux dnsity will rduc th ffctivnss of tim-stpping mthod. In this papr th harmonic balanc finit lmnt mthod [5] (HBFEM) is introducd to solv th nonlinar magntic fild undr DC bias conditions, considring coupling of lctric circuits and magntic filds. Du to th DC bias, zro ordr, odd ordr and vn ordr harmonic componnts constitut th xcitation currnt in transformr windings. All harmonic componnts of currnt and vctor potntial can b solvd dirctly in th harmonic domain by th HBFEM approach. HBFEM can b an ffctiv mthod of analysing th DC biasd problm in transformrs by comparing th calculatd rsults with xprimntal data basd on th Epstin Fram tst. II. HARMONIC BALANCE FINITE ELEMENT METHOD A. Basic Formulation Th nonlinar magntic fild can b prsntd by th following quation, ν A+ σ ( A/ t+ ϕ) J = () A is magntic vctor potntial, B rprsnts flux dnsity, v and σ ar rluctivity and conductivity rspctivly, J is xciting currnt. For two-dimnsional problms in nonlinar magntic filds, quation (2) can b writtn as, A A A ϕ υ + υ + σ + = J x x y y t z (2) It is assumd that th currnt flows in Z dirction ovr an infinitly long distanc. This mans th lctric scalar potntial is constant, so th trm including ϕ can b nglctd in (2). In light of Galrin s mthod, th wightd rsidual is, //$26. 2 IEEE 494

3 Ni A Ni A G = υ + υ dxdy+ Ω x x y y A N σ i dxdy J s N i dxdy Ω t Ω ( ) () whr N is th intrpolation function for a linar triangular i lmnt. In th DC biasd phnomnon w focus on th tim-priodic solutions. This is actually a harmonic problm whn altrnating currnt and dirct currnt ar applid togthr. According to Fourir transformation thory all variabls such as vctor potntial A, flux dnsity B, xcitation currnt J and rluctivity v can b approximatd as a triangl sris basd on th harmonic balanc mthod. { ns ω nc A = A + A sin( n t) + A cos( n t) (4 a) i i i i { ω { ω { ω J = J + J sin( n t) + J cos( n t) (4 b) s ns nc B = B + B sin( n t) + B cos( n t) (4 c) x x xns xnc B = B + B sin( n t) + B cos( n t) (4 d) y y yns ync υ = υ + υ sin( nωt) + υ cos( nωt) (4 f ) { } ns Considring th orthogonal charactristics of trigonomtric functions, th HBFEM matrix quation for a singl lmnt can b obtaind from () and (4), G = [ S ]{ A } + [ N ]{ A } { K } ( bb + cc ) D ( bb 2+ cc 2) D ( bb + cc ) D = ( bb 2 + cc 2 ) D ( bb 2 2+ cc 2 2) D ( bb 2 + cc 2 ) D 4Δ ( bb + cc ) D ( bb 2 + cc 2) D ( bb + cc ) D A 2N N N A K σωδ A 2 N 2N N + 2 A2 K2 (5) A N N 2N A K = b in which i j i j rluctivity matrix and N is th harmonic matrix, D nc = y y, c = x x, D is th d d d d d d d d d d d d d d d = d4 d42 d4 d44 d45 d5 d52 d5 d54 d 55 (6) 2ν ν s ν c ν2s ν2c 2ν 2ν ν ν ν ν ν + ν 2ν 2ν + ν ν + ν ν + ν 2ν Symmtry 2ν + ν = 2 2 s 2c 2s c c s s c 2c s s c c = 2 2ν2s 2ν ν4c ν4s N 2c 4c B. Coupling btwn lctric circuits and th magntic fild Whn lctrical dvics ar xcitd by voltag, such as transformrs undr DC bias, th xcitation currnt is unnown and quation () is no longr applicabl. In this cas th coupling btwn th lctric circuit and th magntic fild should b tan into account [6]. According to Kirchhoff s law th applid voltag on th xtrnal port can b dfind as follows, (7) di Vin = V + R I + L + Idt dt C = [ C ]{ A} + S [ Z ]{ J } (8) c Vin is th input voltag of circuit, V is th corrsponding inducd lctromotiv forc, Sc and Z ar th cross aras and impdanc of winding rspctivly, C and L ar th capacitanc and inductanc rspctivly in circuit. Th inducd lctromotiv forc can b gaind basd on Faraday s law, V dψ d = = B ds dt dt d d = AdS = Adl dt dt f { A } ωnd 2 = [ N N N] A Sc { } (9) { A } Th xprssion of th impdanc matrix is givn by () and () rspctivly, Z Z = [ Z ] () Z2 495

4 Zn = ZRn + ZCn + ZLn R+ nω L n ω C = R+ nω L n ω C () Th systm matrix quation () can b rwrittn if th right trm rprsnting th xcitation charactristic is movd to th lft hand sid, [ S]{ A} + [ M]{ A} [ G ]{ J } = (2) Th nw systm matrix quation considring th applid voltag can b obtaind by combining (7) with (), Fig.2 Excitation currnt undr 5% DC bias [ H] [ G] [ G2] [ G ] { A} [ C] S [ Z] { J} { V } c in [ C2] Sc2[ Z2] { J2} { V 2} in = [ C] Sc[ Z] { J} { Vin} () Th vctor potntials and currnt dnsitis can b solvd simultanously by th abov quation. Fig. Excitation currnt undr 75% DC bias III. CALCULATION RESULTS AND FIELD ANALYSIS An Epstin Fram-li cor modl mad by th TianWi Group, Baoding, has bn tstd undr diffrnt DC bias conditions. Th xcitation currnt without a DC bias is slctd as th bnchmar, which mas th flux dnsity in silicon stl rach ratd valu in opration of transformrs. Th pa valu of th bnchmar currnt I is.68 ampr. A. Excitation Currnt Th DC bias is applid in incrmntal proportions of th bnchmar currnt: 25%, 5%, 75% and %. Figurs -4 compar calculatd valus with xprimntal data. B. Harmonic Analysis Fig.4 Excitation currnt undr % DC bias Fig. Excitation currnt undr 25% DC bias Fig.5 Pa valu of xcitation currnt against harmonic componnts undr diffrnt DC biass 496

5 Unli th Fourir transforming procss in th tim-stpping solution, all harmonic componnts in th magntizing currnt can b obtaind dirctly from th solvd currnt dnsity. Th histogram in Fig.5 shows th contribution of diffrnt harmonic componnts in th xcitation currnt undr diffrnt DC biass. Thr is consistncy in th computation and masurmnt rsults obtaind from th magntizing currnt wavforms. This provs th ffctivnss of th harmonic balanc finit lmnt mthod in solving th DC biasd problm in transformrs. Th xcitation currnt includs odd harmonics whn th transformr oprats without DC bias. Howvr, vn componnts will appar bcaus th dirct currnt invads th transformr windings. Th DC bias rsults in srious magntic saturation and half-cycl saturation of th magntizing currnt. Thrfor, th rlationship btwn DC bias and othr harmonic componnts should b considrd by using harmonic analysis. a. DC componnt b. Fundamntal componnt c. Scond harmonic componnt d. Third harmonic componnt Fig.7 Magntic flux distribution of DC and AC componnts IV. CONCLUSIONS Th HBFEM has bn introducd to solv th DC biasd problms in transformrs. It is provn to b an ffctiv and fficint approach to th analysis of stady-stat nonlinar magntic filds in th harmonic domain. Harmonic analysis of magntizing currnt and magntic fild is prsntd. Th ffctivnss of th HBFEM for th DC biasd problm is indicatd in th consistncy btwn th calculatd and xprimntal rsults. Fig.6 Pa valu of harmonic componnts undr diffrnt DC biass Fig.5 shows that whil th siz of all harmonic componnts incrass whn additional DC bias is applid, th growth rat varis in diffrnt componnts. Th growth tndncy of ach harmonic is shown in Fig.6. Th numbrs, 2,, 4 in th horizontal coordinat rprsnt diffrnt proportions (25%, 5%, 75%, %) of th DC bias bnchmar rspctivly. It is obvious that th fundamntal and scond harmonic componnts incras nar-linarly, whil highr ordr harmonics (such as th third and fourth) grow fastr rathr than linarly. Highr ordr componnts grow fastr than lowr ordr componnts. C. Flux Distribution Th magntic induction in ach lmnt is rprsntd in th form of a harmonic componnt in quation (). It can b calculatd dirctly from th fild quation (2), so th flux distribution in th stl rgion can b givn dirctly from th solution. Fig.7 (a) through (d) show th flux distribution of th harmonic componnts in th magntic cor. Thr ar significant diffrncs btwn DC and AC componnts in trms of flux distribution. ACKNOWLEDGEMENT This wor was supportd by National Natural Scinc Foundation of China undr Grant No and th Australian Rsarch Council (ARC) Foundation undr Grant No.DP Th DC biasd tst was carrid out in th EMLAB of R&D Cntr, Baoding Tianwi Group Co., LTD. Th authors than th anonymous rviwrs for thir commnts that improv th prsntation in this papr. REFERENCES [] O. Biro, S. Aussrhofr, G. Buchgrabr, K.Pris and W. Sitlingr, Prdiction of magntizing currnt wavform in a singl-phas powr transformr undr DC bias, Scinc, Masurmnt & Tchnology, IET, vol., pp. 2 5, Jan. 27. [2] Xiaoping Li, Xishan Wn, Cixuan Chng, Simulating analysis of xcitation currnt of singl phas transformr on DC bias, High Voltag Enginring, vol., pp. 8, Sp. 27. [] Lin Cao, Ji Zhao, and Jinliang H, Improvd powr transformr modl for DC biasing analysis considring transint laag rluctanc, in Proc. ICPST, 26, p.. [4] Yingying Yao, Rsarch on th DC bias phnomna of larg powr transformrs, D. Eng. thsis, Shnyang Univrsity of Tchnology, Shnyang, China, Nov. 2. [5] S. Yamada, K. Bssho, Harmonic fild calculation by th combination of finit lmnt analysis and harmonic balanc mthod, IEEE Transactions on Magntics, vol. 24, pp , Nov [6] Junwi Lu, S. Yamada, K.Bssho, Harmonic balanc finit lmnt mthod taing account of xtrnal circuit and motion, IEEE Transactions on Magntics, vol. 27, pp , Nov