Clcultion of Off-Core Inductnce in Dul-Circuit Model of Trnsformer As Lotfi NTNU Trondheim, Norwy s.lotfi@ntnu.no Hns Kr. Hoidlen NTNU Trondheim, Norwy hns.hoidlen@ntnu.no Nicol Chies Sttoil Trondheim, Norwy nchie@sttoil.com Erhim Rhimpour ABB AG Bd Honnef, Germny erhim.rhimpour@de..com Astrct Zero sequence mgnetic flux, generted in trnsformer core in different opertion cses, is forced out of the core including the oil gp nd the tnk. These off-core flux pths cn e represented y inductnces in dulity trnsformtion sed electricl trnsformer model. These off-core inductnces minly determine the zero sequence impednce tht is vitl in nlyzing trnsformer sujected to the GIC event or unlnced voltges. Estimtion of these inductnces is one of the chllenges in identifiction of the electric equivlent circuit. The min contriution of this pper is to present n pproch for clcultion of the mentioned off-core inductnces sed on 2D- FEM. Since the trnsformer structure is not symmetric for offcore flux pth, 3D-FE nlysis is lso used to evlute nd improve the presented method. The results clculted for 3-leg 3-phse trnsformer hve good greement with the vlues otined from empiricl equtions typiclly dopted y mnufcturer. trnsformers since they provide the min pth of zero sequence (ZS) flux tht determines ZS rectnce. For 5-leg core designs, these inductnces seem not to e s importnt s in 3-leg trnsformers since the lterl legs provide lowreluctnce pth for ZS flux. However, in some specil cses like Geomgneticlly Induced Current (GIC) event [10], the lterl legs my sturte cusing some flux to flow in the offcore pths. In this cse, off-core inductnces ply significnt role in flux distriution within the trnsformer nd electricl quntities such s rective power consumption nd differentil current protection settings [11]. Keywords Trnsformer Modeling, Off-core Inductnces, Finite Element Method I. INTRODUCTION Trnsformer modeling is one of the importnt topics in power system trnsient studies nd lots of models hve een presented for trnsient simultion of power trnsformers. Models of trnsformers cn e ctegorized in two lrge groups; low frequency nd high frequency models [1]. Low frequency models minly used up to 10 khz, represent trnsformer ehvior during slow trnsients nd switching trnsients. Exmples of slow trnsients re torsionl oscilltions, fst us trnsfers, controller interctions, hrmonic interctions, nd resonnces. Switching trnsients re cused y energiztion nd de-energiztion of system components. A review of trnsformer models for low frequency trnsients is given in [2-3]. These references suggest tht trnsformer model for trnsient studies should e sed on topologiclly correct representtion of the mgnetic structure. The electric equivlent circuit derived y dulity trnsformtion of mgnetic circuit is one of these models [4-8]. Fig. 1 shows electric equivlent circuit of 3-leg 3-phse trnsformer which is otined y the dulity trnsformtion [9]. According to Fig. 1, ech of the flux pths in min legs, yokes, lekge chnnels nd off-core re represented y L Leg, L Yoke, L HL nd L off-core respectively. Offcore flux pths include the oil gp (etween ctive prt nd tnk) nd the tnk. The inductnces corresponding to these pths ply min role in nlysis of 3-leg, 3-phse Fig. 1: Equivlent circuit of 3-leg 3-phse trnsformer From mesurement point of view, inductnces of min legs, yokes nd lekge chnnels cn e otined through positive sequence open nd short circuit tests [12-14]. However, there is chllenge in mesuring off-core inductnces depending on different core designs nd terminls connections. For 3-leg, YNyn trnsformers the off-core inductnces cn e found directly from the mesured open circuit ZS rectnce. While for the YNd or Dyn trnsformers the delt connected winding mkes short circuit for ZS currents tht ffects ZS flux pth. So the mesured ZS rectnce is not relted directly to off-core inductnces. It is even worse for 5-leg core design which hs low reluctnce iron pth for ZS flux through lterl legs tht leds to lessen the influence of the off-core pth in the vlue of ZS
impednce. In this cse, it needs to sturte the lterl legs mking the flux to flow outside the core in off-core pth. From clcultion point of view, there is lso lck of nlyticl or numericl clcultion technique for off-core inductnces ssuming tht trnsformer design informtion is ville. The min contriution of this pper is to present n pproch for off-core inductnce clcultion sed on two dimensionl finite element method (2D-FEM) tht hs dvntges of less simultion time nd memory requirement compred to three dimensionl finite element method (3D- FEM). Since the trnsformer structure is not symmetric for off-core flux pth, the use of 2D nlysis is not stright forwrd. Hence, in order to evlute nd improve the 2Dsed clcultions, 3D-FE nlysis s presented in section II, is lso used. II. 3D-FINITE ELEMENT ANALYSIS A. Full 3D model For the purpose of this pper, 3-leg 3-phse trnsformer of 5 MVA, 33/6.3 kv, Dyn11 is considered. All design informtion of this trnsformer is ville in [15]. The mesured ZS rectnce is 0.55 ohm referred to side. On one hnd, since ZS current cn circulte in the delt winding, there is lnced mp-turn induced in the delt for the ZS impednce mesurement. On the other hnd, the delt winding (HV) is the outer one, so it cts like shield ginst ZS flux preventing it to flow through the off-core pth. It mens tht the off-core pth is not involved in the mesured ZS rectnce whose vlue is pproximtely equl to positive sequence short circuit rectnce. According to technicl documents from the mnufcturer the open-circuit ZS inductnce of this trnsformer seen from the winding is 12.7 mh. Considering the dul circuit of trnsformer depicted in Fig. 1 nd Eq (1)-(2), it cn e shown tht the verge of off-core inductnces is seen from the excited windings. penetrtion depth in the tnk mteril. For the tnk iron considered in these simultions ( 1000, 103 10 /, 314.16 / 50 ), the electromgnetic penetrtion depth (skin depth ) is equl to 0.7 mm ( ). Therefore the elements size of the first lyers on the inner surfce of the tnk wlls, top nd ottom must e two to three times smller thn 0.7 mm. This leds to very lrge numer of elements nd long simultion time s well. Since the windings nd the core re crefully designed for 50 Hz, the effect of eddy currents is ignored in those prts implying no need to hve the meshes s fine s in the tnk. For the model in this study, n utomtic dptive meshing tht reduces the elements size t ech solution pss sed on n initil mesh settings, hs een used. The mesh refinement per pss cn e djusted y the softwre nd is set to 20%. The convergence criterion hs een set to energy error of 1%. Totl numer of elements fter the lst pss is 9,597,470 with 5,390,643 only in the tnk leding to out 38 hours of simultion time. Running the 3D-FEM, the inductnce mtrix of the -HV windings (,, 1 6) is clculted using the energy method. Short circuit nd open circuit (off-core) ZS inductnce cn e clculted using Eq. (3) (5). L, HV L, L, i 1..3 (3) L, i 4..6 (4) L SC L L L, i 1..3, k 4..6 (5) L where, is the off-core inductnce of the winding i where 1..3 stnds for windings nd 4..6 stnds for HV windings nd L SC is short circuit ZS inductnce when winding i is excited while winding k is shorted. The clcultion results re seen in tle 1 nd the flux distriution in core is lso shown in Fig. 2. L, L, L, λ 1λ 2 λ 3 mmf 0 (1) if λ λ λ λ then: 3 L offcore,v L offcore,1 L offcore,2 L offcore,3 L, L, L, L, Where is the off-core inductnce, λ is the linkge flux nd mmf is the mgneto-motive force. In these documents the equivlent reltive permeility nd conductivity of the tnk mteril re considered s 1000 nd 10.3 10 S/m, respectively. As first step, full 3D model of the trnsformer is uilt in Ansoft Mxwell [16] (Fig. 2). The type of solution so clled Eddy Current used in the 3D- FEM is formultion where the effect of eddy currents is tken into ccount. In order to otin dequte ccurcy, it is necessry to hve very fine elements in the tnk. As mtter of fct, the size of elements must e smller thn electromgnetic (2) Fig. 2 ) 3D model of the trnsformer ) Flux distriution Tle 1. Results of 3D-FE simultions L SC SC SC L L L, L, (outer leg) (centrl leg) 3D-FEM 1.69 mh 1.7 mh 13.35 mh 13.0 mh Averge off-core inductnce 13.35 13.0 13.35 13.23 3 Multiplying y the ngulr frequency ( 314.16 /) the short circuit inductnces gives vlue of 0.53 ohm. This is in good greement with the test result of 0.55 ohm (reltive error of 3.6%).
Concerning off-core inductnces, the verge vlue of inductnces on three legs is 13.23 mh tht hs good greement with 12.7 mh provided y the trnsformer mnufcturer (reltive error of 4.2%). B. Size Reduction of 3D model Investigting the flux distriution in the previous full model simultion (Fig. 2), it is relized tht there is prllel flux condition t the middle of the yokes. This flux condition mkes it possile to cut the full model into two su-models s shown in Fig. 3, c. The su-model 1 nd su-model 2 correspond to the outer legs nd centrl leg, respectively. Then in order to hve the sme results it is necessry to put the fluxprllel oundry condition on the cutting surfces. Running 3D-FEM with much more fine mesh sizes thn the size used in full-model meshing, the inductnces re otined. As cn e seen in tle 2, the vlues for su-models hve rther good greement with the full model simultion. In ddition, verge off-core inductnce is in etter conformity with the 12.7 mh provided y the trnsformer mnufcturer thn full-model simultion. Cross sections of flux region 1 in su-models 1 nd 2 which is perpendiculr to XY plne nd out Z-xis (see Fig. 3 nd 3c) re similr. The only difference of the cross sections is the distnce from outer winding to the tnk wll. A generl cut clled C1, depicted in Fig. 6, is used for these cross sections. Similr properties re seen for regions 2 (Fig. 4 nd Fig. 5) nd generl cut for this region is clled C2 nd shown in Fig. 6. Fig. 4 Flux regions (gry) of su-model 1, ) Region 1, ) Region 2 Fig. 5 Flux regions (gry) of su-model 2, ) Region 1, ) Region 2 c Fig. 3 -) Su-model 1, c-d) Su-model 2 d The significnce of these su-models is esides llowing higher numer of elements for the sme solution time s the full-model tht they cn e used to clculte off-core inductnces sed on 2D nlysis method. Tle 2. Inductnces clculted on su-models Sumodel 1 Full outer legmodel Off-core inductnces [mh] Averge offcore inductnce Sumodel 2 centrl leg- Full model 13.64 13.35 10.32 13 13.64 13.64 10.32 12.53 3 The next section discusses the mentioned 2D-FEM sed clcultion pproch. III. 2D-Finite Element Anlysis A. Introducing cuts for the sumodels Considering the su-models in Fig. 3, off-core flux pths cn e divided in two regions (regions 1 nd 2). The top view of these flux regions colored gry re shown in Fig. 4 nd Fig. 5. Fig. 6 ) Cut C1 corresponding to regions 1 ) Cut C2 corresponding to regions 2 It is importnt to mention tht the 3D-FEM simultions performed on full model of the trnsformer without considering the yokes (see Appendix 2) demonstrte tht the yokes hve insignificnt effect on the vlues of the off-core inductnces. Accordingly, the influence of yokes in the flux regions 2 hs een ignored.
B. Inductnce clcultion using 2D-FEM In the following, 2D nlysis pproch is mde on sumodel 1 to clculte off-core inductnce of the winding. Since the sme procedure is used for inductnce clcultion in su-model 2, only the finl result is shown for centrl leg. Ech of the flux regions cn e creted y mens of cuts C1 nd C2. Ech of cuts hs vrile distnce identified y ds in Fig. 4 nd Fig. 5 which is the distnce etween the excited winding nd the tnk wll. Vrile ds is prmeter tht is vrying while constructing the su-models. Assuming the winding eing excited, minimum vlue of ds in region 1 of su-model 1 is equl to 168 mm (middle of tnk wll etween points A nd B in Fig. 4) nd mximum vlue which is relted to point B t the corner of tnk wll, equls 394.6 mm. Likewise for the region 2 in su-model 1, the prmeter ds is vrying in the rnge of 103 to 354 mm. In order to clculte the inductnce of the winding in su-model 1, profile of the inductnces in cuts C1 nd C2 (Fig. 6) s function of ds is estlished nd used s look-up tle in the clcultion process. The profiles for regions 1 nd 2, s shown in Fig. 7 nd Fig. 8, re otined using 2D-FEM where Crtesin XY formultion is used with flux-prllel oundry conditions set to the symmetry lines. It is worth to mention tht the tnk hs two different influences. On one hnd, it provides low reluctnce pth for mgnetic flux cusing n increse in inductnce, nd on the other hnd, the tnk is closed conductive pth surrounding the excited windings. Being included y ZS flux, n electric current is induced in the tnk wll, top nd ottom. Therefore, the off-core inductnce cn e interpreted s kind of lekge inductnce etween excited winding nd the tnk. Tht is why the relevnt inductnces in Fig. 7 nd 8 increse y incresing the distnce, ds. Now, it remins to ggregte the inductnces of cuts for region 1 nd 2 of ech su-model in order to otin the totl inductnce. Fig. 9 illustrtes typicl vrition of mgnetic flux intensity (H) in the cut C1 for the minimum nd mximum vlues of ds (168 nd 394.6 mm). As cn e seen, the mgnetic flux intensity rises up from the inner surfce of the excited winding (the in this cse) nd flls down to zero in the tnk. In order to simplify the prolem the thickness of the winding nd the mgnetic penetrtion depth of the tnk ginst ds re ignored. Consequently, it is worth to sy tht the mgnetic flux intensity is minly concentrted in the gp etween ctive prt nd the tnk. Hence the integrtion of the inductnces relted to regions 1 nd 2 is done long the verge depth of the mentioned gp. As illustrted in Fig. 10, the line --c-d-e-f-g is the verge depth of region 1, nd -og is the verge depth of region 2 relted to su-model 1. Fig. 7 Inductnce profile of the winding in cut C1 s function of distnce to the tnk wll Fig. 9 Vrition of mgnetic flux intensity in cut C1 Fig. 10 Averge depth lines of regions 1 nd 2 Fig. 8 Inductnce profile of the winding in cut C2 s function of distnce to the symmetry line Since ech point on the verge depth line is ssocited with specific ds, vrition of the winding inductnces cn e plotted versus these verge depth lines s depicted in
Fig. 11 nd Fig. 12 using the inductnce profiles in Fig. 7 nd Fig. 8 s look-up tle. Only hlf of plots re shown, s the lines re symmetricl with respect to the line o-o-d. It is importnt to note tht there is discontinuity t the points nd g in Fig. 10 due to different oundry conditions in the cuts C1 nd C2. This results in difference in inductnce vlues t the points g nd g+ in Fig. 11 nd Fig. 12. the differentil length of the verge depth lines nd is the numer of turns in winding. The length of the verge depth line is clculted using trigonometry reltions s shown in ppendix 1. Doing the sme clcultion for region 1 nd 2 in su-model 2, the inductnce of the winding in centrl leg is otined s well. The results from the 2D nlysis re summrized in Tle 3 nd compred to those otined with 3D clcultions. As cn e seen the vlues otined y the presented method re it less thn ones clculted y 3D-FEM. Tle 3. Results of 2D-FE simultions mh 2D-method 3D-Method error 12 13.64 12%,, 9.3 10.3 10.7% Fig. 11 Vrition of inductnces in pth d-e-f-g of region 1 Fig. 12 Vrition of inductnces in pth -o of region 2 Integrting the curves long the verge depth nd multiplying the result y 2 (to consider the other hlf) nd squred numer of turns ( ) nd finlly dding the vlues relting to regions 1 nd 2, the totl inductnce of su-model 1 is otined y: L, 2N L. dl L. dl (6) where, is the off-core inductnce of winding,. is the inductnce of winding in cut A (Fig. 11),. is the inductnce of winding in cut B (Fig. 12), is IV. CONCLUSION The min contriution of this pper is to present n pproch sed on 2D-FEM for clcultion of off-core inductnces in topologiclly correct models of trnsformer. These inductnces represent the reluctnce of off-core flux pth. The clcultions performed on rel trnsformer s cse study shows tht the presented 2D-FEM sed pproch hs mximum devition of 12% compred to the 3D-FEM simultion. One of the resons for this error cn e ssocited with the simplifiction considered for the verge depth. By tking the thickness of the winding into ccount, the verge depth line depicted in Fig. 10 displces it closer to the tnk wll leding to lrger length of the depth line nd n increse in totl vlue of the inductnce. The verge vlue of outer nd centrl inductnces for 2D- FEM sed nlysis is 11.1 mh tht is 12.6% less thn the vlue 12.7 mh provided y the mnufcturer showing good greement. It is interesting to mention tht the time tken for 3D-FEM simultions is out 38 hours while it tkes only out 15 minutes to clculte the inductnces y 2D-FEM. Besides hving less simultion time nd memory requirement, the min dvntge of the 2D sed nlysis is tht it cn e further pproximted y the mgnetic circuit theory to extrct detiled model for off-core flux pth. This detiled circuitl model cn e dded to overll model of trnsformers which re otined y mens of dulity trnsformtion. Exmple of these models is the hyrid trnsformer model tht is lredy implemented in ATPDrw [13]. It must e mentioned tht the purpose of this pper is to demonstrte the possiility of the off-core inductnce clcultion using 2D-nlysis sed pproch insted of 3D simultions. So the non-linerity of tnk mteril is not tken into considertion. Once the circuitl model of 2D-models is extrcted, influence of non-linerity cn e tken into considertion using non-liner inductnces.
REFERENCES [1] Jun A. Mrtinez-Velsco, Power System Trnsients Prmeter Determintion, CRC Press Tylor & Frncis Group, Ch. 4, ISBN: 978-1-4200-6529-9, 2010. [2] J. A. Mrtinez nd B. A. Mork, Trnsformer modeling for low- nd mid-frequency trnsients - review, IEEE Trns. Power Del., vol. 20, no. 2 II, pp.1625 1632, 2005. [3] J. A. Mrtinez, R.Wlling, B. A. Mork, J. Mrtin-Arnedo, nd D. Durk, Prmeter determintion for modeling system trnsients - prt III: Trnsformers, IEEE Trns. Power Del., vol. 20, no. 3, pp. 2051 2062, Jul. 2005. [4] E. C. Cherry, The dulity etween interlinked electric nd mgnetic circuits nd the formtion of trnsformer equivlent circuits, Proceedings of the Physicl Society. Section B, vol. 62, pp. 101 111, Fe. 1949. [5] S. Lev nd A. P. Morndo, Topologicl trnsition from mgnetic networks to the electric equivlent ones when iron losses re present, in Proc. 43rd IEEE Midwest Symposium on Circuits nd Systems, vol. 2, pp. 642 645. Aug. 8 11, 2000. [6] G. R. Slemon, Equivlent circuits for trnsformers nd mchines including non-liner effects, Proceedings of the Institution of Electricl Engineers, vol. 100, no. 1, pp. 129 143, Jul. 1953. [7] B. A. Mork, F. Gonzlez, D. Ishchenko, D. L. Stuehm, nd J. Mitr, Hyrid trnsformer model for trnsient simultion: Prt I: development nd prmeters, IEEE Trns. Power Del., vol. 22, no. 1, pp. 248 255, Jn. 2007. [8] N. Chies, B. A. Mork, H. K. Hoidlen, Trnsformer model for inrush current clcultions: simultions, mesurements nd sensitivity nlysis IEEE Trns. Power Delivery, Vol. 25, Issue: 4, pp. 2599 2608, 2010. [9] N. Chies, Power Trnsformer Modeling for Inrush Current Clcultion, PhD Thesis, Norwegin University of Science nd Technology (NTNU), 2010. [10] R. Pirjol, Geomgneticlly induced currents during mgnetic storms, IEEE Trns. Plsm Sci., vol. 28, no. 6, pp. 1867 1873, Dec. 2000. [11] N. Chies, A. Lotfi, H. Høidlen, B. Mork, Ø. Rui, T. Ohnstd, Five-leg trnsformer model for GIC studies Interntionl Conference on Power Systems Trnsients (IPST2013) in Vncouver, Cnd, June 18-20, 2013. [12] B. A. Mork, F. Gonzlez, D. Ishchenko, D. L. Stuehm, nd J. Mitr, Hyrid trnsformer model for trnsient simultion: Prt II: lortory mesurements nd enchmrking, IEEE Trns. Power Del., vol. 22, no. 1, pp. 256 262, Jn. 2007. [13] H. K. Hoidlen, B. A. Mork, F. Gonzlez, D. Ishchenko, N. Chies, Implementtion nd verifiction of the Hyrid Trnsformer model in ATPDrw, Interntionl Conference on Power Systems Trnsients (IPST 07) in Lyon, Frnce, June 4-7, 2007. [14] H. K. Høidlen, N. Chies, A. Avendño, B. A. Mork, Developments in the hyrid trnsformer model Core modeling nd optimiztion, Interntionl Conference on Power Systems Trnsients (IPST2011) in Delft, the Netherlnds June 14-17, 2011. [15] A. Lotfi, Zero Sequence Impednce Clcultion of Power Trnsformers Using 3D-FEM, Mster Thesis, University of Znjn (ZNU), 2007. [16] ANSYS Inc, User s guide Mxwell 3D/ANSYS Mxwell v.16, Cnonsurg, PA, USA, www.nsys.com APPENDIX 1 The length of verge depth line cn e clculted y the expressions elow: (A1) wheres: 0 (A2) 0 tn 0 74, 260.5 Fig. A1 Outer leg of the trnsformer of top view (A3) Strting from point D ( 0), s re clculted for ech y step size of 0.5 degree ( 0.5 ). Once ll s re clculted, the length of verge depth is:, (A4) where n is the numer of steps. APPENDIX 2 In order to investigte the influence of the yokes on the off-core inductnces, 3D-FEM model of the trnsformer shown in Fig. A2 is uilt. Fig. A2 3D-FEM model of the trnsformer without Yokes
Off-core inductnces clculted for this model in comprison with the full model shown in Fig. 2 re gthered in Tle A1. L, L, Full 3D-Model (outer leg) With Yokes 13.35 mh 13.0 mh Without Yokes 13.2 mh 12.95 mh (centrl leg)