Harmonic Modeling of Inrush Current in Core Type Power Transformers using Hartley Transform

Transcription

1 Harmonc Modelng of Inrush Current n Core Type Power Transformers usng Hartley Transform M. A. Taghkhan* (C.A.), A. Shekholeslam** and Z. Taghkhan*** Abstract: Ths paper presents a new method for evaluaton and smulaton of nrush current n varous transformers usng operatonal matrces and Hartley transform. Unlke most of the prevous works, tme and frequency doman calculatons are conducted smultaneously. Mathematcal equatons are frst represented to compute the nrush current based on reteraton and then Hartley transform s used to study harmonc effects n the frequency doman. Beng a real valued functon and accordngly gvng results wth the hgher speed of calculatons are the man features of Hartley transform. The nrush problem s ntally solved for sngle-phase transformers for swtchng at dfferent angles of the voltage waveform usng ths method and then the results of harmonc doman are compared wth that of Fourer transform. The methodology s also appled to three-phase three-lmb transformers snce the analyss of ther transent behavor s sgnfcant owng to the flux couplng nteractons n mult-leg core structures. The feasblty and effcacy of the method s llustrated wth approprate crcuts and MATLAB code s developed to get the tme and frequency doman waveforms wth hgh accuracy. The results are helpful to dentfy and evaluate nrush current harmonc effects n varous transformers and hence the effcency of the method s verfed. Keywords: Harmonc Doman, Hartley Transform, Operatonal Matrces, Transformer Inrush Current. Introducton Power transformers are senstve components n power dstrbuton and transmsson systems. When a transformer s energzed, t wll often draw a nonsymmetrcal magnetzng current, referred to as nrush current whch has partcularly undesrable effects on the wndngs and may gradually run the transformer. Large nrush current wll be created when the transformer operates on no-load energzng condton. It nvolves a large and long lastng dc component, whch s rch n harmoncs, assumes large peak values at the begnnng about 6 to tmes of the rated value. The magntude of the nrush current drawn by a transformer depends on the source strength, the leakage mpedance Iranan Journal of Electrcal & Electronc Engneerng, 5. Paper frst receved Dec. 4 and n revsed form 7 Apr. 5. * The Author s wth the Department of Engneerng, Imam Khomen Internatonal Unversty, Qazvn, Iran. ** The Author s wth the Department of Electrcal and Computer Engneerng, Babol (Noshrvan) Unversty of Technology, Babol, Iran. *** The Author s wth the Department of Electrcal and Computer Engneerng, Mazandaran Unversty of Scence and Technology, Babol, Iran. E-mals: taghkhan@eng.ku.ac.r, ashekh@nt.ac.r and zahra.taghkhan@ustmb.ac.r. and desgn of the transformer, the resdual flux stored n the transformer s core, and the angle of the appled voltage at the tme of energzaton [-]. Inrush phenomenon s of nonlnear nature and can only be reproduced by actual tests and computer smulatons. Due to the non-lnear nature of nrush phenomenon t must be solved by teraton [4]. The authors of [5] used a Newton-type algorthm to solve the nrush problem and a sngle evaluaton of the Jacoban matrx was shown to be most effcent. The method can represent the couplng effects between dfferent harmonc frequences and the full nrush soluton s obtaned n a sngle one harmonc doman teratve soluton,.e., 4-5 Newton teratons. The effects of varous parameters such as swtchng angle and resdual flux on the nrush current of a sngle-phase transformer are nvestgated n [6] and the second harmonc content of the nrush current s also evaluated. Dfferent Fourer technques were proposed n [7, 8] to obtan the magntude and phase angles of the nrush current harmonc component at dfferent voltage angles. The methods are helpful to estmate harmonc effects for nrush current but as the Fourer transform s a complex tool, t requres numerous complex quanttes and multplcatons. An effectve and full frequency 74 Iranan Journal of Electrcal & Electronc Engneerng, Vol., No., June 5

2 doman soluton technque whch solves the nrush phenomenon usng operatonal matrces has been developed, assumng that the overall transent s part of a perodc tran of transents [4]. Ths s shown n Fg. where the transent nrush has a perod of l n the tme doman. The methodology uses operatonal matrces and a gven orthogonal set, e.g., Hartley seres but t takes consderable computaton tme usng large matrces and consequently reduces memory effcency. Actually, Hartley seres are frequently used for perodc sgnals whch are nfnte, but the Hartley transform can be appled to approxmate the contnuous transform of a non-perodc sgnal of fnte duraton. It also appears to be a good method of transformng data nto the frequency doman, accordngly t s an deal tool to solve nrush current problem. A MATLAB code s developed n ths paper to get the tme and frequency doman waveforms of snglephase and three-phase three-lmb transformers nrush currents and ther hysteress loops usng operatonal matrces and Hartley transform. Requrng less number of computatons and performng smultaneous calculatons n the tme and frequency doman are the man features of ths technque. The paper s organzed as follows: Secton s dedcated to descrpton of operatonal matrces and explanaton of ts applcatons. evant equatons for Hartley transform and seres are gven n secton. The method s appled to the nvestgaton of sngle phase transformers and three-phase three-lmb transformers respectvely n secton 4 and the smulaton results are presented. Secton 5 concludes the paper lastly. Operatonal Matrces Algebrac methods can be appled to calculate steady state solutons of lnear, tme-varyng and nonlnear systems. They have been establshed for the soluton of problems descrbed by lnear dfferental equatons, such as analyss, model reducton, optmal control and system dentfcaton. These methods provde dentcal solutons but they have dfferent numercal propertes [4, 9]. In the last four decades, Numercal methods based on operatonal matrces (especally for orthogonal polynomals and functons) have receved consderable attenton for dealng wth a huge sze of appled mathematcs problems. Fg. Transent response seen as a perodc functon. The am of these technques s to smplfy the soluton process of the problem and obtan effectve algorthms that are sutable for dgtal computers []. Operatonal matrces can be appled to varous problems such as tme and frequency doman analyss. Usng algebrac ntegraton of dfferental equatons based on orthogonal calculatons of functons smplfes the soluton process of the problem n ths paper. Hartley Transform and Seres The Hartley transform s an alternate means of analyzng a gven functon n terms of ts snusods. Ths transform s ts own nverse and an effcent computatonal tool where the data are purely real []. The Hartley transform of a functon s a spectral transform and can be obtaned from the Fourer transform by replacng the exponental kernel exp(-jωt) by cas(νt). Fourer transform for g(t) s: { } F g(t) = G(ω) + - jωt G(ω) = g(t) e dt π Hartley transform for g(t) s: { } H g(t) = G( v) + G( v) = g(t) cas( vt) dt π The Hartley transform of the dervatve of g(t) s: dg(t) { } () () H = - v G(- v ) () dt Ths transform doesn t convert nput sgnals nto ther complex exponental and works bascally on the prncpal of even and odd part of sgnal. A perodc functon f(t) of perod l can be approxmated by Hartley functons as, f(t) = C cas(nv t) (4) where n = n cas(nv t) = cos(nv t) + sn(nv t) (5) v = πf (6) f = l [ Hz] s the frequency n hertz and: l C = f(t) cas(nv t) dt n l (7) 4 Inrush Current Calculaton The methods been used for smulaton of transformer nrush current have been almost done for the frst peak and most of these procedures have been also performed n the tme doman, but due to the nature Taghkhan et al: Harmonc Modelng of Inrush Current n Core Type Power Transformers 75

3 of nrush current and exstence of harmoncs wth dfferent orders, t s necessary to consder the nrush current harmonc components. Equatons are frst represented n the tme doman and then the Hartley transform s appled to the equatons n ths method. The problem should be solved by teraton due to the nonlnear nature of nrush phenomenon and ths procedure contnues untl an acceptable soluton s acheved. When the wndng s energzed at a pont dfferent from the voltage peak, a flux equvalent to the remanent flux wll appear. Consderng ths prncple, nrush current s calculated for sngle-phase and threephase three-lmb transformers n ths paper. 4. Sngle-Phase Transformers The unloaded sngle-phase transformer equvalent crcut used to calculate nrush current s shown n Fg.. The nrush current ( nr ) may flow n the prmary crcut under no load condton. The nrush current transent durng the energzaton of a transformer s a nonlnear phenomenon and the nonlnear characterstc s represented by the followng polynomal [4], 7 9 (t) =.7576 (t) +. (t) (8) Equatons below descrbe the crcut n Fg.. d (t) nr v (t) = r (t) + l + v(t) s nr (9) dt v(t) (t) = + (t) () nr rmag In the nonlnear element, d(t) v(t) = () dt where v(t) s the nstantaneous voltage appled to the transformer prmary and (t) s the nstantaneous core flux of the wndng. Dscretzaton of the above equatons n the tme doman under no-load condton gves: (t +Δt) - (t) nr nr v(t) = r (t +Δt) + l + s nr Δt () (t +Δt) - (t) Δt (t +Δt) - (t) (t) = (t +Δt) - () r Δt nr mag rδt +l v(t)δt + l (t) + (t) s nr (t +Δt) - nr = (t) (t +Δt) (4) (t) - r Δt mag r Δt mag Representaton of these equatons n the frequency doman and assumng that nr () =, gves: Fg. Sngle-phase transformer equvalent crcut. Fg. Flowchart for smulaton of nrush current. r + lv v v (t) (cos( t) + sn( t)) dt + () v v s I ( v) v nr = () - Ψ () v (t) (cos( vt) + sn( vt)) dt - rmag rmag (5) The flowchart of the proposed algorthm s llustrated n Fg.. Both of tme and harmonc doman procedures usng the above equatons are depcted n ths flowchart. The nonlnear characterstc s consdered perfectly n the unloaded sngle-phase transformer model and core losses can also be evaluated. MATLAB code s used to perform smulatons. For ths study the 76 Iranan Journal of Electrcal & Electronc Engneerng, Vol., No., June 5

4 Tme(s) Fg. 4 Inrush current n the tme doman at dfferent swtchng angles. Swtchng at the voltage peak (α = ) causes no saturaton n the transformer, and thus, we don t detect the nrush current. Only rated magnetzng current exsts n ths case. The nrush currents obtaned for 4.97 usng the method presented n [4] and the proposed method are also shown n Fg. 5 and so the effcacy of the proposed method can be observed. When the angle s 7, results are the same as 9 but the nrush current possesses the maxmum negatve value as the swtchng takes place n the next half cycle. Due to the deep saturaton of transformer n these two angles, consderable asymmetres n ther hysteress loops are observed and shown n Fg. 6. The second curve n ths fgure shows the hysteress loop for, where the loop s perfectly symmetrcal. Ths method s also compared wth the result of nrush current versus tme n Matlab Smulnk for 9 (whch s the worst angle for nrush current) n Fg. 7. It can be observed that both results are exactly the same but smulatng va Matlab code gves results wth much hgher speed. Flux at 9 (wb).4. Flux at (wb) (a) Tme(s) (b) Fg. 5 Inrush current at α = 4.97 usng (a) method presented n [4], (b) proposed method. followng parameters are selected [4], t max =.5Sec, S = 5 VA, f = 6 Hz, V s = cos(ω t-α), r =.9 Ω, l =.9 mh, r mag = 6.86 Ω. Fg. 4 shows nrush current n the tme doman at dfferent voltage angles under zero resdual flux condton. When the swtchng takes place at zero of the voltage waveform (α = 9 ), the nrush current of the transformer s prmary wndng s at the maxmum value. Flux at 4.97 (wb).4. Flux at 7 (wb) Fg. 6 Hysteress loops for swtchng at α =9,, 4.97 and Tme(s) matlab code matlab smulnk Fg. 7 Comparson between nrush currents obtaned at 9 usng Matlab code and Smulnk. Taghkhan et al: Harmonc Modelng of Inrush Current n Core Type Power Transformers 77

5 The harmonc spectrums obtaned usng operatonal matrces and Hartley transform are shown n Fg. 8. The fundamental harmonc content of nrush current s the only term whch can be observed n ths fgure for. A sgnfcant pont about transformer nrush current s the presence of harmoncs wth varous components such as second harmonc. Inrush Current Magntude(A) (a) Inrush Current Magntude(A) (a) Hartley Fourer Inrush Current Magntude(A) Inrush Current Magntude(A) Inrush Current Magntude(A) (b) 4 6 (c) 4 6 (d) Fg. 8 Inrush current magntude n the frequency doman for α equal to (a) 9, (b), (c) 4.97, (d) 7. (b) Fg. 9 Inrush current magntude comparson at 9 between (a) Hartley and Fourer transforms results va Matlab code and (b) FFT result n Smulnk. Table Maxmum nrush current magntudes obtaned usng Hartley and Fourer transforms. α (deg.) Hartley Transform Fourer Transform The results for 9 usng Hartley transform s compared wth Fourer transform va Matlab code and also wth FFT n Smulnk as shown n Fg. 9. We can conclude that Harley transform gves the best curve wth hgh accuracy and the hghest speed of calculaton as the tme of smulaton for ths transform s about 6 seconds less than Fourer transform. Table shows the maxmum magntude of nrush current at dfferent voltage angles for both of transforms. The results show neglgble dfference n the two magntudes obtaned for each angle. 78 Iranan Journal of Electrcal & Electronc Engneerng, Vol., No., June 5

6 4. Three-Phase Three-Lmb Transformers The method s now appled to the analyss of threephase three-lmb transformers, where the magnetc crcuts shown n Fgs. and are used. The majorty of three phase transformers are mostly n the form of core-type constructon. Three-lmb core-type transformer conssts of a sngle three-phase transformer whch s wound on a common magnetc core and uses the least amount of core materal n comparson wth transformer banks. Under balanced condtons, the three phases have ther three respectve currents whch are dsplaced from each other. Accordngly, the flux vectors n three phases are dsplaced apart and summed to zero n the yoke. There s no need for a return path for the flux. Ths condton s true when the supply voltage s balanced and hence resdual flux (.e. the sum all the three phases) s zero. The electromagnetc behavor of three-phase mult-lmb transformers reles on the magnetc nteractons and nonlneartes n the ferromagnetc ron-core structure [ 6]. Fg. shows one phase of transformer wth open crcuted secondary. Calculatons are complcated owng to the mutual couplng between the dfferent lmbs. Consderng the mutual nducton of fluxes n three-lmb transformer, the equaton, vectors and matrces can be wrtten as: di(t) dψ(t) V (t) = RI(t) + L + s dt dt where: [ ] [ nr nr nr ] T [ ] V (t) = v (t) v (t) v (t) s s s s I(t) = (t) (t) (t) Ψ(t) = (t) (t) (t) T T (6) (7) represent the vectors of nput voltages, currents and fluxes, respectvely, of the transformer. Matrces of the wndng resstances and nductances are defned as: r L m m R = r, L = m L m (8) r m m L The nductance matrx for a three-lmb transformer s derved here wth the help of ts reluctance model n Fg., so the relatonshp between the fluxes ({φ}) and the resultant mmfs ({F}) should be explctly derved n terms of the legs reluctances ({}). Ar flux paths and correspondng nductances are not consdered for smplcty whle dervng the reluctance model. The equatons for the reluctance model are: -F + φ + F - φ = (9) -F + φ + F - φ = () φ + φ + φ = () Fg. Three-phase three-lmb transformer. Fg. Representaton of three-lmb transformer usng reluctance network. Fg. Per-phase of three-lmb transformer. where F = NI, F = NI, F = NI and N, N, and N are the turns. Solvng Eqs. (9), (), () and assumng that, = N φ, = N φ, = N φ () yelds: N ( + )I - NN I - NNI = () + + N ( + )I - NN I - NNI = (4) + + N ( + )I - NNI - NNI = (5) + + Taghkhan et al: Harmonc Modelng of Inrush Current n Core Type Power Transformers 79

7 Assumng that, N = N = N = N, = (6) and = L I + m I + m I = m I + L I + m I = m I + m I + L I gves: (7) N ( + ) L = L = (8) + N L = + (9) - N m = m = m = m = + () - N m = m = () + In the followng equatons L, L and L have been wrtten l, l and l respectvely for smplcty. After obtanng the values of nductances and dscretzaton of the electrcal crcut equatons under no-load condton we have the followng matrx: r Δt + l m m m r Δt + l m m m rδt + l - rmagδt - rmagδt - rmagδt nr (t + Δt) () nr (t + Δt) nr (t + Δt) = (t + Δt) (t + Δt) (t + Δt) v s(t) Δt + l nr(t) + m nr (t) + m nr(t) + (t) v s(t) Δt + m nr(t) + l nr (t) + m nr (t) + (t) v s(t) Δt + m nr(t) + m nr(t) + l nr(t) + (t) ( t) (t) - rmagδt (t) (t)- r mag Δt (t) (t) - r mag Δt Representaton of the above matrces n the frequency doman, usng Hartley transform, gves: r+ lv m v m v v m v r + l v m v v m v m v r + l v v I ( ) v nr - v I ( v) nr r I ( v) nr mag = Ψ () v - v Ψ () v r mag Ψ () v - v r mag v (t) (cos( vt)+sn( vt)) dt + () s v (t) (cos( vt)+sn( vt)) dt + () s v (t) (cos( vt)+sn( vt)) dt + () s () () (t) (cos( vt)+sn( vt)) dt - r mag () (t) (cos( vt)+sn( vt)) dt - r mag () (t) (cos( vt)+sn( vt)) dt - r mag Equatons for the fluxes are as follows: φ = Φ cos (ωt) a max π φ = Φ cos (ωt - ) (4) b max π φ = Φ cos (ωt + ) c max Parameters of the transformer are consdered as, S = 4.5 kva, f = 6 Hz, r =.9 Ω, l = l =.9 mh, l =. mh, m = m = -.6 mh, m = -. mh, r mag = 6.86 Ω. Transformer voltages for an angle of α degree are: V = cos (ωt - α) A π V = cos (ωt - ( + α)) (5) B 4π V = cos (ωt - ( + α)) C Swtchng at zero of the voltage waveform (α = 9 ) for one phase, results n voltage magntudes whch are.866 and of the maxmum voltage for the other phases, flux wll have the maxmum value n one phase and half of that for the other phases. If swtchng occurs at the voltage peak (α = ), there wll be no nrush current for that phase. Dstrbuton of the fluxes n transformer lmbs s sgnfcant. The transformer secondary s assumed to be open wthout any load, accordngly the transformer nrush currents n the tme doman under zero resdual flux condton for swtchng at α = 9 are exposed n Fg.. Low Saturaton n the phases B and C results n the less current value. Hysteress loops for dfferent phases at α = 9 are shown n Fg. 4 and t can be seen that asymmetres n 8 Iranan Journal of Electrcal & Electronc Engneerng, Vol., No., June 5

8 the hysteress loops of phases B and C consderably dffers from phase A Tme(s) Fg. Three-lmb transformer nrush currents for swtchng at α = 9 for phases A, B and C. Phase A Flux(wb) Phase B Flux(wb) Phase C Flux(wb) Phase A (a) Phase B (b) Phase C (c) Fg. 4 Hysteress loops for swtchng at α = 9 for phases A, B and C. A B C Fg. 5 shows the transformer nrush current magntude n the frequency doman usng Hartley and Fourer transforms. It can be observed that the magntude of the nrush current harmonc components vares for phases A, B and C and t has the maxmum value n phase A. The second harmonc component s the most domnant one owng to the asymmetrcal nature of the magnetzng nrush current. The rse of nrush current harmonc contents n ths transformer s more than transformer banks due to the mutual nducton of fluxes between the dfferent lmbs. The results derved from the fgure can also show slght and neglgble dfferences n the nrush current magntudes for Hartley and Fourer methods. Table shows the tme of smulaton va Matlab code for both of the transforms whch ndcates that Harley transform possesses less tme compared to Fourer transform for sngle and three-phase transformers. Phase A Inrush Current Magntude(A) Phase B Inrush Current Magntude(A) Phase C Inrush Current Magntude(A) Hartley Fourer 4 6 (a)..5 Hartley Fourer (b) Hartley Fourer 4 6 (c) Fg. 5 Three-lmb transformer nrush current magntude comparson between Hartley and Fourer transforms for swtchng at α = 9 for phases (a) A, (b) B and (c) C. Taghkhan et al: Harmonc Modelng of Inrush Current n Core Type Power Transformers 8

9 Table Smulaton tme for Hartley and Fourer methods. Transformer Type Sngle-Phase Three-Phase Three-Lmb Hartley Transform 9 (sec) 4 (sec) Fourer Transform 5 (sec) (sec) 5 Concluson Calculaton and smulaton of nrush current based on operatonal matrces and Hartley transform are dscussed n ths paper, consderng the problems of the prevous methods. The proposed approach smplfes the soluton process of a complcated technque wthout losng accuracy and solves the nrush transents effcently. It s also approprate for dgtal computers and the soluton s a mult-resoluton type. Performng smultaneous calculatons n the tme and frequency doman wth hgh effcacy and decreasng the number of computatons are the other features of ths method. Proper crcuts whch can consder the nonlnear characterstc and core loss effects are used n ths paper. To calculate nrush current usng ths method, dfferental equatons are frst converted nto algebrac ones and then Hartley transform whch nvolves no complex quanttes or calculatons s used to represent equatons n the frequency doman. In order to dentfy the nrush phenomenon owng to swtchng operaton, analyss and smulaton of the problem n sngle-phase and three-phase three-lmb transformers are provded n ths paper. From the results, t s obvous that the nrush current ampltude ncreases by lessenng the angle to zero of the voltage waveform (or ncreasng the swtchng angle α) n sngle-phase transformers. The results obtaned usng Hartley transform are also compared wth Fourer transform va Matlab code and also wth FFT n Matlab Smulnk. The method has been executed successfully n three-lmb transformer smulatons under no-load condton and the results ndcate that the nrush current harmonc contents n ths transformer depend on the mutual nducton of fluxes between the dfferent lmbs. Ths approach can be mplemented for computng nrush current n fve-lmb transformers n future works. It can also provde the bass for reducng nrush va some strateges such as magnetc modfcatons consderng the resdual flux. References [] D. I. Taylor, J. D. Law, B. K. Johnson and N. Fscher, Sngle-Phase transformer nrush current reducton usng prefluxng, IEEE Transactons on Power Delvery, Vol. 7, No., pp. 45 5, Jan.. [] J. Faz, B. M. Ebrahm and T. Noor, Three and two dmensonal fnte element computaton of nrush current and short-crcut electromagnetc forces on wndngs of a three-phase core-type power transformer, IEEE Transactons on Magnetcs, Vol. 44, No. 5, pp , May. 8. [] L. C. Wu, C. W. Lu, S. E. Chen and C. S. Chen, The effect of nrush current on transformer protecton, 8th North Amercan Power Symposum (NAPS 6), pp , Sep. 6. [4] J. J. Rco, E. Acha and M. Madrgal, The study of nrush current phenomenon usng operatonal matrces, IEEE Transactons on Power Delvery, Vol. 6, No., pp. 7, Apr.. [5] A. Semlyen, E. Acha and J. Arrllaga, Newtontype algorthms for the harmonc phasor analyss of nonlnear power crcuts n perodcal steady state wth specal reference to magnetc nonlneartes, IEEE Transactons on Power Delvery, Vol., No., pp. 9 98, July 988. [6] M. Jamal, M. Mrzae and S. A. Gholaman, Calculaton and analyss of transformer nrush current based on parameters of transformer and operatng condtons, Electroncs and Electrcal Engneerng (Elektronka Ir Elektrotechnka), Vol. 9, No., pp. 7, Mar.. [7] T. Srdev, K. R. Reddy and N. L. J. Syamala, Harmonc analyss of nrush current usng fast fourer transform, Internatonal Conference on Power, Energy and Control (ICPEC ), pp. 5-54, Feb.. [8] D. Jandong, W. Chang and Y. Janmng, Study of the nrush current dentfcaton usng the mproved half-cycle Fourer analyss, Asa- Pacfc Power and Energy Engneerng Conference, pp. 4, Mar. 9. [9] J. J. R. Melgoza, G. T. Heydt, A. Keyhan, B. L. Agrawal and D. Seln, Synchronous machne parameter estmaton usng the Hartley seres, IEEE Transactons on Energy Converson, Vol. 6, No., pp , Mar.. [] A. K. Sngh, V. K. Sngh and O. P. Sngh, The Bernsten operatonal matrx of ntegraton, Appled Mathematcal Scences, Vol., No. 49, pp , 9. [] N. Sundararajan, Fourer and Hartley transforms-a mathematcal twn, Indan Journal of Pure and Appled Mathematcs, Vol. 8, No., pp. 6 65, Oct [] J. Wang and R. Lascu, Zero sequence crcut of three-legged core type transformers, Annual Conference for Protectve ay Engneers, pp. 88, Apr. 9. [] M. Persson and W. Bag, Modelng and measurements of transformer behavor at dfferent voltages and frequences, Chalmers Unversty of Technology, pp. 58,. 8 Iranan Journal of Electrcal & Electronc Engneerng, Vol., No., June 5

10 [4] P. S. Moses, M. A. S. Masoum and M. Moghbel, Effects of ron-core topology on nrush currents n three-phase mult-leg power transformers, IEEE Power and Energy Socety General Meetng, pp. 6, Jul.. [5] S. G. Abdulsalam, W. Xu and V. Dnavah, Modellng and smulaton of three phase transformers for nrush current studes, IEE Proc. Generaton, Transmsson and Dstrbuton, Vol. 5, No., pp. 8, May 5. [6] M. Jamal, M. Mrzae and S. Gholaman, Dscrmnaton of nrush from fault currents n power transformers based on equvalent nstantaneous nductance technque coupled wth fnte element method, Iranan Journal of Electrcal and Electronc Engneerng, Vol. 7, No., pp. 97-, Sep.. Mohammad Al Taghkhan was born n Tehran, Iran, n 974. He receved the B.Sc. and M.Sc. degrees n electrcal engneerng from Amrkabr Unversty of Technology, Tehran, Iran, n 997 and, respectvely and receved the Ph.D. degree n electrcal engneerng from Iran Unversty of Scence and Technology, Tehran, Iran, n 8. He s currently an assstant professor n the department of engneerng, Imam Khomen Internatonal Unversty, Qazvn, Iran. Hs nterests are power transformers, electrcal machnes, numercal analyss, fnte element method, heat transfer and flud mechancs. Abdolreza Shekholeslam was born n Iran. He receved the B.Sc. degree from Mazandaran Unversty n 978. He receved the M.Sc. and the Ph.D. degrees from Strathclyde Unversty, UK n 989. Snce 9 he has been an Assocate Professor n the department of Electrcal and Computer Engneerng, Noshrvan Unversty n Babol. Hs research nterests nclude Power Electronc, Power Qualty, Harmoncs, Smart Grds and Renewable Energy. Zahra Taghkhan was born n Tehran, Iran. She receved her B.Sc. degree n electrcal engneerng from Zanjan Unversty, Zanjan, Iran, n and her M.Sc. degree n electrcal engneerng from Mazandaran Unversty of Scence and Technology, Babol, Iran, n 4. Her research nterests nclude electrcal machnes and power transformers. Taghkhan et al: Harmonc Modelng of Inrush Current n Core Type Power Transformers 8