Phasor Estimation Algorithm Based on the Least Square Technique during CT Saturation

Transcription

1 Journal of Elecrical Engineering & Technology Vol. 6, No. 4, pp. 459~465, DOI: 1.537/JEET Phasor Esiaion Algorih Based on he Leas Square Technique during CT Sauraion Dong-Gyu Lee*, Sang-Hee Kang and Soon-Ryul Na* Absrac A phasor esiaion algorih based on he leas square curve fiing echnique for he disored secondary curren due o curren ransforer (CT) sauraion is proposed. The aheaical for of he secondary curren during CT sauraion is represened as he scaled priary curren wih agneizing curren. The inforaion on he scaled priary curren is esiaed using he leas square echnique, wih he easured secondary curren in he sauraed secion. The proposed ehod can esiae he phasor of a fundaenal frequency coponen during he sauraed period. The perforance of he algorih is validaed under various faul and CT condiions using a C4 CT odel. A series of perforance evaluaions shows ha he proposed phasor esiaion algorih can esiae he phasor of he fundaenal frequency coponen wih high accuracy, regardless of faul condiions and CT characerisics. Keywords: CT sauraion, Phasor esiaion, Leas square fiing 1. Inroducion The curren flowing hrough a power line is easured by a curren ransforer (CT). As CT is a kind of ransforer, he sauraion of he agneic flux in he core ay occur when a large priary curren flows, which disors he secondary curren of a CT and causes probles in power syses. Several ineresing resuls on he copensaion of secondary curren disorion have been published. One srea on his opic is based on he CT odel [1]-[4]. Alhough hese algorihs are valid under various faul condiions, hey sill require he agneizaion curve or hyseresis curve based on he given CT paraeers. The oher approach is based on he regression echnique [5]-[8]. The regression ehod in [5] deecs he sauraion using discree wavele ransfor and copensaes he disored secion of a secondary curren wih feaures exraced fro he healhy secion using a leas square fiing ehod. The suggesed algorih in [6] only uses a leas ean square fiing ehod wihou a sauraion deecion echnique. A hybrid phasor esiaion ehod in [7] uilizes parial su-based and leas square-based ehods before and afer CT sauraion. This ehod can deal wih he effec of DC offse and CT sauraion. However, hese ehods require he daa afer he firs sauraion is finished; his drawback ay resul in Corresponding Auhor: Deparen of Elecrical Engineering, Myongji Universiy, Yongin, , Korea. (shkang@ju.ac.kr) * Deparen of Elecrical Engineering, Myongji Universiy, Yongin, , Korea. (d.g.lee@e.co, psouh@ju.ac.kr) Received: Noveber 9, 1; Acceped: April 5, 11 he delayed operaion of proecion relays. In [8], a ehod eploying he orphological lifing schee based on orphological waveles is used o deec CT sauraion. A slope of he agneizing curve, which is regarded as a sraigh line in he sauraed secion, is hen calculaed o esiae and copensae he agneizing curren of he CT. This ehod can esiae he scaled down and unsauraed priary curren in real ie. However, he slope of he agneizing curve is deerined by only a few saples a he sar of sauraion. The accuracy of his ehod highly depends on he esiaed value of he slope. In his paper, a phasor esiaion algorih based on he leas square fiing echnique for he disored secondary curren due o CT sauraion is proposed. The aheaical for of he secondary curren during CT sauraion is represened as he scaled priary curren wih agneizing curren. The inforaion on he scaled priary curren is esiaed using he leas square echnique, wih he easured secondary curren in he sauraed secion. The proposed ehod can esiae he phasor of a fundaenal frequency coponen during he sauraed period. The perforance of he algorih is validaed under various faul and CT condiions using a C4 CT odel. To deec he sar and end of sauraion in real-ie, he algorih based on he hird difference funcion in [9] is used. A series of perforance evaluaions shows ha he proposed phasor esiaion algorih can esiae he phasor of he fundaenal frequency coponen wih high accuracy, regardless of faul condiions and CT characerisics.

2 46 Phasor Esiaion Algorih Based on he Leas Square Technique during CT Sauraion.1 Basic CT heory. Phasor Esiaion Algorih Fig. 1 shows a siplified equivalen circui of a CT, where L is he agneizaion inducance, R is he oal secondary resisance, i 1 is he priary curren referred o he secondary, is he agneizing curren and i is he secondary curren. As he value of ( ) is near o zero before sauraion, (4) can be rewrien as R i () = i () d L (5). Phasor esiaion during CT sauraion The faul curren flowing hrough a priary circui can be generally considered as he cobinaion of an exponenially decaying DC offse and sinusoidal coponens. Thus i 1 () can be expressed as M / τ 1 k k = 1 i () = Ae + A sin( kw + ϕk ) (6) Fig. 1. Siplified equivalen circui of a CT The relaionship beween i 1 (), () and i () is i () = i () i () (1) 1 The core flux ϕ() is relaed o i () by he expression N dφ() = Ri () () d where N is he nuber of urns in he secondary winding. Inegraing () fro o yields ( φ φ ) N () ( ) = R i () d (3) The core flux ϕ() is deerined by he agneizaion curren () according o he agneizing curve. To siplify, he agneizing curve in he sauraed secion is considered a sraigh line wih a slope L, as shown in Fig.. Hence, (3) can be rewrien as ( ) L i () i ( ) = R i () d (4) where τ and A are he ie consan and he agniude of a DC offse coponen, A k and ϕ k are he apliude and he phase angle of he k h haronic coponen, and M is he highes order of haronic coponen presen in he signal. By subsiuing (5) and (6) ino (1) and rearranging a ie = 1, he secondary curren i () during he CT sauraion period can be rewrien as Expanding e follows: M 1 / τ 1 k 1 k = 1 R 1 i ( ) d L i ( ) = A e + A sin( kw + ϕ ) 1 τ k (7) using he Taylor series is possible as e Δ 1 Δ τ Δ Δ = + + (8) τ! τ 3! τ where is he sapling inerval. The nuerical approxiaion of he inegral of he secondary curren in (7) can be ipleened using a rapezoidal ehod as follows: +Δ Δ ( ) i () d = i () + i ( +Δ) (9) φ L Higher-order haronics can be blocked by he signal condiioning equipen such as an analog low-pass filer. Thus, assuing ha he haronic coponens higher han he fifh order are effecively blocked by a low-pass filer, and ha he even haronics are hardly presen in he power syse signals, (7) can be represened as Fig.. Magneizaion curve of a CT i ( ) = a x + a x + a x + a x + a x a x + a x + a x + a x + a x (1)

3 Dong-Gyu Lee, Sang-Hee Kang and Soon-Ryul Na 461 where x1 = A, x = A(1/ τ ), x3 = A(1/ τ ) x4 = A1cos ϕ1, x5 = A1sin ϕ1, x6 = R/ L x7 = A3cos ϕ3, x8 = A3sin ϕ3 x9 = A5sin ϕ5 x1 = A5sinϕ5 a = 1, a =, a = / = sin 1, 51 = cos 1, 61 = ( ) ( ) ( ) ( ) ( ) ( ) ( ) a w a w a i d a71= sin 3 w1, a81= cos 3 w1 a = sin 5 w, a = cos 5w If he curren is sapled wih a specific ie inerval,, (1) can be expanded wih each curren saple o he necessary exen. The equaions can be wrien in he arix fors as follows: I = A X (11) The eleens of arix A depend on he ie reference and he sapling inerval, which can be predeerined in an off-line ode. Marix I ade up of he sapled curren daa during he CT sauraion period is also known. If he nuber of curren saples,, is greaer han he nuber of unknown variables, arix X coposed of he unknown variables can be deerined using he leas square echnique as follows: for various fauls on a 345 kv, 1 k siple overhead ransission line, as shown in Fig. 3. Fig. 3. Single line diagra of he odel syse The siulaed faul cases are single phase-o-ground fauls locaed k fro he S bus. The CT odeling described in [1] is used o siulae he reanen flux and a resisive burden of 4.3 Ω is conneced o a C4 CT (:5). The sauraion poin of (.47 A, 1.51 Vs) is seleced o generae hyseresis daa using HYSDAT, which is an auxiliary progra in EMTP. In he proposed algorih, he agneizaion curren is assued o be zero before sauraion. To consider he effec of he assupion, wo kinds of hyseresis loops are used, as shown in Fig. 4. The sauraion poins of he wo hyseresis loops are he sae, bu he hyseresis losses are differen. In Case 1, he original hyseresis loop generaed by he HYSDAT subrouine is used in he CT odeling, whereas in Case, a revised loop is used o increase he agneizaion curren before sauraion. Case 1 Case T 1 T = X A A A I (1) Flux (Vs) Finally, he phasor of he fundaenal frequency coponen a ie 1 can be obained as (13) - x ( ) + jx ( ) = A cosϕ + ja sinϕ (13) Afer he firs sauraion is copleed, he phasor can be esiaed using any ehod suggesed in [5]-[7]. Unil hen, he phasor is given by x4( 1 + pδ ) + jx5( 1+ pδ) = x + jx e ( ( ) ( )) j π p/ N where N is he nuber of saples per cycle. 3.1 Syse configuraion 3. Perforance Evaluaion (14) The perforance of he proposed algorih is evaluaed - Magneizing Fig. 4. Hyseresis loops used in he CT odeling The sapling frequency is se o 3,84 Hz, i.e., 64 saples per cycle in 6 Hz syses. The nuber of curren saples,, for he leas square fiing is se o. The curren signal is preprocessed by a second-order Buerworh low-pass filer wih he gain of.1 a he sopband cuoff frequency of 1,9 Hz o reove he haronics and preven aliasing errors. To deec he sar and end of sauraion in real-ie, he algorih based on he hird difference funcion in [9] is used. 3. Case sudies Figs. 5 and 6 show he es resuls of he proposed algorih for Case 1. To deonsrae he accuracy of he

4 46 Phasor Esiaion Algorih Based on he Leas Square Technique during CT Sauraion proposed algorih, he esiaed resuls using he scaled priary curren are also shown in each figure. Figs. 5(a) and 6(a) show he scaled priary curren referred o he secondary side and he easured secondary curren. Figs. 5(b) and 6(b) show he acual and esiaed agneizing currens and he deeced sauraion secion by he hird difference funcion. The esiaed agneizing curren is calculaed using a 61 and x 6 in (1). In his case, he agneizing curren is close o zero before sauraion, as shown in each enlarged figure. Figs. 5(c) and 6(c) show he esiaed dc coponen using (1). Figs. 5(d) and 6(d) show he esiaed real and iaginary pars of he fundaenal frequency coponen using (1). Even if he esiaed dc coponen conains soe errors, he esiaed real and iaginary pars of he fundaenal frequency coponen are sill nearly equal o he resuls esiaed using he priary curren. Figs. 5(e) and 6(e) show he real and iaginary pars of he esiaed phasor using (13). The error of he esiaed dc coponen can coe fro he agneizaion curve as a sraigh line wih a slope. However, he agneizaion curve is no a sraigh line around he sauraion poin. Alhough his assupion ay lead o soe difference beween he inpu signal and he aheaical for in (7), he difference exponenially decreases as he agneizing curren increases. Therefore, he difference only affecs he esiaed agneizing curren and he dc coponen. Moreover, he ore he CT is severely sauraed, he ore he agneizaion curve is close o a sraigh line. This explains why he esiaed agneizing curren and he dc coponen by he proposed algorih are ore correc when he CT is severely sauraed. Figs. 7 and 8 show he es resuls of he proposed easured sec. easured sec esiaed sauraed secion esiaed sauraed secion (b) agneizing currens (b) agneizing currens x 4 pri x 4 pri I sec I sec (e) real and iaginary pars of he esiaed phasor Fig. 5. Tes resuls for Case 1 (Faul incepion angle, Reanen flux %) (e) real and iaginary pars of he esiaed phasor Fig. 6. Tes resuls for Case 1 (Faul incepion angle, Reanen flux 8%)

5 Dong-Gyu Lee, Sang-Hee Kang and Soon-Ryul Na easured sec. 1 easured sec esiaed sauraed secion esiaed sauraed secion (b) agneizing currens (b) agneizing currens x 4 pri (e) real and iaginary pars of he esiaed phasor I sec Fig. 7. Tes resuls for Case (Faul incepion angle, Reanen flux %) x 4 pri (e) real and iaginary pars of he esiaed phasor I sec Fig. 8. Tes resuls for Case (Faul incepion angle, Reanen flux 8%) algorih for Case. In his case, he agneizing curren before sauraion is greaer han ha in Case 1, as shown in he enlarged inner figures in Fig. (b). However, he agneizing curren er ignored in (4) is a consan value, hus is frequency response ay belong o he dc coponen. As a resul, he esiaed dc coponen and he agneizing curren are slighly differen fro hose in Case 1. However, he real and iaginary pars of he esiaed phasor are nearly he sae wih he resuls of Case 1. Based on hese resuls, i is clear ha he proposed algorih can esiae he accurae phasor of he fundaenal frequency coponen before he CT leaves he sauraion period. Moreover, he proposed algorih can quickly esiae he phasor when he CT is deeply sauraed. Fig. 9 shows he ie responses of he proposed algorih and he leas square (LS)-based algorih in [6] for he faul wih 8% reanen flux. The LS-based algorih uses he saples obained fro no only he firs unsauraed wavefor porion bu also he second unsauraed wavefor porion. However, he proposed algorih only uses he firs sauraed wavefor porion, which explains why he proposed algorih can esiae he accurae phasor faser han he LS-based algorih. 4. Conclusion A phasor esiaion algorih based on he leas square curve fiing echnique for he disored secondary curren due o CT sauraion was described. The aheaical for of he secondary curren during CT sauraion was represened as he scaled priary curren wih he

6 464 Phasor Esiaion Algorih Based on he Leas Square Technique during CT Sauraion easured sec. sauraed secion (b) esiaed phasor by he proposed algorihh agneizing curren. The inforaion on he scaled priary curren was esiaed using he leas square echnique wih he easured secondary curren in he sauraed secion. Then, he proposed ehod can esiaed he phasor of he fundaenal frequency coponen of a curren signal, especially during he sauraed period. The perforance of he algorih was validaed under various faul and CT condiions using a C4 CT odel. A series of perforance evaluaions showed ha he proposed phasor esiaion algorih can esiae he phasor of he fundaenal frequency coponen wih high accuracy, regardless of faul condiions and CT characerisics. The proposed algorih is ore effecive when a CT is deeply sauraed wih a large reanen flux in he CT core. Therefore, he proposed algorih is considered a useful ehod for he phasor esiaion of he faul curren signals in power syse applicaions. For exaple, i can decrease he unnecessary ie delay of he proecion relays. Acknowledgeens This work was suppored by he Naional Research Foundaion of Korea (NRF) gran funded by he Korea governen (MEST) (No ) and he nd Brain Korea 1 Projec. References Re I Mag (c) esiaed phasor by he LS-based algorih in [6] Re I Mag Fig. 9. Tie response coparison in Case 1 (Faul incepion angle, Reanen flux 8%) K. Aggarwal, "An Algorih for Copensaing he Secondary Curren of Curren Transforers", IEEE Trans. Power Delivery, vol. 1, no. 1, pp , Jan [] N. Locci and C. Muscas, "A Digial Copensaion Mehod for Iproving Curren Transforer Accuracy", IEEE Trans. Power Delivery, vol. 15, no. 4, pp , Oc. [3] N. Locci and C. Muscas, "Hyseresis and Eddy Curren Copensaion in Curren Transforer", IEEE Trans. Power Delivery, vol. 16, no., pp , April 1 [4] Y. C. Kang, U. J. Li, S.H. Kang and P.A. Crossley, "Copensaion of he Disorion in he Secondary Curren Caused by Sauraion and Reanence in a CT", IEEE Trans. Power Delivery, vol. 19, no. 4, Oc. 4. [5] F. Li, Y. Li and R. K. Aggarwal, "Cobined Wavele Transfor and Regression Technique for Secondary Curren Copensaion of Curren Transforers", IEE Proc. Generaion, Transission and Disribuion, vol. 149, no. 4, pp , July. [6] J. Pan, K. Vu and Y. Hu, "An Efficien Copensaion Algorih for Curren Transforer Sauraion Effecs", IEEE Trans. Power Delivery, vol. 19, no. 4, pp , Oc. 4. [7] Soon-Ryul Na, Jong-Young Park, Sang-Hee Kang and Mladen Kezunovic, "Phasor Esiaion in he Presence of DC Offse and CT Sauraion", IEEE Trans. Power Delivery, vol. 4, no. 4, pp , Oc. 9. [8] Z. Lu, J. S. Sih and Q. H. Wu, "Morphological Lifing Schee for Curren Transforer Sauraion Deecion and Copensaion", IEEE Trans. Circuis and Syses I: Regular Papers, vol. 55, no. 1, pp , Nov. 8 [9] Y. C. Kang, S. H. Ok and S. H. Kang, "A CT Sauraion Deecion Algorih", IEEE Trans. Power Delivery, vol. 19, no. 1, pp , Jan. 4 [1] M. Kezunovic, Lj. Kojovic, A. Abur, C. W. Froen, D. R. Sevcik and F. Phillips, "Experienal evaluaion of EMTP-based curren ransforer odels for proecive relay ransien sudy", IEEE Trans. Power Delivery, vol. 9, no. 1, pp , Jan Dong-Gyu Lee received he B.S., M.S. and Ph.D. degrees in elecrical engineering fro Myongji Universiy, Korea, in, 4 and 1, respecively. His ain research ineress are power syse proecion. [1] Y. C. Kang, J. K. Park, S. H. Kang, A. T. Johns and R.

7 Dong-Gyu Lee, Sang-Hee Kang and Soon-Ryul Na 465 Sang-Hee Kang received he B.S., M.S. and Ph.D. degrees fro Seoul Naional Universiy, Korea in 1985, 1987 and 1993, respecively. He was a visiing fellow and a visiing scholar a he Universiy of Bah, UK in 1991 and He is a professor a Myongji Universiy, Korea. He has been also wih Nex-generaion Power Technology Cener, Korea since 1. He was an honorary acadeic visior a he Universiy of Mancheser, UK in 7. His research ineres is o develop digial proecion syses for power syses using digial signal processing echniques. Soon-Ryul Na received he B.S., M.S., and Ph.D. degrees in elecrical engineering fro Seoul Naional Universiy, Seoul, Korea in 1996, 1998 and, respecively. He is an associae professor a Myongji Universiy, Yongin, Korea. His research ineress are he proecion, conrol, and auoaion of power syses.