Ground Fault Current Distribution on Overhead Transmission Lines

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1 FACTA UNIVERSITATIS (NIŠ) SER.: ELEC. ENERG. vol. 19, April 2006, Ground Fault Currnt Distribution on Ovrhad Transmission Lins Maria Vintan and Adrian Buta Abstract: Whn a ground fault occurs on an ovrhad transmission lin in a powr ntwork with groundd nutral, th fault currnt rturns to th groundd nutral through th towr structurs, ground rturn paths and ground wirs. This papr prsnts an analytical mthod in ordr to valuat th ground fault currnt distribution in an ffctivly groundd powr ntwork. Th ffct of soil rsistivity, ground rsistanc of towrs and powr lin configuration, on th magnitud of rturn currnts, has bn xamind. Kywords: Ovrhad transmission lin, fault currnt distribution, powr ntwork. 1 Introduction High-voltag systms hav an ffctivly groundd nutral. Whn a ground fault occurs on an ovrhad transmission lin, in a powr ntwork with groundd nutral, th fault currnt rturns to th groundd nutral through th towr structurs, ground rturn paths and ground wirs [1],[2]. In this papr, basd on Kirchhoff s thorms, an analytical mthod in ordr to dtrmin th ground fault currnt distribution in ffctivly groundd powr ntwork is prsntd. It is possibl to find th valus of th currnts in towrs, ground wir and th currnts that rturn to th stations [3],[4],[5],[6]. Th approach usd in this papr, basd on mthod prsntd by Rudnbrg [7] and Vrma [8], can b applid to short lins, rspctivly in cas whr ar only fw spans btwn th fding station and th faultd towr. Manuscript rcivd April 14, M. Vintan is with Univrsity Lucian Blaga of Sibiu, Dpartmnt of Elctrical Enginring, E. Cioran Str. No. 4, Sibiu, , Romania (-mail: A. Buta is with Univrsity Polithnica of Timisoara, Dpartmnt of Elctrical Powr Systm, B-dul V. Parvan, no. 2, Timisoara, Romania (-mail: 71

2 72 M. Vintan and A. Buta: A phas-to-ground fault that appars on a phas of a transmission lin, divids th lin into two sctions, ach xtnding from th fault towards on nd of th lin. Dpnding on th numbr of towrs btwn th faultd towr and th stations, rspctivly of th distanc btwn th towrs, ths two sctions of th lin may b considrd infinit, in which cas th ground fault currnt distribution is indpndnt from th trmination of th ntwork, othrwis, thy must b rgardd as finit, in which cas th ground fault currnt distribution may dpnd gratly on th trmination of th ntwork [5],[6]. In this papr w will b considr th cas whn th fault appars to th last towr of th transmission lin. Thn, it will b considrd th fault that appars at any towr of th transmission lin, th two sctions of th lin ar finit and it is assumd that th fault is fd from both dirctions [9],[10]. Th calculation mthod is basd on th following assumptions: impdancs ar considrd as lumpd paramtrs in ach span of th transmission lin, lin capacitancs ar nglctd; th contact rsistanc btwn th towr and th ground wir, and th towr rsistanc btwn th ground wir and th faultd phas ar nglctd; th ntwork is considrd linar in th sinusoidal stady-stat and only th powr frquncy is considrd. 2 Ground Fault Currnt Distribution W prsntd two cass: th first cas is th fault at th trminal towr of th lin and th scond cas th fault at any towr of th lin. Cas 1. Figur 1 prsnts th connction of a ground wir connctd to arth through transmission towrs, ach transmission towr having its own grounding lctrod or grid,. It is assumd that all th transmission towrs hav th sam ground impdanc and th distanc btwn towrs is long nough to avoid th influnc btwn thr grounding lctrods. Th slf-impdanc of th ground wir connctd btwn two groundd towrs, calld th slf-impdanc pr span, is. Considring th sam distanc I d btwn two conscutiv towrs and that is th sam for vry span, thn Z cp = I d, whr th slf-impdanc of th ground wir is rprsntd in Ω/km. Z cpm rprsnts th mutual-impdanc btwn th ground wir and th faultd phas conductor, pr span. Whn a fault appars, a part of th ground fault currnt will gt to th ground through th faultd towr, and th rst of th fault currnt will gt divrtd to th ground wir and othr towrs. Th currnt I n flowing to ground through th n-th towr, countd from th trminal towr whr th fault is assumd to tak plac, is

3 Ground Fault Currnt Distribution on Ovrhad Transmission Lins 73 qual with th diffrnc btwn th currnts i n and i n+1 [7],[8] I n = i n i n+1 (1) Fig. 1. Fault currnt distribution, cas 1. Th nxt rlation givs th loop quation for th n-th msh I n I n 1 + i n νi d = 0 (2) In quation (2) ν = Z cpm / rprsnts th coupling factor btwn th ovrhad phas and ground wir and I d rprsnts th fault currnt. Th quation (2) could b writtn in th following form and similarly i n = (I n 1 I n ) + νi d (3) i n+1 = (I n I n+1 ) + νi d (4) Substituting quations (3) and (4) in quation (1), for th currnt in th faultd towr th following quation will b obtaind, which is a scond ordr diffrnc quation According to [7], th solution of this quation is I n = I n+1 2I n + I n 1 (5) I n = A αn + B αn (6) In accordanc with th solution (6) that contains th arbitrary paramtrs A and B, th currnt flowing to ground through th succssiv towrs, has an xponntial variation. Th arbitrary paramtrs A and B will b obtaind latr, from th boundary conditions.

4 74 M. Vintan and A. Buta: Paramtr α in th solution (6) could b obtaind by substituting th solution (6) in quation (5). For this purpos, n is substituting with (n + 1), rspctivly with (n 1) in quation (6) Th quation (5) bcam I n+1 = A α(n+1) + B α(n+1) (7) I n 1 = A α(n 1) + B α(n 1) (8) Bcaus, it can b writtn = α + α 2 = 2(sinh a 2 )2 (9) α = Zcpd (10) By applying quation (1) to th (n 1) towr, th following xprssion will b obtaind I n 1 = i n 1 i n (11) By substituting th quations (1) and (11) in (2), th following quation with a constant trm will b obtaind i n = i n+1 2i n + i n 1 + νi d (12) Similar with quation (5), th currnt in th ground conductor is givn by th nxt solution i n = a αn + b αn + νi d (13) a, b rprsnts th arbitrary paramtrs. Bcaus of th link btwn currnts i n and I n, th arbitrary paramtrs A, B and a, b ar not indpndnt. By substituting th solutions (6) and (13) in quation (1), it will b obtaind A αn + B αn = a αn (1 α )+b αn (1 α ) (14) Bcaus ths rlations ar th sam for vry valu of n, th following xprssions will b obtaind A = a(1 α ) (15) B = b(1 α ) (16)

5 Ground Fault Currnt Distribution on Ovrhad Transmission Lins 75 Th currnt in th ground wir will b thn givn by th following xprssion ( ) ( ) αn αn i n = A 1 α + B 1 α + νi d (17) Th boundary condition (condition for n = 0) at th trminal towr of Figur 1, which mans that th fault currnt is givn by th sum btwn th currnt in th faultd towr and th currnt in th first span of th ground wir, is I d = I 0 + i 1 (18) Long Lin. If th lin is sufficintly long so that, aftr som distanc, th varying portion of th currnt xponntially dcays to zro, A 0, and according to (6) and (17) I n = B αn (19) ( ) αn i n = B 1 α +νi d (20) Substituting ths xprssions in (18), with n = 0 for I n and n = 1 for i n, it will b obtaind ( ) ( ) α B I d = B+B 1 α + νi d = 1 α + νi d (21) For B th following xprssion will b obtaind B = (1 ν)(1 α )I d = (1 ν)(2tanh α 2 ) (1+tanh α 2 ))I d (22) Th currnt in th faultd towr will gt th xprssion I 0 = B = (1 ν)(2tanh α 2 ) (1+tanh α 2 ))I d (23) Th currnt in th first span, countd from th faultd towr, will b Th voltag ris at th trminal towr is i 1 = I d I 0 = I d [ α + ν(1 α )] (24) U 0 = I 0 = (1 ν)(1 α )I d = (1 ν)(2tanh α 2 ) (1+tanh α 2 ) I d (25) =(1 ν)zi d

6 76 M. Vintan and A. Buta: whr Z rprsnts th quivalnt impdanc of th ntwork looking back from th fault. Usually, th trminal towr is connctd, through an xtra span Z cp d, to th station-grounding grid (Figur 2). Consquntly, a rsistanc rprsnting th grounding systm of th station rsistanc must clos th laddr ntwork rprsnting such a lin. In this cas, a part of th total ground fault currnt will flow through th station ground rsistanc R p. In ordr to us th prvious rsults, it is nough to rplac th currnt I d with I d = I d I p, and thus th valu of th currnt in th faultd towr will b [8] I 0 = (1 ν)(1 α )I d (26) Noting th sum btwn Z cp d and R p with Z p = R p + Z cp d, th currnt I p through th station grounding grid rsistanc is givn by th following xprssion I p = I dz Z p + Z (27) I d Fig. 2. Fault currnt distributions. In cas th valus of Z p and Z ar known, I p can b found out from (27) abov. is givn by th nxt xprssion I d = I d I p = I 0 + i 1 (28) Now, it will b found th n-th towr, as countd from th trminal towr, whr th currnt gts rducd to 1% thn that travrsing th trminal towr. From quation (19) B αn = 1 100B n = 1 α ln100 4,6 (29) For / = 0.03, will gt n = 26.5, so it taks at last 26 towrs from th fault to gt a currnt rducd to 1% thn that travrsing th trminal towr. Taking into account this considration, it is possibl to s if on could considr A = 0

7 Ground Fault Currnt Distribution on Ovrhad Transmission Lins 77 in th xprssions of th currnts in ground wir and towrs. Th numbr of th towrs should b at last qual with th numbr givn by xprssion (29). Short lin. If th lin cannot b considrd long nough rgarding to th xprssion (29), thn paramtrs A and B will b found from th boundary conditions. With R p, rspctivly R p, ar notd th rsistancs of th grounding systms of th two stations (Figur 3), which dos not includ th grounding ffcts of th ground wir of th considrd lin. Th stations ar connctd to th trminal towrs, at both sids, through an xtra span Z cp d, and th sum btwn Z cp d and grounding systm of th stations rsistancs was notd with Z p = R p + Z cp d, rspctivly Z p = R p + Z cp d. Th boundary condition at th rciving nd of th lin is I d = I p + I 0 + i 1 = i 1 + I 0 + I 0 Z p (30) At th snding nd of th lin w hav I d = I p + i N+1 (31) I N + I p R p i N+1 Z cp d + νi d Z cp d = 0 (32) Substituting I p from (32) to (31), w will ( ) I d 1+ν Z cp d = i N+1 (1+Z cp R d R p )+ I N (33) p R p Fig. 3. Fault currnt distributions. Substituting I 0, I N, i N+1 and i 1 into (30) and (33), according to (6) and (17), a systm with two linar quations will b obtaind ] [ ] I d (1 ν) = A[ Z α st 1 + B Z 1 + Z p α st Z p [ I d (1 ν) = A αn α 1 (1+ Z α cp d R p ] [ ) R p + B αn α 1 (1+ Z α cp d R p ] ) R p (34)

8 78 M. Vintan and A. Buta: (34) givs whr: A 1, B 1, A 2, B 2 ar B 2 B 1 A 1 A 2 A = I d (1 ν),b = I d (1 ν) (35) A 1 B 2 B 1 A 2 A 1 B 2 B 1 A 2 A 1 = 1 1 α + 1,B Z 1 = p 1 α + Z p [ ( ) A 2 = αn α 1 α 1+ Z cp d Z [ ( ) st ],B 2 = αn α R p R p 1 α 1+ Z cp d Z ] st R p R p Th currnt in th faultd towr (obtaind for n = 0 in all xprssions, including A 2, B 2 ) will b [ B 2 B 1 I 0 = A+B = I d (1 ν) (1 α )+ A ] 1 A 2 (1 α ) (36) A 1 B 2 B 1 A 2 A 1 B 2 B 1 A 2 Th voltag ris of th trminal towr in this cas is U 0 = I 0 [ B 2 B 1 = I d (1 ν) (1 α )+ A ] 1 A 2 (1 α ) A 1 B 2 B 1 A 2 A 1 B 2 B 1 A 2 (37) = (1 ν)i d Z N With Z N was notd th quivalnt impdanc of th ntwork looking back from th fault in this cas [ B 2 B 1 Z N = (1 α )+ A ] 1 A 2 (1 α ) (38) A 1 B 2 B 1 A 2 A 1 B 2 B 1 A 2 Cas 2. In Figur 4 th fault occurs at th towr numbr 0. Thr ar N towrs btwn this towr and th lft station and rspctivly M towrs btwn towr numbr 0 and th right station. Th rsistancs of th grounding systms of th lft and right stations (Figur 4) ar R p, rspctivly R p. Th impdanc of th sction of ground wir btwn on station and th first towr (towr N or M) is Z cp d, and Z p = R p + Z cp d, rspctivly Z p = R p + Z cp d. Th total fault currnt I d is givn by th sum btwn th currnt I d from on sid, and I d from th othr sid of th transmission lin. In this cas, considring first th lft part from th faultd towr, th situation is idntically with that prsntd in Figur 1. So th sam quations could b writtn

9 Ground Fault Currnt Distribution on Ovrhad Transmission Lins 79 Fig. 4. Ground fault currnt distributions. (1-17), and th solutions for th currnt in th towrs, rspctivly in th ground wir ar I n s = A s αn + B s αn (39) αn αn i n s = A s 1 α + B s 1 α + νi d (40) Th subscript s is usd for th lft part from th faultd towr. I n s rprsnts th currnt in th towr numbr n, countd from th faultd towr to th lft part of th transmission lin. In ordr to find th constants A s, B s, it is ncssary to writ th boundary conditions. For th faultd towr, th following formulas can b writtn { I0 I 1 i 1 + ν I d = 0 I 1 I 2 i 2 + ν I d = 0 (41) Also th xprssion can b writtn I 1 = i 1 i 2 (42) Substituting i 1 and i 2 from quations (41) in quation (42), it will b obtaind ( I 1 2+ Z ) cp d = I 0 + I 2 (43) For th lft trminal it can b writtn I N + i N+1 = i N (44) I N + I p R p i N+1 Z cp d + I dz m = 0 (45) (I N 1 I N ) i N + ν I d = 0 (46)

10 80 M. Vintan and A. Buta: i N+1 = I d I p (47) whr Z m rprsnts th mutual coupling btwn th ground wir and th faultd phas, in th last span. Rplacing i N+1 and i N from (45), (46) in (44), by taking into account (47), th following xprssion will b obtaind I N ( 1+ + Z cp d + R p ) ( = I N 1 + I d ν R p + Z ) m Z cp d + R p Exprssions (43) and (48) rprsnts th boundary conditions. By rplacing I 1, I 2, I N and I N 1 from th solutions (39) and (40) in (43) and (48), it will b obtaind two systm quations from which rsults A s and B s )( (2+ α Z cp d α ν R p+z m )I R p +Z d I 0 αn L s cp A s = ) d ) (49) (2+ α(n 1) Z cp d α R s (2+ α(n 1) Z cp d α L s whr B s = (48) )( I 0 αn R s (2+ α Z cp d α ν R p+z m )I R p +Z d cp ) d ) (50) (2+ α(n 1) Z cp d α R s (2+ α(n 1) Z cp d α L s L s = 1+(1 α ) +,R R p + Z s = 1+(1 α) + cp d R p + Z cp d Similar xprssions ar obtaind for th currnts from th right part of th faultd towr { In d = A d αn + B d αn i n d = A d 1 αn + B α d 1 αn + νi (51) α d Th constants A d and B d ar givn by th following xprssions A d = B d = )( (2+ α Z cp d α ν R p+z m )I R p+z d I 0 αm L d cp ) d ) (52) (2+ α(m 1) Z cp d α R d (2+ α(m 1) Z cp d α L d )( I 0 αm R d (2+ α Z cp d α ν R p+z m )I R p+z d cp ) d ) (53) (2+ α(m 1) Z cp d α R d (2+ α(m 1) Z cp d α L d

11 Ground Fault Currnt Distribution on Ovrhad Transmission Lins 81 whr L d = 1+(1 α ) +,R R p + Z d = 1+(1 α ) + cp d R p + Z cp d At th faultd towr th total ground fault is (Figur 4) Th currnt in th faultd towr I 0 bcam I 0 = I d = I 0 i 1s i 1d (54) (1 ν)i d +( Ws T s I d + W d T d I d )( ) V1 1 V α 2 1 α ( ) ( ) (55) 1 α Ls αn 1 α T s + L d αm T d + α Rs αn 1 α T s + R d αm T d whr W s = ν R p + Z m,w R p + Z d = ν R p + Z m cp d R p + Z cp d V 1 = 2+ Z cp d ( T s = α(n 1) 2+ Z cp d α ( T d = α(m 1) 2+ Z cp d α 3 Numrical Rsults α,v 2 = 2+ Z cp d α ( )R s α(n 1) 2+ Z cp d )R d α(m 1) ( 2+ Z cp d ) α L s, ) α L d In ordr to illustrat th thortical approach outlind in sction abov, w ar considring that th lin that conncts two stations is a 110 kv transmission lin with aluminium-stl 185/32 mm 2 and on aluminium-stl ground wir 95/55 mm 2 (Figur 5). Lin impdancs pr on span ar dtrmind on th bass of th following assumptions: avrag lngth of th span is 250 m; th rsistanc pr unit lngth of ground wir is 0.3 Ω/km and its diamtr is 16 mm. Ground wir impdanc pr on span and th mutual impdanc Z m btwn th ground wir and th faultd phas ar calculatd for diffrnt valus of th soil rsistivity ρ with formulas basd on Carson s thory of th ground rturn path, givn in th Appndix 1. Impdanc Z m is calculatd only in rlation to th faultd phas conductor, bcaus it could not b assumd that a lin sction of a fw spans ar transposd. Th fault was assumd to occur on th phas which is th furthst from

12 82 M. Vintan and A. Buta: Fig. 5. Disposition of lin conductors. th ground wir, bcaus th lowst coupling btwn th phas and ground wir will produc th highst towr voltag. Figur 6 shows th valus for th impdanc of finit lin in cas of a fault at th last towr of th lin, as a function of th soil rsistivity and for diffrnt valus of towrs impdancs. Th valus ar calculatd with th proposd xprssion (38). Finit lin impdanc [Ω] =50Ω =10Ω =4Ω Ground wir currnt [ka] ρ=500ωm ρ=100ωm ρ=10ωm Soil rsistivity [Ωm] Fig. 6. Impdanc of finit lin for diffrnt valus of soil rsistivity Span numbr Fig. 7. Ground wir currnt for diffrnt valus of ground wir and mutual impdanc in cas of th fault at th last towr. = 10Ω. Figur 7 shows th currnts flowing in th ground wir in cas of a fault at th last towr of th lin. It was assumd that th lin has 15 towrs. Th valus of ground wir and mutual impdanc btwn th faultd phas and th ground wir givn in Tabl 1, wr calculatd for diffrnt valus of th soil rsistivity. For thos valus and for that numbr of towrs, according to th xprssion (29), th lin should b considrd a short lin. In Tabl 1 w prsntd th valus of th ground wir impdanc and th mutual impdanc Z m, pr on span, btwn th ground wir and th faultd phas, calculatd with Carson s formulas, for diffrnt valus of th soil rsistivity.

13 Ground Fault Currnt Distribution on Ovrhad Transmission Lins 83 Tabl 1. Ground wir impdanc and mutual impdanc, pr on span for diffrnt valus of th soil rsistivity. ρ[ωm] R cpd [Ω] Z m [Ω] Ground wir currnt [ka] =500Ω =100Ω =10Ω Ground wir currnt [ka] =100Ω =50Ω =10Ω Span numbr Fig. 8. Currnt flowing in th ground wir in cas of diffrnt valus of soil rsistivity; N = M = 15 towrs btwn faultd towr and th both nds of th lin; = 10Ω Span numbr Fig. 9. Currnt flowing in th ground wir in cas of diffrnt valus for towrs impdancs; N = M = 15 towrs btwn faultd towr and th both nds of th lin. Using th sam valus of th ground wir and mutual impdanc calculatd for diffrnt valus of soil rsistivity givn in th Tabl 1, th currnts flowing in th ground wir in cas of th fault at any towr of th lin ar shown in Figur 8. Figur 9 shows th currnts flowing in th ground wir for diffrnt valus of th towrs impdancs. Th total fault currnt from both stations was assumd to b I d = 15000A and I d = I d = 7500A. Thos valus ar valid for a soil rsistivity of 100Ωm. It was assumd that th fault appars at th middl towr of th lin, so thr ar N = 15 towrs and rspctivly M = 15 towrs btwn th faultd towr and th trminals. 4 Conclusions This papr dscribs an analytical mthod in ordr to dtrmin th ground fault currnt distribution in powr ntworks, whn th fault appars at th last towr of th lin, rspctivly at any towr of th transmission lin. Th author dvlopd this modl basd on th arlir approach of Rudnbrg [7] and Vrma [8]. Rudnbrg dvlopd a modl that was valid only for long lins, without taking into account th mutual coupling btwn th faultd phas and th ground conductor. Vrma dvlopd that modl by taking into account th mutual coupling,

14 84 M. Vintan and A. Buta: but h trats only th cas of long lin to. Th authors improvd thir modl by considring th cas of a short lin, whn it has to considr th station from th nd of th lin. Rfrncs [1] A. Buta, Transmission and Distribution of Elctricity. Timisoara: Publishing Hous of Tchnical Univrsity of Timisoara, [2] E. Clark, Circuit Analysis of A-C Powr Systms. Bucharst: Tchnical Publishing Hous, [3] F. Dawalibi and G. B. Nils, Masurmnts and computations of fault currnt distribution on ovrhad transmission lins, IEEE Transactions on Powr Apparatus and Systms, vol. PAS-103, no. 3, pp , Mar [4] J. Endrnyi, Analysis of transmission towr potntials during ground faults, IEEE Transactions on Powr Apparatus and Systms, vol. PAS-103, no. 10, pp , Oct [5] H. B. Goci and S. A. Sbo, Distribution of ground fault currnts along transmission lins - an improvd algorithm, IEEE Transactions on Powr Apparatus and Systms, vol. PAS-104, no. 3, pp , Mar [6] L. M. Popović, Practical mthod for valuating ground fault currnt distribution in station, towrs and ground wir, IEEE Transactions on Powr Dlivry, vol. 13, no. 1, pp , Jan [7] R. Rudnbrg, Transint Prformanc of Elctric Powr Systms. Bucharst: Tchnical Publishing Hous, [8] R. Vrma and D. Mukhdkar, Ground fault currnt distribution in sub-station, towrs and ground wir, IEEE Transactions on Powr Apparatus and Systms, vol. PAS-98, no. 3, pp , May/Jun [9] M. Vintan, Diffrnt factors influnc on th slf and mutual impdancs of th ovrhad wirs with ground rturn, in Procdings of Th Fourth Intrnational Powr Systms Confrnc, Timisoara, 2001, pp [10] M. Vintan and M. Bogdan, A practical mthod for valuating ground fault currnt distribution on ovrhad transmission lins, Scintific Bulltin of th Polithnica Univrsity of Timisoara, vol. Tom 48(62), pp , Nov

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