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A new technique for location of transmission line faults using single-terminal voltage and current data
Electric Power Systems Research, 23 (1992) 123-128
123
A new technique for location of transmission line faults using single-terminal voltage and current data Omar A. S. Youssef College of Telecommunication, Jeddah (Saudi Arabia) (Received September 4, 1991)
Abstract This p a p e r presents a new technique for location of transmission line faults using voltage and c u r r e n t measurements from one end of the faulted line. The method differs from past techniques in many respects. The main feature of the new technique is t h a t it considers the influence of the remote-end infeed of the t r a n s m i s s i o n line, the effect of the transmission line capacitance, fault resistance, prefault loading conditions, a n d the effect of m u t u a l coupling between different phases of the line, and then computes with a microprocessor the distance to the fault point without any approximations. It uses recorded phase voltages and c u r r e n t s at the n e a r end, then the fundamental components of the measured signals are e x t r a c t e d using microprocessor filters. The modal values of the e x t r a c t e d signals are computed then processed to indicate the precise location of the fault.
Introduction
Rapid power system repair and service restoration subsequent to a permanent fault on a transmission line are essential conditions of utility operation. With the advent of microprocessor based recording devices, utilities have shown increased interest in implementing accurate fault location techniques. Consequently, several methods have been introduced and implemented during recent years. Takagi et al. [1], Eriksson et al. [2], Cook [3] and Wiszniewski [4] proposed techniques which use fault current distribution factors, and prefault and postfault currents and voltages from one line terminal; impedances of equivalent sources connected to the line terminals are used in the computation. In practice, from time to time the distribution factors are changed. Several approximations have also been introduced during the computation process. Richards and Tan [5] proposed an iterative technique which uses several system parameters for estimating the location of faults. They developed a sequence network approach that treats both the fault location and the fault resistance as unknown quantities. Shunt reactance is not modeled, limiting the application of the algorithm to short lines. Schweitzer [6, 7] performed an iterative calcula0378-7796/92/$5.00
tion of the difference in phase angle between the total fault current and the remote-end feed current contribution. A sequence model of the line and remote-end source is used, but the mutual coupling between sequence component networks is ignored, thereby assuming that the phase impedances are balanced. Jeyasurya and Rahman [8, 9] presented an algorithm which is based on using current and voltage information at both line ends and by solving the sequence network of the equivalent power system the location of the fault can be estimated. Johns et al. [10] proposed a technique which is based on modal analysis using voltage and current data available at both ends of the protected line section. Sachdev and Agarwal [11] used local digital impedance and relay current data as well as corresponding data from the remote end to calculate the distance to the fault point. This method gives considerable errors for certain locations and the fault resistance estimates are inaccurate for locations near the midpoint of the line where the fault current contributions from both line terminals are equal. Lawrence and Waser [12] developed an algorithm based on Z-transform techniques using time domain representation of voltage and current signals and the phase network model. This paper presents an accurate algorithm to estimate the location of a transmission line fault. © 1992
Elsevier Sequoia. All rights reserved
124
The method uses the fundamental frequency voltages and currents measured at one line terminal and takes into consideration the following factors: (1) remote-end current infeed; (2) exact transmission line representation (distributed parameter representation) is considitiveeredcurrents;in the computation to account for capac(3) prefault and postfault line currents and voltages are measured and stored and used for computation; (4) the contribution of current from relay location side to the total fault current at the fault location, assumed to be equal to an impedance ratio [2], is corrected; (5) no major approximations are carrried out during computation; (6) a modal analysis is used, permitting the inclusion of unbalanced system impedances caused by untransposed lines. Development
of the algorithm
The method to compute the distance to a fault point is based on the following equations in modal form [13, 14]: VF = Vs cosh(G/) - Z s I s sinh(G/) I'~F = ( V ~ / Z s ) s i n h ( G l )
- I'~ cosh(G/)
where l is the distance to the fault point from the sending end of the line; Vs is the modal voltage at the sending end (postfault value), Is the modal current at the sending end (postfault value), VF the modal voltage at the fault point (postfault value), Ir the modal fault current (postfault value), V~ the superimposed modal voltage at the sending end (modal voltage difference between prefault and postfault), and I~ the superimposed modal current at the sending end (modal current difference between prefault and postfault), and I~r the modal fault current feeding the fault from the sending end side; Zs is the modal surge impedance, and Z the transmission line impedance per unit length; G - - T + j P is the propagation constant. Referring to Fig. 1, for the postfault condition, the modal voltages and currents can be expressed as V~ = Vs cosh(G/) - I s Z s sinh(G/) and the superimposed components of the modal voltages and currents can be expressed as
I~ ~ ) JS-end -(a) ISF IRF
/L'Y'x ~
~)t~ l~ (b)' ~v~ [_~
IR
I~F r ~
t
L_
r~j~_ v~
-I
(c) Fig. 1. S i n g l e - l i n e d i a g r a m of t h e p o w e r system: (a) p r e f a u l t c o n d i t i o n ; (b) p o s t f a u l t c o n d i t i o n ( G F = f a u l t c o n d u c t a n c e ) ; (c) s u p e r i m p o s e d c o m p o n e n t of f a u l t c u r r e n t .
I'~r = ( V s / Z s ) sinh(G/) - I~ cosh(G/)
In matrix form we can write VF = [ cosh(G/)] Vs - [ sinh(G/)]Is
(1)
Vs~- = [sinh(G/)] Ys V~ - [ c o s h ( G l ) ] I ~
(2)
where
vF'= IVy,, v~2, VF:J [cosh(Gl)] = diag[cosh(G1 l), cosh(G2l), cosh(G3l)] VS t =
[ V s 1 , VS2,
Vs3]
[sinh(Gl)] = diag[sinh(Gll), sinh(G2l), sinh(G,l)] Zs -- diag[Zsl, Zs2, Zs3] Ys = diag[Ys,, Ys2, Ys3] rt
ISF tp ~
Is
~
it
tt
rt
[IsF~, ISF2, ISF3] rr
tt
it
[Is~, Is2, Is3]
The voltage at the fault location, V~,, is related to the fault current I~ by the fault constraint matrix G~ (where the superscript p indicates phasor values) as follows: I~
p p = GFV v
The fault constraint matrices for different types of faults are indicated in Appendix A, whereas the distribution of the sequence components of currents at the fault location is indicated in Appendix B. Substituting eqn. (B-l) into eqns. (1) and (2), we get [sinh(G/)] Ys V~ - [ cosh(G/)]I~ = k GF {[ cosh(G/)] Vs - [ sinh(G/)] Zsls }
125
[cosh(G/)] {[tanh(G/)] Ys V~ - UI~} = kG~[cosh(Gl)]
{UVs - [ t a n h ( G l ) ] Z s I s }
EAD MEASUREDVALUES1
r
OF PHASE VOLTAGES AND|
k {UVs - [tanh(V/)] Zsls}
CURRENTS
= [ G~ - ~]~{[ tanh(G l )] Ys V~ - UI"ss~
CALCULATE
Selecting one of the three modes (e.g. mode 1), and letting F
[GF-1] t
=
I
PHASEI
ISUPERIHPOSED ~OLTAGES AND CURRENTSJ
CALCULATE MODAL ALUES OF VOLTAGES ND CL~RENT$
i
FAULTY PHASE1
I
, SELECTION
j
OR ES ONDING
1
then CALCULATE [ G ~ ] - I
k[Vsl - Zs, tanh(V, l) Ira] [tanh(V~ l) Ys~ Vs~ . . . -. Ira= [f~152 f,3] | t a n h ( V j ) Ys2 V~2 - I~2 k t a n h ( G j ) Ys3 V~3 - I~3
IEC,]'CG~]~EC~]
i ALOOLA.E
FAULT LOCAT,N~
CALD :TION J
k[Vsl - Zs~ tanh(G~/) Is1] = f~ [tanh(G~/) Ysl V ~ - I ~ ]
,SPLA. PRI.TO0
+ f~2[tanh(G2/) Ys2 V~2 - I~2] + 53 [tanh(G3/) Ys2 V~3 - I~3]
(3)
Equation (3) is a complex equation. It can be split into two separate equations which can be solved to get the two real u n k n o w n s k and l. Eliminating k, we get one equation in l, the distance to the fault location from the sending end, in the form f[tanh(T1 l), tan(P,/), tanh(T2/), tan(P2/), tanh(T3/), tan(P3/)] = 0 This equation can be solved by the N e w t o n Raphson m e t h o d for successive substitutions, w h e r e an initial approximation is assumed for l, t h a t is, lo, and an iterative process is carried out until the required c o n v e r g e n c e is achieved. This iterative process is governed by
Fig. 2. Flowchart of the fault location algorithm.
TABLE 1. Basic transmission line data No. of circuits No. of conductors per phase No. of earth wires Conductor position symmetry Conductor resistivity (~ m) Earth wire resistivity (~) Conductor strand diameter (cm) Earth wire strand diameter (cm) Outer diameter of earth wire (cm) No. of effective strands in phase conductor No. of effective strands in earth wire Earth resistivity (~ m)
2 3 2 None 3.2 x 10 -s 2.69 x 10 -s 0.32 0.32 2.86 54 54 20
lh +, = lk -- f ( l k ) / f ' ( l k )
The main problem now lies in computing the matrix F. The flowchart of the fault location algorithm is given in Fig. 2.
Test results
The sample power system selected for the studies is a 500 kV, 236 km line section. The basic
transmission line data are given in Table 1. Two equivalent power systems were c o n n e c t e d to buses R and S. Three types of fault were considered (phase to ground, phase to phase, and double phase to ground). Faults were applied at different locations on the line u n d e r different operating conditions. The voltage and c u r r e n t phasors were c a l c u l a t e d by the program, t h e n the equivalent modal values were computed. Mode 2 was used in the study. These data were used by
126 TABLE 2. Results of the study Fault type
Assumed fault location (p.u.)
Calculated fault location (p.u.)
System condition
Phase to earth Phase to phase Double phase to earth
0.6 0.6 0.6
0.6 0.6 0.6006
R F 50
Phase to phase Phase to phase Double phase to earth
0.7 0.7 0.7
0.7 0.7 0.7005
Light loading R~. = l0 Power flow S to R
Phase to earth Phase to phase Double phase to earth
0.9 0.9 0.9
0.9 0.9 0.8999
Light loading Rv = 20 ~ Power flow R to S
Phase to earth Phase to phase Double phase to earth
0.2 0.2 0.2
0.2 0.2 0.1998
Heavy loading RrPower flow S to R
Unloaded
the fault locating algorithm to e v a l u a t e the dist a n c e to the fault from the sending end. Table 2 summarizes the results of this study.
C
RF
L-G fault
L-L fau[t
3-L fault
Conclusion This p a p e r has presented an a c c u r a t e algorithm for locating transmission line faults. The algorithm has been tested for a v a r i e t y of simulated fault conditions, including double-circuit m u t u a l coupling effects, various remote source equivalent conditions, various prefault steadystate conditions, and in the presence of appreciable fault resistance. These simulation studies indicate t h a t the proposed method is a c c u r a t e for all types of s h u n t fault at different locations u n d e r different o p e r a t i n g conditions. The maximum error was found to be 0.1%.
Appendix A Fault
~R F Z-L-G
constraint
fault
3
matrices
0 0
-1 -1
2 -1
(d) for a double line to g r o u n d fault
0 0
R F+ -Rg
Rg
-Rg
R F + Rg
(e) for a three-phase to ground fault p 1 GF =
=
0
1 [oo
(b) for a line to line fault G~
fault
(c) for a three-phase fault
G~. - R~ 2 + 2RFRg G~ = yg
3-L-G
Fig. A-1. Types of fault. Rg is the arc resistance from phase to tower plus the effective tower footing resistance for a single line to ground fault, and the effective tower footing resistance for other types of ground fault. R F is the arc resistance between faulted phases.
v-
Referring to Fig. A-l, the fault c o n s t r a i n t matrix G~ is given by (a) for a line to g r o u n d fault
Rg
o
-1
-
Y0 + 2yF
YO -- Y v
Ye -- Y v
7
Yo-Yv
yo+2yv
Yo-YF
Yo - YF
Yo - Y~"
Yo + 2 y v
A
]
127
where
aFSFb=15Fbo ÷IsFb1*15Fb2
al'SFa_~'SFaodSFal ÷IsFa2
Yo = 1/(RF + 3Rg)
b
The phase q u a n t i t i e s V p a n d I p are r e l a t e d to the modal q u a n t i t i e s V and I by the eigenvector matrices C~ a n d C2:
I lFa=IF°+Ipi+IF2
[~
L-L fault
L-G fault ar~Fa=l~Fao+rsFat÷I~Fa2
V p = Cl V
IFb=IFo IFI+IF2
a [SFb=15Fbo+[.~Fbl+[~'b2
c
I p = C2I
[Fa IFao+
IFal+ IFa2
Since we have p
3-L fault
L-L-G
fault
p
I p = $1GF VF a ISFa= I~Fao÷~Fal ÷I~Fa 2,
where S: is a real q u a n t i t y representing the fault resistance, yg $1 = yF/2 [yF/3
I
for a line to g r o u n d fault for a line to line fault for a three-phase fault
For a double line to g r o u n d fault, S:
=
c IFa =IFao*
lira I+
IFa2 3-L-G
fault
Fig. B-1. D i s t r i b u t i o n of s e q u e n c e c o m p o n e n t s of c u r r e n t s a t t h e fault location.
1/(RF 2 + 2RF2Rg)
X£~IIF = (X~o + I'~, + I ~ ) 1 ( I ~ o + I~, + z ~ ) G~ = RF
1 0
+ Rg
1 -1
----- [ k o I F 0 -'[- k l ( I F 1
-
In this paper Rg is t a k e n to be equal to RF. For a three-phase to g r o u n d fault,
"~- Z ~ ) ] l I ~
= k, + (ko -- kl)IFoIIF (a) For a single line to g r o u n d fault, IF = 3IF0
and
$ 1 : WF
I~F./IF = k + (ko - k:)13 = kLo
G p _ Yo - YF 3Yr
1 1
+
1 0
(b) For a line to line fault, IFo = 0 I~rbllrb
Appendix B Distribution of sequence components of currents at fault location Referring to Fig. B-l, the phase r e l a t i o n between the f a u l t c u r r e n t IF and the fault c u r r e n t from the sending end is governed by the c u r r e n t d i s t r i b u t i o n factors ko. ki, and k2 of the sequence components:
k0
=
it r / SF0/IF0
k, -- ISF1/IF:
k2 = I~F2/IF2 We can assume to a high degree of a c c u r a c y t h a t k: = k2, so t h a t
= k l -'- k L L
(c) For a three-phase fault, IF0 = 0 and I~F./IFa = kl -- k3L (d) For double line to g r o u n d and three-phase to g r o u n d faults, I~Fa/IF. = k: + (ko - k:)IFo./IF. I~Fb/Ivb
= k I + (ko -
kl)IFoblIFb
I~Fc/IFc = k~ + (ko - k~)IFoc/IFc But IFo. = 1Fob =/Foe = I~o
128
and
References
tt
t!
J!
/SFOa = ISFOb = /SFOc =
kolFo
Let (ko - k, )I;~o/ko = Co
then
ISFb I~F~.
Co-- klJ/~u] Co EIF(-
or
In the case of double line to ground and threephase line to ground faults, k~ is assumed to be equal to ko, that is, Co = 0. Hence, Is~ can be represented by a real fraction, kF, of the total fault current I~ (e.g. kLc, kLL , hal , k2LG, kaLG) , that is, the angle of the fault current I~ and the fault current from the sending end, I~F, are assumed to be equal. In phasor form IsP~ = kFI~ Is~ = & k~G~ V~
In modal form I~F = &kFC~-~G~C~ VF = kGvVv
where GF = C2 1G~.C1 = CltGP.C1
(B-l)
1 T. Takagi et al., Development of a new type fault locator using one terminal voltage and current data, IEEE Trans., PAS-101 (1982) 2892 2898. 2 L. Eriksson, et al,, An accurate fault locator with compensation for apparent reactance in the fault resistance resulting from remote-end infeed, IEEE Trans., PAS-104 (1985) 424 436. 3 V. Cook, Fundamental aspect of thult location algorithm used in distance protection, Proc. Inst. Electr. El~g., Part C, 133 (1986) 359 368. 4 A. Wiszniewski, Accurate fault impedance locating algorithm, Proc. Inst. Electr. Eng., Part C. 130 (1983) 311 314. 5 G. C. Richards and O. T. Tan, An accurate fault location estimation for transmission lines. IEEE Trans., PAS-101 (1982) 945 950. 6 E. Schweitzer and J. Jachinowski, A prototype microprocessor-based system for transmission line protection and monitoring, 8th Annu. Western Protective Relay Conf., Spokane. WA, USA 1981. 7 E. Schweitzer, Evaluation and development of transmission line fault-locating techniques which use sinuosidal steadystate information, 9th Annu, Western Protective Relay Conf., Spokane, WA, USA, 1982. 8 B. Jeyasurya and M. A. Rahman, Accurate fault location o[' transmission lines using microprocessors, IEE Conf. Publ. No. 249, (1989) 13 17. 9 B. Jeyasurya and M. A. Rahman. A comparative study of transmission line fault locating algorithm, Proc. Nat. Conf. Microprocessor Applications in Pou,er Systems, Nagpur, India, 1988, pp. 131 146. 10 A. T. Johns, S. Jamali and S. M. Hadan, New accurate transmission line fault location equipment, IEE Conf. Publ. No. 249, (1989) 1 5. 11 M. S. Sachdev and R. K. Agarwa]. Accurate fault-location estimates from digital impedance relay measurements, IEE Conf. Publ. No. 249, (1989) 180 184. 12 D. J. Lawrence and D. L. Waser, Transmission line tault location using digital fault recorders, IEEE Trans., PWRD-3 (2) (1988) 496 502. 13 L. M. Wedepohl, Application of matrix methods to the solution of travelling-wave phenomena in polyphase systems, Proc. Inst. Electr. Eng., 110 (1963) 2200 2212. 14 R. H. Galloway and L. M. Wedepohl. Calculation of electrical parameters for short and long polyphase transmission lines, Proc. Inst. Electr, Eng., I l i (1964) 2051 2059.
123
A new technique for location of transmission line faults using single-terminal voltage and current data Omar A. S. Youssef College of Telecommunication, Jeddah (Saudi Arabia) (Received September 4, 1991)
Abstract This p a p e r presents a new technique for location of transmission line faults using voltage and c u r r e n t measurements from one end of the faulted line. The method differs from past techniques in many respects. The main feature of the new technique is t h a t it considers the influence of the remote-end infeed of the t r a n s m i s s i o n line, the effect of the transmission line capacitance, fault resistance, prefault loading conditions, a n d the effect of m u t u a l coupling between different phases of the line, and then computes with a microprocessor the distance to the fault point without any approximations. It uses recorded phase voltages and c u r r e n t s at the n e a r end, then the fundamental components of the measured signals are e x t r a c t e d using microprocessor filters. The modal values of the e x t r a c t e d signals are computed then processed to indicate the precise location of the fault.
Introduction
Rapid power system repair and service restoration subsequent to a permanent fault on a transmission line are essential conditions of utility operation. With the advent of microprocessor based recording devices, utilities have shown increased interest in implementing accurate fault location techniques. Consequently, several methods have been introduced and implemented during recent years. Takagi et al. [1], Eriksson et al. [2], Cook [3] and Wiszniewski [4] proposed techniques which use fault current distribution factors, and prefault and postfault currents and voltages from one line terminal; impedances of equivalent sources connected to the line terminals are used in the computation. In practice, from time to time the distribution factors are changed. Several approximations have also been introduced during the computation process. Richards and Tan [5] proposed an iterative technique which uses several system parameters for estimating the location of faults. They developed a sequence network approach that treats both the fault location and the fault resistance as unknown quantities. Shunt reactance is not modeled, limiting the application of the algorithm to short lines. Schweitzer [6, 7] performed an iterative calcula0378-7796/92/$5.00
tion of the difference in phase angle between the total fault current and the remote-end feed current contribution. A sequence model of the line and remote-end source is used, but the mutual coupling between sequence component networks is ignored, thereby assuming that the phase impedances are balanced. Jeyasurya and Rahman [8, 9] presented an algorithm which is based on using current and voltage information at both line ends and by solving the sequence network of the equivalent power system the location of the fault can be estimated. Johns et al. [10] proposed a technique which is based on modal analysis using voltage and current data available at both ends of the protected line section. Sachdev and Agarwal [11] used local digital impedance and relay current data as well as corresponding data from the remote end to calculate the distance to the fault point. This method gives considerable errors for certain locations and the fault resistance estimates are inaccurate for locations near the midpoint of the line where the fault current contributions from both line terminals are equal. Lawrence and Waser [12] developed an algorithm based on Z-transform techniques using time domain representation of voltage and current signals and the phase network model. This paper presents an accurate algorithm to estimate the location of a transmission line fault. © 1992
Elsevier Sequoia. All rights reserved
124
The method uses the fundamental frequency voltages and currents measured at one line terminal and takes into consideration the following factors: (1) remote-end current infeed; (2) exact transmission line representation (distributed parameter representation) is considitiveeredcurrents;in the computation to account for capac(3) prefault and postfault line currents and voltages are measured and stored and used for computation; (4) the contribution of current from relay location side to the total fault current at the fault location, assumed to be equal to an impedance ratio [2], is corrected; (5) no major approximations are carrried out during computation; (6) a modal analysis is used, permitting the inclusion of unbalanced system impedances caused by untransposed lines. Development
of the algorithm
The method to compute the distance to a fault point is based on the following equations in modal form [13, 14]: VF = Vs cosh(G/) - Z s I s sinh(G/) I'~F = ( V ~ / Z s ) s i n h ( G l )
- I'~ cosh(G/)
where l is the distance to the fault point from the sending end of the line; Vs is the modal voltage at the sending end (postfault value), Is the modal current at the sending end (postfault value), VF the modal voltage at the fault point (postfault value), Ir the modal fault current (postfault value), V~ the superimposed modal voltage at the sending end (modal voltage difference between prefault and postfault), and I~ the superimposed modal current at the sending end (modal current difference between prefault and postfault), and I~r the modal fault current feeding the fault from the sending end side; Zs is the modal surge impedance, and Z the transmission line impedance per unit length; G - - T + j P is the propagation constant. Referring to Fig. 1, for the postfault condition, the modal voltages and currents can be expressed as V~ = Vs cosh(G/) - I s Z s sinh(G/) and the superimposed components of the modal voltages and currents can be expressed as
I~ ~ ) JS-end -(a) ISF IRF
/L'Y'x ~
~)t~ l~ (b)' ~v~ [_~
IR
I~F r ~
t
L_
r~j~_ v~
-I
(c) Fig. 1. S i n g l e - l i n e d i a g r a m of t h e p o w e r system: (a) p r e f a u l t c o n d i t i o n ; (b) p o s t f a u l t c o n d i t i o n ( G F = f a u l t c o n d u c t a n c e ) ; (c) s u p e r i m p o s e d c o m p o n e n t of f a u l t c u r r e n t .
I'~r = ( V s / Z s ) sinh(G/) - I~ cosh(G/)
In matrix form we can write VF = [ cosh(G/)] Vs - [ sinh(G/)]Is
(1)
Vs~- = [sinh(G/)] Ys V~ - [ c o s h ( G l ) ] I ~
(2)
where
vF'= IVy,, v~2, VF:J [cosh(Gl)] = diag[cosh(G1 l), cosh(G2l), cosh(G3l)] VS t =
[ V s 1 , VS2,
Vs3]
[sinh(Gl)] = diag[sinh(Gll), sinh(G2l), sinh(G,l)] Zs -- diag[Zsl, Zs2, Zs3] Ys = diag[Ys,, Ys2, Ys3] rt
ISF tp ~
Is
~
it
tt
rt
[IsF~, ISF2, ISF3] rr
tt
it
[Is~, Is2, Is3]
The voltage at the fault location, V~,, is related to the fault current I~ by the fault constraint matrix G~ (where the superscript p indicates phasor values) as follows: I~
p p = GFV v
The fault constraint matrices for different types of faults are indicated in Appendix A, whereas the distribution of the sequence components of currents at the fault location is indicated in Appendix B. Substituting eqn. (B-l) into eqns. (1) and (2), we get [sinh(G/)] Ys V~ - [ cosh(G/)]I~ = k GF {[ cosh(G/)] Vs - [ sinh(G/)] Zsls }
125
[cosh(G/)] {[tanh(G/)] Ys V~ - UI~} = kG~[cosh(Gl)]
{UVs - [ t a n h ( G l ) ] Z s I s }
EAD MEASUREDVALUES1
r
OF PHASE VOLTAGES AND|
k {UVs - [tanh(V/)] Zsls}
CURRENTS
= [ G~ - ~]~{[ tanh(G l )] Ys V~ - UI"ss~
CALCULATE
Selecting one of the three modes (e.g. mode 1), and letting F
[GF-1] t
=
I
PHASEI
ISUPERIHPOSED ~OLTAGES AND CURRENTSJ
CALCULATE MODAL ALUES OF VOLTAGES ND CL~RENT$
i
FAULTY PHASE1
I
, SELECTION
j
OR ES ONDING
1
then CALCULATE [ G ~ ] - I
k[Vsl - Zs, tanh(V, l) Ira] [tanh(V~ l) Ys~ Vs~ . . . -. Ira= [f~152 f,3] | t a n h ( V j ) Ys2 V~2 - I~2 k t a n h ( G j ) Ys3 V~3 - I~3
IEC,]'CG~]~EC~]
i ALOOLA.E
FAULT LOCAT,N~
CALD :TION J
k[Vsl - Zs~ tanh(G~/) Is1] = f~ [tanh(G~/) Ysl V ~ - I ~ ]
,SPLA. PRI.TO0
+ f~2[tanh(G2/) Ys2 V~2 - I~2] + 53 [tanh(G3/) Ys2 V~3 - I~3]
(3)
Equation (3) is a complex equation. It can be split into two separate equations which can be solved to get the two real u n k n o w n s k and l. Eliminating k, we get one equation in l, the distance to the fault location from the sending end, in the form f[tanh(T1 l), tan(P,/), tanh(T2/), tan(P2/), tanh(T3/), tan(P3/)] = 0 This equation can be solved by the N e w t o n Raphson m e t h o d for successive substitutions, w h e r e an initial approximation is assumed for l, t h a t is, lo, and an iterative process is carried out until the required c o n v e r g e n c e is achieved. This iterative process is governed by
Fig. 2. Flowchart of the fault location algorithm.
TABLE 1. Basic transmission line data No. of circuits No. of conductors per phase No. of earth wires Conductor position symmetry Conductor resistivity (~ m) Earth wire resistivity (~) Conductor strand diameter (cm) Earth wire strand diameter (cm) Outer diameter of earth wire (cm) No. of effective strands in phase conductor No. of effective strands in earth wire Earth resistivity (~ m)
2 3 2 None 3.2 x 10 -s 2.69 x 10 -s 0.32 0.32 2.86 54 54 20
lh +, = lk -- f ( l k ) / f ' ( l k )
The main problem now lies in computing the matrix F. The flowchart of the fault location algorithm is given in Fig. 2.
Test results
The sample power system selected for the studies is a 500 kV, 236 km line section. The basic
transmission line data are given in Table 1. Two equivalent power systems were c o n n e c t e d to buses R and S. Three types of fault were considered (phase to ground, phase to phase, and double phase to ground). Faults were applied at different locations on the line u n d e r different operating conditions. The voltage and c u r r e n t phasors were c a l c u l a t e d by the program, t h e n the equivalent modal values were computed. Mode 2 was used in the study. These data were used by
126 TABLE 2. Results of the study Fault type
Assumed fault location (p.u.)
Calculated fault location (p.u.)
System condition
Phase to earth Phase to phase Double phase to earth
0.6 0.6 0.6
0.6 0.6 0.6006
R F 50
Phase to phase Phase to phase Double phase to earth
0.7 0.7 0.7
0.7 0.7 0.7005
Light loading R~. = l0 Power flow S to R
Phase to earth Phase to phase Double phase to earth
0.9 0.9 0.9
0.9 0.9 0.8999
Light loading Rv = 20 ~ Power flow R to S
Phase to earth Phase to phase Double phase to earth
0.2 0.2 0.2
0.2 0.2 0.1998
Heavy loading RrPower flow S to R
Unloaded
the fault locating algorithm to e v a l u a t e the dist a n c e to the fault from the sending end. Table 2 summarizes the results of this study.
C
RF
L-G fault
L-L fau[t
3-L fault
Conclusion This p a p e r has presented an a c c u r a t e algorithm for locating transmission line faults. The algorithm has been tested for a v a r i e t y of simulated fault conditions, including double-circuit m u t u a l coupling effects, various remote source equivalent conditions, various prefault steadystate conditions, and in the presence of appreciable fault resistance. These simulation studies indicate t h a t the proposed method is a c c u r a t e for all types of s h u n t fault at different locations u n d e r different o p e r a t i n g conditions. The maximum error was found to be 0.1%.
Appendix A Fault
~R F Z-L-G
constraint
fault
3
matrices
0 0
-1 -1
2 -1
(d) for a double line to g r o u n d fault
0 0
R F+ -Rg
Rg
-Rg
R F + Rg
(e) for a three-phase to ground fault p 1 GF =
=
0
1 [oo
(b) for a line to line fault G~
fault
(c) for a three-phase fault
G~. - R~ 2 + 2RFRg G~ = yg
3-L-G
Fig. A-1. Types of fault. Rg is the arc resistance from phase to tower plus the effective tower footing resistance for a single line to ground fault, and the effective tower footing resistance for other types of ground fault. R F is the arc resistance between faulted phases.
v-
Referring to Fig. A-l, the fault c o n s t r a i n t matrix G~ is given by (a) for a line to g r o u n d fault
Rg
o
-1
-
Y0 + 2yF
YO -- Y v
Ye -- Y v
7
Yo-Yv
yo+2yv
Yo-YF
Yo - YF
Yo - Y~"
Yo + 2 y v
A
]
127
where
aFSFb=15Fbo ÷IsFb1*15Fb2
al'SFa_~'SFaodSFal ÷IsFa2
Yo = 1/(RF + 3Rg)
b
The phase q u a n t i t i e s V p a n d I p are r e l a t e d to the modal q u a n t i t i e s V and I by the eigenvector matrices C~ a n d C2:
I lFa=IF°+Ipi+IF2
[~
L-L fault
L-G fault ar~Fa=l~Fao+rsFat÷I~Fa2
V p = Cl V
IFb=IFo IFI+IF2
a [SFb=15Fbo+[.~Fbl+[~'b2
c
I p = C2I
[Fa IFao+
IFal+ IFa2
Since we have p
3-L fault
L-L-G
fault
p
I p = $1GF VF a ISFa= I~Fao÷~Fal ÷I~Fa 2,
where S: is a real q u a n t i t y representing the fault resistance, yg $1 = yF/2 [yF/3
I
for a line to g r o u n d fault for a line to line fault for a three-phase fault
For a double line to g r o u n d fault, S:
=
c IFa =IFao*
lira I+
IFa2 3-L-G
fault
Fig. B-1. D i s t r i b u t i o n of s e q u e n c e c o m p o n e n t s of c u r r e n t s a t t h e fault location.
1/(RF 2 + 2RF2Rg)
X£~IIF = (X~o + I'~, + I ~ ) 1 ( I ~ o + I~, + z ~ ) G~ = RF
1 0
+ Rg
1 -1
----- [ k o I F 0 -'[- k l ( I F 1
-
In this paper Rg is t a k e n to be equal to RF. For a three-phase to g r o u n d fault,
"~- Z ~ ) ] l I ~
= k, + (ko -- kl)IFoIIF (a) For a single line to g r o u n d fault, IF = 3IF0
and
$ 1 : WF
I~F./IF = k + (ko - k:)13 = kLo
G p _ Yo - YF 3Yr
1 1
+
1 0
(b) For a line to line fault, IFo = 0 I~rbllrb
Appendix B Distribution of sequence components of currents at fault location Referring to Fig. B-l, the phase r e l a t i o n between the f a u l t c u r r e n t IF and the fault c u r r e n t from the sending end is governed by the c u r r e n t d i s t r i b u t i o n factors ko. ki, and k2 of the sequence components:
k0
=
it r / SF0/IF0
k, -- ISF1/IF:
k2 = I~F2/IF2 We can assume to a high degree of a c c u r a c y t h a t k: = k2, so t h a t
= k l -'- k L L
(c) For a three-phase fault, IF0 = 0 and I~F./IFa = kl -- k3L (d) For double line to g r o u n d and three-phase to g r o u n d faults, I~Fa/IF. = k: + (ko - k:)IFo./IF. I~Fb/Ivb
= k I + (ko -
kl)IFoblIFb
I~Fc/IFc = k~ + (ko - k~)IFoc/IFc But IFo. = 1Fob =/Foe = I~o
128
and
References
tt
t!
J!
/SFOa = ISFOb = /SFOc =
kolFo
Let (ko - k, )I;~o/ko = Co
then
ISFb I~F~.
Co-- klJ/~u] Co EIF(-
or
In the case of double line to ground and threephase line to ground faults, k~ is assumed to be equal to ko, that is, Co = 0. Hence, Is~ can be represented by a real fraction, kF, of the total fault current I~ (e.g. kLc, kLL , hal , k2LG, kaLG) , that is, the angle of the fault current I~ and the fault current from the sending end, I~F, are assumed to be equal. In phasor form IsP~ = kFI~ Is~ = & k~G~ V~
In modal form I~F = &kFC~-~G~C~ VF = kGvVv
where GF = C2 1G~.C1 = CltGP.C1
(B-l)
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