Fault-location algorithm for multiphase power lines

A.O. IbeB.J. Cory

Indexing terms: Fault location, Power transmission and distribution, Mathematical techniques

Abstract: Improved mathematical models oftransmission lines are used to develop a fault loca-tion method which copes with high-frequencytransients. The telegraph equations used for a linemodel are solved by the method of characteristicsusing voltage and current samples as boundaryconditions taken at one end of the line within thefirst few milliseconds of fault inception. Estimatesof voltage and current profiles for the entire lengthof the transmission line during the fault areobtained. Criteria for fault location are based on aformulation involving these voltage and currentestimates. For fault location on multiphasesystems the concept of modal analysis is used inenhancing the fault-locating capability of thisalgorithm.

1 Introduction

There has been quite considerable research into faultlocation on power lines. Most of the more recent fault-location algorithms involve two main approaches. One isto inject an electrical pulse into the faulted line segmentand take recordings of subsequent reflected voltage andcurrent signals. The other approach is to record voltageand current at one or more points on an energisedsystem within the first few milliseconds after fault incep-tion. The recordings in both approaches form the initialdata for an estimation problem leading to fault location.

Many algorithms [1-5] employ power-frequencywaveform evaluation and then use iterative techniques tosolve a set of nonlinear equations containing fault dis-tance as a variable. Wiszniewski [6] identified the errorwith impedance methods of fault location as a phase shiftbetween the current measured at one end of the line andthat through the fault resistance. An algorithm that cor-rects this error was presented.

Kohlas [7] proposed the use of improved mathemati-cal models of the transmission line to develop a fault-location method which copes with high-frequencytransients. These models have been used by the authorsfor single-phase fault location in a companion paper [8]but in this paper, the concept is extended by the use ofmodal analysis to cover fault location in three-phasesystems.

Paper 4993C (PI 1, P9), received 27th May 1986Dr. Ibe is with the Department of Electrical Engineering, AnambraState University, PMB 01660, Enugu, Nigeria, and Dr. Cory is with theDepartment of Electrical Engineering, Imperial College of Science andTechnology, Exhibition Road, London, SW7 2BT, United Kingdom

2 Mathematical model and method of solution

The 3-phase transmission-line model is characterised bythe following set of hyperbolic partial differential equa-tions:

dx dt

dV dlC— + — = 0

dt dx

(1)

(2)

where, for a three-phase system, R, L and C are 3 x 3resistance, inductance, and capacitance matrices respec-tively; V and / are third-order column vectors represent-ing voltages and currents.

Eqns. 1 and 2 are solved in two stages: first thematrices are diagonalised by applying the concept ofmodal analysis [9, 10], then the resulting componentequations are solved by the method of characteristics asin the single-phase model presented in Reference 8.

3 Phase-to-modal transformation

Transformation matrices My and M2 are determinedfrom a solution of the eigenvalue problem on the matrixproduct LC [10]. Mx is a matrix whose elements are theeigenvectors of LC, and M2 is the transpose of theinverse of M1(M2 = (A/)t~

1). Mx and M2 are employed totransform the variables in eqns. 1 and 2 from phase tomodal domain as follows:

Let m and p superscripts denote modal and phase vari-ables respectively, then

V(m) = MllV{p) (3)

I^ = M2lI{p) (4)

1 ( (5)

(6)

(7)

Replacing R, L, C, V and / of eqns. 1 and 2 with theirmodal equivalents defined in eqns. 3-7 we obtain

d_

dxL"2j

la 0+••

0 / .dt

M.

2J -h-\(8)

and

,V2_

d_

dx.h.

= 0 (9)

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On expansion, eqns. 8 and 9 will yield three pairs of inde-pendent equations of the form

dx k dt = -rk

and

dvk dik

(10)

(11)

where k = 0, 1, 2.Eqns. 10 and 11 refer to a single mode and are similar

in all respects to the single transmission-line model inReference 8. A similar solution procedure is thereforeadopted (see Appendix 8). When the three pairs of equa-tions have been solved, phase voltages and currents areobtained by modal-to-phase transformation as given by

V abc —

'abc —

1*012

2*012

(12)

(13)

where again p and m refer to phase and modal quantitiesrespectively while Mx and M2 are transformationmatrices defined earlier. Hence, solving eqns. 10 and 11with the fault data (i.e. voltage and current recorded atone end of the line) as boundary conditions, the postfaultvoltage and current profiles can be obtained.

4 Criteria for fault location

As explained in Appendix 8, the algorithm uses thevoltage and current recorded at reference point x0 on theline, over time period T referred to as the data window(see Fig. 1), to compute voltage and current at discrete

t

Fig. 1 Computation of voltage and current from the initial data

points xl3 x2, . . . , Xi along the line. These subsequentcomputations are done over time periods (T — 2yx)where x is the distance from the reference point x0.

The time average functions Fu F2 and F3 in eqns.14-16 are designed to produce a picture of the variationof voltage and current following fault inception. Sincethese signals are either positive or negative depending onthe point on wave of fault, their squares are employed togive a reasonable indication. The time average functionsare:

x<T/2y (15)

F3(x) =l —

V\x, t) * P(x, t)\ dt

x<T/2y (16)

Ft gives the average of the squared voltage over thechosen data window at each discrete point x,. F2 pro-duces a similar picture for current and F3 gives theproduct of both current and voltage.

The effect of a fault on the voltage is a maximum atthe fault point (for a solid fault the voltage will be zero).This influence however gradually diminishes as distancefrom the fault increases. The Fx function therefore pro-duces a V-shaped curve with the minimum (turning)point indicating the fault point (see Fig. 4). From manytests, F3 exhibits a similar characteristic but F2 producesa continuously decreasing function with a point of inflec-tion corresponding to the fault position, requiring afurther test to confirm the fault position. The criterioninvolving Fx is preferred because it has been shown tohave the greatest sensitivity to the fault position.

In 3-phase systems V and / in eqns. 14-16 are prefer-ably those of the faulted phase. Considerable computa-tion time can be saved if this can be identified beforecalculation begins, otherwise Ft for each phase must becomputed. For further ease of computation, the factor 1/(T — 2yx) can be replaced by l/T since T has a typicalvalue of 5 ms and for a 100 km line yx has a maximumvalue of about 0.67 ms.

In a further development, the variation of the tangentto the time average functions was calculated and the faultposition was found to be indicated by the peak variationin the tangent to any of the functions in eqns. 14-16. Thislatter criterion is particularly useful for the F2 functionand for cases where the turning point in F t and F3 is notwell defined (e.g. for resistive faults and teed networks).The G (or tangent) functions are:

G2(x) =

G3(x) =

d'F.jx)

dx2

d2F2(x)

dx2

d2F3(x)

dx2

(17)

(18)

(19)

which show the variation in tangent to various points onthe F-function curves. The turning or inflectionary pointson the F-function curves, which should indicate the faultposition, are further amplified by the remarkable changein the tangent to the F-function curves at such points.The plot of the G function typically produces a scatterdiagram in which the maximum point uniquely corre-sponds to the fault position (see Fig. 4b).

5 Simulation examples

Single line-to-ground faults were simulated on two- andthree-terminal networks of a typical 400 kV three-phasesystem using a transient simulation program. The testsystems are shown in Figs. 2 and 3.

60km 90kmAA y^ X B

Fig. 2 400 kV 3-phase two-terminal network

, = 1.0%

~ IB

A data window of 5 ms was used for each simulation.In addition to data recording at the local terminal of athree-terminal network, recordings from any of the tworemote terminals may be necessary for confirmation ofthe faulted branch. Details of fault location in three-

44 IEE PROCEEDINGS, Vol. 134, Pt. C, No. 1, JANUARY 1987

terminal network have been presented in an earlier paper[8].

60 km 40km

50km

Fig. 3 400 kV 3-phase three-terminal networkXA = 0.29%; XB = 0.5%; XD = 0.67%

Further tests were performed using a laboratorymodel of a 132 kV line. A phase shifter included in thecircuit was used to vary the fault switching angle, i.e. thepoint-on-wave (POW) of fault. The magnitude and rateof rise of the wave travelling away from a fault reduces asthe POW approaches a zero crossing. This weakness inthe signal makes fault location by the F function difficultfor POW below 30° (see Fig. 11). The G function,however, is unaffected by the POW as shown in Fig. 12.This is because the G function only needs to detect theslight deviation in the shape of the F-function curve.

Table 1 : Results from simulation examples

System model

Two-terminalnetwork

Three-terminalnetwork

132kVlaboratory model

Fault position,km

60

90 (from A)50 (from D)

16.0919.3122.53

Position located,km

60.24

90.0650.10

15.8119.4022.27

Accuracy,%

0.4

0.070.2

1.740.471.15

The results of the tests are summarised in Table 1 andthe plots of the functions are reproduced in Figs. 4 to 12.

6 Conclusion

The fault-location algorithm presented in this paperdraws primarily from travelling-wave principles. The tele-graph equations were used as the mathematical model ofthe transmission line, and solved by the method of char-acteristics. The peak variation in the tangent to a time-average integral function was used as a criterion for faultlocation.

The complex matrix equations inherent in the mathe-matical model for three-phase systems were simplified byemploying the concept of modal analysis. Once themodal transformations are carefully handled the rest ofthe method is as straightforward as in the singletransmission-line case.

Location accuracies lie between about 0.1 and 2.0%. Apractical fault locator based on the method is being con-structed.

7 References

1 WESTLIN, S.E., and BUBENKO, J.A.: 'Newton-Raphson tech-nique applied to fault location problem'. Paper No. A76 334-3,IEEE PAS summer meeting, Portland, July 18-23 1976

2 WESTLIN, S.E., and BUBENKO, J.A.: 'An accurate method forfault location on electrical power lines'. IFAC symposium on Auto-matic Control and Protection of Electric Power Systems, Mel-bourne, Australia, 21-25 Feb., 1977, Institution of Engineers,Australia, National Conference Publication 77/1, pp. 261-266

3 TAKAGI, R., TAMAKOSHI, Y, BABA, J., UEMURA, K., andSAKAGUCHI, T.: 'A new algorithm of an accurate fault locationfor EHV/UHV transmission lines, part 1—Fourier transformationmethod', IEEE Trans., 1981, PAS-100, pp. 1316-1323

4 TAKAGI, T., TAMAKOSHI, Y., BABA, J., UEMURA, K., andSAKAGUCHI, T.: 'A new algorithm of an accurate fault locationfor EHV/UHV transmission lines, part II—Laplace transformmethod', ibid., 1982, PAS-101, pp. 564-573

80.00r 16r 60.24

20.00 40.00 60.00 80.00distance,km

a

100.00 120.00 20.00 40.00 60.00 80.00 100.00distance, km

b

120.00

Fig. 4 Criterion functions F, and Gt for a fault at 60 km

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 1, JANUARY 1987 45

21.50

20.70

19.90

^19.10u.c"o

c 18.30

0)enO£ 1 7.50o

1 6.70

15.90

15.10

3.2 h

2.8

2.4

2.0

1.2

20.00 40.00 60.00 80.00 100.00 120.00distance, km

a

0.8

0.4

,60.24

aa a

a _aa

20.00 40.00 60.00 80.00 100.00 120.00distance,km

b

Fig. 5 Criterion functions F2 and G2 for a fault at 60 km

160.00

140.00

120.00

3 100.00

c 80.00

> 60.00

40.00

20.00

0

320.00

280.00

° 240.00

- 160.00

o120.00

80.00

40.00 -a

a ~ 60.24

Q

a

0 20.00 40.00 60.00 80.00 100.00 120.00distance, km

a

Fig. 6 Criterion functions F3 and G3 for a fault at 60 km

46

20.00 40.00 60.00 80.00 100.00 120.00distance.km

b

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 1, JANUARY 1987

80.00

00 40.00 80.00 120.00 160.00 200.00 240.00

distance, kma

Fig. 7 Criterion functions Fx and Gt for a fault at 90 km {teed line)

320.00

280.00

240.00

200.00

-160.00c

§,1 20.00ca

80.00

40.00

a —90.06

a

D

°aDo

40.00 80.00 120.00 160.00 200.00distance, km

b

18.80

18.40

18.00

17.60

17.20

2£ 16.80o

I16.40

16.00

15.6040.00 80.00 120.00 160.00 200.00 240.00

distance, kma

Fig. 8 Criterion finctions F2 and G2 for a fault at 90 km {teed line)

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32

28

24

20

Z 16c

?12

a— 90.06

a a

40.00 80.00 120.00 160.00 200.00distance, km

b

47

160.00

140.00

120.00

100.00

oZ 80.00c3

0/

2 60.00

<n

J 40.00

20.00

0 40.00 80.00 120.00 160.00 200.00 240.00distance, km

a

Fig. 9 Criterion functions F3 and G3 for a fault at 90 km (teed line)

o

6 4 r

56

48

40

42

24

—90.06

40.00 80.00 120.00 160.00distance, km

b '

3.60

3.20

2.80

co~. 2.40

* 2.00od>

E

1.60

1.20

0.80I

Fig. 10

48

8.00 16.00 24.00distance, km

a

32.00

0.14

0.12

0.10

oc 0.08o

£ 0.06

co

0.04

0.02

40.00

Q— 18.68

Criterion functions Ft and Gl for a fault at 19.31 km

2.00 7.00 12.00 17.00 22.00 27.00distance, km

b

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5 RICHARDS, G.G., and TAN, O.T.: 'An accurate fault location esti-mator for transmission lines', ibid., 1982, PAS-101, pp. 945-950

6 WISZNIEWSKI, A.: 'Accurate fault impedance locating algorithm',IEE Proc. C, Gen. Trans. & Distrib., 1983, 130, (6), pp. 311-314

4.00 r

0.800.00 5.00 20.00 25.00 30.00

Fig. 11angle(i) POW = 0°

(ii) POW = 30°(iii) POW = 90°(iv) POW = 60°

0.16r

0.14

0.12

10.00 15.00distance, km

Sensitivity of the time-average function to fault switching

~ 0.10X

O

2 0.08uc

I 0.06

0.04

0X32

0.0C

Q-19.40

Du OD , O O O QO° D » D n n

7 K.OHLAS, J.: 'Estimation of fault location on power lines'. Pro-ceedings of the 3rd IFAC Symposium, the Hague/Delft, the Nether-lands, June 1973

8 IBE, A.O., and CORY, B.J.: 'A travelling wave-based fault locatorfor two- and three-terminal networks'. Proceedings of PowerIndustry Computer Applications (PICA) Conference, San Fransisco,California, USA, 6-10th May 1985

9 WEDEPOHL, L.M.: 'Application of matrix methods to the solutionof travelling-wave phenomena in polyphase systems', Proc. IEE,1963,110, (12), pp. 2200-2212

10 IBE, A.O.: 'Travelling wave-based fault location algorithm forpower systems'. Ph.D. Thesis, Imperial College of Science and tech-nology, London,1984

11 COLLATZ, L.: 'The numerical treatment of differential equations'.(Springer-Verlag, 1960)

8 Appendix: Mathematical model for asingle-phase system

The equations for single-phase systems are:

dv , di

dx

(20)

(21)

v and i are voltage and current at a point on the line; r, /and c are respectively the resistance, inductance andcapacitance per unit length of the line.

The above equations are solved by the method ofcharacteristics, as follows:

Let u = —cv

rj = re

Eqns. 20 and 21 then become

du , di

(22)

(23)

— - — = 0 (24)

dt dx v '

The characteristics for the set of eqns. 23 and 24 are:

dt= ±y dx (25)

which are straight lines similar to those in Fig. 13.

Fig. 13 Characteristics for a set of first-order equations {eqns. 23 and24)

Along the characteristics, u and i are related by thefollowing differential equations [10]:

du di rji

0.00 5.00 10.00 15.00 20.00 25.00 30.00distance, km —A- —

Fig. 12 CriterionfunctionGiforafaultat 19.31 km (POW = 0°) dr dr

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— r]i

(26)

(27)

49

where r and s are arc lengths along the two character-istics (Fig. 13). Eqns. 26 and 27 are solved numerically[8].

Let the distance and time axes be discretised by pointsxf and tj respectively as in Fig. 14 such that

A i + 1 xi — £ i J t V^°/

fy+i - tj = Ax (28)

arithmetic means, we obtain

Ax Ax

and

(31)

t j

Ax Ax

t - v x

Rearranging eqns. 31 and 32 we obtain

t + yx = con stant

Ax) .

Fig. 14 Illustration of the numerical solution techniques

where

At = y Ax

4y

(2y - rj Ax) .

4y

and

(2y + rj Ax)

(32)

(33)

, - l , j+l

(30) (2y - r\ Ax) .(34)

Let the point (x{, tj) be denoted by (i, j) so that u(x,-, tj)and i(Xi, tj) become u(i, j) and i(i, j) respectively.

If the differential quotients of eqns. 26 and 27 arereplaced by central differences and the function values by

From eqns. 33 and 34, itj and M0 can be computed recur-sively from the initial data ioj and uOj at each discretepoint x, at intervals of Ax.

50 1EE PROCEEDINGS, Vol. 134, Pt. C, No. I, JANUARY 1987