Double-Circuit Transmission-Line Fault Location With the Availability of Limited Voltage Measurements

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 1, JANUARY 2012 325

Double-Circuit Transmission-Line FaultLocation With the Availability of

Limited Voltage MeasurementsNing Kang, Member, IEEE, and Yuan Liao, Senior Member, IEEE

Abstract—This paper presents a new approach for locatingshort-circuit faults on a double-circuit transmission line. Variousalgorithms have been proposed previously that usually requiremeasurements recorded from one or two buses of the faultedline. However, such measurements may not always be available insome scenarios, rendering inapplicability of existing methods. Tocomplement existing methods, this paper proposes a novel, generalfault-location method by harnessing voltage measurements at oneor more buses, which may not be taken from the faulted line. Thebus impedance matrix of each sequence network with the additionof a fictitious bus at the fault point can be derived as a function ofthe fault location. The fault location can then be obtained basedon the bus impedance matrix and voltage measurements. Thedistributed parameter line model is utilized. The network data areassumed to be available so that the bus impedance matrix can beconstructed. When multiple voltage measurements are available,an optimal estimator capable of identifying bad measurement datais also proposed for enhanced fault location. Since field data arenot available at this time, simulated data are utilized for evaluationstudies, and quite accurate results have been achieved.

Index Terms—Bus impedance matrix, distributed parameterline model, double-circuit transmission line, fault location, optimalestimator.

I. INTRODUCTION

D OUBLE-CIRCUIT transmission lines, also knownas parallel lines, have been adopted more in modern

power systems for improved reliability and security of energytransmission, although double-circuit lines may impose moreprotection challenges than single-circuit lines. As is known,following the occurrence of a fault, it is important to promptlyand accurately locate the fault and repair the faulted componentto reduce outage time and loss of revenue [1], [2].

A lot of fault-location algorithms that are focused on single-circuit lines have been developed in the past several decades[3]–[7]. On the other hand, diverse fault-location techniquesfor double-circuit lines have been studied extensively as well.The authors of [8] propose an algorithm utilizing one-terminal

Manuscript received May 20, 2011; revised August 01, 2011; acceptedSeptember 08, 2011. Date of publication October 18, 2011; date of currentversion December 23, 2011. This work was supported by the National ScienceFoundation under Grant ECCS-0801367. Paper no. TPWRD-00425-2011.

N. Kang is with ABB Corporate Research Center, Raleigh, NC 27606 USA(e-mail: [email protected]).

Y. Liao is with the Department of Electrical and Computer Engineering, Uni-versity of Kentucky, Lexington, KY 40506 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TPWRD.2011.2168547

voltage and current data, with the estimation accuracy being in-fluenced by the fault resistance. Eriksson et al. employ phasevoltages and currents from the near end of the faulted sectionas input signals and fully compensate the error introduced bythe fault resistance [9]. Based on the assumption that the lineis homogeneous, [10] makes use of modal transformation andthe local terminal voltage and current to locate the fault. Sixvoltage equations are constructed around the parallel loops forpositive-, negative- and zero-sequence networks in [11], fromwhich the fault location is solved. By building voltage equa-tions along parallel loops, the authors in [12] address fault loca-tion for the nonearth fault. Reference [13] constructs the voltageequations but does not include the current phasors of the adja-cent sound line local end information based on the considera-tion that in some practical systems, these current phasors arenot available. Izykowski et al. [14] formulate the generalizedmodel for fault loops by using local-end voltages and completecurrents from healthy and faulted double-circuit lines. Using thetechnique similar to [11]–[14], the authors of [15] have devel-oped a fault-location algorithm applicable to untransposed lines,which utilizes the lumped line model ignoring the shunt capac-itance. The common characteristic of [11]–[15] is that they areall one-end algorithms and independent of the fault resistance.

A two-terminal method by using unsynchronized voltage andcurrent phasors based on the distributed parameter model isdiscussed by Johns et al. [16]. The authors in [17] have pro-posed an iterative approach to improve the accuracy of the faultdistance estimate and it does not require synchronization ofmeasurements. Reference [18] relates the synchronized voltageand current phasors of the sending end and receiving end with

parameters, from which the fault location is derived.In [19], based on the differential component net decomposedfrom the original net, two voltage distributions along the lineare calculated from the unsynchronized two-terminal currents.The fault location is determined based on the fact that these twovoltage distributions have the least difference at the fault point.Chen et al. [20] have proposed a new protection scheme by usingsynchronized phasors at both terminals of the line for transposedand untransposed parallel lines. By decoupling the parallel linesusing eigenvalue theory, both the fault detection and location in-dices have been obtained. Based on a three-terminal fault loca-tion algorithm and an equivalent conversion from an n-terminalsystem to a three-terminal system, the method proposed in [21]utilizes unsynchronized current phasors of all terminals to locatethe fault, neglecting shunt capacitances. Funabashi et al. [22]present two multiterminal lumped parameter model-based algo-rithms: “impedance calculation” and “current diversion ratio.”

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326 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 1, JANUARY 2012

Synchronization for all of the terminal voltages and currentsis needed. An ANN-based approach has been proposed in [23]with the ability to perform fault classification and fault location.

In general, existing algorithms require voltages and currentsfrom one or two terminals of the faulted line, or from all termi-nals of a multiterminal line. For the scenario where only severalmeasurements, which may be far away from the faulted line, areavailable, those methods are not suitable anymore. For example,the utility companies only install a limited number of recordingdevices to reduce costs. Moreover, the recording devices maynot always be triggered to record a fault event.

To fill the gap, a possible method based on the bus impedancematrix technique is proposed in [24]. The distinctive character-istic of this method is that it only demands voltage measure-ments from one or two buses, which may be distant from thefaulted line. The method is based on the lumped parameter linemodel without considering the shunt capacitance. To further in-crease accuracy for long transmission lines, this paper aims atdeveloping fault-location algorithms based on the distributedparameter line model, which fully takes the shunt capacitanceof the line into consideration. Voltage measurements are em-ployed and no current measurements are required. Although thederivation assumes that the network is transposed, the impact ofuntransposition of lines is studied.

In addition, to make the most of multiple measurements, anoptimal estimator for the fault location is proposed that is ca-pable of detecting and identifying bad measurements and, thus,further enhancing fault-location accuracy.

The remainder of this paper is outlined as follows. Section IIpresents the fault-location basis, based on which the fault-loca-tion method is detailed in Section III. An optimal estimator thatis able to detect and identify bad measurements is delineated inSection IV. Evaluation studies are reported in Section V, fol-lowed by the conclusion.

II. FAULT-LOCATION BASIS

The basic methodology of the proposed fault-locationmethod is described as follows. Two fictitious buses are addedto the network. One is at the assumed fault point. The other isat the point corresponding to the assumed fault point, but onthe healthy branch of the double-circuit line. Then, the drivingpoint impedance of the fault bus and the transfer impedancesbetween this bus and other buses are revealed as functions ofthe unknown fault distance. According to the definition of thebus impedance matrix, the voltage changes during the fault atany bus can be formulated with respect to the correspondingtransfer impedance and fault current. In conjunction withboundary conditions of different fault types, the fault locationcan be obtained.

In this section, our objective is to decouple the original net-work into three independent sequence component networks andconstruct the bus impedance matrix with additional buses foreach sequence network separately with respect to the unknownfault distance.

First of all, the notations used in this work are summarized asfollows:

total number of buses of the prefault network;

buses of the faulted section;

fictitious bus and ;

fictitious bus representing the fault point and;

unknown per-unit fault distance from bus ;

length of the line between buses and

symmetrical component index; , 1, 2for zero-, positive- and negative-sequence,respectively, inserted in parentheses as asuperscript throughout the paper;

sequence total equivalent self-series impedanceof the branch between buses and ; incase of a double-circuit line sharing bothterminals and , an extra subscript is usedto distinguish the first and second parallel lines(i.e., ;

zero-sequence total equivalent mutual-seriesimpedance of the branches of the double-circuitline between buses and ;

th sequence total equivalent self-shuntadmittance of the branch between buses and

; in case of a double-circuit line sharing bothterminals and , an extra subscript is usedto distinguish the first and second parallel lines(i.e., );

zero-sequence total equivalent mutual-shuntadmittance of the branches of the double-circuitline between buses and ;

zero-sequence total equivalent mutual-seriesimpedance and mutual-shunt admittancebetween branches and , respectively;

zero-sequence total equivalent mutual-seriesimpedance and mutual-shunt admittancebetween branches and , respectively;

bus impedance matrix of the prefaultsequence network; it has a size of by , whoseelement on the th row and th column isdenoted as ;

bus impedance matrix of the th sequencenetwork with the addition of fictitious busesand ; it has a size of by , whoseelement on the th row and th column isdenoted by .

The construction of bus impedance matrix with addition ofthe fault bus for the zero-sequence network is first considered.The prefault zero-sequence network of a sample power systemis shown in Fig. 1, whose bus impedance matrix can bereadily developed, with detailed steps being referred to in [25,pp. 299–305]. Fig. 2 depicts the zero-sequence network with theaddition of two fictitious buses and .

KANG AND LIAO: DOUBLE-CIRCUIT TRANSMISSION-LINE FAULT LOCATION 327

Fig. 1. Prefault zero-sequence network.

Fig. 2. Zero-sequence network with two additional fictitious buses.

Note that the studied parallel lines in our paper have identicalparameters, i.e., , , and

. The parameters in Fig. 2 in terms of are asfollows [26]:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

where

(9)

(10)

(11)

(12)

and

zero-sequence per-unit length self-series impedanceof the line between buses and ;

zero-sequence per-unit length self-shunt admittance(S/mile) of the line between buses and ;

zero-sequence per unit length mutual-seriesimpedance between the two lines of buses

and ;

zero-sequence per-unit length mutual-shuntadmittance (S/mile) between the two lines of buses

and .

To formulate , suppose there is only one current sourceinjected into a single bus , then the resultingvoltages at bus will be the same for the net-works shown in Figs. 1 and 2. According to the definition ofthe bus impedance matrix, it is obtained that ,

. We have

.... . .

......

. . ....

......

.... . .

......

. . ....

......

(13), , and are derived with respect to net-

work parameters as in (14) and (15) with the detail referencedto Appendix A.

(14)

(15)

where , , , and are formulated with , , , ,, , and ; and in (15) can be obtained

by letting and in (14).Substituting (1)–(8) into (14) and (15) leads to the formula-

tions of and with respect to andknown network parameters.


The construction of the bus impedance matrix with theaddition of the fault bus for the positive-sequence networkcan be obtained similarly, except there is no mutual couplingbetween the two circuits. Without showing details, it has beenobtained, as shown in (16) and (17), at the bottom of the page,that where

(18)

(19)

and

positive-sequence per-unit length self-seriesimpedance of the line between buses and;

positive-sequence per-unit length self-shuntadmittance (S/mile) of the line between buses and .

It is assumed that the parameters are the same for positive-and negative-sequence networks; thus, we have .

III. PROPOSED FAULT-LOCATION METHOD

In this section, fault-location algorithms employing voltagemeasurements from one bus or two buses are derived. At bus

, the following formulas hold:

(20)

(21)

(22)

where

prefault positive-sequence voltage at bus;

positive-, negative-, and zero-sequencevoltage at bus during fault, respectively;

positive-, negative-, and zero-sequencefault current at the fault point, respectively.

Note that all of the sequence voltages and currents are forphase A.

Equations (20)–(22) demonstrate that the sequence voltagechange during the fault at any bus can be expressed with re-spect to the corresponding transfer impedance and sequencefault current.

A. Two-Bus Fault-Location Algorithm

1) Fault Location With Synchronized Measurements FromTwo Buses: Suppose the voltage measurements at buses and

are available. For bus , similar to (20),the following formula exists:

(23)

Eliminating from (20) and (23) results in

(24)

Defining and substituting(16) into (24) gives

(25)

(16)

(17)


Fault location is obtained by simplifying (25) based on theidentity

(26)

The fault-location formula in (26) is applicable only if a pathexists, which passes through the faulted line and does not passany bus more than once, between buses and [24]. Otherwise,the ratio of voltage changes at these two buses will be constantand independent of the fault-location variable. Since most of thepower network is interconnected, most combinations are able toproduce the fault-location estimate.

Negative- or zero-sequence voltage measurements, where ap-plicable, can also be employed for fault location. However, pos-itive-sequence voltages are preferred due to the fact that nofault-type classification is needed.

2) Fault Location With Unsynchronized Measurements FromTwo Buses: Taking the absolute value of (25) leads to

(27)

The Newton–Raphson approach can be utilized here to itera-tively solve for the unknown fault location . An initial value of0.5 p.u. is adopted for the iteration process and the convergencecriterion is that the update for the fault location becomes smallerthan 1.0e-7. As will be illustrated in the evaluation studies, ofall the cases we have tested, all of them have converged within10 iterations.

B. One-Bus Fault-Location Algorithms

For the scenario where measurements are only available froma single bus, we have developed one-bus fault-location algo-rithms for different types of fault. Supposedly, the voltage mea-surements are from a single bus .

1) LG Fault: For a phase-A-to-ground fault, the boundarycondition exists. Eliminating andfrom (21) and (22) yields

(28)

Replacing the transfer impedance terms in (28) by (14) and(16), a nonlinear equation involving one unknown variablecan be formulated, which can be separated into real and imag-inary part to formulate two real equations. To solve it, leastsquares method can be utilized. An initial value of 0.5 p.u. for

can be adopted.2) LLG Fault: For the phase-B-to-C-to-ground fault, the fol-

lowing condition is satisfied:

(29)

Using (21) and (22), (29) becomes

(30)

By employing (14), (15), (16), and (17), we can formulatea nonlinear equation with two unknowns and from(30), which can be separated into two real equations. TheNewton–Raphson approach can be adopted to solve for the twounknown variables. We have chosen [0.5; 0.01] p.u. for and

, respectively, as initial values. The convergence criterion isthat the variable updates become smaller than 1.0e-7 for bothvariables. Our evaluation studies show that within 10 iterations,the algorithm can reach convergence.

3) LL Fault: For the phase-B-to-C fault, the followingboundary conditions hold:

(31)

(32)

Substituting (21) and (32) into (31) results in

(33)

where denotes the prefault positive-sequence voltage atthe fault point and can be calculated as

(34)

where and are the prefault positive-sequence volt-ages at the two terminals of the faulted line and are assumed tobe known.

Using (16), (17), and (34), together with (33), will produce anonlinear equation with and as unknowns, the solution ofwhich is similar to (30).

4) LLL Fault: For the three-phase balanced fault, we have

(35)

Replacing in (35) with (20) gives rise to

(36)

A nonlinear formulation with two unknowns and canbe derived by substituting (16), (17), and (34) into (36), whichcan be solved similar to (30).

Fault-location formulas involving other phases can be de-duced similarly.

It should be pointed out that the proposed one-bus methodrequires the fault type to be known while the two-bus methodutilizing positive-sequence voltages does not. In addition, it isassumed that the faulted section can be decided based on relay


Fig. 3. Schematic diagram of the studied system.

operations; otherwise, a list of possible faulted sections can beattempted.

As for the one-bus method, note that for LLG, LL, and LLLfaults, multiple solutions may be obtained. A solution is de-fined as a pair of fault-location estimates and fault resistanceestimates. A solution could be a valid solution (and ) or an invalid solution ( or or

). An invalid solution can be easily identified and re-moved. In cases where two or more valid solutions arise, oneis the true solution and the others are erroneous solutions. Theidentification of erroneous solutions will be illustrated in casestudies.

IV. OPTIMAL FAULT-LOCATION ESTIMATION CONSIDERING

MEASUREMENT ERRORS

When synchronized voltage measurements at multiple busesare available, this section presents an optimal fault-location esti-mator with the ability to detect and identify bad measurements.

A. Proposed Optimal Estimator

Suppose the positive-sequence superimposed voltages causedby the fault at buses are available, which formthe following vector:

(37)

where is the vector transpose operator, and is the totalnumber of measurements set.

For any two sets of measurements from buses and, the following equation is yielded based on

(25):

(38)Define the unknown variables as

(39)

where

, variables to represent thepositive-sequence superimposed voltage causedby the fault (i.e., );

fault-location variable.

The combinations of any two sets of measurements out ofsets include , the total

number of which is . Here, represents the combi-nation calculation. For any possible combination, say , byemploying the defined variables, (38) can be written as

(40)


In total, we can have equations in the form of (40).Let us introduce and as the measurement vector

and function vector, respectively. is formed as shown

(41)

(42)

(43)

where and produce the magnitude and angle in radianceof the input argument, respectively.

is formulated as follows:

(44)

where Re(.) and Im(.) yield the real and imaginary part of theinput argument, respectively.

The measurement vector and function vector are related by

(45)

where is a vector representing measurement errors and de-pendent on the meter characteristic.

The optimal estimate of is obtained by minimizing thecost function defined as

(46)

where

(47)

And signifies the variance of measurement , indicatesthe expected value, and means a diagonal matrix con-sisting of the values contained in the square bracket.is the total number of measurements. A smaller value of in-dicates a more accurate meter.

Equation (46) can be solved iteratively [27]. During the thiteration, the unknown vector is updated as

(48)

(49)

(50)

where

iteration number starting from 0;

variable vector before and after the th iteration;

variable update during the th iteration.

To initiate the iteration process, we can choose 0.1 p.u. andfor the magnitude and angle of positive-sequence superim-

posed voltage, respectively, and 0.5 p.u. for the fault-locationestimate. When the variable update is smaller than the specifiedtolerance, the iteration process can be terminated. After isobtained, we can use (44) to compute the estimated values ofmeasurements.

B. Detection and Identification of Bad Measurements

To detect the presence of bad measurement data, the methodbased on the chi-square test, as illustrated in [27], can be uti-lized. In this method, the expected value of the cost functionis calculated first, which is equal to the number of degrees offreedom designated as . Then, the estimated value of the costfunction is obtained. If , then the presence of baddata is suspected with probability . The value of canbe calculated for a specific and based on chi-square distri-bution. If bad data exist, the measurement corresponding to thelargest standardized error will be identified as the bad data. Inour study, we choose to be 0.01, indicating a 99% confidencelevel on the detection [27].

V. EVALUATION STUDIES

This section presents the evaluation studies for the developedfault-location algorithms. The Electromagnetic Transients Pro-gram (EMTP) has been utilized to simulate the studied powersystem with fault conditions of different types, locations, andfault resistances [28]. The studied power system is a 27-bus,345-kV, 60-Hz transmission-line system, as shown in Fig. 3with the line length labeled in miles. The detailed parameters,including the transmission-line data, generator data, and loaddata are listed in Appendix B. The section between buses 9 and10 possesses the double-circuit line structure, and the fault oc-curs on one of the parallel lines, with the cross denoting thefault point. The length of the faulted line is 168.2 miles. Thedouble-circuit line is modeled in EMTP based on the distributedparameter line model. The diagram depicting different types offaults is referred to in [25, p. 478].

The estimation accuracy is evaluated by the percentage errorcalculated as

ErrorActual Location Estimated Location

Total Length of Faulted Line(51)

where the location of the fault is defined as the distance betweenthe fault point and bus 9.

A. Cases Without Bad Measurements

The developed fault-location algorithms are tested under var-ious fault conditions. In this section, the fault-location resultswithout bad measurements are presented.

1) Results Considering Transposed Lines: Our fault-locationmethod is based on the assumption that the lines are fully trans-posed. In this part, the results when the transmission lines areperfectly balanced are reported.

Table I shows the fault-location result produced by thetwo-bus algorithm. The first three columns represent the actualfault type, fault location, and fault resistance, respectively. Therest indicate the percentage errors of fault-location estimatesutilizing synchronized voltage measurements from two buses.Table II exhibits the percentage fault-location errors usingunsynchronized measurements from two buses. Various faultresistances have been utilized in the simulation studies, andrepresentative results are reported here.

In Tables I and II, positive-sequence voltage measurementsare used to carry out two-bus fault location. The fault-locationresults are quite satisfactory. It can be observed that quite close


TABLE ITWO-BUS SYNCHRONIZED FAULT-LOCATION RESULTS

fault-location estimates are produced by using synchronized andunsynchronized data. It must be noticed that there are restric-tions in choosing the two buses on account of the reasons ex-plained in Section III. For example, it is impossible to producea fault-location estimate by employing voltage measurementsfrom bus combinations, such as 4 and 5, 10 and 11, 11 and 22,etc.

Table III presents the one-bus fault location results for AG andBCG faults. Columns 4–7 display the percentage errors of fault-location estimate, employing the voltage measurements from asingle bus. It can be seen that the fault-location estimates arequite accurate.

For the BCG fault, under certain fault conditions, multiplevalid solutions might arise, where only one solution is true.The erroneous solutions can be identified by the followingmethod. We can calculate the voltages of the bus with measure-ments from all of the valid solutions by making use of the busimpedance matrix and compare them with the actual voltagemeasurements. The bus voltages computed from the erroneoussolution differ from the original measurements and, thus, theerroneous solution can be recognized.

Table IV conveys the one-bus fault location results for BC andABC faults. Columns 3 to the end list the estimated fault loca-

TABLE IITWO-BUS UNSYNCHRONIZED FAULT-LOCATION RESULTS

TABLE IIIONE-BUS FAULT-LOCATION RESULTS FOR AG AND BCG FAULTS

tion and fault resistance by utilizing voltage measurements froma single bus. The actual fault resistance is 1 (0.00084 p.u.) andthe base value of the impedance is 1190.25 .


TABLE IVONE-BUS FAULT-LOCATION RESULTS FOR BC AND ABC FAULTS

As observed from Table IV, the one-bus algorithms for BCand ABC faults are able to yield a quite accurate fault-locationestimate; however, under certain fault conditions, two valid so-lutions can be produced. Only one of them is the correct solu-tion and the erroneous one is indicated by “ .” For example,when the actual fault location of a BC fault is 0.5 p.u., usingthe voltages from bus 6, two valid solutions (0.50, 0.00088) p.u.and (0.83, 0.00295) p.u. are yielded with the first element rep-resenting the fault-location estimate and the second one repre-senting the fault resistance estimate. Here, the second solutionis erroneous, whose estimated fault location is followed by “ .”

When two valid solutions are yielded, our studies indicatethat it is not possible to tell which one is the true solution byutilizing only the voltage measurements at one bus. This is be-cause based on short-circuit analysis, both fault conditions cor-responding to the two solutions will yield the same voltage pha-sors as the measured ones. Hence, unless more measurementsare available, there may be more than one likely fault-loca-tion estimate. Adequacy of measurement data to pinpoint theunique true fault location for a given network may be deter-mined through an analysis called fault-location observabilityand meter placement, which will be performed in the future.

2) Results Considering Untransposed Lines: In this part, theimpact of untransposition of lines is studied. For the untrans-posed double-circuit line, we have averaged the relevant termsin the series impedance matrix to obtain the self impedance, themutual impedance of the phases within the same single circuit,and the mutual impedance between the phases from two singlecircuits. From these parameters, we can further calculate the se-quence parameters which will be utilized in the fault-locationalgorithm [28]. Shunt admittance is dealt with similarly.

Tables V and VI show the fault-location results using syn-chronized two-bus and one-bus algorithms for the AG fault, re-spectively. The faulted double-circuit line is untransposed in thesimulation model. Compared to the results in Tables I and III,we can observe that the estimates are generally not as accurateas transposed line cases but are still quite accurate.

3) Impact of Zero-Sequence Line Parameter Error: The ac-curacy of zero-sequence line parameters may be of concern dueto the uncertainty of earth resistivity. We have applied a cer-tain error to the zero-sequence impedance of the double-cir-cuit line between buses 9 and 10. The AG fault location re-sults with 3% error in zero-sequence impedances are reported

TABLE VTWO-BUS SYNCHRONIZED FAULT-LOCATION RESULTS

FOR UNTRANSPOSED LINES

TABLE VIONE-BUS AG FAULT-LOCATION RESULTS FOR UNTRANSPOSED LINES

TABLE VIIONE-BUS AG FAULT-LOCATION RESULTS WITH 3%

ERROR IN ZERO-SEQUENCE IMPEDANCES

in Table VII. The fault-location accuracy has decreased com-pared to Table III. However, the errors are still quite small. Notethat only the one-bus algorithm involving ground is affected byzero-sequence parameter errors, since two-bus methods will usepositive-sequence parameters.

B. Case With Bad Measurement

This case study illustrates how to detect and identify bad mea-surements with the proposed optimal estimator. A value of 1e-6is chosen as variance for the first measurements and 1e-4for the variance for the voltage measurements. In our studiedcase, the voltage measurements at buses 4, 6, 8, and 19 are uti-lized to obtain the fault location.

Suppose there is a BCG fault with the actual fault locationbeing 0.3 p.u. and the fault resistance as 50 . Suppose thatthere is an error of 50% in the superimposed positive-sequencevoltage at bus 4.

The optimal estimation result is shown in Table VIII. Thereare 20 equations and 9 state variables; therefore, we have11 and 24.73. The estimated value of cost function

is computed as 34.58, which is greater than . Thus,the presence of bad measurements is suspected. Following themethod outlined in Section IV-B, the normalized error vector


TABLE VIIIOPTIMAL ESTIMATES WITH BAD MEASUREMENT

TABLE IXOPTIMAL ESTIMATES WITH BAD MEASUREMENT REMOVED

is obtained and the largest value corresponds to the magni-tude of superimposed positive-sequence voltage at bus 4, whichis hence identified as a bad measurement.

After the bad measurement is removed, a new set of optimalestimates is calculated as shown in Table IX. In this scenario,the expected value of cost function is equal to 5 and15.09. The estimated value of cost function is 6e-4. Since

is much less than , all of the data are considered fairlyaccurate and the estimates are regarded as satisfactory. A com-parison of Table VIII indicates that the fault-location accuracyhas considerably improved.

VI. CONCLUSION

In this paper, novel one-bus and two-bus fault-location algo-rithms applicable to double-circuit transmission lines are devel-oped. The distinctive feature of the proposed method is that onlyvoltage measurements from one or two buses are needed whichmay be a distance away from the faulted section. The distributedparameter line model is utilized which fully takes the shunt ca-pacitance of long lines into account.

Simulation studies have shown that the proposed algorithmscan yield quite accurate estimates under various fault condi-tions. For the two-bus fault-location method, a unique fault-lo-cation estimate is produced by using synchronized and unsyn-chronized voltage measurements, and the fault-type classifica-tion is not required. For one-bus fault-location algorithms, thefault type is a prerequisite. For LG and LLG faults, a uniquefault-location estimate can be obtained. For LL and LLL faults,prefault measurements at the two terminals of the faulted lineare demanded and, in certain cases, two possible fault-locationestimates may be produced, both of which will be treated as alikely fault location.

When synchronized voltage measurements from multiple(more than three) buses are available, which may not nec-essarily be captured from the buses of the faulted line, anoptimal fault-location estimator is proposed that is capable ofidentifying bad measurement data and, thus, enhancing thefault-location estimate. This approach is preferred over the

one-bus and two-bus fault-location algorithms if more thanthree sets of measurements are available. If only two sets ofmeasurements are captured, the two-bus fault-location algo-rithm should be chosen. However, the one-bus fault-locationalgorithm still provides an effective solution for the scenariowhen measurements are only available from a single bus.

We should emphasize that the existing fault-location methodscertainly provide good estimates if measurements are availablefrom terminals of the faulted line. The proposed methods serveto complement existing approaches in cases where existingmethods are not applicable. In comparison to the authors’previous work [24], the contributions of this paper includefully considering shunt capacitances by using a distributed linemodel and presenting an optimal fault-location estimator forenhanced accuracy.

APPENDIX

A. Derivation of and

Referring to Fig. 2, let us inject a current source of 1 A to bus. Let denote the currents flowing from bus to and ,

respectively, and denote the currents flowing from bus toand , respectively. Making use of the bus impedance matrix

in (13), we obtain

(A1)

It follows from Fig. 2 that:

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

Based on (A2)–(A7), there are six unknownsand six equations, from which

can be solved as follows:

(A8)

where

(A9)

(A10)


TABLE XTRANSMISSION-LINE DATA

(A11)

Substituting (A1) into (A8) will result in (14).To derive , let us inject one current source of 1 A into

bus . Based on the bus impedance matrix in (13), the voltagesat buses , , and in Fig. 2 are

(A12)

From Fig. 2, the following equations hold:

(A13)

(A14)

(A15)

(A16)

TABLE XIGENERATOR DATA

TABLE XIILOAD DATA

(A17)

(A18)

With six unknown variables , solving(A13)–(A18) reaches the expression of as follows:

(A19)

where

(A20)

Substituting (A12) into (A19) yields (15).

B. Studied 27-Bus System Data

This section provides the model data of the studied 27-bussystem. The per-unit system is adopted with a base voltage of345 kV and base volt-ampere of 100 MVA. The transmission-line data, generator data, and load data are listed in Tables X–XII, respectively.

In Table X, the first two columns are the two bus numbers foreach branch. The per-unit length positive-sequence impedance,


zero-sequence impedance, positive-sequence susceptance andzero-sequence susceptance for each branch are listed.

The per-unit length zero-sequence mutual series impedanceof the parallel lines between bus 9 and 10 is

The per-unit length zero-sequence mutual-shunt susceptanceof the parallel lines between buses 9 and 10 is 0.0027 p.u.

In Table XI, the first column represents the bus number thatthe generator is connected to. Columns 2–3 show the positive-and zero-sequence source impedance.

In Table XII, the first column represents the bus number thatthe load is connected to. The second column exhibits the equiv-alent load impedance.

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Ning Kang (M’10) received the B.Sc. degree inelectrical engineering from Xi’an Jiaotong Univer-sity, Xi’an, China, in 2004 and the Ph.D. degree fromthe University of Kentucky, Lexington, in 2010.

Currently, she is a Senior R&D Engineer withABB Corporate Research Center, Raleigh, NC. Herresearch interests include protection, power-qualityanalysis, and large-scale resource schedulingoptimization.

Yuan Liao (S’98–M’00–SM’05) is an AssociateProfessor with the Department of Electrical andComputer Engineering, University of Kentucky,Lexington. He was an R&D Consulting Engineerand then Principal R&D Consulting Engineer withABB Corporate Research Center, Raleigh, NC. Hisresearch interests include protection, power-qualityanalysis, large-scale resource scheduling optimiza-tion, and network-management system/supervisorycontrol and data-acquisition system design.

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Double-Circuit Transmission-Line Fault Location With the Availability of Limited Voltage Measurements