on transmission lines revised 5th june 2017 yu liu dynamic...

IET Generation, Transmission & Distribution

Research Article

Dynamic state estimation-based fault locatingon transmission lines

ISSN 1751-8687Received on 10th March 2017Revised 5th June 2017Accepted on 23rd June 2017E-First on 3rd November 2017doi: 10.1049/iet-gtd.2017.0371www.ietdl.org

Yu Liu1 , A.P. Sakis Meliopoulos1, Zhenyu Tan1, Liangyi Sun1, Rui Fan1

1Department of Electrical and Computer Engineering, Georgia Tech, Atlanta, GA, USA E-mail: [email protected]

Abstract: Accurate fault locating minimises labour and outage time. Efforts to increase the accuracy of fault locating methodsare on-going. In this study, a dynamic state estimation-based fault locating (EBFL) method is proposed. Best implementationrequires GPS synchronised sample value measurements at both ends of the line. The dynamic state estimator operates on thesampled values using a detailed dynamic model of the line, which includes the fault location as a state. Specifically, the dynamicline model is represented as a multi-section transmission line model, inclusive of phase conductor type and size, shield wire(s)type and size, and tower ground impedances, integrated with the fault model. This study presents extensive numericalexperiments demonstrating that the method has higher accuracy than traditional fault locating methods for different fault types,locations and impedances. Additionally, the method works for both two-terminal and three-terminal transmission lines.

1 IntroductionAccurate fault locating on transmission lines is valuable foroperators and utility crews, since the amount of time spentsearching for the fault can be minimised. Thus, the outage time,labour and costs can be reduced. Legacy fault locating techniquescan be mainly classified into two groups: fundamental frequencyphasor-based methods and travelling wave-based methods. A briefreview of the two classes of methods is provided for two-terminallines with commentary on the challenges when applied to three-terminal lines.

1.1 Fundamental frequency phasor-based methods

These methods use voltage and current phasors of fundamentalfrequency to calculate the impedance and afterwards the distancebetween the fault location and the terminals of the line [1–3]. Theycan be further classified into single-ended and dual-endedalgorithms. (i) Single-ended algorithms calculate the distance to thefault from voltage and current phasors at one end. The mainadvantage is that it does not require communication channels fromthe remote end. The accuracy of these methods largely depends onthe fault type and fault impedance, as well as the line modelaccuracy, presence of mutually coupled circuits, and groundingparameters of the line [4]. (ii) Dual-ended algorithms use voltageand current phasors at both ends of the line. They can be furthersubdivided into methods that use GPS synchronised measurementsor non-synchronised measurements. These methods are, in general,more accurate than single-ended methods as the influence of faultimpedance and mutually coupled circuits is mitigated [5, 6]. Thealgorithms are also characterised on the basis of the model theyuse: sequence model or phase-based asymmetric model. Anothercharacteristic is the data used (three-phase voltages only, three-phase voltages and currents) [7–9]. Disadvantages of fundamentalfrequency phasor-based single-ended or dual-ended methods are asfollows. First, since the fundamental frequency phasors filter outtransients, the phasors may contain inaccuracies which propagateto the fault locating results if the system is experiencing transients.Second, most of these methods use a sequence line model(positive, negative and zero sequence) which is an approximatemodel resulting in increased fault locating error. Third, thegrounding of the line is typically neglected (i.e. consideredeffectively grounded) and this can potentially generate large errorsdepending on the fault type.

1.2 Travelling wave-based methods

When a fault initiates, high-frequency travelling waves aregenerated and propagate away in both directions withapproximately the speed of light. At each discontinuity, forexample a bus with multiple lines, the fault etc. they reflect andtransmit generating more travelling waves. Travelling wave-basedmethods [10, 11] monitor the travelling waves at one or both endsof the line and estimate the travelling time of these waves todetermine the fault location. They can be classified into twogroups: single-ended algorithms and dual-ended algorithms. (i)Single-ended algorithms use subsequent (reflected) travellingwaves at one terminal to identify the fault location [12, 13]. Theyare classified as types A, C, E and F fault locators according tohow the travelling waves can be generated. (ii) Dual-endedalgorithms use the time differences between the arrival of the firstwavefront at both terminals to determine fault location [13–15].They are classified as types B and D fault locators according tohow the two terminals can be synchronised with each other. Sincethe travelling waves are modulated by the frequency-dependentcharacteristics of the line, digital signal processing is required toidentify the actual arrival time of the wavefront. Several methodshave been applied, including wavelet transformation [15].Travelling wave-based methods have the following disadvantages.First, the intensity of travelling waves is greatly influenced by thefault initiation time relative to the power frequency cycle. Thisgenerates the issue of detection reliability of the travelling waves.Second, they require special instrumentation since the usual currenttransformers (CTs) will filter out the high-frequency content of thetravelling waves. Third, assuming the instrumentation does notalter the signal, very high sampling rates are needed in order todetect the wavefront with sufficient accuracy [16]. For example, asystem with a 100 kilosamples/s sampling rate could cause up to0.93 miles systematic error with type D fault locator.

1.3 Additional challenges for three-terminal lines

Further, researchers have been studying fault locating methods forthree-terminal lines. These methods can also be categorised intofundamental frequency phasor-based methods and travelling wave-based methods.

i. Fundamental frequency phasor-based methods: One methodtransforms the three-terminal line into a two-terminalequivalent line, and then directly applies standard two-terminal

IET Gener. Transm. Distrib., 2017, Vol. 11 Iss. 17, pp. 4184-4192© The Institution of Engineering and Technology 2017

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line fault locating algorithms [17, 18]. The method firstdetermines the faulted branch by estimating the voltage phasorat the tap point from terminal voltage and current phasors; thefaulted branch is the one providing a different estimated tapvoltage from the other two branches. Next, the three-terminalline is transformed into a two-terminal line with measuredvoltage and current phasor at one terminal and the calculatedvoltage and current phasors at the tap point. Finally, usual two-terminal methods (single-ended, dual-ended) are applied toestimate the location of the fault. These methods can be furthercategorised according to data synchronisation and availability,including three-ended GPS synchronised data [19], non-synchronised terminal voltages and currents [20], dual-endedsynchronised measurements [21], three-ended synchronisedcurrents and one-terminal voltages [22] etc. The fault locationerror of these methods is, in general, larger than that of two-terminal lines.

ii. Travelling wave-based methods: They also first determine thefaulted branch: the intensity of the travelling wave obtained atthe terminal of the faulted branch will be significantly higherthan those obtained at the terminals of the non-faulted branches[23]. Next, the fault location is computed by either single-ended or three-ended algorithms, with principles similar totwo-terminal lines. It is noted that algorithmic challenges aregenerated by the additional reflections and transmissions at thetap point of the fault initiated waves, making it difficult todetermine the arrival time of the wavefronts [24].

We present a method that uses existing instrumentation, CTs,voltage transformers (VTs) and GPS synchronised sampled valueswith improved accuracy in locating faults on transmission lines.The method uses a high-fidelity dynamic line model (physicallybased three-phase model with explicit grounding representation),and dynamic state estimation (DSE). We named the method EBFL(DSE-based fault locating). The main idea of EBFL is to treat thefault location as a state of the dynamic line model and estimate thefault location via a DSE algorithm [25, 26]. The paper describesthe basics of the method, the line dynamic model and presentsnumerous numerical experiments for quantifying the performanceof the method. While the method is equally applicable to cabletransmission systems, the discussions are limited to overhead lines.

2 Transmission line dynamic modelThe EBFL method requires a dynamic line model, a model of themeasurements and a DSE algorithm. We present an object-orientedapproach for the overall implementation of the method. The objectorientation starts with an object that represents the dynamic modelof a transmission line. The object has a specific syntax. In general,the object of the dynamic model of a component is developed byfirst expressing the mathematical description of the component as aset of algebraic and differential equations in terms of the states (x),the controls (u) and the parameters (p) of the component.Subsequently, we convert these equations into a set of algebraic

and first-order differential equations with non-linearities not >2. Ingeneral, this is achieved by introducing additional variables. Werefer to this as the quadratisation procedure. If the model is linearor quadratic, the quadratisation procedure is not needed. In the caseof a faulted line, the model is quadratic and therefore thequadratisation procedure is not needed. The equations are cast intoa specific syntax, which has been named device state, control andparameter quadratised dynamic model (SCPQDM). The syntax forthis model is as follows: (see (1)) where x(t), u(t) and p(t)represent the state, control and parameter vectors of thecomponents, respectively; terminal currents are represented by i(t),and terminal voltages are included in the state x(t). The second andthird equations correspond to internal constraints that describe therelationship among states, controls and parameters of thecomponents.

For the proposed fault locating application, the detailedSCPQDM of a faulted transmission line, including two- and three-terminal lines, is provided in Sections 9.1 and 9.2 of the Appendix,respectively. Note that a multi-section time-domain transmissionline model is adopted. This model is a close approximation of thefully distributed transmission line model with explicit groundingrepresentation [27], as shown in Section 9.3 of the Appendix. Themulti-section model is computational preferable for fault locatingpurposes.

The device SCPQDM is integrated using the quadraticintegration method [28] converting it into an algebraic companionform. The integration converts the SCPQDM into a set of algebraicequations of degree not higher than two in terms of the state,control and parameter variables. We refer to this model as thedevice state, control and parameter algebraic quadratic companionform (SCPAQCF) model.

The device SCPAQCF model has the format shown in (2) whereall the matrices can be expressed by the matrices defined in deviceSCPQDM, Δt is the sampling interval and tm = t − Δt. It is notedthat in our application, the transmission line does not have controlsand therefore control variables are not present.

Any measurement on the line can be expressed as a function ofthe state, control and parameter vectors using the SCPAQCFmodel. For example, the model of phase A current measurement atthe first terminal of the line at time t will be the first equation of theSCPAQCF model in (2). We refer to this as the measurementSCPAQCF object. The measurement SCPAQCF object has theformat as shown in (3): (see (2) and (3)) (see (3))

Measurements z(t, tm) are categorised into the following threetypes: actual, pseudo and virtual measurements. (i) Actualmeasurements are those obtained by actual metres such as terminalthree-phase voltages and currents. They include a measurementerror depending on the metre used which is quantified with astandard deviation. (ii) Pseudo measurements are introduced toensure observability of the system, such as terminal neutralvoltages. These measurements are often chosen as a best guess(e.g. zeros for neutral voltages) with large standard deviation. (iii)Virtual measurements correspond to internal constraints of the

i(t) = Yeqx1x(t) + Yequ1u(t) + Yeqp1 p(t) + Deqxd1dx(t)

dt + Ceqc1

0 = Yeqx2x(t) + Yequ2u(t) + Yeqp2 p(t) + Deqxd2dx(t)

dt + Ceqc2

0 = Yeqx3x(t) + Yequ3u(t) + Yeqp3 p(t) +⋮

x(t)T Feqxx3i x(t)⋮

+⋮

u(t)T Fequu3i u(t)⋮

+⋮

p(t)T Feqpp3i p(t)⋮

+⋮

u(t)T Fequx3i x(t)⋮

+⋮

p(t)T Feqpx3i x(t)⋮

+⋮

u(t)T Fequp3i p(t)⋮

+ Ceqc3

(1)


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system such as the second, third, fifth and sixth sets of equations in(2). Virtual measurements are treated as constraints in the DSEalgorithm or the constraints are relaxed and treated asmeasurements with very small standard deviation.

3 DSE with fault locationIn this section, we present the DSE algorithm where the distance tothe fault is introduced as a state variable to be estimated. We referto the resulting state as the extended state,xp(t, tm) = x(t, tm), p(t, tm) T, where p(t, tm) is the fault location at

times t and tm. The DSE algorithm provides the best estimate of theextended state from the measurements z(t, tm) described in (3).Assuming that the fault location is invariant during the fault, thefollowing additional constraints are introduced to the DSE:

0 = p(t) − p(tm)0 = p(t) − p(t − 2Δt)

(4)

The detail DSE algorithm can be found in [25, 26]. As anexample, the unconstraint weighted least-square algorithm is

xp(t, tm)ν + 1 = xp(t, tm)ν − (HTWH)−1HTW(h(xp(t, tm)ν) − z(t, tm)) (5)

where the Jacobian matrix H = ∂h(xp(t, tm))/∂xp(t, tm).The quality of the estimated xp^(t, tm) is validated by the

confidence level Pconf(t) from the chi-square test [29]. A highconfidence level (near 100%) means the estimated results aretrustworthy, i.e. the measurements fit the model well. Theconfidence level Pconf(t) can be calculated by

Pconf(t) = P(χ2 ≥ ζ(t)) = 1 − P(ζ(t), m) (6)

ζ(t) = (h(xp^(t, tm)) − z(t, tm))TW(h(xp^(t, tm)) − z(t, tm)) (7)

where P ζ(t), m is the probability of χ2 distribution given χ2 ≤ ζ(t)with degree of freedom m, W = diag 1/σ1

2, 1/σ22, … , where σi is

the measurement error standard deviation, and h(xp^(t, tm)) isdefined as in (3).

The overall flowchart of the DSE is provided in Fig. 1.

i(t)00

i(tm)00

= Yeqxx(t, tm) + Yequu(t, tm) + Yeqp p(t, tm) +⋮

x(t, tm)T Feqxxi x(t, tm)⋮

+⋮

u(t, tm)T Fequui u(t, tm)⋮

+⋮

p(t, tm)T Feqppi p(t, tm)⋮

+⋮

u(t, tm)T Fequxi x(t, tm)⋮

+⋮

p(t, tm)T Feqpxi x(t, tm)⋮

+⋮

u(t, tm)T Fequpi p(t, tm)⋮

− b(t − 2Δt)

b(t − 2Δt) = − Neqx ⋅ x(t − 2Δt) − Nequ ⋅ u(t − 2Δt)−Neqp ⋅ p(t − 2Δt) − Meq ⋅ i(t − 2Δt) − Keq

(2)

z(t, tm) = h(x(t, tm), u(t, tm), p(t, tm))

= Ym, xx(t, tm) + Ym, uu(t, tm) + Ym, p p(t, tm) +⋮

x(t, tm)T Fm, xxi x(t, tm)⋮

+⋮

u(t, tm)T Fm, uui u(t, tm)⋮

+⋮

p(t, tm)T Fm, ppi p(t, tm)⋮

+⋮

u(t, tm)T Fm, uxi x(t, tm)⋮

+⋮

p(t, tm)T Fm, pxi x(t, tm)⋮

+⋮

u(t, tm)T Fm, upi p(t, tm)⋮

− bz(t − h)

bz(t − 2Δt) = − Nm, x ⋅ x(t − 2Δt) − Nm, u ⋅ u(t − 2Δt)−Nm, p ⋅ p(t − 2Δt) − Mm ⋅ i(t − 2Δt) − Km

(3)

Fig. 1 Flowchart of DSE with fault location

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4 Simulation resultsWe compare the performance of the proposed EBFL method withlegacy fault locating schemes using two example test systems: (i)two-terminal transmission line (Section 4.1) and (ii) three-terminaltransmission line (Section 4.2). The legacy fault locating schemesare referring to the IEEE standard in [12]. A number of fault eventshave been simulated and the results have been stored inCOMTRADE files for experimentation and performanceevaluation of the proposed method. The data have been stored witha sampling rate of 80 samples/cycle or 4.8 kilosamples/s (samplinginterval of 208.33 ms). Note that the IEC 61850–9-2LE definestwo sampling rates, i.e. 80 and 256 samples/cycle.

4.1 Example test system 1: two-terminal transmission line

The first example test system is partially illustrated in Fig. 2. Thefull network used for the simulations is not shown. The line ofinterest (A1–A2) is a 500 kV, 135.22 mile-long transmission line.Three-phase voltage and current measurements are installed at bothterminals of the transmission line.

We assumed that legacy fundamental frequency phasor-basedsingle-ended and dual-ended fault locating schemes are applied tothis line to estimate the fault distance m from side A1. Note we donot consider travelling wave-based methods because the samplingrate is too low for estimating the fault location with reasonableaccuracy. The two legacy methods for comparison are as follows:(i) single-ended method, m = Im(VA1/IA1)/Im(Z1L); (ii) dual-endedmethod, m = (VA1 − VA2 + Z1LIA2)/(Z1L(IA1 + IA2)); where Z1L is thepositive (negative) sequence impedance of the line; the values ofVAi and IAi (i = 1, 2 for sides A1 and A2, respectively) are chosenaccording to different types of faults, as shown in Table 1. InTable 1, for side Ai (i = 1, 2) VAi, a, VAi, b, VAi, c, IAi, a, IAi, b and IAi, care voltage and current phasors of each phase,IAi, 0 = (IAi, a + IAi, b + IAi, c)/3 is the zero-sequence current;k0 = (Z0L − Z1L)/Z1L is the zero-sequence compensation factor; andZ0L is the zero-sequence impedance of the line.

We compare the performance of the above legacy fault locatingschemes to the EBFL method with different test cases as describedbelow. Due to space limitations, here we will only present resultsfor phase A to neutral faults and phases A–B faults. Results forother fault types are similar.

4.1.1 Test case 1: bolted phase A–N faults: A 0.01 Ω boltedphase A to neutral fault occurred on lines A1–A2, 50 miles frombus A1. Figs. 3a and b depict the fault locating results of legacy

single-ended and dual-ended fundamental frequency phasor-basedmethods. The fault locating results are 51.2722 miles from bus A1for the legacy single-ended method and 49.6947 miles from bus A1for the legacy dual-ended method.

Fig. 4a shows the fault locating results of the proposed EBFLmethod. The fault is computed to be at 50.0461 miles from bus A1.Note that the average error is 0.0461 miles, which is much smallercompared to legacy methods (1.2722 miles for legacy single-endedmethod and 0.3053 miles for legacy dual-ended method). Fig. 4bdemonstrates the χ2 test results of the DSE process. A 100%confidence level proves that all the states and parameters of thesystem are consistent with the model, and the fault locating resultsare trustworthy. Fig. 4c provides the standard deviation of thecalculated fault location (0.0878 miles), which proves the stabilityof the method. From these figures, we can see that for this fault theproposed EBFL method is more accurate than both legacymethods.

Fig. 2 Five hundred kilovolts, two-terminal line example test system

Table 1 Values of VAi and IAi for different fault typesFault type VAi IAia–n VAi,a IAi,a + k0IAi,0b–n VAi,b IAi,b + k0IAi,0c–n VAi,c IAi,c + k0IAi,0a–b, a–b–n VAi,a–VAi,b IAi,a–IAi,bb–c, b–c–n VAi,b–VAi,c IAi,b–IAi,ca–c, a–c–n VAi,a–VAi,c IAi,a–IAi,ca–b–c, a–b–c–n VAi,a IAi,a

Fig. 3 Two-terminal line fault locating results, legacy methods, 0.01 Ωbolted phase A to neutral fault, 50 miles from bus A1 (a) Single-endedmethod, (b) Dual-ended method

Fig. 4 Two-terminal line fault locating results, EBFL methods, 0.01 Ωbolted phase A to neutral fault, 50 miles from bus A1 (a) Estimated faultlocation, (b) Chi-square test, (c) Standard deviation of fault location


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To further validate the effectiveness of the proposed EBFLmethod, 0.01 Ω bolted phase A to neutral faults at different faultlocations are tested. Fig. 5a provides the fault locating error oflegacy single-ended method, legacy dual-ended method and theproposed EBFL method. It is noted that the EBFL method is moreaccurate than both legacy methods.

4.1.2 Test case 2: impedance phase A–N faults: 10 ohmsimpedance phase A to neutral faults at different fault locations aretested. The results are provided in Fig. 5b. It is observed that theEBFL method is more accurate than both legacy methods.

4.1.3 Test case 3: bolted phases A–B faults: 0.01 Ω boltedphases A–B faults at different fault locations are tested. The resultsare provided in Fig. 5c. It is observed that EBFL method is moreaccurate than both legacy methods.

4.1.4 Test case 4: impedance phases A–B faults: 10 ohmsimpedance phases A–B faults at different fault locations are tested.The results are provided in Fig. 5d. It is observed that the EBFLmethod is more accurate than both legacy methods.

4.2 Example test system 2: three-terminal transmission line

The second example test system is partially shown in Fig. 6. Theline of interest is a 500 kV three-terminal transmission line. Thelengths of branch 1 (A1-T), branch 2 (A2-T) and branch 3 (A3-T)are shown in the figure. Here three-phase voltage and currentmeasurements are installed at all three terminals of thetransmission line (A1, A2 and A3). The rest of the network is notshown.

We assumed that legacy fundamental frequency phasor-basedfault locating schemes are applied to this line to estimate thelocation of the fault. The legacy method first estimates the branchthat the fault locates in, and next finds the location of the faultinside that specific branch. It simply estimates the voltage phasor attap T from the terminal voltage and current phasors of two healthybranches; the branch with fault is the branch which has differentestimation of the voltage phasors at tap T. Afterwards, the three-terminal line fault locating problem is equivalently transferred intoa two-terminal line fault locating problem. Similarly as in Section4.1, single-ended and dual-ended methods can be used to determinethe location of the fault.

The reason we do not consider travelling wave-based legacymethods is that (i) the sampling rate here is too low for anestimated fault location with reasonable accuracy, and (ii)travelling wave-based methods have limitations in three-terminallines since they cannot differentiate the reflection from the tappoint and from the terminal of lines.

For our proposed EBFL on three-terminal lines, one importantstep is to determine the branch that the fault locates in. This isachieved by building three different models where each modelcorresponds to the fault inside each branch. Three models arerunning simultaneously with all available measurements and two ofthe fault locating results will generally converge to unrealisticvalues (less than zero or larger than the whole length of the branch)or simply diverge.

We compare the performance of the legacy fault locatingschemes to EBFL method with different test cases as describedbelow. Due to space limitations, here we will only present resultsfor phase A to neutral faults and phases A–B faults, inside branch1. Nevertheless, the results are similar for other fault types andfault locations.

4.2.1 Test case 5: bolted phase A–N faults in branch 1: 0.01 Ω bolted phase A to neutral faults at different fault locations in

Fig. 5 Two-terminal line fault locating results comparison, variable faultlocation (a) 0.01 Ω bolted phase A to neutral faults, (b) Impedance phase Ato neutral faults, 10 Ω fault impedance, (c) 0.01 Ω bolted phases A–Bfaults, (d) Impedance phases A–B faults, 10 Ω fault impedance

Fig. 6 Five hundred kilovolts, three-terminal line example test system


branch 1 are tested. The results are provided in Fig. 7a. It isobserved that EBFL method is more accurate than both legacymethods.

4.2.2 Test case 6: impedance phase A–N faults in branch1: 10 ohms impedance phase A to neutral faults at different faultlocations in branch 1 are tested. The results are provided in Fig. 7b.

It is observed that the EBFL method is more accurate than bothlegacy methods.

4.2.3 Test case 7: bolted phases A–B faults in branch1: 0.01 Ω bolted phases A–B faults at different fault locations inbranch 1 are tested. The results are provided in Fig. 7c. It isobserved that the EBFL method is more accurate than both legacymethods.

4.2.4 Test case 8: impedance phases A–B faults in branch1: 10 ohms impedance phases A–B faults at different faultlocations in branch 1 are tested. The results are provided in Fig. 7d.It is observed that the EBFL method is more accurate than bothlegacy methods.

5 DiscussionsTo further validate the proposed method, additional demonstrativeresults are provided in this section, including the performance ofthe method with different sampling rates and high faultimpedances.

5.1 Performance with different sampling rates

In this section, the performance of the proposed EBFL algorithmwith different sampling rates is demonstrated via the followingexample case: a group of 0.01 Ω phase A to neutral fault throughthe line in the system is shown in Fig. 2. The tested sampling ratesinclude 300, 600, 1200, 2400, 3600, 4800 and 6000 samples/cycle.The results of the fault locating absolute error versus the actualfault location are shown in Fig. 8. Also, the fault locating results ofthe fundamental frequency phasor-based dual-ended legacy methodare provided for comparison. Here the single-ended legacy methodresults are not included in the figure since they have much largererrors than the dual-ended method. It can be observed that (i) theabsolute error of the proposed method is smaller with highersampling rate; (ii) the EBFL method is more accurate than thedual-ended legacy method with sampling rate higher than 600samples/s. It is recommended that the sampling rate of 4800samples/s is used as it is consistent with the standard IEC61850-9-2LE and in this case the error of the proposed method issufficiently small.

5.2 Performance with high fault impedances

In this section, the performance of the proposed EBFL algorithmwith high fault impedances is demonstrated via the followingexample case: a group of phase A to neutral fault through the linein the system is shown in Fig. 2, with sampling rate 4800samples/s. The tested fault impedances include 0.01, 10, 40 and100 Ω. The results of the fault locating absolute error versus theactual fault location and the fault impedance are shown in Fig. 9.

Fig. 7 Three-terminal line fault locating results comparison, variable faultlocation in branch 1 (a) 0.01 Ω bolted phase A to neutral faults, (b) 10 Ωimpedance phase A to neutral faults, (c) 0.01 Ω bolted phases A–B faults,(d) 10 Ω impedance phases A–B faults

Fig. 8 Fault locating results comparison, different sampling rates,variable fault location


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Similarly, here only the dual-ended legacy method is introduced forcomparison since the single-ended legacy method has much largererrors. From the figure, it can be concluded that (i) the proposedEBFL method has more accurate fault locating results than dual-ended legacy method, for all fault impedances; (ii) unlike the dual-ended legacy method, the fault locating errors do not increase withhigher fault impedances.

6 ConclusionsA DSE EBFL algorithm has been presented to accuratelydetermine the fault location in transmission lines, for both two-terminal and three-terminal lines. First, the dynamic model of thefaulted line (multi-section transmission line with fault) is built inan object-oriented SCPQDM/SCPAQCF syntax. A dynamic stateestimator is used to compute the best estimation of the extendedstates of the faulted line. The extended states consist of the states ofthe line and the fault location. The proposed algorithm is validatedvia numerical simulations with different fault types, positions andimpedances on two example test systems. Additional case studiesare also provided with different sampling rates and high faultimpedances.

The main advantages of the algorithm include: (i) it uses a high-fidelity multi-section transmission line dynamic model; (ii) it usesinstantaneous values (sampled values) instead of phasors enablingestimation of the fault location in the presence of transients, whichis very important when the fault duration is short; (iii) itdemonstrates higher accuracy compared with legacy fault locatingschemes; (iv) the sampling rate it requires is relatively low (in thiscase 4.8 kilosamples/s) and therefore it can be used with existinginstrumentation in substations.

7 AcknowledgmentsThis work was supported by EPRI, PSERC and NYPA. Theirsupport was greatly appreciated.

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[13] Azizi, S., Sanaye-Pasand, M., Abedini, M., et al.: ‘A traveling wave-basedmethodology for wide-area fault location in multiterminal DC systems,’ IEEETrans. Power Deliv., 2014, 29, (6), pp. 2552–2560.

[14] Dewe, M.B., Sankar, S., Arrillaga, J.: ‘The application of satellite timereferences to HVDC fault location’, IEEE Trans. Power Deliv., 1993, 8, (3),pp. 1295–1302

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[16] Suonan, J., Gao, S., Song, G.: ‘A novel fault-location method for HVDCtransmission lines’, IEEE Trans. Power Deliv., 2010, 25, (2), pp. 1203–1209

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[18] He, Z.Y., Li, X.P., He, W., et al.: ‘Dynamic fault locator for three-terminaltransmission lines for phasor measurement units’, IET Gener. Transm.Distrib., 2013, 7, (2), pp. 183–191

[19] Aggarwal, R.K., Coury, D.V., Johns, A.T., et al.: ‘A practical approach toaccurate fault location on extra high voltage teed feeders’, IEEE Trans. PowerDeliv., 1993, 8, (3), pp. 874–883

[20] Girgis, A.A., Hart, D.G., Peterson, W.L.: ‘A new fault location technique fortwo-and three-terminal lines’, IEEE Trans. Power Deliv., 1992, 7, (1), pp. 98–107

[21] Lin, Y., Liu, C., Yu, C.: ‘A new fault locator for three-terminal transmissionlines using two-terminal synchronized voltage and current phasors’, IEEETrans. Power Deliv., 2002, 7, (3), pp. 452–459

[22] Izykowski, J., Rosolowski, E., Saha, M.M., et al.: ‘A fault-location methodfor application with current differential relays of three-terminal lines’, IEEETrans. Power Deliv., 2007, 22, (4), pp. 2099–2107

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9 Appendices 9.1 SCPQDM of two-terminal transmission line with fault

This appendix describes the SCPQDM of a two-terminal multi-section transmission line integrated with a fault. The overall modelis given in Fig. 10a. Here multi-section transmission line modelsare adopted for both the left and the right side parts, with m and nsections, respectively, where each section represents a π-equivalentshort line model. The number of sections is chosen in such a waythat the travelling distance of the electromagnetic wave during onesampling interval is comparable to the length of each section. Theoverall model can be obtained by combining the models of allsections. The π-equivalent models for each section inside the leftand the right parts are provided in Figs. 10b and c. The matrices ofSCPQDM for section k, left part are (all other matrices are zeros ornull): (see equation below) where yvk

1L (t), yv k + 11L (t) and yLk

1L (t) arenew states introduced to quadratise the model; I4 is the identitymatrix with the dimension of 4.

Fig. 9 Fault locating results comparison, different fault impedances,variable fault location


The matrices of SCPQDM for section k, right part are (all othermatrices are zeros or null): (see equation below) where yvk

1R (t),

yv k + 11R (t) and yLk

1R (t) are new states introduced to quadratise themodel.

9.2 SCPQDM of three-terminal transmission line with fault

This appendix describes the SCPQDM of a three-terminal multi-section transmission line integrated with a fault. The overall modelin Fig. 11 consists of three parts: a two-terminal transmission linemodel with fault (same as in Section 9.1 of the Appendix) and twomulti-section transmission line models without faults, with sectionnumber m + n, p and q, respectively.Here the section numbers areselected according to the same criteria as in Section 9.1 of theAppendix. Detail model for section k of the multi-sectiontransmission line without fault can be found in [25, 26]. Theoverall model can be obtained by combining the models of allsections.

9.3 Validation of multi-section transmission line model

This appendix provides a comparison between the multi-sectiontransmission line model and the fully distributed, frequency-dependent line model [27]. Owing to space limitations, thecomparison is limited to an energisation event. The test systemconsists of a source, a line and a load as shown in Fig. 12a. Theline is modelled with the multi-section model in one case and withthe distributed, frequency-dependent model in the other case. Theresults during line energisation for both cases are shown for theperiod 0 s to 2 ms. The currents at the output of the source and thevoltages at the middle point of the transmission line, for both cases,are depicted in Figs. 12b and c. It can be observed that the currentshave the maximum difference of <1.3% while the voltages have themaximum difference of <2%. It is concluded that even for thissevere transient event, the accuracy of the model is very good ascompared to the distributed, frequency-dependent line model.

Fig. 10 Model of two-terminal faulted line (a) Overall model; (b) π-equivalent model for section k, left part; (c) π-equivalent model for sectionk, right part

x(t) = vk1L (t) vk + 1

1L (t) iLk1L (t) yvk

1L (t) yv k + 11L (t) yLk

1L (t) T;

Yeqx1 = 0 0 I4 G1/m 0 00 0 −I4 0 G1/m 0 , Deqxd1 = 0 0 0 C1/m 0 0

0 0 0 0 C1/m 0 ;

Yeqx2 = −I4 I4 0 0 0 R1/m , Deqxd2 = 0 0 0 0 0 L1/m ;

Yeqx3 =0 0 0 I4 0 00 0 0 0 I4 00 0 0 0 0 I4

, i(t) = iak1L (t) ibk

1L (t) T;

p(t) = l f (t), Feqpx3i : f eq[1][i] = − 1, othersare0, for i = 1 − 12


4191

x(t) = vk1R (t) vk + 1

1R (t) iLk1R (t) yvk

1R (t) yv k + 11R (t) yLk

1R (t) T;

Yeqx1 = G1 ⋅ l/n 0 I4 −G1/n 0 00 G1 ⋅ l/n −I4 0 −G1/n 0 ;

Deqxd1 = C1 ⋅ l/n 0 0 −C1/n 0 00 C1 ⋅ l/n 0 0 −C1/m 0 ;

Yeqx2 = −I4 I4 R1 ⋅ l/n 0 0 −R1/n ;

Deqxd2 = 0 0 L1 ⋅ l/n 0 0 −L1/n , Yeqx3 =0 0 0 I4 0 00 0 0 0 I4 00 0 0 0 0 I4

;

p(t) = l f (t), Feqpx3i : f eq[1][i] = − 1, othersare0, fori = 1 − 12;

Fig. 11 Model of three-terminal line with fault

Fig. 12 Comparison between the multi-section model and the fullydistributed model (a) Example test system, (b) Currents at the output of thesource, (c) Voltages at the middle point of the line


on transmission lines revised 5th june 2017 yu liu dynamic...

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