Received: 27 January 2021 Revised: 3 May 2021 Accepted: 19 May 2021 IET Generation, Transmission & Distribution
DOI: 10.1049/gtd2.12215
ORIGINAL RESEARCH PAPER
An improved natural frequency based transmission line fault
location method with full utilization of frequency spectrum
information
Yuan Nie1,2,3,4 Yu Liu1 Dayou Lu1 Binglin Wang1
1 School of Information Science and Technology,ShanghaiTech University, Shanghai, China
2 Shanghai Advanced Research InstituteChineseAcademy of Sciences, Shanghai, China
3 Shanghai Institute of Microsystem andInformation TechnologyChinese Academy ofSciences, Shanghai, China
4 University of the Chinese Academy of Sciences,Beijing, China
Correspondence
Yu Liu, School of Information Science and Technol-ogy (SIST), ShanghaiTech University, 393 HuaxiaMiddle Road, Pudong District, Shanghai, China201210.Email: [email protected];[email protected]
Funding information
National Natural Science Foundation of China,Grant/Award Number: 51807119
Abstract
An improved natural frequency based method is proposed to accurately locate faults intransmission lines. The method only requires three phase instantaneous current measure-ments at the local terminal of the line and is compatible with protective relays using high-speed tripping techniques. First, the theoretical peak frequencies along the entire frequencyspectrum of the measured currents are derived, for different fault types, locations and resis-tances. Next, the fault location is determined by finding the minimum average distancebetween the theoretical and the measured peak frequencies. Compared to the existing nat-ural frequency based fault location methods, the proposed method solves the mode mixingproblem during single phase to ground faults and systematically considers the effect of faultresistances. In addition, the proposed method fully utilizes the frequency spectrum infor-mation instead of only the dominant frequency, and therefore overcomes the difficulty toextract the dominant frequency. Numerical experiments have shown that, compared to theexisting method, the proposed method presents much higher fault location accuracy dur-ing single phase to ground faults, and slightly higher (comparable) fault location accuracyfor other types of faults, regardless of fault locations and fault resistances.
1 INTRODUCTION
Accurate fault location method is beneficial for reducing thepower outage time, as well as, operational costs of power sys-tems. Fault location methods of transmission lines are stud-ied in many literatures. The most widely adopted fault locationmethods are the fundamental frequency phasor based meth-ods, which utilize fundamental frequency phasors to calculatethe impedance and afterwards the distance between the faultand one terminal of the line [1–6]. The main limitation of thefundamental frequency phasor based methods is the depen-dency on the accurate extraction of fundamental frequency pha-sors, which could be challenging especially during system tran-sients. Therefore, for transmission lines equipped with protec-tive relays using high-speed tripping techniques, the availabledata window to extract the phasors during faults could be veryshort (e.g. fraction of a cycle), and these fault location methodsmay present compromised fault location results. To overcome
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the above limitation, researchers proposed advanced fault loca-tion methods that are compatible with the short data window,including time domain model based methods, traveling wavebased methods and natural frequency based methods.
Time domain model based methods determine the locationof the fault with the help of the accurate time domain trans-mission line model before and during the fault. Specifically,in “solving equation” methods, the fault location is obtainedby solving analytical equations that describe the physical lawsbetween the instantaneous measurements and the fault location[7–9]. However, to ensure analytical expressions, these trans-mission line models are usually limited to lumped parametermodels or multi-section π models, which may increase the faultlocation errors. To further utilize more accurate transmissionline models, the voltage methods calculate the voltage distribu-tions along the transmission line at either terminal and then findthe intersection point of the two voltage distributions to obtainthe fault location. These methods do not need to formulate
IET Gener. Transm. Distrib. 2021;1–17. wileyonlinelibrary.com/iet-gtd 1
the analytical equations and therefore more accurate distributedparameter transmission line models can be applied such asthe Bergeron model (with assumptions of lumped resistances)[10,11] or even models described via the matrix form partialdifferential equations (fully distributed parameter models) [12].Nevertheless, these methods usually require dual ended voltageand current measurements, and synchronization among themeasurements are typically required.
Traveling wave based methods determine the fault locationby detecting the arrival time of wavefronts at line terminals afterthe occurrence of the fault, including single ended and dualended algorithms. Single ended methods can be further classi-fied into types A, C, E and F methods (according to the gener-ation of the traveling waves). The effectiveness of single endedalgorithms [18–20] relies on the successful detection of multiplereflections of traveling waves between line terminal and faultlocation. Dual ended algorithms [13–17] observe the arrivaltime of first wavefronts at both line terminals and then obtainthe fault location by the time difference, which can be furtherclassified into types B and D methods (according to synchro-nization techniques). However, the reliable detection of wave-fronts is challenging with small fault inception angles or highfault resistances, which increases difficulty to accurately capturethe fault location.
To avoid reliability issues in detecting wavefronts of travelingwaves, researchers proposed the natural frequency based meth-ods. Instead of accurately detecting the arrival time of wave-fronts of traveling waves, these methods adopt the frequencycharacteristics of the voltages/current measurements to locatefaults. The preliminary relationship between the fault locationand the natural frequency was studied by Swift in 1979 [21],where only special system conditions are considered: the systemimpedance is assumed to either infinite or zero. In recent years,researchers largely improve the performance of the natural fre-quency based fault location method by considering more gen-eral system boundary conditions. These natural frequency basedfault location approaches have been applied in both HVAC[24,25] and HVDC [22,23] systems. The main idea of thesemethods is to first extract the dominant frequency from thesingle ended instantaneous current or voltage measurements ofa certain transmission line mode, and afterwards use the wavespeed and the reflection coefficient of a certain transmissionline mode at the local terminal to locate faults. In fact, for threephase AC systems, the natural frequency based fault locationmethods work well for phase to phase faults, double phase toground faults and three phase faults. Nevertheless, when a sin-gle phase to ground fault occurs, there exists the “mode mixingphenomenon” [23,24], where the frequency spectrum of a cer-tain mode is mixed with that of other modes. Therefore, sincethe derivations of the existing methods are usually within onetransmission line mode, the mode mixing phenomenon maycause difficulty to extract the dominant frequency and also com-promised fault location results. In addition, the existing meth-ods typically assume zero fault resistance, which may furtherincrease fault location errors especially during high resistancefaults.
In this paper, a novel natural frequency based method is pro-posed. It only needs three phase instantaneous current measure-ments at the local terminal of the line, and no communicationchannel or time synchronization is required. Unlike the existingmethod which only uses the dominant frequency, the proposedmethod fully exploits the frequency spectrum information andutilizes a series of peak frequencies of the frequency spectrum.First, the frequency spectrum of the measured currents for dif-ferent fault types, locations and resistances is derived in detail.Afterwards, the relationship between the fault location and thepeak frequencies is studied, with full consideration of the modemixing phenomenon and the fault resistance. Finally, the faultlocation is determined by solving an optimization problem thatfinds the best match between the theoretical and the measuredpeak frequencies. The original contributions of this paper arehighlighted as follows:
∙ The method only needs single ended current measurementsand a relatively short time window; compared to the travel-ing wave based methods, the method does not require reli-able wavefront detection and works with zero fault inceptionangle.
∙ Instead of only using the dominant frequency, the proposedmethod strictly and systematically derives all the peak fre-quencies along the frequency spectrum of the measured cur-rents for different fault types, locations and resistances; theproposed method fully utilizes the information embeddedinside the frequency spectrum, and therefore avoids the dif-ficulty to extract the dominant frequency.
∙ During single phase to ground faults, the proposed methodsystematically considers the coupling among different linemodes as well as the fault resistance compared to the existingmethod, and therefore fully solves the mode mixing problem.
∙ During other types of faults (phase to phase, double phase toground, and three phase faults), the proposed method con-siders the fault resistance compared to the existing method;mathematical proofs illustrate that the existing method is aspecial case of the proposed method with zero fault resis-tance.
∙ Numerical experiments prove that the proposed methodpresents much higher fault location accuracy for single phaseto ground faults, and slightly higher (comparable) fault loca-tion accuracy for other types of faults, compared to existingnatural frequency based method.
The rest of the paper is organized as follows. Section 2 intro-duces the fundamental principles and reviews in detail the exist-ing natural frequency based fault location method as well as thelimitations. Sections 3 and 4 provide the strictly mathematicalderivation to obtain the peak frequencies of the frequency spec-trum for different types of faults. Section 5 introduces proposedthe fault location algorithm. Section 6 compares the proposedmethod and the existing method via numerical experiments toshow the advantages of the proposed method. Section 7 dis-cusses the effects of different factors on the fault location accu-racy. Section 8 draws a conclusion.
NIE ET AL. 3
2 FUNDAMENTAL PRINCIPLES
2.1 Review of the healthy three phaselossless line model
The partial differential equations representing a healthy threephase lossless transmission line model are [26],{
𝜕2uabc(x, t )∕𝜕x2 = LC𝜕2uabc(x, t )∕𝜕t 2
𝜕2iabc(x, t )∕𝜕x2 = CL𝜕2iabc(x, t )∕𝜕t 2, (1)
where matrices L and C are the inductance and capacitancematrices per unit length,
uabc(x, t ) =[ua (x, t ), ub(x, t ), uc (x, t )
]T(2)
iabc(x, t ) =[ia (x, t ), ib(x, t ), ic (x, t )
]T, (3)
um (x, t ) and im (x, t )(m = a, b, c) are per phase voltages and cur-rents on the transmission line at location x and time t.
To solve Equation (1), the modal decomposition is usuallyadopted to transform phase components abc into mode compo-nents 0αβ (here the Clarke transformation is taken as an exam-ple),
u0𝛼𝛽 (x, t ) = T ⋅ uabc(x, t ), i0𝛼𝛽 (x, t ) = T ⋅ iabc(x, t ), (4)
where the Clarke transformation matrix T is defined as T =
[1∕3, 1∕3, 1∕3; 2∕3, −1∕3, −1∕3; 0, 1∕√
3, −1∕√
3].From Equations (1) and (4),
⎧⎪⎨⎪⎩𝜕2u0𝛼𝛽 (x, t )∕𝜕x2 =
(T LT−1
)(TCT−1)𝜕2u0𝛼𝛽 (x, t )∕𝜕t 2
𝜕2i0𝛼𝛽 (x, t )∕𝜕x2 =(
TCT−1)
(T LT−1)𝜕2i0𝛼𝛽 (x, t )∕𝜕t 2,
(5)where
TCT−1= diag([C0 C𝛼 C𝛽 ]), (6)
T LT−1= diag([ L0 L𝛼 L𝛽 ]), (7)
and diag([⋅]) is a diagonal matrix with the diagonal vector[⋅].
From Equation (5), the mode voltages and currents arefully decoupled for healthy lines. The solution to Equation (5)is [26],
⎧⎪⎨⎪⎩u j (x, t ) = ( f j
+(t −√
L jCj x ) + f j−(t +
√L jCj x ))
i j (x, t ) = ( f j+(t −
√L jCj x ) − f j
−(t +√
L jCj x ))∕z j
,
(8)
where j = 0, α, β represents different modes. For each mode
j, z j =√
L j∕Cj is the surge impedance, f j+ and f j
− are the
forward and backward waves, with speed v j = 1∕√
L jCj .
2.2 Review of the existing natural frequencybased fault location method
The frequency spectrum of the fault voltage/current measure-ments is called the natural frequency. After modal decomposi-tion, the natural frequency corresponding to each mode can berepresented by roots of Equation (9) [22],
H (s) = 1/
(1 − P (s)Γ1(s)P (s)Γ f (s)), (9)
where P (s) = e−sT , Γ1(s) and Γ f (s) are reflection coefficients atthe line end and fault location in Laplace domain respectively,which can be calculated by the parameters of the system. Thespectrum of the signal can be extracted using MUSIC algorithm[22]. There are peaks in the frequency spectrum, and the fre-quency corresponding to the first peak is selected as the dom-inant frequency to calculate the fault location. The dominantfrequency has the largest amplitude other than the fundamen-tal frequency, and can be identified easily. The existing methodfirstly extracts the frequency spectrum of the terminal currentof a certain mode, and then finds the dominant frequency inthe spectrum. With the calculated wave speed and the reflectioncoefficient at the line end of a certain mode, the fault location isdetermined as,
d = (𝜃1 + 𝜋)v∕(4𝜋 fd ) (10)
where fd is the dominant frequency of the current of the certainmode measured at the line end after the occurrence of the fault,v is the wave speed of the certain mode (or average wave speedof several modes if mode mixing phenomenon occurs), and θ1is the angle of reflection coefficient of the certain mode at theline terminal. Note that the fault resistances are assumed to bezero during the derivation.
The limitations of the existing method are described in detailas follows. First, as mentioned in refs. [23,24], when a singlephase to ground fault occurs on the transmission line, thereexists the mode mixing phenomenon, which increases the diffi-culty of accurate fault location. For example, for a three phasetransmission line with Clarke transformation for modal decom-position, mode α and mode 0 are coupled (mixed) together dur-ing single phase to ground faults, that is, the spectrum extractedby the mode α current will consist of the frequency informa-tion of both mode α and mode 0, vice versa. Consequently, thespectrum of the mode current is rather complicated, resultingin challenge to find the correct dominant frequency. Moreover,even with the correct dominant frequency, it is also challengingto determine the accurate wave speed for the mixed modes. Inrefs. [23,24], the existing methods approximate the two mixedmodes as one mode, with the wave speed as the average speed
4 NIE ET AL.
A
B
C
Fault
source 1 source 2
Line1 Line2
Measured current (1) (0, )ji t ( j = 0αβ )
d l - d
l
FIGURE 1 Example power system of interest
of mode α and mode 0. Nevertheless, this approximation maystill generate errors, especially during high resistance faults. Thelimitations of the existing natural frequency based method arefurther summarized as follows:
1. The existing method only utilizes the dominant frequency;other information of the frequency spectrum is not fully uti-lized;
2. Due to mode mixing phenomenon especially during singlephase to ground faults, accurate extraction of dominant fre-quency and the wave speed of the mixed mode is challenging;
3. Derivations are within one transmission line mode, and thefault resistances are assumed to be zero.
To overcome the limitations of the existing natural frequencybased method, a novel natural frequency based fault locationmethod is proposed in this section. The proposed method fullyutilizes the information of the frequency spectrum instead ofonly one dominant frequency. The mathematical relationshipamong the frequency spectrum, the fault location and the faultresistance is rigorously derived for different types of faults.Afterwards, the fault location is determined with full utilizationof the information in the frequency spectrum. Specifically, themode mixing problem for single phase to ground faults as wellas the effect of fault resistances are systematically consideredusing the proposed fault location methodology.
3.1 Frequency spectrum during singlephase to ground faults
Here phase A to ground faults are taken as examples to showthe frequency spectrum as functions of fault location and faultresistances. Also, the relationship between the existing methodand the proposed method during single phase to ground faultsis explained. The example power system of interest is shownin Figure 1. Some definitions of variables are as follows. Theentire transmission line during faults is divided into two healthytransmission line sections as Line 1 (left side of the fault) andLine 2 (right side of the fault). For mode j (j = 0, α, β) or phase
j (j = a, b, c), the voltages and currents on Line 1 and Line 2are defined as u j
(1)(x1, t ), i j(1)(x1, t ), u j
(2)(x2, t ) and i j(2)(x2, t ),
respectively, where x1 and x2 denote the distance to the left andthe right terminal, respectively; the voltage of the equivalentsources at the left and the right side of the line is u j
s1(t ) andu j
s2(t ), respectively; the voltage at the location of the fault andthe current flowing into the fault are defined as u j
f (t ) and
i jf (t ). The voltage and current at the corresponding voltage
and current vectors are defined as wabc = [wa, wb, wc ]T and
w0𝛼𝛽 = [w0, w𝛼, w𝛽]T , where w can be replaced by u(1), i (1),u(2), i (2), u f , i f , us1(t ) and us2(t ). The fault is at distance d fromthe left terminal. The length of the entire transmission line is l .
3.2 Boundary conditions at the faultlocation and line terminal
For a phase A to ground fault, at the location of the fault,
iabcf (t ) = Y abc
f uabcf (t ), (11)
where
Y abcf = diag
([1∕R f 0 0
]), (12)
with fault resistance R f .For the mode network, from Equation (4), at the location of
the fault,
i0𝛼𝛽f (t ) = Y 0𝛼𝛽
f ⋅ u0𝛼𝛽f (t ), (13)
where
Y 0𝛼𝛽f = TY abc
f T−1
=[1∕R f , 1∕R f , 0; 2∕R f , 2∕R f , 0; 0, 0, 0
]∕3. (14)
The boundary conditions at the fault location can be obtainedby observing that Line 1 and Line 2 share the same node. First,the voltage at the right terminal of Line 1 and the left terminalof Line 2 are the same (Equation (15)). Second, the Kirchhoff ’sCurrent Laws at the fault location should be obeyed (Equa-tion (16)). Therefore,
u0𝛼𝛽(1)(d , t ) = u0𝛼𝛽
(2)(l − d , t ) = u0𝛼𝛽f (t ), (15)
i0𝛼𝛽(1)(d , t ) + i0𝛼𝛽
(2)(l − d , t ) = i0𝛼𝛽f (t ) =Y 0𝛼𝛽
f ⋅ u0𝛼𝛽f (t ).(16)
The boundary conditions at the line ends can be obtained byobserving Kirchhoff’s Voltage Laws from the equivalent sourcesto the transmission line terminals.
u0𝛼𝛽(1)(0, t ) = u0𝛼𝛽
s1(t ) − R0𝛼𝛽s1 ⋅ i0𝛼𝛽
(1)(0, t )
−L0𝛼𝛽s1𝜕i0𝛼𝛽
(1)(0, t )∕𝜕t , (17)
NIE ET AL. 5
u0𝛼𝛽(2)(0, t ) = u0𝛼𝛽
s2(t ) − R0𝛼𝛽s2 ⋅ i0𝛼𝛽
(2)(0, t )
−L0𝛼𝛽s2𝜕i0𝛼𝛽
(2)(0, t )∕𝜕t , (18)
where
R0𝛼𝛽s1 = T Rabc
s1T−1= diag([ R0
s1R𝛼
s1R𝛽
s1 ]), (19)
and
L0𝛼𝛽s1 = T Labc
s1T−1= diag([ L0
s1 L𝛼s1 L𝛽
s1 ]), (20)
are the left side source resistance and inductance matrices forthe mode network, Rabc
s1 and Labcs1 are the corresponding
matrices for the phase networks.
u0𝛼𝛽s1(t ) = [0, u𝛼
s1(t ), u𝛽s1(t )]T . (21)
The definitions are similar for the right side source (with super-script ‘s2’).
3.3 Solving the peak frequencies of thefrequency spectrum
In this section, since the line with fault consists of two healthylines (Line 1 and Line 2) as well as, the fault, the frequencyspectrum can be obtained using the solution in Section 2.1 andboundary conditions in Section 3.2. Here the α mode and 0mode are taken as examples. Through the derivation, the wavespeeds of each mode are defined in Equation (8).
Substitute Equation (8) into Equation (15) and take the firsttwo rows,
⎧⎪⎪⎨⎪⎪⎩
f0+(1)(t − d∕v0) + f0
−(1)(t + d∕v0)
= f0+(2)(t − (1 − d ) ∕v0) + f0
−(2)(t + (l − d )∕v0)
f𝛼+(1)(t − d∕v𝛼 ) + f𝛼
−(1)(t + d∕v𝛼 )
= f𝛼+(2)(t − (1 − d ) ∕v𝛼 ) + f𝛼
−(2)(t + (l − d )∕v𝛼 )
.
(22)
Substitute Equations (8) and (15) into Equation (16) and takethe first two rows,
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
[f0+(1)(t − d∕v0) − f0
−(1)(t + d∕v0) + f0+(2)(t − (l − d )∕v0)
− f0−(2)(t + (l − d )∕v0)
]∕z0 = 1∕
(3R f
)×[
f𝛼+(1)(t − d∕v𝛼 )
+ f𝛼−(1)(t + d∕v𝛼 ) + f0
+(1)(t − d∕v0) + f0−(1)(t + d∕v0)
][f𝛼+(1)(t − d∕v𝛼 ) − f𝛼
−(1)(t + d∕v𝛼 ) + f𝛼+(2)(t − (l − d )∕v𝛼 )
− f𝛼−(2)(t + (l − d )∕v𝛼 )
]∕z𝛼 = 2∕
(3R f
)×[
f𝛼+(1)(t − d∕v𝛼 )
+ f𝛼−(1)(t + d∕v𝛼 ) + f0
+(1)(t − d∕v0) + f0−(1)(t + d∕v0)
]
.
(23)
Substitute Equation (8) into Equations (17) and (18) and takethe first two rows,
⎧⎪⎪⎪⎨⎪⎪⎪⎩
f0+(1)(t ) + f0
−(1)(t ) = −[R0(1) × ( f0
+(1)(t ) − f0−(1)(t ))
+L0(1) × 𝜕( f0
+(1)(t ) − f0−(1)(t ))∕𝜕t ]∕z0
f𝛼+(1)(t ) + f𝛼
−(1)(t ) = u𝛼s1(t ) − [R𝛼
(1)
× ( f𝛼+(1)(t ) − f𝛼
−(1)(t ))
+L𝛼(1) × 𝜕( f𝛼
+(1)(t ) − f𝛼−(1)(t ))∕𝜕t ]∕z𝛼
.
(24)
⎧⎪⎪⎨⎪⎪⎩
f0+(2)(t ) + f0
−(2)(t ) = −[R0(2) × ( f0
+(2)(t ) − f0−(2)(t ))
+L0(2) × 𝜕( f0
+(2)(t ) − f0−(2)(t ))∕𝜕t ]∕z0
f𝛼+(2)(t ) + f𝛼
−(2)(t ) = u𝛼s2(t ) − [R𝛼
(2) × ( f𝛼+(2)(t )
− f𝛼−(2)(t )) +L𝛼
(2) × 𝜕( f𝛼+(2)(t ) − f𝛼
−(2)(t ))∕𝜕t ]∕z𝛼
.
(25)To derive the frequency spectrum, we transform Equations
(22)–(25) into Laplace domain in Equations (26)–(29),
⎧⎪⎪⎨⎪⎪⎩
F0+(1)(s)q(s)−1 + F0
−(1)(s)q(s) = F0+(2)(s)n(s)−1
+ F0(s)− (2)n(s)
F𝛼+(1)(s)p(s)−1 + F𝛼
−(1)(s)p(s) = F𝛼+(2)(s)m(s)−1
+ F𝛼−(2)(s)m(s)
, (26)
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
b(F0+(1)(s)q(s)−1 − F0
−(1)(s)q(s) + F0+(2)(s)n(s)−1
−F0−(2)(s)n(s)) = F𝛼
+(1)(s)p(s)−1 + F𝛼−(1)(s)p(s)
+ F0+(1)(s)q(s)−1 + F0
−(1)(s)q(s)
a(F𝛼+(1)(s)p(s)−1 − F𝛼
−(1)(s)p(s) + F𝛼+(2)(s)m(s)−1
−F𝛼−(2)(s)m(s)) = F𝛼
+(1)(s)p(s)−1 + F𝛼−(1)(s)p(s)
+ F0+(1)(s)q(s)−1 + F0
−(1)(s)q(s)
, (27)
{F0+(1)(s)+F0
−(1)(s) = −d1(s)(F0+(1)(s) − F0
−(1)(s))
F𝛼+(1)(s)+F𝛼
−(1)(s) = U𝛼s1(s) − c1(s)(F𝛼
+(1)(s) − F𝛼−(1)(s))
,
(28){F0+(2)(s) + F0
−(2)(s)=−d2(s)(F0+(2)(s) − F0
−(2)(s))
F𝛼+(2)(s) + F𝛼
−(2)(s)=U𝛼s2(s) − c2(s)(F𝛼
+(2)(s) − F𝛼−(2)(s))
,
(29)where s is the Laplace operator, and the definitions of coeffi-cients in Equations (26)–(29) are shown in Table 1. Fj
+(k)(s)
and Fj−(k)(s) (j= 0, α; k= 1, 2) are the Laplace domain forward
and backward waves for mode j and Line k, respectively; U𝛼s1(s)
and U𝛼s2(s)are the Laplace domain source voltages of mode α,
respectively.From Equations (26)–(29), there are 8 unknowns (Fj
+(k)(s)
and Fj−(k)(s), j = 0, α; k = 1, 2) and 8 equations. Therefore,
these 8 unknowns can be directly solved. Afterwards, the ter-minal mode voltages or currents in Laplace domain can be
6 NIE ET AL.
TABLE 1 Definitions of variables, single phase to ground faults
Variables Value Variables Value
a 3R f ∕(2z𝛼 ) n(s) es(l−d )∕v0
b 3R f ∕z0 c1(s) (R𝛼(1) + sL𝛼
(1) )∕z𝛼
p(s) esd∕v𝛼 c2(s) (R𝛼(2) + sL𝛼
(2) )∕z𝛼
q(s) esd∕v0 d1(s) (R0(1) + sL0
(1) )∕z0
m(s) es(l−d )∕v𝛼 d2(s) (R0(2) + sL0
(2) )∕z0
formulated according to the solution of these 8 unknowns. Dueto space limitations, here we only present the α mode current atthe left terminal i𝛼
(1)(0, s) (other voltages and currents can besimilarly obtained). After converting Equation (8) into Laplacedomain and substituting the solutions,
In Equation (30), one can observe that A(s) includes thesource voltages U𝛼
s1(s) and U𝛼s2(s), denoting that A(s) forms
the frequency peak at the fundamental frequency (source fre-quency: 50 or 60 Hz). Therefore, if we let s = j𝜔 = j2𝜋 f ,where ω and f are the angular frequency and the frequency, thepeaks of the frequency spectrum (peak frequencies) except thepeak of fundamental frequency can be represented as the localminimum of B( j2𝜋 f ) or local maximum of
H ( f ) = 1∕B( j2𝜋 f ), (33)
peak frequencies = maxf
||1∕B( j2𝜋 f )|| , (34)
where the function B(⋅) is defined in the expression of the modecurrent. It can be observed that there could be many local max-ima of 1∕B( j2𝜋 f ) (corresponds to the peak frequencies of themode current). These peak frequencies are functions of the
pre-defined coefficients, and therefore are functions of the faultlocation d and the fault resistance Rf.
3.4 Relationship between the existing andthe proposed method during single phase toground faults
For the proposed method, from Equations (30) and (34), it canbe seen that the peak frequencies through the frequency spec-trum are related to both mode α and mode 0 networks (forexample in Equation (30), variables a, p(s), m(s), c1(s) and c2(s)are related to mode α; variables b, q(s), n(s), d1(s) and d2(s) arerelated to mode 0), proving the correctness of the mode mix-ing phenomenon (mode α and mode 0 are mixed together andcannot be simply decoupled). One can also observe that the fre-quency spectrum is a function of the fault resistance (variables a
and b). On the other hand, for the existing method, from Equa-tion (10), the dominant frequency (corresponding to one of thepeak frequencies in the proposed method) is derived accord-ingly to network of only one certain mode and the fault resis-tance is not considered. Even if the wave speed in Equation (10)is approximated as the average speed in mode α and mode 0 forsingle phase to ground fault [23,24], the existing method will stillpresent errors. Therefore, during single phase to ground faults,the proposed method systematically considers the mode mixingphenomenon and the fault resistance compared to the existingmethod.
Next, the frequency spectrum is derived during other types offaults (including phase to phase, double phase to ground andthree phase faults). For these faults, at least one certain modeis not coupled with the rest of the modes, and therefore thederivation of the frequency spectrum is much simplified. Also,the relationship between the existing method and the proposedmethod during these faults is explained. The example power sys-tem of interest is shown in Figure 1. Note that the definitions ofvariables are consistent with those in Section 3.
4.1 Phase to phase faults
Here phase B to phase C faults are taken as examples. At thelocation of the fault, the expressions are the same as Equations(11) and (13) except that Y abc
f and Y 0𝛼𝛽f are updated as
Y abcf = [0, 0, 0; 0, 1
/R f , −1
/R f ; 0, −1
/R f , 1
/R f ],
(35)and
Y 0𝛼𝛽f = TY abc
f T−1= diag([0, 0, 2
/R f ]). (36)
NIE ET AL. 7
TABLE 2 Definitions of variables, other types of faults
Variables Value Variables Value
a R f ∕2z𝛽 c1(s) (R𝛽(1) + sL𝛽
(1) )∕z𝛽
p(s) esd∕v𝛽 c2(s) (R𝛽(2) + sL𝛽
(2) )∕z𝛽
m(s) es(l−d )∕v𝛽
The boundary conditions are the same as Equations (15)–(18). From the diagonal structure of Y 0𝛼𝛽
f , it can be seen thatthe mode 0αβ are fully decoupled. Here take mode β as anexample. Similarly, substitute Equation (8) into Equation (15),substitute Equations (8) and (15) into Equation (16), substituteEquation (8) into Equations (17) and (18), and take the thirdrow (corresponding to mode β). Afterwards, transform theminto Laplace domain,
F𝛽+(1)(s)p(s)−1 + F𝛽
−(1)(s)p(s) = F𝛽+(2)(s)m(s)−1
+F𝛽−(2)(s)m(s), (37)
a(F𝛽+(1)(s)p(s)−1 − F𝛽
−(1)(s)p(s) + F𝛽+(2)(s)m(s)−1
−F𝛽−(2)(s)m(s)) = F𝛽
+(1)(s)p(s)−1 + F𝛽−(1)(s)p(s),
(38)
F𝛽+(1)(s) + F𝛽
−(1)(s) = U𝛽s1(s) − c1(s)(F𝛽
+(1)(s) − F𝛽−(1)(s)),
(39)
F𝛽+(2)(s) + F𝛽
−(2)(s) = U𝛽s2(s) − c2(s)(F𝛽
+(2)(s) − F𝛽−(2)(s)),
(40)where the definitions of coefficients in Equations (37)–(40) areshown in Table 2.
After solving the 4 unknowns F𝛽+(1)(s), F𝛽
−(1)(s), F𝛽+(2)(s)
and F𝛽−(2)(s), the β mode current at the left terminal i𝛽
Finally, the peak frequencies can be found throughEquation (34).
4.2 Double phase to ground faults
Here phase BC to ground faults are taken as examples. At thelocation of the fault, the expressions are the same as Equations
(11) and (13) except that Y abcf and Y 0𝛼𝛽
f are updated as
Y abcf = diag([0, 1
/R f , 1
/R f ]) (44)
and
Y 0𝛼𝛽f = TY abc
f T−1= [2
/R f , −1
/R f ] (45)
The boundary conditions are the same as Equations (15)–(18). Similarly, one can observe from the structure of Y 0𝛼𝛽
f
that mode β is not coupled with the rest of the modesand therefore mode β is selected. Similarly, the expression ofi𝛽
(1)(0, s) can be derived and the result is the same as Equa-tion (41) except that the definition of a is updated as a =
R f ∕z𝛽 . Finally, the peak frequencies can be found throughEquation (34).
4.3 Three phase faults
At the location of the fault, the expressions are the sameas Equations (11) and (13) except that Y abc
f and Y 0𝛼𝛽f are
updated as
Y abcf = diag([1
/R f , 1
/R f , 1
/R f ]), (46)
and
Y 0𝛼𝛽f = TY abc
f T−1= (diag([1
/R f , 1
/R f , 1
/R f ]). (47)
The boundary conditions are the same as Equations (15)–(18). Similarly, the expression of i𝛽
(1)(0, s) and the peak frequen-cies can be derived. The results are exactly the same as those ofdouble phase to ground faults in Section 4.2.
4.4 Relationship between the existing andthe proposed method during other types offaults
For the proposed method, when dealing with the faults otherthan single phase to ground faults, the peak frequencies throughthe frequency spectrum can be found from Equations (41) and(34), and are only related to mode β, proving that there isno mode mixing phenomenon for these types of faults. Nev-ertheless, one can observe that the peak frequencies are stillfunctions of the fault resistance (variable a). On the otherhand, for the existing method, from Equation (10), the expres-sion of the dominant frequency (corresponding to one ofthe peak frequencies in the proposed method) does not con-sider the fault resistance. Therefore, during faults other thansingle phase to ground faults, the proposed method consid-ers the fault resistance compared to the existing method. Infact, the existing method is a special case of the proposedmethod when the fault resistance is zero. Next, a brief proof isprovided.
8 NIE ET AL.
Proof. If Rf = 0, from the definition of the variable a, a = 0.Therefore, Equation (41) can be simplified as,
i𝛽(1)(0, s)
= (1 + p(s)2)U𝛽s1(s)
/{z𝛽[(−1 + c1(s) + (1 + c1(s))p(s)2]}. (48)
Therefore, the peak frequencies can be found when,
(−1 + c1(s) + (1 + c1(s))p(s)2) = 0. (49)
Substitute s = j𝜔 = j2𝜋 f into Equation (49), we have,
e2× j2𝜋 f ×d∕v𝛽 = |Γs| e j𝜋e j𝜃1 . (50)
where the reflection coefficient
Γs = (z𝛽(1) − z𝛽 )
/(z𝛽
(1) + z𝛽 ). (51)
𝜃1 is the angle of Γs , and
z𝛽(1)=R𝛽
(1) + sL𝛽(1), (52)
The phase angles at both sides of Equation (50) should beequal or with differences 2kπ,
f = (𝜃1 + 𝜋 + 2k𝜋)v𝛽/
(4𝜋d )(k = 0, 1, 2…). (53)
Therefore, the first peak frequency corresponds to k = 0. Wecan conclude that the expression is equivalent to Equation (10).Q.E.D.
5 PROPOSED FAULT LOCATIONALGORITHM
The peak frequencies of the measurements during faults areextracted. Here the three phase instantaneous current measure-ments at the local terminal are utilized. The three phase currentsare first transformed into mode currents. Afterwards, the spec-trum of one certain mode current can be calculated by MUSICalgorithm [22]. Within a certain frequency range, in the spec-trum of the current, there exist many peak frequencies denotedas fi
measured (i = 1, 2, …, n) other than the fundamental fre-quency of the system (50 or 60 Hz).
Next, the extracted peak frequencies of the measurementsare compared to the theoretical peak frequencies to obtain thefault location. These theoretical peak frequencies can be eas-ily extracted offline (even before the installation of the faultlocation device) by first substituting different frequencies intothe objective function in Equation (34) and afterwards applyinga numerical peak-searching algorithm. The derivations in Sec-tions 3 and 4 demonstrate that the theoretical peak frequenciesfi
theoretical (i = 1, 2, …, n) are functions of the fault location d
and the fault resistance Rf. The validity of the proposed method
is based on the following fact: The correct fault location andfault resistance should correspond to the best match betweenfi
measured and fitheoretical . Therefore, this is equivalently to solve
an optimization problem,
[d ,R f ] = mind ,R f
g(d ,R f ), (54)
where
g(d ,R f ) =n∑
i=1
||| fitheoretical − fi
measured ||| ∕n, (55)
is the objective function that describes the average differencebetween the theoretical and measured peak frequencies.
In practice, the number of theoretical and measured peakfrequencies may not be exactly the same within a certain fre-quency band (due to measurement errors, parameter errors etc.).Assume that the number of theoretical peak frequencies is m
while the number of measured peak frequencies is still n. Toimprove the robustness of the proposed fault location method,the proposed method first finds the closest theoretical and mea-sured peak frequency pairs and calculates their absolute fre-quency differences |fj
theoretical−fkmeasured|. Afterwards, these two
frequencies are removed from the two peak frequency datasets,and the same procedure is executed. In order to further improvethe robustness of this fault location algorithm and reject baddata/outliers, the procedure is executed until a certain percent-age (e.g. 80%) of either the theoretical or measured peak fre-quency dataset is covered. The procedure of calculating g(d, Rf)with given d and Rf is summarized as follows.
There are many optimization tools to solve Equation (54).Since here the theoretical frequencies with different d and Rf
can be calculated offline (even prior to the fault), the main cal-culation burden is to find the minimum value of the matrix Aas well as some summations and multiplications after extract-ing the measured peak frequencies, which are not quite high.Therefore, a very effective and straightforward way is to drawa mesh with different d and Rf, calculate all the values of g(d,
Rf), and directly find the minimum. Here we utilize a two-stepalgorithm to reduce the computational burden. For the firststep, the estimated location lf,step1 and fault resistance Rf,step1are obtained throughout the entire span of transmission line[0, l] and fault resistance [0, Rf,max], with resolution Δlf,step1 andΔRf,step1. For the second step, the accurate fault location lf,step2
NIE ET AL. 9
FIGURE 2 Flow chart of the proposed fault location algorithm
and fault resistance Rf,step2 are refined inside the small span oftransmission line [lf,step1−Δlf,step1, lf,step1 + Δlf,step1] and faultresistance [Rf,step1−ΔRf,step1, Rf,step1 + ΔRf,step1], with a higherresolutionΔlf,step2 andΔRf,step2. The flow chart of the proposedfault location algorithm is summarized in Figure 2.
6 NUMERICAL EXPERIMENTS
The proposed method is validated via a 50 Hz system with a500 kV, 200 km three phase ideally transposed transmission linebuilt in PSCAD/EMTDC. The example test system is the sameas Figure 1. For the tower structure, the phase conductors of theline are arranged horizontally (with the same height of 30 m),and the distances between phases AB, BC and AC are 10 m,10 m, 20 m, respectively. The source resistances and inductancesper phase of source 1 and source 2 are 0.9 ohm, 0.05 H and0.2 ohm, 0.01 H, respectively. The phase angle of left source isleading that of right source for 30◦. The available measurementsare three phase current sampled value measurements at the local(left) terminal of the line. The spectrum of the current duringthe fault is extracted using the MUSIC algorithm with signalspace parameter of 300. The sampling rate and data windoware selected as 100 kilo-samples per second and 10 ms after theoccurrence of the fault respectively according to ref. [22]. Thenthe absolute error of fault location is defined as,
Absolute Error =||||Estimated Location − Actual Location
Total Length ofthe Line
||||× 100%. (56)
FIGURE 3 Three phase current at the local terminal of 0.01 ohm 50 kmA-G fault
The theoretical peak frequencies are calculated offline priorto the fault. Here we consider the frequency spectrum with themaximum frequency of 10 kHz. The corresponding variablesare selected as follows: Δlf,step1 = 0.5 km, ΔRf,step1 = 5 ohm,Δlf,step2 = 0.1 km, ΔRf,step2 = 1 ohm. Also, for single phase toground faults, the theoretical peak frequencies can be calculatedby Equations (30) and (34), with Rf,max = 600 ohm; for othertypes of faults, the theoretical peak frequencies can be calcu-lated by Equations (41) and (34), with Rf,max = 50 ohm (theseselection of maximum fault resistances can cover extreme sit-uations). For the computational burden of the algorithm, herewe take the single phase to ground faults (with more compu-tational burden since Rf,max is larger) as examples. The maxi-mum calculation time without parallel computing is less than19 seconds using Matlab R2018b on a personal computer withIntel i7-7700 CPU (with 48,100 values of g(d, Rf); on averagefor each g(d, Rf) the calculation time is less than 400 μs). Thiscalculation time for fault location is acceptable in practice. Inaddition, the calculation time can be further minimized if thevalues of g(d, Rf) with different d and Rf are calculated usingparallel computing techniques, since the values of g(d, Rf) withdifferent d and Rf are independent and can be computed inparallel.
Next, the performances of the proposed method are com-pared to those of the existing natural frequency based faultlocation method (as described in Section 2.2 in detail) via dif-ferent test cases. Specifically, for single phase to ground faults,low and high resistance faults (0.01 ohm, 1 ohm, 10 ohm; 100ohm, 300 ohm, 500 ohm) are studied to demonstrate the effectof mode mixing phenomenon as well as the fault resistance onboth methods. For other types of faults, a specific fault resis-tance of 10 ohm is provided to demonstrate the effect of faultresistance on both methods.
6.1 Low resistance single phase to groundfaults
A 0.01 ohm A-G fault at 50 km from the local terminal is takenas an example. For this fault event, the three phase currentinstantaneous measurements at the local terminal are shown inFigure 3 (the current measurements are not shown for otherfault events due to space limitations). The values of g(d, Rf)in step 1 and step 2 of the proposed algorithm are shownin Figure 4(a),(b). The fault location and fault resistance that
10 NIE ET AL.
49.5km
(b) Step 2
(a) Step 1
FIGURE 4 Average difference of peak frequencies, different faultlocations and fault resistances, 50 km 0.01 ohm A-G fault
FIGURE 5 Comparison between the theoretical and measured frequencyspectrum, 50 km 0.01 ohm A-G fault
minimize g(d, Rf) are 49.6 km and 1 ohm, respectively, which areclose to actual fault location and fault resistance (the absolutefault location error is 0.2%). Figure 5 depicts the comparisonbetween the theoretical and the measured frequency spectrumsin detail. The theoretical frequency spectrum is with the opti-mal fault location d = 49.6 km and fault resistance Rf = 1 ohm.One can observe that the peak frequencies of the theoretical fre-quency spectrum are very consistent with those of the measured
TABLE 3 Average and max absolute errors for low resistance A-G faultswith different fault locations
Existing method Proposed method
Fault resistance Average error Max error Average error Max error
0.01 ohm 2.53% 11.20% 0.18% 0.30%
1 ohm 2.55% 11.20% 0.18% 0.25%
10 ohm 2.54% 11.07% 0.19% 0.30%
frequency spectrum, proving the effectiveness of the proposedcalculation method of the theoretical peak frequencies as well asthe proposed fault location method.
Next, the performance of the existing method is also stud-ied. From Figure 5, there are many peak frequencies in themeasured frequency spectrum and it is not trivial to extractthe dominant natural frequency. Here the first peak frequency(other than the fundamental frequency of 50 Hz) of the mea-sured frequency spectrum is not the dominant frequency: Thefirst peak frequency is 542.9 Hz, resulting in 204.34 km faultlocation according to Equation (10) (with wave speed approx-imated as the average speed of mode α and mode 0 [23,24]).In fact, if we determine the actual dominant frequency by find-ing the peak frequency that results in the most accurate faultlocation, the dominant frequency is 1822.9 Hz, as marked inFigure 5. It can be observed that it is very hard to determinethis dominant frequency. This fact is caused by the mode mix-ing phenomenon. Second, even we assume that the dominantfrequency can somehow be found, the fault location result ofthe existing method according to Equation (10) is 48.38 km (theabsolute fault location error is 0.81%). Therefore, the fault loca-tion accuracy of the proposed method is improved compared tothe existing method.
The effectiveness of proposed method is further validatedusing a group of low fault resistance (0.01 ohm, 1 ohm, 10 ohm)A-G faults at different fault locations through the line (21 faultlocations, 2 km, 198 km and every 10 km from ref. [10], 190km from local terminal). The results are shown in Figure 6(a)and Table 3. Here we assume that the dominant frequency cansomehow be accurately extracted for the existing method, toshow the best performance of the existing method for com-parison. The results demonstrate that compared to the existingmethod, fault location accuracy of the proposed method is alsosignificantly improved.
6.2 High resistance single phase to groundfaults
A 500 ohm A-G fault at 90 km from the local terminal is takenas an example. The values of g(d, Rf) in step 1 and step 2 ofthe proposed algorithm are shown in Figure 7(a),(b). The faultlocation and fault resistance that minimize g(d, Rf) are 90.0 kmand 470 ohm, respectively, which are close to actual fault loca-tion and fault resistance (the absolute fault location error is 0%).Figure 8 depicts the comparison between the theoretical and
NIE ET AL. 11
Proposed method
Existing method
(a).Low fault resistance
Proposed method
Existing method
(b).High fault resistance
FIGURE 6 Fault location results, A-G faults with different fault locations
the measured frequency spectrums in detail. Similarly, one canobserve that the peak frequencies of the theoretical frequencyspectrum are very consistent with those of the measured fre-quency spectrum.
Next, the performance of the existing method is also stud-ied. Similarly, there are many peak frequencies in the measuredfrequency spectrum and it is not trivial to extract the dominantnatural frequency (similarly, if we determine the actual dominantfrequency by finding the peak frequency that results in the mostaccurate fault location, the dominant frequency is 1009.8 Hz,as marked in Figure 8). Even we assume that the dominant fre-quency can somehow be found, the fault location result of theexisting method according to Equation (10) is 98.99 km (theabsolute fault location error is 4.95%). Compared to the exist-ing method, fault location accuracy is significantly improved byproposed method.
90.0km
(a) Step 1
90.0km
(b) Step 2
FIGURE 7 Average difference of peak frequencies, different faultlocations and fault resistances, 90 km 500 ohm A-G fault
FIGURE 8 Comparison between the theoretical and measured frequencyspectrum, 90 km 500 ohm A-G fault
The effectiveness of proposed method is further validated viaa group of high fault resistance (100 ohm, 300 ohm, 500 ohm)A-G faults at different fault locations through the line (21 faultlocations, 2 km, 198 km and every 10 km from ref. [10], 190km from local terminal). The results are shown in Figure 6(b)and Table 4. Similarly, here we assume that the dominant fre-quency can somehow be accurately extracted for the existing
12 NIE ET AL.
TABLE 4 Average and max absolute errors for high resistance A-G faultswith different fault locations
Existing method Proposed method
Fault resistance Average error Max error Average error Max error
100 ohm 2.28% 9.10% 0.18% 0.50%
300 ohm 8.14% 26.87% 0.28% 0.60%
500 ohm 7.72% 27.82% 0.31% 1.05%
method, to show the best performance of the existing methodfor comparison. The results demonstrate that compared tothe existing method, fault location accuracy of the proposedmethod is still significantly improved during high fault resistancefaults.
6.3 Other types of faults
Next, phase to phase faults, double phase to ground faults andthree phase faults are considered. Due to space limitations, herewe only take a 10 ohm phase B to phase C fault occurs at 100 kmas an example (for other types of faults the results are similar).The values of g(d, Rf) in step 1 and step 2 of the proposed algo-rithm are shown in Figure 9(a),(b). The fault location and faultresistance that minimize g(d, Rf) are 100.1 km and 9 ohm, respec-tively, which are close to actual fault location and fault resistance(the absolute fault location error is 0.05%). Figure 10 depictsthe comparison between the theoretical and the measured fre-quency spectrums in detail. Similarly, one can observe that thepeak frequencies of the theoretical frequency spectrum are veryconsistent with those of the measured frequency spectrum.
The performance of the existing method is also studied. Inthis case, it is straightforward to determine the dominant naturalfrequency: it is the first peak frequency 1123.1 Hz, and the faultlocation result of the existing method according to Equation(10) is 100.16 km (the absolute fault location error is 0.08%).The existing method has similar fault location accuracy com-pared to the proposed method. This is because for other typesof faults (other than single phase to ground faults) there doesnot exist the mode mixing phenomenon and therefore the faultlocation result is rather accurate for the existing method.
The effectiveness of proposed method is further validatedusing phase B to phase C faults, phase BC to ground faults,and three phase faults with 10 ohm fault resistance and differ-ent fault locations through the line (21 fault locations, 2 km,198 km and every 10 km from ref. [10], 190 km from local termi-nal). The results are shown in Figure 11 and Table 5. The resultsdemonstrate that the proposed method presents slightly higher(comparable) fault location accuracy than the existing method.This is because the proposed method considers the effect offault resistance. Nevertheless, for these types of faults, boththe proposed method and the existing method present accuratefault location results, and the improvements while consideringthe fault resistance using the proposed method are not signifi-cant.
FIGURE 9 Average difference of peak frequencies with different faultlocations and fault resistances, 100 km 10 ohm phase B to phase C fault
FIGURE 10 Comparison between the theoretical and measuredfrequency spectrum, 100 km 10 ohm phase B to C fault
TABLE 5 Average and max absolute errors for other types of faults with10ohm resistance and different fault locations
Existing method Proposed method
Fault types Average error Max error Average error Max error
B to C 0.074% 0.17% 0.055% 0.1%
BCG 0.095% 0.22% 0.055% 0.15%
3 phase 0.067% 0.19% 0.050% 0.1%
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FIGURE 11 Absolute fault location errors, other types of faults, 10 ohmfault resistance, variable fault location
TABLE 6 Average and max absolute errors, 0.01 ohm A-G faults withdifferent loading conditions, variable fault locations
Existing method Proposed method
Angle
difference
Average
error Max error
Average
error Max error
10◦ 2.53% 11.20% 0.18% 0.30%
20◦ 2.52% 11.19% 0.18% 0.30%
30◦ 2.53% 11.20% 0.18% 0.30%
40◦ 2.53% 11.20% 0.18% 0.30%
50◦ 2.51% 11.20% 0.18% 0.30%
7 DISCUSSION
In this section, the effectiveness of proposed method is furtherdiscussed with different factors. All the conditions of exampletest system are the same as those in Section 6. For all the fol-lowing studies (if not specially mentioned), 0.01 ohm A-G faultswith different fault locations (21 fault locations, 2 km, 198 kmand every 10 km from [10, 190] km from the local terminal) aretaken as examples.
7.1 Effect of loading conditions
The effect of loading conditions (phase angle of source 1 lead-ing that of source 2 for 10◦, 20◦, 30◦, 40◦ and 50◦) is studied.Figure 12(a) and Table 6 summarize the results of fault location.The fault location results of both the proposed and the existingmethods do not change much with different loading conditions.
7.2 Effect of measurement noises
The effect of measurement noises (Gaussian distribution, with0.2%, 0.5% and 1% standard deviations) is studied. Figure 12(b)
FIGURE 12 Absolute fault location errors, different loading conditions,measurement noises and parameter errors, 0.01 ohm fault resistance, variablefault locations
and Table 7 summarize the results of fault location. The faultlocation errors generally increase with higher measurementnoises. The fault location accuracy of the proposed method isstill higher compared to the existing method.
14 NIE ET AL.
TABLE 7 Average and max absolute errors, 0.01 ohm A-G faults withdifferent measurement noises, variable fault locations
Existing method Proposed method
Meas.errors
Average
error Max error
Average
error Max error
0.2% 2.52% 11.20% 0.13% 0.35%
0.5% 2.55% 11.15% 0.13% 0.45%
1% 2.57% 11.20% 0.14% 0.65%
TABLE 8 Average and max absolute errors, 0.01 ohm A-G faults withdifferent parameter errors, variable fault locations
Existing method Proposed method
Parameter
errors
Average
error Max error
Average
error Max error
0.2% 2.60% 11.40% 0.03% 0.20%
0.5% 2.73% 11.87% 0.35% 0.55%
1% 2.96% 12.32% 0.92% 1.45%
7.3 Effect of parameter errors
The effect of parameter errors (all parameter matrices of theline with parameter errors of 0.2%, 0.5% and 1%) is studied.Figure 12(c) and Table 8 summarize the results of fault loca-tion. Similarly, the fault location errors generally increase withhigher parameter errors. The fault location accuracy of pro-posed method is still higher compared to the existing method.
7.4 Effect of sampling rates
The effect of sampling rate (20 kHz, 50 kHz, 100 kHz and200 kHz) is studied. Here a group of A-G faults with 0.01 ohmfault resistance is selected as examples. The fault location resultsare shown in Figure 13 and Table 9. From the figure, one canobserve that when the sampling rate is relatively low (20 kHz or50 kHz), the existing method does not have solution for faultsclose to the local terminal (in this case 2 km, since the corre-sponding dominant frequency is too high to be captured). In
TABLE 9 Average and max absolute errors, 0.01 ohm A-G faults withdifferent sampling rates, variable fault locations
comparison, during 0.01 ohm A-G faults, the proposed methodcan still accurately locate faults through the line. Note that dur-ing extreme conditions (e.g. faults close to terminals, high faultimpedances low sampling rates), both the existing and the pro-posed method could possibly encounter accuracy issues. Withextensive experiments, the 100 kHz sampling rate (which isconsistent with ref. [22]) can ensure accurate extraction of fre-quency spectrum and accurate fault location results through thefault location procedure.
7.5 Effect of line lengths
The effect of different line length (300 km, 400 km and 500 km)is studied. The fault location results are summarized in Figure 14and Table 10. One can observe that the fault location accuracyof the proposed method does not change much, and the pro-posed method still presents high fault location accuracy com-pared to the existing method.
NIE ET AL. 15
TABLE 10 Average and max absolute errors, 0.01ohm A-G faults withdifferent line lengths, variable fault locations
Existing method Proposed method
Line length
Average
error Max error
Average
error Max error
300 km 2.26% 8.99% 0.15% 0.27%
400 km 3.02% 13.37% 0.12% 0.25%
500 km 4.71% 13.23% 0.12% 0.42%
FIGURE 15 Fault location results, A-G faults with different inceptionangles and fault resistances, variable fault locations
7.6 Effect of inception angles andadvantages towards traveling wave based faultlocation methods
First, the effect of fault inception angles (including 0◦, 10◦, 30◦)is studied and the fault location results are shown in Figure 15(a)and Table 11. One can observe that the fault location accuracydoes not change much for both the existing and the proposedmethod, and the proposed method still presents high fault loca-tion accuracy compared to the existing method.
To further evaluate the performance of the proposed methodduring zero inception angle, a group of A-G faults with differ-ent fault resistances (1 ohm, 100 ohm, 500 ohm) is studied. The
TABLE 11 Average and max absolute errors, 0.01 ohm A-G faults withdifferent inception angles, variable fault locations
Existing method Proposed method
Inception
angle
Average
error Max error
Average
error Max error
0◦ 2.53% 11.20% 0.18% 0.30%
10◦ 2.55% 11.15% 0.18% 0.30%
30◦ 2.53% 11.20% 0.18% 0.30%
TABLE 12 Average and max absolute errors for zero inception angle A-Gfaults with different fault resistances, variable fault locations
Existing method Proposed method
Fault
resistance
Average
error Max error
Average
error Max error
1 ohm 2.53% 11.20% 0.17% 0.30%
100 ohm 2.27% 9.07% 0.17% 0.45%
500 ohm 9.39% 27.83% 0.36% 1.05%
results are shown in Figure 15(b) and Table 12. One can observethat the proposed method can accurately locate faults even dur-ing extreme conditions, with zero inception angle and very highfault resistances.
In fact, the validity of the traveling wave based methods ishighly dependent on the reliable detection of the wavefronts.During faults with small inception angles or high fault resis-tances, the intensity of fault-created traveling wave is weak andthe detection of the wavefronts could be problematic. For exam-ple, if the fault inception angle is close to zero, the wavefrontscannot be detected and the traveling wave based method willfail. On the contrary, the proposed method can still accuratelylocate faults even with zero fault inception angle. This is becausethe derivation procedure of the proposed method is based onfrequency spectrum of the measurements, and is independentof the intensity of wavefronts. These results demonstrate clearadvantages of the proposed method towards traveling wavebased methods.
7.7 Advantages towards conventionalsingle-ended impedance based method
Comparing to the conventional single-ended impedance basedfault location (IBFL) methods [27], the advantages of the pro-posed method are summarized as follows. (a) Short data win-dow: Since the IBFL methods are based on fundamental fre-quency phasors, accurate extraction of phasors is mandatoryfor accurate IBFL results. However, if the transmission line istripped with high speed after the fault, the available data win-dow is short (e.g. 10 ms) and in this case the phasor estimationaccuracy will be limited by the system transients and the decay-ing DC offset. In comparison, the proposed method is based on
16 NIE ET AL.
instantaneous measurements, does not require phasor extrac-tion and is immune to decaying DC offset. Therefore, the pro-posed method can accurately locate faults with the short timewindow of 10 ms after the fault and is compatible with trans-mission lines using high-speed tripping techniques. (b) Accu-rate transmission line model: IBFL methods are usually basedon lumped parameters of the transmission line, and thereforeresult in compromised fault location accuracy. In comparison,the derivation of the proposed method is based on distributedparameter model and the modelling accuracy of the transmis-sion line is much improved. As a result, the proposed methodpresents higher fault location accuracy compared to the conven-tional IBFL methods especially for relatively long transmissionlines.
7.8 Other discussions and future work
In section 7.2, the measurement noises are assumed to be Gaus-sian distributed. In practice, if the system operates with con-ventional instrumentation current transformers (CTs) instead ofoptical CTs, the current measurements during faults may experi-ence systematic errors due to CT saturations etc. In this case, themeasurement errors could possibly compromise the fault loca-tion accuracy. Therefore, detailed modelling of CTs might berequired to enable accurate recovery of primary currents duringthe fault [28] before applying the proposed natural frequencybased fault location method.
Besides, the proposed fault location methodology couldpotentially be applied to other transmission lines. Detail veri-fications of the proposed methodology for other transmissionlines will be covered in future publications.
8 CONCLUSION
This paper proposes an improved natural frequency based faultlocation method for transmission lines. Only three phase instan-taneous current measurements at the local terminal are required.The relationship between the theoretical peak frequencies andthe fault location is first strictly derived. The fault location canbe afterwards obtained by minimizing the average differencesbetween the theoretical peak frequencies and measured peakfrequencies. Compared to the existing natural frequency basedfault location method, the proposed method systematically con-siders the mode mixing phenomenon and fault resistances, andfully utilizes the information embedded in the frequency spec-trum of traveling waves instead of only the dominant frequency,resulting in improved fault location accuracy. Numerical exper-iments prove that compared to the existing method, the pro-posed method has much higher fault location accuracy duringsingle phase to ground faults, and slightly higher (comparable)fault location accuracy during other types of faults, with differ-ent fault types, fault locations and fault resistances. The methodalso works with zero fault inception angle, where the system iswithout any fault-induced transient traveling wave.
ACKNOWLEDGEMENTS
This work was sponsored by National Natural Science Founda-tion of China (No. 51807119). The support is greatly appreci-ated.
REFERENCES
1. Brahma, S.M., Girgis, A.A.: Fault location on a transmission line using syn-chronized voltage measurements. IEEE Trans. Power Del. 19(4), 1619–1622 (2004)
2. Liu, Y., et al.: Fault location algorithm for non-homogeneous transmissionlines considering line asymmetry. IEEE Trans Power Del. 35(5), 2425–2437 (2020)
3. Das, S., et al.: Impedance-based fault location in transmission networks:Theory and application. IEEE Access. 2, 537–557 (2014)
4. Kawady, T., Stenzel, J.: A practical fault location approach for double cir-cuit transmission lines using single end data. IEEE Trans on Power Del.18(4), 1166–1173 (2003)
6. Kang, N., Chen, J., Liao, Y.: A fault-location algorithm for series-compensated double-circuit transmission lines using the distributedparameter line model. IEEE Trans. Power Del. 30(1), 360–367(2015)
7. Terzija, V.V., Radojevic, Z.M.: Numerical algorithm for adaptive autoreclo-sure and protection of medium-voltage overhead lines. IEEE Trans. PowerDel. 19(2), 554–559 (2004)
8. Liu, Y., et al.: Dynamic state estimation-based fault locating ontransmission lines. IET Gen. Transm. Distrib. 11(17), 4184–4192(2017)
9. Fan, R., et al.: Precise fault location on transmission lines using ensemblekalman filter. IEEE Trans Power Del. 33(6), 3252–3255, (2018)
10. Suonan, J., et al.: A novel fault-location method for HVDC transmissionlines. IEEE Trans. Power Del. 25(2), 1203–1209, (2010)
11. Davoudi, M., Sadeh, J., Kamyab, E.: Transient-based fault location onthree-terminal and tapped transmission lines not requiring line parameters.IEEE Trans Power Del. 33(1), 179–188, (2018)
12. Lu, D., et al.: Time-domain transmission line fault location method withfull consideration of distributed parameters and line asymmetry. IEEETrans. Power Del. 35(6), 2651–2662, (2020)
13. Azizi, S., et al.: A traveling-wave-based methodology for wide-area faultlocation in multiterminal DC systems. IEEE Trans. Power Del. 29(6),2552–2560, (2014)
14. Liu, Y., et al.: Protection and fault locating method of series compensatedlines by wavelet based energy traveling wave. IEEE Power Energy Soc.Gen. Meet. (PESGM). Chicago, IL, USA, (2017), 1–5
15. Naidu, O., Pradhan, A.K.: A traveling wave-based fault locaton methodusing unsynchronized current measurements. IEEE Trans. Power Del.34(2), 505–513 (2019)
16. Lopes, F.V., et al.: Accurate two-terminal transmission line fault locationusing traveling waves. IEEE Trans. Power Del. 33(2), 873–880 (2018)
17. Wang, D., Hou, M., Guo, Y.: Travelling wave fault location of HVAC trans-mission line based on frequency-dependent characteristic. IEEE Trans.Power Del., early access.
18. Evrenosoglu, C.Y., Abur, A.: Travelling wave based fault location for teedcircuits. IEEE Trans. Power Del. 20(2), 1115–1121 (2005)
19. Zhang, C., et al.: Single-ended traveling wave fault location method in DCtransmission line based on wave front information. IEEE Trans. PowerDel. 34(5), 2028–2038 (2019)
20. Spoor, D., Jian, G.: Improved single-ended traveling-wave fault-locationalgorithm based on experience with conventional substation transducers.IEEE Trans. Power Del. 21(3), 1714–1720 (2006)
21. Swift, G.W.: The spectra of fault-induced transients. IEEE Trans. PowerApp. Syst. PAS-98(3), 940–947 (1979)
22. He, Z., et al.: Natural frequency-based line fault location in HVDC lines.IEEE Trans. Power Del. 29(2), 851–859 (2014)
NIE ET AL. 17
23. Song, G., et al.: A fault-location method for VSC-HVDC transmissionlines based on natural frequency of current. Int. J. Elect. Power EnergySyst. 63, 347–352 (2014)
24. Wu, L., He, Z., Qian, Q.: A new single ended fault location techniqueusing traveling wave natural frequencies. IEEE Power Energy Soc. AsiaPacific Power Energy Eng. Conf. (APPEEC), Wuhan, China, (2009),pp. 1–5
25. He, Z., Li, X., Chen, S.: A traveling wave natural frequency based singleended fault location method with unknown equivalent system impedance.Int. Trans. Elect. Energy Syst. 26, 509–524 (2016)
26. Gopalakrishnan, A., et al.: Fault location using the distributed parametertransmission line model. IEEE Trans Power Del. 15(4), 1169–1174 (2000)
27. IEEE guide for determining fault location on AC transmission and distri-bution lines. IEEE Standard C37.114, (2014)
28. Kong, Y., Meliopoulos, A.P., Cokkinides, G., On-line current instrumenta-tion channel error correction within merging units using constraint WLSdynamic state estimation method. in North American Power Symposium(NAPS), Fargo, ND, (2018), pp. 1–5
How to cite this article: Nie Y., Liu Y., Lu D., Wang B..An improved natural frequency based transmission linefault location method with full utilization of frequencyspectrum information. IET Gener. Transm. Distrib.,1–17. (2021) https://doi.org/10.1049/gtd2.12215.
