A New Theory for Locating Line Fault in Power System: Simulation Part

Three time-reversal-based fault location methods are derived from the theoretical part of this paper, which is named currents-in-time-domain method, forward-currents-in-frequency-domain method, and forward-currents-in-frequency-domain-with-compensation method, respectively. The first proposed method utilizes only the decomposed transient currents in the time domain to calculate the RMS values of the fault currents of assumed fault in a lossless mirror line. The location where the peak value is located is the calculated fault location. The second method uses the forward current in the frequency domain to calculate the RMS values of the fault currents of assumed fault in a loss mirror line. The location where the minimum value is located is the calculated fault location. The third method first calculates the RMS values of fault currents of assumed fault in lossless mirror line, and the fault location is selected from the peak value. Then the final fault location result is compensated by the pre-calculated error for each assumed fault location. A two-terminal AC transmission system is established for testing these three methods. The frequency-dependent lines including homogeneous and inhomogeneous lines are taken into the simulated system for a large number of tests.

SECTION I. Introduction

Time reversal theory brings up a brand new idea for locating the electromagnetic sources. [4] When a fault occurs in a line, an electromagnetic wave is inserted into the power network. The transmission of the wave causes the variation of the voltage and the current along the line.

Researchers in Switzerland applied the electromagnetic time reversal (EMTR) theory into the fault location in the AC power distribution network. They utilize the transients at the point of common coupling to locate the fault segment. The theoretical proof firstly writes out the transfer function between the assumed fault location and the measurement in the actual line, as well as in the mirror line. Then, the phase frequency curve of the ratio between the two functions is acquired. Finally, the phase value would exceed the rated value in the high frequency part at the actual fault location. This method actually uses the ultrahigh frequency part of the transients to locate the fault. [5]–​[8] The method needs high sampling frequency which limits its development.

Our research team applied the EMTR theory into the voltage source converter based high voltage direct current (VSC-HVDC) system. Reference [9] locates the fault in multi-terminal VSC-HVDC system. The transients are filtered by elliptic filter and the 1-mode currents are used for locating the line-to-line fault while the 0-mode currents are applied for locating line-to-ground fault. Reference [10] is derived from [9]. It utilizes the wavelet technology to filter the 1-mode currents to locate the fault. This paper proves that the theory of EMTR-based fault location is correct. The proof process is similar to that in the theoretical part of our article. Based on the method in [9] and [10], the fault location can be calculated precisely with low sampling frequency.

In order to improve the credibility of the theoretical proof, different simulations of fault locations in AC systems are tested.

SECTION II. Three Fault Location Algorithms

Three time reversal-based methods are derived from the theoretical part in this paper. In this section, the detailed procedures of these methods are introduced.

A. Currents-in-Time-Domain Method

The method that only uses measured currents at the ends of the line to locate fault is named currents-in-time-domain method. Fig. 1 gives the flow chart of this method. The steps are as follows:

• Step1:

  the fault currents at the two ends of the line are recorded. The sampling start time is set at the arbitrary time during normal operation of the system and the sampling terminal time is set at the time when the breakers operate. The fault currents are indicated as $i_{M}(t)$ and $i_{N}(t)$ .

• Step2:

  filter the fault currents and extract the initial forward currents generated by the additional voltage source at the fault location. The wavelet technology can decompose the currents and express the break variable of the currents well with its excellent time and frequency feature. The decomposed currents are indicated as $i_{Mf}(t)$ and $i_{Nf}(t)$ .

• Step3:

  suppose that the time window is $T$ , $i_{Mf}(t)$ and $i_{Nf}(t)$ are time reversed in this time window, which are expressed as $i_{Mf}(T-t)$ and $i_{Nf}(T-t)$ .

• Step4:

  by assuming a metallic fault at $l_{Z}$ from end $M$ , its fault current can be calculated in (1) without considering the reflections:\begin{equation*} i_{lz}=i_{Mf}(T-t-\Delta t_{1})+i_{Nf}(T-t-\Delta t_{2})\tag{1}\end{equation*} where, \begin{align*} \begin{cases} \displaystyle \Delta t_{1} =\sum \nolimits _{i1} {{l_{i1}} / {v_{i1}}} \left ({{\sum \nolimits _{i1} {l_{i1}} =l_{Z}} }\right) \\[4pt] \displaystyle \Delta t_{2} =\sum \nolimits _{i2} {{l_{i2}} / {v_{i2}}} \left ({{\sum \nolimits _{i2} {l_{i2}} =l-l_{Z}} }\right) \end{cases}\tag{2}\end{align*}

  $l_{i}$ represents the length of the line whose wave velocity is $v_{i}$ . If the line is a homogeneous line, $\Delta t_{1}=l_{Z}/v$ and $\Delta t_{2}=(l-l_{Z})/v$ .

• Step5:

  calculate the RMS value of (1), and its discrete expression is shown in (3).\begin{equation*} \left |{ {i_{l_{Z}}} }\right |=\sqrt {{\left [{ {\sum \nolimits _{n=1}^{T \mathord {\left /{ {\vphantom {T {f_{s}}}} }\right. } {f_{s}}} {i_{l_{Z}}^{2} \left ({{nf_{s}} }\right)}} }\right]} \mathord {\left /{ {\vphantom {{\left [{ {\sum \nolimits _{n=1}^{T \mathord {\left /{ {\vphantom {T {f_{s}}}} }\right. } {f_{s}}} {i_{l_{Z}}^{2} \left ({{nf_{s}} }\right)}} }\right]} {\left ({{T \mathord {\left /{ {\vphantom {T {f_{s}}}} }\right. } {f_{s}}} }\right)}}} }\right. } {\left ({{T \mathord {\left /{ {\vphantom {T {f_{s}}}} }\right. } {f_{s}}} }\right)}}\tag{3}\end{equation*} where, $f_{s}$ is the sampling frequency.

FIGURE 1.

Flow chart of currents-in-time-domain method.

Finally, by assuming a fault at every point of the mirror line, we can obtain a series of RMS values of the fault current of assumed faults and select the peak value, which is $\max \left \{{\left |{i_{l z}}\right |}\right \}$ . The location where the peak value is located is the calculated fault location.

B. Forward-Currents-in-Frequency-Domain Method

The method that uses measured voltages and currents at the ends of the line to locate fault is named forward-currents-in-frequency-domain method. Fig. 2 gives the flow chart of this method. The steps are as follows:

• Step1:

  the voltages and currents are real time recorded, which are indicated as $u_{M}(t)/i_{M}(t)$ and $u_{N}(t)/i_{N}(t)$ .

• Step2:

  through differential and low pass filtering, the sequence voltages and currents of power frequency, indicated as $U_{M}/I_{M}$ and $U_{N}/I_{N}$ , are calculated based on fast Fourier transform (FFT). The sliding window of FFT is 20ms. Then, the sequence forward currents are calculated as follows:\begin{equation*} I_{Msf}=U_{M}/Z_{C}+I_{M} I_{Nsf}=U_{N}/Z_{C}+I_{N}\tag{4}\end{equation*}

• Step3:

  calculate the complex conjugate of (4), which are expressed as $I_{M s f}^{*}$ and $I_{N s f}^{*}$ . As is shown in (5), the fault current of assumed fault at $l_{Z}$ in the mirror line are calculated.\begin{equation*} I_{l_{Z}}^{\ast } =\frac {1}{I_{Msf}^{\ast }}\frac {e^{-2\gamma ^{\ast }l_{Z}}}{1+\Gamma _{M}^{\ast } e^{-2\gamma ^{\ast }l_{Z} }}-\frac {1}{I_{Nsf}^{\ast }}\frac {e^{-2\gamma ^{\ast }\left ({{l-l_{Z}} }\right)}}{1+\Gamma _{N}^{\ast } e^{-2\gamma ^{\ast }\left ({{l-l_{Z}} }\right)}}\tag{5}\end{equation*} where, $\Gamma _{M}^{*}$ and $\Gamma _{N}^{*}$ are the reflection coefficients at two ends of the mirror line, and $\gamma ^{\ast }$ is the propagation coefficient of the mirror line

FIGURE 2.

Flow chart of forward-currents-in-frequency-domain method.

Finally, by assuming a fault at every point of the mirror line, we can obtain a series of RMS values of the fault current of assumed faults and select the minimum value, which is $min \left \{{\left |{i_{lz}}\right |}\right \}$ . The location where the minimum value is located is the calculated fault location.

C. Forward-Currents-in-Frequency-Domain-With-Compensation Method

The method is derived from the method in the last subsection that uses the pre-calculation errors to compensate the fault location result, so the method is called forward-currents-in-frequency-domain-with-compensation method. Fig. 3 gives the flow chart of this method. The first steps are the same as the last proposed method.

• Step2:

  select a specific frequency and all the parameters in further calculation are based on this frequency.

• Step3:

  the error expression of fault location result is described in (41) in the theoretical part of this work. The function $f(l_{Z})$ at the specific frequency is pre-calculated.

• Step4:

  if the phase coefficient of the line at the specific frequency can satisfy the inequality in (42) in the theoretical part of this work, continue the further calculation. If not, the frequency should be selected again.

• Step5:

  after $I_{M s f}^{*}$ and $I_{N s f}^{*}$ are calculated, formula (6) is used for calculating the fault current of assumed fault.\begin{equation*} I_{l_{Z}}^{\ast } =I_{Msf}^{\ast } e^{-j\sum \nolimits _{i1} {l_{i1} \beta _{i1}}}+I_{Msf}^{\ast } e^{-j\sum \nolimits _{i2} {l_{i2} \beta _{i2}}}\tag{6}\end{equation*} where, $\beta _{i}$ is the phase coefficient of the line. If the line is a homogeneous line, formula (6) would be simplified into (7).\begin{equation*} I_{l_{Z}}^{\ast } =I_{Msf}^{\ast } e^{-j\beta l_{Z}}+I_{Msf}^{\ast } e^{-j\beta \left ({{l-l_{Z}} }\right)}\tag{7}\end{equation*}

• Step6:

  the RMS value of the fault current can be calculated by assuming a fault at every point of the mirror line. We can achieve a series of RMS values and select the peak value which is $\max \left \{{\left |{I_{l z}^{*}}\right |}\right \}$ . The location where the peak value is located is the initial fault location. However, the result needs to be further compensated and revised in the next step.

FIGURE 3.

Flow chart of forward-currents-in-frequency-domain-with- compensation method.

Finally, min{$\vert f(l_{Z})$ -initial fault location$\vert $ } is calculated, and the $l_{Z}$ which corresponds to the minimum value is the final calculated fault location.

SECTION III. Simulation Tests for a Homogeneous Line

A single-line graphic of two-terminal AC transmission system is simulated in PSCAD as is shown in Fig. 4. The simulated line adopts the frequency-dependent line model. The length of the line is 500km whose detailed parameters are in appendix A.

FIGURE 4.

Two-terminal AC transmission system.

The self-impedance and self-admittance of the line is $(0.123125\times 10^{-3}+j0.663124\times 10^{-3})\Omega $ /m and $(0.100000\times 10^{-9}+j0.246251\times 10^{-8})\text{S}$ /m, respectively.

The mutual impedance and admittance is $(0.884491\times 10^{-4}+j0.239759\times 10^{-3}\Omega $ /m and $(j0.263477\times 10^{-9})\text{S}$ /m, respectively. The positive sequence impedance equals to the negative sequence impedance for the two AC sources at the ends of the line.

The faults simulated in PSCAD in this paper are permanent faults. Suppose that an A-phase ground fault with fault resistance of $30\Omega $ occurred at 100km from end $M$ , the transients at two terminals of the line are recorded. Different methods uses different sampling frequencies and different transients. The calculation steps are implemented in MATLAB in this paper. The applications of the proposed three methods for locating fault will be introduced as follows.

A. Method I

The three-phase fault currents at two terminals of the line are recorded. This method does not need detect the precise fault occurrence time. The start time of recorders can be any time in the normal operation and the stop time of recorders can be at the time when the breaks operate.

Firstly, the three-phase currents are recorded from 0.396s to 0.408s and the sampling frequency is 50kHz when the method I is used for fault location. secondly, the aerial mode currents are calculated from the recorded three-phase currents by using phase-mode transformation. Then, the aerial mode currents are decomposed by using wavelet function of ‘db2’ and the further processed decomposed currents are shown in Fig. 5.

FIGURE 5.

Decomposed aerial mode fault currents with wavelet technology: (a) decomposed current at end $M$ , (b) decomposed current at end $N$ .

After the currents in Fig. 5 are time reversed, the fault current of assumed fault can be calculated by using (1). Finally, suppose a fault occurred at every 1m along the lossless mirror line, and the wave velocity is $3\times 10^{8}\text{m}$ /s. the RMS values of all fault currents of assumed faults are shown in Fig. 6.

FIGURE 6.

RMS values of fault current of assumed faults for lossless mirror line.

The filtered currents in Fig. 5 contain many reflected current waves. The superposition of the initial current waves and the reflected current waves will cause many extreme values in Fig. 6. However, there is only one peak value in Fig. 6 because the amplitude of reflected current waves are smaller than the initial current waves. The peak value is only at 100.584km from end $M$ in Fig. 6 so the fault location result is 100.584km.

B. Method II

The positive sequence voltages and currents at two ends of the line are calculated by using FFT technology with a sampling frequency of 12.8 kHz based on a sliding window of 20ms. Fig. 7 shows the positive sequence voltages and currents that are calculated from the the three-phase voltages and currents at two ends of the line by using differential and low pass filters. The sequence forward currents can be calculated from the the positive sequence voltages and currents by using (4). The RMS values of fault currents of assumed faults at every 1m along the loss mirror line can be finally calculated by using (5), which are shown in Fig. 8. The parameters in (5) are as follows: $\Gamma ^\ast _{M}=-0.3376-0.1672j, \Gamma ^\ast _{N}=-0.8967-0.2187j, \gamma ^{\ast }=6.3683\times 10^{-8}-1.0746\times 10^{-6}j$ .

FIGURE 7.

Positive sequence voltages and currents at the ends of the line: (a) Positive sequence voltage at end $M$ , (b) Positive sequence current at end $M$ , (c) Positive sequence voltage at end $N$ , (b) Positive sequence current at end $N$ .

FIGURE 8.

RMS values of fault current of assumed faults for loss mirror line.

The minimum value of Fig. 8 only exits at 99.983km from end $M$ , so the fault location result is 99.983km.

C. Method III

The curve of (41) in the theoretical part of this paper is shown in Fig. 9(a). It is the sum of the assumed fault location and the error caused by the calculation at the assumed fault location. Function $f(l_{Z})$ can be expressed as:\begin{equation*} f(l_{Z})=A1+A2\tag{8}\end{equation*} where A1 represents the assumed fault location, and A2 represents the inherent errors caused by angle difference.

FIGURE 9.

Fault location process of method III: (a) Fault location errors at the assumed fault location, (b) RMS values of fault current of assumed fault, (c) Final fault location result.

Then, the inherent errors can be expressed as:\begin{equation*} A2 =A3-A1\tag{9}\end{equation*} where A3 represents the theoretical result by using method III to locate the fault.

So function $f(l_{Z})$ equals to A3, which represents theoretical result by using method III to locate the fault.

The fault location can be firstly calculated in the lossless mirror line by using (7) based on the sequence forward currents which are calculated from the the positive sequence voltages and currents in Fig. 7. The fault currents of assumed faults are shown in Fig. 9(b). It should be pointed out that the assumed fault locations of this calculation are within the range of $f(l_{Z})$ , that is $\text {min}\{f(l_{Z})\}\le l_{Z}\le \text {max}\{f(l_{Z})\}$ . Then, the peak value of Fig. 9(b) is selected and the fault location is indicated as ‘location’.

In order to ensure the uniqueness of the fault location result, the value of the phase coefficient should satisfy the constraints in (10).\begin{equation*} \pi / \beta >\max \left \{{{f\left ({{l_{Z}} }\right),l_{Z} \in \left [{ {0,l} }\right]} }\right \}\tag{10}\end{equation*}

Since $\pi /\beta =2923.499$ km and $\text {max}\{f(l_{Z})\}=134.362$ km, it can be seen that the redundant solutions are not in the correct interval. Therefore, the fault location result will be unique.

Finally, we find the minimum value of $\vert $ location-$f(l_{Z})\vert $ , as is shown in Fig. 9(c), the minimum value appears at 98.547km from end $M$ so it is the fault location result.

D. Simulated Cases for Different Line Faults

In this section, 7 kinds of faults are simulated for different case studies. They are A-phase ground fault with fault resistance of $0\Omega $ (AGF-$0\Omega$ ), A-phase ground fault with fault resistance of $30\Omega $ (AGF-$30\Omega$ ), A-phase ground fault with fault resistance of $300\Omega $ (AGF-$300\Omega$ ), AB-phase ground fault with fault resistance of $300\Omega $ (ABGF-$300\Omega$ ), AB-phase fault with fault resistance of $30\Omega $ (ABF-$30\Omega$ ), ABC-phase ground fault with fault resistance of $300\Omega $ (ABCGF-$300\Omega$ ), and ABC-phase fault with fault resistance of $30\Omega $ (ABCF-$30\Omega$ ).

Suppose these faults occurred at every 20km of the line, respectively, Tab. 1–​3 give the fault location results by using the three methods described in the previous sections. The mean relative error (MRE) of results in these tables is 0.2216%, 0.07476% and 0.09332%, respectively. The fault location results for the three methods are of high precision.

TABLE 1 Fault Location Results by Using Method I for a Homogeneous Line

TABLE 2 Fault Location Results by Using Method II a Homogeneous Line

TABLE 3 Fault Location Results by Using Method III for a Homogeneous Line

SECTION IV. Simulation Tests for an Inhomogeneous Line

A single-line graphic of two-terminal AC transmission system is shown in Fig. 10. The simulated line adopts the frequency-dependent line model. The line is an inhomogeneous line consisting of two mediums. The detailed parameters of L₁ and L₂ are in appendix A and B, respectively.

The self-impedance and self-admittance of L₂ is $(0.119981\times 10^{-3}+j0.664752\times 10^{-3})\Omega /\text {m}$ and $(0.100000\times 10^{-10}+j 0.245690\times 10^{-8})\text{S}$ /m, respectively.

FIGURE 10.

Two-terminal AC transmission system with lines consisting of two lines with different parameters.

The mutual impedance and admittance is $(0.856480\times 10^{-4}+j 0.245939\times 10^{-3})\Omega $ /m and $(j0.287197\times 10^{-9})\text{S}$ /m, respectively. The positive sequence impedance equals to the negative sequence impedance for the two AC sources at the ends of the line.

In order to ensure the uniqueness of the result by using method III to locate fault, the value of phase coefficient of L₁ and L₂ should satisfy the constraints in (11).\begin{align*}&\left [{\left [{ \pi / \beta _{1}>\max \left \{{f\left ({l_{z}}\right), l_{Z} \in \left [{0, l_{1}+l_{2}}\right]}\right \} }\right]}\right]\quad \text {and} \\&\Biggl [{\Biggl [{ \pi / \beta _{2}>\max \left \{{f\left ({l_{z}}\right), l_{z} \in \left ({l_{1}+l_{2}, l_{1}+l_{2}+l_{3}}\right]}\right \}}} \\&or~{{\pi / \beta _{2}\!< \!\min \left \{{f\left ({l_{z}}\right), l_{z} \in \left ({l_{1}\!+\!l_{2}, l_{1}\!+\!l_{2}\!+\!l_{3}}\right]}\right \} }\Biggr]}\Biggr] {}\tag{11}\end{align*}

Because $\pi /\beta _{1}=2923.499$ km and $\pi /\beta _{2}=2624.710$ km, $\max \left \{{{f\left ({{l_{Z}} }\right),l_{Z} \in \left [{ {0,l_{1} +l_{2}} }\right]} }\right \}=\textrm {202.510km}$ and $\max \left \{{{f\left ({{l_{Z}} }\right),l_{Z} \in \left ({{l_{1} +l_{2},l_{1} +l_{2} +l_{3}} }\right]} }\right \}=\textrm {262.471km}$ , it can be seen that the redundant solutions are not in the correct interval. Therefore, the fault location result will be unique.

Tab. 4–​5 give the fault location results by using the method I and method III. The MRE of results in the two tables is 0.1697% and 0.2227% respectively.

TABLE 4 Fault Location Results by Using Method I for an Inhomogeneous Line

TABLE 5 Fault Location Results by Using Method III for an Inhomogeneous Line

SECTION V. Comparison With Travelling Wave Based Method

In practice, the line fault location is based on the two-ends travelling wave based method. As is shown in Fig. 11, the two curves are the wavelet transform modulus maxima (WTMM) of the measured currents at the ends of the line in Fig. 10 with sampling frequency of 1MHz. $t_{1}$ represents the arrival time of the initial wave front at end $M $ generated by the fault occurrence, and $t_{2}$ represents the arrival time of the initial wave front at end $N$ .

FIGURE 11.

WTMM for two measured currents in Fig. 10.

The fault location can be calculated by using (12).\begin{align*} l_{Z} \!=\!\begin{cases} \dfrac {v_{1}}{2}\left ({{t_{1} -t_{2} +\dfrac {L_{1}}{v_{1}}+\dfrac {L_{2} }{v_{2}}} }\right) &0\le l_{Z} < L_{1} \\ \dfrac {v_{1} v_{2}}{v_{1} +v_{2}}\left ({{t_{1} -t_{2} +\dfrac {L_{1} +L_{2} }{v_{2}}} }\right) &l_{Z} =L_{1} \\ \dfrac {v_{2}}{2}\left ({{t_{1} -t_{2} -\dfrac {L_{1}}{v_{1}}+\dfrac {2L_{1} +L_{2}}{v_{2}}} }\right) &L_{1} < l_{Z} \le L_{1} +L_{2} \\ \end{cases}\!\!\!\!\!\! \\ {}\tag{12}\end{align*} where, $v_{1}$ and $v_{2}$ is respectively the wave velocity of line L₁ and L₂.

The fault location results based on travelling wave based method are listed in Tab. 6. The MRE of the results is 0.2026%, which is a little higher than that of Tab. 5 but lower than that of Tab. 4. Therefore, the precision of the proposed method is similar to that of travelling wave based method. However, the sampling frequency of the proposed method is much lower than that of the travelling wave based method.

TABLE 6 Fault Location Results by Using Travelling Wave Based Method for an Inhomogeneous Line

In addition, the travelling wave based method needs time synchronization while the proposed method II and III do not. For the proposed method I, three situations of time asynchronizations are considered for testing the fault location results.

Suppose that an AGF-$0\Omega $ occurred at every 20km along the line in Fig. 10, the fault location results for different time errors between end $M $ and end $N$ is are listed in Tab. 7. The MRE increases as the increase of the time error. The MRE is 0.5776% when the time error is $15\mu \text{s}$ , but it is lower than the inherent MRE for travelling wave based method of 1.125%. Therefore, the proposed method I has some capability against time asychronization.

TABLE 7 Fault Location Results by Using Method I With Time Asynchronizations

SECTION VI. Effects on Accuracy of Fault Location

The accuracies of the fault location are affected by many factors in actual. In this section, the simulations under four different conditions are implemented. The results of the fault locations are calculated by using the proposed three methods and the effects of the four conditions are analyzed.

A. Fault Inception Angle

The AGF-$0\Omega $ faults in Fig. 4 are simulated in PSCAD with different fault inception angles. The fault inception angles are 0, $\pi /4$ and $\pi /2$ , respectively. The results of fault location are calculated by using the three proposed methods. The relative errors of the fault locations are shown in Fig. 12. Different fault inception angles do not affect the accuracy of the fault location because the relative errors in Fig. 12(a-c) are approximately the same.

FIGURE 12.

Relative errors of fault location by using three methods: (a) Fault inception angle is 0, (b) Fault inception angle is $\pi /4$ , (c) Fault inception angle is $\pi /2$ .

B. Noise Effect

The measured transients may contain much noise in practice. The lower the SNR (signal to noise ratio) is, the greater the influence of noise on the signal is.

Suppose that a AGF-$0\Omega $ fault occurred at 80km in Fig. 4, the three phase currents at terminal $N$ are recorded. The three phase currents are mixed with Gaussian White Noise in MATLAB in order to simulate the actual noise situation. As shown in Fig. 13, the SNR of the three phase currents is 30dB.

FIGURE 13.

Three phase currents that contains the noise.

The AGF-$0\Omega $ faults in Fig. 4 are simulated in PSCAD with different SNRs. The relative errors of different fault locations by using three methods are shown in Fig. 14. If the SNRs of the transients that are used by the three methods are lower than 70dB, 30dB and 30dB, respectively, the relative errors would be much higher. The noise severely affects the detection of the initial forward current in method I, while it slightly affects spectrum calculation in method II and III. Therefore, the SNR of transients used in method I is higher than those in method II and III.

FIGURE 14.

Relative errors of fault locations by using different methods in different background noisy enviroments.

The SNR is higher than 70dB in many cases so the accuracies of the three methods are not affected by the noise to some extent.

C. Line Parameter Uncertainty

The line parameters may change with the temperature and air pressure. In this section, the fault locations are recalculated after the parameters are changed in order to analyze the effect of the line parameter uncertainty on the accuracy of the fault location.

The AGF-$0\Omega $ faults in Fig. 4 are simulated in PSCAD and the transients at two terminals of the line are recorded. The fault locations are recalculated by using method II when the resistance increase 5%, −5%, 10% and −10%, respectively. The relative errors of fault locations are shown in Fig. 15(a). The fault locations are also recalulated and the results are shown in Fig. 15(b-d) after inductance, conductance and capacitance changes, respectively.

FIGURE 15.

Relative errors of fault locations by using method II: (a) Resistance change, (b) Inductance change, (c) Conductance change, (d) Capacitance change.

In addition, the relative errors of fault location by using method III are shown in Fig. 16 with the parameter variations.

FIGURE 16.

Relative errors of fault locations by using method III: (a) Resistance change, (b) Inductance change, (c) Conductance change, (d) Capacitance change.

As is shown in Fig. 15 and Fig. 16, the accuracies of the fault locations decrease with the changes of the parameters. The inductance variations most affect the accuracy while the variations of other parameters slightly affect the results of the fault location. Besides, the accuracy of method III is more susceptible to the line parameter uncertainty than that of method II.

Because the wave velocity of the aerial mode current always remains constant, the accuracy of method I can be unacted on the line parameter uncertainty.

D. Current Transformer Saturation

The saturation of CT will cause the distortion of measured currents. A CT model based on the Jiles-Atheton theory of ferromagnetic hysteresis is used for simulation in the in Fig. 4.

Suppose that a AGF-$0\Omega $ fault occurred at 40km in Fig. 4, the A-phase currents at terminal $M$ is recorded. The parameters of the CT model is shown in appendix C. As is shown in Fig. 17, the bigger the current ratio (K) is, the bigger the aberration rate is.

FIGURE 17.

A-phase current at terminal $M$ that are measured by current transformer under different current ratios.

The AGF-$0\Omega $ faults in Fig. 4 are simulated in PSCAD with different current ratios. The relative errors of different fault locations by using three methods are shown in Fig. 18. The accuracies of fault locations by using method I are not affected by CT saturation as is shown in Fig. 18(a) because method I is based on the singularities of transients at the fault occurrence time. The accuracies of method II and method III are affected by the CT saturation as is shown in Fig. 18(b-c). the higher the saturation of the CT, the higher the accuracies of method II and III are.

FIGURE 18.

Relative errors of fault locations by using different CTs of different current ratio: (a) Method I, (b) Method II, (c) Method III.

SECTION VII. Conclusion

Three practical methods based on time reversal theory are applied into fault location of homogeneous line and inhomogeneous line consisting of two mediums. The currents-in-time-domain and forward-currents-in-frequency-domain-with-compensation methods are suitable for all situations, while the forward-currents-in-frequency-domain method is only practical for the homogeneous line. The simulation results show that these three methods are robust to different fault resistances, fault types and fault inception angles. The effects of CT saturation and parameter uncertainty on accuracy of method I are lower than those of method II and III, while the effects of noise on accuracy of method I is larger than that of method II and III. By comparing with the travelling wave based method, the three methods are of low sampling frequency. Moreover, the proposed frequency domain methods do not need time synchronization and the proposed time domain method has certain ability to conquer time asynchronization.

Appendix

SECTION A. Geometry of Overhead Line for L₁ in Fig. 10

SECTION B. Geometry of Overhead Line for L₂ in Fig. 10

SECTION C. Parameters of CT Model