Abstract
The single-ended fault location based on travelling waves (TWs) is commonly used for long-distance high-voltage AC transmission lines. However, it relies on high sampling frequency and accurate capturing of the TW head arrival time. Accordingly, this study establishes a transient analytical method for fault location based on the similarity between the transient recorded waveform and output waveforms of analytical calculation model. In the proposed method, fuzzy constraints of fault features are constructed through time-distance and waveform-scaling correlations while considering the deviation factors of the frequency-dependent wave velocity and TW head arrival time. Accordingly, the high-dimensional space of the fitting problem is transformed into a one-dimensional implicit function fitting problem containing only the fault distance, thereby enabling the waveform comparison problem to be quickly solved based on fault TW features. Under the fuzzy constraints proposed in this study, the proposed method requires only a relatively vague identification of the TW head, and the requirements for sampling frequency are also more lenient. In addition, a sliding window scheme is adopted for enhancing the TW morphology characteristics. Finally, the proposed method is tested using PSCAD, and the simulations validate the fault location accuracy of the proposed method.
FAST and accurate fault locations of transmission lines are crucial for the safe and reliable operation of power systems [
TW-based methods can be classified into single- and double-ended methods. Double-ended TW-based methods utilize the difference in the TW head arrival time between the two ends for fault location and exhibit high accuracy under the conditions of high-speed sampling, data synchronization, and reliable communication. By contrast, single-ended TW-based methods require only TW information at one end of the line, making them easier to implement in engineering. However, traditional single-ended TW-based methods rely on the reflected TW at the fault point, and because of the greater distortion of the reflected TW head compared with the initial TW, it is difficult to accurately capture the arrival time of the reflected TW. This in turn affects the accuracy of the fault location. To address this issue, some researchers have focused on TW-head singularity recognition schemes based on signal processing methods. Reference [
To improve the accuracy of single-ended fault locations, researchers have utilized fault characteristics such as TW dispersion to enhance the feature information for fault locations. References [
Under the description of complex-domain fault boundary conditions for complex-domain analytical calculation models of AC transmission systems, the fault inception angle of the fault point differs from the reference angle of the measurement point due to the delay caused by TW propagation. This angle deviation is related to the fault distance, propagation velocity, and TW head arrival time, and it is difficult to accurately measure the moment at which the initial TW reaches the measurement point. Consequently, the output waveform of the complex-domain analytical calculation exhibits significant errors without constraints on the phasor-angle deviation for the delay in TW propagation. Therefore, fitting a complex-domain analytical calculation model to fault recorded data is a challenging task for AC transmission systems. In some fault cases under the waveform matching method, the fault feature waveforms of TW may be sparsely scattered over the timeline. Differential features are masked directly using the waveform similarity index. This drawback is not conducive to accurate fault locations.
Therefore, based on the complex-domain analytical calculation model, this study proposes a transient analytical method for single-end fault location of AC transmission lines considering fuzzy constraints of fault features. The main contributions of the proposed method are as follows.
1) The fuzzy constraints of fault features are constructed through time-distance and waveform-scaling correlations while considering the deviation factors of the frequency-dependent wave velocity and TW head arrival time. A characteristic sensitivity analysis reveals that under the maximum total angular deviation of the fuzzy constraints, the sinusoidal variation difference is small. Thus, the proposed fuzzy constraints effectively suppress errors caused by unmeasurable factors, and when combined with linear least-squares fitting, the high-dimensional space of the fitting problem can be transformed into a one-dimensional implicit function-fitting problem that includes only the fault distance.
2) A window sliding method for enhancing the waveform similarity features is proposed. First, the length of the sliding window is shortened to make the waveform similarity features more focused on the TW features and to suppress the effects of feature sparsity. Second, the window sliding method enhances the time-axis dimension of the fault characteristics and increases the sensitivity of the TW morphology characteristics associated with the fault distance.
The remainder of this paper is organized as follows. Section II presents the complex-domain analytical calculation model for FTTW. Section III analyzes the characteristic sensitivities of all fault factors on the FTTW waveform based on the analytical calculation model. Section IV presents the derivation of the fuzzy constraints of fault features that correlate with the analytical calculation model and recorded data. Section V describes the transient analytical method for single-ended fault location based on TW morphology characteristics. Section VI describes the validation of the effectiveness of the proposed method through case studies. Finally, conclusions are drawn in Section VII.
After a fault occurs on a transmission line during the transient initial stage, the fault current exhibits multi-characteristic frequency signals. Accordingly, the frequency-dependent characteristics of the analytical calculation model for FTTW must be considered. The frequency-dependent parameters of transmission lines such as the per-unit-length series impedance and parallel admittance, are typically derived using a vector fitting algorithm with a rational function approximation [
The transmission line model of the complex-domain analytical calculation model for the FTTW can then be constructed based on the long-line equation with frequency-dependent parameters [
(1) |
(2) |
(3) |
where and are the complex-domain currents at terminals J and K of the line, respectively; and are the complex-domain voltages at terminals J and K of the line, respectively; and are the complex-domain self-admittance and mutual admittance of the line, respectively; l is the length of the line; T is the Clarke transformation matrix; is the per-unit-length propagation coefficient of the line in the form of a complex domain; and y(s) and are the per-unit-length parallel admittance and series impedance of the line, respectively.
Then, based on the augmented incidence matrix of the faulty and adjacent lines and the equivalent system parameters of the adjacent remote nodes, the complex-domain nodal admittance matrix of the transmission system is established and inverted into the complex-domain nodal impedance matrix of the transmission system. Thus, if a fault occurs at a fault distance df from the terminal J of the line, the complex-domain nodal impedance matrix of the extended fault node network can be determined by appending the tree and link branches to the initial complex-domain nodal impedance matrix, which is set as in this study. Consequently, selecting a different df changes the self-impedance of the fault point and its mutual impedance at both terminals of the line.
For a fault port f, the time-domain model of the fault excitation voltage source can be expressed as:
(4) |
where is the fault excitation voltage source; the subscript represents each phase; Uf is the voltage amplitude at the fault point before the fault occurs; is the power angular frequency; t0 is the time when a short circuit occurs; is the fault inception angle; and is a unit-step signal.
The time-domain models of fault boundary conditions for different fault types can be established as follows.
1) Single-phase-to-ground short circuit:
(5) |
where and are the voltage and outflow-node current at fault port f in the fault additional network, respectively; the subscript represents the fault phase; the subscripts and represent two non-fault phases; the superscript (1) represents the fault type of single-phase-to-ground short-circuit; and Rf is the fault resistance.
2) Phase-to-phase short circuit:
(6) |
where the subscripts and represent two fault phases; the subscript represents the non-fault phase; and the superscript (2) represents the fault type of phase-to-phase short circuit.
3) Two-phase-to-ground short circuit:
(7) |
where the superscript (1,1) represents the fault type of two-phase-to-ground short circuit.
4) Three-phase short circuit:
(8) |
where the superscript (3) represents the fault type of three-phase short circuit.
The complex-domain expression of fault boundary conditions can be established by applying a Laplace transformation to the electrical quantities of each phase at the fault port. By utilizing symmetrical component transformation and constructing the connection of sequence networks, this study derives the complex-domain expression of the short-circuit current at the fault port as:
(9) |
where Iff is the complex-domain expression of iff; the superscript (type) represents the fault type; is the positive-sequence complex-domain self-impedance of the fault port in ; and , m1, m2, and are the parameters related to the fault types, with their expressions summarized in Supplementary Material A Table SAI.
The complex-domain expression of the transient current at terminal J can be obtained through the use of and as:
(10) |
where and are the mutual impedance elements between nodes f and J (or K) in , respectively.
The solution to (10) in the complex domain can be converted into a numerical solution in the time domain by applying a numerical inverse Laplace transform as [
(11) |
where is the time step of the numerical solution in the time domain; N is the number of samples; k is the sampling point and ; is the corresponding discrete angular frequency for the calculation; c is a stability constant; is a concrete causal complex-domain function; is the corresponding numerical solution of F(s) in the time domain; and is the Hanning function.
The calculation for the current FTTW from (1)-(11) can be expressed in an implicit function form as:
(12) |
where is the numerical solution of the line-terminal fault current; and is the implicit function of calculation for the current FTTW.
According to the established analytical calculation model, if the variables in (12) are equal to those under the actual fault conditions, the output waveform of the analytical calculation model should be consistent with the recorded FTTW waveform.
When the complete transmission system is modeled in detail, the output waveform of the proposed complex-domain analytical calculation model is fully consistent with the simulation waveform. However, from the practical engineering perspective, this modeling is not feasible. Typically, only the target line and its adjacent lines can be mathematically modeled in detail. In addition, the impedance parameter of the system changes during operation, which can result in calculation errors in the reflectivity between adjacent lines and remote buses.

Fig. 1 Comparison between output waveform of analytical calculation model and simulation waveform.
The fault example presented in
The fault factor variables in (12), including the pre-fault voltage amplitude at the fault point Uf, fault occurrence time t0, fault inception angle θf, fault resistance Rf, and fault distance df, are all unknown when a fault occurs. Conducting a calculation from these five dimensions is extremely time-consuming and not easily feasible. Therefore, this section presents a sensitivity analysis of these unknown variables in the analytical calculation model to explore fault characteristic description methods that enhance the sensitivity of df while reducing the sensitivity of other variables.
The magnitude of the steady-state short-circuit current is determined by Uf. The amplitude of the pre-fault voltage at the line terminal can be obtained from the recorded data. However, Uf cannot be estimated in advance because of the unknown fault location.
Equations (

Fig. 2 Comparison of current FTTW waveforms under different values of each variable related to fault excitation source. (a) Uf. (b) t0. (c) ( within 3 ms). (d) ( within 3 ms). (e) ( within 15 ms). (f) ( within 15 ms).
Although t0 and θf are closely related variables, their relationship is not well-defined until the fault distance is confirmed. To analyze the effects of their deviation values on the output waveforms, different values are selected for these two variables, and analytical calculations are conducted to observe the resulting waveforms.
When θf is held constant and t0 is changed, based on (4), (9), and (10), the output waveforms of the analytical calculation model shift along the time axis, as shown in
However, when t0 is held constant (where the reference zero time is used as an example) and is changed, the output waveforms of the analytical calculation model are shown in
After TW attenuation, the differences among these output waveforms include only the differences in sine waves with different phasor angles, as shown in
When a fault point contains , the value of Rf determines the refractive and reflective indices of the TW passing through the fault point. This directly affects the amplitudes of TW at different orders. Based on the assumption that the
(13) |
where is the influence coefficient of Rf on the amplitude of TW; the subscript n represents the order of TW; Zc is the wave impedance of the fault line; and p1 and p2 are the reflection and refraction times of the TW passing through the fault point, respectively.
From (13), it can be observed that the effect of Rf on the amplitudes of TW at different orders is nonlinear. The relationship curve between and for TW at different orders (TW1-TW5) is shown in

Fig. 3 Effect of Rf on amplitudes of TW at different orders and comparison of current FTTW waveforms with different fault resistances and fault distances. (a) Effect of Rf on amplitudes of TW at different orders. (b) Rf. (c) df ( km). (d) df ( km).
The most critical variable in determining fault locations is df. From (9) and (10), when different values of df are selected in the analytical calculation model, the output waveforms exhibit variations in amplitude and time differences for TW at different orders, as shown in
IV. Fuzzy Constraints of Fault Features Correlating with Analytical Calculation Model and Recorded Data
The output waveform of the analytical calculation model depends on several factors, including the chosen fault distance, fault resistance, and fault excitation source. Given the numerous variables involved, establishing correlation constraints among the recorded data, fault distance, fault excitation source signal, and fault resistance is critical. A single-ended fault location model is then constructed using a comparison index between the fault recorded data and the output waveforms of the analytical calculation model.
Because of the effects of factors such as the recording sampling interval and inaccurate calibration of the initial TW head arrival time, constructing fuzzy constraints that consider errors in the transient recorded data, fault distance, fault excitation source, and fault resistance is necessary.
The instantaneous voltage and current before and after the fault can be obtained from the recorded data. When the single-ended positive-sequence voltage and current phasors prior to the fault are utilized, the pre-fault voltage phasor expression related to variable df can be established using the long-line equation:
(14) |
where and are the power-frequency sinusoidal voltage phasor and current phasor measured at terminal J at the moment of abrupt waveform, respectively; is the per-unit-length power-frequency propagation coefficient of the transmission line; and is the corresponding power-frequency sinusoidal voltage phasor of the fault point. The amplitude and phasor angle of are described as:
(15) |
In fact, the reference zero time of fault recording is usually the starting moment when the initial TW reaches the measurement point, meaning that the fault occurrence time should precede the reference zero time by a corresponding delay in the TW propagation. Therefore, to obtain the phasor angle of the fault excitation voltage at the fault point, the propagation delay of the fault voltage phasor derived from (14) must be compensated for:
(16) |
where v is the propagation velocity of TW; and is the total angular deviation caused by the detection error and frequency-dependent wave velocity deviation.
In (16), denotes the delay angle of TW propagation. Because is unknown, it is considered a fuzzy variable, and its error must be further evaluated. Considering the actual engineering scenarios, for example, if the length of the AC transmission line does not exceed 300 km, the ultimate propagation velocity of TW does not exceed 31
(17) |
Thus, the variable Uf is precalculated from (15), and both t0 and are related to (17) with respect to . Therefore, the current FTTW from (12) can be rewritten as a function of Rf and df as follows:
(18) |
B. Correlation Constraint on Fault Distance and Fault Resistance for Fault Characterization Enhancement
According to the analysis of the correlation between the output waveform of the analytical calculation model and Rf as presented in Section III-C, the amplitudes of TW at different orders show different proportions of scaling effects under different values of Rf. From (13), for the first-order TW in which and , the influence coefficient of its amplitude on Rf can be derived as:
(19) |
Therefore, the relationship between the analytical outputs of different resistances in the shape of the first-order TW is proportional and can be described as:
(20) |
where Rassume is the assumed fault resistance that can be set as any constant; T1 is the time window of the initial TW; and is the numerical solution of the line-terminal fault current in the time series of the initial TW.
If df is consistent with the actual fault distance, from (18), the scaling relationship between the recorded waveform and analytical output in the shape of the first-order TW can be described as:
(21) |
where is the recorded data of the line-terminal fault current.
Thus, Rf can be solved by converting the coincidence degree of (20) into a linear least-squares fitting of each time series of the first-order TW as:
(22) |
If the selected df in (22) deviates from the actual fault distance, the resulting Rf deviates from the actual value. This enhances the correlation between the analytical output waveforms and the variable df, which helps to improve the variable sensitivity of df. Therefore, when (22) is substituted into (18), the analytical calculation model for the single-ended fault location can be transformed into a function that contains only the variable df as follows:
(23) |
Under (23), a series of undetermined variables df is scanned to obtain the unique corresponding output waveforms of the analytical calculation model. These waveforms are then matched with the recorded waveforms, as shown in

Fig. 4 Comparison of current FTTW waveforms corresponding to different values of df. (a) df ( km). (b) df ( km). (c) df ( km). (d) Localized enlarged view ( km).
Clearly, within a 1 km range, distinguishable morphological differences exist in the output waveforms for different df. Consequently, the fault location can be determined by a single scan of df and by calculating the waveform similarity index for waveform matching. The cosine similarity index is chosen as the waveform similarity index in this study [
(24) |
where T2 is the time window for calculating the waveform similarity index.
The similarity feature map is expanded into three dimensions as the data window slides along the time axis. The highest similarity index results can only be maintained during the window sliding process if the value of df is consistent with the actual fault distance. However, if the value of df differs from the actual fault distance, the similarity index results exhibit a low similarity valley at the data segments of the respective TW head arrival, resulting in a “cliff” phenomenon. This phenomenon enhances the feature differences in the waveform similarity for different fault distances during the waveform matching process, as illustrated in

Fig. 5 Three-dimensional presentation of cosine similarity index results based on sliding windows with short data window. (a) Isometric view. (b) Top view.

Fig. 6 Flowchart of transient analytical method for single-ended fault location.
For transient fault location methods, using a shorter data window results in harsher fault location requirements. To locate faults accurately, ensuring that the recorded data exhibit significant TW morphology characteristics at any distance is crucial. This can be accomplished by using an effective data window for a fault location that includes at least two TWs. Notably, the second TW, which reaches the line-terminal measurement point, experiences the longest delay when a fault occurs in the middle of the line. Considering redundancy, we set the minimum length of the data window used in the proposed method to at least 1.2l/v.
To evaluate the effectiveness of the proposed method, electromagnetic transient simulation models of a 500 kV AC transmission system are constructed using PSCAD/EMTDC, which include a simplified double-circuit network topology (topology 1) and an IEEE 30-node network topology (topology 2), as shown in

Fig. 7 Schematics of network topology in PSCAD. (a) Schematic of simplified double-circuit network topology. (b) Schematic of IEEE 30-node network topology.
For simulation purposes, different fault types, fault inception angles, and fault resistances are set at various positions along the line. The pre-fault voltage and current signals as well as the current recorded data are collected at terminal M at a sampling frequency of 100 kHz. Because the measured line has a total length of 200 km, a redundant data window is set to be 0.8 ms after the recording starts. The variable df is scanned for analytical calculations from terminals M to N along the transmission line.
Fault location methods based on waveform matching rely mainly on the morphology characteristics of the wave head. When a fault occurs in the middle or near the end of the line, TW exhibits a sparse or dense distribution on the time axis. In addition, when a fault occurs at a lower fault inception angle and higher fault resistance, the characteristics of TW are relatively weak. Therefore, the location results for three typical fault cases are demonstrated.
1) In the case of a fault occurring in the middle of the line, the TW head arrival time interval is long, and the TW morphology characteristics are sparsely distributed on the time axis. As an example, a single-phase-to-ground fault with a fault resistance of 200 and a fault inception angle of 90° occurring at 102.46 km is selected, and the fault location process is shown in

Fig. 8 Fault location process ( Ω and km). (a) Simulated and partially analyzed FTTW waveforms. (b) Cosine similarity index with sliding window data. (c) Average value of cosine similarity index.
The time-domain FTTW waveforms at different fault distances can then be obtained using (23), as shown by dashed lines in
2) In the case of a fault near the end of the line, multiple TW reflections occur in a short period, and the TW morphology characteristics are concentrated in the first half of the time axis. As an example, a single-phase-to-ground fault with a fault resistance of 20 and a fault inception angle of 90° occurring at 10.04 km is selected, and the fault location process is shown in

Fig. 9 Fault location process ( Ω and km). (a) Simulated and partially analyzed FTTW waveforms. (b) Cosine similarity index with sliding window data. (c) Average value of cosine similarity index.
3) In the case of a fault occurring under extreme conditions, the TW morphology characteristics become even weaker due to the effects of propagation losses. As an example, a single-phase-to-ground fault with a fault resistance of 400 and a fault inception angle of 6° occurring at 180.51 km is selected, and the fault location process is shown in

Fig. 10 Fault location process ( Ω and km). (a) Simulated and partially analyzed FTTW waveforms. (b) Cosine similarity index with sliding window data. (c) Average value of cosine similarity index.
Under this fault condition, the proposed method can still accurately determine the fault location.
This study conducted multiple simulations to evaluate the effectiveness of the proposed method. The simulations involve different fault distances, inception angles, resistances, and types. The specific values for each fault factor are as follows. The actual fault distances are distributed along the entire line with values of 5.89, 10.04, 20.39, 30.35, 50.28, 75.42, 102.46, 149.67, and 190.72 km. The fault inception angles are , 60°, 30°, and 6°. The fault resistance values are 0, 20, and 200 . The fault types include four types of AC transmission line short-circuit faults. Tables I-IV list the location results of the proposed method under different fault conditions. In these tests, the average time consumed for a fault location calculation does not exceed 10 s with the i5-12400 processor.
Topology | Actual fault distance (km) | Location result (km) | Error (km) |
---|---|---|---|
1 | 5.89 | 5.9 | +0.01 |
10.04 | 10.1 | +0.06 | |
20.39 | 20.4 | +0.01 | |
30.35 | 30.3 | -0.05 | |
50.28 | 50.3 | +0.02 | |
75.42 | 75.4 | -0.02 | |
102.46 | 102.5 | +0.04 | |
149.67 | 149.6 | -0.07 | |
190.72 | 190.7 | -0.02 | |
2 | 5.89 | 5.9 | +0.01 |
10.04 | 10.1 | +0.06 | |
20.39 | 20.5 | +0.11 | |
30.35 | 30.3 | -0.05 | |
50.28 | 50.3 | +0.02 | |
75.42 | 75.3 | -0.12 | |
102.46 | 102.5 | +0.04 | |
149.67 | 149.6 | -0.07 | |
190.72 | 190.6 | -0.12 |
Topology | Actual fault distance (km) | Fault inception angle (°) | Location result (km) | Error (km) |
---|---|---|---|---|
1 | 10.04 | 90 | 10.1 | +0.06 |
60 | 10.1 | +0.06 | ||
30 | 10.1 | +0.06 | ||
6 | 9.9 | -0.14 | ||
30.35 | 90 | 30.3 | -0.05 | |
60 | 30.3 | -0.05 | ||
30 | 30.3 | -0.05 | ||
6 | 30.2 | -0.15 | ||
102.46 | 90 | 102.5 | +0.04 | |
60 | 102.5 | +0.04 | ||
30 | 102.5 | +0.04 | ||
6 | 102.3 | -0.16 | ||
2 | 10.04 | 90 | 10.1 | +0.06 |
60 | 10.1 | +0.06 | ||
30 | 10.1 | +0.06 | ||
6 | 9.9 | -0.14 | ||
30.35 | 90 | 30.3 | -0.05 | |
60 | 30.3 | -0.05 | ||
30 | 30.3 | -0.05 | ||
6 | 30.2 | -0.15 | ||
102.46 | 90 | 102.5 | +0.04 | |
60 | 102.5 | +0.04 | ||
30 | 102.5 | +0.04 | ||
6 | 102.6 | +0.14 |
Topology | Actual fault distance (km) | Fault resistance (Ω) | Location result (km) | Error (km) |
---|---|---|---|---|
1 | 5.89 | 0 | 5.9 | +0.01 |
20 | 5.9 | +0.01 | ||
200 | 5.9 | +0.01 | ||
75.42 | 0 | 75.4 | -0.02 | |
20 | 75.4 | -0.02 | ||
200 | 75.4 | -0.02 | ||
149.67 | 0 | 149.6 | -0.07 | |
20 | 149.6 | -0.07 | ||
200 | 149.6 | -0.07 | ||
2 | 5.89 | 0 | 5.9 | +0.01 |
20 | 5.9 | +0.01 | ||
200 | 6.0 | +0.11 | ||
75.42 | 0 | 75.3 | -0.12 | |
20 | 75.3 | -0.12 | ||
200 | 75.3 | -0.12 | ||
149.67 | 0 | 149.6 | -0.07 | |
20 | 149.6 | -0.07 | ||
200 | 149.5 | -0.17 |
Topology | Actual fault distance (km) | Fault type | Location result (km) | Error (km) |
---|---|---|---|---|
1 | 20.39 | AG | 20.4 | +0.01 |
BC | 20.5 | +0.11 | ||
BCG | 20.4 | +0.01 | ||
ABC | 20.5 | +0.11 | ||
50.28 | AG | 50.3 | +0.02 | |
BC | 50.2 | -0.08 | ||
BCG | 50.3 | +0.02 | ||
ABC | 50.3 | +0.02 | ||
190.72 | AG | 190.7 | -0.02 | |
BC | 190.8 | +0.08 | ||
BCG | 190.6 | -0.12 | ||
ABC | 190.7 | -0.02 | ||
2 | 20.39 | AG | 20.5 | +0.11 |
BC | 20.5 | +0.11 | ||
BCG | 20.4 | +0.01 | ||
ABC | 20.4 | +0.01 | ||
50.28 | AG | 50.3 | +0.02 | |
BC | 50.3 | +0.02 | ||
BCG | 50.4 | +0.12 | ||
ABC | 50.3 | +0.02 | ||
190.72 | AG | 190.6 | -0.12 | |
BC | 190.5 | -0.22 | ||
BCG | 190.5 | -0.22 | ||
ABC | 190.6 | -0.12 |
Note: AG, BC, BCG, and ABC represent single-phase-to-ground, phase-to-phase, two-phase-to-ground, and three-phase short-circuit faults, respectively.
Results in Tables I-IV indicate that the proposed method is minimally affected by various factors and is applicable to different topological structures. The maximum error observed does not exceed 0.3 km, with a relative error not exceeding 0.15%, which demonstrates a high accuracy level.
To further verify the adaptability and superiority of the proposed method, tests are conducted on other influencing factors, excluding the fault factors, using the fault data of topology 1.
The proposed method requires a detailed mathematical model of the target line and its adjacent lines. Based on the potential geometric deviations between the established model and the actual transmission line, tests are performed to analyze the effects of deviations of transmission line parameters. The corresponding location results are listed in
Actual fault distance (km) | Deviation degree (%) | Location result (km) | Error (km) |
---|---|---|---|
30.35 | +5 | 30.0 | -0.35 |
+3 | 30.2 | -0.15 | |
0 | 30.3 | -0.05 | |
-3 | 30.6 | +0.25 | |
-5 | 30.8 | +0.45 | |
149.67 | +5 | 150.1 | +0.43 |
+3 | 149.9 | +0.23 | |
0 | 149.6 | -0.07 | |
-3 | 149.5 | -0.17 | |
-5 | 149.2 | -0.47 |
Regarding noise interference, white noise with different signal-to-noise ratios (SNRs) is added to the original signal to verify the effectiveness of the proposed method. The corresponding location results are listed in
Actual fault distance (km) | SNR (dB) | Location result (km) | Error (km) |
---|---|---|---|
20.39 | 20.4 | +0.01 | |
70 | 20.4 | +0.01 | |
60 | 20.4 | +0.01 | |
50 | 20.6 | +0.21 | |
40 | 20.8 | +0.41 | |
102.46 | 102.4 | -0.06 | |
70 | 102.4 | -0.06 | |
60 | 102.4 | -0.06 | |
50 | 102.2 | -0.26 | |
40 | 102.0 | -0.46 |
As
The implementation of the fault location method depends on the on-site fault locators, and the sampling frequency is a critical parameter of the fault locator. Therefore, testing the effects of different sampling frequencies on the proposed method is essential. The corresponding location results are presented in
Actual fault distance (km) | Sampling frequency (kHz) | Location result (km) | Error (km) |
---|---|---|---|
75.42 | 20 | 76.5 | +1.08 |
50 | 75.2 | -0.22 | |
100 | 75.4 | -0.02 | |
500 | 75.4 | -0.02 | |
1000 | 75.4 | -0.02 | |
190.72 | 20 | 191.5 | +0.78 |
50 | 190.8 | +0.08 | |
100 | 190.8 | +0.08 | |
500 | 190.7 | -0.02 | |
1000 | 190.7 | -0.02 |
As
This study presents a transient analytical method based on TW morphology characteristics. In extreme fault situations in which the fault TW features are weak, such as when the fault inception angle is close to zero, it is not conducive to accurate location by the proposed method.
However, increasing the sampling frequency can enrich the fault TW features within the data window. Therefore, tests of the proposed method under extreme fault conditions at different sampling frequencies are conducted, and the corresponding results are presented in
Sampling frequency (kHz) | Fault inception angle (°) | Location result (km) | Error (km) |
---|---|---|---|
50 | 6 | 50.6 | +0.32 |
4 | 51.1 | +0.82 | |
2 | 51.1 | +0.82 | |
100 | 6 | 50.5 | +0.22 |
4 | 50.6 | +0.32 | |
2 | 50.8 | +0.52 | |
500 | 6 | 50.3 | +0.02 |
4 | 50.3 | +0.02 | |
2 | 50.6 | +0.32 | |
1000 | 6 | 50.3 | +0.02 |
4 | 50.3 | +0.02 | |
2 | 50.4 | +0.12 |
As
To verify the superiority of the proposed method, tests are conducted to compare the proposed method and other fault location methods, including the wavelet transform-based fault location method (method 1) [
Actual fault distance (km) | Method | Location result (km) | Error (km) |
---|---|---|---|
50.28 | Proposed | 50.3 | +0.02 |
1 | 50.8 | +0.52 | |
2 | 51.3 | +1.02 | |
102.46 | Proposed | 102.5 | +0.04 |
1 | 103.2 | +0.74 | |
2 | 103.3 | +0.84 | |
149.67 | Proposed | 149.6 | -0.07 |
1 | 149.3 | -0.37 | |
2 | 150.5 | +0.83 |
In this study, a transient analytical method for single-ended fault location of AC transmission lines considering fuzzy constraints of fault features is proposed. Complex-domain analytical calculations are first applied to AC transmission systems, and a mathematical analytical calculation model is then established between the single-ended FTTW of an AC transmission line fault and various fault conditions. Fuzzy constraints of fault features are also proposed for fault excitation sources, fault resistances, and fault distances. Fuzzy constraints transform the high-dimensional space of the fitting problem into a one-dimensional implicit function fitting problem containing only the fault distance. To further improve the similarity features between the output waveforms of the analytical calculation model and fault recorded waveforms at fault points, a waveform similarity index under a short sliding window is proposed. Simulation results demonstrate that this index effectively differentiates the fault point and is unaffected by fault resistance. In addition, the proposed method accounts for fuzzy constraints by considering errors in the TW head arrival time. Consequently, precise identification of the TW head is not required. Another advantage is that the proposed method does not require a pre-established FTTW dataset because the analytical calculation model establishes a correlation between the fault distance and recorded data. Thus, the proposed method can adapt to different system operation modes and various fault situations within a short data window based on FTTW.
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