A New Method of Fault Localization for 500 kV Transmission Lines Based on FRFT-SVD and Curve Fitting

: The paper presents the Fractional Fourier Transform-Singular Value Decomposition (FRFT-SVD) method for the localization of various power system faults in a 200 km long, 500 kV Egyptian transmission line using sent end-line current signals. Transient simulations are carried out using Alternating Transient Program/Electromagnetic Transient Program (ATP-EMTP), and the outcomes are then examined in MATLAB to carry out a sensitivity analysis against measurement noises, sampling frequency, and fault characteristics. The proposed work employs current fault signals of ﬁve distinct kinds at nineteen intermediate points throughout the length of the line. The approach utilized to construct the localizer model is FRFT-SVD. It is much more effortless, precise, and effective. The FRFT-SVD is utilized in this technique to calculate 19 sets of indices of the greatest S value throughout the length of the line. The FRFT-SVD localizer model is also designed to be realistic, with power system noise corrupting fault signals. To generate fault curves, the curve ﬁtting technique is applied to these 19 sets of indices. Reduced chi-squared and modiﬁed R-squared criteria are used to choose the best-suited curve. The proposed work results in a very precise localization, with only a 0.0016% average percentage error for fault localization and a maximum percentage error of 0.002% for the 200 km Egyptian transmission line. Finally, this work can be employed as a proper link between the nuclear power plant and the grid. The proposed method is an efﬁcient fault distance estimation method that might contribute to creating a dependable transient-based approach to power system protection.


Introduction
In deregulated situations, accurate fault location in transmission lines saves power system recovery time, associated costs, and financial losses.Three techniques are used to locate faults in the electrical grid: approaches based on impedance [1,2], traveling wave fault location [3][4][5], and artificial intelligence [6][7][8][9][10].While traveling wave-based methods employ high-frequency transient components produced by fault or switching operations, impedance-based methods utilize power frequency components of voltages and currents.Artificial intelligence (AI) and machine learning (ML) techniques are typically the foundation of soft computing.Singular value decomposition (SVD) techniques have drawn greater attention in recent years thanks to developments in signal processing, the ability to sample signals at high frequencies, and the creation of optical sensors.Voltage, current, power, frequency, etc., are only a few of the many variable factors of the power system as a whole.The application of SVD can reduce the size of these electrical characteristics, making it easier, faster, and more accurate to determine fault features.Furthermore, SVD is extremely efficient when dealing with noisy data.Consequently, it reduces the Energies 2023, 16, 758 2 of 23 unpredictability of noise, making SVD more suitable for application in a noisy setting such as the power system.
The FRFT-SVD approach, which is exceedingly effective, accurate, and simple to use, was employed to construct the localizer model.To determine the characteristics of a signal in terms of the maximum S value, which is employed in this approach to calculate the indices, FRFT-SVD is a straightforward technique.In particular, for more decomposition, wavelet transform-based analysis becomes more challenging.On the other hand, neural networks need a lot of training data that are spread out over a long period of time.As a result, the proposed fault localizer is both very precise and fast.The proposed analysis might be enhanced to include fault localization in a transmission network or a system with several interconnected buses.In the event of such a system, it is first necessary to determine the bus from which the line with the problem originated.After conducting the following fault localization study, that bus might be taken as the source point.This study's goal is to provide a technique for the efficient use of curve fitting analysis and FRFT-SVD for the sole purpose of fault site prediction.

Related Work
Scientists have used quick fault detection, classification, and location identification techniques to guarantee system stability and safety [11].
To restore the system stability in a power transmission network, the faulty phase or phases must be removed.Numerous computational tools for defect diagnostics have been created by researchers.As previously noted, the proposed work investigates the function of SVM as an upgrading simulation tool and uses it to create a fault location technique.
For the detection, categorization, and localization of faults, researchers have created numerous mathematical and computational algorithms.Nowadays, researchers use artificial intelligence (AI) widely in the study of power systems and fault analysis.One of the most popular and important techniques for studying the protection of transmission lines in power systems has been the Artificial Neural Network (ANN) and its various variations [12].
One of the most recent developments in this area is the use of neural networks for analysis powered by extreme learning machines (ELMs) [13].As a conventional approach to fault signal analysis, Wavelet Transform (WT) has been crucial in numerous studies of fault analysis, even using contemporary compensating devices [14].
Another useful tool for fault analysis is the fuzzy inference system, which is frequently employed as a primary analytical technique alone [15], as well as in a hybrid model with wavelet analysis [16] and neural networks, known as the (ANFIS) model.To create waveletbased ANFIS models and as a useful means of comprehensive analysis, this hybrid model has frequently been supported by WT analysis [17].As a significant standalone method of study, (SVM) has also been utilized in numerous studies pertaining to power system protection algorithms [18].
A simple approach to classifying and locating power system defects was developed using principal component analysis (PCA).For the quick location of the faulty line, this work solely employs 14-cycle pre-fault and 12-cycle post-fault receiving side current waveforms.
Although SVM-based algorithms are also highly popular for fault analysis, they nevertheless have the issue of intensive training because of noise contamination to some extent.
Both [19] and [20] authors explore faults in OHTL with various sources linked to the system, whereas [19] authors present a method for fault identification intended for busbar zone protection, as demonstrated by the authors of [21], who provided polynomial and Gaussian radial basis functions (RBF) for fault classification, or [22], who used dyadic WT-based SVM for fault classification.
The authors of [23] use time-synchronized fault signals as modern research tools.
Several more methodologies can help with the development of fault analysis strategies.Considering this, the purpose of this study is to provide a technique for the efficient use of curve fitting analysis and FRFT-SVD only for fault site prediction.Consequently, the analysis is conducted with a variable fault resistance, various fault inception angles, and noise.The paper proposes the Fractional Fourier Transform-Singular Value Decomposition (FRFT-SVD) approach for localizing various power system problems in a 200 km long, 500 kV Egyptian transmission line under varied operating conditions utilizing received end-line current data.The proposed method has all the potential to become an efficient way of predicting the distance to a fault, which may help in the creation of a dependable transient-based power system protection strategy.

Motivation and Contributions
The numerous AT-detection technologies that have recently been introduced each have drawbacks.For instance, the choice of the mother wavelet affects DWT performance.Due to the aliasing phenomena, the Empirical Mode Decomposition (EMD) process has the potential to affect HHT performance.Wavelet transform-based analysis becomes increasingly complicated as the depth of decomposition increases.On the other hand, neural networks need extensive training time and widely dispersed training data.
In this paper, a new method for fault location based on FRFT-SVD and curve fitting of 500 kV long transmission lines is proposed.The purpose of this work is to provide a strategy for the efficient use of curve fitting analysis and FRFT-SVD for fault site prediction only.This method lacks the shortcomings of the methods listed above because it was designed based on algebraic operations in the time-frequency domain.The suggested algorithm addresses the finding of defects for all fault kinds under various operational scenarios.Simulations are run in the EMTP/ATP program, and the results are analyzed in MATLAB to assess how well the proposed technique performs.Through comprehensive simulations, the sensitivity analysis of measurement noises, sample frequency, and fault parameters is examined.The results of FRFT-SVD and curve fitting are examined, and the suggested method's proper operation under various circumstances is demonstrated.Following is a summary of the suggested method's key characteristics: (1) It is sufficiently resilient against noise.(2) It is simple to implement and has a straightforward structure.
(3) Because it uses straightforward algebraic calculations, it runs quickly enough for online applications.(4) It responds appropriately without the need for structural adjustments or training in various systems and circumstances.

Organization
The organization of the paper is as follows.Section 2 describes the details of the system under study.The study of 500 kV-long transmission line fault signals is presented in Section 3. Section 4 outlines the basic tenets of the suggested detection method.In Section 5, thorough simulations are used to assess how sensitive the suggested strategy is to the system characteristics; a summary of the outcomes of contrasting the suggested method with traditional methods is also provided in Section 5.In Section 6, conclusions are presented.

System Description and Modelling
Figure 1 depicts the system under analysis, which is based on a typical 500 kV transmission line in Egypt with a 200 km-long line.

System Description and Modelling
Figure 1 depicts the system under analysis, which is based on a typical 500 kV transmission line in Egypt with a 200 km-long line.The most sophisticated JMARTI model, which is a frequency-dependent model and suitable for transients research [31,32], is used to simulate the considered transmission line.Table 1 lists the key characteristics of the transmission system in operation.There are two ground wires with direct tower grounding, and phase conductors are presumed to be perfectly transposed.It is assumed that the soil resistivity is 100 Ωm [31].Figure 2 depicts the conductor arrangements and tower configuration [31].The most sophisticated JMARTI model, which is a frequency-dependent model and suitable for transients research [31,32], is used to simulate the considered transmission line.Table 1 lists the key characteristics of the transmission system in operation.There are two ground wires with direct tower grounding, and phase conductors are presumed to be perfectly transposed.It is assumed that the soil resistivity is 100 Ωm [31].Figure 2 depicts the conductor arrangements and tower configuration [31].

System Description and Modelling
Figure 1 depicts the system under analysis, which is based on a typical 500 kV transmission line in Egypt with a 200 km-long line.The most sophisticated JMARTI model, which is a frequency-dependent model and suitable for transients research [31,32], is used to simulate the considered transmission line.Table 1 lists the key characteristics of the transmission system in operation.There are two ground wires with direct tower grounding, and phase conductors are presumed to be perfectly transposed.It is assumed that the soil resistivity is 100 Ωm [31].Figure 2 depicts the conductor arrangements and tower configuration [31].

Analysis of 500 kV OHTL Fault Signals
A practical Egyptian 500 kV OHTL ATP/EMTP simulation uses a 200 km transmission line model.Twenty identical blocks, each 10 km in length, are connected in a cascade to develop the 200 km OHTL model.This designed system is shown in Figure 3. Faults have been conducted at the intermediate junctions of each consecutive block, and the fault current waveforms are recorded at the sending side only.Faults depend on five main fault parameters: (fault type, fault distance, fault resistance, noise, and inception angle).
Energies 2023, 16, 758 5 of 23 to develop the 200 km OHTL model.This designed system is shown in Figure 3. Faults have been conducted at the intermediate junctions of each consecutive block, and the fault current waveforms are recorded at the sending side only.Faults depend on five main fault parameters: (fault type, fault distance, fault resistance, noise, and inception angle).to develop the 200 km OHTL model.This designed system is shown in Figure 3. Faults have been conducted at the intermediate junctions of each consecutive block, and the fault current waveforms are recorded at the sending side only.Faults depend on five main fault parameters: (fault type, fault distance, fault resistance, noise, and inception angle).@10km @50km @100km @150km @190km

FRFT and SVD Overview
Since they are used in our suggested fault diagnosis technique, FRFT and SVD schemes will be briefly covered in this section.

Fractional Fourier Transform (FRFT)
One effective method for time-varying signal analysis is the fractional Fourier transform (FRFT).The Fourier Transform (FT) is generalized by the FRFT.FT defines a signal's spectral content, not the timing of its spectral components [33][34][35].Signals are rotated in the time-frequency domain by the FRFT method.As a consequence, the FRFT may transform a signal from the time domain X(t) to the frequency domain Xα(u) of the signal.The following mathematical formula can be used to define the αth order FRFT of X(t): A fractional factor (α) with a range of 0 to 1 determines the FRFT coefficients.Therefore, in the suggested technique, we choose eigenvector FRFT as a feature extractor because of its superior advantages (with a factor between 0 and 1).

Singular Value Decomposition (SVD)
Singular value decomposition (SVD) is a matrix factorization technique that is particularly useful for a wide range of applications, including pattern identification, data dimension reduction, matrix approximation, pseudo inverse computation, and linear equation solving.As a data processing approach, SVD has been successfully used in signal processing and has been demonstrated to be effective in preventing modal aliasing.It may divide any matrix into three matrices as follows: where UU′ = 1 and VV′ = 1, which are referred to as the left and right singular vectors, respectively, and U and V are unitary matrices.The singular values of A, which are determined by determining the eigenvalues of AA′, are represented by the diagonal matrix S. It has the following representation:

FRFT and SVD Overview
Since they are used in our suggested fault diagnosis technique, FRFT and SVD schemes will be briefly covered in this section.

Fractional Fourier Transform (FRFT)
One effective method for time-varying signal analysis is the fractional Fourier transform (FRFT).The Fourier Transform (FT) is generalized by the FRFT.FT defines a signal's spectral content, not the timing of its spectral components [33][34][35].Signals are rotated in the time-frequency domain by the FRFT method.As a consequence, the FRFT may transform a signal from the time domain X(t) to the frequency domain X α (u) of the signal.The following mathematical formula can be used to define the αth order FRFT of X(t): A fractional factor (α) with a range of 0 to 1 determines the FRFT coefficients.Therefore, in the suggested technique, we choose eigenvector FRFT as a feature extractor because of its superior advantages (with a factor between 0 and 1).

Singular Value Decomposition (SVD)
Singular value decomposition (SVD) is a matrix factorization technique that is particularly useful for a wide range of applications, including pattern identification, data dimension reduction, matrix approximation, pseudo inverse computation, and linear equation solving.As a data processing approach, SVD has been successfully used in signal processing and has been demonstrated to be effective in preventing modal aliasing.It may divide any matrix into three matrices as follows: where UU = 1 and VV = 1, which are referred to as the left and right singular vectors, respectively, and U and V are unitary matrices.The singular values of A, which are determined by determining the eigenvalues of AA , are represented by the diagonal matrix S. It has the following representation: where ρ is the rank of the matrix A. Note that S 1 > S 2 > . . .> S ρ , i.e., S 1 is the largest singular value.Because SVD has the ability to describe the feature matrix as a collection of values (singular values), it has a dimension-reduction technique.The solitary values also have high stability.In other words, there isn't a significant variation in the singular values of the feature matrix element as it changes.The block diagram for the proposed method for power system fault localization using the Fractional Fourier Transform-Singular Value Decomposition (FRFT-SVD) method is shown in Figure 5.

Results and Discussion
A diagonal matrix produced by SVD over the fault current transients exhibits many of the fundamental properties of the original matrix.The vectors consist of 1500 data points, each with a pre-fault length of quarter cycles and a post-fault length of half cycles, and a sampling frequency of 2000 samples per cycle.As a result, the data matrix is produced as follows:

Data Preparation for Testing
The preparation of test data is identical as well.The result is the test data matrix, which has an unknown fault location but a known fault class t: Hence, T nt = [ia nt ia nt ia nt ] 1500×3 , where T nt is the test data matrix.

The Impact of Noise, Fault Resistance, and Inception Angle
The fault signals have been combined with Gaussian white noise to create noisecontaminated fault signals.By adjusting the SNR level, the fault waveform noise level can also be adjusted in four steps.The more significant point is that the proposed model is tested at a high noise level of 15 dB SNR, which is higher than the typical noise level used in most research.The impact of this undesirable noise is considered even when variations in fault type, location, and fault resistance occur concurrently.FRFT creates a signal's intermediate time-frequency representations.SVD has a dimension reduction strategy because it expresses the feature matrix as a collection of singular values.Additionally, the singular values are stable.The maximum SVD of the FRFT for a single phase yields a single feature (maximum value S matrix).Three features are chosen for each fault state in OHTL.FRFT-SVD thus eliminates the impact of noise on discrimination.In this work, the noise immunity property is also studied.A comparison of the maximum singular value for direct standardized fault signals and that of its filtered form is shown in Table 2.Additional results are declared in Tables A1-A4.The observations demonstrate that filtering has no discernible impact on the FRFT-SVD algorithm's results, as there is no noticeable change in the magnitude of the max singular value.Using the suggested FRFT-SVD based fault analyzer has this as a major benefit.By doing away with the need for filtering, FRFT-SVD can lessen the computational load.The SNR is varied for this purpose to observe the variation in max singular value, and the proposed algorithm is then run under more challenging conditions with higher noise levels.The maximum singular values in Tables 2 and A1, Tables A2-A4 show how the results of analyzing the filtered and unfiltered signals using maximum singular values are extremely similar.Filtering thus becomes unnecessary at the maximum singular value, saving vital computation and processing time.This demonstrates the inherent ability of FRFT-SVD to largely ignore the effect of noise.To demonstrate the fault localizer method, the proposed work carefully examines five different instances.As shown in Tables 4 and A9, Tables A10-A12, the maximum singular value of each faulty signal corresponding to the 19 separate fault locations throughout the length of the line is ascertained by utilizing a singular value to analyze the three-phase working signals to demonstrate this analytically.The maximum singular values obtained for the five faults in Tables 4 and A9, Tables A10-A12 are scaled in relation to the maximum values.For each phase independently, the curve fitting technique uses these 19 scaled S values of the faulty lines as training points.
The five different fit models were progressively applied to the same set of maximum scaled singular values in order to determine which model provided the best fit with the least chi-squared and the highest adjusted R square.For the LG, LL, LL-G, LLL, and LLL-G faults, respectively, these best fit curves with the lowest reduced chi-squared and highest adjusted R-square.The best curve-fitting formula is accounted for in the fifth one.It is the most effective strategy for fault location determination.This last fitting expression employs an exponential decay with a third degree.The lowest possible error between the actual location and the predicted location is estimated using the proposed fitting formula at LG, LL, LL-G, LLL, and LLL-G faults, as depicted in Figure 6, Figure 7, Figure 8, Figure 9, and Figure 10, respectively.The difference between the predicted and real fault distances (P and A) was utilized to quantify this inaccuracy.This estimated error is equivalent to the algorithm's accuracy.As the distance deviation rises, the algorithm's accuracy falls.The overall accuracy is defined as the greatest estimate error throughout the whole length range of the line as well as for all conceivable fault types represented as a percentage of the total line length, C in Equation ( 12), and the average error (AE), is defined in Equation (13).
error (e) = predicted location(P) − actul location(A) total line length(C) × 100 ( 12) Tables 5-14 show the locations of five faults that were predicted based on current line data and fitted using various five-curve fitting methods.Furthermore, the average error (AE) and five different fault prediction errors using line current signals and various five curve fittings are shown in these tables.The proposed work produces a localization that is extremely accurate, with a maximum percentage error of 0.002% and an average percentage error for fault localization of just 0.0016%.The proposed approach is validated with other approaches, as shown in Table 15.Both the maximum and average percentage errors are computed and compared with literature work [9,11,12,35].A comparison between the proposed work and that experimentally published is demonstrated.This comparison was made using the same data in the form of current signals.These simulated data are generated using ATP-EMTP that was tested with the mentioned methods on [9,11,12,35].In addition, this data corresponds to changes in power system faults in a 200 km long, 500 kV Egyptian transmission line using sent end-line current signals.The proposed work is executed using a laptop with Intel(R) Core(TM) i7-10750H and a 2.59 GHz CPU.One of the main strengths of the suggested work is the ability to use it as a link between the nuclear power plant and the grid.Moreover, the proposed method is an effective fault distance estimating method that might help to develop a reliable transient-based strategy for power system protection with minimal error.Tables 5-14 show the locations of five faults that were predicted based on current line data and fitted using various five-curve fitting methods.Furthermore, the average error (AE) and five different fault prediction errors using line current signals and various five curve fittings are shown in these tables.The proposed work produces a localization that is extremely accurate, with a maximum percentage error of 0.002% and an average percentage error for fault localization of just 0.0016%.The proposed approach is validated with other approaches, as shown in Table 15.Both the maximum and average percentage errors are computed and compared with literature work [9, 11, 12, and 35].A comparison between the proposed work and that experimentally published is demonstrated.This comparison was made using the same data in the form of current signals.These simulated data are generated using ATP-EMTP that was tested with the mentioned methods on [9, 11, 12, and 35].In addition, this data corresponds to changes in power system faults in a 200 km long, 500 kV Egyptian transmission line using sent end-line current signals.The proposed work is executed using a laptop with Intel(R) Core(TM) i7-10750H and a 2.59 GHz CPU.One of the main strengths of the suggested work is the ability to use it as a link between the nuclear power plant and the grid.Moreover, the proposed method is an effective fault distance estimating method that might help to develop a reliable transientbased strategy for power system protection with minimal error.

Conclusions
The FRFT-SVD method, which is very effective, precise, and straightforward, was used to develop the localizer model.Here is a practical power system protection approach for long-distance power system issue prediction for a 500 kV, 50 Hz, 200 km overhead transmission line.FRFT-SVD was used to develop the suggested protective mechanism.Using the best-fit curve method, sending end current signals are evaluated after being subjected to FRFT-SVD analysis to obtain fault characteristics expressed in terms of the highest S value.Only around 0.001614% of the typical scheme is inaccurate.The lowest estimate for an LLL-G defect was 0.002%, which is also quite precise for a 200-km long line.This paper aims to provide a technique for predicting the location of faults using curve fitting analysis and FRFT-SVD.For this reason, the analysis makes use of a variety of noise, inception angle, and fault resistance variables.The proposed method may prove to be an effective way to gauge how far a problem would spread, which might be useful in creating a dependable transient-based power system security strategy.In future work, we will implement the new model in hardware and apply it in real systems with different configurations.

Figure 1 .
Figure 1.A 500 kV transmission line in Egypt with a 200 km-long line.

3 . 8 Figure 1 .
Figure 1.A 500 kV transmission line in Egypt with a 200 km-long line.

Figure 1 .
Figure 1.A 500 kV transmission line in Egypt with a 200 km-long line.

3 .
Analysis of 500 kV OHTL Fault SignalsA practical Egyptian 500 kV OHTL ATP/EMTP simulation uses a 200 km transmission line model.Twenty identical blocks, each 10 km in length, are connected in a cascade

Figure 3 .
Figure 3. Simulated OHTL model.Five different fault types (Line to Ground (LG), Line to Line (LL), Line to Line to Ground (LL-G), Line to Line to Line (LLL), and Line to Line to Line to Ground (LLL-G)) are simulated in this regard, together with five fault resistances (R), five inception angles (Ɵ) (including 0°-180°), and 19 distances of the fault from the recording site (including 10 km-190 km).Following the application of LG fault type to the phases at 190 km (before the line's end) from the transmitting side, Figure4a-dshow the faulted-phase current waveforms at different locations, fault resistance, and inception angles.Figure4bshows the sending side signals under different locations (with fault resistance = 10 Ω, inception angle = 0° and fault location = (10 km, 50 km, 100 km, 150 km, 190 km) for the one-phase (a) to ground fault.It is evident that the faulted phase current signal peak reduced from 25 kA to 5.8 kA, or by about 77%.Figure4cshows the sending end signals under different R (with R = (1 Ω, 10 Ω, 25 Ω, 35 Ω, 50 Ω), inception angle = 0° and fault location = 190 Km for the one-phase (a) to ground fault.It is clear that the faulted phase current amplitude reduced from 6.4 kA to 4.1 kA, or by about 36%.Figure4dshow the sending end signals under different inception angle (with fault resistance = 10Ω, Inception Angle Ɵ = (0° to 180°), and fault location = 190 km for the one-phase (a) to ground fault.Moreover, the beginning current amplitude in the case of fault at 0°, and 45° is much more than 90°, 135°, and 180°.
Figure 4b shows the sending side signals under different locations (with fault resistance = 10 Ω, inception angle = 0° and fault location = (10 km, 50 km, 100 km, 150 km, 190 km) for the one-phase (a) to ground fault.It is evident that the faulted phase current signal peak reduced from 25 kA to 5.8 kA, or by about 77%. Figure 4c shows the sending end signals under different R (with R = (1 Ω, 10 Ω, 25 Ω, 35 Ω, 50 Ω), inception angle = 0° and fault location = 190 Km for the one-phase (a) to ground fault.It is clear that the faulted phase current amplitude reduced from 6.4 kA to 4.1 kA, or by about 36%. Figure 4d show the sending end signals under different inception angle (with fault resistance = 10Ω, Inception Angle Ɵ = (0° to 180°), and fault location = 190 km for the one-phase (a) to ground fault.Moreover, the beginning current amplitude in the case of fault at 0°, and 45° is much more than 90°, 135°, and 180°.

Figure 3 .
Figure 3. Simulated OHTL model.Five different fault types (Line to Ground (LG), Line to Line (LL), Line to Line to Ground (LL-G), Line to Line to Line (LLL), and Line to Line to Line to Ground (LLL-G)) are simulated in this regard, together with five fault resistances (R), five inception angles (θ) (including 0 • -180 • ), and 19 distances of the fault from the recording site (including 10 km-190 km).Following the application of LG fault type to the phases at 190 km (before the line's end) from the transmitting side, Figure 4a-d show the faulted-phase current waveforms at different locations, fault resistance, and inception angles.Figure 4b shows the sending side signals under different locations (with fault resistance = 10 Ω, inception angle = 0 • and fault location = (10 km, 50 km, 100 km, 150 km, 190 km) for the one-phase (a) to ground fault.It is evident that the faulted phase current signal peak reduced from 25 kA to 5.8 kA, or by about 77%. Figure 4c shows the sending end signals under different R (with R = (1 Ω, 10 Ω, 25 Ω, 35 Ω, 50 Ω), inception angle = 0 • and fault location = 190 Km for the one-phase (a) to ground fault.It is clear that the faulted phase current amplitude reduced from 6.4 kA to 4.1 kA, or by about 36%. Figure 4d show the sending end signals under different inception angle (with fault resistance = 10 Ω, Inception Angle θ = (0 • to 180 • ), and fault location = 190 km for the one-phase (a) to ground fault.Moreover, the beginning current amplitude in the case of fault at 0 • , and 45 • is much more than 90 • , 135 • , and 180 • .
Figure 4b shows the sending side signals under different locations (with fault resistance = 10 Ω, inception angle = 0 • and fault location = (10 km, 50 km, 100 km, 150 km, 190 km) for the one-phase (a) to ground fault.It is evident that the faulted phase current signal peak reduced from 25 kA to 5.8 kA, or by about 77%. Figure 4c shows the sending end signals under different R (with R = (1 Ω, 10 Ω, 25 Ω, 35 Ω, 50 Ω), inception angle = 0 • and fault location = 190 Km for the one-phase (a) to ground fault.It is clear that the faulted phase current amplitude reduced from 6.4 kA to 4.1 kA, or by about 36%. Figure 4d show the sending end signals under different inception angle (with fault resistance = 10 Ω, Inception Angle θ = (0 • to 180 • ), and fault location = 190 km for the one-phase (a) to ground fault.Moreover, the beginning current amplitude in the case of fault at 0 • , and 45 • is much more than 90 • , 135 • , and 180 • .

Figure 3 .
Figure 3. Simulated OHTL model.Five different fault types (Line to Ground (LG), Line to Line (LL), Line to Line to Ground (LL-G), Line to Line to Line (LLL), and Line to Line to Line to Ground (LLL-G)) are simulated in this regard, together with five fault resistances (R), five inception angles (Ɵ) (including 0°-180°), and 19 distances of the fault from the recording site (including 10 km-190 km).Following the application of LG fault type to the phases at 190 km (before the line's end) from the transmitting side, Figure4a-dshow the faulted-phase current waveforms at different locations, fault resistance, and inception angles.Figure 4b shows the sending side signals under different locations (with fault resistance = 10 Ω, inception angle = 0° and fault location = (10 km, 50 km, 100 km, 150 km, 190 km) for the one-phase (a) to ground fault.It is evident that the faulted phase current signal peak reduced from 25 kA to 5.8 kA, or by about 77%. Figure 4c shows the sending end signals under different R (with R = (1 Ω, 10 Ω, 25 Ω, 35 Ω, 50 Ω), inception angle = 0° and fault location = 190 Km for the one-phase (a) to ground fault.It is clear that the faulted phase current amplitude reduced from 6.4 kA to 4.1 kA, or by about 36%. Figure 4d show the sending end signals under different inception angle (with fault resistance = 10Ω, Inception Angle Ɵ = (0° to 180°), and fault location = 190 km for the one-phase (a) to ground fault.Moreover, the beginning current amplitude in the case of fault at 0°, and 45° is much more than 90°, 135°, and 180°.
Figure 4b shows the sending side signals under different locations (with fault resistance = 10 Ω, inception angle = 0° and fault location = (10 km, 50 km, 100 km, 150 km, 190 km) for the one-phase (a) to ground fault.It is evident that the faulted phase current signal peak reduced from 25 kA to 5.8 kA, or by about 77%. Figure 4c shows the sending end signals under different R (with R = (1 Ω, 10 Ω, 25 Ω, 35 Ω, 50 Ω), inception angle = 0° and fault location = 190 Km for the one-phase (a) to ground fault.It is clear that the faulted phase current amplitude reduced from 6.4 kA to 4.1 kA, or by about 36%. Figure 4d show the sending end signals under different inception angle (with fault resistance = 10Ω, Inception Angle Ɵ = (0° to 180°), and fault location = 190 km for the one-phase (a) to ground fault.Moreover, the beginning current amplitude in the case of fault at 0°, and 45° is much more than 90°, 135°, and 180°.

Figure 5 .
Figure 5.The block diagram for the proposed method.Figure 5.The block diagram for the proposed method.

Figure 5 .
Figure 5.The block diagram for the proposed method.Figure 5.The block diagram for the proposed method.

Figure 6 .
Figure 6.The best fitting curve at 19 fault locations for LG fault using scaled maximum singular value.

Figure 7 .Figure 6 .Figure 6 .
Figure 7.The best fitting curve at 19 fault locations for LL fault using scaled maximum singular value.

Figure 7 .Figure 7 .
Figure 7.The best fitting curve at 19 fault locations for LL fault using scaled maximum singular value.

Figure 8 .
Figure 8.The best fitting curve at 19 fault locations for LL-G fault using scaled maximum singular value.

Figure 9 .Figure 8 . 24 Figure 8 .
Figure 9.The best fitting curve at 19 fault locations for LLL fault using scaled maximum singular value.

Figure 9 .Figure 9 .
Figure 9.The best fitting curve at 19 fault locations for LLL fault using scaled maximum singular value.

Figure 10 .
Figure 10.The best fitting curve at 19 fault locations for LLL-G fault using scaled maximum singular value.

Table 1 .
The main parameters of the power line.

Table 1 .
The main parameters of the power line.

Table 1 .
The main parameters of the power line.

Table 2 .
Max singular value of FRFT-SVD results at various SNRs for direct and filtered signals underLG fault scenario.

Table 3 .
Max singular values of FRFT-SVD results for direct and filtered current signals at various faults and inception angles with R = 10 Ω.

Table 4 .
The maximum singular and its scaled values for LG fault at 19 locations.

Table 5 .
Actual (A in km) and predicted (P in km) fault locations under LG fault scenario.

Table 6 .
LG fault prediction error utilizing line current signals.

Table 5 .
Actual (A in km) and predicted (P in km) fault locations under LG fault scenario.

Table 6 .
LG fault prediction error utilizing line current signals.

Table 7 .
Actual (A in km) and predicted (P in km) fault locations under LL fault scenario.

Table 9 .
Actual (A in km) and predicted (P in km) fault locations under LL-G fault scenario.

Table 11 .
Actual (A in km) and predicted (P in km) fault locations under LLL fault scenario.

Table 13 .
Actual (A in km) and predicted (P in km) fault locations under LLL-G fault scenario.

Table A2 .
Max singular value of FRFT-SVD results at various SNRs for direct and filtered signals under LL-G fault scenario.

Table A3 .
Max singular value of FRFT-SVD results at various SNRs for direct and filtered signals under LLL fault scenario.

Table A7 .
Max singular values of FRFT-SVD results for direct and filtered current signals at various faults and inception angles with R = 150 Ω.

Table A8 .
Max singular values of FRFT-SVD results for direct and filtered current signals at various faults and inception angles with R = 200 Ω.

Table A9 .
The maximum singular and its scaled values for LL fault at 19 locations.

Table A10 .
The maximum singular and its scaled values for LL-G fault at 19 locations.

Table A11 .
The maximum singular and its scaled values for LLL fault at 19 locations.

Table A12 .
The maximum singular and its scaled values for LLL-G fault at 19 locations.