Fault Detection Based on the Generalized S-Transform With a Variable Factor for Resonant Grounding Distribution Networks

Fault detection in resonant grounding (RG) distribution networks remains a challenge due to weak fault signals, extremely complex fault conditions, and unstable intermittent arc grounding faults. This paper addresses this issue by applying generalized S-transform (GST) with a variable factor to conduct denoising of transient zero-sequence currents based on threshold filtering followed by time-frequency distribution filtering in sequence. Meanwhile, this paper proposes a comprehensive multi-criteria faulty feeder detection method based on the transient zero-sequence current polarity (criterion 1), the energy relative entropy (criterion 2), and the total transient current energy (criterion 3). Here, criteria 2 and 3 are based on the time-frequency representation of the GST. The performances of the proposed denoising and faulty feeder detection methods are evaluated under single line to ground faults based on simulations conducted using a modeled 10 kV RG networks with overhead and cable mixed lines in addition to reasonably sophisticated permanent and intermittent arc discharge models to ensure that the simulations faithfully represent actual complex working conditions. Compared with existing method, simulation experiments and field test show that the method proposed in this paper provide a better denoising effect with stronger self-adaptability, higher detection accuracy, and a faster calculation speed.


I. INTRODUCTION
Most of the distribution networks in China operating in the range of 10-35 kV are grounded by arc suppression coils. Such systems are typically denoted as resonant grounding (RG) distribution networks. Approximately 80% of faults that occur in RG networks are single line to ground (SLG) faults. Such faults are particularly serious because they can damage feeder insulation, and thereby lead to multi-point grounding that can critically detract from system stability and power supply reliability. The seriousness of SLG faults is indicated by the newest technical guidelines for distribution networks published in 2016 [1] instructing operators to identify faulty feeders as soon as possible after an SLG The associate editor coordinating the review of this manuscript and approving it for publication was Francesco Tedesco . fault occurs, which represents a substantial upgrade in concern from the previous instruction to ''continue operating for 2 hours''. However, the detection of SLG faults in RG networks remains a challenge due to fault signals that are too weak to identify, frequent intermittent arc grounding faults with unstable signals, and complex working conditions such as noisy environments due to extensive cabling and limitations in measurement accuracy, and high impedance faults [2]- [5]. Hence, selecting the faulty feeder quickly and correctly is essential for ensuring the stability and reliability of RG networks.
The WT provides a time-frequency window that varies according to the frequency so that it can extract time and frequency information from transient fault signals. Wang et al. [7] constructed a mixed atom dictionary by a cosine packet and a wavelet and adopted the matching pursuit algorithm to extract characteristic atoms of each feeder to extract transient signals. Guo et al. [8] used wavelet packet to make five-layer decomposition of main loop current firstly, and then, selected the effective coefficients based on the change rate of a wavelet packet energy entropy to reconstruct an effective signal of the arc fault. Guo et al. [9] acquired time-frequency gray scale images for all feeders via the continuous wavelet transform (CWT), and distinguished images associated with the faulty feeder from those associated with healthy feeders using a trained convolutional neural network (CNN). Fault detection has also been facilitated by applying the wavelet packet transform to segregate the high-frequency components of three-phase current signals during power swings [10].
The HHT employs empirical mode decomposition (EMD) and Hilbert transform, and is applicable for the analysis of nonlinear and non-stationary signals. Chen et al. [11] used the EMD to resolve the current into several intrinsic mode function (IMF), and then calculated Hilbert transform for IMF to obtain the Hilbert spectrum of arc fault current signals to recognize the series arc fault. Wang et al. [12] utilized complete ensemble EMD with adaptive noise (CEEMDAN) to extract the first intrinsic mode function (IMF) of the zero-sequence current and used the Hilbert transform to obtain a fault detection index for identifying the faulty feeder.
For the ST, Stockwell et al. [13] presented a lossless and reversible multiresolution time-frequency analysis method in 1996, where the resolution was adjustable according to changes in the frequency based on a moving and scalable localizing Gaussian window. Pinnegar and Mansinha [14] modified the ST into a generalized ST (GST) form to obtain time-frequency spectra with different time-frequency resolutions by manually changing the value of an adjustable parameter. However, the process was not adaptive. Zhou and Chen [15] addressed this limitation in the local window resolution of the time-frequency spectrum by adopting a variable factor that improves the scalability of the localizing Gaussian window. He et al. [16] obtained the time-frequency matrix of transient zero-sequence currents via the GST, and realized fault section location by comparisons of the relative entropy of the transient current energies (i.e., transient current energy relative entropy). Fault detection has also been facilitated by extracting features from three-phase current signals by the ST, and distinguishing features associated with the faulty feeder from those associated with healthy feeders using a feedforward neural network (FFNN) [17]. Peng et al. [18] utilized ST to extract the standard amplitude ratios of the frequency components in zero-mode transients of each feeder and proposed a faulty feeder detection criterion by utilizing the amplitude information of the zero-mode transients captured at both terminals.
The Prony algorithm is a method for fitting data sampled in equal intervals using a linear combination of a set of exponential terms. Faulty feeder detection was conducted based on a linear relationship between the fault location and the inverse of the damping coefficient of the fundamental frequency component in the Prony model [19]. Liu et al. [20] utilized Prony algorithm to extract the transient information of main frequency components at each feeder to select faulty feeder.
The MM approach is a mature method of nonlinear signal analysis that was originally applied in the field of image signal processing. Barik et al. [21] was the first to utilize MM for identifying faulty feeders in RG systems. Shu et al. [22] detected faulty feeders according to the morphological peak-valley features of the constructed wavelet coefficients under characteristic bands. The morphological median filter is exploited to wrest unique fault features which are then fed as an input to a decision tree classifier to classify the fault type [23].
The FCM clustering algorithm automatically classifies sample data to obtain the membership of each sample point to sample centers. Guo and Yang [24] applied FCM clustering to the amplitude-polarity feature matrix (APFM) of transient fault signals to detect faulty feeders according to matrix elements lying outside of a threshold setting. Amplitude correlation coefficient matrix (ACCM) of transient zero-sequence currents was obtained via improved dynamic time warping (DTW), and then the faulty feeder was selected by FCM cluster method without setting threshold [25].
The SVM is a supervised learning model that serves as a kind of generalized linear classifier for the binary classification of data. An SVM classification algorithm has been employed for single-phase arc-fault detection in distribution networks based on a combination of transient zero-sequence current and two-phase current difference characteristics [26]. Chaitanya et al. [27] presented an high-impedance faults detection technique for distribution lines incorporating the distributed generators, where variational mode decomposition was used to obtain the analytic signal and then used this data as the input into the support vector machine to create an intelligent classifier.

B. PROBLEMS OF EXISTING FAULTY FEEDER DETECTION METHODS FOR RG SYSTEMS 1) FAULT SIGNAL PROCESSING
The presence of noise distorts the characteristics of fault signals. However, most existing methods extract fault features directly without first filtering the signals to remove noise.

2) FAULT FEATURE EXTRACTION
For WT-based methods, the wavelet basis function is difficult to select, and this seriously restricts the adaptability of WT signal decomposition to accommodate variations in signal features. The HHT has three problems that distort the signal decomposition results. These include the end effect of EMD, the EMD mode mixing problem, and the residual noise. The specific mother wavelet of the original ST proposed by Stockwell renders adjustments to the time-frequency resolution inflexibly because of the fixed window width of Gaussian window function, and the later proposed modifications to the GST are not effectively applied in fault detection. Signal feature extraction conducted by the Prony algorithm is susceptible to noise, and its fitting effect for high frequency signals is not good.

3) FAULTY FEEDER DETECTION CRITERIA
Most existing faulty feeder detection methods apply a single detection criterion. As a result, these methods are prone to incorrectly detect faulty feeders due to variations in signal features. Although multi-criteria methods have been developed to improve the accuracy of faulty feeder detection, existing methods are difficult to utilize because of the vast sample libraries required, complicated procedures, and long calculation times.

C. CONTRIBUTIONS
The above problems are addressed in the present study, and thereby makes the following contributions.

1) THE PURITY OF RECORDED FAULT DATA
The GST is modified to include a variable factor σ for self-adaptively adjusting the local window resolution of the VOLUME 8, 2020 time-frequency spectrum, rather than simply narrowing the window with increasing frequency. Then, the modified GST, denoted as GST σ , is employed to obtain the time-frequency matrix of a transient zero-sequence current signal. Denoising is then applied to the time-frequency matrix, and is conducted using a sequence of threshold filtering followed by time-frequency distribution filtering, which has been demonstrated to provide accurate denoising [28]. Most existing methods detect faulty feeder by recorded fault data without denoising, but this proposed method applies threshold filtering and time-frequency distribution filtering based on the generalized S-transform with a variable factor, which can detect correct faulty feeder in most noisy environments.

2) THE RELIABILITY OF MULTI-CRITERIA
A comprehensive multi-criteria faulty feeder detection method is established. Compared with existing singlecriterion detection method, this proposed method includes three criteria (transient zero-sequence current polarity, transient current energy relative entropy, and total transient current energy) to reduce misjudgment. The second and third criteria are based on the time-frequency representation of the GST σ approach. The method considers both the direction and energy of transient signal currents, which provides more comprehensive and accurate detection results than single-criterion methods. Meanwhile, the method is immune to overcompensation degrees, fault locations, fault resistances, initial fault angles, and noise intensities.

3) THE REALITY OF SIMULATION SYSTEM
The performances of the proposed denoising and faulty feeder detection methods are evaluated under both constant resistance faults and arc grounding faults based on simulations conducted using a modeled 10 kV RG distribution network with an overhead and cable mixed line structure in addition to a reasonably sophisticated arc discharge model to ensure that the simulations faithfully represent actual complex working conditions.
The proposed method is verified to provide a better denoising performance with stronger self-adaptability, higher detection accuracy, and a faster calculation speed compared with existing methods.

II. ANALYSIS OF TRANSIENT ZERO-SEQUENCE CURRENTS
An equivalent circuit of a transient SLG fault process in an RG system is illustrated schematically in Fig. 1, where C 0 is three-phase equivalent capacitance to ground, R 0 and L 0 are the respective equivalent resistance and inductance of the three-phase lines and all components in the zero-sequence circuit, L and r L are the respective inductance and resistance of the arc suppression coil, u 0 is the equivalent zero-sequence supply voltage at the bus bar, i d is the transient ground current, and i L and i C are the transient inductance and capacitance ground currents respectively produced by the arc suppression coil.  When an SLG fault occurs in an RG system, i C is larger in magnitude and changes faster than i L , which can therefore be ignored. As a result, i d = i C . Therefore, only transient capacitance currents are considered in this paper.
According to the simplified equivalent circuit presented in Fig. 2, i C can be obtained as follows.
Here, t is time, U m is the voltage amplitude of the m-th phase, ω is the angular frequency of the steady-state power frequency components, and ϕ is its phase angle. In addition, we can define the amplitude of i C as follows.
The transient capacitance current i C (t) consists of the transient free oscillation component i C (t) and the steady-state power frequency component i C (t), which adhere to the following relationship.
According to (2) and (3), we can obtain the following equation by applying the Laplace transform.
Here, ω f and δ are the angular frequency and the attenua- , with τ C represents a time constant. Therefore, when ϕ = π 2, the transient zero-sequence current can be calculated as follows.
For a three-phase line, the grounding current i k (t) is three times greater than i 0 (t), which yields the following expression.
Meanwhile, when an SLG fault occurs, the transient zero-sequence currents of the feeder lines adhere to the following characteristics in the selected frequency band (SFB) [29], [30].
(i) The amplitude of i C (t) is larger for a faulty feeder than that of any healthy feeder.
(ii) The flow direction of i C (t) (i.e., its polarity) is different for a faulty feeder than that for a healthy feeder, where i C (t) flows from the line to the bus bar for a faulty feeder, while i C (t) flows from the bus to the line for a healthy feeder.

III. DENOISING BASED ON THE MODIFIED GST A. MODIFIED GST
As discussed above, Zhou and Chen [15] improved the scalability of the localizing Gaussian window used to define the local window resolution in GST by applying a variable factor σ f , as follows.
Here, f is the frequency and σ f is dependent on the value of f , such as where a, b, and γ are constants. For the linear variation σ f (f ) = af + b, the GST of a signal h (t) is given as follows.
When h [mT ], m = 0, 1, . . . , N − 1 denote a discrete time series corresponding to h (t) with a time sampling interval of T . The discrete Fourier transform is given as follows.
And for the n = 0 voice, the discrete GST can be defined as follows.
In the discrete signal analysis, the maximum frequency is denoted as f max and σ f is given as where α = 0.3, β = 8, and σ f is a monotonically increasing function of f . Finally, the two-dimensional (2-D) time-frequency matrix (GST(k, n)) of the discrete signal based on h (t) is obtained by the GST as follows.
Here, A (k, n) is the amplitude matrix and ϕ (k, n) is the phase matrix. The rows of A (k, n) and ϕ (k, n) correspond to the amplitude and phase vary with the time at a certain sampled frequency point, respectively, and the columns of A (k, n) and ϕ (k, n) correspond to the amplitude and phase vary with the frequency at a certain sampled time point, respectively.

B. DENOISING METHOD
The proposed denoising method is schematically illustrated by the flow chart shown in Fig. 3. Here, threshold filtering and time-frequency distribution filtering are applied in sequence to GST(k, n), which is obtained by applying the discrete GST σ to the discrete time series based on h (t).
Finally, the discrete denoised signal is obtained by applying the discrete inverse GST σ . The denoising process is described as follows. Firstly, threshold filtering is applied to GST(k, n) based on a determined threshold η. The appropriate value of η is obtained by first determining the overall range of GST(k, n), which is given as the difference between the maximum GST H = max (GST (k, n)) and the minimum GST L = min (GST (k, n)). Then, the overall range is divided into K = 2 n (i.e., n = 7, 8) intervals as follows.
The range of each interval is Then, the appropriate value of η is obtained by analyzing data in each interval. The threshold function is given as follows.
After conducting threshold filtering, GST(k, n) is transformed as follows. Secondly, time-frequency distribution filtering is applied to GST 1 (k, n). A noisy signal h (t) can be described as the sum of an effective signal x (t) and a noise signal n (t), as follows.
This relationship is preserved in the transformed version of h (t) as follows.
As a result, GST x (k, n) can be obtained if GST n (k, n) = 0. As shown in Fig. 4, time-frequency distribution filtering eliminates GST n (k, n) based on the clustering of GST x (k, n) and GST n (k, n) in the time-frequency domain. Therefore, the range of the noise signal time and frequency can be determined analyzing the time-frequency distribution of noisy signal. The time-frequency distribution function is then given as follows.
After conducting time-frequency distribution filtering, GST 1 (k, n) is transformed as follows.
Finally, applying the discrete inverse of the GST yields the effective signal as follows.

IV. COMPREHENSIVE MULTI-CRITERIA FAULTY FEEDER DETECTION METHOD
A. TRANSIENT ZERO-SEQUENCE CURRENT POLARITY CRITERION As discussed above, the transient zero-sequence current of a faulty feeder has a polarity that is opposite to that of a healthy feeder in the SFB [30]. Therefore, a faulty feeder can be identified if its transient zero-sequence current has the opposite polarity from those of all other feeders. Moreover, the faulty feeder can be identified as the bus if these current polarities are equivalent for all feeders in the RG system. Therefore, for an RG system composed of l feeders (l ≥ 2), the transient zero-sequence current polarity criterion can be formulated according to the following polarity relation between feeder i and feeder j.
If P ij > 0, then i 0i and i 0j have the same polarity, while, if P ij < 0, i 0i and i 0j have the opposite polarity. For example, if i = a, P aj < 0 (j = a), and all the other i = a, P ij > 0 (j = i), then feeder a is the faulty feeder. In addition, if P ij > 0 for all i and j (j = i), then the bus is the faulty feeder.

B. TRANSIENT CURRENT ENERGY RELATIVE ENTROPY CRITERION
Relative entropy, also known as Kullback-Leibler divergence or information divergence, is used to measure the difference between two probability distributions. For example, the relative entropy between P = {P 1 , P 2 , . . . , P n } and Q = {Q 1 , Q 2 , . . . , Q n } is calculated as follows.
Here, D KL (P Q) = 0 when P = Q, and increases as the difference between the two probability distributions increases. This expression is then employed to evaluate the differences between the transient zero-sequence current energies of all feeders in the SFB. Firstly, the transient energy of feeder i (i = 1, 2, . . . , l, l ≥ 2) at frequency f n is defined as follows [16], [31].
The transient energy of all feeders at frequency f n is then given as follows.
Then, the weight coefficient of feeder i at frequency f n , which is defined as where i p i_n = 1, is employed in (24) to evaluate the differences between the transient energies of feeder i and feeder j as follows.
Thus, the current energy relative entropy between feeder i and all other feeders is Clearly, the energy relative entropy between a faulty feeder and healthy feeders is greater than that between healthy feeders in the SFB because the amplitude of the transient current is larger for a faulty feeder than that of any healthy feeder. Therefore, as for a network with more than three feeders, the transient energy relative entropy criterion can be formu- According to the law of energy conservation, the transient energy E i produced by faulty feeder i is equivalent to the transient energy consumed by arc suppression coils E L and all healthy feeders j in the SFB [32]. This relationship can be expressed as follows.
To take advantage of this relationship, we define the difference between E i and the sum of all other E j as follows. Consequently, the faulty feeder can be identified as the feeder i for which E i is smallest. This can be formalized as follows.
Therefore, the total transient energy criterion identifies the faulty feeder as the feeder i meeting the condition E i = E f .

D. MULTI-CRITERIA FAULTY FEEDER DETECTION PROCESS
The above three criteria were employed for detecting faulty feeders according to the process illustrated by the flow chart in Fig. 5. Firstly, the polarities of all feeders are assessed, and, if all feeders have the same polarity (i.e., P all > 0), the detection result is a bus fault, and the process terminates. Otherwise, detection accuracy is enhanced by applying both the current polarity criterion (criterion 1) and the transient energy relative entropy criterion (criterion 2) simultaneously to determine if feeder a is faulty, and accepting a positive decision only if the decisions of both criteria are positive. However, the process yields no detection result if the decision of criterion 1 is negative. Otherwise, the total transient energy criterion (criterion 3) is applied to determine if feeder a is faulty. If so, then a positive decision is obtained; otherwise, the process yields no detection result. VOLUME 8, 2020

V. MODELS AND SIMULATIONS A. 10 KV RG DISTRIBUTION NETWORK MODEL
A model of the 10 kV RG distribution network illustrated in Fig. 6 was established in PSCAD. This model is based on a section of an actual medium-voltage distribution network in Jiangsu Province, China. A frequency dependent (phase) model was adopted for establishing the line model in PSCAD because the line component characteristics are related to the system frequency [33]. The system frequency is 50 Hz. A typical overhead and cable mixed line structure was adopted, where the overhead line includes single-circuit transmission lines and common-tower double-circuit transmission lines. The model adopted the parameters of JKLYJ-240 transmission cable for the common-tower double-circuit transmission lines, which are 2.26 km long in total, while the parameters of JKLYJ-150 transmission cable were adopted for the single-circuit transmission lines, which are 9.73 km long in total. The overall length of the overhead line is 31.43 km.
The underground cable was modeled as YJV22-3*400-400, YJV22-3*300-300, and YJV22-3*150-150 cable, which is 15.5 km long in total. The network includes 5 feeders, which are given as L 1 = 7.13 km, L 2 = 6.14 km, L 3 = 4.74 km, L 4 = 5.02 km, and L 5 = 8.4 km. Also shown in Fig. 6 is that an SLG fault has occurred at feeder L 5 . For a 10% overcompensation degree of the distribution network, the inductance L and resistance R L of the arc suppression coil can be calculated as L = 2.47 H and R L = 23.28 , respectively. In addition, i 01 , i 02 , i 03 , i 04 , and i 05 are the zero-sequence currents of feeders L 1 , L 2 , L 3 , L 4 , and L 5 , respectively.

B. ARC DISCHARGE MODEL
A standard arc discharge model is regarded as a description of the dynamic relationship between arc voltage and current. In the present study, a number of arc models have been subjected to comparison [33]. Kilicay arc model based on arc energy balance theory and control theory has been widely used, which was found to reflect arc grounding faults more intuitively and accurately. According to this model,  the change in the instantaneous conductance of the arc g (t) with respect to t is derived as follows [34].
Here, τ is the time constant of arc discharge, and G (t) is the static conductance of the arc. According to the standard relationship, g (t) can be expressed according to the arc resistance R arc (t) as follows.
In addition, G (t) can be expressed as a function of the arc current i arc (t) and the static arc voltage u st (t) as follows.
Here, u st (t) is given as where u arc (t) and r arc (t) are the characteristic voltage and resistance of the arc, respectively. Consequently, the dynamic mathematical arc model is described as follows.
It is noted that τ , u arc (t), and r arc (t) are characteristic parameters of arc discharge, and are usually considered constants. The values adopted in this paper are τ = 0.0002 s, u arc (t) = 1500 V, and r arc (t) = 0.005 . The arc discharge model illustrated in Fig. 7 with a constant resistance R in series with R arc (t) was employed in the present work owing to the characteristics of overhead and underground cable lines and the physical processes of line faults. The mathematical form of the model is where u f (t) is the arc grounding fault voltage. The arc discharge model established in PSCAD is shown in Fig. 8. The waveforms of i arc (t), u f (t) and R arc (t) obtained by arc ground fault simulations in PSCAD are shown in Fig. 9.

C. DIFFERENT FAULT CONDITIONS TESTING
The performance of the proposed comprehensive multicriteria faulty feeder detection method was evaluated for the modeled 10 kV RG distribution network under the constant resistance fault experiments with different fault conditions. It is verified from the aspects of different fault feeders, noise intensity, overcompensation degree, initial fault phase, fault resistance and fault distance, and specific fault conditions are shown in Table 2. Here, noisy environments with signal to noise ratio (SNR) values ranging from −10 dB to 10 dB were introduced in the simulations by adding white Gaussian noise with a zero mean and unit variance to the feeder signals according to the standard noisy signal model given in (18) using the awgn function of MATLAB.
In order to reflect the good performance of denoising method based on threshold filtering and time-frequency distribution filtering of GST with a variable factor proposed above, Fig. 10 shows the simulated transient zero-sequence current original waveforms, the current waveforms with added white Gaussian noise under low SNR (−5 dB) and denoised current waveforms of all feeders obtained during the first 1/4 industrial frequency period after a constant resistance SLG fault occurs in the feeder L 1 , L 3 , L 5 , and bus respectively. The figures indicate that the denoising method has a perfect denoising effect even if the excessive noise nearly covers the signals. In addtion, the transient zero-sequence current polarity of faulty feeder is opposite to those of other feeders except that the current polarities of all feeders are equivalent when an SLG fault occurs in the bus.
Meanwhile, Fig. 11 presents the transient current energy relative entropy of all feeders in the SFB under −5 dB SNR condition after a constant resistance SLG fault occurs in the feeder L 1 , L 3 , L 5 , and bus respectively. It indicates that the transient current energy relative entropy of faulty feeder significantly larger than that of any healthy feeder except when the fault occurs in the bus. However, the fault has already been detected in the bus by criterion 1 of the multi-criteria detection method. Therefore, the detection method can avoid misjudgment by comprehensive multi-criteria.
The detection results obtained during testing for all criteria are listed in Table 3. The detection results show that the criterion 2 has failed to detect the faulty feeder L 4 , and the criterion 3 is applied to determine the faulty feeder. The results demonstrate that the proposed comprehensive multi-criteria detection method accurately detected the faulty feeder in all cases and indicate that the proposed detection method is robust against signal noise, overcompensation degree, initial fault phase, fault resistance, and fault location.

D. ARC GROUNDING FAULT TEST
Arc grounding faults are usually analyzed using high frequency theory and fundamental frequency theory [35]. While fundamental frequency theory provides analyses more proximate to practical networks, suggesting that arc discharges are immediately extinguished when discharge currents cross zero, actual arc grounding faults may last several cycles before extinguishing. Therefore, the accuracy and   Table 2.
adaptability of the proposed comprehensive multi-criteria faulty feeder detection method were tested under the 22 kinds of permanent and intermittent arc grounding fault conditions listed in Table 4 with different faulty feeders, noise intensity, overcompensation degree, arc resumption intervals and fault distance. The permanent arc grounding fault is the arc grounding fault without arc resumption intervals and the intermittent arc grounding faults with 10 ms and 30 ms resumption intervals were simulated to resume and extinguish three times. Fig. 12(a) and (b) present the transient zero-sequence currents obtained for all feeders in the 10 kV RG network during PSCAD simulations after a permanent arc grounding fault  occurs in the feeder L 1 and an intermittent arc grounding fault with the arc resumption intervals of 10 ms occurs in the feeder L 2 . Fig. 13 presents the comparisons of original current waveforms, current waveforms with noise, and denoised current waveforms of all feeders obtained during the first 1/4 indus-  under −5 dB SNR condition, but the multi-criteria detection method can still identify the correct faulty feeder. However, the transient zero-sequence currents can be denoised perfectly to recover the original signals from noisy signals corrupted by additive noise under 0 dB and above SNR conditions whether permanent or intermittent arc grounding faults occur.
Meanwhile, Fig. 14 presents the transient current energy relative entropy of all feeders in the SFB after an arc grounding fault occurs in the feeder L 1 , L 2 , L 3 , and L 4 respectively. Fig. 14 shows that the criterion 2 has failed to detect the faulty feeder L 3 , and the criterion 3 is applied to determine the faulty feeder. It is noted that arc grounding faults exhibit more obvious transient signal characteristics than constant resistance faults. Moreover, the respective conditions considered by criteria 1 and 2 are obviously apparent in Figs. 13 and 14.
The detection results obtained during testing for all criteria are listed in Table 5. The results demonstrate that the proposed faulty feeder detection method provides accurate detection under a wide range of arc grounding fault conditions.

A. DENOISING EFFECT
Existing denoising methods include adaptive MM filtering, WT threshold filtering, and adaptive time-frequency filtering using the ST. Therefore, the performance of the proposed GST σ -based denoising method was compared with the method without denoising and those of these three existing methods based on the SNR of the denoised signal and the mean squared error (MSE) between the denoised signal and the known effective signal.
Here, N is the number of sampling points, x (n) is the denoised signal and s (n) is the origin signal without noise. It can be seen that the denoising performance increases as the post-filtering SNR value increases and as the MSE decreases.
As was conducted previously, white Gaussian noise was added to simulate noisy environments with SNR values ranging from −10 dB to 10 dB. The denoising performance of the simulated signals obtained for feeder L 1 are listed in Table 6 under different constant resistance faults and arc grounding faults occurring in feeder L 1 of the 10 kV RG system. The effective, noisy, and denoised current waveforms of L 1 are also presented in Fig. 15 under constant resistance faults and arc grounding faults with different SNR conditions. We note from Table 6 that the denoising performance of the proposed GST σ filtering is substantially greater than those of MM adaptive filtering and ST adaptive filtering below 0 dB SNR condition for both constant resistance faults and arc grounding faults. We also note that the differences between the denoising performances of WT filtering and GST σ filtering are not substantial above 0 dB SNR condition, particularly when the noise is very small relative to the effective signal at SNR values of 5 dB and 10 dB. However, a subjective appraisal of Fig. 15 indicates that the waveform denoised by GST σ filtering appears to be much closer to the waveform of the transient zero-sequence current of L 1 without noise than the waveforms denoised by other three methods. As such, we can TABLE 6. Denoising performance results of the simulated signals obtained for L 1 during L 1 faults in the 10 kV RG system.  [16], [31], [36] and this paper. conclude that the denoising performance of the proposed method is better than the other methods considered.

B. DETECTION ACCURACY EVALUATION AND CALCULATION TIME
The detection accuracy of the proposed comprehensive multi-criteria faulty feeder detection method was evaluated by comparisons between the detection accuracies obtained for the modeled 10 kV RG system by the proposed method and those of existing methods [16], [32], [36] that uniformly rely on a single criterion, which are criterion 2 based on the GST, criterion 3 based on the ST, and criterion 2 based on the WT, respectively. The results are presented in Table 7. From the table, it can be seen the single-criterion detection methods are prone to misjudgment when the noise is very large relative to the effective signal or when an arc grounding fault occurs.  These results indicate that the method proposed in this paper provides superior faulty feeder detection accuracy than the existing methods considered.
In addition, the calculation times required by the compared detection methods were evaluated for the detection process results discussed above for the SLG fault, and the times are listed in Table 8. Here, the methods of paper [26] and paper [31] are added to test the compared calculation times. Paper [26] applied the single-criterion method based on the difference of two-phase current with SVM, and the method of paper [31] also applied single criterion based on ST energy from short window data. We note from the table that the proposed comprehensive multi-criteria detection method requires less calculation time than the other methods, particularly that of paper [26]. Therefore, in practical applications, the proposed method has a faster judgment speed.

VII. FIELD TEST OF SLG FAULT
In order to verify the performance of the proposed method, we carried out a field test at 10 kV distribution power line of Jiangsu in China. The 10 kV RG distribution network topology is shown in Fig. 16  Meanwhile, the faulty feeder detection results by multi-criteria are listed in Table 9, which can be seen that the detection results are all correct. Therefore, it is verified that the proposed multi-criteria faulty feeder detection method can also produce accurate judgment results in the actual 10 kV RG distribution network.

VIII. CONCLUSION
This paper addressed the issues surrounding the detection of faulty feeders in RG power distribution networks by proposing a modified GST with a variable factor for self-adaptively adjusting the local window resolution of the time-frequency spectrum, and then applied the time-frequency representation of the GST to conduct denoising of transient zero-sequence currents based on threshold filtering followed by time-frequency distribution filtering in sequence. In addition, a comprehensive multi-criteria faulty feeder detection method was proposed. The results obtained by both a large number of simulation experiments and field test under different constant resistance faults and arc grounding faults have demonstrated the following conclusions.
(i) The GST provides a powerful time-frequency analysis proficiency compared with those of the ST and WT. The introduction of a variable factor into the GST has greatly improved its self-adaptability. As a result, the sequence of threshold filtering and time-frequency distribution filtering based on the time-frequency representation of the modified GST provides superior denoising performance compared to existing methods. Therefore, the detection method can identify faulty feeders accurately even in extremely noisy environments.
(ii) The proposed comprehensive multi-criteria detection method considers both the direction of transient zero-sequence signals (criterion 1) as well as the transient current energies of the feeders using two criteria based on the time-frequency representation of the modified GST (i.e., the current energy relative entropy [criterion 2] and the total current energy [criterion 3]), which provides faulty feeder detection that is robust to widely varying fault conditions, including constant resistance faults and arc grounding fault conditions. The proposed detection method was demonstrated to provide higher faulty feeder detection accuracy and reduce calculation time compared to existing methods.
(iii) The modeled 10 kV RG distribution network with an overhead and cable mixed line structure and arc discharge fault model established in this paper ensured that the simulations faithfully represented actual complex working conditions. The detection results of field test can verify the reality and effectiveness of models and simulations established in this paper. Therefore, the simulation testing conducted in this paper can be expected to provide realistic results. From 2015 to 2016, he was a Visiting Scholar with North Carolina State University, Raleigh, NC, USA. He is currently an Associate Professor with the College of Energy and Electrical Engineering, Hohai University. His research interests include power system analysis, economic dispatch, and optimal control of integrated energy systems.
HAIXIANG ZANG received the B.S. degree in electrical engineering from Nanjing Normal University, in 2009, and the Ph.D. degree in electrical engineering from Southeast University, in 2014.
He is currently an Associate Professor with the College of Energy and Electrical Engineering, Hohai University, Nanjing, China. His research interests include generation of renewable energy, operation, and control of power systems, and so on. VOLUME 8, 2020