Abstract
When high-impedance faults (HIFs) occur in resonant grounded distribution networks, the current that flows is extremely weak, and the noise interference caused by the distribution network operation and the sampling error of the measurement devices further masks the fault characteristics. Consequently, locating a fault section with high sensitivity is difficult. Unlike existing technologies, this study presents a novel fault feature identification framework that addresses this issue. The framework includes three key steps
①utilizing the variable mode decomposition (VMD) method to denoise the fault transient zero-sequence current (TZSC); ② employing a manifold learning algorithm based on t-distributed stochastic neighbor embedding (t-SNE) to further reduce the redundant information of the TZSC after denoising and to visualize fault information in high-dimensional 2D space; and ③ classifying the signal of each measurement point based on the fuzzy clustering method and combining the network topology structure to determine the fault section location. Numerical simulations and field testing confirm that the proposed method accurately detects the fault location, even under the influence of strong noise interference.
AMONG all fault types of distribution networks, the probability of a single phase-to-ground (SPG) accounts for more than 70% [
Currently, the fault detection technologies can be roughly categorized into two groups: methods based on steady-state signals and methods based on transient signals [
The fault detection methods based on steady-state signals are mainly applicable to isolated neutral distribution systems [
The fault detection methods based on transient signals are not affected by the neutral grounding mode and contain rich fault information. Combined with appropriate signal analysis approaches, these methods have been widely favored by researchers [
A common issue encountered by these transient methods is that the fault location may fail because of the gradually increasing noise introduced by the measurement devices as the fault resistance increases, which eventually overwhelms the transient fault characteristic signal. Reference [
To enhance the accuracy and reliability of transient methods, researchers have realized the significance of extracting and analyzing weak fault signals in the presence of background noise. Reference [
To address this problem, this study proposes a fault detection method for HIF based on variable mode decomposition (VMD) and t-distributed stochastic neighbor embedding (t-SNE) with the aim of compensating for the insufficient sampling accuracy of current sensors using signal analysis methods. The main contributions include:
1) For constructing fault criteria, based on an analysis of the fault mechanism, we find that the capacitor current is not immediately compensated by the inductor current after an HIF occurs within a very short period. Therefore, a fault criterion is proposed based on the TZSC amplitude feature.
2) For signal preprocessing, the VMD method is proposed to decompose the noisy TZSC signals after the faults. Combined with the fault mechanism analysis, the optimal decomposition parameter is determined to be 3 to ensure the effective extraction of weak fault features under strong noise interference.
3) For fault feature extraction, to further reduce redundant information and better preserve HIF features, a manifold learning algorithm based on t-SNE and a fuzzy clustering method are used to visualize the denoised TZSC in 2D space and to classify the measurement points, respectively. Under the strong noise interference background of -3 dB, the precise fault section location of 3000 fault resistance is achieved.

Fig. 1 Resonant grounding distribution network containing DG.
The distribution of ZSC in the entire network when HIF occurs on feeder j is shown in

Fig. 2 Distribution of ZSC in entire network with HIF.
Because the initial stored energy of the dynamic component is zero, the following differential equation can be obtained from
(1) |
where is the zero-sequence capacitance to ground of the entire distribution network.
For a practical HIF, R generally satisfies:
(2) |
In this case, the equivalent circuit corresponding to
(3) |
(4) |
(5) |
can be obtained by:
(6) |
Thus, can be obtained by:
(7) |
When , it holds that
(8) |
(9) |
Without consideration of the overcompensation effect of the arc-suppression coil during the transient period, it can be assumed that:
(10) |
Combining (8)-(10), we can obtain:
(11) |
This implies that after a sufficiently long period of oscillation decay, the oscillation direction of the ZSC at the faulty feeder terminal tends to be the same as that at the non-faulty feeder terminal, and the ratio of their amplitudes depends on the zero-sequence capacitance ratio of the respective feeders. At this point, neither the oscillation direction nor the amplitude of the ZSC at each feeder can be used for fault discrimination.
Considering the case of , we have:
(12) |
In fact, (12) can be not only obtained from (6) and (7), but also explained by the principle that the energy of the inductive element cannot undergo a sudden change. Because the electric current is a representation of the energy stored in the inductive components, we have:
(13) |
In some special cases such as when impulse excitation occurs in the circuit, (13) may not hold. However, for a practical distribution network, there is no need to consider such special cases. Therefore, we can assume that (13) always holds.
For the fault point shown in
(14) |
Clearly, the conclusions of (12) and (14) are identical, both of which prove that at the transient moment of the HIFs, i.e., at time , the ZSC flowing out of the faulty feeder is the sum of all non-faulty feeder-to-ground capacitive currents, but with opposite directions. As the transient process progresses, the inductive current flowing through the arc-suppression coil gradually increases, slowly compensating for the capacitive current flowing through the fault point, which is manifested as the ZSC at the faulty feeder exits as it slowly decays from its maximum amplitude. This suggests that this transient process can be used to construct fault criteria, and that the decay factor determines the duration. Based on the actual distribution network feeder parameters, this paper sets that this conclusion holds true during the period of [0, T/2], where s represents the power system period. The physical essence of this process is that the feeder-to-ground capacitance of the entire system releases the stored energy, and the faulty feeder forcibly absorbs this part of the energy.
The aforementioned theoretical analysis and conclusions indicate that it is possible to directly utilize the TZSC for criterion construction, without the need to exclude the fundamental current components in (6) and (7) nor to obtain voltage signals from various measurement points. This method has obvious advantages over the existing ones [
The VMD method demonstrates the excellent performance in noise reduction and processing of nonstationary signals [
The VMD method constructs a variational problem as expressed by:
(15) |
where represents the various mode components; represents the central frequency of each mode component; is the impulse function; is the partial derivative with respect to t; and f is the original signal.
Introducing the secondary penalty factor and Lagrange multiplier operator , (15) is transformed into an unconstrained variational problem, where the secondary penalty factor ensures the reconstruction accuracy of the signal in the presence of Gaussian noise. The extended Lagrange expression is:
(16) |
The expression for each mode component and its central frequency can be obtained by utilizing the alternating direction method of multipliers to solve the extended Lagrange expression.
(17) |
(18) |
When (17) and (18) are iteratively computed until the convergence is achieved, the Fourier inverse transform can be applied to , and its real part is used to obtain an analytical solution for .
(19) |
The setting for parameter K directly affects the effect of VMD, and determining the optimal K value has always been a challenge when using the VMD method [
(20) |
(21) |
where A1-A6 are the system-related constants of the distribution network.
According to (20) and (21), the central frequency of the time-domain waveform of ZSC for each feeder is significantly correlated with and . Combined with (4), we can infer that the attenuation factor during HIF is generally small, resulting in a longer transient attenuation period and a smoother attenuation process. Therefore, we can assume that the transient transition process of the HIF corresponds to two central frequencies and . In addition, when the high-frequency noise introduced by the current sensors during the sampling process is considered, an additional higher central frequency can be introduced, and can be set to decompose the noisy ZSC. Note that for most HIFs in practical engineering, is approximately satisfied, which means that the main energy of each feeder is concentrated near the fundamental frequency. Therefore, in practical applications, the modal component corresponding to can be utilized to reconstruct the denoised fault features, that is, the intrinsic mode function (IMF) associated with .

Fig. 3 Typical ZSC of HIF. (a) Original signal. (b) Polluted signal with noise interference.

Fig. 4 Polluted signal decomposed by VMD.
According to the fault mechanism analysis in Section II, the transient duration of HIF is related to the attenuation factor , which can vary within a certain range due to the influence of fault conditions and system parameters. Because this paper focuses on the TZSC within T/2 after the fault, there will be corresponding levels of redundant data under specific fault conditions. Therefore, it is crucial to eliminate the redundant features within these data and simultaneously reduce the dimensionality of the high-dimensional fault dataset in order to extract valuable fault state information that reflects the HIF. This reduces the difficulty of fault classification and improves the accuracy of fault identification.
t-SNE is a nonlinear manifold deep learning algorithm that can effectively achieve the visualization and dimensionality reduction of high-dimensional data [
For a high-dimensional dataset , the probability distribution is defined to represent the conditional probability that xi will choose xj as its neighbor:
(22) |
where is the Gaussian variance centered at xi, which is obtained based on the specified perplexity.
The perplexity is related to Shannon’s information entropy, and its expression is given by:
(23) |
As holds here, a joint probability distribution satisfying the symmetry property is defined as:
(24) |
To map the high-dimensional dataset to a low-dimensional space, let be the initial coordinate set in the low-dimensional space, where S should satisfy . Here, the t-distribution is used to model the medium distances in the high-dimensional space as long distances in the low-dimensional space. As a result, the joint probability distribution of modeled in the low-dimensional space follows the Cauchy distribution and is given by:
(25) |
To minimize the difference in probability distribution between the datasets and , a loss function is constructed using the Kullback-Leibler divergence as:
(26) |
To obtain the target dataset , is minimized using gradient descent starting from a random initialization.
(27) |
The corresponding iterative update process is:
(28) |
where represents the learning rate; r is the iteration step; and is the iteration coefficient.
Thus, we can obtain N S-dimensional vectors that correspond to N actual measurement points. To determine the fault section location, these N measurement points must be further divided into two categories: fault paths and non-fault paths. In this study, the FCM algorithm is used as a reference to construct the objective function:
(29) |
where gij is the element of membership matrix G; and h is the weighted index. The optimal solution to (29) can be obtained using the Lagrange multiplier method.
(30) |
(31) |
When the difference between the new and old membership matrices is less than the set threshold value, the final classification result {c1,c2} can be obtained, and the actual fault section can be determined by combining it with the structure of the distribution network.
To provide a clear illustration of the entire detection process, a flow chart of the HIF section location algorithm is presented in

Fig. 5 Flow chart of HIF section location algorithm.
As shown in

Fig. 6 Radial distribution network model established in PSCAD/EMTDC.
A 1 MW constant impedance load is used for all feeders. Thirteen groups of ZSC transformers are installed to obtain the ZSC at each measurement point (A-M) with a sampling frequency of 10 kHz, which is equal to 200 samples per cycle.
Based on the assumption that an SPG fault occurs in the section C-D at with the corresponding initial fault angle of and fault resistance of , the system is in an under-damped state.

Fig. 7 Time-domain waveform of faulty ZSCs with Rf = 3000 Ω (original signals without considering noise interference).
1) When , the value of ZSC at point A is significantly greater than those at points D and L, which is consistent with the mechanism analytical results derived from (12)-(14). At this time, the ZSCs at each measurement point are all capacitive.
2) As the fault progresses, the inductive current flowing through the arc-suppression coil begins to increase slowly. However, due to the relatively high value of , the growth rate is relatively slow. This leads to the conclusion that within after the fault occurs, the ZSC at point A is assumed to be significantly greater than those at points D and L.
3) When , the system is still in a transient process, during which the ZSC at point A does not have a unchanged relationship with those at points D and L in terms of the wave amplitude and phase angle. This transient process also compensates for the arc-suppression coil inductive current of the system capacitive current.
4) When , i.e., , the system enters a steady state. During this period, the phases of the ZSCs at points A, D, and L are the same, and the relationship between the amplitudes is unchanged. This is consistent with the analysis and conclusions of the mechanisms derived from (8)-(11). During this period, the measured ZSCs cannot be used to construct an effective location criterion.
In practical distribution networks, ZSCs are not as clear as those shown in

Fig. 8 Polluted signals with Gaussian white noise at an SNR of dB.

Fig. 9 Comparison among original signals, polluted signals, and reconstructed results. (a) Point A. (b) Point D. (c) Point L.
After the VMD denoising process is completed for all 13 ZSCs shown in

Fig. 10 t-SNE visualization results. (a) Under VMD denoising step. (b) Without VMD denoising step.
In contrast,
Considering the actual network topology, the fault section location can be determined as the section C-D, as shown in

Fig. 11 Fault section location results of HIF.
Fault section | Fault condition | SNR (dB) | Result | |
---|---|---|---|---|
(°) | Rf () | |||
B-C | 0 | 500 | 30 | B-C (√) |
B-C | 0 | 1000 | 20 | B-C (√) |
B-C | 45 | 1500 | 10 | B-C (√) |
L-M | 45 | 2000 | 5 | L-M (√) |
I-J | 90 | 2500 | 0 | I-J (√) |
C-D | 90 | 3000 | -3 | C-D (√) |
E-F | 0 | 3000 | -10 | Error (×) |
C-D | 0 | 2000 | -10 | Error (×) |
C-D | 90 | 4000 | -6 | Error (×) |
Note: the symbols “√” and “×” represent that the results are correct and incorrect, respectively.
From
1) The proposed method can achieve accurate fault section location for all fault conditions within a -3 dB SNR, with a maximum transition resistance value of 3000 .
2) When the SNR is too high such as -6 dB or -10 dB, it results in erroneous fault section location, indicating that the noise interference level is a major factor affecting the accuracy of the location results.
3) It should be noted that the actual sampling accuracy of the current sensors is limited, within a measurement error of approximately ±1 A [
During the verification phase, it is necessary to consider the influence of impulsive or non-normally distributed noise on the reliability of the algorithm. Based on the case shown in

Fig. 12 Analysis of VMD results. (a) Under five random impulsive noise pulses and Gaussian distributed noise interference. (b) Under only Gaussian noise interference.
In addition, in practical applications, the sampling frequency has a significant impact on the data volume for faulty signal transmission. Therefore, we must also consider this possibility.
Sampling frequency (kHz) | Probability distribution of noise | ||
---|---|---|---|
Exponential distribution | Uniform distribution | Gaussian distribution | |
0.5 | × | × | × |
3.0 | √ | √ | √ |
5.0 | √ | √ | √ |
8.0 | √ | √ | √ |
10.0 | √ | √ | √ |
20.0 | √ | √ | √ |
100.0 | √ | √ | √ |
It can be observed that the proposed algorithm has a lower limit requirement for the sampling frequency, whereas higher sampling rates do not affect the reliability of the algorithm. In terms of reducing faulty data transmission, the lower the sampling frequency, the higher the engineering practicality of the algorithm. According to test results, the actual sampling frequency should not be lower than 3 kHz. Regarding the different probability distributions of noise, although they may have a certain effect on the VMD results, accurate results can still be obtained after a series of processes using the proposed t-SNE is conducted. This indicates that when the SNR is constant, different probability distributions of noise do not significantly affect the computational results of the algorithm.
Considering the aforementioned analysis, the proposed method exhibits satisfactory performance in terms of its applicability in the sampling frequency range and its immunity to noise under different probability distributions.
A comprehensive comparison is presented in
Reference | Technical scheme | Noise (dB) |
---|---|---|
This paper | VMD denoising followed by t-SNE manifold learning | 30, 20, 10, 5, 0, and |
[ | Adaptive FDM and density-distance based FCM | 32.76, 1.71, , and |
[ | Piecewise linear fitting based on least square | 10, 8, and 5 |
[ | Compressed sensing algorithm | 60, 40, 35, and 30 |
[ | Inner product transformation of extreme values | 5 |
[ | Optimized bistable system |
To evaluate the adaptability of the proposed method in practical applications, a manual SGP test is conducted on a 10 kV urban distribution network in Wuxi, China. As shown in

Fig. 13 Structure of distribution network for field tests.

Fig. 14 Original recorded signal and its corresponding IMF1 obtained by VMD. (a) Point F1. (b) Point F2.
Clearly, the sampling accuracy of F1 is significantly higher than that of F2. Moreover, because both F1 and F2 belong to the same upstream fault point, their ZSC waveforms should be similar. Therefore, F1 could be regarded as the original fault signal and F2 could be considered as a noisy signal polluted by measurement errors. After the proposed method is applied to both signals and their respective IMF1 values are obtained, the decomposition results are found to be very similar to the original fault signal of F1, thus demonstrating the effectiveness of the proposed method.
Method/institution | Judgment result | Accuracy rate (%) | |||
---|---|---|---|---|---|
200 | 500 | 1000 | 2000 | ||
Proposed | √ | √ | √ | √ | 100 |
Institution 1 | √ | √ | √ | × | 75 |
Institution 2 | √ | √ | √ | × | 75 |
Institution 3 | √ | √ | × | × | 50 |
This study proposes a novel method for identifying noisy HIF features. An analysis of the fault mechanism reveals that a significant difference exists in the TZSC amplitude between faulty and non-faulty feeders shortly after fault occurrence.
To improve the signal processing, we utilize a VMD method to preprocess the sampled signal and a t-SNE method to remove redundant signals during . The proposed method effectively localizes HIF sections under the operating conditions of and dB, and we are able to verify its reliability through successful field tests. This study provides a reference for the future exploration of the HIF mechanism and highlights the importance of studying the effects of noise interference on algorithm reliability.
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