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Feeder selection method for full cable networks earth faults based on improved K‐means

Feeder selection method for full cable networks earth faults based on improved K‐means INTRODUCTIONAfter an earth fault occurs in full cable networks, the grounding current is the capacitive current of the system. A small current loop is formed through the grounding capacitor, which will not excessively damage the equipment and system, and will not have an impact on the continuous power supply within a short period. However, after the fault occurs, due to the continuation of time, the fault may further expand, causing a two‐phase short circuit, arc discharge, and system overvoltage [1, 2]. Because of this, prompt and correct fault feeder selection in the event of earth faults in full cable networks is critical to ensure that full cable networks continue to operate as intended.Currently, earth fault feeder selection methods in full cable networks can be divided into three ways, mainly signal injection method [3, 4], fault signal features comparison method [5, 6], and artificial intelligence fault feeder selection method [7–9]. By injecting signals of a specified frequency into the system and then recognizing problematic feeders based on feeder characteristic signals, the signal injection technique detects faulty feeders. In [10], a method for selecting faulty feeders with small current grounding is presented, which combines injected signals with wavelet energy. It selects feeders by comparing the injected signal energy per feeder; however, high transition resistance results in feeder selection failure. The fault signal feature comparison method uses a mathematical algorithm to extract the steady or transient components of the fault current and then selects the feeder according to the difference in current characteristics between the normal feeder and the faulty feeder. Since the transient feature information of fault is obvious when earth faults occur in full cable networks, the effect of the feeder selection algorithm using the transient component is more significant. The main feeder selection methods include S transform [11–13], Prony algorithm [14], EMD algorithm [15, 16], wavelet transform [17], and active power method [18]. In [19], a method of earth faults feeder selection in full cable networks based on the fundamental amplitude of the transient current is proposed. In [20], a feeder selection algorithm for low‐current grounding systems is proposed through zero‐sequence current harmonics. In [21], a method of using the average active component of the transient current to realize earth faults line selection of full cable networks is proposed. In [22], fault feeder selection is realized by wavelet transient energy value. The above research uses transient information such as base wave amplitude, fifth harmonic amplitude, the energy value of wavelet packet, and average active power component of feeder zero‐sequence to improve the feeder selection ability, but all of them use a transient component, which has a single eigenvalue and poor anti‐jamming ability. The majority of them are incompatible with high resistance grounding faults.Due to the advancements in artificial intelligence and big data technologies in recent years, fault feeder selection by artificial intelligence has become a popular study area. Artificial intelligence fault feeder selection mainly converts the problem of fault feeder selection into a pattern recognition problem to solve. In [23], an earth faults feeder selection method for full cable networks based on deep confidence networks is presented. The algorithm is not influenced by the change of system‐neutral grounding mode and has a strong anti‐noise ability. In [24], a single‐phase grounding fault diagnosis method for distribution networks based on the K‐means is presented. The method is unstable due to the randomness of the initial cluster centres, which affects the accuracy of feeder selection. At present, there are many studies on fault line selection methods based on artificial intelligence, but the accuracy of the algorithm depends on a large amount of data samples. In actual situations, in order to improve the speed of the algorithm, it is necessary to screen a large amount of fault data, which will lose part of the fault information and make it difficult to obtain complete fault samples. When the sample data is less, it will be affected by fault conditions. For the fault identification method based on artificial intelligence, the more eigenvalues selected, the more accurate the identification result, but it will increase the time of fault identification, and misjudgment will occur when there are fewer eigenvalues. Literature [19–22] are fault diagnosis methods of single eigenvalue, which are vulnerable to the influence of fault conditions and have low feeder selection accuracy. Therefore, three to five fault features are usually selected, and four fault features are selected in this paper.In summary, this paper decomposes the transient zero‐sequence current of each feeder under different fault conditions through Fourier transform, active power method and wavelet packet transform, and fuses the four characteristics of fundamental wave amplitude, fifth harmonic amplitude, average active power component, and wavelet energy value of each feeder through principal component analysis, extracts the principal component and establishes the feature database. Then, a method of earth fault feeder selection using the improved K‐means is proposed to realize fault feeder selection. The results demonstrate that the technique is not affected by fault conditions, noise, CT saturation, different cable models, and sampling frequency and can effectively solve the problem of low accuracy of single‐phase grounding fault feeder selection in full cable networks.FEATURE ANALYSIS OF EARTH FAULTSThe change of working conditions of earth faults weakens the fault features and brings difficulties to the feeder selection in full cable networks. The method of fault feeder selection based on the steady‐state cannot satisfy the changing fault conditions. The transient process can be used to analyse earth faults to obtain more obvious fault features. Usually, when earth faults occur, the transient value of the current is much larger than that of the steady‐state value, which can be several times or tens of times that of the steady‐state value. Figure 1 shows diagram of typical earth faults in 10 kV full cable networks. Among them, the full cable networks include n feeders, L1–Ln in sequence, and C01–C0n are the zero‐sequence capacitance of each feeder. ĖA, ĖB, and ĖC represent a three‐phase symmetrical power supply. When switches K1, K2, and K3 are closed, respectively, the neutral point is grounded through the arc suppression coil system, grounded through a small resistance system and ungrounded system, and Lp and Rg are inductance and resistance under the mode of grounded through arc suppression coil and grounded through small resistance.1FIGURETypical diagram of earth fault of 10 kV full cable networksFor a full cable network with n feeders, when an earth fault occurs in feeder Ln, the system equivalent zero‐sequence network is depicted in Figure 2.2FIGUREEquivalent zero‐sequence network of an earth faultThe moment of fault is equivalent to connecting the zero‐sequence voltage source uf to the fault point, uf = Umsin(ω0t+θ), ω0 and θ are the power frequency angular velocity and fault initial phase angle, respectively, Um is the fault phase voltage amplitude, u0f is the neutral point voltage, i0Rg is the branch current of small resistance, i0Lp is the branch current of arc suppression coil, i0f is the current at the fault point, Ri, Li, C0i, i0i, iC0i (where i = 1, 2, …, n) are the equivalent resistance, equivalent inductance, distributed capacitance to ground, current, and capacitive current of per feeder, respectively. In the case of earth faults, the resistance and inductance of the feeder itself are far less than the feeder to ground capacitive reactance; as a result, only the feeder to ground capacitive reactance is considered in the calculation of feeder zero‐sequence impedance.When the system is grounded through the arc suppression coil, K1 is closed. The transient zero‐sequence current of the normal feeder could be represented as:1i0i=iC0i=C0idu0fdt\begin{equation}{i}_{0i}{\rm{ = }}{i}_{C0i}{\rm{ = }}{C}_{0i}\frac{{{\rm{d}}{u}_{0f}}}{{{\rm{d}}t}}\end{equation}By Kirchhoff's current law, i0f and i0n can respectively be rewritten as:2i0f=i0Lp+∑i=1niC0i=i0Lp+du0fdt∑i=1nC0i\begin{equation} {i}_{0f}{\rm{ = }}{i}_{0Lp}{\rm{ + }}\displaystyle\sum\limits_{i = 1}^n {{i}_{C0i}} = {i}_{0Lp}{\rm{ + }}\frac{{{\rm{d}}{u}_{0f}}}{{{\rm{d}}t}}\displaystyle\sum\limits_{i = 1}^n {{C}_{0i}} \end{equation}3i0n=iC0n−i0f=−(i0Lp+du0fdt∑i=1n−1C0i)\begin{equation}{i}_{0n}{\rm{ = }}{i}_{C0n}{\rm{ - }}{i}_{0f}{\rm{ = - (}}{i}_{0Lp}{\rm{ + }}\frac{{{\rm{d}}{u}_{0f}}}{{{\rm{d}}t}}\displaystyle\sum\limits_{i = 1}^{n - 1} {{C}_{0i}} )\end{equation}For further studying the features of earth faults in full cable networks, Figure 2 is simplified to Figure 3.3FIGUREFault transient equivalent circuitR is the sum of the equivalent feeder resistance from the bus to the fault point and three times the transition resistance Rf, L is the equivalent feeder inductance from the bus to the fault point, C0Σ is the sum of the ground capacitance of the whole system. Since L < < Lp, ignoring the influence of L, the voltage differential equation is:4uf=R(C0∑du0fdt+i0Lp)+u0fu0f=Lpdi0Lpdt\begin{equation}\left\{ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{u}_f = R({C}_{0\displaystyle\sum }\frac{{{\rm{d}}{u}_{0f}}}{{{\rm{d}}t}} + {i}_{0Lp}) + {u}_{0f}}\\ {{u}_{0f} = {L}_p\frac{{{\rm{d}}{i}_{0Lp}}}{{{\rm{d}}t}}} \end{array} } \right.\end{equation}Substitute (4) into (1) to obtain:5i0i=iC0i=C0idu0fdt=C0iLpd2i0Lpdt2\begin{equation}{i}_{0i}{\rm{ = }}{i}_{C0i}{\rm{ = }}{C}_{0i}\frac{{{\rm{d}}{u}_{0f}}}{{{\rm{d}}t}} = {C}_{0i}{L}_p\frac{{{{\rm{d}}}^2{i}_{0Lp}}}{{{\rm{d}}{t}^2}}\end{equation}It can be seen that i0i is linearly related to the second derivative d2i0Lp/dt2 of the arc suppression coil transient current, and the amplitude is only influenced by the capacitance parameters of the feeder itself.Substitute (4) into (2) and (3) to obtain:6i0f=i0Lp+∑i=1niC0i=i0Lp+Lpd2i0Lpdt2∑i=1nC0i\begin{equation}{i}_{0f}{\rm{ = }}{i}_{0Lp}{\rm{ + }}\displaystyle\sum\limits_{i = 1}^n {{i}_{C0i}} = {i}_{0Lp}{\rm{ + }}{L}_p\frac{{{{\rm{d}}}^2{i}_{0Lp}}}{{{\rm{d}}{t}^2}}\displaystyle\sum\limits_{i = 1}^n {{C}_{0i}} \end{equation}7i0n=iC0n−i0f=−(i0Lp+Lpd2i0Lpdt2∑i=1n−1C0i)\begin{equation}{i}_{0n}{\rm{ = }}{i}_{C0n}{\rm{ - }}{i}_{0f}{\rm{ = - (}}{i}_{0Lp}{\rm{ + }}{L}_p\frac{{{{\rm{d}}}^2{i}_{0Lp}}}{{{\rm{d}}{t}^2}}\displaystyle\sum\limits_{i = 1}^{n - 1} {{C}_{0i}} )\end{equation}Specifically, the transient zero‐sequence current of faulted feeder is concerned with i0Lp and the zero‐sequence current of the normal feeder of the whole system, and the polarity is opposite to that of the sound feeder. The compensation of the arc suppression coil to the transient capacitive current is very weak in the early stage of an earth fault, so the grounding current is primarily a transient capacitive current. The results of the aforementioned investigation indicate that the transient zero‐sequence current of the faulty feeder is greater than the normal feeder and the polarity of the current is opposite when earth faults occur in full cable networks.ESTABLISH FEATURE DATABASEThe occurrence of earth faults in full cable networks has led to the conclusion [19–22] that the fundamental amplitude of zero‐sequence current of faulted feeder is equal to the total of all sound feeders, the higher harmonic is not influenced by arc suppression coil compensation, the fifth harmonic amplitude of zero‐sequence current of faulted feeder is equal to the total of all sound feeders, the zero‐sequence average active power component of faulted feeder is greater than per sound feeder, the zero‐sequence current wavelet energy value of the faulted feeder is greater than per sound feeder. Therefore, transient zero‐sequence current per feeder is decomposed by Fourier transform, active power method, and wavelet packet transform, and four feature indexes of fundamental wave amplitude, fifth harmonic amplitude, average active power component, and wavelet energy value can be extracted as feeder selection fault features, as depicted in Table 1. Suppose a full cable network contains n feeders, L1–Lnn in sequence. The steps of establishing the earth faults feature database of full cable networks are divided into four steps: data collection, fault features extraction, fault features fusion, and establishing a feature database.1TABLEFeatures of earth faults feeder selection in full cable networksFeatureSymbolThe fundamental amplitude of zero‐sequence currentI1The fifth harmonic amplitude of zero‐sequence currentI5The zero‐sequence average active power componentPWavelet energy value of zero‐sequence currentEData collectionThe earth faults simulation model of full cable networks containing n feeders is built, and the simulation is carried out for m different fault conditions of full cable networks (earth faults occur under different feeders, different initial phase angles, different neutral grounding modes, different points, and different transition resistors). After an earth fault occurs in full cable networks, the transient zero‐sequence current lasts about one power frequency cycle (0.02 s) [25]. Following that, the transient zero‐sequence voltage and zero‐sequence current at the beginning of per feeder within a power frequency cycle under different fault conditions are collected. The sampling frequency is 10 kHz, and 200 points are sampled in one cycle.Fault features extractionAfter an earth fault in full cable networks, transient zero‐sequence current and zero‐sequence voltage within the first power frequency cycle at the beginning of n feeders are extracted in each simulation experiment.I1 and I5 are obtained after the Fourier transform of transient zero‐sequence current per feeder. I1 and I5 of feeder i (i = 1, 2, 3, …, n) are I1i and I5i, respectively. Then, the eigenvectors formed by I1 and I5 of n feeders are respectively:8I1=[I11,I12,…,I1i,…,I1n]T\begin{equation}{{\bm{I}}}_1 = {[{I}_{11},{I}_{12}, \ldots ,{I}_{1i}, \ldots ,{I}_{1n}]}^T\end{equation}9I5=[I51,I52,…,I5i,…,I5n]T\begin{equation}{{\bm{I}}}_5 = {[{I}_{51},{I}_{52}, \ldots ,{I}_{5i}, \ldots ,{I}_{5n}]}^T\end{equation}The zero‐sequence average active power component P of per feeder is calculated by the active power method, and the Pi of feeder i is:10Pi=1200∑j=1200Ui(j)Ii(j)\begin{equation}{P}_i{\rm{ = }}\frac{1}{{200}}\displaystyle\sum_{j = 1}^{200} {{U}_i(j){I}_i(j)} \end{equation}where Ii(j) and Ui(j)are zero‐sequence current amplitude and zero‐sequence voltage amplitude of feeder i at j (j = 1, 2, 3, …, 200) sampling points, respectively. Then, the eigenvector formed by the zero‐sequence average active power component P of n feeders is:11P=[P1,P2,…,Pi,…,Pn]T\begin{equation}{\bm{P}} = {[{P}_1,{P}_2, \ldots ,{P}_i, \ldots ,{P}_n]}^T\end{equation}Wavelet packet transform decomposes the transient zero‐sequence current in different frequency bands, and the sampling frequency is 10 kHz. The Db6 wavelet packet decomposes the transient zero‐sequence current per feeder into five layers [26]. After decomposition, the energy corresponding to the wavelet packet coefficients S (5,0), S (5,1), S (5,2), …, S (5, j), …, S (5,31) of the fifth layer is:12E5,j=S(5,j)2\begin{equation}{E}_{5,j} = {\left\| {S(5,j)} \right\|}^2\end{equation}where j = 0, 1, …, 31. Take the maximum band energy of layer 5 except the (5,0) band where the power frequency is located as the E of the feeder, and Ei of feeder i is:13Ei=max(E5,1,E5,2,…,E5,j,…,E5,31)\begin{equation}{E}_i{\rm{ = max(}}{E}_{{\rm{5,1}}},{E}_{{\rm{5,2}}}, \ldots ,{E}_{5,j}, \ldots ,{E}_{{\rm{5,31}}}{\rm{)}}\end{equation}Then, the eigenvector formed by E of n feeders is:14E=[E1,E2,…,Ei,…,En]T\begin{equation}{\bm{E}} = {[{E}_1,{E}_2, \ldots ,{E}_i, \ldots ,{E}_n]}^T\end{equation}Take I1, I5, P, and E of n feeders as the feature input x of a fault condition, with:15x=I11,I12,…,I1i,…,I1n,I51,I52,…,I5i,…,I5n,P1,P2,…,Pi,…,Pn,E1,E2,…,Ei,…,EnT\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\bm{x}} = \left[{I}_{11},{I}_{12}, \ldots ,{I}_{1i}, \ldots ,{I}_{1n},{I}_{51},{I}_{52}, \ldots ,{I}_{5i}, \ldots ,{I}_{5n},\right.\\ \left.{P}_1,{P}_2, \ldots ,{P}_i, \ldots ,{P}_n,{E}_1,{E}_2, \ldots ,{E}_i, \ldots ,{E}_n \right]^{\rm{T}} \end{array} \end{equation}where I11–I1n are I1 of n feeders, respectively, I51–I5n are I5 of n feeders, respectively, P1–Pn are P of n feeders respectively and E1–En are E of n feeders, respectively.Carry out m simulation experiments for m different fault conditions, and m feature inputs are obtained to form the fault feature matrix X which can be represented as:16X=[x1,x2,x3,…,xp,…,xm]T\begin{equation}{\bm{X}} = {[{{\bm{x}}}_1,{{\bm{x}}}_2,{{\bm{x}}}_3, \ldots ,{{\bm{x}}}_p, \ldots ,{{\bm{x}}}_m]}^{\rm{T}}\end{equation}where xp is the feature input obtained from the pth simulation experiment, x1– xm are the feature inputs obtained from m simulation experiments.Fault features fusionPrincipal component analysis transforms a given set of related variables into another set of unrelated variables through linear transformation [27]. These new variables are arranged in the order of decreasing variance. They can be used to extract the main feature components of data. They are often used to reduce the dimension of high‐dimensional data and realize feature fusion. Compared with other feature selection methods, principal component analysis can retain the information contained in the original variables. Only the first few principal component components can represent the fault features of samples, which can greatly improve the accuracy and operation speed of the algorithm. In this paper, the feature matrix Xm×4n is extracted by principal component analysis (there are m fault conditions, and each fault condition contains 4n fault features) to realize fault feature fusion.To eliminate the dimensional influence, I1, I5, P, and E of per feeder in x are normalized to the effective data between [0,1] by discrete standardization. Taking E as an example, the normalization method is:17Ei′=Ei−EminEmax−Emin\begin{equation}E_i^{\prime} = \frac{{{E}_i - {E}_{\min }}}{{{E}_{\max } - {E}_{\min }}}\end{equation}where Ei and E'i are E before and after normalization of feeder i, respectively, Emax and Emin are the maximum and minimum values of E in n feeders, respectively. Similarly, the normalized I'1i, I'5i, and P'i of the feeder i are obtained. Record the normalized feature input as x' which can be represented as:18x′=I11′,I12′,…,I1i′,…,I1n′,I51′,I52′,…,I5i′,…,I5n′,P1′,P2′,…,Pi′,…,Pn′,E1′,E2′,…,Ei′,…,En′T\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\bm{x}}^{\prime} = \left[I_{11}^{\prime},I_{12}^{\prime}, \ldots ,I_{1i}^{\prime}, \ldots ,I_{1n}^{\prime},I_{51}^{\prime},I_{52}^{\prime}, \ldots ,I_{5i}^{\prime}, \ldots ,I_{5n}^{\prime},\right.\\[2pt] \left.{P_1}^{\prime},P_2^{\prime}, \ldots ,P_i^{\prime}, \ldots ,P_n^{\prime},E_1^{\prime},E_2^{\prime}, \ldots ,E_i^{\prime}, \ldots ,E_n^{\prime} \right]^{\rm{T}} \end{array} \end{equation}where I'11–I'1n are the normalized values of I1 of n feeders, respectively, I'51–I'5n are I5 normalized values of n feeders, respectively, P'1–P'n are P normalized values of n feeders, respectively, and E'1–E'n are the normalized values of E of n feeders, respectively.Thus, the normalized feature matrix X'm×4n can be represented as:19X′=[x1′,x2′,x3′,…,xp′,…,xm′]T\begin{equation}{{\bm{X}}}^{\prime} = {[{\bm{x}}_1^{\prime},{\bm{x}}_2^{\prime},{\bm{x}}_3^{\prime}, \ldots ,{\bm{x}}_p^{\prime}, \ldots ,{\bm{x}}_m^{\prime}]}^{\rm{T}}\end{equation}where x'p is the normalized feature input of the pth simulation experiment, and x'1–x'm are the normalized feature input obtained from the m simulation experiment, respectively.Find the covariance matrix R4n×4n of X'm×4n and compute the eigenvalues λ1, λ2, …, λ4n of R (where λ1≥λ2≥…≥λ4n≥0) and the corresponding eigenvectors a1, a2, …, a4n. The matrix A formed by the eigenvectors can be represented as:20A=[a1,a2,…,a4n]\begin{equation}{\bm{A}} = [{{\bm{a}}}_1,{{\bm{a}}}_2, \ldots ,{{\bm{a}}}_{4n}]\end{equation}Let Y = X'A and get the matrix Y as:21Y=[y1,y2,…,yi,…,y4n]\begin{equation}{\bm{Y}} = [{{\bm{y}}}_1,{{\bm{y}}}_2, \ldots ,{{\bm{y}}}_i, \ldots ,{{\bm{y}}}_{4n}]\end{equation}where yi (i = 1, 2, …, 4n) is the vector composed of the ith principal component of X. Variance contribution rate of the ith principal component αi and cumulative variance contribution rate βi can be described as:22αi=λiλi∑j=14nλj∑j=14nλjβi=∑k=1iλk∑k=1iλk∑j=14nλj∑j=14nλj\begin{equation}\left\{ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{\alpha }_i = {{{\lambda }_i} \mathord{\left/ {\vphantom {{{\lambda }_i} {\displaystyle\sum_{j = 1}^{4n} {{\lambda }_j} }}} \right. \kern-\nulldelimiterspace} {\displaystyle\sum_{j = 1}^{4n} {{\lambda }_j} }}}\\ {{\beta }_i = {{\displaystyle\sum_{k = 1}^i {{\lambda }_k} } \mathord{\left/ {\vphantom {{\displaystyle\sum_{k = 1}^i {{\lambda }_k} } {\displaystyle\sum_{j = 1}^{4n} {\lambda {}_j{\rm{ }}} }}} \right. \kern-\nulldelimiterspace} {\displaystyle\sum_{j = 1}^{4n} {\lambda {}_j{\rm{ }}} }}} \end{array} } \right.\end{equation}Generally, the principal component components whose cumulative variance contribution rate reaches 85% are taken [27]. Assuming that the contribution rate of the cumulative variance of the preceding k (k ≤ 4n) principal components have already met the requirements, the feature matrix Z after feature fusion can be expressed as:23Z=[y1,y2,…,yi,…,yk]\begin{equation}{\bm{Z}} = [{{\bm{y}}}_1,{{\bm{y}}}_2, \ldots ,{{\bm{y}}}_i, \ldots ,{{\bm{y}}}_k]\end{equation}where yi (i = 1, 2, …, k) is the vector composed of the ith principal component of each fault condition (m fault conditions in total).Establish feature databaseThe feature database is formed by using the matrix Z after feature fusion. The k principal component components of each fault condition in the feature database form a principal component vector. Eight per cent and 20% of the database data are used as training sets and test sets respectively.IMPROVED K‐MEANS FEEDER SELECTION PRINCIPLEThe term ‘K‐means clustering’ refers to the process of randomly selecting the number of classes and initial cluster centre, to minimize the sum of squares of the distance between the selected class centre and per feature vector [28].K‐means feeder selection principleThe step of the K‐means feeder selection algorithm is to randomly select n principal component vectors from the feature database as the initial clustering centre, and then the principal component vector samples which need to be classified in the training set are assigned to one of n cluster centres through the criterion of minimum distance. The Euclidean distance is adapted to calculate the distance from each sample to the clustering centre. The calculation formula is described as:24d(x,y)=∑i=1k(yi−xi)2\begin{equation}{\rm{d}}({\bm{x}},{\bm{y}}) = \sqrt {\displaystyle\sum_{i = 1}^k {{{({y}_i - {x}_i)}}^2} } \end{equation}where x and y are the principal component vectors of any two fault conditions, respectively, xi and yi are, respectively, the ith principal component of x and y and k is the number of principal component components meeting the requirements of the cumulative contribution rate. After all cluster samples are allocated, the average value of all samples in each class is taken as the new cluster centre, and cycled until the cluster centre converges. Each class represents a feeder with an earth fault. After the best clustering centre is obtained from the training set, the accuracy of the method is tested by the sample data in the test set.Improved K‐means feeder selection principleWhen the first clustering centre is chosen at random, the K‐means algorithm becomes unstable, the outcome of feeder selection varies with the initial cluster centre, and the accuracy of the method will be affected when the iterative termination conditions are unreasonable. As a result, by refining the selection of initial clustering centres and terminating termination criteria, this study provides an enhanced K‐means method to overcome the issue of poor accuracy of feeder selection in the K‐means algorithm. The improvement parts are as follows:(1) Select n principal component vectors in the feature database as the initial cluster centres. The n principal component vectors are obtained when n feeders have earth faults, respectively.(2) The iterative process is optimized by introducing a threshold and setting the maximum number of iterations. The minimum value of the distance between each initial cluster centre is set as the threshold. After the cluster centre converges, the distance between per cluster centre and other cluster centres is calculated cyclically, and its relationship with the threshold is judged. If it is greater than or equal to the threshold, it exits the cycle. If it is less than the threshold, the two clusters will be merged, and the two principal component vectors with the largest dispersion after merging will be used as the new cluster centre for reiteration until the conditions are met. The threshold will be adjusted for recalculation if the iteration count exceeds the maximum iteration count.The flow chart of feeder selection based on the improved K‐means algorithm is shown in Figure 4.4FIGUREFeeder selection flow chart based on the improved K‐means algorithmSIMULATION EXAMPLESimulation modelBased on MATLAB/Simulink, this paper establishes a simulation model of 10 kV full cable networks with five feeders, as demonstrated in Figure 5. The cable route parameters [29] are shown in Table 2. Feeders L1, L2, L3, L4, and L5 are full cable feeders with lengths of 10 km, 20 km, 25 km, 5 km, and 8 km, respectively. Rg is taken as 10 Ω. The compensation degree of the arc suppression coil is taken as 8%. The distributed capacitance CΣ of the system to the ground from feeder parameters is calculated, then the equivalent inductance Lp of the arc suppression coil could be counted by:25Lp=11.08×13ω2C∑=0.164H\begin{equation} {L}_p = \frac{1}{{1.08}} \times \frac{1}{{3{\omega }^2 {C}_{\sum}}} = 0.164{\rm{H}}\end{equation}The active power loss of the arc extinguishing coil is 2.5–5% of the inductive loss, here 3% is taken, calculated as:26RL=0.03ωL=1.545Ω\begin{equation}{R}_L = 0.03\omega L = 1.545\Omega \end{equation}5FIGURESimulation system diagram2TABLECable feeder parameterPhase sequenceR (Ω/km)C (μF/km)L (mH/km)Zero‐sequence2.7000.2801.019Positive sequence0.2700.3390.255Simulation analysisThe following are the individual fault conditions for full cable networks based on various fault circumstances: (1) change the faulted feeder, and earth faults occur in L1, L2, L3, L4, and L5, respectively, (2) change the grounding method of the neutral point, namely, ungrounded, grounded through arc suppression coil and grounded via small resistance, (3) Change the transition resistance, which is 0 Ω, 5 Ω, 10 Ω, 50 Ω, 100 Ω, 500 Ω, and arc grounding, respectively, (4) Change the position of feeder fault point, and take 10%, 50%, and 90% of the feeder length respectively, (5) The initial phase angle is changed to 0°, 30°, 45°, 60°, and 90°, respectively. After an earth fault in full cable networks, the transient zero‐sequence voltage and current within the first power frequency cycle at the beginning of per feeder under each fault condition are collected with a sampling frequency of 10 kHz. The transient zero‐sequence current per feeder is decomposed through Fourier transform, active power method, and wavelet packet transform, and the four‐fault features of I1, I5, P, and E of per feeder are extracted. There are a total of 1,575 (5 × 3 × 7 × 3 × 5 = 1,575) fault conditions, and each fault condition includes 20 (5 × 4 = 20) fault features, as depicted in Table 3.3TABLETwenty kinds of fault feature data under different fault conditionsFault condition serial numberFaulty feederFault featuresI11 (A)I12 (A)I13 (A)I14 (A)I15 (A)I51 (A)…E5 (J)1L1122.54426.84225.01436.29533.7855.258…1,418.2442L229.787118.35523.07233.44231.1351.254…1,121.9563L328.64423.809115.77032.15429.9391.193…1,043.6904L433.70827.99026.039123.75135.2411.443…1,591.6845L532.86627.29525.39436.914123.1011.401…24,232.064…………………………1,575L516.41113.24712.13918.64261.2330.550…2,834.079The feature data of each fault condition can form a 1 × 20 dimensional fault feature vector; there are 1,575 fault conditions, so a 1,575 × 20 dimensional fault feature matrix is formed, the feature matrix is normalized by discrete normalization, and then the eigenvalues, variance contribution rate, and cumulative variance contribution rate of the normalized feature matrix are obtained by using principal component analysis, as illustrated in Table 4.4TABLEThe eigenvalue, variance contribution rate, and cumulative variance contribution rate of the eigenmatrixPrincipal componentEigenvalueVariance contribution rate (%)Cumulative variance contribution rate (%)10.67423.7223.7220.66123.2546.9730.64122.5569.5240.59921.0990.6150.1645.7696.3760.0351.2597.6270.0260.9298.5480.0240.8399.3790.0170.6199.98102.738×10−40.0199.99…………205.572×10−70100Table 4 demonstrates that the cumulative contribution rate of variance of the first four principal components reaches 90.61%, greater than 85%, meeting the requirements. Therefore, the first four principal components of each fault condition are taken as the fault features after fusion. The eigenvectors corresponding to the first four eigenvalues are shown in Table 5.5TABLEThe eigenvectors corresponding to the first four eigenvaluesEigenvector 1Eigenvector 2Eigenvector 3Eigenvector 4−0.309−0.108−0.290−0.1250.399−0.210−0.148−0.0810.0670.478−0.030−0.057−0.055−0.0650.0660.445−0.083−0.1090.388−0.198−0.241−0.087−0.201−0.0930.301−0.110−0.082−0.0520.0870.287−0.022−0.033−0.004−0.0200.0500.280−0.038−0.0680.300−0.134−0.327−0.106−0.308−0.1350.407−0.205−0.151−0.0830.0670.478−0.030−0.057−0.057−0.0610.0710.485−0.087−0.1080.416−0.215−0.327−0.105−0.312−0.1370.406−0.205−0.150−0.0820.0650.477−0.031−0.057−0.054−0.0590.0700.492−0.085−0.1060.419−0.219The normalized feature matrix is mapped to the first four eigenvectors to obtain the principal component matrix. The matrix is composed of 1,575 fault conditions. Each fault condition contains four principal component components (y1, y2, y3, y4). The feature database is composed of the principal component matrix, as demonstrated in Table 6.6TABLEFeature databaseFault condition serial numbery1y2y3y41−1.264−0.467−1.122−0.52421.414−0.784−0.557−0.34130.1971.615−0.145−0.2494−0.231−0.2920.2151.5665−0.364−0.4701.437−0.7836−1.259−0.459−1.121−0.51371.419−0.772−0.551−0.33080.2011.613−0.139−0.2379−0.237−0.2730.2151.56910−0.357−0.4521.436−0.776……………1,575−0.343−0.4331.435−0.766The four principal component components of each fault condition in the feature database form a principal component vector, and 1,260 (1,575× 80% = 1,260) are randomly selected from the feature database principal component vectors as training set sample data, 315 (1,575 × 20% = 315) principal component vectors are used as the sample data of the test set, and fault feeder selection is realized by using improved K‐means and K‐means.The principal component vectors of serial numbers 1, 2, 3, 4, and 5 in the feature database represent the earth faults of feeders L1, L2, L3, L4, and L5, respectively. The above five principal component vectors are chosen as the initial clustering centres, and the minimum distance between the initial cluster centres is set as the initial threshold. The maximum number of iterations is 200. The sample data in the training set is clustered. After the cluster centre has reached convergence, the process to compute the distance between each cluster centre and the other cluster centres to determine its connection with the threshold is repeated. If it is less than the threshold, the two clusters are merged and the two samples with the largest dispersion after merging as a new cluster centre are reiterated, until the conditions are met. If the iteration count is greater than the maximum iteration count, reduce the threshold and recalculate until clustering can be carried out smoothly. After the best clustering centre is obtained from the training set, the accuracy of the algorithm is tested through the data in the test set, and the results of the improved K‐means feeder selection method are demonstrated in Figure 6.6FIGUREThe test set actual fault feeders and the improved K‐means feeder selection method result graphAs shown in Figure 6, the faulty feeders predicted by test set data are completely consistent with the actual faulty feeder, and the feeder selection accuracy reaches 100%.The K‐means feeder selection algorithm randomly selects five groups of principal component vectors from the training set as the initial clustering centre. Iterate repeatedly, output the clustering result when the cluster center converges, and test the line selection accuracy through the data in the test set. Figure 7 depicts the outcomes of the K‐means feeder selection method in terms of feeder selection results.7FIGUREThe test set actual fault feeders and K‐means feeder selection method feeder selection result diagramIn Figure 7, it can be seen that 46 out of 315 sets of test data have been incorrectly classified, and the feeder selection accuracy is only 85.4%. Moreover, the feeder selection outcomes are affected by the selected initial cluster centre, and the accuracy of feeder selection is not great.In this paper, the fused fault features are trained by K‐means and improved K‐means feeder selection algorithms to realize fault feeder selection. A comparison of time and accuracy of feeder selection between the two methods is depicted in Table 7.7TABLEComparison of time and accuracy of feeder selection between the two methodsFeeder Selection AlgorithmFeeder Selection Time (s)Feeder Selection Accuracy (%)K‐means1.0185.4Improved K‐means1.19100The feeder selection time of the two feeder selection algorithms is similar; however, the accuracy of the feeder selection is significantly different. In the feeder selection results of the K‐means feeder selection algorithm, some faulty feeders are misjudged, and the improved K‐means feeder selection method can accurately distinguish the faulted feeders, resulting in the realization that samples with the same feeder fault are grouped in the same category. This verifies the effectiveness of the fault feature fusion and the improved K‐means feeder selection method proposed. When compared to the K‐means feeder selection method, improved K‐means can significantly improve the clustering performance as well as the accuracy of the feeder selection.APPLICABILITY ANALYSIS OF FEEDER SELECTION METHODInfluence of different fault conditionsConsidering that fault line selection results may be related to different neutral grounding modes, different fault lines, different fault initial phase angles, different fault locations, different transition resistances and other factors, the applicability of the method is verified by setting different fault conditions. Table 8 shows the selection results.8TABLEThe selection results under different fault conditionsFaulty feederNeutral grounding modeFault initial phase angleFault location (%)Transition resistancey1y2y3y4Feeder selection resultsL1Ungrounded0°105 Ω−1.259−0.459−1.121−0.513L1L1Arc suppression coil0°105 Ω−1.254−0.469−1.123−0.517L1L1Small resistance0°105 Ω−1.255−0.461−1.124−0.518L1L1Arc suppression coil30°1010 Ω−1.264−0.470−1.108−0.500L1L2Arc suppression coil30°1010 Ω1.413−0.793−0.544−0.316L2L3Arc suppression coil30°1010 Ω0.1921.618−0.131−0.224L3L4Arc suppression coil30°1010 Ω−0.248−0.2650.2211.573L4L5Arc suppression coil30°1010 Ω−0.367−0.4451.446−0.759L5L3Arc suppression coil0°1050 Ω0.2631.575−0.098−0.163L3L3Arc suppression coil30°1050 Ω0.2451.582−0.098−0.166L3L3Arc suppression coil45°1050 Ω0.2331.583−0.096−0.166L3L3Arc suppression coil60°1050 Ω0.2201.585−0.094−0.165L3L3Arc suppression coil90°1050 Ω0.1841.594−0.095−0.172L3L4Arc suppression coil90°1010 Ω−0.310−0.3300.2431.532L4L4Arc suppression coil90°5010 Ω−0.309−0.3250.2441.530L4L4Arc suppression coil90°9010 Ω−0.309−0.3250.2441.530L4L5Arc suppression coil0°100 Ω−0.364−0.4511.441−0.787L5L5Arc suppression coil0°105 Ω−0.356−0.4431.438−0.778L5L5Arc suppression coil0°1010 Ω−0.342−0.4271.437−0.768L5L5Arc suppression coil0°1050 Ω−0.191−0.2661.422−0.676L5L5Arc suppression coil0°10100 Ω−0.155−0.2201.290−0.605L5L5Arc suppression coil0°10500 Ω−0.142−0.1991.200−0.562L5L5Arc suppression coil0°10Arc fault−0.296−0.4081.396−0.830L5The results show that the improved K‐means method is unaffected by neutral grounding modes, fault feeders, fault initial phase angles, fault locations, and transition resistances, and can realize the accurate feeder selection of earth faults in full cable networks.Disadvantages of individual featureWhen an individual feature is used, the anti‐interference ability of the algorithm is poor. Most of them are not suitable for high resistance grounding faults, and the feeder selection accuracy is not high. Taking the single‐phase grounding fault of L1 as an example, the initial phase angle of the fault is 90°, the fault resistance is 100 Ω, the fault location is 1 km away from the bus, and the neutral point is grounded through the arc suppression coil. The feature values of each feeder are shown in Table 9.9TABLEThe feature values of each feederFeederI1 (A)I5 (A)P (W)E (J)Feeder conditionL113.5570.0857.87014,101.165FaultL22.9010.0643.9094,463.469NormalL32.6560.1153.3323,746.385NormalL44.0940.0977.6898,562.780NormalL53.7830.0636.5507,253.690NormalIt can be seen from Table 9 that the fifth harmonic component of L3 is greater than that of L1. If the fifth harmonic component is used only for fault feeder selection, misjudgment will occur. The active component of L4 is similar to that of L1. If only the active component is used for fault feeder selection, misjudgment will also occur. The fault feeder selection algorithm based on feature fusion proposed in this paper can accurately identify the faulted feeder, which is not affected by the fault conditions.Influence of noiseWhen a single‐phase grounding fault occurs in L1, the initial phase angle of the fault is 30°, the fault resistance is 5 Ω, the fault location is 5 km away from the bus, and the neutral point is grounded through the arc suppression coil. In order to verify the anti‐noise ability of the feeder selection method proposed in this paper, Gaussian white noise signals with a signal‐to‐noise ratio (SNR) of 50 dB, 10 dB, and −10 dB are added to the transient zero‐sequence current signal. The feeder selection results are shown in Table 10.10TABLEThe selection results under influence of noiseFaulty feederSNR/dBy1y2y3y4Feeder selection resultsL150−1.249−0.444−1.122−0.503L1L110−1.110−0.228−1.104−0.396L1L1−10−0.921−0.163−0.954−0.333L1According to the results in Table 10, the proposed feeder selection method can correctly identify the fault feeder when injecting noise signals of different intensities into the transient zero‐sequence current signal.Influence of different upstream short circuit levelsTaking the single‐phase grounding fault of L2 as an example, the fault resistance is 10 Ω, the initial phase angle of the fault is 30°, the neutral point is grounded through the arc suppression coil, and the fault distances are 2 km, 4 km, 6 km, 8 km, and 10 km, respectively. The feeder selection results are shown in Table 11.11TABLEThe selection results under different upstream short circuit levelsFaulty feederFault distance/kmy1y2y3y4Feeder selection resultsL221.463−0.557−0.474−0.219L2L241.336−0.514−0.431−0.194L2L261.283−0.487−0.445−0.179L2L281.414−0.702−0.575−0.386L2L2101.424−0.767−0.539−0.306L2It can be seen from the feeder selection results in Table 11 that the fault feeder selection method proposed in this paper can still be applied under different upstream short circuit conditions by changing the fault distance.Influence of different cable characteristicsTaking the single‐phase grounding fault of L3 as an example, the initial phase angle of the fault is 0°, the fault resistance is 1 Ω, the fault location is 5 km away from the bus, and the neutral point is grounded through arc suppression coil. Four different types of cables are used for fault simulation, and the cable parameters are shown in Table 12. The feeder selection results are shown in Table 13.12TABLEParameters of different types of cable feedersCable typePhase sequenceR (Ω/km)C (μF/km)L (mH/km)1Zero‐sequence2.7000.2801.019Positive sequence0.2700.3390.2552Zero‐sequence0.3070.0083.300Positive Sequence0.1030.0511.2003Zero‐sequence0.3160.0013.530Positive sequence0.0270.0280.9044Zero‐sequence0.4120.1530.153Positive sequence0.0240.0280.08913TABLEThe selection results under different cable characteristicsFaulty feederCable typey1y2y3y4Feeder Selection ResultsL310.2931.550−0.193−0.308L3L320.1851.602−0.086−0.161L3L330.1971.595−0.080−0.150L3L340.2101.588−0.075−0.142L3According to the feeder selection results in Table 13, the faulted feeder selection method proposed in this paper can still be applied by changing the cable model.Influence of different sampling frequencyTaking the single‐phase grounding fault of L4 as an example, the initial phase angle of the fault is 0°, the fault resistance is 1 Ω, the fault location is 2.5 km away from the bus, and the neutral point is grounded through the arc suppression coil. The sampling frequencies are 5 kHz, 10 kHz, 20 kHz, and 50 kHz respectively. The feeder selection results are shown in Table 14.14TABLEThe selection results under different sampling frequencyFaulty feedersampling frequency/kHzy1y2y3y4Feeder selection resultsL45−0.248−0.2790.2141.575L4L410−0.238−0.2640.2171.569L4L420−0.223−0.2450.2191.562L4L450−0.104−0.1090.2311.511L4According to the feeder selection results in Table 14, changing the sampling frequency will not affect the feeder selection results.Influence of CT saturationIn order to verify the anti‐CT saturation performance of the algorithm, a single‐phase ground fault on L5 is taken as an example. The initial phase angle of the fault is 90°, the fault resistance is 5 Ω, 10 Ω, and 50 Ω, respectively, the fault location is 4 km away from the bus, and the neutral point is grounded through the arc suppression coil. CT saturation occurs at the head end of L5, and the feeder selection results are shown in Table 15.15TABLEThe selection results under influence of CT saturationFaulty FeederTransition Resistance/Ωy1y2y3y4Feeder selection resultsL55−0.168−0.2191.283−0.598L5L510−0.158−0.2001.200−0.558L5L550−0.296−0.4311.393−0.824L5It can be seen from Table 15 that the algorithm can still accurately identify faults under CT saturation conditions.CONCLUSIONTo address the issue of poor feeder selection accuracy when earth faults occur in 10 kV full cable networks under different fault conditions combining the processing of large amounts of fault data, a feeder selection method for full cable networks earth faults based on improved K‐means is proposed. By establishing the transient zero‐sequence network of earth faults in full cable networks, it is concluded that the transient zero‐sequence current of faulted feeder is significantly different from the normal feeder in the transient process of earth faults. Using Fourier analysis method, active power method, and wavelet packet method to decompose the transient zero‐sequence current per feeder under different fault conditions, extract multiple fault features, improve the unreliable fault feeder selection of a single feature, integrate the fault features by principal component analysis method, and construct the feature database. Finally, the feature database is trained by the improved K‐means method to realize fault feeder selection. Compared with the K‐means, the improved K‐means greatly improves the feeder selection accuracy. The findings demonstrate the proposed algorithm is not affected by fault conditions, noise, CT saturation, different cable models, and sampling frequency, and can effectively solve the problem of low accuracy of single‐phase grounding fault feeder selection in full cable networks. With its high flexibility, the algorithm is suitable for widespread use in practical operations in combination with the actual situation.ACKNOWLEDGEMENTSThis work was supported by the Fund Project of Beijing Education Commission: Key Technology Research on Intelligent Operation and Maintenance of Big Data for Power Distribution under Grant 110052972027/067, by the Simulation Analysis of Small Current Grounding Faults Based on Full Cable Feeder under Grants SGSXDT00YCJS2100298, and by the Collaborative Innovation Center of Key Power Energy‐Saving Technologies in Beijing.CONFLICT OF INTERESTAll authors declare that no conflict of interest exists.DATA AVAILABILITY STATEMENTThe data used to support the findings of this study are available from the corresponding author upon request.REFERENCESWang, X., Gao, J., Wei, X., Zeng, Z., Wei, Y., Kheshti, M.: Single line to earth faults detection in a non‐effectively grounded distribution network. IEEE Trans. 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Feeder selection method for full cable networks earth faults based on improved K‐means

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Abstract

INTRODUCTIONAfter an earth fault occurs in full cable networks, the grounding current is the capacitive current of the system. A small current loop is formed through the grounding capacitor, which will not excessively damage the equipment and system, and will not have an impact on the continuous power supply within a short period. However, after the fault occurs, due to the continuation of time, the fault may further expand, causing a two‐phase short circuit, arc discharge, and system overvoltage [1, 2]. Because of this, prompt and correct fault feeder selection in the event of earth faults in full cable networks is critical to ensure that full cable networks continue to operate as intended.Currently, earth fault feeder selection methods in full cable networks can be divided into three ways, mainly signal injection method [3, 4], fault signal features comparison method [5, 6], and artificial intelligence fault feeder selection method [7–9]. By injecting signals of a specified frequency into the system and then recognizing problematic feeders based on feeder characteristic signals, the signal injection technique detects faulty feeders. In [10], a method for selecting faulty feeders with small current grounding is presented, which combines injected signals with wavelet energy. It selects feeders by comparing the injected signal energy per feeder; however, high transition resistance results in feeder selection failure. The fault signal feature comparison method uses a mathematical algorithm to extract the steady or transient components of the fault current and then selects the feeder according to the difference in current characteristics between the normal feeder and the faulty feeder. Since the transient feature information of fault is obvious when earth faults occur in full cable networks, the effect of the feeder selection algorithm using the transient component is more significant. The main feeder selection methods include S transform [11–13], Prony algorithm [14], EMD algorithm [15, 16], wavelet transform [17], and active power method [18]. In [19], a method of earth faults feeder selection in full cable networks based on the fundamental amplitude of the transient current is proposed. In [20], a feeder selection algorithm for low‐current grounding systems is proposed through zero‐sequence current harmonics. In [21], a method of using the average active component of the transient current to realize earth faults line selection of full cable networks is proposed. In [22], fault feeder selection is realized by wavelet transient energy value. The above research uses transient information such as base wave amplitude, fifth harmonic amplitude, the energy value of wavelet packet, and average active power component of feeder zero‐sequence to improve the feeder selection ability, but all of them use a transient component, which has a single eigenvalue and poor anti‐jamming ability. The majority of them are incompatible with high resistance grounding faults.Due to the advancements in artificial intelligence and big data technologies in recent years, fault feeder selection by artificial intelligence has become a popular study area. Artificial intelligence fault feeder selection mainly converts the problem of fault feeder selection into a pattern recognition problem to solve. In [23], an earth faults feeder selection method for full cable networks based on deep confidence networks is presented. The algorithm is not influenced by the change of system‐neutral grounding mode and has a strong anti‐noise ability. In [24], a single‐phase grounding fault diagnosis method for distribution networks based on the K‐means is presented. The method is unstable due to the randomness of the initial cluster centres, which affects the accuracy of feeder selection. At present, there are many studies on fault line selection methods based on artificial intelligence, but the accuracy of the algorithm depends on a large amount of data samples. In actual situations, in order to improve the speed of the algorithm, it is necessary to screen a large amount of fault data, which will lose part of the fault information and make it difficult to obtain complete fault samples. When the sample data is less, it will be affected by fault conditions. For the fault identification method based on artificial intelligence, the more eigenvalues selected, the more accurate the identification result, but it will increase the time of fault identification, and misjudgment will occur when there are fewer eigenvalues. Literature [19–22] are fault diagnosis methods of single eigenvalue, which are vulnerable to the influence of fault conditions and have low feeder selection accuracy. Therefore, three to five fault features are usually selected, and four fault features are selected in this paper.In summary, this paper decomposes the transient zero‐sequence current of each feeder under different fault conditions through Fourier transform, active power method and wavelet packet transform, and fuses the four characteristics of fundamental wave amplitude, fifth harmonic amplitude, average active power component, and wavelet energy value of each feeder through principal component analysis, extracts the principal component and establishes the feature database. Then, a method of earth fault feeder selection using the improved K‐means is proposed to realize fault feeder selection. The results demonstrate that the technique is not affected by fault conditions, noise, CT saturation, different cable models, and sampling frequency and can effectively solve the problem of low accuracy of single‐phase grounding fault feeder selection in full cable networks.FEATURE ANALYSIS OF EARTH FAULTSThe change of working conditions of earth faults weakens the fault features and brings difficulties to the feeder selection in full cable networks. The method of fault feeder selection based on the steady‐state cannot satisfy the changing fault conditions. The transient process can be used to analyse earth faults to obtain more obvious fault features. Usually, when earth faults occur, the transient value of the current is much larger than that of the steady‐state value, which can be several times or tens of times that of the steady‐state value. Figure 1 shows diagram of typical earth faults in 10 kV full cable networks. Among them, the full cable networks include n feeders, L1–Ln in sequence, and C01–C0n are the zero‐sequence capacitance of each feeder. ĖA, ĖB, and ĖC represent a three‐phase symmetrical power supply. When switches K1, K2, and K3 are closed, respectively, the neutral point is grounded through the arc suppression coil system, grounded through a small resistance system and ungrounded system, and Lp and Rg are inductance and resistance under the mode of grounded through arc suppression coil and grounded through small resistance.1FIGURETypical diagram of earth fault of 10 kV full cable networksFor a full cable network with n feeders, when an earth fault occurs in feeder Ln, the system equivalent zero‐sequence network is depicted in Figure 2.2FIGUREEquivalent zero‐sequence network of an earth faultThe moment of fault is equivalent to connecting the zero‐sequence voltage source uf to the fault point, uf = Umsin(ω0t+θ), ω0 and θ are the power frequency angular velocity and fault initial phase angle, respectively, Um is the fault phase voltage amplitude, u0f is the neutral point voltage, i0Rg is the branch current of small resistance, i0Lp is the branch current of arc suppression coil, i0f is the current at the fault point, Ri, Li, C0i, i0i, iC0i (where i = 1, 2, …, n) are the equivalent resistance, equivalent inductance, distributed capacitance to ground, current, and capacitive current of per feeder, respectively. In the case of earth faults, the resistance and inductance of the feeder itself are far less than the feeder to ground capacitive reactance; as a result, only the feeder to ground capacitive reactance is considered in the calculation of feeder zero‐sequence impedance.When the system is grounded through the arc suppression coil, K1 is closed. The transient zero‐sequence current of the normal feeder could be represented as:1i0i=iC0i=C0idu0fdt\begin{equation}{i}_{0i}{\rm{ = }}{i}_{C0i}{\rm{ = }}{C}_{0i}\frac{{{\rm{d}}{u}_{0f}}}{{{\rm{d}}t}}\end{equation}By Kirchhoff's current law, i0f and i0n can respectively be rewritten as:2i0f=i0Lp+∑i=1niC0i=i0Lp+du0fdt∑i=1nC0i\begin{equation} {i}_{0f}{\rm{ = }}{i}_{0Lp}{\rm{ + }}\displaystyle\sum\limits_{i = 1}^n {{i}_{C0i}} = {i}_{0Lp}{\rm{ + }}\frac{{{\rm{d}}{u}_{0f}}}{{{\rm{d}}t}}\displaystyle\sum\limits_{i = 1}^n {{C}_{0i}} \end{equation}3i0n=iC0n−i0f=−(i0Lp+du0fdt∑i=1n−1C0i)\begin{equation}{i}_{0n}{\rm{ = }}{i}_{C0n}{\rm{ - }}{i}_{0f}{\rm{ = - (}}{i}_{0Lp}{\rm{ + }}\frac{{{\rm{d}}{u}_{0f}}}{{{\rm{d}}t}}\displaystyle\sum\limits_{i = 1}^{n - 1} {{C}_{0i}} )\end{equation}For further studying the features of earth faults in full cable networks, Figure 2 is simplified to Figure 3.3FIGUREFault transient equivalent circuitR is the sum of the equivalent feeder resistance from the bus to the fault point and three times the transition resistance Rf, L is the equivalent feeder inductance from the bus to the fault point, C0Σ is the sum of the ground capacitance of the whole system. Since L < < Lp, ignoring the influence of L, the voltage differential equation is:4uf=R(C0∑du0fdt+i0Lp)+u0fu0f=Lpdi0Lpdt\begin{equation}\left\{ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{u}_f = R({C}_{0\displaystyle\sum }\frac{{{\rm{d}}{u}_{0f}}}{{{\rm{d}}t}} + {i}_{0Lp}) + {u}_{0f}}\\ {{u}_{0f} = {L}_p\frac{{{\rm{d}}{i}_{0Lp}}}{{{\rm{d}}t}}} \end{array} } \right.\end{equation}Substitute (4) into (1) to obtain:5i0i=iC0i=C0idu0fdt=C0iLpd2i0Lpdt2\begin{equation}{i}_{0i}{\rm{ = }}{i}_{C0i}{\rm{ = }}{C}_{0i}\frac{{{\rm{d}}{u}_{0f}}}{{{\rm{d}}t}} = {C}_{0i}{L}_p\frac{{{{\rm{d}}}^2{i}_{0Lp}}}{{{\rm{d}}{t}^2}}\end{equation}It can be seen that i0i is linearly related to the second derivative d2i0Lp/dt2 of the arc suppression coil transient current, and the amplitude is only influenced by the capacitance parameters of the feeder itself.Substitute (4) into (2) and (3) to obtain:6i0f=i0Lp+∑i=1niC0i=i0Lp+Lpd2i0Lpdt2∑i=1nC0i\begin{equation}{i}_{0f}{\rm{ = }}{i}_{0Lp}{\rm{ + }}\displaystyle\sum\limits_{i = 1}^n {{i}_{C0i}} = {i}_{0Lp}{\rm{ + }}{L}_p\frac{{{{\rm{d}}}^2{i}_{0Lp}}}{{{\rm{d}}{t}^2}}\displaystyle\sum\limits_{i = 1}^n {{C}_{0i}} \end{equation}7i0n=iC0n−i0f=−(i0Lp+Lpd2i0Lpdt2∑i=1n−1C0i)\begin{equation}{i}_{0n}{\rm{ = }}{i}_{C0n}{\rm{ - }}{i}_{0f}{\rm{ = - (}}{i}_{0Lp}{\rm{ + }}{L}_p\frac{{{{\rm{d}}}^2{i}_{0Lp}}}{{{\rm{d}}{t}^2}}\displaystyle\sum\limits_{i = 1}^{n - 1} {{C}_{0i}} )\end{equation}Specifically, the transient zero‐sequence current of faulted feeder is concerned with i0Lp and the zero‐sequence current of the normal feeder of the whole system, and the polarity is opposite to that of the sound feeder. The compensation of the arc suppression coil to the transient capacitive current is very weak in the early stage of an earth fault, so the grounding current is primarily a transient capacitive current. The results of the aforementioned investigation indicate that the transient zero‐sequence current of the faulty feeder is greater than the normal feeder and the polarity of the current is opposite when earth faults occur in full cable networks.ESTABLISH FEATURE DATABASEThe occurrence of earth faults in full cable networks has led to the conclusion [19–22] that the fundamental amplitude of zero‐sequence current of faulted feeder is equal to the total of all sound feeders, the higher harmonic is not influenced by arc suppression coil compensation, the fifth harmonic amplitude of zero‐sequence current of faulted feeder is equal to the total of all sound feeders, the zero‐sequence average active power component of faulted feeder is greater than per sound feeder, the zero‐sequence current wavelet energy value of the faulted feeder is greater than per sound feeder. Therefore, transient zero‐sequence current per feeder is decomposed by Fourier transform, active power method, and wavelet packet transform, and four feature indexes of fundamental wave amplitude, fifth harmonic amplitude, average active power component, and wavelet energy value can be extracted as feeder selection fault features, as depicted in Table 1. Suppose a full cable network contains n feeders, L1–Lnn in sequence. The steps of establishing the earth faults feature database of full cable networks are divided into four steps: data collection, fault features extraction, fault features fusion, and establishing a feature database.1TABLEFeatures of earth faults feeder selection in full cable networksFeatureSymbolThe fundamental amplitude of zero‐sequence currentI1The fifth harmonic amplitude of zero‐sequence currentI5The zero‐sequence average active power componentPWavelet energy value of zero‐sequence currentEData collectionThe earth faults simulation model of full cable networks containing n feeders is built, and the simulation is carried out for m different fault conditions of full cable networks (earth faults occur under different feeders, different initial phase angles, different neutral grounding modes, different points, and different transition resistors). After an earth fault occurs in full cable networks, the transient zero‐sequence current lasts about one power frequency cycle (0.02 s) [25]. Following that, the transient zero‐sequence voltage and zero‐sequence current at the beginning of per feeder within a power frequency cycle under different fault conditions are collected. The sampling frequency is 10 kHz, and 200 points are sampled in one cycle.Fault features extractionAfter an earth fault in full cable networks, transient zero‐sequence current and zero‐sequence voltage within the first power frequency cycle at the beginning of n feeders are extracted in each simulation experiment.I1 and I5 are obtained after the Fourier transform of transient zero‐sequence current per feeder. I1 and I5 of feeder i (i = 1, 2, 3, …, n) are I1i and I5i, respectively. Then, the eigenvectors formed by I1 and I5 of n feeders are respectively:8I1=[I11,I12,…,I1i,…,I1n]T\begin{equation}{{\bm{I}}}_1 = {[{I}_{11},{I}_{12}, \ldots ,{I}_{1i}, \ldots ,{I}_{1n}]}^T\end{equation}9I5=[I51,I52,…,I5i,…,I5n]T\begin{equation}{{\bm{I}}}_5 = {[{I}_{51},{I}_{52}, \ldots ,{I}_{5i}, \ldots ,{I}_{5n}]}^T\end{equation}The zero‐sequence average active power component P of per feeder is calculated by the active power method, and the Pi of feeder i is:10Pi=1200∑j=1200Ui(j)Ii(j)\begin{equation}{P}_i{\rm{ = }}\frac{1}{{200}}\displaystyle\sum_{j = 1}^{200} {{U}_i(j){I}_i(j)} \end{equation}where Ii(j) and Ui(j)are zero‐sequence current amplitude and zero‐sequence voltage amplitude of feeder i at j (j = 1, 2, 3, …, 200) sampling points, respectively. Then, the eigenvector formed by the zero‐sequence average active power component P of n feeders is:11P=[P1,P2,…,Pi,…,Pn]T\begin{equation}{\bm{P}} = {[{P}_1,{P}_2, \ldots ,{P}_i, \ldots ,{P}_n]}^T\end{equation}Wavelet packet transform decomposes the transient zero‐sequence current in different frequency bands, and the sampling frequency is 10 kHz. The Db6 wavelet packet decomposes the transient zero‐sequence current per feeder into five layers [26]. After decomposition, the energy corresponding to the wavelet packet coefficients S (5,0), S (5,1), S (5,2), …, S (5, j), …, S (5,31) of the fifth layer is:12E5,j=S(5,j)2\begin{equation}{E}_{5,j} = {\left\| {S(5,j)} \right\|}^2\end{equation}where j = 0, 1, …, 31. Take the maximum band energy of layer 5 except the (5,0) band where the power frequency is located as the E of the feeder, and Ei of feeder i is:13Ei=max(E5,1,E5,2,…,E5,j,…,E5,31)\begin{equation}{E}_i{\rm{ = max(}}{E}_{{\rm{5,1}}},{E}_{{\rm{5,2}}}, \ldots ,{E}_{5,j}, \ldots ,{E}_{{\rm{5,31}}}{\rm{)}}\end{equation}Then, the eigenvector formed by E of n feeders is:14E=[E1,E2,…,Ei,…,En]T\begin{equation}{\bm{E}} = {[{E}_1,{E}_2, \ldots ,{E}_i, \ldots ,{E}_n]}^T\end{equation}Take I1, I5, P, and E of n feeders as the feature input x of a fault condition, with:15x=I11,I12,…,I1i,…,I1n,I51,I52,…,I5i,…,I5n,P1,P2,…,Pi,…,Pn,E1,E2,…,Ei,…,EnT\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\bm{x}} = \left[{I}_{11},{I}_{12}, \ldots ,{I}_{1i}, \ldots ,{I}_{1n},{I}_{51},{I}_{52}, \ldots ,{I}_{5i}, \ldots ,{I}_{5n},\right.\\ \left.{P}_1,{P}_2, \ldots ,{P}_i, \ldots ,{P}_n,{E}_1,{E}_2, \ldots ,{E}_i, \ldots ,{E}_n \right]^{\rm{T}} \end{array} \end{equation}where I11–I1n are I1 of n feeders, respectively, I51–I5n are I5 of n feeders, respectively, P1–Pn are P of n feeders respectively and E1–En are E of n feeders, respectively.Carry out m simulation experiments for m different fault conditions, and m feature inputs are obtained to form the fault feature matrix X which can be represented as:16X=[x1,x2,x3,…,xp,…,xm]T\begin{equation}{\bm{X}} = {[{{\bm{x}}}_1,{{\bm{x}}}_2,{{\bm{x}}}_3, \ldots ,{{\bm{x}}}_p, \ldots ,{{\bm{x}}}_m]}^{\rm{T}}\end{equation}where xp is the feature input obtained from the pth simulation experiment, x1– xm are the feature inputs obtained from m simulation experiments.Fault features fusionPrincipal component analysis transforms a given set of related variables into another set of unrelated variables through linear transformation [27]. These new variables are arranged in the order of decreasing variance. They can be used to extract the main feature components of data. They are often used to reduce the dimension of high‐dimensional data and realize feature fusion. Compared with other feature selection methods, principal component analysis can retain the information contained in the original variables. Only the first few principal component components can represent the fault features of samples, which can greatly improve the accuracy and operation speed of the algorithm. In this paper, the feature matrix Xm×4n is extracted by principal component analysis (there are m fault conditions, and each fault condition contains 4n fault features) to realize fault feature fusion.To eliminate the dimensional influence, I1, I5, P, and E of per feeder in x are normalized to the effective data between [0,1] by discrete standardization. Taking E as an example, the normalization method is:17Ei′=Ei−EminEmax−Emin\begin{equation}E_i^{\prime} = \frac{{{E}_i - {E}_{\min }}}{{{E}_{\max } - {E}_{\min }}}\end{equation}where Ei and E'i are E before and after normalization of feeder i, respectively, Emax and Emin are the maximum and minimum values of E in n feeders, respectively. Similarly, the normalized I'1i, I'5i, and P'i of the feeder i are obtained. Record the normalized feature input as x' which can be represented as:18x′=I11′,I12′,…,I1i′,…,I1n′,I51′,I52′,…,I5i′,…,I5n′,P1′,P2′,…,Pi′,…,Pn′,E1′,E2′,…,Ei′,…,En′T\begin{equation} \def\eqcellsep{&}\begin{array}{l} {\bm{x}}^{\prime} = \left[I_{11}^{\prime},I_{12}^{\prime}, \ldots ,I_{1i}^{\prime}, \ldots ,I_{1n}^{\prime},I_{51}^{\prime},I_{52}^{\prime}, \ldots ,I_{5i}^{\prime}, \ldots ,I_{5n}^{\prime},\right.\\[2pt] \left.{P_1}^{\prime},P_2^{\prime}, \ldots ,P_i^{\prime}, \ldots ,P_n^{\prime},E_1^{\prime},E_2^{\prime}, \ldots ,E_i^{\prime}, \ldots ,E_n^{\prime} \right]^{\rm{T}} \end{array} \end{equation}where I'11–I'1n are the normalized values of I1 of n feeders, respectively, I'51–I'5n are I5 normalized values of n feeders, respectively, P'1–P'n are P normalized values of n feeders, respectively, and E'1–E'n are the normalized values of E of n feeders, respectively.Thus, the normalized feature matrix X'm×4n can be represented as:19X′=[x1′,x2′,x3′,…,xp′,…,xm′]T\begin{equation}{{\bm{X}}}^{\prime} = {[{\bm{x}}_1^{\prime},{\bm{x}}_2^{\prime},{\bm{x}}_3^{\prime}, \ldots ,{\bm{x}}_p^{\prime}, \ldots ,{\bm{x}}_m^{\prime}]}^{\rm{T}}\end{equation}where x'p is the normalized feature input of the pth simulation experiment, and x'1–x'm are the normalized feature input obtained from the m simulation experiment, respectively.Find the covariance matrix R4n×4n of X'm×4n and compute the eigenvalues λ1, λ2, …, λ4n of R (where λ1≥λ2≥…≥λ4n≥0) and the corresponding eigenvectors a1, a2, …, a4n. The matrix A formed by the eigenvectors can be represented as:20A=[a1,a2,…,a4n]\begin{equation}{\bm{A}} = [{{\bm{a}}}_1,{{\bm{a}}}_2, \ldots ,{{\bm{a}}}_{4n}]\end{equation}Let Y = X'A and get the matrix Y as:21Y=[y1,y2,…,yi,…,y4n]\begin{equation}{\bm{Y}} = [{{\bm{y}}}_1,{{\bm{y}}}_2, \ldots ,{{\bm{y}}}_i, \ldots ,{{\bm{y}}}_{4n}]\end{equation}where yi (i = 1, 2, …, 4n) is the vector composed of the ith principal component of X. Variance contribution rate of the ith principal component αi and cumulative variance contribution rate βi can be described as:22αi=λiλi∑j=14nλj∑j=14nλjβi=∑k=1iλk∑k=1iλk∑j=14nλj∑j=14nλj\begin{equation}\left\{ { \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {{\alpha }_i = {{{\lambda }_i} \mathord{\left/ {\vphantom {{{\lambda }_i} {\displaystyle\sum_{j = 1}^{4n} {{\lambda }_j} }}} \right. \kern-\nulldelimiterspace} {\displaystyle\sum_{j = 1}^{4n} {{\lambda }_j} }}}\\ {{\beta }_i = {{\displaystyle\sum_{k = 1}^i {{\lambda }_k} } \mathord{\left/ {\vphantom {{\displaystyle\sum_{k = 1}^i {{\lambda }_k} } {\displaystyle\sum_{j = 1}^{4n} {\lambda {}_j{\rm{ }}} }}} \right. \kern-\nulldelimiterspace} {\displaystyle\sum_{j = 1}^{4n} {\lambda {}_j{\rm{ }}} }}} \end{array} } \right.\end{equation}Generally, the principal component components whose cumulative variance contribution rate reaches 85% are taken [27]. Assuming that the contribution rate of the cumulative variance of the preceding k (k ≤ 4n) principal components have already met the requirements, the feature matrix Z after feature fusion can be expressed as:23Z=[y1,y2,…,yi,…,yk]\begin{equation}{\bm{Z}} = [{{\bm{y}}}_1,{{\bm{y}}}_2, \ldots ,{{\bm{y}}}_i, \ldots ,{{\bm{y}}}_k]\end{equation}where yi (i = 1, 2, …, k) is the vector composed of the ith principal component of each fault condition (m fault conditions in total).Establish feature databaseThe feature database is formed by using the matrix Z after feature fusion. The k principal component components of each fault condition in the feature database form a principal component vector. Eight per cent and 20% of the database data are used as training sets and test sets respectively.IMPROVED K‐MEANS FEEDER SELECTION PRINCIPLEThe term ‘K‐means clustering’ refers to the process of randomly selecting the number of classes and initial cluster centre, to minimize the sum of squares of the distance between the selected class centre and per feature vector [28].K‐means feeder selection principleThe step of the K‐means feeder selection algorithm is to randomly select n principal component vectors from the feature database as the initial clustering centre, and then the principal component vector samples which need to be classified in the training set are assigned to one of n cluster centres through the criterion of minimum distance. The Euclidean distance is adapted to calculate the distance from each sample to the clustering centre. The calculation formula is described as:24d(x,y)=∑i=1k(yi−xi)2\begin{equation}{\rm{d}}({\bm{x}},{\bm{y}}) = \sqrt {\displaystyle\sum_{i = 1}^k {{{({y}_i - {x}_i)}}^2} } \end{equation}where x and y are the principal component vectors of any two fault conditions, respectively, xi and yi are, respectively, the ith principal component of x and y and k is the number of principal component components meeting the requirements of the cumulative contribution rate. After all cluster samples are allocated, the average value of all samples in each class is taken as the new cluster centre, and cycled until the cluster centre converges. Each class represents a feeder with an earth fault. After the best clustering centre is obtained from the training set, the accuracy of the method is tested by the sample data in the test set.Improved K‐means feeder selection principleWhen the first clustering centre is chosen at random, the K‐means algorithm becomes unstable, the outcome of feeder selection varies with the initial cluster centre, and the accuracy of the method will be affected when the iterative termination conditions are unreasonable. As a result, by refining the selection of initial clustering centres and terminating termination criteria, this study provides an enhanced K‐means method to overcome the issue of poor accuracy of feeder selection in the K‐means algorithm. The improvement parts are as follows:(1) Select n principal component vectors in the feature database as the initial cluster centres. The n principal component vectors are obtained when n feeders have earth faults, respectively.(2) The iterative process is optimized by introducing a threshold and setting the maximum number of iterations. The minimum value of the distance between each initial cluster centre is set as the threshold. After the cluster centre converges, the distance between per cluster centre and other cluster centres is calculated cyclically, and its relationship with the threshold is judged. If it is greater than or equal to the threshold, it exits the cycle. If it is less than the threshold, the two clusters will be merged, and the two principal component vectors with the largest dispersion after merging will be used as the new cluster centre for reiteration until the conditions are met. The threshold will be adjusted for recalculation if the iteration count exceeds the maximum iteration count.The flow chart of feeder selection based on the improved K‐means algorithm is shown in Figure 4.4FIGUREFeeder selection flow chart based on the improved K‐means algorithmSIMULATION EXAMPLESimulation modelBased on MATLAB/Simulink, this paper establishes a simulation model of 10 kV full cable networks with five feeders, as demonstrated in Figure 5. The cable route parameters [29] are shown in Table 2. Feeders L1, L2, L3, L4, and L5 are full cable feeders with lengths of 10 km, 20 km, 25 km, 5 km, and 8 km, respectively. Rg is taken as 10 Ω. The compensation degree of the arc suppression coil is taken as 8%. The distributed capacitance CΣ of the system to the ground from feeder parameters is calculated, then the equivalent inductance Lp of the arc suppression coil could be counted by:25Lp=11.08×13ω2C∑=0.164H\begin{equation} {L}_p = \frac{1}{{1.08}} \times \frac{1}{{3{\omega }^2 {C}_{\sum}}} = 0.164{\rm{H}}\end{equation}The active power loss of the arc extinguishing coil is 2.5–5% of the inductive loss, here 3% is taken, calculated as:26RL=0.03ωL=1.545Ω\begin{equation}{R}_L = 0.03\omega L = 1.545\Omega \end{equation}5FIGURESimulation system diagram2TABLECable feeder parameterPhase sequenceR (Ω/km)C (μF/km)L (mH/km)Zero‐sequence2.7000.2801.019Positive sequence0.2700.3390.255Simulation analysisThe following are the individual fault conditions for full cable networks based on various fault circumstances: (1) change the faulted feeder, and earth faults occur in L1, L2, L3, L4, and L5, respectively, (2) change the grounding method of the neutral point, namely, ungrounded, grounded through arc suppression coil and grounded via small resistance, (3) Change the transition resistance, which is 0 Ω, 5 Ω, 10 Ω, 50 Ω, 100 Ω, 500 Ω, and arc grounding, respectively, (4) Change the position of feeder fault point, and take 10%, 50%, and 90% of the feeder length respectively, (5) The initial phase angle is changed to 0°, 30°, 45°, 60°, and 90°, respectively. After an earth fault in full cable networks, the transient zero‐sequence voltage and current within the first power frequency cycle at the beginning of per feeder under each fault condition are collected with a sampling frequency of 10 kHz. The transient zero‐sequence current per feeder is decomposed through Fourier transform, active power method, and wavelet packet transform, and the four‐fault features of I1, I5, P, and E of per feeder are extracted. There are a total of 1,575 (5 × 3 × 7 × 3 × 5 = 1,575) fault conditions, and each fault condition includes 20 (5 × 4 = 20) fault features, as depicted in Table 3.3TABLETwenty kinds of fault feature data under different fault conditionsFault condition serial numberFaulty feederFault featuresI11 (A)I12 (A)I13 (A)I14 (A)I15 (A)I51 (A)…E5 (J)1L1122.54426.84225.01436.29533.7855.258…1,418.2442L229.787118.35523.07233.44231.1351.254…1,121.9563L328.64423.809115.77032.15429.9391.193…1,043.6904L433.70827.99026.039123.75135.2411.443…1,591.6845L532.86627.29525.39436.914123.1011.401…24,232.064…………………………1,575L516.41113.24712.13918.64261.2330.550…2,834.079The feature data of each fault condition can form a 1 × 20 dimensional fault feature vector; there are 1,575 fault conditions, so a 1,575 × 20 dimensional fault feature matrix is formed, the feature matrix is normalized by discrete normalization, and then the eigenvalues, variance contribution rate, and cumulative variance contribution rate of the normalized feature matrix are obtained by using principal component analysis, as illustrated in Table 4.4TABLEThe eigenvalue, variance contribution rate, and cumulative variance contribution rate of the eigenmatrixPrincipal componentEigenvalueVariance contribution rate (%)Cumulative variance contribution rate (%)10.67423.7223.7220.66123.2546.9730.64122.5569.5240.59921.0990.6150.1645.7696.3760.0351.2597.6270.0260.9298.5480.0240.8399.3790.0170.6199.98102.738×10−40.0199.99…………205.572×10−70100Table 4 demonstrates that the cumulative contribution rate of variance of the first four principal components reaches 90.61%, greater than 85%, meeting the requirements. Therefore, the first four principal components of each fault condition are taken as the fault features after fusion. The eigenvectors corresponding to the first four eigenvalues are shown in Table 5.5TABLEThe eigenvectors corresponding to the first four eigenvaluesEigenvector 1Eigenvector 2Eigenvector 3Eigenvector 4−0.309−0.108−0.290−0.1250.399−0.210−0.148−0.0810.0670.478−0.030−0.057−0.055−0.0650.0660.445−0.083−0.1090.388−0.198−0.241−0.087−0.201−0.0930.301−0.110−0.082−0.0520.0870.287−0.022−0.033−0.004−0.0200.0500.280−0.038−0.0680.300−0.134−0.327−0.106−0.308−0.1350.407−0.205−0.151−0.0830.0670.478−0.030−0.057−0.057−0.0610.0710.485−0.087−0.1080.416−0.215−0.327−0.105−0.312−0.1370.406−0.205−0.150−0.0820.0650.477−0.031−0.057−0.054−0.0590.0700.492−0.085−0.1060.419−0.219The normalized feature matrix is mapped to the first four eigenvectors to obtain the principal component matrix. The matrix is composed of 1,575 fault conditions. Each fault condition contains four principal component components (y1, y2, y3, y4). The feature database is composed of the principal component matrix, as demonstrated in Table 6.6TABLEFeature databaseFault condition serial numbery1y2y3y41−1.264−0.467−1.122−0.52421.414−0.784−0.557−0.34130.1971.615−0.145−0.2494−0.231−0.2920.2151.5665−0.364−0.4701.437−0.7836−1.259−0.459−1.121−0.51371.419−0.772−0.551−0.33080.2011.613−0.139−0.2379−0.237−0.2730.2151.56910−0.357−0.4521.436−0.776……………1,575−0.343−0.4331.435−0.766The four principal component components of each fault condition in the feature database form a principal component vector, and 1,260 (1,575× 80% = 1,260) are randomly selected from the feature database principal component vectors as training set sample data, 315 (1,575 × 20% = 315) principal component vectors are used as the sample data of the test set, and fault feeder selection is realized by using improved K‐means and K‐means.The principal component vectors of serial numbers 1, 2, 3, 4, and 5 in the feature database represent the earth faults of feeders L1, L2, L3, L4, and L5, respectively. The above five principal component vectors are chosen as the initial clustering centres, and the minimum distance between the initial cluster centres is set as the initial threshold. The maximum number of iterations is 200. The sample data in the training set is clustered. After the cluster centre has reached convergence, the process to compute the distance between each cluster centre and the other cluster centres to determine its connection with the threshold is repeated. If it is less than the threshold, the two clusters are merged and the two samples with the largest dispersion after merging as a new cluster centre are reiterated, until the conditions are met. If the iteration count is greater than the maximum iteration count, reduce the threshold and recalculate until clustering can be carried out smoothly. After the best clustering centre is obtained from the training set, the accuracy of the algorithm is tested through the data in the test set, and the results of the improved K‐means feeder selection method are demonstrated in Figure 6.6FIGUREThe test set actual fault feeders and the improved K‐means feeder selection method result graphAs shown in Figure 6, the faulty feeders predicted by test set data are completely consistent with the actual faulty feeder, and the feeder selection accuracy reaches 100%.The K‐means feeder selection algorithm randomly selects five groups of principal component vectors from the training set as the initial clustering centre. Iterate repeatedly, output the clustering result when the cluster center converges, and test the line selection accuracy through the data in the test set. Figure 7 depicts the outcomes of the K‐means feeder selection method in terms of feeder selection results.7FIGUREThe test set actual fault feeders and K‐means feeder selection method feeder selection result diagramIn Figure 7, it can be seen that 46 out of 315 sets of test data have been incorrectly classified, and the feeder selection accuracy is only 85.4%. Moreover, the feeder selection outcomes are affected by the selected initial cluster centre, and the accuracy of feeder selection is not great.In this paper, the fused fault features are trained by K‐means and improved K‐means feeder selection algorithms to realize fault feeder selection. A comparison of time and accuracy of feeder selection between the two methods is depicted in Table 7.7TABLEComparison of time and accuracy of feeder selection between the two methodsFeeder Selection AlgorithmFeeder Selection Time (s)Feeder Selection Accuracy (%)K‐means1.0185.4Improved K‐means1.19100The feeder selection time of the two feeder selection algorithms is similar; however, the accuracy of the feeder selection is significantly different. In the feeder selection results of the K‐means feeder selection algorithm, some faulty feeders are misjudged, and the improved K‐means feeder selection method can accurately distinguish the faulted feeders, resulting in the realization that samples with the same feeder fault are grouped in the same category. This verifies the effectiveness of the fault feature fusion and the improved K‐means feeder selection method proposed. When compared to the K‐means feeder selection method, improved K‐means can significantly improve the clustering performance as well as the accuracy of the feeder selection.APPLICABILITY ANALYSIS OF FEEDER SELECTION METHODInfluence of different fault conditionsConsidering that fault line selection results may be related to different neutral grounding modes, different fault lines, different fault initial phase angles, different fault locations, different transition resistances and other factors, the applicability of the method is verified by setting different fault conditions. Table 8 shows the selection results.8TABLEThe selection results under different fault conditionsFaulty feederNeutral grounding modeFault initial phase angleFault location (%)Transition resistancey1y2y3y4Feeder selection resultsL1Ungrounded0°105 Ω−1.259−0.459−1.121−0.513L1L1Arc suppression coil0°105 Ω−1.254−0.469−1.123−0.517L1L1Small resistance0°105 Ω−1.255−0.461−1.124−0.518L1L1Arc suppression coil30°1010 Ω−1.264−0.470−1.108−0.500L1L2Arc suppression coil30°1010 Ω1.413−0.793−0.544−0.316L2L3Arc suppression coil30°1010 Ω0.1921.618−0.131−0.224L3L4Arc suppression coil30°1010 Ω−0.248−0.2650.2211.573L4L5Arc suppression coil30°1010 Ω−0.367−0.4451.446−0.759L5L3Arc suppression coil0°1050 Ω0.2631.575−0.098−0.163L3L3Arc suppression coil30°1050 Ω0.2451.582−0.098−0.166L3L3Arc suppression coil45°1050 Ω0.2331.583−0.096−0.166L3L3Arc suppression coil60°1050 Ω0.2201.585−0.094−0.165L3L3Arc suppression coil90°1050 Ω0.1841.594−0.095−0.172L3L4Arc suppression coil90°1010 Ω−0.310−0.3300.2431.532L4L4Arc suppression coil90°5010 Ω−0.309−0.3250.2441.530L4L4Arc suppression coil90°9010 Ω−0.309−0.3250.2441.530L4L5Arc suppression coil0°100 Ω−0.364−0.4511.441−0.787L5L5Arc suppression coil0°105 Ω−0.356−0.4431.438−0.778L5L5Arc suppression coil0°1010 Ω−0.342−0.4271.437−0.768L5L5Arc suppression coil0°1050 Ω−0.191−0.2661.422−0.676L5L5Arc suppression coil0°10100 Ω−0.155−0.2201.290−0.605L5L5Arc suppression coil0°10500 Ω−0.142−0.1991.200−0.562L5L5Arc suppression coil0°10Arc fault−0.296−0.4081.396−0.830L5The results show that the improved K‐means method is unaffected by neutral grounding modes, fault feeders, fault initial phase angles, fault locations, and transition resistances, and can realize the accurate feeder selection of earth faults in full cable networks.Disadvantages of individual featureWhen an individual feature is used, the anti‐interference ability of the algorithm is poor. Most of them are not suitable for high resistance grounding faults, and the feeder selection accuracy is not high. Taking the single‐phase grounding fault of L1 as an example, the initial phase angle of the fault is 90°, the fault resistance is 100 Ω, the fault location is 1 km away from the bus, and the neutral point is grounded through the arc suppression coil. The feature values of each feeder are shown in Table 9.9TABLEThe feature values of each feederFeederI1 (A)I5 (A)P (W)E (J)Feeder conditionL113.5570.0857.87014,101.165FaultL22.9010.0643.9094,463.469NormalL32.6560.1153.3323,746.385NormalL44.0940.0977.6898,562.780NormalL53.7830.0636.5507,253.690NormalIt can be seen from Table 9 that the fifth harmonic component of L3 is greater than that of L1. If the fifth harmonic component is used only for fault feeder selection, misjudgment will occur. The active component of L4 is similar to that of L1. If only the active component is used for fault feeder selection, misjudgment will also occur. The fault feeder selection algorithm based on feature fusion proposed in this paper can accurately identify the faulted feeder, which is not affected by the fault conditions.Influence of noiseWhen a single‐phase grounding fault occurs in L1, the initial phase angle of the fault is 30°, the fault resistance is 5 Ω, the fault location is 5 km away from the bus, and the neutral point is grounded through the arc suppression coil. In order to verify the anti‐noise ability of the feeder selection method proposed in this paper, Gaussian white noise signals with a signal‐to‐noise ratio (SNR) of 50 dB, 10 dB, and −10 dB are added to the transient zero‐sequence current signal. The feeder selection results are shown in Table 10.10TABLEThe selection results under influence of noiseFaulty feederSNR/dBy1y2y3y4Feeder selection resultsL150−1.249−0.444−1.122−0.503L1L110−1.110−0.228−1.104−0.396L1L1−10−0.921−0.163−0.954−0.333L1According to the results in Table 10, the proposed feeder selection method can correctly identify the fault feeder when injecting noise signals of different intensities into the transient zero‐sequence current signal.Influence of different upstream short circuit levelsTaking the single‐phase grounding fault of L2 as an example, the fault resistance is 10 Ω, the initial phase angle of the fault is 30°, the neutral point is grounded through the arc suppression coil, and the fault distances are 2 km, 4 km, 6 km, 8 km, and 10 km, respectively. The feeder selection results are shown in Table 11.11TABLEThe selection results under different upstream short circuit levelsFaulty feederFault distance/kmy1y2y3y4Feeder selection resultsL221.463−0.557−0.474−0.219L2L241.336−0.514−0.431−0.194L2L261.283−0.487−0.445−0.179L2L281.414−0.702−0.575−0.386L2L2101.424−0.767−0.539−0.306L2It can be seen from the feeder selection results in Table 11 that the fault feeder selection method proposed in this paper can still be applied under different upstream short circuit conditions by changing the fault distance.Influence of different cable characteristicsTaking the single‐phase grounding fault of L3 as an example, the initial phase angle of the fault is 0°, the fault resistance is 1 Ω, the fault location is 5 km away from the bus, and the neutral point is grounded through arc suppression coil. Four different types of cables are used for fault simulation, and the cable parameters are shown in Table 12. The feeder selection results are shown in Table 13.12TABLEParameters of different types of cable feedersCable typePhase sequenceR (Ω/km)C (μF/km)L (mH/km)1Zero‐sequence2.7000.2801.019Positive sequence0.2700.3390.2552Zero‐sequence0.3070.0083.300Positive Sequence0.1030.0511.2003Zero‐sequence0.3160.0013.530Positive sequence0.0270.0280.9044Zero‐sequence0.4120.1530.153Positive sequence0.0240.0280.08913TABLEThe selection results under different cable characteristicsFaulty feederCable typey1y2y3y4Feeder Selection ResultsL310.2931.550−0.193−0.308L3L320.1851.602−0.086−0.161L3L330.1971.595−0.080−0.150L3L340.2101.588−0.075−0.142L3According to the feeder selection results in Table 13, the faulted feeder selection method proposed in this paper can still be applied by changing the cable model.Influence of different sampling frequencyTaking the single‐phase grounding fault of L4 as an example, the initial phase angle of the fault is 0°, the fault resistance is 1 Ω, the fault location is 2.5 km away from the bus, and the neutral point is grounded through the arc suppression coil. The sampling frequencies are 5 kHz, 10 kHz, 20 kHz, and 50 kHz respectively. The feeder selection results are shown in Table 14.14TABLEThe selection results under different sampling frequencyFaulty feedersampling frequency/kHzy1y2y3y4Feeder selection resultsL45−0.248−0.2790.2141.575L4L410−0.238−0.2640.2171.569L4L420−0.223−0.2450.2191.562L4L450−0.104−0.1090.2311.511L4According to the feeder selection results in Table 14, changing the sampling frequency will not affect the feeder selection results.Influence of CT saturationIn order to verify the anti‐CT saturation performance of the algorithm, a single‐phase ground fault on L5 is taken as an example. The initial phase angle of the fault is 90°, the fault resistance is 5 Ω, 10 Ω, and 50 Ω, respectively, the fault location is 4 km away from the bus, and the neutral point is grounded through the arc suppression coil. CT saturation occurs at the head end of L5, and the feeder selection results are shown in Table 15.15TABLEThe selection results under influence of CT saturationFaulty FeederTransition Resistance/Ωy1y2y3y4Feeder selection resultsL55−0.168−0.2191.283−0.598L5L510−0.158−0.2001.200−0.558L5L550−0.296−0.4311.393−0.824L5It can be seen from Table 15 that the algorithm can still accurately identify faults under CT saturation conditions.CONCLUSIONTo address the issue of poor feeder selection accuracy when earth faults occur in 10 kV full cable networks under different fault conditions combining the processing of large amounts of fault data, a feeder selection method for full cable networks earth faults based on improved K‐means is proposed. By establishing the transient zero‐sequence network of earth faults in full cable networks, it is concluded that the transient zero‐sequence current of faulted feeder is significantly different from the normal feeder in the transient process of earth faults. Using Fourier analysis method, active power method, and wavelet packet method to decompose the transient zero‐sequence current per feeder under different fault conditions, extract multiple fault features, improve the unreliable fault feeder selection of a single feature, integrate the fault features by principal component analysis method, and construct the feature database. Finally, the feature database is trained by the improved K‐means method to realize fault feeder selection. Compared with the K‐means, the improved K‐means greatly improves the feeder selection accuracy. The findings demonstrate the proposed algorithm is not affected by fault conditions, noise, CT saturation, different cable models, and sampling frequency, and can effectively solve the problem of low accuracy of single‐phase grounding fault feeder selection in full cable networks. With its high flexibility, the algorithm is suitable for widespread use in practical operations in combination with the actual situation.ACKNOWLEDGEMENTSThis work was supported by the Fund Project of Beijing Education Commission: Key Technology Research on Intelligent Operation and Maintenance of Big Data for Power Distribution under Grant 110052972027/067, by the Simulation Analysis of Small Current Grounding Faults Based on Full Cable Feeder under Grants SGSXDT00YCJS2100298, and by the Collaborative Innovation Center of Key Power Energy‐Saving Technologies in Beijing.CONFLICT OF INTERESTAll authors declare that no conflict of interest exists.DATA AVAILABILITY STATEMENTThe data used to support the findings of this study are available from the corresponding author upon request.REFERENCESWang, X., Gao, J., Wei, X., Zeng, Z., Wei, Y., Kheshti, M.: Single line to earth faults detection in a non‐effectively grounded distribution network. IEEE Trans. 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Journal

"IET Generation, Transmission & Distribution"Wiley

Published: Oct 1, 2022

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