Faulty Feeder Detection Based on Grey Correlation Degree of Adaptive Frequency Band in Resonant Grounding Distribution System

Abstract

Due to the particularity of their process, petrochemical enterprises have high requirements for the reliability of power supply. If a large-scale blackout occurs due to a grounding fault, it will pose a huge threat to safe production. When the resonant grounding system of petrochemical enterprises faults, due to the complex fault process and weak fault signal, it is difficult to accurately detect the faulty feeder by traditional methods. This paper presents a new method of grey correlation degree based on adaptive frequency band. Firstly, the transient zero-sequence current of each feeder is decomposed by coif5 wavelet, and the low frequency band a5 (power frequency component) and high frequency band d1, d2 (noise signal) are removed. By stacking all of the remaining frequency band signals to construct the wavelet area matrix, the faulty feeder detection characteristic scale and the first faulty feeder detection result are obtained. Secondly, based on the faulty feeder detection characteristic scale, the second faulty feeder detection result is obtained by the average grey correlation degree matrix, which detects the faulty feeder according to the waveform correlation degree. Finally, the final faulty feeder detection result is obtained by equal weight voting. In MATLAB/Simulink, the 10 kV resonant grounding system of petrochemical enterprises is modeled. A large number of simulation results show that the faulty feeder detection method is not affected by the initial phase angle (0°, 45° and 90°), transition resistance (10 Ω, 100 Ω and 1000 Ω), fault distance (1 km, 8 km and 15 km) and overcompensation degree (5%, 8% and 10%), and has good sensitivity.

1. Introduction

As we all know, petrochemical enterprises have very high requirements for the reliability of power supply. If a large-scale blackout occurs due to a single-phase grounding fault, the production unit will be shut down and a major safety accident will even be caused, causing unpredictable losses to the enterprise. Most of the 3~66 kV power supply systems in China’s petrochemical enterprises adopt neutral point non-effective grounding. Among them, due to the compensation effect of the arc suppression coil, the resonant grounding system has the disadvantages of the fault characteristic quantity not being obvious, and the fault detection being difficult after the single-phase grounding fault occurs [1,2,3,4,5], resulting in a worse faulty feeder detection effect [6,7].

The existing faulty feeder detection methods are mainly divided into a steady-state faulty feeder detection method and a transient faulty feeder detection method [8,9,10]. Since the electromagnetic environment of the fault site is extremely complex, and after the arc suppression coil is grounded to compensate for the current of the grounding point, the fault signal is weak and more difficult to capture. The transient signal is not affected by the compensation current and contains rich ‘fault information’. Therefore, this paper utilizes the transient signal to detect the faulty feeder. One study [11] makes full use of the transient traveling wave fault information and realizes fault location by calculating the comprehensive similarity coefficient of traveling wave waveform and comparing it to the former. In another study [12], the transient characteristics of the high resistance grounding fault are studied, and the fault location is realized by using the difference of the projection coefficient of transient zero-sequence current on transient zero-sequence voltage. In another work [13], considering the difference of transient zero-sequence current waveforms during fault, a matrix reflecting the amplitude and polarity of transient zero-sequence current is constructed and used as the input of fuzzy clustering to realize fault location. In another paper [14], the transient zero-sequence current is subjected to wavelet transform, and the morphological peak-valley detection is performed on the reconstruction coefficient. The fault location is realized by comparing the polarity relationship of the peak-valley shape of the zero-sequence current of each feeder. However, the above method is not ideal under the influence of high resistance grounding and an initial phase angle of 0°, so there is an urgent need to find a more reliable localization method.

In this paper, a faulty feeder detection method based on the grey correlation degree of the adaptive frequency band is proposed. Firstly, the method removes the interference of the power frequency component and high frequency component (noise signal) by wavelet decomposition, constructs the wavelet area matrix by reconstruction coefficient of transient zero-sequence current at each decomposition scale, obtains the faulty feeder detection characteristic scale and carries out the first faulty feeder detection. Secondly, the second faulty feeder detection based on transient zero-sequence current at the characteristic scale is carried out by GRA (Grey Relational Analysis). Finally, the faulty feeder is identified by equal weight voting based on wavelet decomposition waveform area matrix and GRA faulty feeder detection results. The sensitivity of the method is verified by simulink simulation analysis.

2. Fault Transient Characteristics of Resonant Grounding System

Figure 1 is the fault transient equivalent circuit [15], where $C$, $L_0$, $R_0$, $L$, $r_L$ and u₀ = U_m sin(ωt + φ) correspond to various operating parameters and the zero-sequence voltage of the feeder. $i_C$ is the transient capacitive current, $i_L$ is the transient inductive current, $i_k$ is the transient grounding current.

Write KVL equation from the column of Figure 1:

$$\left\{\begin{array}{c}{R}_0{i}_C+L_0\frac{d{i}_C}{dt}+\frac{1}{C}{{\displaystyle \int }}_0^t{i}_{C}dt=U_m\sin (\omega t+\phi )\\ r_L{i}_L+W\frac{d{\psi }_L}{dt}=U_m\sin (\omega t+\phi )\end{array}\right..$$

(1)

In Formula (1), $W$ and ${\psi }_L$ are the number of turns of the inductor coil and the magnetic flux flowing through, respectively. Using the initial conditions, $i_{C1}+i_{C2}=0$, $I_{Cm}=U_m\omega C$, ${\psi }_L=0$ and $I_{Lm}=U_m\omega L$ when t = 0. By using Laplace transform, transient grounded current $i_k$ can be obtained [16]:

$$i_k=i_C+i_L=\left(I_{Cm}-I_{Lm}\right)\cos \left(\omega t+\phi \right)+I_{Cm}(\frac{{\omega }_f}{\omega }\sin \phi \sin \omega t-\cos \phi \cos \omega t)e^{-\delta t}+I_{Lm}\cos \phi e^{-\frac{t}{{\tau }_L}}.$$

(2)

In Formula (2), $i_{\mathit{Cm}}$ and $i_{\mathit{Lm}}$ are the amplitude of $i_C$ and $i_L$. ${\omega }_f=\frac{2\pi }{T_f}$ are the angular frequency of free oscillation component, $\delta =\frac{1}{{\tau }_C}=\frac{R_0}{2L_0}$ are the attenuation coefficient. ${\tau }_L=\frac{L}{r_L}$ is the time constant of the inductance loop.

In analysis of Formula (2), the power frequency component of transient zero-sequence current becomes complex due to the influence of arc suppression coil overcompensation. Therefore, eliminating the power frequency component (a5) in the process of wavelet decomposition can greatly reduce the interference of power frequency with other high frequency components, which can improve the reliability of feeder detection. For the transient component, the decay rate of the transient capacitance current component is smaller than that of the transient inductance current component, so the transient zero-sequence current characteristics of the first half cycle after the fault mainly depend on the characteristics of the transient capacitance current component [17].

The transient capacitive current of the non-faulty feeder is proportional to the distributed capacitance of the feeder itself to the ground. The distribution of the feeders of the high-voltage power supply system is mostly network-like, and the feeders are generally short. Therefore, the amplitude of the transient capacitive current of the non-faulty feeder is generally small, and the direction flows from the bus to the end of the feeder. The transient zero-sequence current detected at the head end of the faulty feeder is the sum of the transient capacitive current of all non-faulty feeders. Therefore, the amplitude is extremely large, and the direction flows from the feeder to the bus.

Therefore, this paper sets up the basis of a single-phase grounding faulty feeder detection method from two angles: the transient zero-sequence current amplitude of the faulty feeder is obviously larger than that of the non-faulty feeder, and the transient zero-sequence current of all non-faulty feeders is generally similar and, obviously, different from the waveform of the faulty feeder.

3. Faulty Feeder Detection Theory

3.1. Wavelet Transform

From the above analysis, it is known that the frequency component of fault signal is extremely complex. If the same time-frequency resolution is used to process the signal, the feature information will be insufficiently utilized. The wavelet transform can use the change of the time-frequency window to obtain the local time-frequency signal, so it is very suitable to deal with the fault signal to help with fault location [18,19].

Next, using Multi Resolution Analysis (MRA), the wavelet mother function is selected as the standard orthogonal basis with which to improve the speed of wavelet decomposition and decrease the redundancy of information.

According to the principle of MRA, the original fault signal can be continuously decomposed into two spatial components according to Formula (3), where $V_j$ contains the general part of the original fault signal $A_{j}f\left(t\right)$. The $W_j$ contains the details of the original fault signal $D_{j}f\left(t\right)$.

$$V_0=V_1\oplus W_1=V_2\oplus W_2\oplus W_1=\cdots$$

(3)

Each decomposition will decompose the general and detailed parts of the signal on the general part of the previous layer, and the intensity and direction of the mutation under each decomposition represent the magnitude and polarity of the modulus maximum of the fault signal at this scale.

Cofi5 is a wavelet basis function in wavelet analysis. Compared with other wavelet basis functions, it has the characteristics of a higher order of vanishing moment, longer support length and longer filter length. It has better performance in processing high frequency signals. Due to the fact that the modulus maximum amplitude of the fault mutation point after coif5 decomposition is the largest, the fault characteristics are more obvious, and the anti-interference ability is stronger. Therefore, this paper selects the coif5 wavelet to decompose the original signal, which has better singularity detection effect.

The fault signal is decomposed by five layers of wavelet; the high frequency detail part d1~d5 and the low frequency profile part a5 are obtained. The five scales of d1~d5 correspond to the five high frequency detail parts of the original signal, which can reflect the characteristics of different frequencies of the fault signal. The frequency band of the low frequency profile part a5 is 0~3200/(2⁵ Hz), that is, 0~100 Hz (sampling frequency is 6.4 kHz), which contains 50 Hz power frequency component. After thorough consideration of enhancing the speed and reliability of faulty feeder detection, we conducted extensive simulation experiments. It has been determined that the fault signal is decomposed by five-layer wavelet to obtain the high-frequency detail part d1~d5 and the power frequency general part a5; as shown in the Figure 2, feeder 5 is faulty feeder.

Since the faulty feeder detection method in this paper uses the capacitive component in the transient signal, the power frequency component is an interference signal for the criterion. Therefore, ignoring a5 in the faulty feeder detection method can eliminate power frequency interference. Furthermore, as the scale decreases, the wavelet decomposition result of the feeder zero-sequence current becomes more severely disturbed by the noise signal. Therefore, two high-frequency bands (d1 and d2) at small scales can be removed in the faulty feeder detection method to reduce the degree of noise interference.

3.2. Grey Relational Analysis

GRA can reflect the similarity of geometric shapes between curves. The more similar the curves are, the smaller the correlation between the two is [20,21].

Denote the reference sequence y = {y(k)|k = 1, 2, …, n}; comparison sequence x_i = {x_i(k)|k = 1, 2, …, n}, i = 1, 2, ..., m. Then, the grey correlation coefficient of y(k) and x_i(k) is:

$${\xi }_i=\frac{\underset{i}{\min }\underset{k}{\min }\left|y(k)-\right.\left.x_i(k)\right|+\rho \underset{i}{\max }\underset{k}{\max }\left|y(k)-\right.\left.x_i(k)\right|}{\left|y(k)-\right.\left.x_i(k)\right|+\rho \underset{i}{\max }\underset{k}{\max }\left|y(k)-\right.\left.x_i(k)\right|}.$$

(4)

In Formula (4), ρ is the resolution coefficient, which is usually set to 0.5 for better resolution.

The average value of the correlation coefficient is concentrated into a grey correlation degree, as shown in Formula (5).

$$r_i=\frac{1}{n}\sum _{k=1}^n{\xi }_i(k).$$

(5)

Due to the geometric shape of the curve of the fault signal, it is very different from that of the non-fault signal. The faulty feeder can be detected by calculating the grey correlation degree.

4. Faulty Feeder Detection Method

4.1. Faulty Feeder Detection Method Based on Wavelet Decomposition Waveform Area Matrix

By calculating the absolute value of the area of the reconstruction coefficient of each decomposition scale, the area matrix S can be obtained. In S, the faulty feeder detection characteristic scale dj₀ is obtained by considering the anti-interference and the mutation relationship of transient quantity between faulty feeder and non-faulty feeder. The first faulty feeder detection result is then obtained from dj₀.

The area of the reconstruction coefficient of the transient zero-sequence current of the feeder Li at the decomposition scale dj s_ij is defined as:

$$s_{ij}={\displaystyle \sum }\left|A(k)\right|,$$

(6)

I, j and k represent the feeder I, the decomposition scale dj and the kth sampling point, respectively. A(k) represents the reconstruction coefficient of the kth sampling point.

The wavelet area matrix S is constructed by s_ij:

$$S=\left[\begin{array}{ccccc}{s}_{11}& s_{12}& \cdots & s_{1b}& s_{1j}\\ s_{21}& s_{22}& \cdots & s_{2b}& s_{2j}\\ ⋮& ⋮& & ⋮& ⋮\\ s_{a1}& s_{a2}& \cdots & s_{ab}& s_{aj}\\ s_{i1}& s_{i2}& \cdots & s_{ib}& s_{ij}\end{array}\right].$$

(7)

In order to reduce the degree of noise interference, two high-frequency bands at small scales are removed, and Formula (7) is changed into Formula (8):

$$S=\left[\begin{array}{ccccc}\backslash & \backslash & \cdots & s_{1b}& s_{1j}\\ \backslash & \backslash & \cdots & s_{2b}& s_{2j}\\ ⋮& ⋮& & ⋮& ⋮\\ \backslash & \backslash & \cdots & s_{ab}& s_{aj}\\ \backslash & \backslash & \cdots & s_{ib}& s_{ij}\end{array}\right].$$

(8)

The maximum area under dj is s_max, and the three minimum values of the remaining feeder area in the same frequency band are s_min1, s_min2 and s_min3. In order to avoid misjudgment of bus fault, a certain faulty feeder detection margin is set up as Formula (9), which is used to represent the amplitude relationship satisfied by the feeder fault.

$${\mathrm{s}}_{\max }>s_{\min 1}+s_{\min 2}+s_{\min 3}.$$

(9)

Taking advantage of the large amplitude of the modulus maximum, the scale with the largest reconstruction coefficient area is defined as the primary faulty feeder detection characteristic scale dj₁, which enhances the anti-interference of faulty feeder detection. If the area under dj₁ satisfies the Formula (9), then dj₁ is called the faulty feeder detection characteristic scale dj₀. If the area under dj₁ does not satisfy the Formula (9), the further faulty feeder detection characteristic scale dj₂ is defined according to the principle of maximum area in the remaining scale. At this stage, dj₂ is called the faulty feeder detection characteristic scale dj₀.

The area of each feeder under dj₀ is:

$$S_{dj0}=\left[\begin{array}{ccccc}{s}_{1j0}& s_{2j0}& \cdots & s_{aj0}& s_{ij0}\end{array}\right].$$

(10)

At this stage, if S_dj0 satisfies the Formula (9), the feeder corresponding to s_max faults, and if it still does not satisfy the Formula (9), then the bus faults.

4.2. Faulty Feeder Detection Method Based on GRA

After the wavelet decomposition waveform area matrix is used to obtain the primary faulty feeder detection characteristic scale dj₁ (or even the further faulty feeder detection characteristic scale dj₂), the geometric shape difference between the fault signal curve and the non-fault signal curve is further amplified. The gray correlation degrees between all feeders’ zero-sequence currents at the primary faulty feeder detection characteristic scale dj₁ are calculated (if there is a further faulty feeder detection characteristic scale dj₂, the gray correlation degrees at two scales are calculated at the same time). The gray correlation degree matrix can then be obtained:

$$R=\left[\begin{array}{ccccc}\backslash & r_{12}& \cdots & r_{1a}& r_{1m}\\ r_{21}& \backslash & \cdots & r_{2a}& r_{2m}\\ ⋮& ⋮& & ⋮& ⋮\\ r_{a1}& r_{a2}& \cdots & \backslash & r_{am}\\ r_{m1}& r_{m2}& \cdots & r_{ma}& \backslash \end{array}\right].$$

(11)

In Formula (11), r_ab represents the grey correlation degree of the zero-sequence current of feeder a (reference number column) and feeder b (comparison number column) at the given scale. Then, average grey correlation degree between feeder a and other feeders is defined as:

$$R_a=\frac{1}{m-1}\sum _{b=1,b\ne a}^m{r}_{ab}.$$

(12)

The average grey correlation matrix of each feeder under the decomposition scale dj₁ (or include dj₂) is:

$$R_{d{j}_1}=\left[\begin{array}{ccccc}{R}_1& R_2& \cdots & R_a& R_m\end{array}\right].$$

(13)

The minimum two values of Formula (13) are defined as r_min1 and r_min2 (r_min1 < r_min2), and the following threshold formula is proposed:

$$r_{min2}-r_{min1}>0.1.$$

(14)

If the threshold for Formula (14) is satisfied, the feeder corresponding to r_min1 is the faulty feeder. Otherwise, the bus bar is the faulty feeder.

4.3. Equal Weight Voting Faulty Feeder Detection

The faulty feeder detection based on wavelet decomposition waveform area matrix has a vote. The number of faulty feeder detection votes based on grey correlation analysis is determined by the primary faulty feeder detection characteristic scale dj₁ and the primary faulty feeder detection characteristic scale dj₂. If there is only dj₁, only the grey correlation matrix of the transient zero-sequence current of each feeder at scale dj₁ is calculated, and the faulty feeder detection method has one vote. If there are dj₁ and dj₂, the grey correlation matrix of the transient zero-sequence current of each feeder at scale dj₁ and scale dj₂ is calculated, and the faulty feeder detection method has two votes. The final faulty feeder detection criterion is: if a feeder obtains two or more votes, the feeder faults.

4.4. Faulty Feeder Detection Steps

Combined with the faulty feeder detection method, the flow chart of the faulty feeder detection method is shown in Figure 3. The specific faulty feeder detection steps are as follows:

(1)

If the zero-sequence voltage rise is detected to be 0.3 times the original, the faulty feeder detection starts.

(2)

The transient zero-sequence current of each feeder is collected and decomposed by the coif5 wavelet. Then, the lowest frequency band (a5) and the two high frequency bands (d1 and d2) are removed.

(3)

The remaining frequency band signals are constructed to the area matrix.

(4)

The primary faulty feeder detection characteristic scale dj₁ is obtained by the principle of maximum area of reconstruction coefficient. When dj₁ satisfies s_max > s_min1 + s_min2 + s_min3, dj₁ is defined as the faulty feeder detection characteristic scale dj₀. The feeder corresponding to s_max of dj₀ gets a vote.

(5)

If dj₁ does not satisfy s_max > s_min1 + s_min2 + s_min3, the further faulty feeder detection characteristic scale dj₂ is defined as the faulty feeder detection characteristic scale dj₀. The feeder corresponding to s_max of dj₀ gets a vote.

(6)

The gray correlation matrix under dj₁/dj₁ and dj₂ (if there exists dj₁ and dj₂) is constructed by GRA, and the average gray correlation degree of each feeder is calculated. Then, the second faulty feeder detection is completed according to the threshold formula.

(7)

The first faulty feeder detection result has one vote, and the second faulty feeder detection result has one vote/two votes (if there exists dj₁ and dj₂). The feeder with the largest number of votes is the faulty feeder.

5. Simulation Experiment and Analysis

5.1. Simulation Model

A 10 kV resonant grounding system model of petrochemical enterprises was built using Simulink as shown in Figure 4, where L1, L3 and L5 are overhead feeders, L2 and L4 are cable feeders. The positive-sequence and zero-sequence parameters of the overhead feeders and cable feeders are shown in

Table 1. Feeder parameters per unit length (km).

Circuit Type	R1/Ω	L1/mH	C1/μF	R0/Ω	L0/mH	C0/μF
Overhead Feeder	0.46	0.9337	0.07	0.75	0.2837	0.05
Cable Feeder	0.09	0.2837	0.26	0.14	0.8364	0.07

. The three-phase power source is simulation system default ideal power supply. The main transformer is Y/Y connected with a transformation ratio of 110 kV/10 kV, and the load parameters are set to default values. The calculation shows that the overcompensation is 10% when the inductance L = 0.7388 H is set, and the fault is triggered by the step signal. The simulation sampling frequency is set to 6.4 kHz. The fault condition can be changed by setting a different fault initial phase angle φ (fault time), fault transition resistance R_f and fault distance X_f.

5.2. Simulation Example

(1) Set L1 to have a single-phase ground fault when the fault initial phase angle φ is 45° (the fault time can be set to 0.0225 s), the fault transition resistance R_f is 100 Ω and the fault distance X_f is 5 km. The resulting area matrix S is shown in Formula (15).

$$S=\left[\begin{array}{rrrrr}\hfill \backslash & \hfill \backslash & \hfill 62.69& \hfill 145.61& \hfill 166.09\\ \hfill \backslash & \hfill \backslash & \hfill 12.14& \hfill 32.80& \hfill 54.42\\ \hfill \backslash & \hfill \backslash & \hfill 10.48& \hfill 28.59& \hfill 45.59\\ \hfill \backslash & \hfill \backslash & \hfill 16.63& \hfill 44.09& \hfill 72.50\\ \hfill \backslash & \hfill \backslash & \hfill 42.02& \hfill 44.63& \hfill 65.34\end{array}\right].$$

(15)

Since the area of the reconstruction coefficient of the first feeder at the d5 scale is the largest (data marked red), d5 is selected as the primary faulty feeder detection characteristic scale dj₁ (the fifth column data in Equation (15)). In addition, S_d5 can be obtained as:

$$S_{d5}=\left[166.09\hspace{1em}\hspace{1em}54.42\hspace{1em}\hspace{1em}45.59\hspace{1em}\hspace{1em}72.50\hspace{1em}\hspace{1em}65.34\right].$$

(16)

From the Formula (16), because s_max = 166.09 > s_min1 + s_min2 + s_min3 = 165.35, (s_min1, s_min2 and s_min3 are marked as blue), the faulty feeder detection characteristic scale dj₀ is d5, and the further faulty feeder detection characteristic scale dj₂ does not exist. Therefore, it can be determined that the first faulty feeder detection result is L1 (the feeder corresponding to s_max is L1).

The second faulty feeder detection is carried out under the decomposition scale d5, determined by the wavelet area matrix. By Formula (4), the grey correlation degree of scale d5 of each feeder to other feeders is obtained, which is summarized into matrix R as Formula (17):

$$R=\left[\begin{array}{ccccc}\backslash & 0.573& 0.582& 0.557& 0.563\\ 0.550& \backslash & 0.963& 0.928& 0.954\\ 0.550& 0.962& \backslash & 0.894& 0.918\\ 0.550& 0.933& 0.904& \backslash & 0.973\\ 0.551& 0.957& 0.925& 0.973& \backslash \end{array}\right].$$

(17)

By Formula (12) and Formula (17), the average grey correlation degree between each feeder and other feeders can be obtained as Formula (18):

$$R_{d5}=\left[0.569\hspace{1em}\hspace{1em}0.849\hspace{1em}\hspace{1em}0.831\hspace{1em}\hspace{1em}0.840\hspace{1em}\hspace{1em}0.852\right].$$

(18)

From Formula (18), it is evident that r_min1 = 0.569 (data marked red), and r_min2 − r_min1 = 0.262 > 0.1, r_min2 is marked in blue. Therefore, the second faulty feeder detection result is L1 (the feeder corresponding to r_min1 is L1).

Since the results of the first faulty feeder detection and the second faulty feeder detection are both L1, the feeder L1 has two votes. Finally, it is determined that the feeder L1 is the faulty feeder.

(2) Set bus to have a single-phase ground fault when the fault initial phase angle φ is 90° (the fault time can be set to 0.025 s), and the fault transition resistance R_f is 1000 Ω. The resulting area matrix S is shown in Formula (19),

$$S=\left[\begin{array}{rrrrr}\hfill \backslash & \hfill \backslash & \hfill 1.58& \hfill 3.23& \hfill 6.90\\ \hfill \backslash & \hfill \backslash & \hfill 2.52& \hfill 5.38& \hfill 11.57\\ \hfill \backslash & \hfill \backslash & \hfill 2.55& \hfill 4.61& \hfill 9.72\\ \hfill \backslash & \hfill \backslash & \hfill 3.44& \hfill 7.20& \hfill 15.51\\ \hfill \backslash & \hfill \backslash & \hfill 4.00& \hfill 7.13& \hfill 13.81\end{array}\right].$$

(19)

Since the reconstruction coefficient area of the fourth feeder at the d5 scale is the largest (data marked red), d5 is selected as the primary faulty feeder detection characteristic scale dj₁ (the fifth column data in Equation (19)). Since s_max = 15.51 < s_min1 + s_min2 + s_min3 = 28.19, s_min1, s_min2 and s_min3 are marked as blue. It is necessary to continue to calculate the further faulty feeder detection characteristic scale dj₂.

Since the reconstruction coefficient area of the fourth feeder at the d4 scale is the largest (data marked pink), d4 is selected as the further faulty feeder detection characteristic scale dj₂ (the fourth column data in Equation (19), and the faulty feeder detection characteristic scale dj₀ is d4. S_d4 can be obtained as:

$$S_{d4}=\left[3.23\hspace{1em}\hspace{1em}5.38\hspace{1em}\hspace{1em}4.61\hspace{1em}\hspace{1em}7.20\hspace{1em}\hspace{1em}7.13\right].$$

(20)

From the Formula (20), it is evident that s_max = 7.20 is less than s_min1 + s_min2 + s_min3 = 13.22, (s_max is marked as red, s_min1, s_min2 and s_min3 are marked as blue), therefore, it can be determined that the first faulty feeder detection result is the bus bar.

Since the first faulty feeder detection process exists for the primary faulty feeder detection characteristic scale d5 and further faulty feeder detection characteristic scale d4, the second faulty feeder detection needs to be carried out under the d5 and d4 components. The obtained grey correlation matrix is shown in Formulas (21) and (22).

$$R=\left[\begin{array}{ccccc}\backslash & 0.671& 0.761& 0.544& 0.590\\ 0.542& \backslash & 0.730& 0.575& 0.691\\ 0.686& 0.767& \backslash & 0.543& 0.616\\ 0.544& 0.702& 0.628& \backslash & 0.836\\ 0.543& 0.763& 0.653& 0.801& \backslash \end{array}\right],$$

(21)

$$R=\left[\begin{array}{ccccc}\backslash & 0.685& 0.773& 0.551& 0.559\\ 0.547& \backslash & 0.755& 0.584& 0.602\\ 0.687& 0.787& \backslash & 0.546& 0.559\\ 0.543& 0.709& 0.635& \backslash & 0.955\\ 0.545& 0.719& 0.641& 0.954& \backslash \end{array}\right].$$

(22)

Using Formulas (12), (21) and (22), it can be surmised: the value in Formula (23) represents the average grey correlation degree of the d5 component of each feeder to other feeders. The value in Formula (24) represents the average grey correlation degree of the d4 component of each feeder to other feeders.

$$R_{d5}=\left[0.642\hspace{1em}\hspace{1em}0.635\hspace{1em}\hspace{1em}0.653\hspace{1em}\hspace{1em}0.677\hspace{1em}\hspace{1em}0.690\right],$$

(23)

$$R_{d4}=\left[0.642\hspace{1em}\hspace{1em}0.622\hspace{1em}\hspace{1em}0.645\hspace{1em}\hspace{1em}0.711\hspace{1em}\hspace{1em}0.715\right].$$

(24)

From Formula (23): r_min2 − r_min1 = 0.007 < 0.1 (r_min2, r_min2 are marked in red and blue, respectively). From Formula (24): r_min2 − r_min1 = 0.02 < 0.1 (r_min2, r_min2 are marked in red and blue, respectively). Therefore, the second faulty feeder detection result has two results: both of them are bus bar.

The result of the first faulty feeder detection is one vote for the bus, the result of the second faulty feeder detection is two votes for the bus. Finally, it is determined that the bus bar is the faulty feeder.

5.3. Applicability Analysis of Faulty Feeder Detection Method

The results of faulty feeder detection are affected by the initial phase angle φ, fault transition resistance R_f and fault distance X_f, so the applicability of the faulty feeder detection method in this paper can be verified by setting different fault conditions. Due to space limitations, only some of the results are listed.

(1) Different fault initial phase angle.

Set L2 to have a fault when R_f is 1000 Ω, X_f is 6 km and φ is 0°, 45° and 90°. The bus has a fault when R_f is 1000 Ω and φ is 0°, 45° and 90°. The faulty feeder detection results based on wavelet decomposition waveform area matrix and the faulty feeder detection results based on grey correlation analysis are shown in

Table 2. Results of faulty feeder detection based on wavelet decomposition waveform area matrix.

Faulty Feeder	$\mathbf{\phi }$(°)	dj1	dj2	Sdj0	Faulty Feeder Detection Result	Get the Number of Votes
L2	0	5	\	Sd5 = [2.17 13.56 3.06 4.86 4.36]	L2	1
	45	4	\	Sd4 = [2.60 17.04 3.70 5.79 5.71]	L2	1
	90	5	4	Sd4 = [3.24 21.41 4.62 7.21 7.19]	L2	1
Bus Bar	0	5	4	Sd4 = [0.61 1.03 0.87 1.37 1.27]	Bus Bar	1
	45	5	4	Sd4 = [2.61 4.35 3.71 5.82 5.70]	Bus Bar	1
	90	5	4	Sd4 = [3.23 5.38 4.61 7.20 7.13]	Bus Bar	1

The red data in this table is s_max, the blue data in this table are s_min1, s_min2 and s_min3.

and

Table 3. Faulty feeder detection results based on grey correlation analysis and final results.

Faulty Feeder	$\mathbf{\phi }$(°)	Rdj1 Rdj2	Faulty Feeder Detection Result	Get the Number of Votes	Aggregate Votes
L2	0	Rd3 = [0.801 0.599 0.826 0.822 0.832]	L2	1	2
		\
	45	Rd4 = [0.810 0.572 0.833 0.835 0.837]	L2	1	2
		\
	90	Rd5 = [0.775 0.581 0.808 0.803 0.817]	L2	2	3
		Rd4 = [0.809 0.600 0.832 0.835 0.836]
Bus Bar	0	Rd5 = [0.633 0.632 0.647 0.674 0.685]	Bus Bar	2	3
		Rd4 = [0.606 0.594 0.618 0.661 0.685]
	45	Rd5 = [0.641 0.643 0.657 0.676 0.695]	Bus Bar	2	3
		Rd4 = [0.642 0.622 0.655 0.707 0.720]
	90	Rd5 = [0.642 0.635 0.653 0.677 0.690]	Bus Bar	2	3
		Rd4 = [0.642 0.622 0.644 0.710 0.715]

The red data in this table is r_min1, the blue data in this table is r_min2.

, respectively.

In Table 2 and Table 3, when the faulty feeder is L2 and the initial phase angle of the fault is 0°, s_max, s_min1, s_min2 and s_min3 satisfy the threshold for Formula (9), therefore, the first faulty feeder detection result is a vote for feeder L2. Since there only exists the primary faulty feeder detection characteristic scale d5, the second faulty feeder detection has only one feeder detection result. Since r_min2 and r_min2 satisfy the threshold for Formula (14), the second faulty feeder detection result is a vote for feeder L2. Finally, feeder L2 gets two votes, the feeder L2 is the faulty feeder.

When the faulty feeder is the bus bar and the initial phase angle of the fault is 90°, s_max, s_min1, s_min2 and s_min3 do not satisfy the threshold for Formula (9), and there exists further faulty feeder detection characteristic scale dj₄. However, s_max, s_min1, s_min2 and s_min3 of dj₄ still do not satisfy the threshold for Formula (9). Therefore, the first faulty feeder detection result is a vote for the bus bar. Due to the existence of the primary faulty feeder detection characteristic scale dj₅ and the further faulty feeder detection characteristic scale dj₄, the second faulty feeder detection has two feeder detection results. Since both of them do not satisfy the threshold for Formula (14), the second faulty feeder detection result is two votes for the bus bar. Finally, the bus bar gets three votes, and the bus bar is the faulty feeder.

Through analysis, it can be seen from Table 2 and Table 3 that the faulty feeder detection method can correctly detect the feeder under different fault initial phase angles.

(2) Different fault transition resistance.

Set L3 to have a fault when φ is 45°, X_f is 7 km and R_f is 10 Ω, 100 Ω and 1000 Ω in turn. The bus has a fault when φ is 45° and R_f is 10 Ω, 100 Ω and 1000 Ω. The faulty feeder detection results based on wavelet decomposition waveform area matrix and the faulty feeder detection results based on grey correlation analysis are shown in

Table 4. Results of faulty feeder detection based on wavelet decomposition waveform area matrix.

Faulty Feeder	Rf (Ω)	dj1	dj2	Sdj0	Faulty Feeder Detection Result	Get the Number of Votes
L3	10	4	\	Sd4 = [101.62 167.44 702.07 225.53 228.99]	L3	1
	100	5	\	Sd5 = [32.11 53.72 151.11 71.72 64.48]	L3	1
	1000	4	\	Sd4 = [2.60 4.32 17.77 5.80 5.83]	L3	1
Bus Bar	10	3	5	Sd5 = [46.22 78.13 65.14 104.28 94.05]	Bus Bar	1
	100	5	4	Sd4 = [18.15 30.20 25.89 40.43 40.39]	Bus Bar	1
	1000	5	4	Sd4 = [2.61 4.35 3.71 5.82 5.70]	Bus Bar	1

The red data in this table is s_max, the blue data in this table are s_min1, s_min2 and s_min3.

and

Table 5. Faulty feeder detection results based on grey correlation analysis and final results.

Faulty Feeder	Rf (Ω)	Rdj1 Rdj2	Faulty Feeder Detection Result	Get the Number of Votes	Aggregate Votes
L3	10	Rd4 = [0.807 0.841 0.582 0.845 0.844] \	L3	1	2
	100	Rd5 = [0.788 0.837 0.569 0.827 0.838] \	L3	1	2
	1000	Rd4 = [0.805 0.840 0.572 0.842 0.843] \	L3	1	2
Bus Bar	10	Rd3 = [0.708 0.747 0.772 0.751 0.707] Rd5 = [0.660 0.649 0.665 0.691 0.703]	Bus Bar	2	3
	100	Rd5 = [0.648 0.642 0.660 0.687 0.701] Rd4 = [0.621 0.613 0.630 0.706 0.708]	Bus Bar	2	3
	1000	Rd5 = [0.641 0.643 0.657 0.676 0.695] Rd4 = [0.642 0.622 0.655 0.707 0.720]	Bus Bar	2	3

The red data in this table is r_min1, the blue data in this table is r_min2.

, respectively.

By analyzing Table 2 and Table 3, it can be surmised from Table 4 and Table 5 that the faulty feeder detection method can correctly detect the feeder under different fault transition resistances.

(3) Different fault distance.

Set L4 to have a fault when R_f is 100 Ω, φ is 45°, and X_f is 1 km, 8 km and 15 km. The faulty feeder detection results based on wavelet decomposition waveform area matrix and the faulty feeder detection results based on grey correlation analysis are shown in

Table 6. Results of faulty feeder detection based on wavelet decomposition waveform area matrix.

Faulty Feeder	Xf (km)	dj1	dj2	Sdj0	Faulty Feeder Detection Result	Get the Number of Votes
L4	1	5	\	Sd5 = [32.78 55.28 52.43 133.86 66.34]	L4	1
	8	5	\	Sd5 = [32.26 54.35 52.63 132.93 66.00]	L4	1
	15	5	\	Sd5 = [32.10 53.88 51.57 131.35 65.40]	L4	1

The red data in this table is s_max, the blue data in this table are s_min1, s_min2 and s_min3.

and

Table 7. Faulty feeder detection results based on grey correlation analysis and final results.

Faulty Feeder	Xf (km)	Rdj1 Rdj2	Faulty Feeder Detection Result	Get the Number of Votes	Aggregate Votes
L4	1	Rd5 = [0.80 0.84 0.84 0.55 0.82]	L4	1	2
	8	Rd5 = [0.80 0.84 0.84 0.57 0.82]	L4	1	2
	15	Rd5 = [0.81 0.84 0.84 0.59 0.83]	L4	1	2

The red data in this table is r_min1, the blue data in this table is r_min2.

, respectively.

By analyzing Table 2 and Table 3, it can be concluded from Table 6 and Table 7 that the faulty feeder detection method can correctly detect the feeder under different fault distances.

(4) Different overcompensation degree.

Set L5 to have a fault when R_f is 100 Ω, φ is 45°, X_f is 6 km, overcompensation degree is 5%, 8% and 10%. The faulty feeder detection results based on wavelet decomposition waveform area matrix and the faulty feeder detection results based on grey correlation analysis are shown in

Table 8. Results of faulty feeder detection based on wavelet decomposition waveform area matrix.

Faulty Feeder	v(%)	dj1	dj2	Sdj0	Faulty Feeder Detection Result	Get the Number of Votes
L5	5	5	\	Sd5 = [32.03 53.79 45.09 71.89 135.05]	L5	1
	8	5	\	Sd5 = [32.22 54.12 45.36 72.33 133.96]	L5	1
	10	5	\	Sd5 = [32.33 54.29 45.51 72.56 133.12]	L5	1
Bus Bar	5	5	4	Sd4 = [18.12 30.14 25.85 40.36 40.32]	Bus Bar	1
	8	5	4	Sd4 = [18.15 30.19 25.89 40.41 40.37]	Bus Bar	1
	10	5	4	Sd4 = [18.15 30.20 25.90 40.43 40.39]	Bus Bar	1

The red data in this table is s_max, the blue data in this table are s_min1, s_min2 and s_min3.

and

Table 9. Faulty feeder detection results based on grey correlation analysis and final results.

Faulty Feeder	v(%)	Rdj1 Rdj2	Faulty Feeder Detection Result	Get the Number of Votes	Aggregate Votes
L5	5	Rd5 = [0.801 0.833 0.831 0.809 0.577] \	L5	1	2
	8	Rd5 = [0.800 0.833 0.831 0.808 0.575] \	L5	1	2
	10	Rd5 = [0.800 0.832 0.830 0.803 0.572] \	L5	1	2
Bus Bar	5	Rd5 = [0.647 0.642 0.659 0.686 0.701] Rd4 = [0.621 0.613 0.630 0.706 0.708]	Bus Bar	2	3
	8	Rd5 = [0.648 0.642 0.659 0.687 0.702] Rd4 = [0.621 0.613 0.630 0.706 0.708]	Bus Bar	2	3
	10	Rd5 = [0.648 0.642 0.660 0.687 0.702] Rd4 = [0.621 0.613 0.630 0.706 0.708]	Bus Bar	2	3

The red data in this table is r_min1, the blue data in this table is r_min2.

, respectively.

By analyzing Table 2 and Table 3, it can be concluded from Table 8 and Table 9 that the faulty feeder detection method can correctly detect the feeder under different overcompensation degrees.

6. Discussion of Faulty Feeder Detection Method

6.1. The Advantages and Disadvantages of Faulty Feeder Detection Method

Compared with the traditional modulus maximum method, this method has the following advantages:

(1) The transient zero-sequence current extracted by coif5 wavelet contains abundant ‘fault information’, which can highlight the difference between the faulty feeder and non-faulty feeders.

(2) The interference of high frequency noise (d1 and d2) and power frequency signal (a5) is removed by wavelet decomposition, which enhances the anti-interference ability reliability of the criterion.

Compared with wavelet modulus maximum method, this method has the following advantages:

(1) The faulty feeder detection characteristic scale dj₀ is processed adaptively by the principle of maximum area of reconstruction coefficient and the threshold Formula (9). Therefore, dj₀ contains the main ‘fault information‘. The zero-sequence current of scale dj₀ is used for grey correlation analysis to find out the faulty feeder. It is able to identify the faulty feeder in the extracted ‘fault information’, and the faulty feeder selected is more reliable.

(2) This method organically integrates two faulty feeder detection methods. The method based on wavelet decomposition waveform area matrix can get one vote, and the method based on GRA can get two votes. The final total vote is three votes, and the feeder with more votes is the faulty feeder. Compared with the single wavelet modulus maximum method, it has higher credibility.

The main disadvantage of this method is that the algorithm of the proposed method is relatively complicated, and the wavelet toolbox embedded in MATLAB is needed. This requires higher performance from the faulty feeder detection device. (The performance of the host used in this article: Processors: AMD Ryzen 7 5800 H, Disk: 512 GB, RAM: 16 GB).

6.2. The Possible Implementation of Faulty Feeder Detection Method

The field application of faulty feeder detection technology is generally realized by a wide area measurement system (WAMS). The PMU (Phasor Measurement Unit) device of WAMS takes GPS (Global Position System) as the sampling reference. The voltage, current and important switch protection signals of each feeder are collected synchronously in the whole network. The signal is sent to the ground central system through optical fiber for centralized processing, and the detection of the fault feeder is realized by algorithm. The algorithm proposed in this paper can be realized by WAMS system, but it has higher requirements for the host. (Minimum device requirements for MATLAB with the wavelet toolbox: Processors: Any Intel or AMD x86-64 processor, Disk: 3 GB of HDD space for MATLAB only, RAM: 4 GB)

6.3. Contribution of Faulty Feeder Detection Method to Sustainability

Energy is a necessary resource to maintain social operation. Petroleum will still be the largest energy source over the next two decades, so oil and gas storage and transportation systems with energy-saving design can greatly reduce energy consumption and make great contributions to the development of green environmental protection and a low-carbon economy.

The faulty feeder detection method proposed in this paper can, in a timely manner, detect the faulty feeder of the oil and gas storage and transportation system, so as to remove the fault in time and avoid causing a larger area of power outage, which improves the safety of the oil and gas storage and transportation system with energy-saving design.

7. Conclusions

This paper aims at the transient component of a single-phase grounding fault in the resonant grounding system of petrochemical enterprises. Using the excellent time-frequency localization ability of wavelet analysis, this paper calculates the components of the transient component at different decomposition scales. The area matrix of the reconstruction coefficient of each scale is used to obtain the faulty feeder detection characteristic scale dj₀ and the first faulty feeder detection result. Then, combined with wavelet decomposition waveform area and grey correlation analysis, the second faulty feeder detection results can be obtained. Finally, the faulty feeder detection is realized by equal weight voting. This scheme has the following advantages:

(1)

The faulty feeder detection method based on wavelet decomposition waveform area matrix can avoid the influence of the detection of wavelet decomposition scale. Moreover, combined with the definition of the reconstruction coefficient area, it overcomes the disadvantage of poor anti-interference ability when selecting a single modulus maximum point as the criterion.

(2)

At the characteristic scale dj₁ and dj₂, obtained by the wavelet decomposition waveform area matrix, the transient waveform difference between the faulty feeder and the non-faulty feeder is obviously improved. On this basis, the GRA-based faulty feeder detection method is used to detect the faulty feeder according to waveform similarity, which effectively improves the reliability of the faulty feeder detection criterion.

(3)

The method of equal weight voting, which combines wavelet decomposition waveform area matrix and grey correlation analysis, overcomes the low reliability of the single faulty feeder detection method. The simulation results show that the faulty feeder detection method has good reliability and sensitivity.