Fault Location Method in Nonsolid-Earthed Network Based on Spatial Domain Image Fusion and Convolution Neural Network

Given that the single-terminal traveling wave location method has significant errors, a novel fault location method based on the spatial domain image fusion and convolutional neural network (CNN) is proposed. Firstly, the three-phase traveling wave can be decoupled by the phase-mode transformation matrix for obtaining the line-mode component of the traveling wave. Secondly, the 1D line-mode traveling wave can be converted into a 2D image by the Gramian angular field (GAF). The 1D line-mode component can be mapped into the color, point, line, and other characteristic parameters of the 2D image. In order to expand the invisible information of the line-mode traveling wave, the images obtained by the Gramian angular summation field (GASF) and Gramian angular difference field (GADF) are weighted and fused. Finally, the CNN can be used to autonomously mine the characteristic parameters of the weight-fusion image and realize fault location. The simulation results show that the proposed method does not need to be considered in the traveling wave head and the traveling wave speed. The localization method is not affected by fault time, fault distance, or transition resistance factors. It possesses high reliability with an absolute range error of no more than 200m.


Introduction
The medium-voltage distribution network usually adopts a small current grounding system (SCGS) in China, and the single-phase grounding fault has become one of the crucial factors influencing the reliability of the power supply [1,2]. Fast and accurate fault location technology can reduce power outage range and power outage time [3]. It has an essential significance for improving the safety, reliability, and economy of the distribution network.
The distribution network structures mostly are arborescent radiation patterns. The model has many branches and short line lengths. When the single-phase grounding fault occurs in the small current grounding system, the fault signal presents nonlinear, faint, complexity, and other features. These factors can give the challenge for fault location. At present, the fault location methods can be roughly divided into several ways, such as fault analysis method, artificial intelligence method, signal injection method, traveling wave method, and impedance method. At the beginning of fault location, the distribution network system lines were considered geometrically symmetrical using symmetrical component-based methodologies [4]. However, symmetrical component-based analysis was applied only to the balanced system. Elkalashy and colleagues [5] used a controlled thyristor to generate a disturbance signal, achieving the wave distance measurement. However, the technique must inject a signal into the neutral point. It can only be used in the neutral point grounding by arc suppression coil and cannot be used in the neutral point ungrounded system. Liao [6] used the substation voltage and current quantities to construct the bus impedance matrix, through which the fault location can be yielded. However, the method needs to know the distribution network parameters and topological structures in advance. Sun and colleagues [7] presented an automated outage mapping method for the automatic faulted line section location. The method needs to collect much customer electric information, and the process is relatively complicated. Sapountzoglou et al. [8] used the gradient boosting trees to establish a fault diagnosis for the low-voltage smart distribution network. The fixed number of interpolations replaces the measurements of a particular branch. This model adapts to changes in the network topology to some extent but does not accurately locate the faulty segment. Teng et al. [9] used the relationship matrix to locate the faulted line sections. The matrix method is easy to realize. However, when the topology changes, the network matrix must be regenerated, demanding large-scale computation quality. One of the most effective methods to calculate the distance-to-fault is the traveling wave method [10,11]. The uncertainty of fault traveling velocity and arriving time of fault initial traveling wave can affect the accuracy of the location. Karmacharya and Gokaraju [12] used the waveletbased frequency characteristics to locate the fault sections. But the wrong selection of the decomposition scale and wavelet basis function may obtain erroneous fault location. Machine learning has broad prospects in fault diagnosis. Many intelligent fault diagnosis models have been proposed [13,14]. Luo et al. [15] built the stack autoencoder (SAE) regression model to constitute the relationship between fault features and fault locations, which can realize the end-to-end faulted line location. Mirzaei and colleagues [16] used the discrete wavelet transform to extract the features of fault voltage, and the generated features are used to train the deep neural networks for obtaining the faulted line segments. Luo et al. [17] combined SAE with signal phase information to realize the fault range finding for the multiterminal transmission line.
Rapid and precise fault location in the distribution network can effectively reduce fault detection and outage time, reduce economic losses, and improve power supply reliability. This article focuses on solving the problem of the distribution network fault location. The main contributions of this article are summarized as follows: (1) The 1D traveling wave signal can be converted into the 2D image by Gramian angular field (GAF). We used the spatial domain image fusion method to reinforce the invisible information of the traveling wave signal. The signal-to-image method can be beneficial to the convolutional neural network (CNN) (2) The CNN network structure is designed for fault location in the distribution network. The CNN can mine the image features autonomously, without human extraction of features, avoiding human factor interference (3) The difference from the traditional single-terminal traveling wave location method, the proposed method does not require detecting the wave head and speed of the traveling wave. It can improve the distance-measuring accuracy The paper can be divided into the following sections. The phase-mode transformation is introduced in Section 2. In Section 3, the signal-to-image conversion method is proposed. Section 4 describes the fault location based on GAF-CNN. Section 5 shows the results of the fault-ranging experiments.  [18,19]. The method is heavily dependent on the first traveling wave head [20]. Figure 1 provides the direction traveling waves of the voltage-mode component. As can be seen in Figure 1, the duration time of the initial wave head is extremely short. The traveling wave location methods are easily affected by the high resistance earth fault, noise, and other harsh grounding conditions.   As the traveling wave abruptly changes at the junction point of the hybrid line consisting of cable and aerial wire, we proposed an approach that does not consider the location of the traveling wave head and traveling wave speed. The proposed method only considers the traveling wave per se of characteristics.

Phase-Mode Transformation Matrix of Three-Phase
System. Different phase-mode transformation matrices may affect the effectiveness of the fault location [21]. The diagram of the three-phase system is shown in Figure 3. _ I A , _ I B , and _ I C can denote the three-phase current phase quantity; m and n represent the head-end of the line and the end of the line, respectively; Z S and Z M represent the mutual impedance and the self-impedance, respectively.
For the three-phase system shown in Figure 3, there is a complex electromagnetic coupling between the conductors of each phase of the power system, which needs to be decoupled using the phase mode transformation method in order to simplify the analytical calculations as where ½u m represents the modal voltage vector, ½i m denotes the modal current vector, and T is the mode transformation matrix. The mode transformation matrix T and the inverse matrix T -1 can be defined as [22] T = 1 15 The mode transformation matrix T can transform the current vectors into a modal form, as shown in where _ I 0 , _ I 1 , and _ I 2 represent the current modulus components of 0, 1, and 2. Table 1 provides the current modulus components under all kinds of fault types. In Table 1, it is evident that the modulus values obtained by applying the phase-mode transformation matrix T are the nonnull value for any fault types. Hence, modulus 1 or modulus 2 can accomplish the fault analysis.

Signal-to-Image Conversion
3.1. Imaging Line-Mode Component. In order to obtain the 2D image of the traveling wave component, the Gramian angular field (GAF) [23,24] is used to encode mode 2 obtained by Equation (11). In the traditional Cartesian coordinate system, the time domain signal is a typical 1D signal. The x-axis direction can represent the time value, and the y-axis direction can represent the signal amplitude. The time-domain signal can be represented in a polar coordinate system. Given a time series X = fx 1 , x 2 , x 3 , x n g, the normalization method of ½−1, 1 can be used The normalized time series can be transformed by using the polar coordinate transformation. The magnitude can be calculated as the cosine value of the polar coordinate system, and the timestamp can be used as the radius. The process of polar coordinate conversion can be defined as where θ i represents the polar angle, t i denotes the timestamp of the i th time series, r i is the radius of the polar coordinate, andX is the rescaled time series in polar coordinates.
Since time series has been transformed within polar coordinates, it is easy to mine the correlation between different moment points by the angle of each moment point. The GAF can generate two types of images by using different equations [25,26]. Equation (7) defines the Gramian angular summation field (GASF), and Equation (9) defines the Gramian angular difference field (GADF).
where I is the unit vector. By redefining the vector innerproduct The main diagonal elements of the GAF can store the original values for all moments in the time series. The i th row and j th column of the matrix can represent the correlation coefficient between the i th moment and the j th moment. The GAF can effectively represent the time correlation between different data points. For an original time series with length n, the size of the obtained GAF matrix is n × n. If n is too big, the dimension of the obtained matrix is over large.
The paper used the piecewise aggregate approximation (PAA) to reduce the dimension of the original time series. PAA is to segment the long time series uniformly and then calculate the mean values of each segment. The mean values can become a new data reduction representation [27]. In the case of guaranteeing comprehensive information for the original time series, the length of the time series is reduced by the PAA. Figure 4 provides the procedure of signal-toimage conversion.
The method of the signal-image conversion has two major advantages as follows: (1) The relationship between the 1D signal and the 2D image is double-mapping relation, which does not lose any information of the 1D signal (2) It can maintain the time dependence of the signal. The texture information and color distribution of the 2D image can reflect the invisible information of the 1D signal 3.2. Spatial Domain Image Fusion. Figure 5 provides the density histograms of the outputs of the GASF matrix and GADF matrix. As can be seen in Figure 5, the distribution of the GASF is different from that of the GADF. Hence, in order to make full use of the fault information, the images generated by GASF and GADF can be fused by the weighted average fusion algorithm. The fusion process is shown in Figure 6. By integrating the complementary information, image fusion can reduce or inhibit the impact of the single, Journal of Sensors     cos(θ n + θ n ) cos(θ n + θ 1 ) sin(θ n -θ n ) sin(θ n -θ 1 )

GADF
Weighted average fusion 1 2 sin(θ n + θ 1 ) cos(θ n + θ 1 ) sin(θ n -θ n ) cos(θ n -θ n )   [28]. CNN can have significant potential advantages in both feature extraction and model fitting [29]. It has been applied in object detection, image processing, and natural language processing with notable success [30,31]. The basic structure of the CNN is mainly composed of the input layer, convolution layer, pooling layer, fully connected layer, and output layer. The fault location model based on GAF and CNN is shown in Figure 7. Table 2 provides the parameters of the fault location model based on GAF and CNN. In Table 2, "Valid" represents the valid padding, and "Same" denotes the same padding. In order to cope with the problem that the traditional singleended traveling wave ranging method is affected by wave speed and pseudo wave head, the paper proposes a single-ended traveling wave fault location method for distribution networks that combines traveling wave theory, GAF, and CNN. The method locates the fault location without considering the wave head and wave speed of the traveling waves.     Figure 8 provides the flowchart of the fault location. The step-by-step procedure can be described as follows: (1) The sampling frequency is 1 MHz. In a postfault distribution network, the phase voltage vectors of the head-end distribution line can be collected, which are converted to the mode vectors by the phasemode transformation (2) The line-mode component of mode 2 can be converted into the image by the Gramian angular summation field and the Gramian angular difference field  (4) The fusion images can be used as the input of the CNN. We establish a corresponding relationship between the fault information and feature information by convolution layer, activation layer, normalization layer, pooling layer, fully connected layer, and regression layer. The output of the CNN can achieve the fault location. Some parameters need to be set before training the CNN model: "Solver", adam; "InitialLearnRate", 0.001; "LearnRateDrop-Factor", 0.005; "LearnRateDropPeriod", 10; "L2Regularization", 0.004, "MaxEpochs", 15; and "MiniBatchSize", 10     Journal of Sensors training and iteration. It is simply impossible to complete the preparation of massive datasets by using manual simulation with multiple repetitions in the simulation process. MATLAB/SIMULINK has developed a rich operation mechanism for automation of the simulation process in order to relieve users from the burden of multiple identical repetitions of simulation so that users can realize model customization and simulation batching, and this paper selects MATLAB/SIMULINK for batching and automation of different types of fault simulation. We used the hybrid line consisting of cable and aerial wire to acquire data. The structure map of the hybrid line is shown in Figure 9, with a total line length of 20 km. The sampling frequency is 1 MHz. The simulations and deep learning have been performed using i7-10875H CPU, NVIDIA Geforce RTX 2060 with Max-Q Design, 64-bit operating system.

Experimental Results and Analysis
The zero and positive sequence parameters of the aerial wire are listed as  Table 3 provides the parameters of the training samples, including fault types, sample size, ground resistance, and fault close angle. Figure 10 provides two-dimensional image conversion diagrams of four kinds of fault distances. As seen from Figure 10, the 1D line-mode component can be mapped into the color, point, line, and other characteristic parameters of the 2D image. In the fusion images, texture information and color distribution show different characteristic parameters with different fault distances. The traditional analytical method needs to detect the traveling wave head and needs to know the traveling wave speed. But both are difficult to identify in the traveling wave. The mapping of the GAF coding method on the time series can possess uniqueness. The line-mode component can be encoded by GAF, which can produce the fault graph of obvious characteristics. Then, the fault distance can be obtained by regression prediction of fault feature maps using the convolution neural network.  Figure 11 provides the root mean squared error (RMSE) and loss curve of the CNN model. As seen from Figure 11, RMSE and loss values can decrease rapidly with increasing training steps, and all finally tend to zero, indicating that the fault location model has fast convergence speed, high stability, and high prediction precision. The model has a good training process. Figure 12 provides the visualization of the features of each layer of the convolution. In extracting image features, each convolutional kernel is equipment to a feature extractor. Different levels of convolution kernels can extract different feature information. Different layers have different abilities to learn characteristics and can extract images with more abstract features. The anterior convolution kernel in CNN can extract the bottommost image features, such as color features and texture features. As the number of the convolutional layers increases continuously, the range of feature images becomes smaller. When the convolution layers are accumulated continuously, the convolutional kernels gradually start to extract the abstract features that are more complicated and have a higher identification degree. Table 4 provides fault location results under different fault conditions. From Table 4, it is easy to know that the test results have higher judging accuracy. The range error is relatively small. This method can fully utilize the convolution neural network to extract the features of the image generated by blending the GASF matrix and GADF matrix. This method avoids the problematic points in the wave head identification and traveling-wave velocity estimation. Compared to traditional traveling wave distance measurement, the proposed method has some advantages in the ranging accuracy. Figure 13 provides the evaluation indexes for three input methods under 500 testing samples. From Figure 13, it is easy to know that the explained variance score (EVS) and decision coefficient (DC) are near 1, indicating that three input methods all have good fitting results. However, the root means square error (RMSE) and mean absolute error (MAE) of image fusion is the smallest, indicating that image fusion-CNN has the preferable fault location effect.

Result Analysis of Testing Samples.
Different phase-mode transformation matrixes also influence the effect of fault distance measurement. Table 5 provides the results of fault location under different phase-

Parts of Conv1
Parts of Conv2 Parts of Conv3

Parts of Conv4
Parts of Conv5   Table 5, it is evident that the Clarke transformation has a poor fault location effect when the system occurs in the BG fault. The reason is that the mode 2 component of the Clarke transformation always equals zero when the system occurs in the BG fault. However, the phase-mode transformation matrix used in this paper still has a good effect on the fault location under different fault conditions.
The BP neural network, support vector machine (SVM), and extreme learning machine (ELM) are added for testing experiments. The test samples are 500 fusion images. Figure 14 provides the evaluation results of fault location under the different diagnosis methods. The RMSE and MAE of this paper are the smallest in different diagnosis methods. From Figure 14, it can be seen that this method can adaptively extract feature vectors for fault range finding.
In order to verify the effectiveness of the proposed method, three different kinds of fault location methods are compared. The fault range finding results are shown in Table 6. Compared with other methods, the proposed method has the smallest distance-measuring error. This is because the proposed method does not require detection of the wave head and speed of the traveling wave. This method features intelligence and does not require manual mining of fault features.

Conclusions
In this paper, a new fault distance measuring in the nonsolid-earthed network based on spatial domain image fusion and convolution neural network is proposed and verified: