Arc fault detection method based on improved grey wolf algorithm and probability neural network
By improving the position update strategy and dynamic adaptive step size weight of the Grey Wolf optimization algorithm, a probabilistic neural network model was built, which solved the efficiency and accuracy problems of the Grey Wolf algorithm in arc fault detection, realized efficient and reliable arc fault detection, and improved the safety of the power system.
Patent Information
- Application Number
- CN202310365153.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-04-07
- Publication Date
- 2026-02-13
- Estimated Expiration
- 2043-04-07
AI Technical Summary
The existing gray wolf optimization algorithm suffers from low solution efficiency and unsatisfactory optimization results when optimizing probabilistic neural network parameters, resulting in insufficient accuracy in arc fault detection and failing to meet the high standards required by power systems.
An improved position update strategy for the Grey Wolf optimization algorithm is proposed. By using dynamic adaptive step size weights and nonlinearly decreasing control factors, the global search capability is enhanced. A probabilistic neural network model based on the improved Grey Wolf algorithm is constructed, and the smoothing factor is optimized to improve the accuracy of arc fault detection.
It achieves stable, reliable, and efficient identification of arc faults, with a detection accuracy rate of 97.7%, significantly improving the safety of power systems and the reliability of fault diagnosis.
Smart Images
Figure CN116383630B_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the field of arc fault diagnosis, and particularly to an arc fault detection method based on an improved grey wolf algorithm and a probabilistic neural network. BACKGROUND
[0002] With the gradual expansion of people's demand for electricity, higher standards for the stability of household power lines have been proposed. Under the current background of rapid development of artificial intelligence, the normal and safe operation of the power system oriented by intelligence is the basic premise, and the safe operation of the power system cannot be separated from high-quality fault diagnosis work; and the types of household loads are becoming more and more complex, and the frequency of failure in daily use is also increasing. The aging and damage of the insulation layer can cause the occurrence of arc faults, and arc faults often cause equipment to burn or even cause fire accidents. Rapid and accurate diagnosis can effectively ensure the safe operation of the power supply system. Because in the circuit where the arc fault occurs, the size of the current does not change significantly compared with the current during normal operation, and the waveform characteristics are similar to the current waveform characteristics in the circuit connected with nonlinear loads to some extent, the conventional line protection devices such as circuit breakers and residual current protectors cannot accurately diagnose them, so it has become a hot and difficult point of arc detection technology.
[0003] At present, artificial neural network technology is widely used in fault diagnosis of power systems because it can produce memory and store data in the database accordingly, providing a reference for subsequent work, greatly reducing manpower and resources. Artificial neural network technology includes BP neural network, convolutional neural network, probabilistic neural network, etc. The structure of BP neural network and convolutional neural network is complex, and the convergence speed is not ideal. In comparison, the probabilistic neural network is more suitable for solving fault diagnosis problems due to its simple principle and fast convergence speed. The performance of the probabilistic neural network depends on the value of the smoothing factor parameter inside the model. Many scholars use particle swarm optimization algorithm, genetic algorithm, and seagull optimization algorithm to optimize the parameters of the probabilistic neural network. However, these optimization algorithms still have the problem of being difficult to jump out of the local optimum, resulting in the solving accuracy of the optimized probabilistic neural network still cannot meet the requirements of arc fault detection, and the ideal probabilistic neural network model parameters cannot be obtained. Compared with traditional optimization algorithms, the grey wolf optimization algorithm has obvious advantages in convergence ability. However, when solving complex problems, it has the problems of low solving efficiency and unsatisfactory optimization effect, and is not suitable for directly optimizing the parameters of the probabilistic neural network. Therefore, the grey wolf optimization algorithm needs to be improved appropriately to overcome the above problems. SUMMARY
[0004] The purpose of the present application is to overcome the above-mentioned deficiencies of the prior art and provide an arc fault detection method based on an improved grey wolf algorithm probability neural network, which can stably, reliably and efficiently identify arc fault data.
[0005] The technical solution of the present application is: an arc fault detection method based on an improved grey wolf algorithm probability neural network, comprising the following steps:
[0006] Step S1, obtaining current signal data sets of different load combinations under normal operation and arc fault in household power lines.
[0007] Step S2, preprocessing the obtained current signal data set.
[0008] Step S3, based on the improved grey wolf optimization algorithm, the position parameters of the grey wolves in the wolf pack are optimized to obtain the final position parameters of the optimal wolf.
[0009] Step S4, using the probability neural network model built based on the improved grey wolf optimization algorithm: taking the final position parameters of the optimal wolf obtained based on the improved grey wolf optimization algorithm as the smoothing factor of the probability neural network, building the probability neural network model.
[0010] Step S5, obtaining real-time current signal data during operation in the line, after processing the real-time current signal data according to step S2, inputting the real-time current signal data into the input layer of the probability neural network model built in step S4, and calculating to obtain the classification result of fault diagnosis.
[0011] The further technical solution of the present application is: the current signal data set uses existing databases to collect normal operation and arc fault current signal data, or uses a special data generation machine to generate normal operation and arc fault current signal data.
[0012] The further technical solution of the present application is: the preprocessing includes normalization processing, and extracting time domain characteristic values, frequency domain characteristic values and energy characteristic values, and generating a multi-dimensional feature vector.
[0013] The further technical solution of the present application is: the time domain characteristic values include waveform indicators, peak value indicators, pulse indicators, kurtosis indicators and margin indicators; the frequency domain characteristic values include center of gravity frequency, frequency variance and mean square frequency.
[0014] The further technical solution of the present application is: the parameter optimization of the position parameters of the grey wolves in the wolf pack based on the improved grey wolf optimization algorithm is specifically,
[0015] S31, initializing the wolf pack position, the initial wolf pack position is generated according to the following formula:
[0016] Xi = X L + r1 x (X U - X L ) (1)
[0017] In the formula, X i is the initial wolf pack position, r1 is a random number between [0, 1], X L is the lower limit value of the wolf pack position, which is 0; X U is the upper limit value of the wolf pack position, which is 1.
[0018] S32, calculate the fitness value of each wolf, and the fitness function is calculated as follows:
[0019]
[0020] Where n tp is the number of correct detections in the data with arc fault; n fn is the number of misjudgments in the data with arc fault; n tn is the number of correct diagnoses in the data with normal operation; n fp is the number of misjudgments as arc fault in the data with normal operation.
[0021] S33, calculate the wolf pack position in the hunting process, and the position change of each individual is calculated as follows:
[0022]
[0023] In the formula, t is the current iteration number; T is the maximum iteration number; is the position of gray wolf i in the tth iteration; is the current position of the prey; d is the dimension of the feature vector; ρ is the step weight, ρ max and ρ min are the maximum and minimum values of the step weight, respectively; θ(t) is the step dynamic factor, is the predicted prey position at the t-1th iteration, X(t) is the actual position of the prey at the tth iteration; A and C are coefficients, where r2 is a number between [0, 1]; a is a control factor.
[0024] S34, determine how the gray wolf individual moves to the prey by the distance between the wolf individual ω and the three types of leading wolves, and the calculation formula is as follows:
[0025]
[0026] In the formula, and are the update step lengths of the ith ω wolf towards the gray wolf α, the gray wolf β, and the gray wolf δ at the tth iteration, respectively, and respectively, the position of grey wolf alpha, grey wolf beta, grey wolf delta in this iteration; A1, A2, A3 and C1, C2, C3 are the coefficients generated in this iteration; is the updated position of the i-th omega wolf in the t+1 iteration.
[0027] S35, if the maximum number of iterations is reached, the parameter optimization is ended, and the final position parameter of the optimal wolf is output; otherwise, turn to S33, and continue the iteration optimization using the updated position parameter of the grey wolf.
[0028] The further technical solution of the present application is that the decay rate of the control factor is small in the early iteration, and the decay rate is large in the late iteration.
[0029] The further technical solution of the present application is that the probabilistic neural network model is divided into four layers, which are input layer, hidden layer, summation layer and output layer; the input layer receives data and transmits it to the hidden layer, the hidden layer calculates the matching degree of the feature vector and the class of the training sample, and sends the result to the summation layer after completing the matching degree calculation; the summation layer obtains the estimated probability density function of the class by weighted average of the result; the output layer outputs the class with the highest estimated probability as the classification result.
[0030] The further technical solution of the present application is that the matching degree calculation formula is as follows:
[0031]
[0032] wherein, Φ ij is the matching degree determined by the vector x input to the hidden layer through the neuron j of the i-th class in the hidden layer; i=1, 2, …, M, M represents the number of classes in the training sample; x ij is the j-th center of the i-th sample, the value of j is the same as the number of training samples; and sigma is a smoothing factor.
[0033] Compared with the prior art, the present application has the following characteristics:
[0034] (1) The present application uses a control factor to perform nonlinear decrease, so as to jump out of local optimum and improve the global search ability.
[0035] (2) The present application uses a dynamic adaptive step weight for the position update strategy, which not only enhances the flexibility of the algorithm, but also highlights the leading advantage of the optimal wolf.
[0036] (3) The present application uses an improved grey wolf algorithm to optimize the smoothing factor, and builds a probabilistic neural network model, so that the data classification effect is good and the accuracy is higher.
[0037] The detailed structure of the present application is further described below in combination with the drawings and specific embodiments. BRIEF DESCRIPTION OF DRAWINGS
[0038] FIG. 1 is a flowchart of the arc fault detection method of the present application; Figure 1
[0039] FIG. 2 is a comparison graph of the attenuation curve of the control factor a of the traditional grey wolf algorithm and the improved grey wolf algorithm; Figure 2 FIG. 3 is a comparison graph of the iteration number and step weight change in the iteration process of the value strategy of the present application, the linear attenuation value strategy and the cosine attenuation value strategy;
[0040] Figure 3 FIG. 4 is a comparison graph of the evaluation index of the present application, PNN and GWO-PNN;
[0041] FIG. 5 is a comparison graph of the iteration number and fitness value between the present application, PNN and GWO-PNN; Figure 4 FIG. 6 is a comparison graph of the evaluation index of the present application, PNN and GWO-PNN.
[0042] Figure 5 FIG. 7 is a comparison graph of the evaluation index of the present application, PNN and GWO-PNN.
[0043] FIG. 8 is a comparison graph of the evaluation index of the present application, PNN and GWO-PNN. Figure 6 FIG. 9 is a comparison graph of the evaluation index of the present application, PNN and GWO-PNN.
[0044] Embodiment one, as shown in FIG. 1, the arc fault detection method of the probability neural network based on the improved grey wolf algorithm, specifically includes the following steps: Figures 1-5 Step S1, obtaining the current signal data set of different load combinations under the two conditions of normal operation and arc fault in the household power line: collecting normal operation and arc fault current signal data by using existing database, or generating normal operation and arc fault current signal data by using a special data generator.
[0045] Step S2, preprocessing the obtained current signal data set: mainly including normalization processing, and extracting its time domain characteristic value, frequency domain characteristic value and energy characteristic value, and generating a multi-dimensional feature vector. The time domain characteristic value includes waveform index, peak value index, pulse index, kurtosis index and margin index. The frequency domain characteristic value includes center frequency, frequency variance and mean square frequency.
[0046] Wherein the normalization processing, time domain characteristic value, frequency domain characteristic value and energy characteristic value extraction are all prior art, which will not be described here.
[0047] Step S3, based on the improved grey wolf optimization algorithm, the position parameters of the grey wolves in the wolf group are optimized.
[0048] Step S4, the optimized position parameters of the grey wolves in the wolf group are used as the initial value of the probability neural network, and the probability neural network is trained to obtain the probability neural network model.
[0049] S31, the wolf pack position is initialized, the initial wolf pack position is generated according to the following formula:
[0050] X i = X L + r1x (X U - X L ) (1)
[0051] In the formula, X i is the initial wolf pack position, r1 is a random number between [0, 1], X L is the lower limit value of the wolf pack position, which is 0; X U is the upper limit value of the wolf pack position, which is 1.
[0052] S32, the fitness value of each wolf is calculated, and the fitness function is calculated as follows:
[0053]
[0054] Where n tp is the number of correct detections in the data of arc fault; n fn is the number of misjudgments in the data of arc fault; n tn is the number of correct diagnoses in the data of normal operation; n fp is the number of misjudgments as arc fault in the data of normal operation, and the number of gray wolf population in the embodiment is set to 30. According to the fitness value from large to small, the gray wolf population is divided into α, β, δ and ω a total of 4 levels.
[0055] S33, the wolf pack position in the hunting process is calculated, and the position change of each individual is calculated as follows:
[0056]
[0057] In the formula, t is the current iteration number; T is the maximum iteration number; is the position of gray wolf i in the tth iteration; is the current position of the prey; d is the dimension of the feature vector; ρ is the step weight, ρ max and ρ min are the maximum and minimum values of the step weight, respectively; θ (t) is the step dynamic factor, X(t) is the actual position of the prey at the tth iteration, and here the position of the wolf with the minimum movement is approximated as the actual position of the prey at the current iteration, A and C are coefficients, where r2 is a number with a value between [0, 1]; a is a control factor, which is set to have a smaller decay rate in the early iterations, and the value of A fluctuates greatly, so that the gray wolf group has a wider search range, which is beneficial to global search and helps to jump out of the local optimum; in the later iterations, the decay rate is larger, so as to improve the optimization efficiency and convergence speed of the algorithm, which is beneficial to quickly obtain the optimal solution. In the embodiment, the maximum iteration number T is set to 70, and the values of p and a are 1 and 0.4 respectively. The attenuation curves of the control factors a of the traditional gray wolf algorithm and the improved gray wolf algorithm are shown in the attached max and the value of p min is 0.4. The comparison of the attenuation curves of the control factors a of the traditional gray wolf algorithm and the improved gray wolf algorithm is shown in the attached Figure 2 .
[0058] In S34, the movement of the gray wolf individual to the prey is determined by the distance between the wolf individual and the three types of leader wolves, and the calculation formula is as follows:
[0059]
[0060] wherein, and are the update step lengths of the ith ω wolf towards the gray wolf a, the gray wolf β, and the gray wolf δ at the tth iteration, and are the positions of the gray wolf a, the gray wolf β, and the gray wolf δ at the current iteration; A1, A2, A3 and C1, C2, C3 are the coefficients generated at the current iteration; is the updated position of the ith ω wolf at the t+1th iteration.
[0061] In S35, it is judged whether the maximum iteration number is reached. If yes, the parameter optimization is ended, and the final position parameter of the optimal wolf is output; otherwise, the iteration optimization is continued by using the updated position parameter of the gray wolf in S33.
[0062] In step S4, a probability neural network model is built based on the improved gray wolf optimization algorithm: the final position parameter of the optimal wolf obtained based on the improved gray wolf optimization algorithm is used as a smoothing factor of the probability neural network, and the probability neural network model is built.
[0063] The probability neural network model has four layers, namely an input layer, a hidden layer, a summation layer, and an output layer. The input layer receives data and transmits it to the hidden layer; the hidden layer calculates the matching degree of the feature vector and the class of the training sample, and the matching degree calculation formula is as shown in the following formula:
[0064]
[0065] wherein, Φij The matching degree of the vector x input to the hidden layer, determined by the neuron j of the i-th class in the hidden layer; i = 1, 2, ..., M, where M represents the number of classes in the training samples, and here M = 2, M = 1 and M = 2 represent the two classes, normal operation and arc fault, respectively; x ij σ is the j-th center of the i-th class of samples, and the value of j is the same as the number of training samples; σ is the smoothing factor, which is the final position parameter of the optimal wolf obtained in step S3. It plays a crucial role in the performance of the probabilistic neural network model and can reflect the classification accuracy of the probabilistic neural network model.
[0066] After the matching degree calculation is completed, the result is sent to the summation layer; the summation layer performs a weighted average of the results to obtain the estimated probability density function of the category; the output layer outputs the category with the highest estimated probability as the classification result.
[0067] Step S5: Obtain real-time current signal data during operation of the line. After processing the real-time current signal data in step S2, input it into the input layer of the probabilistic neural network model built in step S4, and calculate the classification result of fault diagnosis.
[0068] As attached Figure 3 As shown in the figure, the arc fault detection method based on the improved gray wolf algorithm of this embodiment is used to detect faults on a validation set of 176 sets of current signal data. The comparison figure shows the identification results of the predicted values and the true values. 0 represents no arc fault, and 1 represents the presence of an arc fault. (See attached figure.) Figure 3 It can be seen that the probabilistic neural network arc fault detection method based on the improved gray wolf algorithm in this embodiment has an identification rate of 97.7% for arc faults in the validation set, with false positives of 2.3% and 3.4%, respectively, which is a very low probability of false positives.
[0069] As attached Figure 4 As shown, to illustrate the superiority of the step size weighting strategy proposed in this embodiment compared to other strategies, a comparison curve of the number of iterations versus the change in step size weights is plotted between the step size weighting strategy proposed in this embodiment, the linear decay strategy, and the cosine decay strategy. (See attached figure.) Figure 4As can be seen, compared to other methods, the step size weighting strategy proposed in this embodiment maintains the maximum value in the early stages of iteration, allowing the wolf pack to perform a longer global search and a larger search range. This, to some extent, avoids missing parts of the search space and prematurely falling into local optima. In the middle of the iteration, the step size weight is set to θ(t), which is related to the predicted prey position and the actual position of the iterated prey. The curve of the step size weighting strategy proposed in this embodiment decays faster, and the value of θ(t) reflects the current prediction accuracy; the smaller the value, the higher the accuracy. When the iteration reaches forty times, the curve of the step size weighting strategy proposed in this embodiment stabilizes at the minimum value.
[0070] When using the step-weighting strategy proposed in this embodiment, the step-weighting in the mid-cycle is correlated with both the predicted prey location and the actual location of the iterated prey, thereby enhancing the alpha wolf's leadership and enabling the pack to conduct a precise search within a small area. In contrast, the other two strategies only consider the current iteration number and cannot be adjusted based on predictions. This results in a sacrifice of some search range in the early stages and an inability to fully leverage the alpha wolf's leadership in the later stages, leading to slow convergence.
[0071] As attached Figure 5 As shown, to further verify the effectiveness and superiority of different detection methods for arc fault identification, curves relating the number of iterations to the fitness value were constructed for three models: a simple probabilistic neural network (PNN), a probabilistic neural network model GWO-PNN optimized based on the traditional gray wolf algorithm, and a probabilistic neural network IGWO-PNN based on the improved gray wolf algorithm. The parameter conditions were as follows: the arc fault detection model was trained with the same training parameters and historical data, with 70 iterations and a learning rate of 0.001. (See attached figure...) Figure 5 As can be seen, compared with the probabilistic neural network model GWO-PNN based on the simple probabilistic neural network PNN and the probabilistic neural network model GWO-PNN based on the traditional gray wolf algorithm, the probabilistic neural network IGWO-PNN based on the improved gray wolf algorithm reaches the maximum fitness value earlier and the maximum fitness value is higher. Therefore, it can be concluded that it has a faster convergence speed and a more accurate classification effect.
[0072] As attached Figure 6 As shown, a comparison chart of evaluation metrics is presented for the methods using simple probabilistic neural networks (PNN), probabilistic neural network models optimized based on the traditional gray wolf algorithm (GWO-PNN), and probabilistic neural networks (IGWO-PNN) based on the improved gray wolf algorithm. (See attached chart.) Figure 6As can be seen, the arc fault detection accuracy of the IGWO-PNN method based on the improved grey wolf algorithm is 97.159%, the recall rate is 97.727%, and the precision rate is 96.629%, all of which are higher than those of the other two arc fault detection methods. Therefore, the IGWO-PNN method based on the improved grey wolf algorithm has strong ability to identify arc faults.
[0073] The above comparison results show the effectiveness of the IGWO-PNN method based on the improved grey wolf algorithm and the necessity of using the improved grey wolf algorithm to determine the value of the smoothing factor. The fault detection model provides effective diagnosis for arc fault detection, which is conducive to the follow-up prevention and control of arc faults.
Claims
1. A probabilistic neural network arc fault detection method based on an improved gray wolf algorithm, characterized in that, Includes the following steps: Step S1: Obtain current signal datasets for different load combinations under two conditions: normal operation and arc fault in household electrical circuits. Step S2: Preprocess the acquired current signal dataset; Step S3: Based on the improved gray wolf optimization algorithm, the position parameters of the gray wolves in the wolf pack are optimized to obtain the final position parameters of the optimal wolf. Specifically, the optimization of the position parameters of the gray wolves in the wolf pack involves... S31, Initialize the wolf pack positions. The initial wolf pack positions are generated according to the following formula: X i =X L +r1×(X U -X L ) (1) In the formula, X i Let r1 be the initial wolf pack position, and X be a random number between [0,1]. L X is the lower bound of the wolf pack's location, taking a value of 0; U This is the upper bound of the wolf pack's location, and its value is 1. S32, calculate the fitness value of each wolf. The fitness function is calculated as follows: Where, n tp n represents the number of correctly detected arc faults from the data; fn n represents the number of data points where arc faults were misclassified. tn n represents the number of correctly diagnosed data points within the normally functioning data set. fp This represents the number of data points that were mistakenly identified as arc faults within normally operating data. S33, calculates the wolf pack's position during the hunting process, including the positional changes of each individual wolf, using the following formula: In the formula, t is the current iteration number; T is the maximum iteration number; Let i be the position of the gray wolf in the t-th iteration; d represents the current position of the prey; d is the dimension of the feature vector; ρ is the step size weight. max and ρ min θ(t) represents the maximum and minimum values of the step size weight, respectively; θ(t) is the step size dynamic factor. Let r2 be the predicted prey position at iteration t-1, and X(t) be the actual prey position at iteration t; A and C are coefficients, where r2 is a number between [0,1]; a is a control factor. S34, the movement of a gray wolf towards its prey is determined by the distance ω between an individual wolf and the three types of lead wolves. The calculation formula is as follows: In the formula, and Let be the update step size of the i-th ω wolf towards the gray wolves α, β, and δ in the t-th iteration, respectively. and These represent the positions of gray wolf α, gray wolf β, and gray wolf δ in this iteration; A1, A2, A3 and C1, C2, C3 are the coefficients generated in this iteration; Let be the updated position of the i-th ω wolf in the (t+1)-th iteration; S35: Determine if the maximum number of iterations has been reached. If it has, the parameter optimization ends and the final position parameters of the optimal wolf are output. Otherwise, go to S33 and continue iterative optimization using the updated position parameters of the gray wolf. Step S4: Using a probabilistic neural network model built on the improved gray wolf optimization algorithm: The final position parameters of the optimal wolf obtained based on the improved gray wolf optimization algorithm are used as the smoothing factor of the probabilistic neural network to build a probabilistic neural network model. Step S5: Obtain real-time current signal data during operation of the line. After processing the real-time current signal data in step S2, input it into the input layer of the probabilistic neural network model built in step S4, and calculate the classification result of fault diagnosis.
2. The probabilistic neural network arc fault detection method based on the improved gray wolf algorithm as described in claim 1, characterized in that: The current signal dataset is collected using existing databases to gather current signal data during normal operation and arc faults, or using a dedicated data generator to generate current signal data during normal operation and arc faults.
3. The probabilistic neural network arc fault detection method based on the improved gray wolf algorithm as described in claim 1, characterized in that: The preprocessing includes normalization, extraction of time-domain feature values, frequency-domain feature values, and energy feature values, and generation of multi-dimensional feature vectors.
4. The probabilistic neural network arc fault detection method based on the improved gray wolf algorithm as described in claim 3, characterized in that: The time-domain feature values include waveform index, peak index, impulse index, kurtosis index, and margin index; the frequency-domain feature values include centroid frequency, frequency variance, and mean square frequency.
5. The probabilistic neural network arc fault detection method based on the improved gray wolf algorithm as described in claim 1, characterized in that: The control factor decays at a low rate in the early stages of the iteration and at a high rate in the later stages of the iteration.
6. The probabilistic neural network arc fault detection method based on the improved gray wolf algorithm as described in claim 1, characterized in that: The probabilistic neural network model consists of four layers: an input layer, a hidden layer, a summation layer, and an output layer. The input layer receives data and passes it to the hidden layer. The hidden layer calculates the matching degree between the feature vector and the training sample category. After completing the matching degree calculation, the result is sent to the summation layer. The summation layer performs a weighted average on the result to obtain the estimated probability density function of the category. The output layer outputs the category with the highest estimated probability as the classification result.
7. The probabilistic neural network arc fault detection method based on the improved gray wolf algorithm as described in claim 6, characterized in that: The formula for calculating the matching degree is as follows: Where, Φ ij The matching degree of the vector x input to the hidden layer determined by the neuron j of the i-th class in the hidden layer; i = 1, 2, ..., M, where M represents the number of classes in the training samples; x ij Let j be the j-th center of the i-th class of samples, where the value of j is the same as the number of training samples; σ is a smoothing factor.