In the single hidden layer feedforward networks (SLFNs), many parameters have to be adjusted because the weights of the neurons in the different layers are interdependent. In the past few decades, the gradient-based learning algorithm has been generally used in feedforward neural networks. The method is slow and easy to fall into the local minimum. Different from the traditional feedforward neural network, the extreme learning machine (ELM) has to adjust all the parameters of the feedforward neural network (for the structure of the ELM, see Figure 1). This method randomly gives the input weight and threshold value of the neuron weight and then calculates the output weight by solving the generalized inverse (Dac and Trung, 2023). It has been proved that random selection of the node parameters of the feedforward neural network of a single hidden layer does not affect the convergence ability of the neural network, which makes the network training speed of the ELM thousands of times higher than that of the traditional network (Ma et al., 2023). Therefore, we first let SLFN have one hidden node. For the feedforward neural network with a single hidden layer, its standard model is ∑ i = 1 N ∼ g ( w i ∙ x j + b i ) β i = o j , j = 1,2,3 , … , N , (1) where w i ∙ x j is the inner product of vector w i a n d   x j , N is the training sample, N ∼ is the number of hidden layer units, o j is the actual input value. g x is the activation function. Using sigmoid function as the activation function, if let Eq. 2 is the infinite approximation at 0 with existing W , β , b : min ∑ j = 1 N ∥ o j − t j ∥ , (2) ∑ i = 1 N ∼ g ( w i ∙ x j + b i ) β i = t j , j = 1,2,3 , … , N . (3) Eq. 3 can be written compactly as H β = T , (4) where T ∈ R N × m and β ∈ R N × m . H = W , b = h i j N × N ∼ (5) where h i j = g w i ∙ x j + b i , and H is the output matrix of the hidden layer of the neural network. When the number of hidden layer elements is the same as the total number of training samples, and the matrix is invertible, Eq. 4 has a unique solution. That is, Eq. 2 is satisfied. However, in many cases, when the number of hidden layer elements is far less than the total number of training samples, H is a rectangular matrix at this time and W , β , b does not necessarily exist and makes Eq. 2 hold, so it can be equivalent to finding the minimum value of Eq. 5 as the solution of Eq. 2. E = ∑ j = 1 N ∥ ∑ i = 1 N ∼ g w i ∙ x j + b i β i − t j ∥ 2 . (6)

If Eq. 5 is solved by the gradient learning method, it can be used to represent all parameters, and the iteration can be written as given in Eq. 7. θ k = θ k − 1 − η ∂ E θ ∂ θ , (7) where η is the learning efficiency. For feedforward neural networks, back-propagation neural networks are generally used. The neural network is a multilayer feedforward network trained according to error (You et al., 2021). The neural network includes input layer nodes, output layer nodes, and one or more hidden layer nodes. First, the input signal reaches the hidden layer node, where it passes through the excitation function and the output signal of the hidden layer node is then transmitted to the output node to finally get the output result. (Fukui and Kawakami, 1986). The learning process is that the neural network constantly changes the connection weight of its own network in the case of external input samples, so as to make the output result of the network closest to the expected output value (Xiong et al., 2018). If the output results differ greatly from the expected values, backpropagation can be carried out, and then the weights of each neuron can be modified again, and finally, a good classifier can be trained through continuous iteration. If the learning rate is too small, the learning speed is very slow. If the selection is too large, it is difficult to obtain network convergence. If Eq. 6 is a non-convex function, it is easy to fall into a local minimum by continuously iterating and adjusting parameters. Repeated iteration is not only time consuming but also easily falls into the situation of learning and fitting (Lcfcbvre, 2014).

In SLFNs, W and b are given at the beginning of the algorithm and can be arbitrarily specified. Then, H is calculated, while the value remains unchanged. In this way, only the parameter β that can be changed is left, and this shows that the given W and b do not affect the results.

When W and b are fixed, Eq. 4 is solved by replacing it with Eq. 8: ∥ H β ^ − T ∥ = min ⁡ ∥ H β − T ∥ . (8)

The least squares solution can be obtained by solving the Eq. 8.

In some large-scale projects, the learning process usually uses all the data and these learning times are very long. If new samples are added at this time, they have to learn together with the original data. In this way, it is a waste of time to relearn all the data. The online sequential learning neural network (OS-LNN) does not have to learn the previous data, but only has to add the new data to the learned network. However, the OS-LNN has to set network weights, and the training speed is also very slow. Although the training is completed, the online sequential learning–extreme learning machine (OS-ELM) can not only learn data one by one but also learn them batch by batch. The least squares solution of H β − T is β ^ = H * T , and we considered that R a n k   H = N ∼ is the number of hidden layer units, and H * is the left pseudo inverse of H, H * H = I N . H * = H T H − 1 H . (9) By substituting equation (9) into equation (8), we get β ^ = ( H T H − 1 H T . (10)

According to previous studies (Dac and Trung, 2023; Ma et al., 2023), the least square root is β 0 = K 0 − 1 H 0 T , where K 0 = H 0 T H 0 . When adding the new data (k + 1), N k + 1 = x i , t i i = ( ∑ j = 0 k N j ) + 1 ∑ j = 0 k + 1 N j , k ≥ 0 , N k + 1 is (k+1)^th data, K k + 1 = K k + H k + 1 T H k + 1 , and β k + 1 = β k + K k + 1 − 1 H k + 1 T T k + 1 − H k + 1 β k . (11) With the help of the Sherman-Morrison-Woodbury (SMW) equation, β k + 1 can be calculated by β k + 1 = β k + P k + 1 H k + 1 T T k + 1 − H k + 1 β k . (12) where P k + 1 = K k + 1 − 1 = P k − P k H k + 1 T I + H k + 1 P k H k + 1 T − 1 × H k + 1 P k .

When the OS-ELM faces the new data, it does not have to relearn the old data, which makes it faster than the other neural network methods. When selecting the network parameters, it becomes only necessary to determine the number of neural units in the hidden layer, which also reduces the dependence on the network layers.

The whale optimization algorithm is mainly divided into three steps: surround prey, spiral bubble net attack method, and randomly search for prey.

Surround prey: because the location of the target prey is unknown a priori, the WOA algorithm treats the location of the best candidate in the current whale group as the location of the target prey, and the other individuals in the whale group update the location according to the location of the best candidate: D = C ∙ X * t − X t , (13) X t + 1 = X * t − A ∙ D , (14) A = 2 a r − a , (15) C = 2 r , (16) where X is the position vector of the current solution; t is the number of iterations; A and C are coefficient vectors; X * is the position vector of the optimal solution in the current whale population. a decreases linearly from 2 to 0 as the number of iterations increases; r is an arbitrary vector between 0 and 1.

Spiral bubble net attack method: the WOA algorithm first calculates the distance between the individual whale and the target prey, and then simulates the spiral movement of humpback whales for hunting behavior: D ′ = X * t − X t , (17) X t + 1 = D ′ e b l ⁡ cos 2 π l + X * t , (18) where b is the constant coefficient defining the spiral shape, and l is a random number in the interval [−1, 1].

Randomly search for prey: in the process of predation, when A is greater than 1 or less than −1, the individuals in the whale group randomly select a prey with reference to each other’s position to improve the global search ability of the algorithm, namely, D = C X r a n d − X , (19) X t + 1 = X r a n d − A D , (20) where X r a n d is a randomly selected position vector for the current whale group.

The steps for WOA to optimize the ELM are as follows:

(1) Parameter initialization. Set the WOA parameters, namely, the number of whales, maximum iterations, variable dimensions, and upper and lower limits of variables;

The structure of the wind solar storage microgrid system is shown in Figure 2.

The internal line faults of the microgrid can be divided into single-phase ground short circuit (AG, BG, and CG), two-phase short circuit (AB, AC, and BC), two-phase ground short circuit (ABG, ACG, and BCG), three-phase short circuit (ABC), and three-phase ground short circuit (ABCG) faults.

When processing the signal, wavelet packet decomposition can decompose the low-frequency component and high-frequency component of the signal at the same time; higher the resolution, more detailed the decomposition and better the effect. The square-integrable function f(t) can be decomposed into a scaling function ϕ t and wavelet function φ t ; ϕ t is low frequency of f(t), φ t is high frequency of f(t), and their relationship includes δ 2 n t = 2 ∑ k ∈ z h k δ n 2 t − k δ 2 n + 1 t = 2 ∑ k ∈ z g k δ n 2 t − k , (21) where h k is low pass filter coefficient, and g k is high pass filter coefficient.

When n = 0 , δ 0 t = ϕ t , δ 1 t = φ t , the set of functions defined above { δ n t } (n = 0,1,2,...) is determined by δ 0 t = ϕ t determined wavelet packet. According to the fast algorithm of orthogonal wavelet transform, the recursive formula of wavelet packet coefficients can be obtained as follows: λ i + 1 2 i = ∑ k ∈ z h k − 2 t λ i j k λ i + 1 2 i + 1 = ∑ k ∈ z g k − 2 t λ i j k , (22) where λ i j k is the kth coefficient corresponding to the jth node of layer i after wavelet packet decomposition.

Wavelet packet energy entropy is a description of signal uncertainty, which can reflect the degree of random change of signals. When a fault occurs in the internal lines of the microgrid, because the voltage signal contains non-stationary signal components, the wavelet packet voltage reconstruction signal waveform will immediately fluctuate at the time of the fault. The wavelet packet decomposition and reconstruction technology can make accurate and rapid localization analysis of the voltage signal, which is reflected in the wavelet packet energy entropy, so the wavelet packet energy entropy can well reflect the fault characteristics of the voltage signal. According to information entropy theory, wavelet packet energy entropy can be defined as W P E E = − ∑ i = 1 L P X i , j log 2 ⁡ P X i , j , (23) where L is the original signal length; X _{i, j} is the jth decomposition signal of layer i; P (X _{i, j} ) is the frequency band energy probability density, and the mathematical expression is P X i , j = E i , j ∑ j = 1 2 i E i , j , (24) where E _i,j is the energy of the jth decomposed signal of the ith layer, defined as E i , j = ∑ k = 1 N λ i j k 2 , (25) where N is the length of the jth frequency band.

According to Figure 2, a microgrid model that includes wind turbine (10 kW), photovoltaic cell (10 kW), and battery (10 Ah) is built in the MATLAB simulink environment. The filter inductances L₁, L₂, and L₃ are 3.6e⁻³ H; the filter capacitors C₁, C₂, and C₃ are 200e⁻⁶ F; the electrical loads Load1, Load2, Load3, and Load4 are 10 kVA, 10 kVA, 5 kVA, and 15 kVA, respectively; the line resistance r is 0.175 Ω/km; and the line reactance x is 0.070 Ω/km. Simulate each type of line fault at 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, and 90% of the lines between P ₁ and P ₂ on the microgrid side. The db6 wavelet is selected as the wavelet base, and the simulated A-phase fault voltage is analyzed by three-layer wavelet packets to obtain 2³ sub signals in different frequency bands, which are reordered from low frequency to high frequency. The wavelet packet signal reconstruction is carried out for each frequency band, and a total of eight wavelet packet reconstruction signals are obtained. The energy entropy of the reconstructed signal of each wavelet packet is calculated, and a set of eigenvectors are constructed from the energy entropy of eight wavelet packets. By the same processing of phase B and phase C voltage signals, a eigenvector containing 24 wavelet packet energy entropy can be obtained X = x 1 , x 2 , … , x 24 T . Taking X = x 1 , x 2 , … , x 24 T as the input sample of the network, and the output sample of the network is T = t 1 , t 2 , … , t 4 T , t₁, t₂ and t₃, respectively, representing the line status of phase A, phase B, and phase C, and t₄ representing whether the fault phase is grounded when the line fails. When it is 0, it means there is no fault or the fault phase is not grounded at this time; when the output is 1, it means the fault phase is grounded.

The data samples at 10%, 20%, 40%, 60%, 80%, and 90% of the line positions are taken as training samples, and the data samples at 30%, 50%, and 70% are taken as test samples. The number of neurons in the input layer of ELM is 24, the number of neurons in the output layer is 4, the number of neurons in the hidden layer is determined to be 35 according to the trial and error method, the number of whales in WOA is 30, the maximum number of iterations is 200, and the variable dimension is 875.

As shown in Table 1, the method proposed in this article is nearly twice (mean error is 0.47%) as good as the method using B2 data only (mean error is 1.48%). The performance based on B2 only is better than that based on B1 (4.58%), it should be noted that the performance using combined B1B2 data (4.0%) is the poorest among the all methods.

We compared the results between actual and predictive in Figure 3, a total of samples of 209 are used for testing. Only 21 samples is not completely positioned accurately, that is, 10% samples have positioning errors, and the predicted values of the remaining test samples are identical to the actual values. The maximum error is that the actual value is 85% of the fault point that away from B1 and the predicted value is 92% of the fault point, with 7% error.

We did a comparative experiment, and the results are presented in Figure 4, when the B1 fault occurs, the data of B1 and B2 are used to locate the fault location. Through comparison, it can be found that the data on the B1 side should be accurate to the data on the B2 side. Table 2 divides the fault distance into three sections: 0%–33%, 33%–66%, and 66%–100%, respectively. Then, only the data of B1 is used to locate the fault, and it is found that the farther away from B1, the greater the test error. We demonstrated that it is more effective to use data close to the fault point. From the experimental results, it is shown that using the data points at the near end has better results. The farther the signal is transmitted, the greater is the resistance affected by various electrical components.

The experiment in Table 3 uses the SVM classifier, random forest (RF), and recurrent neural network (RNN) to test the method proposed in this article. Moreover, we choose to use libsvm and genetic the algorithm to optimize C and gamma parameters. The range of C is 0–1,000, the range of gamma is 0–1,000, the maximum evolutionary algebra is 200, and the maximum data of the population is 20. We can see that, in our methods, the selection of data based on the fault distance coupled with SVM can improve the accuracy of detection of fault in smart grids, with an average error of 0.86%, followed by RF with a mean error of 0.91% and B1B2-SVM with a mean error of 0.98%. Keeping the results of Figure 4 in mind when coupling with the SVM algorithm, the B1B2 method can obtain substantial accuracy with an average error of 0.98%, B1+SVM has the poorest performance with a mean error of 3.87%, B2+SVM has a moderate performance with a mean error of 1.61% (ranking of performance in Figure 4 is B2 > B1B2 > B1), which means that the artificial intelligence algorithm (SVM here) can substantially improve the detection of fault in the smart grid. the SVM can improve accuracy of fault detection more robustly than RF and RNN, for example, B1+RF has an error of 4.3%, followed by B1+RNN with 3.8%, and 1.9% and 2.0% for B2+RF and B2+RNN, respectively; 1.4% and 1.1% for B1B2+RF and B1B2+RNN, respectively. Fault distance–based data selection can substantially improve accuracy of fault detection on the basis of not only SVM but also RF and RNN, for example, 0.8%, 0.9%, and 1.2% for SVM, RF, and RNN, respectively, which means the fault distance–based data selection is useful for fault diagnoses.

After processing the data with Daubechies wavelet, the data output of the first layer is selected as the feature of the input classifier. The results obtained after positioning with ELM and with SVM, RF, and RNN are shown in Figures 5, 6, respectively. It can be seen that the method proposed in this article still has a good effect, with an average error of 0.66%. The method of B1+ELM after wavelet transform is the poorest with a mean error of 5.12%, followed by B1B2+ELM with of mean error of 4.93% and B2+ELM with a mean error of 3.72. Performance of the SVM, RF, and RNN methods after wavelet transform is better than that of the ELM method, and the SVM has the best performance. As we can see in Figure 6, B1+SVM, B2+SVM, or B1B2+SVM has smaller errors in fault detection, specifically, B1+SVM has the poorest performance with a mean error of 4.7%, followed by B2+SVM (1.98%), B1B2+SVM (1.14%), and our method, i.e., fault distance–based data selection + SVM (0.96%), which means that our fault distance–based method to detect faults is robust. The methods of both RF and RNN are poorer than SVM; B1+RF and B1+RNN are the poorest with a mean 5.04% and 3.9%, respectively. However, RF and RNN substantially improve the accuracy of fault detection, which is still poorer than SVM, for example, a mean error of 1.099% and 1.26% for the fault distance–based data selection + RNN and fault distance–based data selection + RF, respectively, while the mean error was 0.959% for our method, which means that improvement of SVM is robust.

In order to verify the superiority of the WOA-ELM diagnostic model and improve the recognition accuracy, the WOA-ELM diagnostic model is compared with the traditional ELM diagnostic model and optimized ELM by Quantum PSO, i.e., PSO-ELM. The results are shown in Figure 7. The error of WOA-ELM and WOA-ELM diagnostic faults are nearly 8.5% and 8.9%; separately, they are higher than the traditional ELM model with an error of 11%. We further compared the results of the three models using the mean square error (MSE), root mean square error (RMSE), and mean absolute error (MAE) (Table 4). The WOA-ELM model is largely better than the PSO-ELM and ELM models both in the training and testing data sets in fault detection. The WOA-ELM model uses WOA algorithm to optimize the input weight and hidden layer node threshold of the ELM, overcomes the shortcomings of random initialization of the input weight and hidden layer node threshold of the ELM, improves the global search ability of the network, and makes the network have better recognition accuracy.

Compared with the research results of others, the literature (Ji et al., 2022) proposes a basic network architecture design, using a simplified residual connection technology, using focal loss as the objective function for supervised training, adding a BatchNormalization layer to the network for optimization, reducing parameters based on ShuffleNet network, and improving accuracy on the basis of the attention mechanism, a process that can automatically determine the appropriate CNN architecture for fault diagnosis problems. Wang et al. (2011) proposed a new method for fault identification on the basis of parameter optimized variational mode decomposition (VMD) and convolutional neural network (CNN). Luo et al. (2014) proposed a real-time deep learning algorithm to classify and localize the faults that occurred in the system based on measured data. Luo et al. (2017) presented a method on the basis of gated graph neural network for automatic fault localization on distribution networks. The method aggregates problem data in a graph where the feeder topology is represented by the graph links and nodes attributes that can encapsulate any selected information such as operated devices, electrical characteristics, and measurements at the point. Kasun et al. (2013), Zhibin et al. (2019), and Xue and Dola (2022) proposed a multi-fault diagnosis model of distribution network on the basis of the fuzzy optimal convolutional neural network. In order to compare their results with ours, we used the RBF neural network and BP neural network to detect fault location in the smart grid and used indicators of MSE, RMSE, and MAE to evaluate accuracy; the results are provided in Table 5.

The three training errors of the WOA-ELM model are about one order of magnitude smaller than those of the BP neural network model and ELM model, while the three training errors of the RBF neural network model are 6–13 orders of magnitude smaller than those of the WOA-ELM model. The training effect of RBF neural network is the best. However, it can be seen from Table 6 that the three test errors of the WOA-ELM model are significantly smaller than those of the RBF neural network model. It shows that the RBF neural network model has a phenomenon of over-fitting, and its generalization ability is weak and cannot accurately identify the untrained fault types.

In substituting the data samples into the WOA-ELM fault diagnosis model for training and testing, the line fault diagnosis results of the test samples at 50% of the line position are shown in Table 6.

It can be seen from Table 6 that the absolute value of the error between the expected output and the actual output of the WOA-ELM fault diagnosis model does not exceed 0.015 at the most. The error is small, the accuracy is high, and the approximation ability is strong. It can accurately identify the fault types of microgrid lines.

In order to verify that WOA-ELM diagnostic model has better performance and higher recognition accuracy than the other models, the BP neural network, RBF neural network, and ELM were selected to establish the diagnostic models for comparative analysis. The expected output and actual output results of all training samples (72) are shown in Figure 8A. It can be seen from Figure 8A that the training error of the BP neural network model is large; the training results of the other three models can well approximate the expected output; the error between the actual value and expected value is small; and the training accuracy is high.

The expected output and actual output results of all test samples (36) are shown in Figure 8B. It can be seen from Figure 8B that the test results of the BP neural network model, RBF neural network model, and the non-optimized ELM model have large errors, while the test accuracy of the WOA-ELM model is the highest.