## 4 Arc Fault Identification
The input used the feature vector \(\mathbf{x}\) including the frequencies \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\). The output was the classification result \(\mathbf{y}\). The status of output contained \(-1\) and \(1\). The \"\(-1\)\" represented the normal state and \"\(1\)\" represented arc fault state. 2. Construction of the training set \((\mathbf{x},\mathbf{y})\). From the experimental data of nine types of typical loads in different working states, three hundred and sixty samples were chosen for further processing. Two hundred and eightty samples of those samples were treated as the training samples and the remainder were treated as testing samples. The training set of recognizer was listed in Table 1.  
(3) Selection of LSSVM parameters. The RBF can be described as  
\[K\left(\mathbf{x}_{i},\mathbf{x}_{j}\right)=\exp\left(-\left\|\mathbf{x}_{i }-\mathbf{x}_{j}\right\|^{2}\Big{/}\sigma^{2}\right) \tag{24}\]  
where \(\sigma^{2}\) is the kernel parameter. At first, the initial kernel parameter \(\sigma^{2}\) and the penalty parameter \(C\) were selected. Then, the ten-fold cross-validation method was applied to optimize the parameters through the training set. According to the method, the training set was equally divided into ten subsets which were mutually disjoint. The entire method would be repeated ten times. Each time, one of the ten subsets was selected as a new test subset and the other nine subsets were placed together to form a new training subset. The nine subsets were used for training and the tenth subset was used for testing. After repeating ten times, the average error of the ten trials was computed. Finally, the optimal parameters were obtained when the error was the minimum [49]. \begin{table}
\begin{tabular}{c c c c c c} \hline Samples & \(\mathbf{1}\) & \(\mathbf{2}\) & \(\mathbf{3}\) & \(\mathbf{\ldots}\) & \(\mathbf{280}\) \\ \hline \(\mathbf{x}_{1}\) & \(0.1521\pi\) & \(0.3534\pi\) & \(0.6171\pi\) & \(\ldots\) & \(0.8051\pi\) \\ \(\mathbf{x}_{2}\) & \(0.2213\pi\) & \(0.2514\pi\) & \(0.3502\pi\) & \(\ldots\) & \(0.1586\pi\) \\ \(\mathbf{y}\) & \(-1\) & \(1\) & \(1\) & \(\ldots\) & \(-1\) \\ \hline \end{tabular}
\end{table} 
Table 1: The training set of recognizer. (4) Arc fault identification. The testing samples were input into the arc fault recognizer. Then the identification results were compared with the real results. Finally, the generalization ability of recognizer was evaluated based on error rate which could be calculated as  
\[e=\frac{\sum_{i=1}^{p}\left|y_{i}^{*}-y_{i}\right|}{2p}\times 100\% \tag{25}\]  
where \(p\) was the number of testing samples and \(y_{i}^{*}\) was the classification result. From the above procedures of arc fault identification, the arc fault recognizer was proposed. The kernel parameter \(\sigma^{2}=0.54\) and the penalty parameter \(C=7.1\) were obtained by the ten-fold cross-validation method. The identification results are shown in Figure 7 where \"*\" represents the normal states and \"\(\Box\)\" represents arc faults of the loads. The testing samples can be identified from the feature vector space. If the testing sample appears on the arc fault feature space in Figure 7, it means an arc fault has occurred in the circuit; otherwise, there is no arc fault. The identification results are the globally optimal solution in that RBF kernel function parameters. The optimal classification plane of arc faults and the normal states can be found in the high dimensional feature space. After completing sample experiments, the actual results and the classified results are listed in Table 2. The results show the average error rate of LSSVM is 2.6%. Hence, the arc fault identification rate is 97.4%. Since some weak arc fault signals are added to the testing samples, the frequency of the arc fault is very close to that of the normal state. Therefore, the LSSVM recognizer fails to classify the arc faults from the normal states. This limitation can be improved in future works. Figure 7: Arc fault identification results.