## 4 Arc Fault Identification  
The SVM algorithm which is based on statistical learning theory has been successfully used to solve linear regression, pattern recognition and classification problems [40]. The LSSVM algorithm which has been developed by [PERSON] and [PERSON] [42] is a new deformation algorithm based on the traditional SVM. It can reduce computing complexity and guarantee the accuracy of data classification simultaneously. The frequencies \(\omega_{1}\) and \(\omega_{2}\) are selected as the input vector \(x\). The feature vector \(x\) is mapped from the original to the high dimensional feature space by nonlinear transformation \(f(x)\). Then, the optimal classification plane is found in the high dimensional feature space. According to the structural risk minimization principle, the constrained optimization problem corresponding to the original classification problem can be expressed as follows:  
\[ \begin{cases}\min\biggl{(}\frac{1}{2}\mathbf{V}^{\top}\mathbf{V}+\frac{C}{2}\sum\limits _{i=1}^{n}\xi_{i}^{2}\biggr{)}\\ \text{s.t. }\ y_{i}(\mathbf{V}^{\top}\ f(\mathbf{x}_{i})+b)=1-\xi_{i}\ \ (i=1,\ldots,\ m)\end{cases}  \tag{19}\]  
where \(\mathbf{V}\) is a weight vector, \(C\) is the penalty parameter, \(m\) is the number of samples, \(\xi_{i}\) is the relaxation coefficient and \(b\) is a bias term. The output result of classification is \(y_{i}= \begin{cases}1,\ \text{if}\ \ \mathbf{x}_{i}\in\text{class}\ 1\\ -1,\ \text{if}\ \ \mathbf{x}_{i}\in\text{class}\ 2\end{cases} \)[40,42,48]. In order to solve the constrained optimization problem, Lagrange multipliers \(\alpha_{i}\) is introduced. Equation (19) is transformed to the unconstrained objective function as  
\[L(\mathbf{V},\xi_{i},\alpha_{i},b)=\frac{1}{2}\mathbf{V}^{\top}\mathbf{V}+\frac{C}{2}\sum \limits_{i=1}^{n}\xi_{i}^{2}-\sum\limits_{i=1}^{n}\alpha_{i}\Bigl{\{}y_{i} \Bigl{[}\mathbf{V}^{\top}\ f(\mathbf{x}_{i})+b\Bigr{]}-1+\xi_{i}^{2}\Bigr{\}} \tag{20}\]  
According to the Karush-Kuhn-Tucker condition, let the derivative to \(\mathbf{V}\), \(b\), \(\alpha_{i}\), \(\xi_{i}\) of Equation (20) equal to 0, then the equation can be expressed as  
\[ \begin{cases}y_{i}(\mathbf{V}^{\top}f(\mathbf{x}_{i})+b)-1+\xi_{i}=0\\ \mathbf{V}=\sum\limits_{i=1}^{n}\alpha_{i}y_{i}f(\mathbf{x}_{i})\\ \sum\limits_{i=1}^{n}\alpha_{i}y_{i}=0\\ \alpha_{i}=C\xi_{i}^{2}\ \ (i=1,\ldots,m)\end{cases}  \tag{21}\]  
Furthermore, Equation (21) can be organized into a matrix form  
\[ \begin{pmatrix}\mathbf{\varDelta}&\mathbf{y}\\ -\mathbf{y}^{\top}&0\end{pmatrix}\begin{pmatrix}\mathbf{\varalpha}\\ b\end{pmatrix} = \begin{pmatrix}\mathbf{R}\\ 0\end{pmatrix}  \tag{22}\]where \(\mathbf{R}=\left(1,\cdots,1\right)^{\mathrm{T}},\mathbf{y}=\left(y_{1},\cdots,y_{m}\right)^ {\mathrm{T}},\alpha=\left(\alpha_{1},\cdots,\alpha_{m}\right)^{\mathrm{T}},\) element in the positive definite matrix \(\mathbf{\varOmega}\) is \(\Omega_{y}=y_{i}y_{j}f^{\mathrm{T}}(\mathbf{x}_{i})f(\mathbf{x}_{j})+\frac{\delta(i,j)} {C}=y_{i}y_{j}K(\mathbf{x}_{i},\mathbf{x}_{j})+\frac{\delta(i,j)}{C}\), \(\delta(i,j)= \begin{cases}1,&i=j=0\\ 0,&\text{others}\end{cases} .\) On the basis of Mercer condition, \(K(\mathbf{x}_{i},\mathbf{x}_{j})=f^{\mathrm{T}}(\mathbf{x}_{i})f(\mathbf{x}_{j})\) is defined as a kernel function. According to the features of arc fault signals, the radial basis function (RBF) is selected as the kernel function which is widely used in classification cases. Hence the optimization problem of Equation (19) can be solved by using Equation (22), where \(\mathbf{\alpha}\) and \(b\) can be solved through least square method [42]. The identification result of the loads states by LSSVM can be described as  
\[y^{*}(\mathbf{x})=\mathrm{sgn}\left(\sum_{i=1}^{m}\alpha_{i}y_{i}K (\mathbf{x},\mathbf{x}_{i})+b\right) \tag{23}\]  
The frequencies \(\omega_{1}\) and \(\omega_{2}\) are selected as the input vector \(\mathbf{x}\). The main procedures for arc fault identification are listed as follows:  
1. Selection of sample set for LSSVM.