## 4 Available Capacity Modeling Based on Terminal Voltage Drop
### Analysis of the SVM  
SVM is a machine learning method for classification, regression, or other learning tasks. This analysis is based on structural risk minimization and it can minimize the empirical risk and confidence interval [44,45]. SVM is chosen for the excellent approximation and generalization capability and its demonstrated potential in the realm of nonlinear system identification. A typical SVM application included two steps:  
(1) Training the sample set data to obtain a model between the input and output data;  
(2) Using the model to predict the test set data.  
Supposed the training sample set is \(\{(x_{1},y_{1}),\ldots,(x_{l},y_{l})\}\). In which, \(x_{i}\in R^{n}\), that is, \(x_{i}\) is an n-dimensional feature vector and \(y_{i}\in R\). \(l\) is the sample number.  
When SVM is used for regression, its purpose is to get a regression model as Equation (6) to minimize the error between model outputs _f(x)_ and desired values y. \(\omega\) and \(b\) are the parameters to be determined. Here, \(\omega\) is also an n-dimensional feature vector, \(\omega^{T}\) denotes the transpose of the vector \(\omega\). b is the deviation and it is a scalar.  
\[f(x)=\omega^{T}x+b \tag{6}\]  
It supposes that the maximum deviation between the model output and the desired value we can tolerate is \(\varepsilon\) (that is, a positive constant). Then, when the estimation error of a sample is less than \(\varepsilon\), the estimate result of the sample is correct.  
According to the principle of structural risk minimization, Equation (7) is used to identify the parameters \(\omega\) and \(b\):  
\[\min_{\omega,b}\frac{1}{2}\
orm{\omega}^{2}+C\cdot\sum_{i=1}^{l}\ell_{ \varepsilon}(f(x_{i})-y_{i}) \tag{7}\]  
Figure 8: Changes in terminal voltage with different rest times.  
When it gets the solution \(a_{i}\) and \(a_{i}^{*}\) of Equation (15), substituting Equation (11) into (6), it can obtain the following Equation (17):  
\[f(x)=\sum_{i=1}^{l}{(a_{i}^{*}-a_{i})x_{i}^{T}x+b} \tag{17}\]  
Combining the solution \(a_{i}\) and \(a_{i}^{*}\) of Equation (15) with Equation (16), it can get the value of \(b\):  
\[b=y_{i}+\varepsilon-\sum_{j=1}^{l}{(a_{j}^{*}-a_{j})x_{j}^{T}x_{i}} \tag{18}\]  
When the linear characteristics of samples are not obvious, the nonlinear mapping function \(\phi(x)\) is adopted to map the low-dimensional feature space \(R^{n}\) to a high-dimensional space \(H\), that is, Hilbert space.  
Then, the x will be replaced with \(\phi(x)\) and the regression model (6) becomes the Equation (19):  
\[f(x)=\omega^{T}\cdot\phi(x)+b \tag{19}\]  
Consequently, Equation (17) becomes the following equation:  
\[\omega=\sum_{i=1}^{l}{(a_{i}^{*}-a_{i})\phi(x_{i})} \tag{20}\]  
Then the regression model (19) can be represented as:  
\[f(x)=\sum_{i=1}^{l}{(a_{i}^{*}-a_{i})k(x,x_{i})+b} \tag{21}\]  
In which, \(k(x_{i},x_{j})=\phi(x_{i})^{T}\cdot\phi(x_{j})\). It is a kernel function. Common kernel functions of the SVM are the linear kernel function, polynomial kernel function, radial basis kernel function (RBF) and two-layer perceptive kernel function. In this paper, we use the RBF, as shown in Equation (22):  
\[k(x_{i},x)=\exp(-g\|x-x_{i}\|^{2}),g>0 \tag{22}\]  
Prior to the SVM training, the penalty factor C and kernel function parameters \(g\) need to be predetermined.  
About the detailed solution process of SVM and the information on kernel function, please refer to the references [46].