## 3 Construction of the DT Model
### Modeling of Power Battery Based on Data-Driven
#### 3.3.2 SVM-Based Data-Driven Modeling  
From the previous section, it can be seen that the internal chemistry of the power cell is complex, with many influencing factors such as temperature uncertainty, aging conditions, and dynamic operating conditions, which is a highly nonlinear system [42]. In the face of this situation, a data-driven modeling approach based on SVM techniques is proposed to solve the current nonlinear problems and improve the prediction accuracy of the model. The temperature \(\tau\), voltage \(V\), SOC value of the battery at the previous moment \(S\), current \(I\), and internal resistance \(R\) were selected as input variables, and the voltage at the battery terminal \(y_{i}\) was the output. The sample points \(X_{i}\) at any moment \(i\) can be expressed as \(X_{i}=\left[S^{i-1},\tau^{i},V^{i},I^{i},R^{i}\right]\).  
By mapping each one with a nonlinear function \(\phi(x)\) to the high-dimensional feature space, and performing linear regression operations in the high-dimensional feature space, the effect of nonlinear regression in the original space can be obtained, and the regression estimation function \(f\) can be expressed as:  
\[f(x)=\omega\cdot\phi(x)+b \tag{10}\]  
where \(w\) is the normal vector of the optimal hyperplane, its dimension is the dimension of the feature space, and \(b\) is the threshold value.  
For a given nonlinear sample set \(T=\{(x_{1},y_{1}),(x_{2},y_{2}),\cdots,(x_{i},y_{i})\}\), \(x_{i}\in R\), \(y_{i}\in R\), the optimization problem is:  
\[\min_{\omega,b}\frac{1}{2}\|w\|^{2}+C\sum_{i=1}^{l}\left(\xi_{i}+ \xi_{i}^{*}\right) \tag{11}\] \[s.t.\left\{\begin{array}{l}y_{i}-\omega\cdot\phi(x_{i})-b\leq \xi_{i}+\varepsilon\\ \omega\cdot\phi(x_{i})-b-y_{i}\leq\xi_{i}^{*}+\varepsilon\\ \xi_{i},\xi_{i}^{*}\geq 0,i=1,2,\cdots,l\\ C>0\end{array}\right.\]  
where \(\xi_{i}\), \(\xi_{i}\)' is the relaxation factor, which represents the degree to which the group of samples does not satisfy the constraint. \(C\) represents the penalty factor, which indicates the  
Figure 6: Internal resistance and open circuit voltage characteristics of the battery: (**a**) Description of what is contained in the first panel; (**b**) Description of what is contained in the second panel.  
strength of the constraint that some of the samples can be allowed not to be completely divided by the hyperplane intact, and \(\epsilon\) is the insensitivity loss factor, which represents the error in the true value that can be allowed within some upper and lower range. The exact expression of the regression estimation function is obtained by solving it using the Lagrange multiplier method:  
\[f(x,\alpha_{i},\alpha_{i}^{*})=\sum_{i=1}^{1}(\alpha_{i}-\alpha_{i}^{*})\phi(x _{i})\phi(x_{j})+b \tag{12}\]  
where \(\alpha_{i},\alpha_{i}^{*}\geq 0\) is the Lagrangian multiplier. Using [PERSON]'s pairwise theory and introducing the kernel function \(k\big{(}x_{i},x_{j}\big{)}=\phi(x_{i})\cdot\phi\big{(}x_{j}\big{)}\), the final regression estimation function is shown in Equation (13):  
\[f(x,\alpha_{i},\alpha_{i}^{*})=\sum_{i=1}^{l}(\alpha_{i}-\alpha_{i}^{*})k\big{(} x_{i},x_{j}\big{)}+b \tag{13}\]  
The radial basis function is chosen as the kernel function because the radial basis kernel function performs better on highly nonlinear problems. The expression of the radial basis kernel function is:  
\[K\big{(}x_{i},x_{j}\big{)}=\exp\Bigl{(}-\gamma\big{|}x_{i}-x_{j}\big{|}^{2} \Bigr{)} \tag{14}\]  
where \(\gamma=\
icefrac{{1}}{{2\
u^{2}}}\) is the kernel function parameter.