## 3 Establishment of the Improved Electromagnetic-Thermal Coupling Model
### Prediction Function of Inductance Gradient  
According to Figure 8, taking 0-0.24 ms as the rising stage and 0.24-0.6 ms as the declining stage, approximate descriptions of the inductance gradient for the two stages are as follows. \[\left\{ \begin{array}{l}L_{\rm up}(t)=k_{1}(k_{2}t)^{p_{1}}+q_{1}\\ L_{\rm down}(t)=p_{2}t+q_{2}\end{array} \right. \tag{25}\]  
where \(L_{\rm up}(t)\) is the inductance gradient in the rising stage, \(L_{\rm down}(t)\) the inductance gradient in the declining stage and \(k_{1}\), \(k_{2}\), \(p_{1}\), \(p_{2}\), \(q_{1}\), \(q_{2}\) are fitting coefficients. In the rising stage, the inductance gradient relations with the armature conductivity and track one at different time are obtained, which are shown in Figure 15. \begin{table}
\begin{tabular}{c c} \hline \hline
**Coefficients** & **Fitting Values** \\ \hline \(w_{0}\) & 0.66 \\ \(w_{1}\) & 7.21 \\ \(w_{2}\) & \(-52.36\) \\ \(w_{3}\) & 216.24 \\ \(w_{4}\) & \(-501.12\) \\ \(w_{5}\) & 599.22 \\ \(w_{6}\) & \(-286.49\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficient fitting values of \(k_{0}(t)\). It can be seen that, in the rising stage, at the same time point, the inductance gradient gradually increases with the armature conductivity \(\sigma_{\text{d}}\), and gradually decreases with the track conductivity \(\sigma_{\text{g}}\). Moreover, the ascending (descending) speed gradually slows down. In the declining stage, the inductance gradient relations with the armature conductivity and track one at different times are obtained, which are shown in Figure 16.  
It can be seen that, in the declining stage, at the same time point, the inductance gradient gradually increases with the armature conductivity \(\sigma_{\text{d}}\), and gradually decreases with the track conductivity \(\sigma_{\text{g}}\), in which the ascending (descending) speed gradually slows down. Figure 16: The inductance gradient relations with the conductivity at different times in the declining stage. (**a**) Inductance gradient with the armature conductivity \(\sigma_{\text{d}}\); (**b**) inductance gradient with the track conductivity \(\sigma_{\text{g}}\). Figure 15: The inductance gradient relations with the conductivity at different times in the rising stage. (**a**) Inductance gradient with the armature conductivity \(\sigma_{\text{d}}\); (**b**) inductance gradient with the track conductivity \(\sigma_{\text{g}}\). Discretizing Equation (30):  
\[\left\{ \begin{array}{l}a_{v}\big{(}\sigma_{\text{d}}(k+1),\sigma_{\text{g}}(k+1),k+1\big{)}\\ =\frac{v\big{(}\sigma_{\text{d}}(k+1),\sigma_{\text{g}}(k+1),k+1\big{)}-v\big{(} \sigma_{\text{d}}(k),\sigma_{\text{g}}(k),k\big{)}}{A_{\text{d}}}\\ a_{L}\big{(}\sigma_{\text{d}}(k+1),\sigma_{\text{g}}(k+1),k+1\big{)}\\ =\frac{l^{2}(k+1)}{2m_{0}}L\big{(}\sigma_{\text{d}}(k+1),\sigma_{\text{g}}(k+1 ),k+1\big{)}\end{array} \right. \tag{31}\]  
According to Equation (31), \(e_{a}\) is defined as follows:  
\[e_{a}=\frac{1}{T}\sum_{k=0}^{T}\left|\frac{a_{v}\big{(}\sigma_{\text{d}}(k), \sigma_{\text{g}}(k),k\big{)}-a_{L}\big{(}\sigma_{\text{d}}(k),\sigma_{\text{ g}}(k),k\big{)}}{a_{v}\big{(}\sigma_{\text{d}}(k),\sigma_{\text{g}}(k),k\big{)}}\right| \tag{32}\]  
\(e_{v},e_{L},e_{a}\) are regarded as the objective functions, and the parameters of the improved model are identified to make the objective functions minimum. Considering that the objective functions contain a first-order differential term and are nonlinear, the general identification method easily falls into the local optimal. Therefore, this paper improves on the basis of the particle swarm optimization algorithm, selects the optimal individual for each iteration through the gene sequence crossover and mutation of the genetic algorithm and saves the information and the PSO-GA hybrid algorithm is obtained, which is suitable for solving optimization problems with nonlinear high-dimensional functions, global convergence is good and it can effectively solve the problem of a population falling into the local optimal due to the calculated stability.