## 3 Establishment of the Improved Electromagnetic-Thermal Coupling Model
### Prediction Function of Inductance Gradient
The specific process is shown in Figure 17:  
The specific computational steps of the algorithm are as follows:  
First, population initialization is performed and random individuals are generated, then, the gene editing part of the GA is run, which specifically includes:  
Figure 17: Proposed PSO-GA hybrid algorithm. (1) Selection operation  
According to the size of the fitness to select the better individuals to form a new population; the larger the fitness value, the higher the probability of being selected. (2) Crossover operation  
This refers to the genetic exchange of any two individuals in the population to form new individuals. (3) Mutation operation  
An individual is selected with a certain probability of carrying out one or more gene mutations and produce a new individual after the mutation. Next, it is determined whether a solution superior to the current population is produced, and if so, it is recorded. If not, the GA algorithm is rerun to obtain a better solution. After generating a better solution, the PSO algorithm section is run to record the local and global optimal solutions under the current iteration and then updates the velocity and position information of the current particles based on the above solutions, which ensures that the particles move in the optimal direction. After this, the particle information and iteration information are recorded. Moreover, the next generation of iterations is performed. The calculated parameter identification values of the improved model are obtained by the algorithm as shown in Table 3:  
To verify the established improved model, another ten sets of finite element calculated values are taken and compared with the predicted values of the improve model, and the relative error can be calculated by Equation (33). \[\left\{ \begin{array}{l}e_{\text{r}\text{r}}=\left|\frac{v_{\text{r}}-v_{ \text{p}}}{v_{\text{f}}}\right|\\ e_{\text{r}L}=\left|\frac{L_{\text{f}}-v_{\text{p}}}{L_{\text{f}}}\right|\end{array} \right. \tag{33}\]  
where, \(v_{\text{f}},L_{\text{f}}\) are the finite element calculated values, \(v_{\text{p}},L_{\text{p}}\) the predicted values of improve model, \(e_{\text{r}\text{r}}\) the relative error between \(v_{\text{f}}\) and \(v_{\text{p}}\), \(e_{\text{r}L}\) the relative error between \(v_{\text{f}}\) and \(v_{\text{p}}\). The relative error is calculated as shown in Tables 4 and 5: \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Number**} & \multicolumn{2}{c}{**Finite Element**} & \multicolumn{2}{c}{**Predicted Values of**} \\  & **Calculated Values** & **Improved Model** & **Relative Error** \\  & **(\(v_{\text{r}}\))/(m/s)** & **(\(v_{\text{p}}\))/(m/s)** & **Relative Error** \\ \hline
1 & 921.63 & 905.04 & 1.80 \\
2 & 1835.66 & 1851.38 & 0.86 \\
3 & 1943.88 & 1932.96 & 0.56 \\
4 & 2011.34 & 2035.08 & 1.18 \\
5 & 2249.60 & 2222.86 & 1.19 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Relative error analysis of the armature velocity. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Parameters**} & **Identification Values** & **Parameters** & **Identification Values** \\ \hline \(r_{1}\) & 0.0550 & \(a_{5}\) & \(-0.2760\) \\ \(r_{2}\) & \(-0.0300\) & \(a_{6}\) & 0.0968 \\ \(s_{0}\) & 0.9641 & \(b_{0}\) & 1.0034 \\ \(k_{1}\) & 2.0023 & \(b_{1}\) & \(-0.1835\) \\ \(k_{2}\) & 0.5013 & \(b_{2}\) & 0.7222 \\ \(a_{1}\) & 0.1350 & \(b_{3}\) & \(-0.7680\) \\ \(a_{2}\) & \(-0.0710\) & \(b_{4}\) & 0.6185 \\ \(a_{3}\) & 0.1702 & \(q_{2}\) & \(-0.0158\) \\ \(a_{4}\) & 0.2900 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter identification values of the improved model. It can be seen that \(e_{rv}\) and \(e_{rL}\) are controlled to within 3%. Therefore, it can be considered that the improved model established in this paper can accurately predict the armature velocity and inductance gradient.