# Clustering Optimization of IPMSM for Electric Vehicles: Considering Inverter Control Strategy
## 3 System-Scale Optimization of Drive Systems
### Optimization Model for Motor Level
#### 3.1.1 Semianalytical Torque Model and Efficacy Model of IPMSMs  
For each RP with a specific torque and speed, motor performance is evaluated by semianalytical models based on FEA results to balance accuracy and computation cost. Considering the saturation, cross coupling, and spatial harmonics of IPMSMs, the torque \(T\) is derived through the co-energy method [26] as follows:  
\[ \begin{array}{c}T\ \ =\frac{\partial W_{\text{m}}^{m}(i_{d}i_{d},\theta)}{ \partial\theta}=i_{\text{s}}\frac{\partial A(i_{d}j_{d}i_{d})}{\partial\theta} -\frac{\partial W_{\text{m}}(i_{d}j_{d}i_{d})}{\partial\theta}\\ \ \ =\frac{3\pi}{T_{1}}(\lambda_{d}i_{d}-\lambda_{3}i_{d})+\frac{3}{2}(i_{d}\frac{ \lambda_{3}}{\lambda_{3}}+i_{d}\frac{\lambda_{3}}{\lambda_{3}})-\frac{\partial W _{\text{m}}(i_{d}j_{d})}{\partial\theta}}{i_{3}}\ \,\end{array}  \tag{6}\]  
where the subscribed d and q represent variables under the d-axis and q-axis. Figure 3: Parameterized model of the investigated IPMSM. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(D_{\text{so}}\) (mm) & 210 & \(l_{\text{e}}\) (mm) & 153 & \(w_{\text{PM}}\) (mm) & \(x_{4}\) \\ \(l_{\text{PM}}\) (mm) & 5 & \(l_{\text{a}}\) (mm) & \(x_{1}\) & \(l_{\text{s}}\) (mm) & \(x_{5}\) \\ \(D_{\text{ri}}\) (mm) & 48 & \(w_{\text{t}}\) (mm) & \(x_{2}\) & \(\alpha\) & \(x_{6}\) \\ \(D_{\text{ro}}\) (mm) & 136 & \(w_{\text{s}}\) (mm) & \(x_{3}\) & \(\beta\) & \(x_{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Design parameters of the investigated IPMSM. It can be noted that there are three torque components in (6), which take torque ripple components \(T_{2}\) and \(T_{3}\) into consideration compared with the traditional torque equation, which only considers the average torque \(T_{1}\). Moreover, the magnetic density inside the driving motor core is usually non-sinusoidal and rich in harmonics. Additionally, the two magnetization modes, alternating and rotating, will make the variation trends of the magnetic density trajectory completely different. In order to consider the effects of alternating magnetization, rotating magnetization, and magnetic field harmonics, the iron losses \(P_{\text{Fe}}\) of each RP are calculated by the magnetic density orthogonal decomposition model [21] as follows:  
\[P_{\text{Fe}}=K_{y}(fh)\sum_{h=1}^{h_{\text{m}}}\left(B_{th}^{2}+B_{th}^{2} \right)+K_{\text{c}}(fh)^{2}\sum_{h=1}^{h_{\text{m}}}\left(B_{th}^{2}+B_{th}^{ 2}\right)+K_{\text{e}}(fh)^{1.5}\sum_{h=1}^{h_{\text{m}}}\left(B_{th}^{2}+B_{ th}^{2}\right). \tag{7}\]  
where \(h\), or the subscript \(h\), represents the number of magnetic density harmonics, and \(h_{\text{m}}\) is the total harmonic number. The \(\lambda_{\text{d}}\), \(\lambda_{\text{q}}\), \(B_{\text{r}}\), and \(B_{\text{t}}\) in analytical expressions (6) and (7) are calculated using computationally efficient FEA [26], which is the so-called semianalytical model. According to the magnetic and electric symmetry and periodicity of the considered IPMSM, when \(n\) position samples are selected and flux linkage of winding and flux density of iron core are calculated by \(n\) magnetostatics FEA of \(1/8\) model, there will be another \(5n\) samples reconstructing by post-processing techniques, which is much more time-saving than FEA. Figure 4 shows the reconstructed results of \(\lambda_{\text{d}}\), \(\lambda_{\text{q}}\), \(B_{\text{r}}\), and \(B_{\text{t}}\), which are used to set up the efficiency model and torque model of IPMSMs. Figure 4a,b presents the variation of \(\lambda_{\text{d}}\) and \(\lambda_{\text{q}}\) with stator phase current amplitude \(I_{\text{s}}\) and rotor angle \(\theta\), where the rainbow color maps the flux linkage value. Figure 4c,d present the variation of \(B_{\text{r}}\) and \(B_{\text{t}}\) with stator current and rotor angle in the stator teeth, where the rainbow color maps the flux density value. Figure 4: Computationally efficient FEA results for semianalytical models. (a) \(\lambda_{\text{d}}\).