# Efficiency Optimization of the Main Operating Points of an EV Traction Motor
## 3 Optimal Design  
### Design Problem Formulation  
In this study, three main operating points were selected for efficient motor optimization, and the optimal design was achieved by combining the metamodels with optimization algorithms, as shown in Figure 7.  
The optimization software PIAnO 2024 was employed, and the objective function was set to maximize the efficiency at the main operating points, as shown in Equation (2). The weights for each point were determined based on the density of the operating points, ensuring that their sum was equal to 1. Here, \(\eta_{i}\) represents the efficiency of the main operating points and \(\mathbf{w_{i}}\) represents the weights of the main operating points. \(T_{Max\_BaseRPM}\) is the maximum torque at the rated speed of 1575 rpm, and \(T_{Max\_maxRPM}\) is the maximum torque at the maximum speed of 5000 rpm, set to 251 Nm and 80 Nm, respectively, to maintain an output of 40 kW and to maintain the values of the initial model. \(T_{ripple}\) is the torque ripple at the rated speed, which was set to be below the allowable level of 10%, and, \(THD_{Back-EMF}\) is the THD of the back electromotive force, which was set not to exceed the value of the initial model.  
Additionally, \(V_{line-to-line}\) is the line-to-line voltage and was constrained to 380 V to ensure it did not exceed the battery voltage, thereby formulating the design problem. During the DOE process, the input current beta angle at the rated speed was fixed, and at the maximum speed, the beta angle was set according to the voltage limit; therefore, \(V_{line-to-line}\) represents the line-to-line voltage at the rated speed.  
Figure 7: Optimal design process.  
\[\begin{array}{ll}\text{to maximize}&\sum_{i=1}^{3}\eta_{i}w_{i}=\left(\eta_{1}w_{1}+ \eta_{2}w_{2}+\eta_{3}w_{3}\right)\\ \text{subject to}&T_{Max\_BaseRPM}\geq 251\text{ Nm}\\ &T_{Max\_maxRPM}\geq 80\text{ Nm}\\ &T_{ripple}\leq 10\%\\ &THD_{Back-EMF}\leq\ 8\%\\ &V_{line-to-line}\leq 380\text{ V}\end{array} \tag{2}\]  
Six design variables were selected based on the objectives and constraints, and their minimum and maximum ranges are presented in Table 4. The configurations of the design variables are shown in Figure 8. X1 represents the open slot, with a lower limit of 0.4 mm, which is the minimum thickness required for mold production considering manufacturability, and an upper limit of 3.6 mm, which is smaller than the winding size to ensure that the winding does not protrude. X2 denotes the Shoe thickness, with a lower limit of 0.5 mm, considering the minimum thickness of 5 mm required for mold production of the shoe area, and an upper limit of 2 mm to ensure that the flux density of the back yoke does not exceed 1.8 T, considering the armature reaction [23]. Considering the armature reaction, X3 represents the Shoe angle, with a lower limit of 0 mm and an upper limit of 2 mm. X4 is the Notch angle, with a lower limit of 5.2\({}^{\circ}\) and an upper limit of 7.5\({}^{\circ}\), ensuring that the surrounding thickness is at least 0.5 mm for die manufacturability. X5 denotes the Rib thickness, with a lower limit of 0.9 mm and an upper limit of 2 mm, ensuring that the surrounding area maintains a thickness of 0.5 mm for mold production. X6 represents the Magnet angle, where holes are placed in rotor areas with low flux density for motor weight reduction, with the range set to ensure that the flux density around these holes remains below the original value of 0.2 T.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Design** & **Unit** & **Lower** & **Initial** & **Upper** \\
**Variables** & & & & \\ \hline X1 (Slot open) & mm & 0.4 & 0.41 & 3.6 \\ X2 (Shoe thickness) & mm & 0.5 & 0.7 & 2 \\ X3 (Shoe angle) & mm & 0 & 0.85 & 2 \\ X4 (Notch angle) & \({}^{\circ}\) & 5.2 & 7.21 & 7.5 \\ X5 (Rib thickness) & mm & 0.9 & 0.9 & 2 \\ X6 (Magnet angle) & \({}^{\circ}\) & 135 & 145 & 155 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Range of variables.  
Figure 8: Shape of design variables.