# Multi-Objective Optimization for Outer Rotor Low-Speed Permanent Magnet Motor
## 4 Multiple Objectives Comprehensive Optimization  
### Multi-Objective Program Design  
In the multi-objective optimization design of the LSPMM, the electromagnetic efficiency and motor temperature are used as the two optimization design goals. First, the maximum efficiency is set as the optimization objective. In addition, the minimum thermal load is set as the optimization objective considering the goal of optimizing the motor temperature. For a given cooling system, both the copper loss and core loss have a great influence on the temperature distribution of the motor, but the electromagnetic simulation results of the initial LSPMM design clearly show that the copper loss is about 10 times larger than the core loss; furthermore, the stator winding is the part with the highest temperature in the motor and mainly depends on the copper loss. Therefore, in this optimization process, in order to obtain the lowest stator winding temperature, the minimum copper loss is set as an optimization objective. The objectives are defined as  
\[\text{Objective}:\left\{\begin{aligned} &\max(\eta)\\ &\min(Q_{thermal})\\ &\min(P_{copper})\end{aligned}\right. \tag{3}\]  
where \(\eta\) is the efficiency, \(Q_{thermal}\) is the thermal load, and \(P_{copper}\) is the copper loss.  
The various constraints include the output torque, no-load back EMF, power factor, current density, output power and core loss.  
1. Output torque: The output torque should be satisfied first in the design. For the LSPMM, its rated torque depends on the rated power and rated speed of the motor; the greater the rated power and the lower the rated speed, the greater the motor's torque. In this paper, the rated torque of the design is 34,107.1 Nm and the output torque is constrained to a minimum of 34,110 Nm.
2. No-load back EMF: The no-load back EMF is an important factor affecting the power factor. When the back EMF point is close to the voltage amplitude and the deviation is within the voltage range of \(\pm\)2%, the motor power factor is greater than 0.9; otherwise, the power factor is low. Therefore, to pursue a high power factor, the no-load back EMF should be limited to the vicinity of the rated voltage, which is 205 V to 220 V.
3. Power factor: The power factor is an important factor to measure the motor efficiency. The higher the power factor, the higher the motor efficiency. Therefore, the minimum power factor is set to 0.95.
4. Current density: The current density also needs to be limited. If the current density is too large, then it will increase the loss and affect the motor temperature; if it is too small, then the amount of coil material will increase. According to the traditional permanent magnet motor theory, the current density of the LSPMM is about 4-6 A/mm\({}^{2}\).
5. Output power: The power selection of the motor should be appropriate. When the power selection is too large, the efficiency and power factor will be reduced, and when the power selection is too small, the motor will be overloaded and the lifespan will be shortened. In this optimization, the LSPMM is rated at 200 kW, so the minimum output power is limited to 200,000 W.
6. Core loss: Although the core loss is about 10 times smaller than the copper loss, the effect of the core loss on the motor temperature is not as large as that of the copper loss, but excessive core loss still affects motor performance. Therefore, the maximum core loss is set to 1500 W.  
Based on the above analysis, the constraints of the LSPMM design are listed as the following:  
\[\text{Constraints}:\left\{\begin{aligned} & 34,110\;\text{Nm}\leq T_{N}\\ & 205\;\text{V}\leq E_{L}\leq 220\;\text{V}\\ & 0.95\leq\varphi_{p}\\ & 4\;\text{A}/\text{mm}^{2}\leq J_{C}\leq 6\;\text{A}/\text{mm}^{2}\\ &(2E+5)\;\text{W}\leq P_{out}\\ & P_{core}\leq 1500\;\text{W}\end{aligned}\right. \tag{4}\]  
where \(T_{N}\) is the output torque, \(E_{L}\) is the no-load back EMF, \(\varphi_{p}\) is the power factor, \(J_{C}\) is the current density, \(P_{out}\) is the output power, \(P_{core}\) is the core loss.