## 5 CGE-SSA Multi-Objective Optimization
### CGE-SSA Multi-Objective Optimization  
In contrast to the single-objective algorithm, the fitness values of individuals in the multi-objective algorithm are determined through multiple indicators, which cannot be directly compared to derive the optimal individual. Therefore, by adding a non-dominated sorting procedure and a congested space calculation procedure to CGE-SSA, the comparison of fitness values between multi-objective individuals is realized, and the non-dominated solution set is output to form an improved multi-objective sparrow search algorithm, the optimization flowchart of which is shown in Figure 12.  
In this study, the second-order regression equation between the parameters of the optimization structure and the optimization objective was fitted through response-surface analysis, the improved multi-objective sparrow algorithm program was written using MATLAB R2018b software, and then the fitted objective function was written into the program to define the upper and lower limits of each parameter, as well as the constraints of the optimization objective. After setting the dimension to 4, the number of sparrow populations to 50, the non-dominated solution set repository to 50, and the maximum number of iterations to 200, the optimization objective was optimized to obtain the Pareto frontier, the result of which is shown in Figure 13.  
Figure 12: Algorithm optimization flowchart. Figure 13: Pareto frontier. According to the optimization objective and its constraints, set in Section 2.1, five optimal solutions were obtained, shown as point A and region B in the Pareto frontier chart. Among them, both point A and region B satisfy the requirement that the output torque is greater than 9.7 Nm, and the output torque of point A is slightly higher than that of region B. In addition, the torque ripple values at neither point A nor region B exceed 4%, which indicates a good output stability. Compared with the torque ripple and the output torque, the back EMF amplitude at point A differs greatly from that in region B, reaching 257.5 V. Therefore, point A was taken as the optimal solution point. As can be seen in Table 8, within the reasonable variation range of the structural parameters, the optimized motor models using the four algorithms meet the requirements of the optimization objectives, and the torque ripple and back EMF amplitude are significantly improved, compared to the pre-optimization results, in terms of satisfying the output. Compared with the optimization results of SSA, GA, and PSO, the motor optimized with CGE-SSA has a lower torque ripple, higher back EMF amplitude, and higher output torque. In Figure 14, it can be seen that the magnetization density amplitude of the stator tooth part is about 1.6 T, and that of the yoke part is about 1.3 T, after optimization with CGE-SSA, and the magnetization density distribution is good; meanwhile, the rotor spacer bridge is saturated with magnetic density near the spacer bridge, which can effectively suppress magnetic leakage. Therefore, CGE-SSA demonstrates better optimization results than SSA, GA, and PSO, resulting in a better output performance for the optimized motor. As can be seen in Figure 15a, the red curve sits above the other curves, demonstrating that the average output torque after optimization using CGE-SSA is higher than those after optimization using the SSA, GA, or PSO, with levels of 10.31 Nm, 10.02 Nm, 9.94 Nm, and  
Figure 14: No-load magnetic density distribution.