# Multi-Objective Optimization for Outer Rotor Low-Speed Permanent Magnet Motor
## 3 Comprehensive Sensitivity Analysis
### Response-Surface Analysis  
The response-surface optimization method takes into account the random error of the experiment, and a reasonable experimental design method is used to obtain certain data through experiments. The response-surface method generally uses a second-order polynomial mathematical model to construct a response-surface model, and its specific expression is as follows [18]:  
\[y(x)=\beta_{0}+\sum_{i=1}^{n}\beta_{i}x_{i}+\sum_{i=1}^{n}\beta_{i}x_{i}^{2}+ \sum_{i=1,j}^{n}\sum_{i=1}^{n}\beta_{ij}x_{i}x_{j}+\epsilon \tag{1}\]  
where \(\beta\) is the fit coefficient for second-order polynomial response surfaces, \(x_{i}\) and \(x_{j}\) are the design variables, \(\epsilon\) is the error, and \(n\) is the number of design variables. The power factor is an important parameter in motor design. A large power factor can reduce the stator current. In the optimization, it is necessary to obtain a high power factor. The power factor of a motor refers to the angle at which the voltage leads the current. In numerical terms, the power factor is the ratio of the motor's active power to the apparent power. Through a finite-element simulation, the voltage and current curves under a load operation can be obtained, and thus the phase difference of the voltage and the current can be obtained, that is, the power factor angle can be obtained. The power factor can be calculated by the cosine value of the power factor angle. The selection of test points plays an important role in the construction of the response surface, which directly affects the accuracy of the response-surface construction. Unreasonable test points can even lead to the failure to construct the response surface. The selection of the experimental points in this paper adopts the CCD (central composite designs) method, which can obtain a lot of information about the experimental errors of the experimental variables with the fewest number of experiments. The test points are obtained by the CCD method, and simulation experiments are individually carried out for each test point. Then, the response value is fitted by Equation (1), the coefficients of the second-order polynomial mathematical model are obtained by the least-square regression analysis, and the response-surface mathematical model is finally obtained. Figure 5 shows a comparison of the predicted and experimental values for the efficiency, thermal load, power factor, core loss, copper loss and no-load back EMF. In Figure 5, the degree of the fit between the predicted value and the experimental value is relatively high, indicating that \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Range** \\ \hline Pole arc coefficient \(\alpha_{\text{P}}\) & [0.65, 0.95] \\ PM thickness \(TH_{\text{P}}\) (mm) & [10; 15] \\ Stator outer diameter \(D_{\text{s}}\) (mm) & [840, 880] \\ Core length \(L_{\text{eff}}\) (mm) & [900, 940] \\ Number of conductors per slot N & [8; 10; 12] \\ Air gap length \(\delta\) (mm) & [2; 5] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimized design parameters and variation range. the influence of the optimized design parameters on the performance parameters can be analyzed according to this model. The influence of optimized parameters on motor performance can be evaluated at the rated speed by FEA, as shown in Figures 6-9. In this analysis, the efficiency, no-load back EMF, copper loss, core loss and thermal load are obtained by the 2D FEA in a Maxwell of Ansys Workbench. Figure 5: Fitted curve of the predicted and calculated values of the response surface. Figure 6: Response surface of pole arc coefficient and PM thickness to electromagnetic performance parameters. (**a**) Efficiency; (**b**) back EMF; (**c**) power factor. Figure 7: Response surface of pole arc coefficient and PM thickness to electromagnetic performance parameters. (**a**) Core loss; (**b**) thermal load; (**c**) copper loss.