# Optimization Design and Control of Six-Phase Switched
## 4 Multiobjective Optimization  
It can be found in Figure 5 that there is a conflict in the solution set between increasing the output torque and reducing the torque ripple; therefore, the optimization algorithm should be utilized to find a multiobjective optimal result based on the data predicted by the Kriging model. NSGA-II is an improved non-dominated sorting genetic algorithm, which has been widely utilized in multiobjective optimization problems [32; 33; 34]. It was first proposed by [PERSON] in 1994 [35]. The procedures of NSGA-II mainly consist of non-dominated sorting and genetic operator operation. The detailed procedures are presented as follows. 1. Initialization of the population. In this step, the population and the offspring population are created randomly. 2. Classification of the population into nondominated levels. The nondominated number \(n_{\text{p}}\) of each individual can be figured out, and \(n_{\text{p}}=0\) if the individual is in the first nondominated level. The rest individuals are repeated to complete the classification of the whole population. 3. Calculation of crowding distance. Each Pareto point is ranked according to its objective \(f_{\text{m}}\). The crowding distance \(n_{\text{d}}\) of the first and last Pareto points are set as infinite, and the crowding distance of the rest Pareto points can be calculated as \[n_{\text{d(i+1)}}=n_{\text{d(i)}}+(f_{\text{m}}(i+1)-f_{\text{m}}(i-1))/(f_{ \text{m}}^{\text{max}}-f_{\text{m}}^{\text{min}})\] (16)
4. Creation of offspring population. The parent population is created by the nondominated levels and crowding distance, and the offspring population is generated through selection, crossover, and mutation. 5. Repetition of steps (2)-(4). The step (2)-(4) are repeated until they reach the appointed generation and the Pareto front can be achieved. The output torque, torque ripple, and the normalization value of phase current are defined as optimal objectives, and the current density is defined as a constraint. The multiobjective model can be defined as  
\[\left\{ \begin{array}{ll}\text{Function}:&\min\{-T(Z),T_{\text{ trip}}(Z),\,I^{*}(Z)\}\\ \text{Constraint}:&J_{\text{A}}\leq 10\,,\,\,J_{\text{B}}\leq 10\\ s.t.&Z=\{x_{1},x_{2},x_{3},x_{4},x_{5}\}\end{array} \right. \tag{17}\]  
where \(J_{\text{A}}\) and \(J_{\text{B}}\) are phase current densities of phases A and B, respectively. After using the multiobjective optimization algorithm, the Pareto front [36; 37] can be obtained, as presented in Figure 7. It can be found that the Pareto front points are scattered when the population size is defined as 200, while it is more concentrated when the population size is 10. The optimal solutions selected from the Pareto front are listed in Table 3 when the number of individuals is defined as 10, and each design satisfies the constraint for current density. The comparison between the optimization results and FEM is presented in Table 4, in which \(T_{\text{-S}}\), \(T_{\text{rip-S}}\), and \(I^{*}_{\text{-S}}\) represent the results from the FEM. It can be found the errors between these are acceptable. Figure 7: Pareto front when population sizes are (a) 200 and (b) 10, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline  & \(\lambda_{\mathbf{hcs}}\) & \(\boldsymbol{\kappa_{\text{Aon}}}\) & \(\boldsymbol{\kappa_{\text{Bon}}}\) & \(\boldsymbol{P_{\text{dA}}}\) & \(\boldsymbol{P_{\text{dB}}}\) \\ \hline Design 1 & 0.909 30 & 13.002 & 18.543 & 15.013 & 15.113 \\ Design 2 & 0.895 98 & 13.000 & 18.564 & 15.027 & 15.065 \\ Design 3 & 0.915 14 & 13.000 & 18.599 & 15.000 & 15.159 \\ Design 4 & 0.893 55 & 13.036 & 18.549 & 15.820 & 15.327 \\ Design 5 & 0.911 77 & 13.015 & 18.704 & 15.216 & 15.274 \\ Design 6 & 0.890 19 & 13.037 & 18.545 & 15.667 & 15.303 \\ Design 7 & 0.906 14 & 13.001 & 18.523 & 15.024 & 15.049 \\ Design 8 & 0.91 508 & 13.000 & 18.601 & 15.028 & 15.166 \\ Design 9 & 0.895 79 & 13.000 & 18.562 & 15.084 & 15.070 \\ Design 10 & 0.909 39 & 13.000 & 18.539 & 15.247 & 15.024 \\ \hline \end{tabular}
\end{table}