# A Hybrid Multi-Objective Optimization Method and Its Application to Electromagnetic Device Designs
## 4 Numerical Examples
### Multi-Objective Design of a SMES Device  
A superconducting magnetic energy storage system (SMES) was optimized to obtain the required storage energy with minimal stray fields. The configuration of the device [22] is shown in Figure 8; two concentric coils, carrying currents in opposite directions and operating under superconducting conditions, provide the conditions to store large amounts of magnetic energy in their magnetic fields while keeping the stray field within certain limits. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{_f_(-)} & \multicolumn{2}{c}{**Hybrid**} & \multicolumn{2}{c}{**NSGA-MOPSO**} & \multicolumn{2}{c}{**MOPSO**} & \multicolumn{2}{c}{**NSGA-II**} \\ \cline{2-9}  & **E(\(\Delta\))** & \(\sigma\)(\(\Delta\))** & **E(\(\Delta\))** & \(\sigma\)(\(\Delta\))** & **E(\(\Delta\))** & \(\sigma\)(\(\Delta\))** & **E(\(\Delta\))** & \(\sigma\)(\(\Delta\))** \\ \hline KUR & 0.5103 & **0.0216** & 0.5019 & 0.02983 & 0.7230 & 0.0249 & **0.4996** & 0.0326 \\ ZDT1 & **0.4831** & **0.0436** & 0.5173 & 0.107 & 0.9513 & 0.1455 & 0.6499 & 0.0557 \\ ZDT2 & **0.5073** & **0.0427** & 1.02 & 0.1157 & 0.9181 & 0.1130 & 0.8497 & 0.1626 \\ ZDT3 & **0.6182** & **0.0257** & 0.6623 & 0.03448 & 0.8655 & 0.0350 & 0.7647 & 0.0407 \\ ZDT4 & **0.4512** & 0.0488 & 0.8297 & 0.07222 & 1.2481 & **0.3920** & 0.7906 & 0.0416 \\ ZDT6 & **0.4590** & 0.1278 & 1.1983 & **0.1032** & 0.9143 & 0.1424 & 1.3307 & 0.1083 \\ \hline \end{tabular}
\end{table} 
Table 4: Statistical results of diversity metric. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{_f_(-)} & \multicolumn{2}{c}{**Hybrid**} & \multicolumn{2}{c}{**NSGA-MOPSO**} & \multicolumn{2}{c}{**MOPSO**} & \multicolumn{2}{c}{**NSGA-II**} \\ \cline{2-9}  & **E(\(\gamma\))** & \(\sigma\)(\(\gamma\))** & **E(\(\gamma\))** & \(\sigma\)(\(\gamma\))** & **E(\(\gamma\))** & \(\sigma\)(\(\gamma\))** & **E(\(\gamma\))** & \(\sigma\)(\(\gamma\))** \\ \hline KUR & **0.0137** & **0.0111** & 0.01523 & 0.01248 & 0.0379 & 0.0349 & 0.0143 & 0.0115 \\ ZDT1 & 0.0231 & 0.0114 & 0.05392 & 0.01327 & **0.0213** & **0.0092** & 0.3370 & 0.0940 \\ ZDT2 & **0.2370** & **0.0045** & 0.8526 & 0.06125 & 0.2691 & 0.1213 & 0.6127 & 0.0879 \\ ZDT3 & **0.0171** & 0.0308 & 0.06385 & 0.05269 & 0.0393 & 0.0296 & 0.0557 & **0.0131** \\ ZDT4 & **0.1103** & **0.0092** & 1.4603 & 0.3753 & 2.7106 & 0.4014 & 2.1833 & 0.376 \\ ZDT6 & **0.0979** & **0.0039** & 1.4917 & 0.9526 & 0.7835 & 0.2724 & 0.9325 & 0.9128 \\ \hline \end{tabular}
\end{table} 
Table 3: Statistical results of convergence metric. A three-parameter MOO problem is defined as  
\[\min\left\{ \begin{aligned} f_{1}(r_{2},h_{2}/2,d_{2})& =\frac{1}{B_{norm}^{2}}B_{Stray}^{2}\\ f_{2}(r_{2},h_{2}/2,d_{2})&=\frac{1}{E_{ref}}\Big{|}E-E _{ref}\Big{|}\\ s.t.&\left|B_{max}\right|\leq 4.92T\end{aligned} \right. \tag{15}\]  
where \(E_{ref}=180\) MJ, \(B_{norm}=3\) uT, and \(B_{stray}^{2}\) and \(E\) are the practical stray fields energy and stored energy, respectively. \(r_{2},h_{2},\text{and }d_{2}\) are the dimension parameters of the outer coils. \(B_{stray}^{2}\) can be calculated by  
\[B_{stray}^{2}=\frac{1}{22}\sum_{i=1}^{22}\bigl{|}B_{stray,i}\bigr{|}^{2} \tag{16}\]  
The proposed hybrid algorithm is applied to solve this case study, and then compared with its two counterparts, NSGA-II and MOPSO. The comparison results of the Pareto solutions for the three algorithms with the same number of iterations are shown in Figure 9. The \(NDR\) metrics for Hybrid, NSGA-II, and MOPSO are 57.72%, 13.79%, and 34.48%, respectively, which indicates that the proposed hybrid algorithm obtains better Pareto solutions. The trajectory of the Pareto solutions obtained by the proposed hybrid algorithm are shown in Figures 10 and 11. The pentagram in Figure 10 denotes the solution with min (\(f_{1}+f_{2}\)). It can be observed that the objective values of \(f_{1}\) and \(f_{2}\) are becoming smaller as the number of iterations increases, which demonstrates that the Pareto solution sets are becoming closer to the true PF.