## 5 Optimization of WPT Coupler
### Results and Discussion  
In the multi-objective optimization process using NSGA-II, the population size is 20 and the number of generations is 60. Through the multi-objective optimization process, 20 Pareto optimal design points are obtained. The corresponding objective functions value are shown in Table 1.  
Figure 14 shows the convergence process of the Pareto front. Since the algorithm optimizes in the direction of the minimum by default, a negative sign is added in front of the mutual inductance. As the number of iterations increases, the Pareto front gradually converges. After 60 iterations, the Pareto front hardly changes, and the iteration stops.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **M/uH** & **ECLI/m\(\Omega\)** \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 1: Pareto optimal design points.  
Table 2 lists the design points with different weights selected from the Pareto optimal points using the TOPSIS method, where W1 is the weight of mutual inductance, W2 is the weight of ECLI. The optimal design point A stands for the maximum mutual inductance ratio, while the optimal design point E stands for the minimal ECLI, respectively. The results of design point A and design point E can also be obtained by the single-objective optimization method. The reason that design points A and B are the same is the limited number of Pareto optimal solutions and the randomness of NSGA-II. There are two ways to obtain different design points; the first is to increase the number of Pareto optimal solutions, which can make the design points more evenly distributed. Another method is to redo the optimization process. Due to the randomness of the genetic algorithm, it may obtain different results if it is re-iterated.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **W1** & **W2** & **M/uH** & **ECLI/m\(\Omega\)** \\ \hline A & 1 & 0 & 24.82117 & 401.69557 \\ B & 0.75 & 0.25 & 24.82117 & 401.69557 \\ C & 0.5 & 0.5 & 16.82043 & 275.73258 \\ D & 0.25 & 0.75 & 16.82043 & 275.73258 \\ E & 0 & 1 & 16.82043 & 275.73258 \\  & Initial point P & & 20.43746 & 353.01086 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Sets of optimization results.  
Figure 14: Pareto front: (**a**) The 1 st iteration. (**b**) The 2 nd iteration. (**c**) The 4 th iteration. (**d**) The 60 th iteration.  
The mutual inductance and ECLI gradually decrease from design points A to E because the weight of mutual inductance is decreasing, and the weight of ECLI is increasing. In practical applications, it may not be necessary to maximize the mutual inductance or minimize the ECLI, and we can adjust the weights W1 and W2 according to our actual needs.  
Compared with the initial point P, the mutual inductance value of design point B has increased by 21.45%, while the ECLI has increased by less than 50 m\(\Omega\), which has little effect on power loss. From (4) and (5), we can see that the mutual inductance directly affects output power and efficiency, so the performance of the coupler at design point B is significantly better than that at design point P. Therefore, the process of multi-objective optimization is effective.