## 3 Design Optimization  
### Rotor Shape Optimization Process  
The proposed optimization process was performed by taking into account the rotor shape of the PMSM as a design variable and by taking into account the efficiency and cogging torque. The proposed optimization process for the design of PMSM is illustrated in Figure 3.  
The proposed optimization process is as follows. Depending on the design specifications of the initial model, design variables are selected that improve the efficiency and reduce cogging torque characteristics. To execute the DOE, the design variables were determined using OLHD. The efficiency and cogging torque characteristics of the sampling models were calculated based on finite element analyses. To ensure the accuracy of the DOE results, metamodeling of multipurpose functions and constraints used the following three techniques: Kriging model, MLP, and EDT. For the evaluation of the predictive performance of the metamodel, the best approximation modeling method was adopted \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Items** & **Unit** & **Value** \\ \hline \multirow{4}{*}{Required specification} & Input voltage & V & 380 \\  & Rated speed & rpm & 6000 \\  & Rated output power & W & 3000 \\  & Operation frequency & Hz & 400 \\ \hline \multirow{4}{*}{Mechanical dimension} & Stator’s outer diameter & mm & 93 \\  & Stator’s inner diameter & mm & 51 \\  & Rotor’s outer diameter & mm & 50.2 \\  & Shaft diameter & mm & 20 \\  & Air gap length & mm & 0.4 \\  & Stack length & mm & 60 \\ \hline \multirow{2}{*}{Electrical dimension} & Coil turns & mm & 40 \\  & Coil thickness & & 0.9 \\ \hline \multirow{2}{*}{Material} & Electrical steel & - & 35 PN230 \\  & Permanent magnet & & N42 UH \\ \hline \multirow{4}{*}{Characteristics} & Back-EMF@1000 rpm) & V\({}_{\text{rms}}\) & 27.8 \\  & Cogging torque(peak to peak) & Nm & 0.3479 \\ \cline{1-1}  & Torque(@rated speed) & Nm & 4.775 \\ \cline{1-1}  & Input current(@rated speed) & A\({}_{\text{rms}}\) & 12.85 \\ \cline{1-1}  & Efficiency(@rated speed) & \% & 91.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of the initial model of permanent magnet synchronous motor (PMSM) (rpm: revolutions per minute, EMF: electromotive force). Figure 2: Shape of conventional permanent magnet synchronous motor (PMSM) for an electric compressor. based on the comparison of the results of RMSE. The objective functions and constraints were set to achieve the required target specifications. To obtain the values of the design variables of the optimal model, metamodeling results were obtained with the optimal GA algorithm. Finally, if the target design result does not appear, the optimization process is restarted by adjusting the design variables. According to this optimization process, the mechanical dimensions of the stator and rotor, i.e., the diameter of the core and the stack length, were fixed. Five design variables of the rotor shape were selected as shown in Figure 4.  
Table 2 shows the range of the five design variables used for the optimization of the rotor shape of the PMSM. \(X_{1}\) is the magnet length, \(X_{2}\) is the magnet width, \(X_{3}\) is the distance between the center and the inner diameter of rotor, \(X_{4}\) is the distance between the center and the magnet, and \(X_{5}\) is the distance between the center and the barrier. These design variables were chosen to optimize the rotor shape because they affect the efficiency and cogging torque of the PMSM. In addition, the ranges of the upper and lower limits for each design variable are also listed. The design variables and ranges were determined based on the consideration of the PMSM's manufacturability. Figure 4: Rotor shape design variables. Figure 3: Proposed optimization process. The objective function used to maximize efficiency is expressed by Equation (1). In addition, the constraint that should be lower than the cogging torque of the initial model of 0.3479 Nm, as indicated by Equation (2). * Objective function: Maximize the efficiency.