# Efficiency Optimization of the Main Operating Points of an EV Traction Motor
## 3 Optimal Design
### Metamodeling  
The optimal metamodeling technique depends on the distribution of the test values; therefore, to develop the optimal metamodel for the objective and constraint functions, 11 metamodels were compared and selected based on the Root Mean Square Error (RMSE) test values [27]. The metamodeling techniques employed include the Ensemble of Decision Trees (EDT), which combines multiple decision trees to enhance the robustness and accuracy of the predictive model, effectively modeling nonlinear relationships [28]; Kriging (KRG), which approximates complex relationships between design variables and objectives based on spatial correlations and contributes to improving reliability by enabling uncertainty evaluation [29]; Multi-layer Perceptron (MLP), which is a type of artificial neural network with one or more hidden layers, that learn nonlinear relationships between input and output variables and effectively solves complex design problems [30]; and Polynomial Regression (PRG) techniques, which model the relationship between one or more independent variables and a dependent variable using polynomials. PRG (Polynomial Regression) techniques approximate the relationship between design variables and objectives and are divided into various sub-models depending on the complexity of the problem and the degree of interaction between design variables. PRG Linear Model (LR) expresses simple linear relationships, while PRG Full Quadratic Model (FQ) incorporates second-order polynomials and interaction terms to model more complex relationships. PRG Simple Quadratic Model (SQ) and Simple Cubic Model (SC) handle second and third-order polynomials, respectively, to address nonlinear characteristics. Furthermore, PRG Forward Step (FG) and Backward Step (BS) utilize variable selection algorithms to adjust model complexity and improve efficiency [31]. Radial Basis Function (RBF) techniques, which transform the input space into a high-dimensional feature space using functions that vary based on the distance from the center, are also employed. RBF techniques use functions that vary with the distance from a center point to model nonlinear relationships between design variables and objectives effectively. These techniques are further divided into RBF Interpolation (Int) and RBF Regression (Reg), depending on the characteristics of these data. RBF Interpolation (Int) generates functions that precisely pass through the given data points and are suitable for cases where exact reproduction of these data is required. Conversely, RBF Regression (Reg) employs approximation methods to provide generalized results even for noisy datasets, allowing for a smoother representation of the relationships between design variables and objectives [32]. An RMSE test value closer to zero indicates a smaller prediction error between the metamodel and the actual FEA experimental point values. To apply the RMSE test, the number of experimental points was determined according to Equation (4).  
\[\text{nEXP}>\text{Min}\ [(\text{nDV}+1)\ (\text{nDV}+2)/2,\ 10\ \times\text{nDV}]+5\ \times\text{nDV} \tag{4}\]  
\[\text{nTS}=\text{min}\ [0.1\ \times\text{nEXP},\ 10\ \times\text{nDV}] \tag{5}\]  
where nTS denotes the number of test points used to evaluate the metamodel performance. According to Equation (5), the experimental points must exceed 46; therefore, 54 points were selected from the OA table that satisfied the condition of being 46 or more, and the RMSE test was conducted using six test points. The results are presented in Table 7.  
KRG was selected as the most suitable metamodeling technique for MOP1, MOP2, MOP3, and \(T_{Max\_max\_PRM}\), while EDT was chosen for \(THD_{Back-EMF}\), RBF for \(T_{Max\_BaseRPM}\) RBF (Int.), and MLP for \(V_{line-to-line}\), \(T_{ripple}\).