## 5 Results of Optimization
Here, the input variables of each datasets are treated as the design variables of the corresponding optimization problem so that the degrees of freedom of the optimization problems using the surrogate models with G-DLB-8-3 and O-DLB-6-4 are 8 and 6, respectively. These optimization problems will be respectively referred to as the optimization problems with G-DLB-8-3 and O-DLB-6-4. Note that both the optimization problems with G-DLB-8-3 and O-DLB-6-4 can be classified further into two independent problems based on the types of surrogate models used in the optimization process: RSM-computed and ANN-computed surrogate models. As a result, the total number of optimization problems solved in the current study is four. The optimal wind turbine airfoil's shapes obtained from the optimization problems with G-DLB-8-3 and O-DLB-6-4 are presented in Figure 7a,b, respectively. In each figure, the wind turbine airfoils plotted in blue and red are the optimal designs using the surrogate models computed from the RSM and ANN, respectively. Here, all eight design variables of the optimization problems with G-DLB-8-3 are geometric parameters. However, only four design variables are geometric parameters for the optimization problems with O-DLB-6-4, while the other two are the variables related to the airfoil's operating conditions; therefore, the effect of these two design variables are not reflected in Figure 7b. The lift-to-drag ratios of the optimal designs are presented in Table 5 with their operating conditions. The lift-to-drag ratio of the optimal design obtained from the optimization problem with O-DLB-6-4 using the RSM-computed surrogate model is relatively lower than the other optimal designs. Note here that the value of \(E\) for the RSM-computed surrogate model with O-DLB-6-4 is the highest comparedwith the other three surrogate models. Judging from the values of \(E\), it is believed that inaccurate prediction of the RSM-computed surrogate model for the optimization problem with O-DLB-6-4 makes the input variables reach the \"wrong destination\" where the wind turbine airfoil's performance is not maximized in reality because of the discrepancy between the surrogate model-computed and XFOIL-computed wind turbine airfoil's performance, resulting in less performance improvement than the cases when the surrogate models predict the output value with a high level of accuracy. This is also supported by the fact that the lift-to-drag ratios of the optimal designs found with the high-accuracy surrogate models (ANN-computed surrogate models) are higher than the ones using the less-accurate surrogate models (RSM-computed surrogate models) for both optimization problems with G-DLB-8-3 and O-DLB-6-4. The pressure coefficients (defined as \(c_{p}\)) of the optimal designs of the wind turbine airfoils shown in Figure 7a,b are presented in Figure 8a,b, respectively, with the baseline wind turbine airfoil's pressure coefficient graph. The pressure difference between the upper and lower cambers grows wider for the optimal designs compared with the baseline design. This indicates high lift force is produced using the optimal designs, which is the reason for the performance enhancement of the optimal wind turbine airfoils compared with the baseline design. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Optimization** & **Surrogate Model** & **Operational Conditions** & **Lift-to-Drag Ratio** & \(E\) \\ \hline \multirow{2}{*}{G-DLB-8-3} & RSM & Re = 1.00 \(\times\) 10\({}^{6}\), AOA = 5.00\({}^{\circ}\) & 139.85 & 3.20 \(\times\) 10\({}^{-2}\) \\  & ANN & Re = 1.00 \(\times\) 10\({}^{6}\), AOA = 5.00\({}^{\circ}\) & 140.60 & 1.47 \(\times\) 10\({}^{-2}\) \\ \hline \multirow{2}{*}{O-DLB-6-4} & RSM & Re = 9.95 \(\times\) 10\({}^{5}\), AOA = 5.48\({}^{\circ}\) & 129.48 & 2.24 \(\times\) 10\({}^{-1}\) \\  & ANN & Re = 9.97 \(\times\) 10\({}^{5}\), AOA = 7.57\({}^{\circ}\) & 140.09 & 6.57 \(\times\) 10\({}^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The values of the lift-to-drag ratios of the optimal designs.