# Technology Identification, Evaluation, Selection, and Optimization of a HALE Solar Aircraft
## 4 General Methodology
### Design Space Exploration
#### 4.4.2 Response Surface Equations  
The response surface method uses mathematical and statistical techniques to build an empirical model, and optimizes the response of various independent variables through design experiments. As in the equation below, optimization is performed using the second model:  
\[\mathrm{R}=b_{0}+\sum_{i=1}^{k}b_{i}k_{i}+\sum_{i=1}^{k}b_{i}k_{i}^{2}+\sum_{ j=1}^{k-1}\sum_{j=i+1}^{k}b_{i}j_{k}k_{j} \tag{26}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Classification** & \(R^{2}\) \\ \hline Wing loading & 0.86 \\ Power to weight & 0.95 \\ Maximum take-off weight & 0.83 \\ Lift to drag ratio & 0.98 \\ \hline \hline \end{tabular}
\end{table}
Table 9: \(R^{2}\) values of screening test.  
Figure 12: Pareto plot of design target variables.  
where \(b_{i}\) is a regression coefficient for linear terms, \(b_{ii}\) is a coefficient for pure quadratic terms, \(b_{ij}\) is a coefficient for cross-product terms, \(k_{i}\) and \(k_{j}\) are the design variables, and \(k_{i}k_{j}\) denotes the interactions between the two design variables [25].  
The central composition design is used to construct an optimization model for the response surface method. The central composition design is a method of minimizing the number of experiments to optimize the second-order model, and consists of \(2^{n}\) factorial points, 2n axis points, and one center point. In this study, the central composition is designed using the JMP, with 144 cases and 2 center points [12]. Then, as in the screening test, design of experiment is performed to derive the performance variables for each case. The graph of the regression analysis results compares the values calculated via the regression equation with the actual values, as shown in Figure 14. The determination coefficient (\(R^{2}\)), as shown in Table 11, had a value close to 1, indicating that the relationship between the design objective parameters and the independent variables was well explained. The regression analysis results under the response surface method using the second model, in comparison with the screening test results using the first model, are shown.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Classification** & **Unit** \\ \hline Aspect ratio (X1) & – \\ Wing area (X2) & m\({}^{2}\) \\ Airframe weight adjustment factor (X3) & – \\ Battery specific energy density (X4) & W/kg \\ Solar cell specific mass (X5) & kg/m\({}^{2}\) \\ Solar cell efficiency (X6) & – \\ Payload weight (X7) & kg \\ \hline \hline \end{tabular}
\end{table}
Table 10: Selected independent design variables.  
Figure 13: \(p\)-values of design target variables. * denotes values less than 0.05.  
Figure 15 shows a surrogate model indicating the relationship between the selected independent design variables and the design objective parameters. The horizontal axis represents the seven selected independent design variables; the vertical axis shows the design objective parameters indicating aircraft performance. Each inclination is sensitive to the independent variables. The larger the inclination, the higher the sensitivity. For instance, when the aspect ratio (X1) rises, the power to weight falls sharply, but the lift-to-drag ratio moves up very much. Moreover, the relationship between solar cell efficiency (X6) and lift to drag ratio has almost 0 inclination, signaling that the solar cell efficiency, an independent design variable, has little impact on the lift-to-drag ratio.