# Technology Identification, Evaluation, Selection, and Optimization of a HALE Solar Aircraft
## 4 General Methodology
### Design Space Exploration
#### 4.4.3 Monte Carlo Simulation  
In the aircraft conceptual design phase, a stochastic model is employed that does not provide precise result prediction. Unlike deterministic models, which predict results precisely with clear variable relationships, the stochastic model makes it impossible to find an analytical solution. To resolve this problem, a series of numerically random numbers are repeatedly generated to perform simulation in a technique called Monte Carlo simulation. The Monte Carlo simulation analyzes the stochastic model of unit variables and estimates a model of unit variable combinations. That is, by integrating the probability distribution of variables, it finds the probability distribution of a target value. In this study, random numbers were created within a set design range and the Monte Carlo simulation was implemented using the model produced through the response surface equation. \begin{table}
\begin{tabular}{c c} \hline \hline
**Classification** & \(\mathbf{R}^{2}\) \\ \hline Wing loading & 0.96 \\ Power to weight & 0.98 \\ Maximum take-off weight & 0.97 \\ Lift to drag ratio & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: \(R^{2}\) values of response surface equations. Figure 14: Regression results of design target variables. Based on the response surface method performed above, a regression equation can be established for the independent design variables and their combinations. Regression coefficients are organized in Table 12. The equation established from the regression analysis can to a great extent save time required for design of experiment for screening test and response surface method, contributing to an efficient determination of the overall design point. For example, the following equation is a regression equation for the wing loading; the intercept is the y-intercept. \(X_{1}\sim X_{7}\) are the independent design variables selected using the screening test above, and \(\beta_{1}\sim\beta_{35}\) are the regression coefficients for each term. The following is an expression representing various combinations of selected independent design variables; the regression coefficient is the effect of the combination. \[\text{Wing loading }=\text{Intercept}+X_{1}\beta_{1}+X_{2}\beta_{2}+\cdots+X_{7} \beta_{7}+X_{1}X_{2}\beta_{8}+X_{1}X_{3}\beta_{9}+\cdots+X_{7}X_{7}\beta_{35} \tag{27}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Term** & **Coefficient (\(\beta\))** & **Wing Loading** & **Power to Weight** & **MTOW** & **L/D** \\ \hline Intercept & & 38.29466 & 0.012175 & 131.8033 & 39.48104 \\ X1 & 1 & \(-1.45885\) & \(-0.00369\) & \(-2.72136\) & 6.619312 \\ X2 & 2 & \(-3.8979\) & \(-0.00103\) & 46.53446 & 1.432357 \\ X3 & 3 & 10.59271 & 0.001857 & 35.43302 & 1.300914 \\ X4 & 4 & \(-8.89292\) & \(-0.00136\) & \(-30.7049\) & \(-1.3019\) \\ X5 & 5 & 4.420539 & 0.000786 & 14.93174 & 0.583766 \\ X6 & 6 & 0.719599 & \(9.49\times 10^{-5}\) & 2.12396 & 0.057537 \\ X7 & 7 & 2.764475 & 0.000503 & 7.604405 & 0.333755 \\ X1 \(\times\) X2 & 8 & 0.445333 & 0.000299 & \(-0.02664\) & 0.180061 \\ X1 \(\times\) X3 & 9 & \(-1.26241\) & \(-0.00067\) & \(-3.79547\) & 0.04719 \\ X2 \(\times\) X3 & 10 & 0.097532 & \(3.18\times 10^{-5}\) & 15.38417 & 0.206605 \\ X1 \(\times\) X4 & 11 & \(-4.86717\) & \(-0.00037\) & \(-15.8155\) & \(-0.7403\) \\ X2 \(\times\) X4 & 12 & \(-0.48553\) & \(-0.00011\) & \(-14.4865\) & \(-0.29344\) \\ X3 \(\times\) X4 & 13 & 0.682484 & 0.000418 & 1.828877 & 0.294346 \\ X1 \(\times\) X5 & 14 & \(-0.59253\) & \(-0.0003\) & \(-1.91813\) & 0.032379 \\ X2 \(\times\) X5 & 15 & 0.144814 & \(2.26\times 10^{-5}\) & 6.716138 & 0.113982 \\ X3 \(\times\) X5 & 16 & 0.109353 & \(-0.00011\) & 0.962494 & \(-0.10835\) \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Regression coefficients for response surface method. Figure 15: Prediction profile for selected design objective variables. After that, random numbers between \(-1\) and \(1\) were generated 10,000 times using the Excel Random function.