## 3 Simulation and Results Discussion
### Result and Discussion  
In this study, the SPA selects the South China Sea (105\({}^{\circ}\)E-116\({}^{\circ}\)E, 4\({}^{\circ}\)N-15\({}^{\circ}\)N) as the target flight area. Besides, the two level flight altitude are 15 km and 23 km, which is obtained according to the environmental characteristics of the flight area. As described in Section 2.6 above, the study firstly use the GPM method to solve the optimal flight path, and then look for the minimum battery mass that meets long flight time constraints. As results, the final optimized battery mass \(m_{bat}\) of Zephyr 7 for satisfying the \(t_{5}-t_{0}=24\) h in Equation (25) is 12.61 kg, which is shown in MATLAB workspace. Compared with the original initial parameter of battery mass, it is equivalent to a 23.1% reduction for Zephyr 7. It also means how much potential energy can be gained by this flight strategy. Meanwhile, The the three-dimensional trajectory optimization, the speed with altitude, the quantity of electricity, the thrust, the flight path pitch angle with attack angle, the yaw angle and the roll angle with minimal battery mass are shown in Figures 6-10, respectively.  
From the Figure 7a, the time of each phase for this flight strategy is 2.73 h, 5.03 h, 4.09 h, 2.06 h and 10.09 h, respectively. Meanwhile, the speed has good following to the altitude, and the highest flight altitude of Zephyr 7 can reach is about 28.47 km at time 18:03. In addition, it can be seen that the flight trajectory of phase 2 from Figure 6 is estimated to have a symmetrical trend at noon time as the same as the trend of yaw angle in Figure 10, which can conclude that the SPA will adjust the trajectory with the solar altitude angle changing symmetrically at the standard noon time. Besides, in order to better analyze the changes of battery charge percentage at noontime, we have performed a simple polynomial interpolation on the data points of the battery percentage at 11:00 to 14:00 by EXCEL. As shown in Figure 8, the battery charge rate has the same trend to the intensity of solar radiation, which is different from the same charge rate in some other studies. More importantly, the unpowered gliding time of the phase 4 is 2.06 h (Figure 8), which did not consume the energy of the battery. This optimization result can be seen as the result of gravity energy reserving, and thus the quality of the battery can be reduced by this strategy.  
Another thing has to note is that there is a phenomenon of oscillation at the connection points over the flight time. The literature [32] has specifically studied the oscillation of this numerical result,  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Minimum Value** & **Maximum Value** \\ \hline \(h\) (km) & 15 & 30 \\ \(v\) (m/s) & 0 & 50 \\ SOC\({}_{0}\) & 0.05 & 0.99 \\ _Lati_ (\({}^{\circ}\)) & 4 & 15 \\ _Long_ (\({}^{\circ}\)) & 105 & 11 \\ \(\alpha\) (\({}^{\circ}\)) & 5 & 10 \\ \(\gamma\) (\({}^{\circ}\)) & -15 & 15 \\ \(\psi\) (\({}^{\circ}\)) & -1 & 1 \\ \(T\) (\(N\)) & 0 & 1000 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The constraints of variables.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Description** \\ \hline \(h_{1}\) (km) & 15 & Cruising altitude \\ \(h_{2}\) (km) & 23 & Battery mass \\ \(SOC_{0}\) & 0.05 & Initial battery percentage \\ \(d_{n}\) (\(\cdot\)) & 266 & Autumn equinox day number \\ _Lati_ (\({}^{\circ}\)) & 4 & Latitude of the starting point \\ _Long_ (\({}^{\circ}\)) & 105 & Longitude of starting point \\ \(t_{sr}\) (h) & 6.8611 & The sunrise time \\ \(t_{ncom}\) (h) & 12.8564 & The standard noon time \\ \(t_{ss}\) (h) & 18.8517 & The sunset time \\ \hline \hline \end{tabular}
\end{table}
Table 3: The parameters of simulation conditions.