# Optimization of Multi-Mission CubeSat Constellations with a Multi-Objective Genetic Algorithm
## 4 Simulation Results and Discussion
### Results and Discussion  
For the regional coverage optimization case, the optimization parameters used are the number of orbital planes \(P\) and the number of satellites in each orbital plane 5. The objective functions for this case are to minimize the number of satellites \(f_{1}\) and the regional target ART \(f_{3}\) as given in Equations (1) and (8), respectively. The results in Figure 5, for three cases of altitude, show that ART decreases as the number of satellites increases. For the four satellites, the ART is 3.93 h, 3.62 h, and 3.34 h for 400 km, 450 km, and 500 km altitudes, respectively. Whereas for 100 satellites, the ART is 0.15 h, 0.13 h, and 0.12 h for 400 km, 450 km, and 500 km altitudes, respectively. ART decreases with an increase in altitude, but the relative change in ART is small as the number of satellites increases. The second simulation case considers the inclination \(i\), the number of orbital planes \(P\) and the number of satellites in each orbital plane 5 as the optimization variables. The altitude is kept constant at 450 km because varying the altitude results in a relatively slight change in the objective function, as demonstrated by the first simulation case. Table 3 shows Pareto optimal solution sets for the multi-mission simulation case by considering four objective functions (\(f_{1}\), \(f_{2}\), \(f_{3}\), and \(f_{4}\)) as given in Equations (1), (7), (8), and (9), respectively. Figure 6a,b show a decrease in regional and global ART as the number of satellites increases. Moreover, we can see an increase in the ART at the same number of satellites as the inclination increases. Consider solutions 1 and 9 in Table 3, which have the same number of satellites at 40\({}^{\circ}\) and 89\({}^{\circ}\) inclination, respectively. Global and regional ART increases as inclination increases. However, for the same solution, the total percentage coverage increases from 70.79% to 99.93%. Figure 6c shows the plot of the number of satellites versus the total percentage coverage, showing that the same number of satellites can have different coverage. Figure 6e,f show the global and regional ART plots with the total percentage coverage. This result indicates that the number of satellites is not the main parameter affecting the total percentage coverage for the assumed simulation period. Furthermore, we can see that both global and regional ARTs depend on the number and pattern of CubeSats in the \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Optimization Parameter**} & \multicolumn{3}{c}{**Range**} \\ \cline{2-4}  & **Regional** & **Multi-Mission** & **Unit** \\ \hline Altitude & 400, 450, 500 & 450 & km \\ Inclination & 40 & 40–90 & deg. \\ Number of orbital planes & 2–10 & 2–10 & – \\ Number of sat per plane & 2–10 & 2–10 & – \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimization parameters with their boundary value for two simulation cases. Figure 5: Pareto front for regional target coverage simulation case at different altitudes. constellation. Figure 7 shows the relationship between the inclination and the total percentage coverage. The result shows that the total percentage coverage is mainly dependent on the inclination of the constellation rather than on the other constellation parameters. Figure 8 shows a 3D plot of the objective functions \(f_{1}\), \(f_{2}\), and \(f_{3}\). The result shows the conflicting nature of the selected objectives, showing that there is no single solution to satisfy these objectives. Let us consider one solution from all possible feasible solutions in the Pareto optimal solution sets that can compromise all objectives. Solution number 20 in Table 3 has a total of 20 satellites, making it less costly to design, launch, and operate compared to the other solutions with a higher number of satellites. The solution also has a regional ART of 1.07 h, sufficient to monitor and manage most of the natural disasters in the target area [7]. The solution provides 98.77% of global coverage with 4.37 h of ART.