## 5 Multiobjective Optimization Approach
### Numerical Results of the Algorithm  
This paper is based on the actual tower line parameters of 750 KV, where the vertical distance between the lowest point of the ground line and the take-off point of the FPTLIR is 50 m. Additionally, a horizontal distance of 6 m is maintained between the take-off point of the FPTLIR and the plane defined by the ground line during both on-line and off-line missions. The algorithm used in this research was implemented using MATLAB, with optimization results being exported for analysis. Several representative generations were selected to compare convergence effects and optimization results. Figure 9 shows these optimization results over different generations.  
It can be observed that the population distributions of the multiobjective optimization for on-line and off-line missions are different in the 1 st, 10 th, 30 th, and 50 th generations in Figure 9. As the number of generations increases, both missions show a similar evolutionary pattern in terms of optimization results. In the first generation, the chromosome distribution is more dispersed with some chromosomes positioned further away from the boundary. In addition, the number of chromosomes does not reach the predetermined number of 50, which nevertheless validates the effectiveness of our improved algorithm. By the 10 th generation, there is a significant increase in both the quantity and quality of chromosomes, which gradually show regularity and are distributed along the pareto bounds; however, a uniform distribution has not yet been achieved at this stage. By the 30 th generation, chromosomes are relatively evenly distributed on Pareto bounds. From the 30 th generation to the 50 th generation, no significant improvement is observed, indicating convergence of the algorithm by the 30 th generation.  
Each data point corresponds to a specific combination of flight state command values and energy consumption over time in Figure 9. Table 2 presents the optimized flight state commands that satisfy all relevant requirements and constraints for both on-line and off-line missions, resulting in the shortest possible flight time and minimum energy consumption. Simulation results and plots have been analyzed to arrive at these commands.  
Figure 9: Optimization results of the improved NSGA-II algorithm with different algebras: (**a**) the 1 st generation; (**b**) the 10 th generation; (**c**) the 30 th generation; (**d**) the 50 th generation.  
The velocity curves of FPTLIR before and after optimization for both on-line and off-line mission intervals can be obtained from the flight state command. These are shown in Figures 10a and 10b, respectively. Comparing the velocity curves before and after optimization, it can be seen that the FPTLIR performs the acceleration and deceleration actions with smaller acceleration and operates at a lower flight velocity after optimization, considering both the flight energy consumption and time constraints. Based on the simulation results and the 50 th generation optimization outcomes, it is determined that the optimal energy consumption for completing the on-line mission in a minimum time of 8.22 s amounts to 1.3750 \(\times\) 10\({}^{5}\) J. After optimization, the shortest total time for completing the on-line mission with a minimum energy consumption of 1.3017 \(\times\) 10\({}^{5}\) J is found to be 17.77 s, resulting in a significant reduction in energy consumption of about 5.33%. The optimum energy consumption for completing the off-line mission in a minimum time of 9.84 s is estimated to be 1.3667 \(\times\) 10\({}^{5}\) J. After optimization, it is found that the total time required to complete the off-line mission with a minimum energy consumption of 1.2983 \(\times\) 10\({}^{5}\) J is about 28.78 s, resulting in an effective reduction in energy consumption of about 5.01%. Thus, the objective of reducing the energy consumption of the system and ensuring the shortest mission execution time for the corresponding energy consumption is achieved.