## 5 Numerical Results
### Performance Evaluation in a Multiple-EHD Scenario
#### 5.2.1 Performance Evaluation in an Urban Environment  
First, we evaluate the various methods when the horizontal distance is 100 m, i.e., \(r_{\text{max}}=100\) m. Figure 4a,b show the average UAV altitudes and their standard deviations obtained using proposed method #1, proposed method #2, and the optimal design. Using proposed method #1, the near-optimal altitude of the design is obtained considering the entire range. The performance gap between proposed method #1 and the optimal design is less than 4 m. The performance gap of proposed method #2 is larger than that of proposed method #1. In addition, the UAV's altitude is within \([70,80]\) m, and it increases as the number of EHDs increases. Moreover, Figure 4b demonstrates that the standard deviation is within \([4,15]\) m, and it increases as the number of EHDs increases. In Figure 5a,b, the average overall charging times and the average execution times of various methods are compared with respect to the number of EHDs. In Figure 5a, the charging time performance for each method is presented, with values of \([279,1062]\) seconds for W-sum EB with \(H_{\text{Opt}}\)[43; 173] seconds for optimal EB with \(H_{\text{min}}\), \([79,267]\) seconds for optimal EB with \(H_{\text{max}}\), \([14.6,53.5]\) seconds for proposed method #1, \([17.6,101.3]\) seconds for proposed method #2, and \([14.2,52.4]\) seconds for the optimal design. As expected, proposed method #1 outperforms the other methods and achieves a near-optimal charging time over the entire range. The performance gap increases as the number of EHDs increases. Specifically, W-sum EB with \(H_{\text{Opt}}\) significantly extends the charging time, as the energy beam is not optimized for minimizing charging time. On the other hand, the optimal design yields a large execution time, increasing from 120 s to 180 s as the number of EHDs increases. By contrast, the execution times range from 7 s to 11 s for proposed method #1, a considerable reduction compared to the optimal design. Meanwhile, proposed method #2, which leverages the weighted-sum EB solution in closed form, yields the shortest execution time of below 0.02 s while yielding lower charging times than the other methods with \(H_{\text{min}}\) and \(H_{\text{max}}\). Figure 4: Average UAV altitude and its standard deviation via various methods when \(r_{\text{max}}=100\) m in an Urban environment. Next, we evaluate the various methods when the horizontal distance is \(150\) m, i.e., \(r_{\text{max}}=150\) m. Figure 5(a,b) show the average UAV altitude and its standard deviation obtained from proposed method #1, proposed method #2, and the optimal design. Proposed method #1 achieves near-optimal performance, with the performance gap staying within \(1.5\) m. In addition, proposed method #1 performs comparably to proposed method #2, with the performance gap being within \(2.5\) m. As expected, the UAV altitude increases as the number of EHDs increases. The UAV altitude is within \([108,122]\) m, which is higher than the average altitude when \(r_{\text{max}}=100\) m, such as in Figure 5(a). In addition, Figure 5(b) exhibits that the standard deviation is within \([8,25]\) m, which is larger than the standard deviation when \(r_{\text{max}}=100\) m, such as in Figure 5(b). Figure 5(a),b show a comparison of the average overall charging time and the average execution time of various methods with respect to the number of EHDs. In Figure 6(a), the charging time performance for each method is presented, such as with values of \([627,2461]\) seconds for W-sum EB with \(H_{\text{Opt}}\), \([1370,13710]\) seconds for optimal EB with \(H_{\text{min}}\), \([85,288]\) seconds for optimal EB with \(H_{\text{max}}\), \([33.1,120]\) seconds for proposed method #1, \([39.8,229]\) seconds for proposed method #2, and \([32.3,119]\) seconds for the optimal design. Proposed method #1 achieves significantly reduced charging time and near-optimal performance, with a considerable gap compared with other methods as the number of EHDs increases.