## 5 Numerical Results
### Performance Evaluation in a Multiple-EHD Scenario  
In this subsection, we evaluate our proposed methods in a multiple-EHD scenario. For the Rician fading channel, the \(u\)-th EHD's deterministic LoS component \(\mathbf{a}_{u}\) is obtained from (4) with \(\frac{d_{s}}{\lambda_{c}}=0.5\). To evaluate the average overall charging time, we take the average of 100 independent realizations, varying the energy requirements \([Q_{u}^{\text{Req}}]_{u=1}^{U}\), horizontal distances \([r_{u}]_{u=1}^{U}\), and AoD \([\phi_{u}]_{u=1}^{U}\) for each realization. For this purpose, the parameters \([Q_{u}^{\text{Req}}]_{u=1}^{U}\), \([r_{u}]_{u=1}^{U}\), and \([\phi_{u}]_{u=1}^{U}\) are uniformly sampled within specified ranges to capture variability. Specifically, the energy requirements \([Q_{u}^{\text{Req}}]_{u=1}^{U}\) are chosen from \([1,10]\) mJ, the horizontal distances \([r_{u}]_{u=1}^{U}\) are chosen from \([0,r_{\text{max}}]_{u=1}^{U}\) where \(r_{\text{max}}\in[100,150]\) m, and the angles of departure (AoD) \([\phi_{u}]_{u=1}^{U}\) are chosen from \([-\pi/2,\pi/2]\) radians. Moreover, our proposed method was evaluated according to the number of EHDs. For performance comparison, we considered the following six methods:  
* A weighted sum-power maximization energy beamforming (W-sum EB) with optimal altitude \(H_{\text{Opt}}\);
* Optimal energy beamforming (EB) for the UAV altitude of \(H_{\text{min}}\);
* Optimal EB for the UAV altitude of \(H_{\text{max}}\);
* Proposed method #1--efficient near-optimal altitude and EB design with low complexity, i.e., Algorithm 4;
* Proposed method #2--suboptimal design with weighted-sum EB in a closed-form solution, i.e., Algorithm 5;
* Optimal design: optimal EB with optimal altitude, i.e., Algorithm 3 (we set the sample grid accuracy to \(\Delta=0.5\) m for the 1D exhaustive line search by considering a practical scenario while avoiding high execution times that make it impossible to use).  
Notably, we consider the widely adopted W-sum EB for a comparative analysis with other methods in terms of energy beamforming performance. The W-sum EB was proposed to maximize the weighted sum-power transferred to the EHDs by an energy beam directed  
\begin{table}
\begin{tabular}{l c} \hline \hline  & **Average Execution Time (Seconds)** \\ \hline Exhaustive 1D line searching (Algorithm 1) & 1 \\ \hline Proposed method (Algorithm 2) & \(7\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of the average execution time (seconds).  
Figure 3: Comparison of the optimal UAV altitude determined by the 1D exhaustive line search method (Algorithm 1) and the proposed method (Algorithm 2).  
toward the dominant eigenvector of the composite weighted channel gain matrix of the EHDs [1; 2]. The optimal altitude for the W-sum EB is obtained using a 1D exhaustive line search method.  
To examine the robustness of our proposed methods across various practical scenarios, we consider both Urban and Dense Urban environments for the A2G Rician fading channel model. In this case, the environment-dependent LoS probability parameters are given by \((a_{1},\ b_{1})=(9.61,\ 0.16)\) for the Urban environment and \((a_{1},\ b_{1})=(12.08,\ 0.11)\) for the Dense Urban environment [8; 9]. First, we evaluate the proposed methods in an Urban environment, and then consider a Dense Urban environment.