## 2 System Model and Problem Formulation
### Formulation of the Energy Harvesting Optimization Problem  
By denoting \(\tau_{u}\) as the charging time of the \(u\)-th EHD, the energy harvested at the \(u\)-th EHD is given by [1; 2; 3; 4; 5]  
\[Q_{u}=\zeta_{u}\tau_{u}\operatorname{Tr}\left(\mathbf{h}_{u}\mathbf{h}_{u}^{H} \mathbf{X}\right),\ \forall u\in\mathcal{U} \tag{13}\]  
where \(\zeta_{k}\) (\(\in(0,1]\)) is the energy-harvesting efficiency, which is determined by the circuit. Without loss of generality, we set \(\zeta_{1}=\cdots=\zeta_{U}=1\) for simplicity. However, any constant efficiency values can be integrated into our energy-harvesting optimization problem. When the energy requirement from the \(u\)-th EHD is represented by \(\mathcal{Q}_{u}^{\text{Req}}>0\), the required charging time at the \(u\)-th EHD to meet the energy requirements is given by \(\tau_{u}=\dfrac{Q_{u}^{\text{Req}}}{\operatorname{Tr}(\mathbf{h}_{u}\mathbf{h }_{u}^{H}\mathbf{X})}\). On the other hand, obtaining perfect channel state information (CSI) at the UAV for energy beamforming design requires additional resources and costs for channel estimation, which are not expected to be feasible in practice due to the high energy consumption and computational complexity [2; 4; 5; 15]. Therefore, we explore channel statistics only in the joint design of UAV altitude and energy beamforming. In the context of optimization for energy harvesting, we focus on the long-term channel statistics, i.e., \(\mathbb{E}\left[\mathbf{h}_{u}\mathbf{h}_{u}^{H}\right]\), rather than the perfect CSI of \(\mathbf{h}_{u}\mathbf{h}_{u}^{H}\)[4; 5; 7]. To this end, we consider the expected required charging time at the \(u\)-th EHD, which is represented as  
\[\tau_{u}=\dfrac{Q_{u}^{\text{Req}}}{\operatorname{Tr}(\mathbb{E}[\mathbf{h}_{ u}\mathbf{h}_{u}^{H}]\mathbf{X})},\ \forall u\in\mathcal{U}. \tag{14}\]  
In consideration of the different energy requirements of EHDs, i.e., \([Q_{u}^{\text{Req}}]_{u=1}^{U}\), the overall charging time required for all EHDs to meet their energy demands becomes \(\max(\tau_{1},...,\tau_{U})\)[3; 6; 7]. Our design objective is to jointly optimize UAV altitude and energy beamforming to minimize the overall charging time required to meet the energy demands of all the EHDs by leveraging only channel statistics. Consequently, our design problem is formulated as  
\[\mathbf{P}_{1}: \min_{h,\mathbf{X}}\max(\tau_{1},...,\tau_{U})\] s.t. \[\tau_{u}=\dfrac{Q_{u}^{\text{Req}}}{\operatorname{Tr}\left( \mathbb{E}[\mathbf{h}_{u}\mathbf{h}_{u}^{H}]\mathbf{X}\right)},\ \forall u\in\mathcal{U}, \tag{15}\] \[\operatorname{Tr}(\mathbf{X})\leq P,\] (16) \[\mathbf{X}\succeq\mathbf{0},\] (17) \[H_{\text{min}}\leq h\leq H_{\text{max}}. \tag{18}\]  
In problem \(\mathbf{P}_{1}\), constraints (16) and (17) follow from the transmit sum power constraint and (2), respectively, and the constraint (18) indicates the UAV altitude range with \(h\in[H_{\text{min}},H_{\text{max}}]\). In practical UAV operations, altitude adjustments should consider regulatory constraints such as maximum allowable altitude, no-fly zones, and airspace restrictions, as well as practical factors like energy consumption and safety management [33; 34; 35]. Additionally, from (3), \(\mathbb{E}[\mathbf{h}_{u}\mathbf{h}_{u}^{H}]\) in (15) is expressed as  
\[\mathbb{E}[\mathbf{h}_{u}\mathbf{h}_{u}^{H}] =\beta_{0}d_{u}^{-a_{u}}\left(\dfrac{\mathcal{K}_{u}}{\mathcal{K} _{u}+\mathbf{1}}\mathbf{a}_{u}\mathbf{a}_{u}^{H}+\dfrac{\dfrac{1}{\mathcal{K} _{u}+1}\mathbf{I}_{N}}{\text{NLoS}}\right)\] \[\triangleq\mathbf{G}_{u}(h), \tag{19}\]where \(d_{u}\), \(\alpha_{u}\), and \(\mathcal{K}_{u}\) are optimization functions for the variable \(h\) as well as the horizontal distance between the UAV and \(u\)-th EHD, i.e., \(r_{u}\). In (19), \(\frac{\mathcal{K}_{u}}{K_{u}+1}\mathbf{a}_{u}\mathbf{a}_{u}^{H}\) and \(\frac{1}{K_{u}+1}\mathbf{I}_{N}\) indicate the LoS and NLoS components in the long-term channel statistics, respectively.