## 3 System Model and Problem Formulation
### UAV-Aided Wireless Power and Information Transfer Model  
As shown in Figure 1, we consider a multi-UAV aided wireless power and information transfer network, where an \(M\) number of UAVs replenish energy and collect the information for an \(N\) number of SDs within given area. The UAVs are indexed by the set \(\mathcal{M}=\{1,2,\cdots\cdots,M\}\), and the SDs are indexed by the set \(\mathcal{N}=\{1,2,\cdots\cdots,N\}\). In a three-dimensional Cartesian coordinate system, the coordinate of the \(i\)th UAV can be expressed as \((x_{i},y_{i},H)\), and the coordinate of the \(j\)-th SD is represented as \(\big{(}w_{j},0\big{)}\). The UAV cruises at an economic speed \(V_{e}\) for energy saving. Let \(\mathfrak{q}(t)=[x_{i}(t),y_{i}(t)]\) denote the coordinate projected by the \(i\)-th UAV on the plane \(\beta\), then the flight distance of the UAV can be given. The main parameters are summarized in Table 1.  
Figure 1: Illustration of multi-UAV aided wireless power and information transfer. Assuming that a UAV starts from and returns to the depot, it should immediately fly back to the depot if the energy is carried below a certain threshold. Let the flight period be \(T\), then  
\[q_{i}(0)=q_{i}(T)=(x_{0},y_{0}),i\in\mathcal{M} \tag{1}\]  
Assuming that the UAV is flying at a fixed altitude H, H is then the minimum height at which obstacles such as buildings, trees, and streetlights can be avoided. Moreover, there are three modes of power consumption: flight, hovering, and power transmission. The first two are driven by propulsion power and shown as follows,  
\[P_{pro}(V)=P_{0}\left(1+\frac{3V^{2}}{{U_{tip}}^{2}}\right)+P_{i}\left(\sqrt{1+ \frac{V^{4}}{4{v_{0}}^{4}}}-\frac{V^{2}}{2{v_{0}}^{2}}\right)^{\frac{1}{2}}+ \frac{1}{2}d_{0}\rho sAV^{3} \tag{2}\]  
where \(P_{0}\) and \(P_{i}\) are two constants representing the blade profile power and induced power in hovering status, respectively; \(U_{tip}\) denotes the tip speed of the rotor; \(v_{0}\) is known as the mean rotor induced velocity in hover; \(d_{0}\) and \(s\) are fuselage drag ratio and rotor solidity, respectively; and \(\rho\) and \(A\) denote the air density and rotor disc area [28]. During the flight of the UAV, if the additional losses caused by acceleration and deceleration are ignored, the moving power is  
\[P_{mov}=P_{pro}(V=V_{e}) \tag{3}\]  
and the hovering power is  
\[P_{hov}=P_{pro}(V=0)=P_{0}+P_{i} \tag{4}\]  
The channel power gain at time \(t\) between the \(i\)-th UAV and \(j\)-th SD is  
\[h_{ij}(t)=\frac{\beta_{0}}{\parallel q_{i}(t)-w_{j}\parallel^{2}+H^{2}} \tag{5}\]  
where \(\beta_{0}\) is the channel power gain at a reference distance 1 m, and \(\parallel\cdot\parallel\) is the Euclidean norm. The receiving power of SD at time \(t\in T\) is  
\[P_{rec}=\eta^{\prime}P_{tra}h_{i,j}(t) \tag{6}\]  
where \(P_{tra}\) is the transmitting power of UAV, and \(0\leq\eta^{\prime}\leq 1\) is the energy efficiency of RF-WPT. The UAV's energy utilization rate is denoted as \(\phi\), which is the ratio of the total energy received by the SDs to the total energy consumed by the UAV. Since the UAV energy consumption consists of flight, hovering, and power transmission, then the UAV energy utilization rate is  
\[\phi=\frac{\sum_{j=1}^{N}E_{j,rec}^{i}}{E_{hov}^{i}+E_{hov}^{i}+E_{tra}^{i}} \tag{7}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(M\) & number of UAVs \\ \(N\) & number of SDs \\ \(P_{nov}\) & moving power of the UAV \\ \(P_{hop}\) & hovering power of the UAV \\ \(P_{tra}\) & transmitting power of the UAV \\ \(E_{(pred}^{i}\) & received energy of the \(j\)th SD from the \(i\)th UAV \\ \(E_{hop}^{i}\) & moving energy of the \(i\)th UAV \\ \(E_{tra}^{i}\) & transmitting energy of the \(i\)th UAV \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main notations. Since the amount of data perceived by a SD is not large, it can be considered that the moment a UAV arrives directly above a SD is the moment the UAV receives the information.