# Optimizing Charging Pad Deployment by Applying a Quad-Tree Scheme
## 3 Optimizing the Charging Pad Deployment Problem by Applying the Quad-Tree Scheme
### Problem, Notation and System Model
#### 3.1.2 Drone Energy Consumption Model  
For the convenience of later discussion, we use the symbols listed in Table 1 to represent some characteristics of the drone and sensors.  
Under the assumption that the drone arrives when the sensor is either at or near depletion of its battery, the drone needs to recharge the sensor with an amount of energy denoted as \(E_{sensor}\). Considering the charging efficiency, the drone can recharge the sensor with a rate of \(\rho\times P_{speed}\) energy per second.  
Thus, the time required for a drone to complete the charging of a sensor upon reaching its position is as follows:  
\[T_{charge}=\frac{E_{sensor}}{\rho\times P_{speed}} \tag{1}\]  
Considering that the drone hovers while charging the sensor, the energy required for the charging task needs to account for the energy consumed for hovering:  
\[E_{charge}=E_{sensor}/\rho+P_{hover}\times T_{charge} \tag{2}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols** & **Meaning** \\ \hline \(E_{sensor}\) & the maximum battery energy of the sensor (J) \\ \(E_{max}\) & the maximum battery energy of the drone (J) \\ \(E_{charge}\) & the energy required for charging a sensor (J) \\ \(P_{fly}\) & the power consumption of the drone during flight (J/s) \\ \(P_{hover}\) & The power consumption for drone hovers during the charging process (J/s) \\ \(P_{speed}\) & the charging speed (J/s) \\ \(T_{charge}\) & the time to charge a sensor (s) \\ \(V_{fly}\) & the flying speed of the drone (m/s) \\ \(D_{maxcharging}\) & The maximum flying distance of the drone for charging tasks (m) \\ \(D_{maxpl}\) & The maximum flying distance of the drone for moving between pads (m) \\ \(\rho\) & the charging efficiency of the drone to the sensor (percentage) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameters used to derive the maximum flight distance of the drone.  
Figure 1: A schematic network layout; the base station is located at the center of the network, and the drone can travel directly or via pads to reach sensor locations for recharging tasks.  
After subtracting the energy used for the recharging task, the drone has a remaining energy of \(E_{max}-E_{charge}\) available for flying. Since the drone must return after completing the recharging task, the flying distance must be divided by 2. Therefore, the maximum distance at which the drone can perform recharging tasks can be calculated as follows:  
\[D_{maxcharging}=\left(E_{max}-E_{charge}\right)/P_{fly}\times V_{fly}\times\frac{ 1}{2} \tag{3}\]  
On the other hand, when the drone flies between pads, it does not need to reserve energy for recharging tasks or the return trip, as it can be fully recharged at the destination pad. Therefore, the maximum distance between two pads is determined by Equation (4):  
\[D_{maxpad}=E_{max}/P_{fly}\times V_{fly} \tag{4}\]  
Assuming the sensor remains in the sleeping state while being charged, its energy consumption is not considered. Furthermore, to further simplify the problem, both the charging efficiency and the energy consumption for hovering are disregarded; in other words, \(\rho\) = 1 and \(P_{hover}=0\).  
Thus,  
\[D_{maxcharging}=\frac{E_{max}-E_{sensor}}{P_{fly}}\times V_{fly}\times\frac{1}{2} \tag{5}\]  
\[D_{maxpad}=\frac{E_{max}}{P_{fly}}\times V_{fly} \tag{6}\]