## 2 Experimental Section
### Problem Statement
#### 2.1.2 Coverage Area and Charging Power Model  
In this section, we will illustrate the different types of chargers and why we have selected the directional chargers for deployment.  
A. Omnidirectional Chargers  
According to previous studies by [PERSON] et al. and [PERSON] et al. [7; 8], in the past, the coverage area of the omnidirectional charging model assumed the same charging ability and the equal charging distances. In fact, however, the received power attenuates when the distance increases and eventually the received power will be decreased to zero when a critical distance is reached. The charging coverage area of an omnidirectional charger is assumed to be a disk and the charger is assumed to be at the center of the charging area.  
B. Directional Chargers  
According to previous studies by [PERSON] [12; 13], the coverage area of the charging model is a sector, and products from different companies have charging angles with different ranges. Due to their relatively stable power coverage and low cost, directional chargers are often used in wireless rechargeable sensor networks.  
Figure 2 is a schematic diagram of the coverage area of a directional charger. As shown in Figure 2, the directional charger \(a_{i}\) with angle \(\theta\) charges the sensor \(s_{i}\), and the radius is \(r\). The arrow is directed out from the face of the charger. \(\overset{\rightarrow}{\mathrm{v}}\) is a unit vector of the charger face, and \(\overset{\rightarrow}{u}\) is the vector formed by the charger and the sensor. According to [PERSON] and [PERSON] [13], the following Equation can be used to determine whether the sensor \(s_{i}\) is in the coverage area.  
\[D(a_{i},\overset{\rightarrow}{\mathrm{v}},s_{j})=\begin{cases}1,&if\overset{ \rightarrow}{\mathrm{v}}\cdot\overset{\rightarrow}{u}-||\overset{ \rightarrow}{u}||_{2}\mathrm{cos}(\frac{\theta}{2})\geq 0\\ 0,&if\overset{\rightarrow}{\mathrm{v}}\cdot\overset{\rightarrow}{u}-|| \overset{\rightarrow}{u}||_{2}\mathrm{cos}(\frac{\theta}{2})<0\end{cases} \tag{1}\]  
Equation (1) determines whether the sensors are in the coverage area, and when the value is 1, the charger's charging power is calculated according to Equation (2).  
\[E(dist(a_{i},s_{j}))=\begin{cases}\frac{\alpha}{(dist(a_{i},s_{i})+\beta)},& dist(a_{i},s_{i})\leq r\\ 0,&dist(a_{i},s_{i})>r\end{cases} \tag{2}\]  
In Equation (2), we define \(\mathrm{dist}(a_{i}S_{j})\) as the Euclidean distance between the location of charger \(a_{i}\) and the location of sensor \(s_{j}\), and \(\alpha=\frac{5W_{0}G_{n}G_{s}}{C_{W}}(\frac{\lambda}{4\pi})^{2}\) where \(\lambda\) is the wavelength, and \(\zeta\) is the efficiency of the rectifier. \(G_{a}\) and \(G_{s}\) are the antenna gain of the charger and the sensor, respectively. \(W_{0}\) is the charging power, \(C_{w}\) is the antenna polarization loss, and \(\beta\) is the compensation parameter for short range transmission [17]. When the distance is larger than the charging radius \(r\), the charging power  
Figure 2: Coverage area of a directional charger.  
is 0. According to [PERSON] et al. [18], although the total power may not be precise, the total power can still be expressed as a function of the following Equation (3).  
\[W(j)=\sum_{i\in M}D(a_{i},\overset{\rightharpoonup}{v},s_{j})E(dist(a_{i},s_{j})) \tag{3}\]