# A Study on Wireless Charging for Prolonging the Lifetime of Wireless Sensor Networks
## 3 Model and Hypothesis  
In this work, we assume that there is a MC, a base station and \(n\) similar sensor nodes in the network. The base station can acquire the remaining energy and the energy consumption rate of the sensor nodes. It generates a charging strategy for the MC. The charging strategy includes the charging path of the MC and the charging time of each sensor node.  
The MC moves with a constant velocity starting from the base station. Therefore, the energy consumption of the MC is only related to the moving distance. The maximum amount of power carried by MC is denoted as \(E_{MC}\). When a charging period is finished, the MC will return to the base station to replenish its power. Assume that the output power of MC is \(P_{out}\), and the wireless output range is fan-shaped, then only one node can be charged. The reception power of sensor node is denoted as \(P_{in}\). The relationship between the output power of MC and the reception power of sensor node is [10]:  
\[P_{in}=P_{out}\times\mu(d) \tag{1}\]  
In Equation (1), \(\mu(d)\) is a function of the distance between the MC and sensor node within a range of [0, 1]. In this work, we assume that the MC moves closely to the sensor node, thus, \(\mu(d)=1\), \(P_{in}=P_{out}\).  
We randomly deployed \(n\) sensor nodes in a rectangular area of dimensions \(L\times L\) in the network. The coordinates of the sensor nodes are denoted as \((x_{i},y_{i})\). Each sensor node is equipped with a rechargeable battery with a maximum capacity of \(E_{t}\). The energy consumption rate of node \(i\) is denoted as \(C_{i}\) with a range of \([C_{\min},C_{\max}]\). The nodes are not equal to probability values. We assume that the energy consumption rate is subject to a Gaussian distribution. In the real scenario, the energy consumption rate of sensor nodes is mostly around the mean value. Only a small part is particularly high or low. Let the energy consumption rate of the sensor node be denoted as \(X\), which is subjected to a Gaussian distribution with mean \(\mu_{x}\) and variance \(\sigma_{x}^{2}\). Table 1 describes the symbols used in this paper and their respective meanings.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Meaning** \\ \hline \(E_{t}\) & total battery capacity of sensor node \\ \(\lambda_{i}\) & remaining energy of sensor node \(i\) \\ \(P_{out}\) & output power rate of MC \\ \(P_{in}\) & received power rate of sensor node \\ \(V\) & set of charging nodes \\ \(C_{i}\) & energy consumption rate of sensor node \(i\) \\ \(s_{i}\) & sensor node \(i\) \\ \(C_{MC}\) & moving energy consumption of MC \\ \(v\) & velocity of MC \\ \(E_{MC}\) & total energy of MC \\ \(n\) & number of sensor node \\ \(n_{up}\) & maximum network size \\ \(n_{low}\) & minimum network size \\ \(T\) & charging period \\ \(T_{charge}\) & charging time of MC \\ \(T_{travel}\) & moving time of MC \\ \(T_{back}\) & rest time of MC at base station \\ \(t_{schar}\) & the time to replenish electricity of MC \\ \(E_{min}\) & remaining energy threshold \\ \(d_{ij}\) & distance between sensor node \(i\) and node \(j\) \\ \(t_{i}\) & charging time of sensor node \(i\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of notations.