# Adaptive Energy Balanced Routing Strategy for Wireless Reschargeable Sensor Networks
## 2 System Model  
As shown in Figure 1, we consider a network with sink node, a WCV and \(n\) nodes that are labeled as \(N=\{1,2,\ldots,n\}\). Nodes are randomly deployed within a circular area of diameter \(L\), and are responsible for sensing and transmitting data to the sink node. The WCV sends wireless signals to nodes through energy transmitting module, and nodes' energy receiving module transforms them into energy. The whole sensing region is divided into different concentric static rings around the sink node. Here, the circumferences with different diameters represent charging paths, and the WCV will stop at the charging points on these paths to charge the nodes in its charging radius of \(R_{C}\). Moreover, the radius of circumference closest to the sink node is \(d\), and each radius of other circumferences increases by \(d\). For the convenience of expression, we marked the charging path of \(d\times v\) away from sink node as \(r_{v}\). Figure 1: Rechargeable Wireless Sensor Network. To ensure that nodes at any location can be charged, the charging points are evenly distributed on the charging path, and the set of charging points \(S_{\rm v}\) is represented as:  
\[S_{\rm v}=\left\{p_{1}^{\rm v},p_{2}^{\rm v},\ldots,p_{u}^{\rm v},\ldots p_{|\rm v |_{\rm v}|}^{\rm v}\right\},1\leq u\leq|S_{\rm v}| \tag{1}\]  
where \(|S_{\rm v}|\) is the number of charging points on the \(r_{\rm v}\), and it meets \(|S_{\rm v}|=l\times v,\ l\geq 4\), in which \(l\) denotes the difference in the number of charging points between adjacent two paths, and its value also shows the number of charging points on \(r_{1}\). When \(l\) becomes larger, charging points on the paths will be denser, and the WCV will stay at each charging point for a shorter time when the charging period is fixed. In addition, \(p_{u}^{\rm v}\) stands for the \(u\)-th charging points on \(r_{\rm v}\). When WCV stays at charging point \(p_{u}^{\rm v}\), it can energize a plurality of sensor nodes within \(R_{\rm C}\). According to [PERSON] propagation model [18; 19], the charging efficiency of node \(I\) can be calculated by Equation (2):  
\[\Pr_{p_{u}^{\rm v},i}=\left\{ \begin{array}{l}\frac{G_{\rm v}G_{\rm v}\eta_{ \rm F}}{L_{\rm p}}(\frac{\lambda}{4\pi(d_{p_{u}^{\rm v},i}+\beta)})^{2},d_{p_{ u}^{\rm v},i}\leq R_{c}\\ 0,d_{p_{u}^{\rm v},i}>R_{c}\end{array} \right. \tag{2}\]  
The equation often serves as a basic model for different applications. In WRSN, WCV transmits RF signals through circularly polarized antennas, which have transmit gain \(G_{\rm s}=8 dBi\). Nodes receive the signal via linearly polarized dipole antennas, which have receive gain \(G_{\rm r}=2 dBi\). \(\lambda\) is the wavelength. \(L_{\rm p}\) is polarization loss, \(\eta\) can be referred as rectifier efficiency and \(\beta\) is a parameter to adjust the Friis' free space equation for short distance transmission. Moreover, \(P_{t}\) is transmitting power and \(d_{p_{u}^{\rm v},i}\) is the distance between node \(i\) and the WCV. When \(d_{p_{u}^{\rm v},i}\leq R_{C}\), the equation is reformulated as \(P_{r}=P_{t}\partial/(d_{p_{u}^{\rm v},i}+\beta)^{2}\) in [19], where \(\partial=4.32\times 10^{-4}\), \(\beta=0.2316\). According to the charging method described above, it is obvious that the nodes may be charged multiple times in a charging cycle due to the overlapping of the charging areas of different charging points, and the energy of the sensor node after charging is:  
\[\hat{E_{i}}=min\{E_{i}+\sum_{\forall p_{u}^{\rm v}}(\Pr_{p_{u}^{\rm v},i} \times t_{\rm v}),E_{max}\} \tag{3}\]  
where \(E_{i}\) is the residual energy of node \(i\) before charging, \(t_{\rm v}\) is the charging time that WCV stays at each charging point on \(r_{\rm v}\), that is, \(\Pr_{p_{u}^{\rm v},i}\times t_{\rm v}\) represents the energy that node \(i\) receives from the WCV at the \(u\)-th charging point on \(r_{\rm v}\). \(E_{\rm max}\) is the maximum storage energy threshold of nodes.