## 3 Network Model  
Similar to [17; 30], the network is regarded as a circular region with a radius \(R\) and divided into \(k\) virtual concentric annuli with the same width. Meanwhile, this circular network is also divided into \(n\) sectors of the same size. That is to say, the network is composed of \(nk\) annular sectors. For example, to illustrate the model more clearly, we divide the network into four virtual concentric annuli, as shown in Figure 3. In addition, this network also consists of six sectors (the red area). Thus, there are 24 annulus-sectors (the yellow area) in the network. It is not difficult to know that, the number of these annulus-sectors will increase with the expansion of network scale, which enhances the flexibility of our model.  
A static base station (BS) is located at the center of the network, and \(N\) rechargeable sensor nodes are uniformly and randomly distributed in the network. Without loss of generality, the density of nodes is denoted as \(\rho\). Moreover, the communication radius as well as the battery capacity of each node is defined as \(r_{t}\) and \(C\), respectively. It is regarded that nodes in each annulus-sector form a cluster, and they periodically send packets to the cluster heads (CHs) within one hop. Meanwhile, the CHs transmit all the collected data to BS in a multi-hop manner. What needs to be pointed out is that the CHs which are closest to the center of the network send their data directly to the BS. The definitions of parameters used in the network are given in Table 1.  
It has been proved that the free space model cannot be completely applied to WSNs. Therefore, the classic energy consumption model for sensor nodes proposed by [PERSON] is adopted in this paper [18]. For simplicity, we ignore the energy consumption for computation and data storage.  
\[E_{t}=\left\{\begin{array}{ll}1E_{elec}+l\varepsilon_{fs}d^{2},&d\leq d_{0}\\ 1E_{elec}+l\varepsilon_{amp}d^{4},&d>d_{0}\end{array}\right. \tag{1}\]  
\[E_{r}=lE_{elec} \tag{2}\]  
\(E_{t}\) and \(E_{r}\) denote the energy consumption for sending and receiving \(l\) bit of data. In addition, \(E_{dec}\) is the unit energy consumption of the transmitting and receiving circuit. \(\varepsilon_{fs}\) and \(\varepsilon_{amp}\) are regarded as the coefficients of energy consumption during communication in free space transmission and multi-path fading transmission, respectively. Moreover, \(d\) is defined as the Euclidean distance between the sender and the receiver, while \(d_{0}\) denotes the threshold distance.  
According to the network model as well as the energy consumption formula, it is known that the CHs close to BS are heavily loaded nodes. They exhaust their energy quickly, and thus energy holes appear, so in this paper, the \"Region for Candidate CHs\" (RCCH) is firstly constructed in each annulus-sector which ensures the balance of energy consumption in CH election. Then, the data packets are forwarded by the collaborative Candidate Cluster Headers (CCHs), which further balance the load of CHs. Finally, \(k\) WCVs are assigned to \(k\) annuli, and the recharging strategy for the RCCH is proposed under the constraint of the maximal network lifetime. The energy consumption of the whole network is effectively balanced, and the \"energy hole problem\" is also alleviated to some extent. Without loss of generality, the moment when a dead node appears is defined as the end of the network lifetime.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Definition** & **Unit** \\ \hline \(R\) & Network Radius & m \\ \(k\) & Number of Annuluses & - \\ \(n\) & Number of Sectors & - \\ \(N\) & Total Number of Nodes & - \\ \(\rho\) & Density of Nodes & \(1/\text{m}^{2}\) \\ \(r_{t}\) & Communication Radius of Node & m \\ \(C\) & Battery Capacity of Node & J \\ \hline \hline \end{tabular}
\end{table}
Table 1: Definitions of parameters used in the network.  
Figure 3: Network Model.