## 5 The Recharging Scheme for the Nodes in the RCCH
### Energy Threshold of the Recharging Request  
As mentioned above, the CH election is carried out after each round of data gathering. Therefore, for the convenience of analysis, the total residual energy of all nodes in the same RCCH is taken as a whole. Moreover, nodes are uniformly and randomly distributed in the circular network, and the energy consumption on nodes in the same RCCH could be balanced due to the collaboration-based data forwarding strategy. Thus, it is regarded that the energy consumption rates of each RCCH are approximately equal to each other. When receiving the recharging request from the first RCCH, the WCV starts the first round of recharging. If the residual energy of WCV is exactly zero when it returns back to the initial position at the end of the first round of recharging, the recharging benefits reaches to the maximal value. In this case, the following equation is met:  
\[\sum_{j=1}^{n}R_{i,j}^{(1)}+l_{i}P_{m}=B \tag{31}\]  
In (31), \(\sum\limits_{j=1}^{n}R_{i,j}^{(1)}\) is the total amount of energy that WCV recharges for nodes in all the RCCH of the \(i\)-th annulus during the first round of recharging, and \(l_{i}P_{m}\) is the total energy consumption of WCV when moving in the \(i\)-th annulus. At the same time, for the nodes in the last RCCH (the \(n\)-th RCCH), it needs to ensure that they still alive when the WCV arrives. Thus, for this region, the minimum value of the total energy that ensures nodes alive is defined as \(E_{i}^{min}\). On the other hand, it is easy to know that the CNs in the \(k\)-th annulus have the highest energy consumption rate among all the non-rechargeable nodes as the average data transmission distance of them is the longest, so if the average residual energy of nodes in a RCCH one moment before recharging is equal to that of the CNs in the \(k\)-th annulus at the same moment, it is regarded that the balance of energy consumption is achieved. Therefore, when the WCV arrives at the \(j\)-th annulus-sector of the \(i\)-th annulus the second time, the following equation should be met:  
\[\frac{R_{i,j}^{(1)}+E_{i,j}^{r(1)}-\left(T_{i}^{r}/t_{r}\right)E_{\text{CH}}^ {i}}{\pi r_{i}^{2}\rho}=\frac{E_{i,j}^{r(1)}(k)-\left(T_{i}^{r}/t_{r}\right)E _{\text{CN}}^{k}}{N_{k}-\pi r_{k}^{2}\rho} \tag{32}\]  
In (32), \((T_{i}^{r}/t_{r})/E_{\text{CH}}^{i}\) is the total energy consumption of nodes in the \(j\)-th RCCH of the \(i\)-th annulus during the waiting time of recharging. Similarly, \((T_{i}^{r}/t_{r})/E_{\text{CN}}^{i}\) is the total energy consumption of CNsin each annulus-sector of the \(k\)-th annulus during the waiting time of recharging. In addition, \(\pi{r_{t}}^{2}\rho\) is the average number of nodes in each RCCH of the \(i\)-th annulus. Thus, \(N_{k}-\pi{r_{t}}^{2}\rho\) is the average number of CNs in each annulus-sector of the \(k\)-th annulus. Moreover, the value of \(E_{i,j}^{r(1)}(k)\) can be further expressed as:  
\[E_{i,j}^{r(1)}(k)=E_{\text{CN}}^{k}\times\left(T_{i}^{r}-\sum_{j=1}^{j}\left(t _{ij}^{c}+t_{ij}^{m}\right)\right)/t_{r}+E_{i,n}^{r(1)}(k) \tag{33}\]  
In (33), \(E_{i,n}^{r(1)}(k)\) is defined as total residual energy of all the CNs in one annulus-sector of the \(k\)-th annulus at the moment when the WCV arrives at the \(n\)-th RCCH of the \(i\)-th annulus the first time. Meanwhile, the total residual energy of nodes in the \(n\)-th RCCH is approximately equal to \(E_{i}^{min}\) at this moment. When the total residual energy of nodes in the \(n\)-th RCCH is equal to \(E_{i}^{min}\), the network has already run for \(\left(\pi{r_{i}^{2}}\rho e_{0}-E_{i}^{\min}\right)/E_{CH}^{i}\) rounds. Therefore, it is not difficult to know that:  
\[E_{i,n}^{r(1)}(k)=\left(N_{k}-\pi{r_{k}^{2}}\rho\right)e_{0}-\left(\left(\pi{r _{i}^{2}}\rho e_{0}-E_{i}^{\min}\right)/E_{CH}^{i}\right)\times E_{CN}^{k} \tag{34}\]  
By combining Equations (31)-(34), the value of \(E_{i,j}^{r(1)}\) can be calculated. Without loss of generality, \(E_{i,1}^{r(1)}\) is regarded as the energy threshold of the recharging request.