## 5 The Recharging Scheme for the Nodes in the RCCH
### The Amount of Energy Provided by the WCV in Each Round of Recharging  
It is known from the above analysis that, for each node being recharged in the \(i\)-th annulus, the time interval between the first and the second recharging is \(T_{i}^{r}\). Furthermore, the energy consumption rates of the common nodes are relatively stable, and the WCV moves at a constant speed. Thus, for different RCCHs of the same annulus, the energy supplemented from the WCV after the first round of recharging are the same in each round of recharging. To further optimize the recharging strategy of WCV, we analyze the waiting time of the rechargeable nodes.  
\(T_{w}{}^{i}(a,\,a+1)\) is defined as the waiting time of the rechargeable nodes in the \(i\)-th annulus from the end of the \(a\)-th round of recharging to the beginning of the \((a+1)\)-th round of recharging. Thus, \(T_{w}{}^{i}(1,\,2)=T_{i}^{r}\), as shown in Figure 9a. However, it is not reasonable if this waiting time is still \(T_{i}^{r}\) after the second round of recharging. In this case, the residual energy of the rechargeable nodes is still high, so there is no need to recharge it again for the time being. On the other hand, the frequent movement of WCV will consume more energy and also result in low recharging efficiency.  
For this reason, we discuss the situation after the second round of recharging. To ensure that the average residual energy of nodes in a RCCH is equal to that of the CNs in the \(k\)-th annulus at the moment when the WCV arrives at this RCCH for the \(a\)-th time (\(a\geq 2\)), Equation (35) should be met:  
\[E_{i,j}^{r(a)}/\pi{r_{t}^{2}}\rho=E_{i,j}^{r(a)}(k)/\left(N_{k}-\pi{r_{k}^{2} }\rho\right) \tag{35}\]  
Similarly, when the WCV arrives at the same RCCH for the \((a+1)\)-th time, the above constraint still needs to be met. According to Equation (32), it is known that:  
\[\frac{R_{i,j}^{(a)}+E_{i,j}^{r(a)}-\left(T_{w}^{i}(a,a+1)/t_{r}\right)E_{CH}^ {i}}{\pi{r_{t}^{2}}\rho}=\frac{E_{i,j}^{r(a)}(k)-\left(T_{w}^{i}(a,a+1)/t_{r} \right)E_{CN}^{k}}{N_{k}-\pi{r_{k}^{2}}\rho} \tag{36}\]By substituting Equation (35) into (36), the amount of energy (denoted as \(R_{i,j}{}^{(a)}\)) that the WCV recharges for the nodes in this RCCH for the \(a\)-th time could be expressed as:  
\[R_{i,j}^{(a)}=\frac{T_{iw}^{i}(a,a+1)\Big{(}(N_{k}-\pi r_{k}^{2}\rho)E_{CH}^{i}- \pi r_{l}^{2}\rho E_{CN}^{k}\Big{)}}{t_{r}\big{(}N_{k}-\pi r_{k}^{2}\rho\big{)}} \tag{37}\]  
From Equation (37), we can find that the value of \(R_{i,j}{}^{(a)}\) is unrelated to the value of \(j\). It means that the amount of energy for recharging the nodes in each RCCH of the \(i\)-th annulus is unchanged after the second round of recharging. On the other hand, after finishing the \(a\)-th round of recharging, the residual energy of WCV should be enough to ensure that it can return back to its initial position. That is:  
\[B\geq R_{i,j}^{(a)}\times n+p_{m}l_{i} \tag{38}\]  
so, the value of \(R_{i,j}{}^{(a)}\) needs to meet Equation (39):  
\[R_{i,j}^{(a)}\leq(B-p_{m}l_{i})/n \tag{39}\]  
In addition, to avoid energy waste, it is necessary to ensure that there is no \"overcharge\". That is, the energy recharging for a node plus its residual energy should not be greater than its maximum battery capacity. It is not difficult to conclude that, during each round of recharging, when the WCV arrives at the first RCCH, the average residual energy of nodes in this RCCH is higher than that of the nodes in other RCCHs when the WCV arrives at them. Therefore, it only need to ensure that the nodes in the first RCCH do not overcharged. That is:  
\[R_{i,1}^{(a)}\leq\pi r_{i}^{2}\rho e_{0}-E_{i,1}^{r(a)} \tag{40}\]  
According to Equations (37), (39) and (40), the value of \(R_{i,j}{}^{(a)}\) can be calculated.