## 5 The Recharging Scheme for the Nodes in the RCCH
### The Number of Recharging Rounds  
In this paper, it is regarded that when the node's residual energy is less than 5% of its initial energy, it is dead. Thus, at the beginning of the \((a+1)\)-th round of recharging, the WCV will judges whether or not one or more CNs in the \(k\)-th annulus will die at the end of this round of recharging. That is to say, we need to judge whether Equation (41) is met:  
\[E_{i,j}^{r(a)}(k)-E_{CN}^{k}\times\Big{(}T_{iw}^{i}(a,a+1)/t_{r}\Big{)}\geq 5 \%\times e_{0}\times\Big{(}N_{k}-\pi r_{k}^{2}\rho\Big{)} \tag{41}\]  
\(E_{CN}^{k}\times(T_{w}{}^{i}(a,a+1)/t_{r})\) is the total energy consumption of CNs in the \(k\)-th annulus during \(T_{w}{}^{i}(a,a+1)\).  
By combining Equations (37), (39) and (40), we can get the upper limit value of \(T_{w}{}^{i}(a,a+1)\), and it is then substituted into the left side of Equation (41).  
1. If Equation (41) is met, it can predict that when the WCV finishes the \((a+1)\)-th round of recharging, no common nodes die. Therefore, the upper limit value of \(T_{w}{}^{i}(a,a+1)\) is regarded as the waiting time of the rechargeable nodes between the \(a\)-th round and the \((a+1)\)-th round of recharging.
2. If Equation (41) cannot be met, the maximal value of \(T_{w}{}^{i}(a,a+1)\) that satisfies Equation (41) is taken as the waiting time. In this case, the total energy recharging for nodes in one RCCH is less than \((B-p_{m}l_{j})/n\) in the \((a+1)\)-th round of recharging. Meanwhile, the left side of Equation (41) equals the right side of it. Therefore, when the WCV arrives at the last RCCH in the \((a+1)\)-th round of recharging, one or more common nodes will die and the whole network lifetime ends. Of course, the WCV will not recharge the last RCCH.  
In Case 2, it just ensures that all the nodes in each RCCH of the \(i\)-th annulus are almost dead at the same time when the WCV arrives at the last RCCH in the \((a+1)\)-th round of recharging. Therefore, the efficiency of the \((a+1)\)-th round of recharging is obviously not high enough. In this case, we add the amount of energy recharging for nodes in the \((a+1)\)-th round to the \(a\)-th round, and cancel the \((a+1)\)-th round of recharging. Moreover, it is still necessary to judge whether the \"overcharge\" occurs in the RCCH of the first annulus-sector. In other words, we need to judge whether Equation (42) is met:  
\[R_{i,1}^{(a)}+\frac{(n-j)\left(t_{ij}^{e}+t_{ij}^{m}\right)\times\left(\left(N _{k}-\pi r_{k}^{2}\rho\right)E_{CH}^{i}-\pi r_{i}^{2}\rho E_{CN}^{k}\right)}{t _{r}\left(N_{k}-\pi r_{k}^{2}\rho\right)}\leq\frac{B-p_{m}l_{i}}{n} \tag{42}\]  
According to Equation (37), it is not difficult to know that, \(\frac{(n-j)\left(t_{ij}^{e}+t_{ij}^{m}\right)\times\left(\left(N_{k}-\pi r_{k}^ {2}\rho\right)E_{CH}^{i}-\pi r_{i}^{2}\rho E_{CN}^{k}\right)}{t_{r}\left(N_{k} -\pi r_{k}^{2}\rho\right)}\) refers to the total amount of energy recharging for nodes of the first RCCH in the \((a+1)\)-th round of recharging. If Equation (42) is met, the WCV only needs to do \(a\) rounds of recharging, otherwise it must carry out the \((a+1)\)-th round of recharging.  
Figure 9: The AEBDC recharging sequence diagram.