## 5 The Recharging Scheme for the Nodes in the RCCH
### The Moving Speed of WCV  
From Section 5.2 to Section 5.4, it can be seen that the total amount of energy recharging for nodes is not always the same in different rounds. Moreover, the value of \(T_{i}{}^{r}\) is unchanged during the network lifetime, so the value of \(v_{WCV}\) needs to be discussed in this Section. It is assumed that the movement of a WCV from one RCCH to another is a uniform linear motion. When \(a=1\), according to Equation (32), it is known that:  
\[R_{i,j}^{(a)}=\left(\frac{E_{i,j}^{r(1)}(k)-\left(T_{i}^{r}/t_{r}\right)E_{CN} ^{k}}{N_{k}-\pi r_{k}^{2}\rho}\right)\pi r_{i}^{2}\rho+-(T_{i}^{r}/t_{r})E_{CH} ^{i}-E_{i,j}^{r(1)} \tag{43}\]  
When a \(>1\),  
* If Equation (41) is met, the value of \(R_{i,j}^{(a)}\) should be calculated by Equation (37).
* If Equation (41) cannot be met but Equation (42) is met, the right expression of Equation (37) should be substituted into Equation (42). In this case, the \(a\)-th round of recharging is the last round, and the value of \(R_{i,j}^{(a)}\) is: \[R_{i,j}^{(a)}=\left(T_{w}^{i}(a,a+1)+(n-j)\Big{(}t_{ij}^{c}+t_{ij}^{m}\Big{)} \right)\times\frac{\left((N_{k}-\pi r_{k}^{2}\rho)E_{CH}^{i}-\pi r_{i}^{2} \rho E_{CN}^{k}\right)}{t_{r}\big{(}N_{k}-\pi r_{k}^{2}\rho\big{)}}\] (44)
* If both Equations (41) and (42) are not met, the amount of energy recharging for each RCCH in the \(a\)-th round is calculated by Equation (37), and the amount of energy recharging for each of them in the \((a+1)\)-th round is: \[R_{i,j}^{(a+1)}=\frac{(n-j)\Big{(}t_{ij}^{c}+t_{ij}^{m}\Big{)}\times\left((N_ {k}-\pi r_{k}^{2}\rho)E_{CH}^{i}-\pi r_{i}^{2}\rho E_{CN}^{k}\right)}{t_{r} \big{(}N_{k}-\pi r_{k}^{2}\rho\big{)}}\] (45)  
As shown in Figure 8(b), the \((a+1)\)-th round of recharging is the last round, while in Figure 8(a), the last round is the \(a\)-th round of recharging, which is one round less than the case in Figure 8(b).  
According to Equation (43), it is easy to see that during the first round of recharging, the amount of energy recharging for each RCCH increases with the increase of \(j\). Thus, the value of \(t_{ij}{}^{c}\) is also increasing. However, the value of \(T_{i}{}^{r}\) is always unchanged, so the moving speed of WCV on different segments should be increased one by one. Moreover, the moving path length of WCV is equal between any two adjacent RCCHs in the same annulus. Thus, in Figure 8(a), the length of \(t_{ij}{}^{m}\) decreases with the increase of \(j\) in the first round of recharging.  
In addition, it can be seen from Equation (37) that from the second round of to the penultimate round of recharging, the amount of energy recharging for each RCCH is equal to each other in the same round. Therefore, the value of \(v_{WCV}\) on different segments should be the same in this case, and the length of \(t_{ij}{}^{m}\) shown in Figure 8(a) is unchanged after the first round of recharging.  
From Equations (44) and (45), it also knows that in the last round of recharging, the amount of energy recharging for each RCCH is constantly decreasing. Furthermore, the value of \(t_{ij}{}^{c}\) is also decreasing with the increase of \(j\). Thus, the moving speed of WCV on different segments should be gradually decreased.  
In summary, the relationship among the moving speed of WCV, the amount of energy for recharging and the time for one round of recharging can be expressed as follows:  
\[R_{i,j}^{(a)}/P_{c}+(l_{i}/n)/v_{WCV}=T_{i}^{r}/n \tag{46}\]That is:  
\[v_{WCV}=P_{c}l_{i}/n\left(P_{c}T_{i}^{r}-R_{i,j}^{(a)}\right) \tag{47}\]  
From Equation (47), it is easy to know that the value of \(v_{WCV}\) increases with the increase of \(R_{i,j}^{(a)}\) which is relevant to \(t_{ij}^{c}\). Therefore, the variation of \(v_{WCV}\) is relevant to the recharging time \(t_{ij}^{c}\), which is consistent with the above analysis. Figure 9 is the sequence diagram about each round of recharging.