## 5 The Optimal Working Strategies for Wireless Rechargeable Sensor Networks
### Problem Formulation  
Assume that there are \(N\) nodes, including sensor nodes and relay nodes, deployed in one layer of the coal mine. Each node is equipped with a wireless rechargeable battery. The initial energy of each node is denoted as \(E_{max}\). The minimum energy level to maintain the routine duty of sensor nodes is denoted as \(E_{min}\), where \(E_{min}=\alpha E_{max}\) (\(0\leq\alpha\leq 1\)).  
Before formulating the optimization problem, we should first introduce some constraints that nodes and recharging locomotives should comply with. First of all, all nodes, including sensor nodes and relay nodes, should satisfy the following equation:  
\[r^{i}+r^{i}_{{}_{R}}(t)=r^{i}_{{}_{T}}(t) \tag{12}\]where \(r^{i}\) is the data rate generated by nodes \(i\). For relay nodes, \(r^{i}\) is zero. \(r^{i}_{{}_{R}}(t)\) is the data rate received by node \(i\) at time instance \(t\). \(r^{i}_{{}_{T}}(t)\) is the data rate transmitted by node \(i\) at time instance \(t\). In other words, Equation (12) is the network flow constraint.  
The second constraint is the power usage constraint. Each sensor node should satisfy:  
\[p^{i}(t)=p^{i}_{{}_{R}}(t)+p^{i}_{{}_{T}}(t)=\rho r^{i}_{{}_{R}}(t)+(\Phi_{1}+ \Phi_{2}d^{i}_{i})r^{i}_{{}_{T}}(t) \tag{13}\]  
We want to design cyclic work strategies for the recharging locomotive, that is it will roam periodically along the traveling path and charge each sensor node with a certain amount of energy. The period of each workload of recharging locomotives is denoted as \(\tau\). In order to meet the periodic requirement, the energy consumed during each period should be the same as that recharged, _i.e._,  
\[\int_{\tau}p_{i}(t)dt=U\tau_{i} \tag{14}\]  
where \(U\) is the power used for recharging nodes and \(\tau\) is the length of a single charging period.  
During each period, in order to keep sensor nodes working properly, the energy level of each node should not fall below \(E_{min}\) and should not exceed \(E_{max}\), which is the upper bound of battery energy. Therefore, we have the following constraint:  
\[E_{min}\leq E_{i}(t)\leq E_{max} \tag{15}\]  
There are also some constraints that the wireless energy transfer device should satisfy. First of all, since the wireless energy transfer device should stop by each node, we denote the visiting order of nodes as \(P=\{\pi_{0},\pi_{1},\cdots,\pi_{0}\}\), where \(\pi_{0}\) stands for the sojourn spot at which the locomotive receives services, such as replacing the energy source, checking mechanical structure status, and so on. \(\pi_{i}\) stands for the i-th node visited by the wireless energy transfer device. The length of the traveling path taken by the wireless energy transfer device is denote as \(D_{P}\), and the time spent on roaming is denoted as \(\tau_{P}\). Then, we have,  
\[\tau_{p}=D_{P}/V \tag{16}\]  
where \(V\) is the moving velocity of the recharging locomotive.  
\(\tau\) could be further divided into three parts, that is,  
\[\tau=\sum_{i}\tau_{i}+\tau_{P}+\tau_{s} \tag{17}\]  
where \(\tau_{s}\) is the time spent on staying at the sojourn spot and \(\tau_{i}\) is the time spent on charging the i-th node along the traveling path.  
In order to minimize the interference introduced by locomotives mounted with wireless energy devices when roaming around the sensor networks, we hope that locomotives could stay at sojourn spots as much as possible while all of the Constraints (12)-(17) will be satisfied. Therefore, we have the following optimization problem, OPT-4,  
\[\max \tau_{s}/\tau\] s.t. (12)-(17)  
The objective function stands for our pursuit of more time spent at sojourn spots in each charging period. The optimization variables of OPT-4 are the traveling path \(P,\tau_{s},\tau_{i},\tau,r^{i}_{{}_{R}}(t)\) and \(r^{i}_{{}_{T}}(t)\), and the rest of the symbols stand for the optimization constants.