# Joint Power Charging and Routing in Wireless
## 2 System Model and Problem Formulation
### Energy Charging Cycle  
We consider the scenario of a mobile charging vehicle periodically traveling inside the sensor network and charging each sensor node's battery wirelessly.  
To prolong the network lifetime, there exists a mobile charger \(v\) with a charging radius \(R_{v}\) and an initial power \(P_{v}\), which we assume can charge multiple sensors within its charging radius simultaneously. Similarly to [12; 13; 29], we assume the charging is only concerned with the distance and we ignore the environmental factors for simplicity. We thus define the charging rate of sensor \(n\) as  
\[U_{n}=\left\{\begin{array}{cc}P_{v}u(d_{n,v})&\text{ if }d_{n,v}\leq R_{v}\\ 0&\text{ otherwise}\end{array}\right. \tag{8}\]  
We assume this mobile charger charges the network with a fixed schedule \(S\) and a predefined traveling path \(P\) with a period of \(\tau\). The path \(P\) consists of a set of resident locations denoted as \(\mathcal{M}=\{1,.....,M\}\). We denote the set of nodes covered by location \(m\) as \(\mathcal{S}_{m}\), which is given by  
\[\mathcal{S}_{m}=\{n|d_{n,m}\leq R_{v},n\in\mathcal{N},m\in\mathcal{M}\} \tag{9}\]  
This should satisfy \(\mathcal{N}=\mathcal{S}_{1}\cup\mathcal{S}_{2}\cup...\cup\mathcal{S}_{M}\), which means that all sensors should be covered by all resident locations.  
The charger starts from the first location and charges those nodes in \(\mathcal{S}_{1}\) with a time \(\tau_{1}\). Then, it leaves the first location and moves to the second location with a traveling speed \(v\). When it reachesthe second location, it remains for time \(\tau_{2}\) to charge the nodes in \(\mathcal{S}_{2}\). These steps are repeated until all the locations are visited once. After that, the charger returns to the first location to replace its battery and ready itself for the next charging tour. Thus, the cycle time \(\tau\) can be written as  
\[\tau=\sum_{m\in\mathcal{M}}\tau_{m}+(\sum_{m=1}^{M-1}\frac{d_{m,m+1}}{v}+\frac {d_{M,1}}{v})+\tau_{vac} \tag{10}\]  
where the first term is the time for charging, the second term is the time for moving, and the third term is the rest time.  
For the resident location \(m\in\mathcal{M}\), the charging time \(\tau_{m}\) depends on the charger's residual energy and the node's energy consumption rate. In order to ensure that all the nodes are still alive, the amount of charged energy for sensor \(i\in\mathcal{S}_{m}\) must be greater that the amount of consumed energy during the next cycle \(\tau\), which is expressed as  
\[\tau P_{i}\leq\tau_{m}P_{m}\,\forall i\in\mathcal{S}_{m} \tag{11}\]  
where \(P_{m}\) denotes the charger's residual energy when it reaches the resident location \(m\), which is given as  
\[P_{m}=P_{v}-\sum_{q=1}^{m-1}\sum_{n\in\mathcal{S}_{q}}\tau_{q}P_{n}-\rho\sum_ {q=1}^{m-1}d_{m,m+1} \tag{12}\]  
Here, the second term is the consumed energy for the charging nodes from set \(\mathcal{S}_{1}\) to \(\mathcal{S}_{m-1}\), the third term is the consumed energy for moving, and \(\rho\) is the energy consumption for a moving unit of meters. One example of the charging model is illustrated in Figure 1 with 16 sensors and 4 resident locations. The mobile periodically visits each resident location and charges those sensors in its charging radius. How to find the optimal resident locations and the optimal routing tree to achieve the optimal charging efficiency is discussed in the next section.