# A Study on Wireless Charging for Prolonging the Lifetime of Wireless Sensor Networks
## 2 Related Work  
Wireless power transfer technology can be divided into three categories: inductive coupling [11], microwave radiation [12] and magnetic resonance coupling [1]. In [13], [PERSON] et al. were the first to employ wireless power conversion technology to prolong the lifetime of network by deploying a charging car into wireless sensor network. Since the wireless energy can be radiated in all directions and in a particular direction, the charging model of MC can be divided into point-to-point model and point-to-multipoint model. Based on the point-to-point charging model, [PERSON] et al. did some work on how to maximize the network lifetime in [3; 14]. In [3], the authors first proved that the charging problem is a non-deterministic polynomial complete (NPC) problem. Then, they proposed two heuristic algorithms known as Greedy and GreedyPlus. The Greedy algorithm takes the remaining energy of sensor node ascending order as the charging sequence, and the required charging time to prolong network lifetime. The GreedyPlus takes a fixed time as the charging time for each sensor node by a binary search. However, these heuristic algorithms do not consider the travelled distance of the MC, which could increase the MC's energy consumption and reduce the charging time for sensor nodes. In [14], the authors computed the charging time by a binary search. As for charging sequence, the authors analyzed three routing protocols: energy-minimum routing, energy-balanced routing and charging-aware routing. According to the routing information, the nodes with large energy consumption in the network are considered as the charging nodes; and the remaining energy ascending order is used as the charging sequence. In [2; 4; 5], the authors minimized the number of MCs needed to ensure that the network could run persistently. In [2], the authors divided the whole network area into several sub-regions by a K-means clustering algorithm. Then, they dispatched a MC for each sub-region to charge the sensor nodes in the sub-region. In [5], the authors designed a Hamiltonian circuit between all nodes as the shortestcharging path for the MC. Then, the charging path was divided into several sub-paths based on the limited energy of the MC. Finally each sub-path dispatched a MC. In [2], [PERSON] et al. proposed a _q-root_ TSP to dispatch multiple MCs, and minimize the number of deployed MCs. The authors first introduced a tree decomposition algorithm with a provable 5-approximation ratio to find \(q\) closed tours. Then, an approximation algorithm was proposed to minimize the number of MCs, according to the total distance of each tour. However, a drawback of this strategy is that the complexity grows exponentially with the number of to-be-charged nodes. There was another study on maximizing network utility which is expressed as the amount of data collected in the network [15; 16]. In [16], the authors first selected a subset of the sensor nodes as anchor. The sencar (called a multifunctional mobile entity) stops at the anchor position to charge the anchor and collect data. Besides, the authors proposed an anchor selection algorithm based on remaining energy to control the travelled distance of the sencar within a given threshold. In [15], the authors combined flow conservation, energy balance and battery capacity as the constraints to maximize the utility function and turn it into a non-convex optimization problem. Finally, the authors transformed the original problem into a convex optimization method by using auxiliary variables. As for point-to-multipoint charging model, researchers [6; 17; 18] maximized charge efficiency. [PERSON] et al. took the charge efficiency as an objective function and formulated a mixed nonlinear program [17; 18]. In [18], authors first addressed an ideal case assuming zero traveling time for the charger. By adopting the discretization and logic point representation techniques, a provably near optimal solution is obtained.