# A Study on Wireless Charging for Prolonging the Lifetime of Wireless Sensor Networks
## 1 Introduction  
In recent years, many works have focused on employing wireless energy conversion technology to charge the nodes in wireless sensor networks [1]. The wireless charging models can be simply divided into point-to-point model [2; 3; 4; 5] and point-to-multipoint model [6; 7; 8; 9]. In the former case, a mobile charger (MC), or a charging car, can only charge a single sensor node. In the latter case, the MC can charge more than one sensor node simultaneously. In this paper, we focus on developing an optimal wireless charging strategy based on the point-to-point model to prolong the lifetime of sensor networks.  
Many works have studied wireless charging based on the point-to-point model [2; 3; 4; 5]. [PERSON] et al. proposed a charging scheme based on the residual energy of sensor nodes [3]. However, such a method ignored the travelled distance of the MC. The extra travelled distance may waste energy of the MC. In [2; 4; 5], researchers employed the traveling salesman problem (TSP) to design the moving path of the MC. They selected the shortest distance between sensor nodes as the variable. However, it did not consider the factor of residual energy of nodes. Some works [2; 3; 10] utilized a charging strategy in which the charging car completely charges the sensor nodes one by one. Such a method did not arrange the charging time for each sensor node. It may result in that the sensor nodes with very low energy at the end of the moving path will turn off before the MC arrives.  
The traveling salesman problem (TSP) is one of the most famous algorithms in the field of mathematics. Assuming that there is a traveler who wants to visit \(N\) cities, he has to choose a path to follow. The path constraint is that the traveler visits each city once, and returns to the starting point. The TSP algorithm aims to find out the shortest distance of all moving paths.  
In this paper, we focus on developing a wireless charging strategy, which is called integrated path and charging time scheme (IPCTS), to prolong the lifetime of the sensor network. The IPCTS is studied based on the analysis of the travelled distance of MC and the residual energy of the sensor nodes. We theoretically analyze the limited size of the sensor network to match the charging capability of a MC. We consider the networked parameters as the inputs of a Hamilton circuit problem under a given networked size. For the charging time allocation problem, we transform the maximum network lifetime into charging time problem by using linear programming. The main contributions of this paper are as follows:  
* We first theoretically analyze the limited size of the sensor network to match the charging capability of a MC.
* We consider two factors to design the moving path of the MC, including the residual energy of nodes and the distance between nodes.
* To maximize the lifetime of network, we utilize the linear programming to compute the charging time of each node.  
The remainder of this paper is organized as follows: in Section 2, we review the related work on wireless sensor networks published in recent years. Section 3 introduces our model and some assumptions. In Section 4, the upper and lower limits of network size are analyzed. Section 5 designs the moving path of a MC by the TSP problem. Section 6 models the charging time and solves the optimization problem to obtain the charging time of each node. Section 7 introduces the charging period. Section 8 presents simulation results. Section 9 concludes this paper.