# Joint Power Charging and Routing in Wireless
## 1 Introduction
For cluster-based WSNs, an energy-efficient cooperative transmission optimization was proposed in [16], for which the power allocation, power splitting and relay selection were formulated as a non-convex problem, and a distributed iteration algorithm based on fractional programming and dual decomposition was proposed. However, the charging model defined in their work can harvest energy to only one sensor each time, which may cause a longer charging time for large-scale applications. Assuming that the mobile charger can also be a data collector, the joint mobile energy replenishment and data gathering are considered [17; 18; 19]. However, because of downlink (DL) WPT and uplink (UL) information transmission, the energy replenishment and data gathering should work in different time slots, which may induce larger data collecting latency, especially for large-scale applications. In addition, combining the data collector and power transfer together has another inherent drawback, called the \"doubly near-far\" problem [20]; that is, the far-away sensors consume more energy in the UL and harvest less energy in the DL as a result of the distance-dependent power loss. To tackle these problems, a separately located power charger and data collector are considered as a more flexible way to balance the energy and information transmissions [21; 22]. In conventional battery-powered WSNs, data-gathering techniques have been widely investigated in the last decade. In such works, as a result of uneven energy depletion with the distance to the predetermined sink, how to extend the network lifetime by designing a routing tree has attracted much attention. Uniform sensor deployment, resulting in the network lifetime decreasing by the sensors at the first hop from the sink, was first discovered in [23], and is also known as the \"energy hole\" problem. The authors in [24; 25; 26] have proposed several approaches to mitigate this problem. In [24], the authors propose an analytic model to estimate the entire network lifetime from network topology. This method can also be used to determine the boundary of the energy hole in a data-gathering WSN. In [25], the authors present a heuristic relay selection algorithm to balance the whole energy consumption, and a routing scheme based on maximum residual energy is proposed to mitigate the energy hole problem during data gathering. The authors in [26] investigated the energy hole problem and designed guidelines for maximizing the lifetime and avoiding energy holes in sensor networks with a non-uniform distribution. However, it should be noted that there exists a major difference between battery-based WSNs and WRSNs. In battery-based WSNs, one common objective is to minimize the total energy consumption of those sensors near the sink node, thus prolonging the network lifetime. However, such an energy balance-orientated design is not necessarily optimal for WRSNs, because of the fact that a high power consumption of any sensors in any location can now be replenished by means of WPT via the mobile chargers. This clear difference indicates that the routing techniques in WRSNs should be redesigned to fully take advantage of WPT. Unlike existing works, the aim of this work is to design a joint energy replenishment and routing mechanism for WRSNs. Specifically, given the velocity of the mobile charger, we aim to find the best traveling path corresponding with the optimal routing tree. In this way, we mitigate the gap between the heterogeneous energy consumption among the routing tree and the charging efficiency. To the best of our knowledge, this is the first study on the joint optimization for both energy charging and routing. The main contributions of this paper are summarized as follows:  
* We present a novel joint optimization model including both charging efficiency and routing, rather than the typical charging optimization problem considering only the predefined data-gathering route.