## 1 Introduction
The proposed scheme constructs some matching pairs sets by the algorithm presented later in this work. The matching pair with the longest distance is selected first, and two paired nodes are removed from the set. The above process iterates until the number of selected pairs achieves the number of drones being carried by the vehicle, or the node set becomes empty or a single node. In each of a matching pair, the node closer to the base station is allocated to the vehicle, and the other node is allocated to one of the drones. After allocating the nodes, the schedule of the assigned nodes for the vehicle is planned with NJNP starting from the base station, which always considers the nearest sensor to be charged first. Therefore, in the nodes allocated to the vehicle, the node closest to the base station is selected first to be charged by the vehicle in the proposed scheduling algorithm. Under the premise of charging as many nodes as possible, LDFM can minimize the moving distance of charging scheduling of the WCV, that is to say, it can reduce energy consumption upon moving, improving charging efficiency. Our previous work [25] proposed a hybrid scheme that allocated charging tasks to different vehicles by dividing the sensing region into three circles: inner, middle and outer. Different from the hybrid scheme, LDFM considers the longest distance to be allocated first to significantly reduce the moving distance of the WCV. Also note that the proposed algorithm allows the vehicle to carry multiple drones to charge far away and urgent nodes so that the total traveling time or distance of the low-speed vehicle can be minimized. Figure 1 depicts the new proposed WRSN system, which integrates one vehicle and multiple drones carried by the vehicle together to construct a more efficient charging system. In the proposed system, the charging requests from sensor nodes are sent to the base station, and the base station computes and plans the charging scheduling tours for the vehicle and drones. When receiving a charging task, the nodes within the flight distance of the base station are charged directly by the drones from the base station, and the residual nodes are allocated to the vehicle or the drones according our proposed LDFM scheme. The vehicle carrying a set of drones departs from the base station and travels along the predefined tour route to visit nodes allocated to it. When the vehicle arrives at a sensor node having a long matching node, a drone takes off and flies toward the target node and hovers around to charge it. After finishing the charging task, the vehicle returns to the base station to wait for the next task. This scheme maximizes the efficiency of drone usage and significantly shares the charging burden of the vehicle. In large WRSN applications, WRSNs are usually deployed in certain constrained conditions and harsh environments, such as hills, natural disaster areas or war zones. It is difficult for vehicles to move close to some sensor nodes and charge them. Nevertheless, the proposed scheme not only enables the vehicle to charge the sensor nodes in safe area, but also makes the drones charge the nodes in remote and harsh areas. Therefore, the proposed scheme expands the application scenarios of WRSNs, and greatly increases the reliability of WRSNs. Extensive simulations were conducted to evaluate the performance of the proposed scheme. The proposed scheme is compared to a typical hybrid scheduling scheme [25]. Simulation results show that the proposed scheme outperforms the hybrid scheduling scheme with varying configurations with respect to network metrics such as number of nodes being charged in time and network charging efficiency. Especially under heavy traffic load, the proposed scheme can charge more than 10% of additional nodes compared to the hybrid. Moreover, the proposed scheme achieved a reduction of over 50% in moving distance compared to the hybrid. The contributions of this work are listed below:  
1.