## 2 Related Work
### Mobile Charging Methods  
The offline method assigns the determined charging sequence to MC according to the specified path when the network state is known. [PERSON] et al. [14] constructed a multi-objective optimization MC charging path planning model, under which MC can supplement energy and collect data simultaneously. They proposed a multi-objective ant colony optimization algorithm to minimize the average data transmission delay. In order to optimize the charging timeliness of nodes, [PERSON] et al. [12] defined a secondary performance index of charging waiting time and used it to evaluate MC charging performance. In order to maximize the mobile charging efficiency, [PERSON] et al. [13] transformed the multi-MC coordination problem into a mixed integer linear programming problem to minimize the MC energy consumption and ensure that each node will not run out of energy. In the multi-node charging scheme, MC moves to the clustered charging location and charges all nodes within the charging range. Although the charging efficiency of MC for a single node is reduced under this scheme, the overall charging efficiency is significantly improved [22]. [PERSON] et al. [23] presented an uneven cluster-based mobile charging algorithm, which determines the nodes to be charged through the uneven network clusters to reduce the number of off-working nodes. [PERSON] et al. [24] took the critical path as the core factor and proposed a decomposition strategy to identify the interacting variables based on the CP. Finally, they combined the grouping strategy and cooperative evolutionary algorithm and proposed a new algorithm, to optimize the power allocation strategy in the system. Using multiple MCs for multi-node charging, [PERSON] et al. [25] determined each MC's independent closed charging path by designing the problem of minimizing the maximum charging delay to accelerate the charging speed and reduce the number of dead nodes.  
The offline method assumes that the energy consumption of networks changes periodically. However, this method is challenging to implement in practical applications because the network state is dynamic. In the online method, when the residual energy of the node is lower than a certain threshold, it will immediately send a charging request to the MC. MC adjusts the charging action according to the real-time state of WRSNs. [PERSON] et al. [26] used a simple and effective nearest-job-next with preemption algorithm to determine the MC's charging decision according to the nodes' position in the request queue. In the same year, [PERSON] et al. [27] proposed a double warning threshold with double preemption algorithm, which uses two warning thresholds to adjust the charging priority of different nodes. In addition, [PERSON] et al. [28] presented a primary and passer-by scheduling algorithm, which considers the urgency and proximity of balanced charging and ignores the effect of selecting inefficient nodes. Considering the influence of network topology changes, node failures and other uncertain factors on the charging performance, [PERSON] et al. [14] proposed a temporal spatial charging scheduling algorithm for large-scale WRSNs which calculated the node's dead time according to the node's energy state and adjusts the charging sequence according to the dead time.