# On-Demand Charging Management Model and Its Optimization for Wireless Renewable Sensor Networks
## 2 Problem Description  
In this section, we describe the model of WMCD behavior as well as the WSN control approach. Abbreviations contain a list of the main acronyms and notations used in this paper. The energy harvesting method not only increases the lifespan of individual sensor nodes but also the overall lifespan of the WSN. We take into account a two-dimensional area with N randomly distributed sensor nodes, which is similar to the scenario used in [13; 14; 15]. Each node has an energy receiving device that allows it to recharge its batteries remotely. A wireless mobile charging device (WMCD) has a static base station (BS) and a rest station (RS), also known as a service station, which is a place that the mobile charger uses to rest and recharge its battery for the next tour. The WMCD renewable energy cycle is known as the Hamiltonian cycle, and it begins and finishes at RS, as seen in Figure 1. The WMCD battery, on the other hand, is recharged on the RS via immobile energy storage and takes the lead from the RS, proceeding to the nearest node. The WMCD process continues as all deployed nodes use WET to wirelessly recharge their batteries. It returns to its rest station after recharging the node batteries to replenish its battery for the next charge cycle. A wireless renewable sensor network (WRSN) is defined as a WSN built with a WMCD. In real world scenario, The WMCD could be a vehicle following a set path. The WMCD must visit and charge some sensor nodes during each cycle (C). \(H_{n}\) denotes the group of nodes that must be visited in the C_th_ cycle. WMCD goes through the smallest Hamiltonian cycle that joins all nodes in \(H_{n}\) and the RS in the C_th_ cycle. \(P_{n}\) represents the shortest Hamiltonian cycle's traveling path. \(D_{n}\) denotes the length of path \(P_{n}\), and \(t_{n}\) denotes the time spent traveling across distance \(D_{n}\). Figure 1: A WSN with a wireless mobile charging device. \(T\) is denoted as the time required for the WMCD's tour cycle and \(t_{\text{tuc}}\) is the vacation time of the WMCD in the _Cth_ cycle. During the _Cth_ cycle, the WMCD travels from RS to \(H_{n}\), visiting and charging nodes before returning to RS for \(t_{\text{tuc}}\). The cycle time \(T\) is calculated as follows:  
\[T=t_{n}+t_{\text{tuc}}+\sum_{j\in h_{n}}t_{j} \tag{1}\]  
\[t_{n}=\frac{D_{n}}{V} \tag{2}\]  
where \(\sum_{j\in h_{n}}t_{j}\) is the entire length of time that the WMCD spends in charging all nodes in \(H_{n}\). We employ a periodic strategy in this paper. As previously stated, a WMCD is used to charge the sensor nodes periodically on a \(T\)-cycle basis. Each node, on the other hand, should be charged on a regular basis to complement its energy use. Previous studies [13; 14; 15] found that the WMCD goes into the network and charges all nodes in every cycle. In real-world networks, however, sensors' energy consumption rates may vary. Some nodes near the base station may spend several times less energy than faraway nodes; therefore, it is not necessary to charge every node in the cycle. This observation introduced a new research direction in wireless energy transfer research. The questions arising include: Do all nodes need to be charged at every charging cycle? If not, then how do we select the nodes to be charged at any given time? Must the energy in the node approach Emin before charging can be initiated? If not, then what energy level should be initiated? As a result, we can use a variety of strategies that consider each node's energy consumption rate. For example, in each cycle, the nodes with the highest energy consumption will be visited, and those with the lowest amount of energy consumption can be visited every two cycles or more. As shown in Figure 2, the charging procedure for node \(i\) (\(i\in\) N) should be replenished regularly to supplement its energy usage over time. The energy conservation principle states that node \(i\)'s energy consumption should be equal to the WMCD's energy supply [12].