Swarm Intelligence Techniques for Mobile Wireless Charging

This paper proposes energy-efficient swarm intelligence (SI)-based approaches for efficient mobile wireless charging in a distributed large-scale wireless sensor network (LS-WSN). This approach considers the use of special multiple mobile elements, which traverse the network for the purpose of energy replenishment. Recent techniques have shown the advantages inherent to the use of a single mobile charger (MC) which periodically visits the network to replenish the sensor-nodes. However, the single MC technique is currently limited and is not feasible for LS-WSN scenarios. Other approaches have overlooked the need to comprehensively discuss some critical tradeoffs associated with mobile wireless charging, which include: (1) determining the efficient coordination and charging strategies for the MCs, and (2) determining the optimal amount of energy available for the MCs, given the overall available network energy. These important tradeoffs are investigated in this study. Thus, this paper aims to investigate some of the critical issues affecting efficient mobile wireless charging for large-scale WSN scenarios; consequently, the network can then be operated without limitations. We first formulate the multiple charger recharge optimization problem (MCROP) and show that it is N-P hard. To solve the complex problem of scheduling multiple MCs in LS-WSN scenarios, we propose the node-partition algorithm based on cluster centroids, which adaptively partitions the whole network into several clusters and regions and distributes an MC to each region. Finally, we provide detailed simulation experiments using SI-based routing protocols. The results show the performance of the proposed scheme in terms of different evaluation metrics, where SI-based techniques are presented as a veritable state-of-the-art approach for improved energy-efficient mobile wireless charging to extend the network operational lifetime. The investigation also reveals the efficacy of the partial charging, over the full charging, strategies of the MCs.


Introduction
Recent technological advances have contributed to the pervasiveness of wireless mobile devices. Currently, charging the batteries of these devices are facing critical challenges, since the majority of these devices are battery-powered using electrical wires and power plugs, which limits their operational usage. A battery-powered sensor-node has limitations in its operational usage with a finite node lifetime [1]. Due to the energy limitations of battery-powered mobile devices, a wireless sensor network (WSN) is limited in its operations. To prolong the operational network lifetime, several research efforts have emerged, most of which have been studied in [2][3][4][5][6][7][8][9][10] and highlight the various techniques of prolonging the operational lifetime of the network, including methods of energy provisioning to the sensors. Recent approaches have revealed the advantages inherent in the use of mobile elements over the traditional schemes using static multi-hop routing. For instance, terms of the performance metrics. The authors in [29] presented a review of SI algorithms for feature selection, which provides solutions to various optimization problems. The survey gave insight to researchers on the possible design approach for a specific feature selection problem. In [30], authors studied the deployment optimization of an unmanned aerial vehicle (UAV) for the data collection platform of IoT devices and proposed a coding scheme based on the SI algorithm. The focus of this approach was to minimize the energy consumption of the UAV. A comprehensive review can be found in [31][32][33]. However, we note that none of the techniques in the literature have exploited SI-based approaches for energy provisioning to sensor-nodes. Our approach investigates the challenges of efficient mobile wireless charging in distributed LS-WSNs, using SI-based techniques.

Energy Provisioning Using MCs in WRSNs
Recently, the field of wireless rechargeable sensor networks (WRSNs) has witnessed many significant research efforts using a single MC. In [34], authors relied on Powercast technology to prolong the network lifetime of a WSN by constructing a wireless charging queue built on the greedy algorithm. The charging behavior of the MC is considered greedy, since the MC maintains the continuous charging of a sensor-node for a long time before charging another sensor-node. The authors studied the charging problem and proposed several heuristic algorithms to plan the charging activities for the MC (i.e., to determine the sequence of nodes to be charged and the amount of energy to be delivered to the sensor-nodes). This approach exploited a single MC, a robot which is used to deliver energy to where it is needed. Hence, the authors built a proof-of-concept prototype and conducted experiments to evaluate its feasibility and performance in smallscale networks. In [35,36], the authors proposed a single mobile charging vehicle that periodically visits the network and wirelessly replenishes every sensor-node in the network based on a near-optimal solution. The authors capitalized on the recent breakthrough in wireless energy transfer technology and proposed a scenario where a single mobile charging vehicle periodically visits the network to wirelessly charge each sensor-node. They studied a practical optimization problem, with the aim of maximizing the ratio of the wireless charging vehicle (WCV)'s vacation time over the cycle time, and subsequently, developed a provable near-optimal solution for flow routing, the total cycle time, and the individual charging time at each node. Their numerical results showed that a sensor network operating under their solution could remain operational for a long time. The authors in [37] studied the problem of maximizing the network lifetime of a single MC (subject to the energy capacity of the MC) and designed a heuristic algorithm, called FACT, to address the problem. Their work focused on locating closed charging tours for the MC, as well as designing an energy allocation scheme for the MC in a way that maximized the minimum battery energy of all the sensor-nodes after charging. However, the above techniques exploring a single MC have scalability problems in the way that the techniques are limited in the context of LS-WSNs containing thousands of sensor-nodes. The singlecharger technique will not be sustainable for LS-WSN scenarios, since the single MC will require a high amount of movement energy to traverse the whole network and it would have depleted its energy before completing its tour of the network. The implication is that several sensor-nodes will be left energy-hungry, thus leading to high node death rates and, consequently, resulting in a network operational breakdown. This is because a single MC has a limited amount of energy that it can provide, which is not feasible in a highly dense network environment. A single MC may not have enough energy to recharge all the sensor-nodes in a large-scale network scenario on a single tour. On the contrary, the use of multiple MCs will undoubtedly handle this problem and will optimize the energy efficiency of the network.
Based on the scalability problem with a single MC in a LS-WSN scenario, the authors in [38,39] investigated the minimum MC problem (i.e., the number of energy-constrained MCs for rechargeable sensor networks to keep every sensor-node working continuously). The authors in [38] approached the problem firstly by proving its computational hardness and then by proposing the use of approximation algorithms, with proven performance bounds, to address the minimum MCs problem. Conversely, the authors in [39] divided the minimum MCs problem into two N-P hard sub-problems: a Tour Construction Problem (TCP) and a Tour Assignment Problem (TAP). They considered a two-step solution to the problem. The first solution employed a greedy charging scheme to solve the TCP, while the second solution proposed a heuristic algorithm to address the TAP. The authors conducted simulations to evaluate the performance of their solutions. However, their solutions were focused on investigating the minimum MCs problem and their routes taken to recharge their sensor-nodes, which may not significantly improve the network lifetime. Our solution is designed to critically evaluate the issues affecting efficient wireless mobile charging in LS-WSNs to maintain a continuous network operation. Moreover, while their solution is limited to sensor-charging, ours can be exploited for both wireless charging and data collection schemes. The work in [9] proposed the use of two different kinds of vehicles, one for data collection and one for wireless charging. Apart from the problem of scheduling two different types of vehicles to a given network, this technique also incurred an increase in the energy consumption of the network, arising from an increase in the vehicles' movement energy consumption. Moreover, this technique was not cost effective. On the contrary, our approach can exploit the same mobile elements for both the data collection and the energy provisioning to the sensor-nodes to further minimize energy consumption and the cost of deployments. Owing to the potential hazards caused by a high exposure to EM radiation, the paper [40] gave safety considerations to the use of multiple MCs and proposed a safe charging algorithm for WRSNs. This algorithm was meant to create a balance in the relationship between safe charging and radiation safety. The authors in [41] introduced collaborative mobile charging, whereby MCs can also charge each other (in contrast to our model, where we do not address chargers charging each other).
Most of the works in the literature are advancing the topic but are failing to address the capability of a network infrastructure supporting more than one MC, thus creating scalability problems. Such a capability is crucial regarding extending the operational lifetime of a large-scale sensor network containing thousands of sensor-nodes through exploiting multiple MCs. Others overlooked the need to comprehensively address some critical issues and important tradeoffs affecting efficient wireless mobile charging, as is currently being investigated in this study. Compared to existing approaches, our method employs a distributed approach that exploits SI-based techniques for mobile wireless charging in a large-scale wireless sensor network, utilizing local and global network knowledge. Moreover, the multiple MCs approach will handle the scalability problem arising from the single MC technique, which is characterized by high latency times and high node death rates in LS-WSN scenarios.

The Network Model
This section presents the proposed network model. The whole network is composed of uniformly and randomly distributed sensor-nodes in a wireless rechargeable sensing field. The sensor-energy provisioning may involve three steps. Since our approach considers a large-scale network environment, the first step will be to partition the network size into smaller sizes using the MCs. The second step is to select a suitable starting point for the MCs (i.e., envisioning the movement trajectory of the MCs). If the MCs decide to start from the same location in the last region after splitting the network, it is good to optimize the MC's movement-energy consumption by locating a suitable starting point in every region.

Network Elements
Sensor data is generated at normal nodes and is aggregated at cluster heads (CHs) in a multi-hop fashion. Figure 1 shows a clear description of the entire network. Since the network is partitioned into multiple clusters inside a particular region, it is appropriate to have some cluster heads (CHs). Here, CHs are sensor-nodes that collect data packets from other nodes and transmit them to the MCs (here, the MCs are also the DCs). Hence, the rechargeable sensor-nodes consist of the normal nodes and CHs. Other nodes can forward their data to CHs using a multi-hop routing mechanism. The base station can be sited at the central position of the network, and it can be used for the data processing and recharging of the MCs to maintain continuous network operations. Finally, we will determine the CHs and the shortest moving distance of the MCs in each small region. The MCs (also serving as DCs) stay at the CHs in each region and collect the data aggregated by the CHs, while also replenishing the energy of the sensors within that vicinity. Moreover, the selection of the CHs is dynamic, as every region reselects the CH to avoid/minimize the hotspot problem and the data request flooding problem for multi-hop clustering. Then, the geometric traveling salesman problem (G-TSP) path of the MCs is structured to minimize the total traveling cost, the movement-energy consumption, and the data collection delays. The MCs can serve as both energy transmitters and data collectors. While traversing the network, the MCs collect the aggregated data from the CHs and recharge the sensor-nodes located within the vicinity of the CHs. Collected data are subsequently uploaded to the base station or the big data collection center. We propose a point-to-point charging of the sensor-nodes based on charging request priorities. Once an MC uses up its energy to charge the sensors, it quickly returns to the base station for energy replenishment.
A detailed list of notations and their descriptions are presented in Table 1. other nodes and transmit them to the MCs (here, the MCs are also the DCs). Hence, the rechargeable sensor-nodes consist of the normal nodes and CHs. Other nodes can forward their data to CHs using a multi-hop routing mechanism. The base station can be sited at the central position of the network, and it can be used for the data processing and recharging of the MCs to maintain continuous network operations. Finally, we will determine the CHs and the shortest moving distance of the MCs in each small region. The MCs (also serving as DCs) stay at the CHs in each region and collect the data aggregated by the CHs, while also replenishing the energy of the sensors within that vicinity. Moreover, the selection of the CHs is dynamic, as every region reselects the CH to avoid/minimize the hotspot problem and the data request flooding problem for multi-hop clustering. Then, the geometric traveling salesman problem (G-TSP) path of the MCs is structured to minimize the total traveling cost, the movement-energy consumption, and the data collection delays. The MCs can serve as both energy transmitters and data collectors. While traversing the network, the MCs collect the aggregated data from the CHs and recharge the sensor-nodes located within the vicinity of the CHs. Collected data are subsequently uploaded to the base station or the big data collection center. We propose a point-to-point charging of the sensor-nodes based on charging request priorities. Once an MC uses up its energy to charge the sensors, it quickly returns to the base station for energy replenishment. A detailed list of notations and their descriptions are presented in Table 1. other nodes and transmit them to the MCs (here, the MCs are also the DCs). Hence, the rechargeable sensor-nodes consist of the normal nodes and CHs. Other nodes can forward their data to CHs using a multi-hop routing mechanism. The base station can be sited at the central position of the network, and it can be used for the data processing and recharging of the MCs to maintain continuous network operations. Finally, we will determine the CHs and the shortest moving distance of the MCs in each small region. The MCs (also serving as DCs) stay at the CHs in each region and collect the data aggregated by the CHs, while also replenishing the energy of the sensors within that vicinity. Moreover, the selection of the CHs is dynamic, as every region reselects the CH to avoid/minimize the hotspot problem and the data request flooding problem for multi-hop clustering. Then, the geometric traveling salesman problem (G-TSP) path of the MCs is structured to minimize the total traveling cost, the movement-energy consumption, and the data collection delays. The MCs can serve as both energy transmitters and data collectors. While traversing the network, the MCs collect the aggregated data from the CHs and recharge the sensor-nodes located within the vicinity of the CHs. Collected data are subsequently uploaded to the base station or the big data collection center. We propose a point-to-point charging of the sensor-nodes based on charging request priorities. Once an MC uses up its energy to charge the sensors, it quickly returns to the base station for energy replenishment. A detailed list of notations and their descriptions are presented in Table 1. The residual energy of the sensor-node j before charging t v The charging time of the MC P t The transmit power R mc The charging radius of the MC P L The polarization loss η The rectifier efficiency 15 of 28 6] used Friis' free space propagation model to compute the charging efficiency of the es according to Equation (9): re are the uth charging points of the MCs, = 8dBi (transmit gain), = 2dBi eive gain), is the wavelength, is the polarization loss, ŋ is the rectifier iency, ϻ is an adjustable parameter that adjusts Friis' free space equation for short ance transmissions, is the transmit power, is the distance between node j and MC, and R is the charging radius of the MC. If ≤ R , Equation (9) is sformed [46] as: re Ծ = 4.32 × 10 , ϻ = 0.2316. The energy of the sensor-node, after being charged by the MC, is given in Equation [47]: re is the residual energy of the sensor-node j before charging, t is the charging of the MC, × t represents the energy received by node j from the MC, and is the maximum storage energy threshold of the sensors. Here, we randomly select 40% of Ḗ as the initial energy of the MC. The results shown in Figure 6. The basic result obtained shows that our partial charging strategy ficient compared to a full charging strategy. Our partial charging scheme outperforms ull charging counterpart after a given number of generated events. The outcome of investigation reveals the fact that the MC expends more energy on sensor recharging n applying the full charging method. Hence, much of the MC's energy is consumed ker, leading to a rise in the node death rate. Initial energy of the sensors other nodes and transmit them to the MCs (here, the MCs are also the DCs). Hence, the rechargeable sensor-nodes consist of the normal nodes and CHs. Other nodes can forward their data to CHs using a multi-hop routing mechanism. The base station can be sited at the central position of the network, and it can be used for the data processing and recharging of the MCs to maintain continuous network operations. Finally, we will determine the CHs and the shortest moving distance of the MCs in each small region. The MCs (also serving as DCs) stay at the CHs in each region and collect the data aggregated by the CHs, while also replenishing the energy of the sensors within that vicinity. Moreover, the selection of the CHs is dynamic, as every region reselects the CH to avoid/minimize the hotspot problem and the data request flooding problem for multi-hop clustering. Then, the geometric traveling salesman problem (G-TSP) path of the MCs is structured to minimize the total traveling cost, the movement-energy consumption, and the data collection delays. The MCs can serve as both energy transmitters and data collectors. While traversing the network, the MCs collect the aggregated data from the CHs and recharge the sensor-nodes located within the vicinity of the CHs. Collected data are subsequently uploaded to the base station or the big data collection center. We propose a point-to-point charging of the sensor-nodes based on charging request priorities. Once an MC uses up its energy to charge the sensors, it quickly returns to the base station for energy replenishment. A detailed list of notations and their descriptions are presented in Table 1.

The Deployment and Energy Model
Three kinds of devices can be identified with the proposed scheme, namely, the static sensors, the mobile chargers (MCs), and the static sink, as shown in Figure 2. Our model assumes that there are N sensors of the wireless communication distance ãrandomly distributed in a circular sensing field with the area Electronics 2022, 11,371 other nodes and transmit them to the MCs (here, the rechargeable sensor-nodes consist of the normal nodes their data to CHs using a multi-hop routing mechanis the central position of the network, and it can be recharging of the MCs to maintain continuous netw determine the CHs and the shortest moving distance o MCs (also serving as DCs) stay at the CHs in each reg by the CHs, while also replenishing the energy of Moreover, the selection of the CHs is dynamic, as avoid/minimize the hotspot problem and the data requ clustering. Then, the geometric traveling salesman pr structured to minimize the total traveling cost, the mo the data collection delays. The MCs can serve as b collectors. While traversing the network, the MCs co CHs and recharge the sensor-nodes located within the are subsequently uploaded to the base station or the big a point-to-point charging of the sensor-nodes based o an MC uses up its energy to charge the sensors, it qui energy replenishment.
A detailed list of notations and their descriptions Battery capacity of th Total energy replenishe and the radius (we investigate both uniform and non-uniform distributions) and n MCs are initially deployed at the center of a circular area along the x-y coordinates in Equation (1): where p = {1, 2, . . . n}. If we assume that the N sensors are uniformly and randomly distributed in a circular area of the radius , then the network density is given by: 11,371 other nodes and transmit them to the MCs (here, the MCs are a rechargeable sensor-nodes consist of the normal nodes and CHs. their data to CHs using a multi-hop routing mechanism. The ba the central position of the network, and it can be used for t recharging of the MCs to maintain continuous network oper determine the CHs and the shortest moving distance of the MCs MCs (also serving as DCs) stay at the CHs in each region and co by the CHs, while also replenishing the energy of the senso Moreover, the selection of the CHs is dynamic, as every reg avoid/minimize the hotspot problem and the data request floodin clustering. Then, the geometric traveling salesman problem (Gstructured to minimize the total traveling cost, the movement-e the data collection delays. The MCs can serve as both energ collectors. While traversing the network, the MCs collect the ag CHs and recharge the sensor-nodes located within the vicinity of are subsequently uploaded to the base station or the big data colle a point-to-point charging of the sensor-nodes based on charging an MC uses up its energy to charge the sensors, it quickly retur energy replenishment. A detailed list of notations and their descriptions are presen  The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the Electronics 2022, 11, 371 9 of 28 network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote ℛ The communication range of the sensors varies underlying routing protocols. Because the algorithms only one initialization, the initial position of the MC also be located at the center of the network. The base the network and is used to replenish the MCs and a model relies on both wireless power transfer and d assume that all sensors can generate data packets at This assumption is typical of the monitoring and repo denote Ḗ as the total available energy in the netw where Ḗ is the amount of energy distributed am the total energy initially possessed by the MCs that charge the sensors. Each sensor and each MC, respect energy representing their initial energy, given as: and Ḗ total as the total available energy in the network, then initially, distributed in a circular area of the radius ℜ , then the network density is given by: The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: distributed in a circular area of the radius ℜ , then the network density is given by: The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: sensors + distributed in a circular area of the radius ℜ , then the network density is given § = ℛ The communication range of the sensors varies according to requiremen underlying routing protocols. Because the algorithms can run for an extended ti only one initialization, the initial position of the MCs may not be important as also be located at the center of the network. The base station or sink can lie anyw the network and is used to replenish the MCs and aggregate data from the M model relies on both wireless power transfer and data collection. For simpli assume that all sensors can generate data packets at the same rate of δ per unit This assumption is typical of the monitoring and reporting applications in a WS denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and the total energy initially possessed by the MCs that can be delivered to the ne charge the sensors. Each sensor and each MC, respectively, have a maximum am energy representing their initial energy, given as: where 2 n 2 n where p = {1, 2, … n}. If we assume that the sensors are uniformly and randomly distributed in a circular area of the radius ℜ , then the network density is given by: The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: sensors is the amount of energy distributed among the sensor-nodes and 2 n 2 where p = {1, 2, … n}. If we assume that the se distributed in a circular area of the radius ℜ , then t § = ℛ The communication range of the sensors varie underlying routing protocols. Because the algorithm only one initialization, the initial position of the MC also be located at the center of the network. The base the network and is used to replenish the MCs and model relies on both wireless power transfer and assume that all sensors can generate data packets at This assumption is typical of the monitoring and rep denote Ḗ as the total available energy in the netw where Ḗ is the amount of energy distributed a the total energy initially possessed by the MCs that charge the sensors. Each sensor and each MC, respec energy representing their initial energy, given as: MC is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: distributed in a circular sensing field with the area ⍲ and the radius ℜ (we investigate both uniform and non-uniform distributions) and n MCs are initially deployed at the center of a circular area along the x-y coordinates in Equation (1): where p = {1, 2, … n}. If we assume that the sensors are uniformly and randomly distributed in a circular area of the radius ℜ , then the network density is given by: The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: distributed in a circular sensing field with the area ⍲ and the radius ℜ (we investigate both uniform and non-uniform distributions) and n MCs are initially deployed at the center of a circular area along the x-y coordinates in Equation (1): where p = {1, 2, … n}. If we assume that the sensors are uniformly and randomly distributed in a circular area of the radius ℜ , then the network density is given by: The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: sensors N and distributed in a circular sensing field with the area ⍲ and the radius ℜ (we inves both uniform and non-uniform distributions) and n MCs are initially deployed center of a circular area along the x-y coordinates in Equation where p = {1, 2, … n}. If we assume that the sensors are uniformly and rand distributed in a circular area of the radius ℜ , then the network density is given by § = ℛ The communication range of the sensors varies according to requirements underlying routing protocols. Because the algorithms can run for an extended time only one initialization, the initial position of the MCs may not be important as it also be located at the center of the network. The base station or sink can lie anywh the network and is used to replenish the MCs and aggregate data from the MCs model relies on both wireless power transfer and data collection. For simplicit assume that all sensors can generate data packets at the same rate of δ per unit of This assumption is typical of the monitoring and reporting applications in a WSN. denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ the total energy initially possessed by the MCs that can be delivered to the netwo charge the sensors. Each sensor and each MC, respectively, have a maximum amo energy representing their initial energy, given as: distributed in a circular sensing field with the area ⍲ and the radius ℜ (w both uniform and non-uniform distributions) and n MCs are initially de center of a circular area along the x-y coordinates in Equation where p = {1, 2, … n}. If we assume that the sensors are uniformly a distributed in a circular area of the radius ℜ , then the network density is g § = ℛ The communication range of the sensors varies according to require underlying routing protocols. Because the algorithms can run for an extend only one initialization, the initial position of the MCs may not be importa also be located at the center of the network. The base station or sink can lie the network and is used to replenish the MCs and aggregate data from t model relies on both wireless power transfer and data collection. For s assume that all sensors can generate data packets at the same rate of δ per This assumption is typical of the monitoring and reporting applications in denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-node the total energy initially possessed by the MCs that can be delivered to th charge the sensors. Each sensor and each MC, respectively, have a maximu energy representing their initial energy, given as: For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a 7-bit transmitted message, the radio expends I For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a ɤ-bit transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which depend on the transmission range ɖ of the sensors and the radio module. Generally, the power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the sensornodes in the network. We show, in Figure 3, a network that is virtually partitioned into ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure  3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to the intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote an arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , selected independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a high relative density. By putting together the knowledge about the total number of sensors in the network with the relative density Ɣ , and the area Ӑ , of every sector, we obtain the number of nodes ᴎ , using the relation: For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a ɤ-bit transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which depend on the transmission range ɖ of the sensors and the radio module. Generally, the power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the sensornodes in the network. We show, in Figure 3, a network that is virtually partitioned into ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure  3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to the intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote an arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , selected independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a high relative density. By putting together the knowledge about the total number of sensors in the network with the relative density Ɣ , and the area Ӑ , of every sector, we obtain the number of nodes ᴎ , using the relation: . Then, ᴎ , nodes are distributed in the area corresponding to sector Ȿ , . The fraction of the actual densities of the two sectors Ȿ , and Ȿ is . In addition, this model considers a uniform deployment, where all sectors have the same relative density ( . . , Ɣ , = Ɣ for all , ). R (7) = recv .7 where trans and recv are constants which depend on the transmission range ãof the sensors and the radio module. Generally, the power needed for transmitting a message at a distance ãis roughly d β where 2 ≤ β ≤ 6 is a constant; for simplicity, we take β = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the N sensornodes in the network. We show, in Figure 3, a network that is virtually partitioned into 2π ∅ slices and 2π d co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = π/4), while Figure 3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes. Let ù m,n represent the sector corresponding to the intersection between a slice m and a ring n as shown in Figure 3b and let k > 1 denote an arbitrary constant. Let G m,n denote the relative density of the sector ù m,n selected independently along G m,n ∈ |1, k| based on a uniform distribution where For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a ɤ-bit transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which depend on the transmission range ɖ of the sensors and the radio module. Generally, the power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the sensornodes in the network. We show, in Figure 3, a network that is virtually partitioned into ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure  3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to the intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote an arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , selected independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a high relative density. By putting together the knowledge about the total number of sensors in the network with the relative density Ɣ , and the area Ӑ , of every sector, we obtain the number of nodes ᴎ , using the relation: . Then, ᴎ , nodes are distributed in the area corresponding to sector Ȿ , . The fraction of the actual densities of the two sectors Ȿ , and Ȿ is . In addition, this model considers a uniform deployment, where all sectors have the same relative density ( . . , Ɣ , = Ɣ for all , ).
= |1, k|. We note that the values of G m,n tend to 1 for a low relative density and tend to k for a high relative density. By putting together the knowledge about the total number of sensors in the network N with the relative density G m,n and the area For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a ɤ-bit transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which depend on the transmission range ɖ of the sensors and the radio module. Generally, the power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the sensornodes in the network. We show, in Figure 3, a network that is virtually partitioned into ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure  3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to the intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote an arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , selected independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a high relative density. By putting together the knowledge about the total number of sensors in the network with the relative density Ɣ , and the area Ӑ , of every sector, we obtain the number of nodes ᴎ , using the relation: . Then, ᴎ , nodes are distributed in the area corresponding to sector Ȿ , . The fraction of the actual densities of the two sectors Ȿ , and Ȿ is . In addition, this model considers a uniform deployment, where all sectors have the same relative density ( . . , Ɣ , = Ɣ for all , ).
m,n of every sector, we obtain the number of nodes For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a ɤ-bit transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which depend on the transmission range ɖ of the sensors and the radio module. Generally, the power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the sensornodes in the network. We show, in Figure 3, a network that is virtually partitioned into ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure  3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to the intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote an arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , selected independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a high relative density. By putting together the knowledge about the total number of sensors in the network with the relative density Ɣ , and the area Ӑ , of every sector, we obtain the number of nodes ᴎ , using the relation: For message transmission and reception, our model assumes that the amount energy dissipated by the radio module is proportional to the message size. For a ɤtransmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤmessage. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants whi depend on the transmission range ɖ of the sensors and the radio module. Generally, t power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the senso nodes in the network. We show, in Figure 3, a network that is virtually partitioned in ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersecti between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figu 3b shows the different partitions of the slice sectors. Here, each slice contains 5 secto thus giving a total of 40 sectors in the network. We assume a case of random non-unifor deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to t intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , select independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. W note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a hi relative density. By putting together the knowledge about the total number of sensors the network with the relative density Ɣ , and the area Ӑ , of every sector, we obta the number of nodes ᴎ , using the relation: . Then, ᴎ , nodes are distributed in the area corresponding sector Ȿ , . The fraction of the actual densities of the two sectors Ȿ , and Ȿ For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a ɤ-bit transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which depend on the transmission range ɖ of the sensors and the radio module. Generally, the power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the sensornodes in the network. We show, in Figure 3, a network that is virtually partitioned into ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure  3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to the intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote an arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , selected independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a high relative density. By putting together the knowledge about the total number of sensors in the network with the relative density Ɣ , and the area Ӑ , of every sector, we obtain the number of nodes ᴎ , using the relation:  For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a ɤ-bit transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which depend on the transmission range ɖ of the sensors and the radio module. Generally, the power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the sensornodes in the network. We show, in Figure 3, a network that is virtually partitioned into ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure  3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to the intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote an arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , selected independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a high relative density. By putting together the knowledge about the total number of sensors in the network with the relative density Ɣ , and the area Ӑ , of every sector, we obtain the number of nodes ᴎ , using the relation: For message transmission and reception, our model assumes t energy dissipated by the radio module is proportional to the messa transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are depend on the transmission range ɖ of the sensors and the radio mod power needed for transmitting a message at a distance ɖ is roughly ɖ is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deploymen nodes in the network. We show, in Figure 3, a network that is virtua ∅ slices and ɖ co-centric rings of small sectors. We define a sector between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = 3b shows the different partitions of the slice sectors. Here, each slice thus giving a total of 40 sectors in the network. We assume a case of ra deployment of the sensor-nodes. Let Ȿ , represent the sector cor intersection between a slice m and a ring n as shown in Figure 3b and arbitrary constant. Let Ɣ , denote the relative density of the se independently along Ɣ , ∈ |1, | based on a uniform distribution wh note that the values of Ɣ , tend to 1 for a low relative density and t relative density. By putting together the knowledge about the total nu the network with the relative density Ɣ , and the area Ӑ , of eve the number of nodes ᴎ , using the relation: For message transmission and reception, our model assumes energy dissipated by the radio module is proportional to the messa transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ ar depend on the transmission range ɖ of the sensors and the radio mod power needed for transmitting a message at a distance ɖ is roughly ɖ is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployme nodes in the network. We show, in Figure 3, a network that is virtua ∅ slices and ɖ co-centric rings of small sectors. We define a sector between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = 3b shows the different partitions of the slice sectors. Here, each slice thus giving a total of 40 sectors in the network. We assume a case of ra deployment of the sensor-nodes. Let Ȿ , represent the sector co intersection between a slice m and a ring n as shown in Figure 3b and arbitrary constant. Let Ɣ , denote the relative density of the s independently along Ɣ , ∈ |1, | based on a uniform distribution w note that the values of Ɣ , tend to 1 for a low relative density and relative density. By putting together the knowledge about the total nu the network with the relative density Ɣ , and the area Ӑ , of eve the number of nodes ᴎ , using the relation: where N = ∑ m,n For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a ɤ-bit transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which depend on the transmission range ɖ of the sensors and the radio module. Generally, the power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the sensornodes in the network. We show, in Figure 3, a network that is virtually partitioned into ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure  3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to the intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote an arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , selected independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a high relative density. By putting together the knowledge about the total number of sensors in the network with the relative density Ɣ , and the area Ӑ , of every sector, we obtain the number of nodes ᴎ , using the relation: For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a ɤ-bit transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which depend on the transmission range ɖ of the sensors and the radio module. Generally, the power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the sensornodes in the network. We show, in Figure 3, a network that is virtually partitioned into ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure  3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to the intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote an arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , selected independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a high relative density. By putting together the knowledge about the total number of sensors in the network with the relative density Ɣ , and the area Ӑ , of every sector, we obtain the number of nodes ᴎ , using the relation: nsmission and reception, our model assumes that the amount of the radio module is proportional to the message size. For a ɤ-bit , the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit s Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which ission range ɖ of the sensors and the radio module. Generally, the nsmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 plicity, we take = 2.
niform Deployment oth cases of uniform and non-uniform deployments of the sensor-. We show, in Figure 3, a network that is virtually partitioned into entric rings of small sectors. We define a sector as the intersection ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure nt partitions of the slice sectors. Here, each slice contains 5 sectors, 40 sectors in the network. We assume a case of random non-uniform sensor-nodes. Let Ȿ , represent the sector corresponding to the a slice m and a ring n as shown in Figure 3b and let > 1 denote an et Ɣ , denote the relative density of the sector Ȿ , selected Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We of Ɣ , tend to 1 for a low relative density and tend to k for a high utting together the knowledge about the total number of sensors in he relative density Ɣ , and the area Ӑ , of every sector, we obtain ᴎ , using the relation: . Then, ᴎ , nodes are distributed in the area corresponding to ction of the actual densities of the two sectors Ȿ , and Ȿ is n addition, this model considers a uniform deployment, where all e relative density ( . . , Ɣ , = Ɣ for all , ). ansmission and reception, our model assumes that the amount of y the radio module is proportional to the message size. For a ɤ-bit e, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit s Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which mission range ɖ of the sensors and the radio module. Generally, the nsmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 plicity, we take = 2.
niform Deployment oth cases of uniform and non-uniform deployments of the sensork. We show, in Figure 3, a network that is virtually partitioned into entric rings of small sectors. We define a sector as the intersection ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure nt partitions of the slice sectors. Here, each slice contains 5 sectors, 40 sectors in the network. We assume a case of random non-uniform sensor-nodes. Let Ȿ , represent the sector corresponding to the a slice m and a ring n as shown in Figure 3b and let > 1 denote an Let Ɣ , denote the relative density of the sector Ȿ , selected Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We of Ɣ , tend to 1 for a low relative density and tend to k for a high utting together the knowledge about the total number of sensors in the relative density Ɣ , and the area Ӑ , of every sector, we obtain ᴎ , using the relation: , . Then, ᴎ , nodes are distributed in the area corresponding to ction of the actual densities of the two sectors Ȿ , and Ȿ is n addition, this model considers a uniform deployment, where all e relative density ( . . , Ɣ , = Ɣ for all , ). For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a ɤ-bit transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which depend on the transmission range ɖ of the sensors and the radio module. Generally, the power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the sensornodes in the network. We show, in Figure 3, a network that is virtually partitioned into ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure  3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to the intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote an arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , selected independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a high relative density. By putting together the knowledge about the total number of sensors in the network with the relative density Ɣ , and the area Ӑ , of every sector, we obtain the number of nodes ᴎ , using the relation:  For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a ɤ-bit transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ-bit message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants which depend on the transmission range ɖ of the sensors and the radio module. Generally, the power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ ≤ 6 is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the sensornodes in the network. We show, in Figure 3, a network that is virtually partitioned into ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Figure  3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to the intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote an arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , selected independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. We note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a high relative density. By putting together the knowledge about the total number of sensors in the network with the relative density Ɣ , and the area Ӑ , of every sector, we obtain the number of nodes ᴎ , using the relation:  For message transmission and reception, our model assumes that the amoun energy dissipated by the radio module is proportional to the message size. For a ɤ transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive the ɤ message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are constants wh depend on the transmission range ɖ of the sensors and the radio module. Generally, power needed for transmitting a message at a distance ɖ is roughly ɖ where 2 ≤ is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the sens nodes in the network. We show, in Figure 3, a network that is virtually partitioned i ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the intersect between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), while Fig  3b shows the different partitions of the slice sectors. Here, each slice contains 5 sect thus giving a total of 40 sectors in the network. We assume a case of random non-unifo deployment of the sensor-nodes. Let Ȿ , represent the sector corresponding to intersection between a slice m and a ring n as shown in Figure 3b and let > 1 denote arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ , selec independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = |1, |. note that the values of Ɣ , tend to 1 for a low relative density and tend to k for a h relative density. By putting together the knowledge about the total number of sensor the network with the relative density Ɣ , and the area Ӑ , of every sector, we obt the number of nodes ᴎ , using the relation: . Then, ᴎ , nodes are distributed in the area corresponding sector Ȿ , . The fraction of the actual densities of the two sectors Ȿ , and Ȿ For message transmission and reception, our model assumes that the energy dissipated by the radio module is proportional to the message size. transmitted message, the radio expends Ѥ (ɤ) = Ԑ . ɤ and to receive message. It expends Ѥ (ɤ) = Ԑ . ɤ where Ԑ and Ԑ are consta depend on the transmission range ɖ of the sensors and the radio module. Gen power needed for transmitting a message at a distance ɖ is roughly ɖ where is a constant; for simplicity, we take = 2.

Uniform and Non-Uniform Deployment
We investigate both cases of uniform and non-uniform deployments of the nodes in the network. We show, in Figure 3, a network that is virtually parti ∅ slices and ɖ co-centric rings of small sectors. We define a sector as the i between a slice and a ring. Figure 3a shows that there are 8 slices (∅ = /4), w 3b shows the different partitions of the slice sectors. Here, each slice contain thus giving a total of 40 sectors in the network. We assume a case of random no deployment of the sensor-nodes. Let Ȿ , represent the sector correspond intersection between a slice m and a ring n as shown in Figure 3b and let > 1 arbitrary constant. Let Ɣ , denote the relative density of the sector Ȿ independently along Ɣ , ∈ |1, | based on a uniform distribution where Ц = note that the values of Ɣ , tend to 1 for a low relative density and tend to k relative density. By putting together the knowledge about the total number of the network with the relative density Ɣ , and the area Ӑ , of every sector the number of nodes ᴎ , using the relation:

Recharge by Adaptive Network Partitioning and Clustering
In order to effectively cover the wide geographical regions of the whole network, multiple MCs are employed to charge the sensor-nodes and to collect data. To solve the complex problem of scheduling multiple MCs in LS-WSNs, we propose to adaptively partition the whole network into several clusters and distribute an MC in each region. Some advantages of a distributed clustering approach in real-world scenarios are provided in [42], which include improving the route connection, minimizing the energy consumption, and increasing the scalability while reducing the traffic. To calculate the minimum number of MCs needed to achieve perpetual network operation, we use Equation (6) [43]. That is, a point-to-point charging of the sensor-n an MC uses up its energy to charge the energy replenishment. A detailed list of notations and thei where n denotes the number of MCs, β −1 is the inverse cumulative distribution function of normal distribution, ∈ is a value approaching 1 but is not equal to 1, T is a large time frame, E c (T) is the total energy consumption of the network up to T, E 0 is the initial energy of the sensor-nodes, L is the length of the sensor field, v is the speed of the MCs, avoid/minimize the ho clustering. Then, the g structured to minimiz the data collection de collectors. While trave CHs and recharge the are subsequently uploa a point-to-point charg an MC uses up its ene energy replenishment. A detailed list of n is the recharge time of the sensor-node's battery, and ψ s is battery capacity of the sensor-nodes. The probability for the energy-neutral condition [43] is modelled as: where E r is the total energy replenished into the network by the MCs, up to T. Since β −1 (1) → ∞ , the network operation is considered perpetual at a high probability of occurrence of ∈, if ∈→ 1 but is not equal to 1, e.g., ∈= 0.99, β −1 (0.99) = 2.33. Though the rate of data generation is dynamic, the process can be modeled as a Poisson process to determine the number of MCs. Since the number of MCs n in the network is known, the number of network partitions ¢ can get through ¢ = n. Hence, each sub-region of the network always has MCs to ensure wireless energy transfer and data collection. The process of network partitioning and clustering is as follows: • The first partition is conducted. This method is like that in [9] and follows a uniform division of the network into c parts. Unlike [9], our method uses a dynamic selection of the cluster centroids, where CHs can be in each region of the network. These nodes are located at the center of each region. Besides, we evaluate the distance r and the shortest hop routing h from each node to the cluster centroids. r jk denotes the distance between node j and cluster centroid k, where j ∈ N, k ∈ c.
• The second partition is conducted. Firstly, we evaluate the weight W jk of each node j to k. Here, W jk = αR jk + ψH jk ; α + ψ = 1, 0 ≤ α, ψ ≤ 1, shortest hop routing ĥ from each node to the cluster centroids. denotes the distance between node j and cluster centroid k, where ∈ , ∈ . • The second partition is conducted. Firstly, we evaluate the weight of each node j to k. Here, = + Ĥ ; + = 1, 0 ≤ , ≤ 1, ⍱ ∈ , ⍱ ∈ [1, ]. We define and as part of the distance and routing hop priorities, respectively, and is the serial number of the distance from node j to cluster centroid k. If = 1 denotes the minimum value of , = denotes the maximum distance .
We denote ⍫ = as the ratio of and . If ⍫ → ∞, our algorithm only considers a routing hop. On the contrary, our algorithm can also consider distance when ⍫ → 0. In this case, we jointly consider the routing hop and distance to make ⍫ = 1. Next, the smallest is chosen, owing to its partitioning, and the node j is assigned to the kth region. This whole process is repeated until all the nodes in the network are partitioned. The proposed network partitioning and clustering method can be summarized in Algorithm 1.

Algorithm 1:
Node-partition algorithm based on cluster centroids Let S sensor-nodes and c cluster centroids be inputted We denote the output as: min[

… , ]
Calculate distance and the shortest hop ĥ between node j and centroid k where ∈ , ∈ .

Arrange
, … , and (ĥ , … , ĥ ) in ascending order denotes the distance of the serial number and Ĥ is the routing hop serial number of node j to centroid k.
], the node j is assigned to the kth region.

= − End
In addition, computing the distance and the routing hop for every node requires at least | |. Every node has a sorting time complexity equal to 0(| |{ln | |}) [9]. Since there are S sensor-nodes, we would need | |{ln | |}. Lastly, to calculate and the least value in … , , we need the time complexity | | and 0(| |{ln | |}), respectively. Hence, the overall complexity of this algorithm is 0(| |{ln | |}). Figure 4 shows the results of the network partition. The red dotted lines equally partition the network into four parts, Q, R, S, and T. This represents the first division. Besides, the red points indicate the cluster heads (CHs) of every region, which also indicate the initial points of each region. Then, a second partition is run based on these CHs. We obtain the same shape of the cluster with the nodes belonging to a partition after operating a partition algorithm. The blue dotted circle represents the result of the second partition. Obviously, some nodes previously belonging to areas Q, R, S, and T are now assigned to the U region, as a result of the second partition. The final partition occurs when all the nodes in the network have been assigned to a particular cluster, having a CH in that region. j ∈ N, Electronics 2022, 11, 371 11 shortest hop routing ĥ from each node to the cluster centroids. denotes distance between node j and cluster centroid k, where ∈ , ∈ . • The second partition is conducted. Firstly, we evaluate the weight of each n j to k. Here, = + Ĥ ; + = 1, 0 ≤ , ≤ 1, ⍱ ∈ , ⍱ ∈ [1, ].
define and as part of the distance and routing hop priorities, respectively, is the serial number of the distance from node j to cluster centroid k. If 1 denotes the minimum value of , = denotes the maximum distance We denote ⍫ = as the ratio of and . If ⍫ → ∞, our algorithm only consi a routing hop. On the contrary, our algorithm can also consider distance when ⍫ 0. In this case, we jointly consider the routing hop and distance to make ⍫ = 1. N the smallest is chosen, owing to its partitioning, and the node j is assigned to kth region. This whole process is repeated until all the nodes in the network partitioned. The proposed network partitioning and clustering method can summarized in Algorithm 1.

Algorithm 1: Node-partition algorithm based on cluster centroids
Let S sensor-nodes and c cluster centroids be inputted We denote the output as: min[

… , ]
Calculate distance and the shortest hop ĥ between node j and centroid k wh ∈ , ∈ .

Arrange
, … , and (ĥ , … , ĥ ) in ascending order denotes the distance of the serial number and Ĥ is the routing hop serial numbe node j to centroid k.
], the node j is assigned to the kth region. = −

End
In addition, computing the distance and the routing hop for every node require least | |. Every node has a sorting time complexity equal to 0(| |{ln | |}) [9]. Since th are S sensor-nodes, we would need | |{ln | |}. Lastly, to calculate and the least v in … , , we need the time complexity | | and 0(| |{ln | |}), respectively. Hence overall complexity of this algorithm is 0(| |{ln | |}). Figure 4 shows the results of the network partition. The red dotted lines equ partition the network into four parts, Q, R, S, and T. This represents the first divis Besides, the red points indicate the cluster heads (CHs) of every region, which indicate the initial points of each region. Then, a second partition is run based on th CHs. We obtain the same shape of the cluster with the nodes belonging to a partition a operating a partition algorithm. The blue dotted circle represents the result of the sec partition. Obviously, some nodes previously belonging to areas Q, R, S, and T are n assigned to the U region, as a result of the second partition. The final partition oc when all the nodes in the network have been assigned to a particular cluster, having a in that region.
We define α and ψ as part of the distance and routing hop priorities, respectively, and R jk is the serial number of the distance r jk from node j to cluster centroid k. If R jk = 1 denotes the minimum value of r jk , R jk = c denotes the maximum distance r jk . We denote 11 of 28 shortest hop routing ĥ from each node to the cluster centroids. denotes the distance between node j and cluster centroid k, where ∈ , ∈ . • The second partition is conducted. Firstly, we evaluate the weight of each node j to k. Here, = + Ĥ ; + = 1, 0 ≤ , ≤ 1, ⍱ ∈ , ⍱ ∈ [1, ]. We define and as part of the distance and routing hop priorities, respectively, and is the serial number of the distance from node j to cluster centroid k. If = 1 denotes the minimum value of , = denotes the maximum distance .
We denote ⍫ = as the ratio of and . If ⍫ → ∞, our algorithm only considers a routing hop. On the contrary, our algorithm can also consider distance when ⍫ → 0. In this case, we jointly consider the routing hop and distance to make ⍫ = 1. Next, the smallest is chosen, owing to its partitioning, and the node j is assigned to the kth region. This whole process is repeated until all the nodes in the network are partitioned. The proposed network partitioning and clustering method can be summarized in Algorithm 1.

Algorithm 1:
Node-partition algorithm based on cluster centroids Let S sensor-nodes and c cluster centroids be inputted We denote the output as: min[

… , ]
Calculate distance and the shortest hop ĥ between node j and centroid k where ∈ , ∈ .

Arrange
, … , and (ĥ , … , ĥ ) in ascending order denotes the distance of the serial number and Ĥ is the routing hop serial number of node j to centroid k.
], the node j is assigned to the kth region.

= − End
In addition, computing the distance and the routing hop for every node requires at least | |. Every node has a sorting time complexity equal to 0(| |{ln | |}) [9]. Since there are S sensor-nodes, we would need | |{ln | |}. Lastly, to calculate and the least value in … , , we need the time complexity | | and 0(| |{ln | |}), respectively. Hence, the overall complexity of this algorithm is 0(| |{ln | |}). Figure 4 shows the results of the network partition. The red dotted lines equally partition the network into four parts, Q, R, S, and T. This represents the first division. Besides, the red points indicate the cluster heads (CHs) of every region, which also indicate the initial points of each region. Then, a second partition is run based on these CHs. We obtain the same shape of the cluster with the nodes belonging to a partition after operating a partition algorithm. The blue dotted circle represents the result of the second partition. Obviously, some nodes previously belonging to areas Q, R, S, and T are now assigned to the U region, as a result of the second partition. The final partition occurs when all the nodes in the network have been assigned to a particular cluster, having a CH in that region. shortest hop routing ĥ from each node to the cluster centroids. denotes the distance between node j and cluster centroid k, where ∈ , ∈ . • The second partition is conducted. Firstly, we evaluate the weight of each node j to k. Here, = + Ĥ ; + = 1, 0 ≤ , ≤ 1, ⍱ ∈ , ⍱ ∈ [1, ]. We define and as part of the distance and routing hop priorities, respectively, and is the serial number of the distance from node j to cluster centroid k. If = 1 denotes the minimum value of , = denotes the maximum distance .
We denote ⍫ = as the ratio of and . If ⍫ → ∞, our algorithm only considers a routing hop. On the contrary, our algorithm can also consider distance when ⍫ → 0. In this case, we jointly consider the routing hop and distance to make ⍫ = 1. Next, the smallest is chosen, owing to its partitioning, and the node j is assigned to the kth region. This whole process is repeated until all the nodes in the network are partitioned. The proposed network partitioning and clustering method can be summarized in Algorithm 1.

Algorithm 1:
Node-partition algorithm based on cluster centroids Let S sensor-nodes and c cluster centroids be inputted We denote the output as: min[

… , ]
Calculate distance and the shortest hop ĥ between node j and centroid k where ∈ , ∈ .

Arrange
, … , and (ĥ , … , ĥ ) in ascending order denotes the distance of the serial number and Ĥ is the routing hop serial number of node j to centroid k.
], the node j is assigned to the kth region. = −

End
In addition, computing the distance and the routing hop for every node requires at least | |. Every node has a sorting time complexity equal to 0(| |{ln | |}) [9]. Since there are S sensor-nodes, we would need | |{ln | |}. Lastly, to calculate and the least value in … , , we need the time complexity | | and 0(| |{ln | |}), respectively. Hence, the overall complexity of this algorithm is 0(| |{ln | |}). Figure 4 shows the results of the network partition. The red dotted lines equally partition the network into four parts, Q, R, S, and T. This represents the first division. Besides, the red points indicate the cluster heads (CHs) of every region, which also indicate the initial points of each region. Then, a second partition is run based on these CHs. We obtain the same shape of the cluster with the nodes belonging to a partition after operating a partition algorithm. The blue dotted circle represents the result of the second partition. Obviously, some nodes previously belonging to areas Q, R, S, and T are now assigned to the U region, as a result of the second partition. The final partition occurs when all the nodes in the network have been assigned to a particular cluster, having a CH in that region.
→ ∞ , our algorithm only considers a routing hop. On the contrary, our algorithm can also consider distance when Electronics 2022, 11,371 shortest hop routing ĥ from each node to the cluster centr distance between node j and cluster centroid k, where ∈ , • The second partition is conducted. Firstly, we evaluate the we j to k. Here, = + Ĥ ; + = 1, 0 ≤ , ≤ 1, ⍱ define and as part of the distance and routing hop prior is the serial number of the distance from node j to clus 1 denotes the minimum value of , = denotes the m We denote ⍫ = as the ratio of and . If ⍫ → ∞, our alg a routing hop. On the contrary, our algorithm can also consid 0. In this case, we jointly consider the routing hop and distance the smallest is chosen, owing to its partitioning, and the n kth region. This whole process is repeated until all the nod partitioned. The proposed network partitioning and clust summarized in Algorithm 1.

Algorithm 1:
Node-partition algorithm based on cluster centroids Let S sensor-nodes and c cluster centroids be inputted We denote the output as: min[

Arrange
, … , and (ĥ , … , ĥ ) in ascending order denotes the distance of the serial number and Ĥ is the routin node j to centroid k.
], the node j is assigned to the kth region = −

End
In addition, computing the distance and the routing hop for least | |. Every node has a sorting time complexity equal to 0(| |{ are S sensor-nodes, we would need | |{ln | |}. Lastly, to calculate in … , , we need the time complexity | | and 0(| |{ln | |}), re overall complexity of this algorithm is 0(| |{ln | |}). Figure 4 shows the results of the network partition. The red partition the network into four parts, Q, R, S, and T. This repres Besides, the red points indicate the cluster heads (CHs) of eve indicate the initial points of each region. Then, a second partition CHs. We obtain the same shape of the cluster with the nodes belong operating a partition algorithm. The blue dotted circle represents t partition. Obviously, some nodes previously belonging to areas Q assigned to the U region, as a result of the second partition. The when all the nodes in the network have been assigned to a particula in that region.
→ 0 . In this case, we jointly consider the routing hop and distance to make Electronics 2022, 11,371 shortest hop routing ĥ from each node to the cluster centroids. distance between node j and cluster centroid k, where ∈ , ∈ . • The second partition is conducted. Firstly, we evaluate the weight j to k. Here, = + Ĥ ; + = 1, 0 ≤ , ≤ 1, ⍱ ∈ , define and as part of the distance and routing hop priorities, re is the serial number of the distance from node j to cluster cen 1 denotes the minimum value of , = denotes the maximu We denote ⍫ = as the ratio of and . If ⍫ → ∞, our algorithm a routing hop. On the contrary, our algorithm can also consider dista 0. In this case, we jointly consider the routing hop and distance to ma the smallest is chosen, owing to its partitioning, and the node j is kth region. This whole process is repeated until all the nodes in t partitioned. The proposed network partitioning and clustering m summarized in Algorithm 1.

Algorithm 1:
Node-partition algorithm based on cluster centroids Let S sensor-nodes and c cluster centroids be inputted We denote the output as: min[

… , ]
Calculate distance and the shortest hop ĥ between node j and ce ∈ , ∈ .

Arrange
, … , and (ĥ , … , ĥ ) in ascending order denotes the distance of the serial number and Ĥ is the routing hop s node j to centroid k.
], the node j is assigned to the kth region. = −

End
In addition, computing the distance and the routing hop for every n least | |. Every node has a sorting time complexity equal to 0(| |{ln | |}) are S sensor-nodes, we would need | |{ln | |}. Lastly, to calculate and in … , , we need the time complexity | | and 0(| |{ln | |}), respecti overall complexity of this algorithm is 0(| |{ln | |}). Figure 4 shows the results of the network partition. The red dotte partition the network into four parts, Q, R, S, and T. This represents th Besides, the red points indicate the cluster heads (CHs) of every regi indicate the initial points of each region. Then, a second partition is run CHs. We obtain the same shape of the cluster with the nodes belonging to operating a partition algorithm. The blue dotted circle represents the resu partition. Obviously, some nodes previously belonging to areas Q, R, S, assigned to the U region, as a result of the second partition. The final when all the nodes in the network have been assigned to a particular clust in that region. = 1 . Next, the smallest W jk is chosen, owing to its partitioning, and the node j is assigned to the kth region. This whole process is repeated until all the nodes in the network are partitioned. The proposed network partitioning and clustering method can be summarized in Algorithm 1.  shortest hop routing ĥ from each node to the cluster centroids. denotes the distance between node j and cluster centroid k, where ∈ , ∈ . • The second partition is conducted. Firstly, we evaluate the weight of each node j to k. Here, = + Ĥ ; + = 1, 0 ≤ , ≤ 1, ⍱ ∈ , ⍱ ∈ [1, ]. We define and as part of the distance and routing hop priorities, respectively, and is the serial number of the distance from node j to cluster centroid k. If = 1 denotes the minimum value of , = denotes the maximum distance .
We denote ⍫ = as the ratio of and . If ⍫ → ∞, our algorithm only considers a routing hop. On the contrary, our algorithm can also consider distance when ⍫ → 0. In this case, we jointly consider the routing hop and distance to make ⍫ = 1. Next, the smallest is chosen, owing to its partitioning, and the node j is assigned to the kth region. This whole process is repeated until all the nodes in the network are partitioned. The proposed network partitioning and clustering method can be summarized in Algorithm 1.

Algorithm 1:
Node-partition algorithm based on cluster centroids Let S sensor-nodes and c cluster centroids be inputted We denote the output as: min[

… , ]
Calculate distance and the shortest hop ĥ between node j and centroid k where ∈ , ∈ .

Arrange
, … , and (ĥ , … , ĥ ) in ascending order denotes the distance of the serial number and Ĥ is the routing hop serial number of node j to centroid k.
], the node j is assigned to the kth region. = −

End
In addition, computing the distance and the routing hop for every node requires at least | |. Every node has a sorting time complexity equal to 0(| |{ln | |}) [9]. Since there are S sensor-nodes, we would need | |{ln | |}. Lastly, to calculate and the least value in … , , we need the time complexity | | and 0(| |{ln | |}), respectively. Hence, the overall complexity of this algorithm is 0(| |{ln | |}). Figure 4 shows the results of the network partition. The red dotted lines equally partition the network into four parts, Q, R, S, and T. This represents the first division. Besides, the red points indicate the cluster heads (CHs) of every region, which also indicate the initial points of each region. Then, a second partition is run based on these j ∈ S, Electronics 2022, 11, 371 11 of 28 shortest hop routing ĥ from each node to the cluster centroids. denotes the distance between node j and cluster centroid k, where ∈ , ∈ . • The second partition is conducted. Firstly, we evaluate the weight of each node j to k. Here, = + Ĥ ; + = 1, 0 ≤ , ≤ 1, ⍱ ∈ , ⍱ ∈ [1, ]. We define and as part of the distance and routing hop priorities, respectively, and is the serial number of the distance from node j to cluster centroid k. If = 1 denotes the minimum value of , = denotes the maximum distance .
We denote ⍫ = as the ratio of and . If ⍫ → ∞, our algorithm only considers a routing hop. On the contrary, our algorithm can also consider distance when ⍫ → 0. In this case, we jointly consider the routing hop and distance to make ⍫ = 1. Next, the smallest is chosen, owing to its partitioning, and the node j is assigned to the kth region. This whole process is repeated until all the nodes in the network are partitioned. The proposed network partitioning and clustering method can be summarized in Algorithm 1.

Algorithm 1:
Node-partition algorithm based on cluster centroids Let S sensor-nodes and c cluster centroids be inputted We denote the output as: min[

… , ]
Calculate distance and the shortest hop ĥ between node j and centroid k where ∈ , ∈ .

Arrange
, … , and (ĥ , … , ĥ ) in ascending order denotes the distance of the serial number and Ĥ is the routing hop serial number of node j to centroid k.
], the node j is assigned to the kth region. = −

End
In addition, computing the distance and the routing hop for every node requires at least | |. Every node has a sorting time complexity equal to 0(| |{ln | |}) [9]. Since there are S sensor-nodes, we would need | |{ln | |}. Lastly, to calculate and the least value in … , , we need the time complexity | | and 0(| |{ln | |}), respectively. Hence, the overall complexity of this algorithm is 0(| |{ln | |}). Figure 4 shows the results of the network partition. The red dotted lines equally partition the network into four parts, Q, R, S, and T. This represents the first division. Besides, the red points indicate the cluster heads (CHs) of every region, which also indicate the initial points of each region. Then, a second partition is run based on these k ∈ [1, c], W jk = min W j 1 . . . , W j c , the node j is assigned to the kth region.
In addition, computing the distance and the routing hop for every node requires at least |Sc|. Every node has a sorting time complexity equal to 0(|c|{ln|c|}) [9]. Since there are S sensor-nodes, we would need |Sc|{ln|c|}. Lastly, to calculate W j and the least value in W j 1 . . . , W j c , we need the time complexity |c| and 0(|c|{ln|c|}), respectively. Hence, the overall complexity of this algorithm is 0(|Sc|{ln|c|}). Figure 4 shows the results of the network partition. The red dotted lines equally partition the network into four parts, Q, R, S, and T. This represents the first division. Besides, the red points indicate the cluster heads (CHs) of every region, which also indicate the initial points of each region. Then, a second partition is run based on these CHs. We obtain the same shape of the cluster with the nodes belonging to a partition after operating a partition algorithm. The blue dotted circle represents the result of the second partition. Obviously, some nodes previously belonging to areas Q, R, S, and T are now assigned to the U region, as a result of the second partition. The final partition occurs when all the nodes in the network have been assigned to a particular cluster, having a CH in that region.

Charging Model and Strategy for MCs
Our model considers point-to-point charging. That is, each sensor-node is charged by an MC at a time when the MC approaches it at a very close range to optimize the charging efficiency. The movement time of the MC to visit each of the sensor-nodes is considered negligible compared to the charging time; nonetheless, the trajectory of the MC is particularly of interest to us, since it may incur some sizeable and reasonable cost implications. We assume that the charging time is equal for all the sensors and is independent of their battery status. Compared to other works, which integrate and couple both phases, our charging model is split into two phases: the coordination phase, and the charging phase. These two phases are discussed in detail under Section 5. One of the objectives of this study is to investigate efficient coordination procedures and charging strategies for the MCs. This will help to avoid/minimize some charging conflicts among the sensor-nodes. We note that charging conflicts may arise in a situation, such as when a charging request from a sensor-node to an MC causes a sudden rise in the RF exposure of the networks above the safety threshold because another MC is already responding to an energy request from another sensor by radiating its energy to the sensor. Again, in the absence of proper coordination among several multiple MCs and receivers, a malicious energy receiver, which is fully charged, may prevent other energy receivers from being replenished by reporting a high RF value above the threshold and, subsequently, prompting energy transmitters to switch off their power transceivers or significantly minimize their rate of power transmission and, consequently, deprive many neighboring nodes of energy replenishment. Because power transmitters always expect requests and feedback from the receivers, a malicious energy receiver can purposely inject malicious feedback to undermine the general transmission efficiency. A situation may also arise where greedy and cheating energy receivers become the largest beneficiaries by continuously sending charging requests to the power transmitters, with the intention of starving other neighboring receivers of energy replenishment. This becomes more problematic in cases where the MCs are equipped with directional antennas.

Recharge Optimization Problem with Multiple Mobile Chargers (ROPMMCs)
This section presents our multiple charger recharge optimization problem (MCROP) and considers some practical challenges for real sensing applications. The first challenge is the palpable increase in MC's 'movement energy' consumption, due to the improper coordination procedures for energy provisioning to the sensors. The 'movement energy' of the MC is the amount of energy spent in traversing the network to recharge the sensors. In this instance, the trajectories of the MCs should be of paramount importance to minimize the movement energy consumption and travelling costs. Thus, the recharge route should be carefully structured to reflect the MC's current energy position and traveling costs. The second constraint has to do with the non-uniformity in energy consumption occasioned by data transmissions. We note that some nodes assume higher energy consumption rates than others and should be given higher priority recharge-request considerations (i.e., they should be allocated a higher recharge frequency) to maintain the network's functionality. This is particularly the case with nodes close to the CHs with higher data transmission activities compared to other nodes. We adopted an adaptive recharge threshold, which is proportional to the energy consumption rates at various cluster-regions. Hence, nodes close to the CHs are given a higher charging frequency than others. The recharge routes also take into consideration all the concerns stated earlier.
Next, we formulate the wireless recharging problem as a multiple charger recharge optimization problem (MCROP) and show that it is N P-complete from the geometric traveling salesman problem (G-TSP). Our aim is to minimize the MCs' total cost of traveling when they traverse the network to charge the sensors. The TSP with profits [44] presents a reward by visiting a city with the aim of maximizing profits. The reward, in our problem, denotes the amount of replenishable energy by a sensor-node, while the cost is a measurement of the energy cost used to traverse the entire network.
The MCROP can be formulated as follows: MCROP: Given a set of MCs, M = {1, 2, . . . m}, and a set of rechargeable sensor-nodes, S r = {1, 2, . . . s r }, where each sensor-node is capable of storing E units of energy for each s r ∈ S r in a list L s of pairs t r s r , e r s r , r ≥ 1, where t r s r represents the time taken to generate the rth message of s r , and e r s r is the energy expended by the sensor-node in its message transmission. We consider a graph, G = (U, V), where U j (j ∈ S r ) is the location of the sensor-nodes j to be visited, and V is the set of edges. Each edge V j,k has a traveling energy cost e j,k which varies directly with the distance between the nodes j and k, and a latency cost c jk = t j + t jk , where t j is the recharge time of node j from its current energy status to its full capacity, and t jk is the traveling time from node j to node k. The recharge capacity R c of an MC is used to determine the maximum number of sensor-nodes to be recharged before the MC's energy is depleted. We note that different MCs could have varying recharge capacities during the run. We also consider an |S r |×|S r | matrix R, where R j,k is the distance between sensors j and k and an MC, M, which can recharge a sensor-node to its initial energy in a timely manner. If we assume that the MC moves at a constant speed v, then the time of travel between j and k can be calculated as The MCROP is to determine the feasibility of scheduling an MC for node-energy replenishment to minimize/avoid packet loss arising from insufficient energy.
We adapt the method in [22] to prove that MCROP is N P-complete. If we consider a certain walk W c of the mobile charger (MC) when it visits the sensors in S r , we can validate whether this walk is sufficient enough to avoid packet loss, i.e., no data packet y is generated on a sensor-node s r such that y denotes the rth message of s r and s r has less than e r s r available energy at time t r s r . This can particularly occur in 0(T E . |W c |) time, where T E is the sum of events generated in the network. Hence, MCROP ∈ N P. To complete the proof and show that MCROP is N P-hard, we use the G-TSP (see [45], pg. 212). If Q ⊆ W × W and A U are the inputs of G-TSP, then this can be transformed into an input for MCROP, as follows. We use a set S r of |S r | = |Q| sensors and R j,k equal to the Euclidean distance between the j th and kth points in Q. Moreover, each sensor s r ∈ S r has a list of events defined by L s = (0, E), A v , 1 where v denotes the MC's speed. This implies that each sensor s r has two event-occurrences. The event at time 0 depletes all the available energy in s r and the other at A v requires energy 1. A solution to this MCROP problem gives an answer to G-TSP, implying that G-TSP ≤ MCROP.

MCs Coordination and Charging Traversal Strategies
In Section 4, we showed that the problem is hard to solve, even with the global knowledge of the energy dissipation. The heuristics presented face a more difficult problem, since the knowledge is restricted to a local level. In contrast to other known approaches, we have split the MC's charging traversal strategies into two phases, namely, the coordination and the charging phases. This splitting will give us room to focus on each aspect more precisely. To thoroughly understand and examine the MC's coordination and charging process, we first analyze some of its characteristics and inherent trade-offs.

Percentage of Energy Available to the MCs
An important tradeoff to be investigated is the optimal amount of energy for the MCs, with respect to the overall network energy,

The Deployment and Energy Model
Three kinds of devices can be identified with the proposed scheme, namely, the static sensors, the mobile chargers (MCs), and the static sink, as shown in Figure 2. Our model assumes that there are sensors of the wireless communication distance ɖ randomly distributed in a circular sensing field with the area ⍲ and the radius ℜ (we investigate both uniform and non-uniform distributions) and n MCs are initially deployed at the center of a circular area along the x-y coordinates in Equation (1): where p = {1, 2, … n}. If we assume that the sensors are uniformly and randomly distributed in a circular area of the radius ℜ , then the network density is given by The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: total . In order to maintain an open-mindedness, we assume that the total available network energy

The Deployment and Energy Model
Three kinds of devices can be identified with the proposed scheme, namely sensors, the mobile chargers (MCs), and the static sink, as shown in Figure 2. O assumes that there are sensors of the wireless communication distance ɖ distributed in a circular sensing field with the area ⍲ and the radius ℜ (we in both uniform and non-uniform distributions) and n MCs are initially deploy center of a circular area along the x-y coordinates in Equation (1): cos n (2 − 1) , where p = {1, 2, … n}. If we assume that the sensors are uniformly and distributed in a circular area of the radius ℜ , then the network density is give § = ℛ The communication range of the sensors varies according to requireme underlying routing protocols. Because the algorithms can run for an extended only one initialization, the initial position of the MCs may not be important a also be located at the center of the network. The base station or sink can lie an the network and is used to replenish the MCs and aggregate data from the M model relies on both wireless power transfer and data collection. For simp assume that all sensors can generate data packets at the same rate of δ per un This assumption is typical of the monitoring and reporting applications in a W denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes an the total energy initially possessed by the MCs that can be delivered to the n charge the sensors. Each sensor and each MC, respectively, have a maximum a energy representing their initial energy, given as: total is finite and remains the same for all cases. This will help to investigate any increase in the energy efficiency (and to what extent) with respect to the presence of the MCs in the network and the charging process. This specific tradeoff defines how much energy (with respect to the total available energy) should be initially provided to the MC. It may appear that increasing the energy of the MC will result in a better energy management in the network. However, this idea is contradicted by Equation (3), i.e., distributed in a circular area of the radius ℜ , then the network density is given by: The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: distributed in a circular area of the radius ℜ , then the network density is given by: § = ℛ The communication range of the sensors varies according to requirements of underlying routing protocols. Because the algorithms can run for an extended time w only one initialization, the initial position of the MCs may not be important as it co also be located at the center of the network. The base station or sink can lie anywhere the network and is used to replenish the MCs and aggregate data from the MCs. T model relies on both wireless power transfer and data collection. For simplicity, assume that all sensors can generate data packets at the same rate of δ per unit of tim This assumption is typical of the monitoring and reporting applications in a WSN. If denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ the total energy initially possessed by the MCs that can be delivered to the network charge the sensors. Each sensor and each MC, respectively, have a maximum amoun energy representing their initial energy, given as: Ḗ = Ḗ and Ḗ = Ḗ sensors + distributed in a circular area of the radius ℜ , then the network density is g § = ℛ The communication range of the sensors varies according to requirem underlying routing protocols. Because the algorithms can run for an extend only one initialization, the initial position of the MCs may not be importan also be located at the center of the network. The base station or sink can lie the network and is used to replenish the MCs and aggregate data from th model relies on both wireless power transfer and data collection. For si assume that all sensors can generate data packets at the same rate of δ per This assumption is typical of the monitoring and reporting applications in a denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes the total energy initially possessed by the MCs that can be delivered to the charge the sensors. Each sensor and each MC, respectively, have a maximum energy representing their initial energy, given as: MC . If we assume that the total energy available to the network, where p = {1, 2, … n}. If we assume that the sensors are uniformly and randomly distributed in a circular area of the radius ℜ , then the network density is given by: The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: total , remains the same, then any increase in the energy of the MCs implies that the sensor-nodes will not be fully charged (i.e., will initially receive partial charging) and will, most likely, run out of energy shortly before being charged by the MC, thus leading to a possible operational network breakdown. We investigate the optimal amount of energy available to MCs as a percentage of ( , ) = The communication range of the sensors varies according underlying routing protocols. Because the algorithms can run fo only one initialization, the initial position of the MCs may not also be located at the center of the network. The base station or the network and is used to replenish the MCs and aggregate d model relies on both wireless power transfer and data collec assume that all sensors can generate data packets at the same r This assumption is typical of the monitoring and reporting appl denote Ḗ as the total available energy in the network, then i where Ḗ is the amount of energy distributed among the s the total energy initially possessed by the MCs that can be deli charge the sensors. Each sensor and each MC, respectively, have energy representing their initial energy, given as: Ḗ = Ḗ and Ḗ = Ḗ total , by selecting the initial energy of the MC to be 20, 40, 60, and 80% of the total energy of the network for the various ratios of both uniform and non-uniform distributions) and n MCs are initially deployed at the center of a circular area along the x-y coordinates in Equation (1): ( , ) = ℜ 2 cos n (2 − 1) , where p = {1, 2, … n}. If we assume that the sensors are uniformly and randomly distributed in a circular area of the radius ℜ , then the network density is given by: The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: where p = {1, 2, … n}. If we assume that the sensors are uniformly and randomly distributed in a circular area of the radius ℜ , then the network density is given by: The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: init MC . Figure 5 shows the result of this investigation. From this result, it is evident that above 40% of the total energy provision to the MC will negatively affect the operations of the network. This result is predicated on two factors. Firstly, from Equation (3) above, an increase in the energy of the MC minimizes the initial energy of the nodes, thus leading to a quicker node death rate. Secondly, increasing the energy of the MC to a high amount may reduce the MC's rate of energy distribution in the entire network, resulting in an increase in the residual energy at the MC when the network is disconnected. A major conclusion shows that using a moderate percentage of energy at the MC is correct, such as the arbitrary value of 20%.

Partial vs. Full Charging
One straightforward strategy about the MC's visit to a sensor-node would be to fully charge the node. By so doing, the MC would have maximized the time interval of revisiting the sensor-node before its energy is depleted. However, the operations of the network are such that energy is dissipated at the sensor-nodes for data transmissions and at the MC for charging activities, thus making it difficult for the MC to deliver energy to more and more sensor-nodes, since the MC will have increasingly less energy to distribute.
Another strategy would be for the MC to judiciously distribute its available energy to as many sensor-nodes as possible to prolong the network lifetime. With this rationale, the amount of energy that the MC can deliver to a node j is proportional to the residual charging energy of the MC. Simply put, the MC charges a node until its energy becomes One straightforward strategy about the MC's visit to a sensor-node would be to fully charge the node. By so doing, the MC would have maximized the time interval of revisiting the sensor-node before its energy is depleted. However, the operations of the network are such that energy is dissipated at the sensor-nodes for data transmissions and at the MC for charging activities, thus making it difficult for the MC to deliver energy to more and more sensor-nodes, since the MC will have increasingly less energy to distribute.
Another strategy would be for the MC to judiciously distribute its available energy to as many sensor-nodes as possible to prolong the network lifetime. With this rationale, the amount of energy that the MC can deliver to a node j is proportional to the residual charging energy of the MC. Simply put, the MC charges a node until its energy becomes To determine the best charging strategy of the MC and the energy that it transmits to the sensor-nodes, we adapt the MC charging model and strategy in [22,46], where the full charging strategy is compared against our adaptive partial charging strategy. The model ≈ ℛ The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: The communication range of the sensors varies according to requirements of underlying routing protocols. Because the algorithms can run for an extended time only one initialization, the initial position of the MCs may not be important as it c also be located at the center of the network. The base station or sink can lie anywhe the network and is used to replenish the MCs and aggregate data from the MCs. model relies on both wireless power transfer and data collection. For simplicity assume that all sensors can generate data packets at the same rate of δ per unit of t This assumption is typical of the monitoring and reporting applications in a WSN. I denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ the total energy initially possessed by the MCs that can be delivered to the networ charge the sensors. Each sensor and each MC, respectively, have a maximum amou energy representing their initial energy, given as: To determine the best charging strategy of the MC and the energy that it transmits to the sensor-nodes, we adapt the MC charging model and strategy in [22,46], where the full charging strategy is compared against our adaptive partial charging strategy. The model in [46] used Friis' free space propagation model to compute the charging efficiency of the nodes according to Equation (9): where are the uth charging points of the MCs, = 8dBi (transmit gain), = 2dBi (receive gain), is the wavelength, is the polarization loss, ŋ is the rectifier efficiency, ϻ is an adjustable parameter that adjusts Friis' free space equation for short distance transmissions, is the transmit power, is the distance between node j and the MC, and R is the charging radius of the MC. If ≤ R , Equation (9) is transformed [46] as: where Ծ = 4.32 × 10 , ϻ = 0.2316. The energy of the sensor-node, after being charged by the MC, is given in Equation (11) [47]: where is the residual energy of the sensor-node j before charging, t is the charging time of the MC, × t represents the energy received by node j from the MC, and Ḗ is the maximum storage energy threshold of the sensors. Here, we randomly select 40% of Ḗ as the initial energy of the MC. The results are shown in Figure 6. The basic result obtained shows that our partial charging strategy is efficient compared to a full charging strategy. Our partial charging scheme outperforms its full charging counterpart after a given number of generated events. The outcome of this investigation reveals the fact that the MC expends more energy on sensor recharging when applying the full charging method. Hence, much of the MC's energy is consumed quicker, leading to a rise in the node death rate.  , d p u ≤ R mc 0, d p u > R mc (9) where p u are the uth charging points of the MCs, G t = 8dBi (transmit gain), G r = 2dBi (receive gain), λ is the wavelength, P L is the polarization loss, η is the rectifier efficiency, 15 of 28 riis' free space propagation model to compute the charging efficiency of the ing to Equation (9): the uth charging points of the MCs, is the wavelength, is the polarization loss, ŋ is the rectifier is an adjustable parameter that adjusts Friis' free space equation for short missions, is the transmit power, is the distance between node j and R is the charging radius of the MC. If ≤ R , Equation (9) is 46] as: 32 × 10 , ϻ = 0.2316. gy of the sensor-node, after being charged by the MC, is given in Equation he residual energy of the sensor-node j before charging, t is the charging C, × t represents the energy received by node j from the MC, and maximum storage energy threshold of the sensors. randomly select 40% of Ḗ as the initial energy of the MC. The results Figure 6. The basic result obtained shows that our partial charging strategy pared to a full charging strategy. Our partial charging scheme outperforms ng counterpart after a given number of generated events. The outcome of tion reveals the fact that the MC expends more energy on sensor recharging g the full charging method. Hence, much of the MC's energy is consumed ng to a rise in the node death rate.
ber of alive nodes over time for different MC initial energy percentages.
is an adjustable parameter that adjusts Friis' free space equation for short distance transmissions, P t is the transmit power, d p u is the distance between node j and the MC, and R mc is the charging radius of the MC. If d p u ≤ R mc , Equation (9) is transformed [46] as: Electronics 2022, 11, 371 15 of 28 in [46] used Friis' free space propagation model to compute the charging efficiency of the nodes according to Equation (9): where are the uth charging points of the MCs, = 8dBi (transmit gain), = 2dBi (receive gain), is the wavelength, is the polarization loss, ŋ is the rectifier efficiency, ϻ is an adjustable parameter that adjusts Friis' free space equation for short distance transmissions, is the transmit power, is the distance between node j and the MC, and R is the charging radius of the MC. If ≤ R , Equation (9) is transformed [46] as: where Ծ = 4.32 × 10 , ϻ = 0.2316. The energy of the sensor-node, after being charged by the MC, is given in Equation (11) [47]: where is the residual energy of the sensor-node j before charging, t is the charging time of the MC, × t represents the energy received by node j from the MC, and Ḗ is the maximum storage energy threshold of the sensors. Here, we randomly select 40% of Ḗ as the initial energy of the MC. The results are shown in Figure 6. The basic result obtained shows that our partial charging strategy is efficient compared to a full charging strategy. Our partial charging scheme outperforms its full charging counterpart after a given number of generated events. The outcome of this investigation reveals the fact that the MC expends more energy on sensor recharging when applying the full charging method. Hence, much of the MC's energy is consumed quicker, leading to a rise in the node death rate. in [46] used Friis' free space propagation model to compute the charging efficiency of the nodes according to Equation (9): where are the uth charging points of the MCs, = 8dBi (transmit gain), = 2dBi (receive gain), is the wavelength, is the polarization loss, ŋ is the rectifier efficiency, ϻ is an adjustable parameter that adjusts Friis' free space equation for short distance transmissions, is the transmit power, is the distance between node j and the MC, and R is the charging radius of the MC. If ≤ R , Equation (9) is transformed [46] as: where Ծ = 4.32 × 10 , ϻ = 0.2316. The energy of the sensor-node, after being charged by the MC, is given in Equation (11) [47]: where is the residual energy of the sensor-node j before charging, t is the charging time of the MC, × t represents the energy received by node j from the MC, and Ḗ is the maximum storage energy threshold of the sensors. Here, we randomly select 40% of Ḗ as the initial energy of the MC. The results are shown in Figure 6. The basic result obtained shows that our partial charging strategy is efficient compared to a full charging strategy. Our partial charging scheme outperforms its full charging counterpart after a given number of generated events. The outcome of this investigation reveals the fact that the MC expends more energy on sensor recharging when applying the full charging method. Hence, much of the MC's energy is consumed quicker, leading to a rise in the node death rate. ive gain), is the wavelength, is the polarization loss, ŋ is the rectifier ency, ϻ is an adjustable parameter that adjusts Friis' free space equation for short nce transmissions, is the transmit power, is the distance between node j and C, and R is the charging radius of the MC. If ≤ R , Equation (9) is formed [46] as: e Ծ = 4.32 × 10 , ϻ = 0.2316. The energy of the sensor-node, after being charged by the MC, is given in Equation  47]: e is the residual energy of the sensor-node j before charging, t is the charging of the MC, × t represents the energy received by node j from the MC, and is the maximum storage energy threshold of the sensors. Here, we randomly select 40% of Ḗ as the initial energy of the MC. The results hown in Figure 6. The basic result obtained shows that our partial charging strategy icient compared to a full charging strategy. Our partial charging scheme outperforms ll charging counterpart after a given number of generated events. The outcome of nvestigation reveals the fact that the MC expends more energy on sensor recharging applying the full charging method. Hence, much of the MC's energy is consumed er, leading to a rise in the node death rate. in [46] used Friis' free space propagation model to compute the charging efficiency of the nodes according to Equation (9): where are the uth charging points of the MCs, = 8dBi (transmit gain), = 2dBi (receive gain), is the wavelength, is the polarization loss, ŋ is the rectifier efficiency, ϻ is an adjustable parameter that adjusts Friis' free space equation for short distance transmissions, is the transmit power, is the distance between node j and the MC, and R is the charging radius of the MC. If ≤ R , Equation (9) is transformed [46] as: where Ծ = 4.32 × 10 , ϻ = 0.2316. The energy of the sensor-node, after being charged by the MC, is given in Equation (11) [47]: where is the residual energy of the sensor-node j before charging, t is the charging time of the MC, × t represents the energy received by node j from the MC, and Ḗ is the maximum storage energy threshold of the sensors. Here, we randomly select 40% of Ḗ as the initial energy of the MC. The results are shown in Figure 6. The basic result obtained shows that our partial charging strategy is efficient compared to a full charging strategy. Our partial charging scheme outperforms its full charging counterpart after a given number of generated events. The outcome of this investigation reveals the fact that the MC expends more energy on sensor recharging when applying the full charging method. Hence, much of the MC's energy is consumed quicker, leading to a rise in the node death rate.
= 0.2316. The energy of the sensor-node, after being charged by the MC, is given in Equation (11) [47]:

The Deployment and Energy Model
Three kinds of devices can be identified with the proposed scheme, na sensors, the mobile chargers (MCs), and the static sink, as shown in Figur assumes that there are sensors of the wireless communication distan distributed in a circular sensing field with the area ⍲ and the radius ℜ both uniform and non-uniform distributions) and n MCs are initially d center of a circular area along the x-y coordinates in Equation (1): where p = {1, 2, … n}. If we assume that the sensors are uniformly distributed in a circular area of the radius ℜ , then the network density is § = ℛ The communication range of the sensors varies according to requi underlying routing protocols. Because the algorithms can run for an exte only one initialization, the initial position of the MCs may not be impor also be located at the center of the network. The base station or sink can l the network and is used to replenish the MCs and aggregate data from model relies on both wireless power transfer and data collection. For assume that all sensors can generate data packets at the same rate of δ p This assumption is typical of the monitoring and reporting applications i denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nod the total energy initially possessed by the MCs that can be delivered to charge the sensors. Each sensor and each MC, respectively, have a maxim energy representing their initial energy, given as: where E j is the residual energy of the sensor-node j before charging, t v is the charging time of the MC, P r p u × t v represents the energy received by node j from the MC, and

The Deployment and Energy Model
Three kinds of devices can be identified with th sensors, the mobile chargers (MCs), and the static s assumes that there are sensors of the wireless distributed in a circular sensing field with the area both uniform and non-uniform distributions) and center of a circular area along the x-y coordinates in The communication range of the sensors var underlying routing protocols. Because the algorith only one initialization, the initial position of the M also be located at the center of the network. The ba the network and is used to replenish the MCs an model relies on both wireless power transfer an assume that all sensors can generate data packets This assumption is typical of the monitoring and r denote Ḗ as the total available energy in the ne where Ḗ is the amount of energy distributed the total energy initially possessed by the MCs th charge the sensors. Each sensor and each MC, resp energy representing their initial energy, given as: and max sensor is the maximum storage energy threshold of the sensors.
Here, we randomly select 40% of

The Deployment and Energy Model
Three kinds of devices can be identified with the proposed scheme, namely, the static sensors, the mobile chargers (MCs), and the static sink, as shown in Figure 2. Our model assumes that there are sensors of the wireless communication distance ɖ randomly distributed in a circular sensing field with the area ⍲ and the radius ℜ (we investigate both uniform and non-uniform distributions) and n MCs are initially deployed at the center of a circular area along the x-y coordinates in Equation (1): where p = {1, 2, … n}. If we assume that the sensors are uniformly and randomly distributed in a circular area of the radius ℜ , then the network density is given by: The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of δ per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote Ḗ as the total available energy in the network, then initially, where Ḗ is the amount of energy distributed among the sensor-nodes and Ḗ is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as: total as the initial energy of the MC. The results are shown in Figure 6. The basic result obtained shows that our partial charging strategy is efficient compared to a full charging strategy. Our partial charging scheme outperforms its full charging counterpart after a given number of generated events. The outcome of this investigation reveals the fact that the MC expends more energy on sensor recharging when applying the full charging method. Hence, much of the MC's energy is consumed quicker, leading to a rise in the node death rate.

Coordination Strategy
As MCs traverse the network to replenish the sensor-nodes, their own energy is also consumed. The rate of their energy consumption may vary, owing to the non-uniform rate of events generation. Hence, some MCs may be more burdened than others. For this reason, it is necessary for the MCs to periodically communicate with themselves for the proper coordination in their charging process to optimize energy efficiencies and their associated costs. For example, a weaker MC, in terms of energy, may be assigned to a smaller network area. The coordination strategy of MCs can be achieved either through a centralized or distributed approach. In the distributed approach, an MC is informed about the energy position of its neighboring chargers, leading to a more secluded coordination between neighboring chargers. The centralized approach is performed using information from all n MCs. For centralized coordination, we assume that the computations are performed by a computationally strong network entity, such as the sink. Generally, centralized coordination is seen to be powerful compared to distributed coordination. Hence, the performance of their protocols serves an upper bound on which their distributed counterparts are compared to.

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Distributed coordination: Here, the MCs perform distributed coordination among themselves and assume limited network knowledge. The network area is split into slices as shown in Figure 3a, and each slice is assigned to an MC. Each MC can adjust their slice limits to suit their region of interest, either by increasing or shrinking the slice. The MCs, in a distributive manner, define their slice limits in accordance with the size of the region of interest. Hence, an MC with a lower energy status provides its neighbor with a portion of its region of interest for charging. Besides, the energy levels of the MCs determine which of them should reduce their region of interest. Figure 7 shows the distributed form of coordination. Specifically, Figure 7a-c depicts distributed coordination before, during, and after the coordination, respectively. • Centralized coordination: We can distinguish two kinds of centralized coordination. (a) The MCs assume local network knowledge (LNK), and (b) the MCs assume global network knowledge (GNK). In (a), the coordination process uses information from all MCs, including their energy status and position. Essentially, each MC is assigned to a network region. Using Equation (1), the initial deployment coordinates are (m, n) = 2 cos π n (2p − 1) , 2 sin π n (2p − 1) where p = {1, 2, . . . n}. The network area is split into slices, as shown in Figure 3a (e.g., for n = 8 chargers) with one MC assigned to each slice. During the initialization of the coordination process, we calculate the region of each MC. An MC is assigned to a regional area equivalent to its present energy status to balance the energy dissipation among the MCs. To calculate the regional area of an MC k we compute the central angle φ k corresponding to the slice of the MC. That is, In the case of global knowledge, the MCs assume global network knowledge and obtain the most updated energy information from sensors, which they use to make real-time decisions. The energy information is aggregated by CHs (special nodes in the network acting as representatives of the partitioned network areas). The MCs communicate with each other to update their energy status and positions. To avoid charging conflicts (already discussed in Section 3.4) where MCs select the same node for charging, the sink can be used to store and update each node's availability, as well as prioritizing their charging requests. This can be done by maintaining a 0-1 node list wherein a node is assigned a value of 1 (locked) when selected for charging and is returned to a value of 0 after being charged. where E curr , E init , and d curr are the current energy, initial energy, and current transmission distances of each sensor, respectively.

The Charging Traversal Strategy
Here, the MC traverses the network area it is assigned to and charges the sensors in that region. Studies on charging traversal strategies are currently limited to a single mobile charger (MC). In our approach, we have extended this to multiple MCs and have also given special attention to the amount of knowledge possessed by the MCs, in terms of its region. Hence, to suitably design the MC's charging traversal strategy, we distinguish the MC's amount of knowledge under the following three major characteristics: no network knowledge (NNK), local network knowledge (LNK), and global network knowledge (GNK).
In the case of NNK, the MC is assumed to be 'blind' and, thus, is limited in using sophisticated charging methods. The trajectory of the MC is also restricted to several naïve options, presented in [22]. Our approach uses the 'blind' scanning of the network area, where the MC begins its tour from the sink and traverses an exhaustive route until it reaches the boundaries of the network region. The advantage of this movement is that the MC tours the entire network to charge almost all the nodes until its energy is completely depleted. However, owing to lack of knowledge, this movement is not adaptive.
In terms of LNK, the MC operates with a local network assumption. The slice relating to charger k is divided into l sectors S kl of the same width, as shown in Figure 3b. The sectors in charger k are prioritized based on the high number of sensors with a low residual energy status. We adapt the theoretical model in [10] for the charging process. Let E min kl represent the lowest residual energy level of a sensor-node in the sector S kl . Let E min+ kl represent an energy level close to E min kl , then E min+ kl = E min kl + ∆ · E max sensor E min kl , ∆ ∈ (0, 1).
Let n(S kl ) denote the number of sensors in Sector S kl with residual energy between . The motive behind this charging process is to group many sensor-nodes according to slices (sectors) and to select a particular group based on their energy requirements. The group that requires more energy than the others is first selected for charging. The GNK charging strategy considered in our model is impacted by two important metrics, energy and distance. These two factors have equal weight, and one is not more dominant than the other. The MC calculates the product of these two metrics for every node in their cluster and chooses to charge the node with the minimized product. For each phase, the MC moves to a sensor in the cluster in a way that reduces the product of each sensor's energy and the distance from the MC. Specifically, the MC charges the node j = arg j∈C k min 1 + d jk where E j is the residual energy of the sensor j. In other words, the charging strategy is prioritized among nodes with a low residual energy and a minimum distance from the MC.
The analytical studies conducted in this section enable us to investigate the traveling distance of the MCs operating under the three scenarios that have already been discussed. Essentially, the traveling distance of the MCs is an indirect reflection of the coordination process and the charging strategy. The traveling distance also has an impact on the relevant movement cost. Figures 8 and 9 depict the results for both uniform and non-uniform cases of node deployment. The results show the total distance travelled by three MCs exploring the three scenarios (GNK, LNK, and NNK) for both uniform and non-uniform deployments. It is very clear that the MCs operating under GNK and LNK achieve the required charging strategy by traveling less distance than NNK. As seen in both Figures, MCs exploring the NNK scenario travel a longer distance, compared to MCs exploring LNK and GNK. The MCs also achieve a balanced distribution of the traveled distance, unlike when the chargers operate under NNK. In Figure 9, the total distance covered by all MCs is higher compared to that in Figure 8. This implies that in non-uniform deployment, MCs travel much longer distances than in the uniform case. This presents a more realistic scenario that occurs in real-life large-scale sensor networks. Nonetheless, the distribution of the total distance covered for both cases are similar. Generally, a non-uniform node deployment incurs a much longer distance (and, by extension, a higher movement energy consumption) than in the uniform case.

The MC's Movement Energy and Charging Costs
The movement energy cost of an MC is the amount of energy expended by the MC for its movement in order to charge the sensor-nodes within the network region. We assume that MCs have only one energy bank used for both the movement and charging of sensors. On the other hand, the charging cost of the MCs is the amount of energy used to charge the sensors. This paper assumes no energy losses during the charging process. Hence, the charging cost equals the beneficial energy cost (i.e., the energy received by the sensors). The paper also does not consider the amount of energy spent for movement, but it has computed the amount of distance traveled by each MC, which provides some estimate of the cost. More precisely, we have applied similar assumptions for the MCs as in [36,48], as well as the battery calculator in [49], and have provided, in Figure 10, an estimate of the MCs' movement energy spent in traversing the network for its uniform node deployment. It is observed that MCs exploring the NNK spend a high amount of energy in movement. On the other hand, MCs exploring LNK and GNK, to minimize their traveling distance, also minimize their movement energy costs.

Simulation Results and Analysis
The proposed approach is simulated in the MATLAB environment using RMASE, an application framework that runs under PROWLER. The simulator is an event driven WSN simulator which makes use of codes that are compatible with MATLAB codes and, thus, is run in a MATLAB environment. RMASE has well-defined metrics whose values are generated as the simulation is being run. Our experiment considers three swarm intelligence-based routing protocols, IEEABR [50,51] BeeSensor [52,53], and FF [51,54], which are compared using evaluation metrics. For the network model under consideration, we place the sink at the center (m, n) = (0, 0) of the circular network with 1000 to approximately 4000 sensor-nodes and two to approximately six MCs. The following evaluation metrics are considered, and the results are shown in Figures 11-17. Overall, the IEEABR, which is an energy-efficient swarm intelligence-based routing protocol, outperformed other protocols in terms of the evaluation metrics. The reason for the high performance of the IEEABR, compared to other routing protocols, is attributed to its ability to easily converge dynamically to achieve an increased packet delivery ratio.

The Packet Delivery Ratio (PDR) and the Total Packets Delivered
The PDR is defined as the ratio of the total transmitted and received packets. The total number of packets delivered at the sink node may be lower than, or equal to, those generated at the sensor-nodes in the network. Figure 11 shows the results of the PDR, showing the high performance of the IEEABR over other SI-based protocols. Initially, at a very minimal node density, the IEEABR performed optimally by delivering 100% of all the generated packets in the network to the sink node via the MCs, which double as mobile data collectors. This is followed by the BeeSensor, which achieved a 99.91% PDR, and then FFR, with a 99.8% PDR. As the node density gradually increased, the PDR for the IEABR still remained high compared to the other protocols. Even up to the highest node density of 4000 sensor-nodes, the IEEABR still maintained a high PDR of 99.96% and is also rated as high compared to the BeeSensor and the FFR. Figure 12 shows the total packets delivered by each of the routing protocols (which shows the actual number of packets transmitted to the sink from different sensor-nodes). The IEEABR achieved the highest PDR compared to the BeeSensor and the FFR, even at a maximum node density. This achievement is significant and critical for surveillance/monitoring and target-tracking applications in large-scale wireless sensor network environments.

The Energy Consumption of the Sensor-Nodes and MCs
The energy consumption of the sensor-nodes denotes the total energy expended by all the sensor-nodes in the network in the data transmission to the MCs and/or to the sink. Generally, energy consumption refers to the average energy utilized by the sensor-nodes in the network for a given simulation round. Conversely, the energy consumption of the MCs refers to the average energy spent by all the MCs in the network in recharging the sensor-nodes and/or transmitting their data packets to the sink. Figures 13 and 14 show the energy consumption of the sensor-nodes and that of the MCs, respectively. In both figures, the IEEABR outperformed other routing protocols and consumed the least amount of energy compared to the BeeSensor and the FFR, even at a high node density. In Figure 14, the average energy consumption of the MCs is evaluated using various node distributions and a number of MCs. Increasing the number of MCs results in an increase in the total energy consumption. However, as clearly observed, the distribution of nodes does not have impact on the energy consumption of the MCs.
In Figure 15, we show the effect of the number of MCs on the total energy consumed by the sensor-nodes. As can be seen, the energy consumed by h sensor-nodes is reduced by increasing the number of MCs. This is because an increase in the number of MCs in the network further minimizes the transmission/communication range of the sensor-nodes and, consequently, their energy consumption.

Throughput
This is the overall number of packets delivered at the sink within a second. This is measured in data packets per second (Kbits/sec). Figure 16 shows the performance of the routing protocols in terms of throughput. Again, the IEEABR outperformed the other routing protocols. At 2000 nodes, the three protocols almost achieved an equal performance. However, the BeeSensor and the FFR protocols lagged behind the IEEABR as the number of nodes increased (i.e., high node density).

Network Lifetime
This is the extended lifetime of the network for different rounds at a particular time. Figure 17 shows the network operational lifetime prediction of the routing protocols. As can be seen, the IEEABR outperformed other routing protocols by showing the highest lifetime. The network is evaluated by considering the number of rounds until the first node has utilized all its energy. The purpose of this evaluation is to show the improvement and efficacy of SI-based techniques in saving energy and prolonging the network lifetime. The results show that the IEEABR achieved the longest network lifetime, followed by the BeeSensor and then by the FFR, thus making the IEEABR suitable for target tracking application scenarios. Unlike other existing schemes, SI-based techniques employing multiple MCs lead to a reduction in the transmission range of sensor-nodes, as well as their energy consumption, thereby resulting in an improved network lifetime. The figure further shows the ability of the IEEABR to support almost 50% of the additional rounds at the first stage of the network. The maximum support strength is reached as the node density increases. Figure 18 shows the charging time of the MCs per unit cycle with different numbers of sensor-nodes. As shown, the IEEABR outperformed the other routing protocols. In general, the result shows the efficacy of SI-based techniques for wireless mobile charging in a dense sensor network environment. The BeeSensor is observed to have the highest charging time among the protocols. This performance may be attributed to its reactive method of route discovery, which tries to recalculate routes each time there is a need for packet transmission to the sink node. This, invariably, creates delays and, consequently, high charging times.

Conclusions
In this paper, we studied swarm intelligence (SI)-based approaches for efficient mobile wireless charging, which utilize special multiple mobile elements that traverse the network for energy-provisioning to the sensor-nodes. We first formulated the wireless charging problem and showed its computational hardness. We also proposed the node-partitioning algorithm, which adaptively splits the network into smaller sizes to enable efficient WPTs at a low transmission range and to solve the complex problem of scheduling multiple MCs in LS-WSNs. Some critical issues and tradeoffs associated with the mobile wireless charging schemes, which have been overlooked by many researchers, were then investigated. The most important among them included: (1) determining the efficient coordination and charging strategies for the MCs, and (2) determining the optimal amount of energy available for the MCs, given the overall network's available energy. Unlike many existing techniques, we envisioned methods employing SI-based approaches for a distributed wireless charging, which utilized the local and global network knowledge to further optimize the network lifetime. To this end, we compared three state-of-the-art SI-based routing protocols, using evaluation metrics. The results of this evaluation show that the proposed approach is a veritable method that can be exploited to further optimize energy consumption and to extend the network operational lifetime. Our investigation also reveals the efficacy of the partial charging, over the full charging, strategies of the MCs.
In the future, we intend to investigate the possibility of collaborative charging among the MCs, whereby a higher energy MC can be used to charge other chargers with lower energy to maintain a continuous network operation. The MCs employed in this study cannot communicate with more than one sensor-node simultaneously. Thus, another future research direction will consider the modelling, analysis, and simulation of a charging model that simultaneously charges more than one sensor-node at a given time, since the current approach is limited to point-to-point mobile charging (whereby each sensor-node is charged by an MC when the MC approaches it at a very close range to optimize the charging efficiency).