Spatiotemporal charging scheduling in wireless rechargeable sensor networks

Computer Communications 152 (2020) 155–170

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Spatiotemporal charging scheduling in wireless rechargeable sensor networks Chuaxin Zhao a , Hengjing Zhang b , Fulong Chen a ,∗, Siguang Chen c , Changzhi Wu d , Taochun Wang a a

School School c School d School b

of of of of

Computer and Information, Anhui Normal University, Wuhu 241003, China Computer Science, University of Science and Technology of China, Suzhou, 215123, China Internet of Things, Nanjing University of Post and Telecommunication, Nanjing 210003, China Management, Guangzhou university, Guanghzou 510006, China

ARTICLE

INFO

Keywords: Wireless sensor network Wireless charging Joint optimization Evolution teaching learning based optimization Hybrid task

ABSTRACT Wireless sensor networks have a wide range of applications. However, the battery-constrained energy limits the scope of network applications and timeliness. In this paper, we study how to schedule charging and allocate charging time simultaneously for a wireless sensor network for the purpose of prolonging network lifetime and improve charging efficiency. To this end, a mixed integer optimization model for simultaneous charging scheduling and charging time allocation is established through the maximization of the charging efficiency. Then, an offline algorithm is developed to solve the problem. Further, an online charging node insertion algorithm is developed for real-time service. The simulation results show that the proposed algorithm can achieve near 100% charging success rate under periodic and hybrid services, which is significantly superior to the results obtained by the compared algorithms.

1. Introduction Wireless sensor networks have been widely used in industry and science [1]. Due to the energy constraints of sensor nodes powered by batteries, the network has limited lifetime while some applications are expected to work indefinitely. For this reason, how to effectively extend the working time of wireless sensor networks has attracted the continuous attention of researchers. Some researchers have proposed energy-saving methods [2,3], but these methods only extend the limited working time [4]. Recent studies have shown that wireless charging technology may effectively prolong the network lifetime. Wireless sensor networks can be recharged by mobile charging vehicles or by deploying static chargers. With the help of wireless energy transfer technology [5], rechargeable batteries can be supplemented by wireless charging, so that the energy can be updated in time and the network lifetime can be prolonged. By carefully optimizing the charging design, the lifetime of the wireless sensor network can be significantly increased [6]. Generally, wireless sensor networks can be recharged by statically deploying chargers or by mobile wireless charging vehicles(WCV). Static chargers can be deployed in the network to continuously replenish the nodes in the coverage area [7]. However, the coverage of static chargers is limited and multiple chargers need to be deployed, which leads to higher costs. Wireless charging vehicles can replenish the

energy of nodes in different positions in the network by mobile vehicles, and achieve the goal of sustainable network lifetime [8]. Obviously, the nodes in the network need to have their energy before they are exhausted. Therefore, how to schedule the charging behavior of the wireless charging vehicles has become an important issue. Recently, some researchers have put forward a variety of approaches, such as maximizing the charging utility [9], minimizing the total length of the charging tour path [10], charger deployment and path planning [11], charging path optimization and stopping point selection [12]. In these studies, the charging scheduling solutions of mobile wireless charging vehicles are proposed from different perspectives. However, the charging time of nodes is seldom considered in the existing scheduling methods. In the case of multiple requires of nodes, it is difficult to satisfy all nodes on time, which leads to the energy exhaustion in the charging cycle and affects the quality of service of the network. To reduce the number of exhausted nodes in the energy replenishment process, this paper not only considers how to schedule the mobile charging vehicles to minimize the path cost, but also optimizes the supply time of each node. Further, the charging time of each sensor node is optimized to maximize the charging efficiency. The main work of this paper is presented as follows: (1) To improve the charging efficiency and reduce the number of exhausted nodes, a collaborative model of node charging scheduling and time allocation is established to minimize the charging cost of

∗ Corresponding author. E-mail address: [email protected] (F. Chen).

https://doi.org/10.1016/j.comcom.2020.01.037 Received 1 May 2019; Received in revised form 2 December 2019; Accepted 18 January 2020 Available online 23 January 2020 0140-3664/© 2020 Elsevier B.V. All rights reserved.

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Computer Communications 152 (2020) 155–170

architecture and then designed a protocol to schedule the paths of multiple mobile chargers to maximize the efficiency. In [23], Ma, Y.et al. consider how to use a mobile charger to charge multiple sensors simultaneously with energy capacity constraints. They formulate a novel charging utility maximization problem to minimize the sensor energy expiration time and the travel distance of the mobile charger for all requested sensors. Wu, T.et al. study a multi-drone wireless charging method for wireless sensor networks [24]. They jointly optimize the route association and charging routes to maximize the total charging 1 utility and devise an efficient approximation algorithm with a 3𝛼 approximation ratio to solve the NP-hard problem. Because of the complexity of wireless sensor networks, a single charging plan can effectively prolong the network lifetime, but the network utility enhancement is limited. Some researchers consider charging planning and data collection jointly. For example, Wang et al. use mobile charging nodes to recharge sensor nodes and collect data packets in the network and return to base station [25]. To reduce the delay caused by data collection of charging nodes, separate mobile charging and mobile data collection vehicles are used to reduce the end-to-end delay. By considering both charging planning and protocol design of sensor networks, the two behaviors can be better coordinated. In [26], Zhao et al. collaboratively optimize charging and data collection to maximize the charging efficiency while taking into account data acquisition delay under the condition of guaranteeing timely data transmission of nodes. In [27], Gao, Y. et al. jointly schedule recharging and sensor activity to save the traveling energy of vehicles and propose greedy algorithm to balance clustering and distribute sensor activation and schedule recharging. In [28], Jiang, G. et al. jointly optimize travel planning and mobile charging depot deployment for large-scale wireless sensor networks. They propose a multi-stage method to solve charging travel planning, depot deployment and travel assignment. Through deploying multiple depots in network, the limitation that mobile charging vehicles have small battery capacities can be overcome. To respond to node charging in real time, some real-time charging schemes have been proposed. For example, in [29], L Khelladi et al. propose an on-demand planning scheduling algorithm. A one-to-many charging model is used to optimize the TSP to find the charging vehicle path on the basis of minimizing the number of charging stops. In [4], Lin, C et al. designed a temporal–spatial charging algorithm to maximize energy efficiency while minimizing the number of dead nodes to prolong network lifetime. They adjust the charging order based on charging efficiency first, then, the less-efficient sensors are removed, further, an excellent node insertion method is integrated to decrease the exhaustion of sensors. Although there have been some researches focus on charging in wireless sensor networks, there are still some shortcomings. (1) Currently, the research on charging schemes deals with single businesses, such as periodic charging, which must select nodes that need to be charged in advance, or on-demand charging, which leads to lowefficient charging. These schemes designed for specific applications cannot effectively deal with the mixed businesses. (2) The existing literature seldom focuses on charging time. Under the constraints of a small number of charging vehicles and limited carrying energy, it is difficult to ensure that all nodes are satisfied in time [4]. To maximize the received energy efficiency and minimize the number of energydepleted nodes, this paper proposes a charging scheduling and time allocation approach for hybrid services in wireless sensor networks to support the quality of service in wireless sensor networks.

wireless charging vehicles and maximize the energy replenishment of nodes. The model takes into account the service cost and energy replenishment utility, and optimizes the charging time at the same time, which can effectively reduce the number of exhausted nodes. (2) For periodic services, an evolutionary teaching-learning base optimization algorithm is proposed to optimize the discrete scheduling and the continuous time allocation problem. Based on hybrid encoding, a novel teaching and learning iteration process is designed. To reduce the occurrence of infeasible solutions, a charging time repair algorithm is designed. (3) For hybrid services, a dynamic node insertion algorithm is proposed to schedule new nodes for real-time requests in the network. The combination of the dynamic insertion method and periodic scheduling algorithm provides a charging scheduling and time allocation scheme for mixed operation. Finally, the proposed approaches are employed in experiments to verify their effectiveness. The rest of this paper is summarized as follows: the related works are organized in the second section, the system model is formulated in the third section, the periodic scheduling and time allocation method is developed in fourth section, the realtime scheduling for the urgent tasks model is proposed in fifth section, the sixth section evaluates the performance of the algorithm through experiments, and the last section summarizes the paper. 2. Related works Traditional wireless sensor networks mainly powered by batteries, however the limited battery capacity limits network lifetime. The methods of saving sensor energy [13] has been proposed in the past decade; however, energy-saving methods can only extend network lifetime to a limited extent. To solve the problem of energy constraints in wireless sensor networks, many researchers have proposed renewable energy technology methods to obtain the corresponding energy supplement, thus network lifetime is prolonged. For example, the sensor harvests solar energy to supply the nodes [14,15]. Due to the limitation of natural conditions, the energy collection is unpredictable so it is difficult to support stable network services. In recent years, breakthroughs have been made in wireless energy transfer technology. In [5], Kurs et al. proposed a promising wireless energy transfer technology that allows energy to be transmitted from charger to terminal device in the air, which has attracted wide attention of scholars [16]. Some works focus on supplementary the nodes through mobile charging devices. In [17], Jennifer et al. study work time of sensors for bridge monitoring, the nodes that need to be charged are selected reasonably, and an unmanned aerial vehicle is used to replenish these nodes to prolong monitoring lifetime. A key issue that needs to be solved for wireless chargeable sensors is charging planning, which maximizes charging efficiency or minimizes charging cost through reasonable charging scheduling [18]. Xie et al. [19] study wireless power transfer for sensor networks; they convert the energy charging problem into a TSP problem based on an energy consumption model. Then they develop an approximation algorithm to construct the shortest Hamiltonian cycle to solve problem. Because the scheduling in charging planning belongs to the class of NPhard problem, researchers have proposed many heuristic methods to optimize the process. In [20], Li et al. formulated an energy-constrained Qi-ferry wireless charging problem to maximize the number of charged sensors; they propose particle swarm optimization to optimize the problem. Some researchers have investigated charging planning with different objectives, such as Liang et al. who studied the problem with the objective of minimizing the number of mobile chargers in [21], they also designed an approximate algorithm to simultaneously schedule multi-vehicle charging paths to replenish the sensor network based on decomposing the tree of charging nodes with charging station as the root node. In [22], Adelina et al. proposed a hierarchical charging

3. Network model and problem formulation To achieve the permanent work of a wireless sensor network, it is necessary to effectively replenish energy for sensors in the network so that all sensors in the network can steadily work in the expected period. Further, how to select and schedule the nodes to replenish is a key 156

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Computer Communications 152 (2020) 155–170 Table 1 Parameters statement. Parameter

Statement

𝑇 𝑒𝑡 𝑒𝑟 𝑠𝑡𝑖 , 𝑑𝑡𝑖 𝑉𝑐 𝐶𝑉 𝐵 𝐶𝑡𝑜𝑡𝑎𝑙 𝑐 𝐿𝑡𝑜𝑢𝑟 𝜌 𝐵𝑚𝑖𝑛 𝐸𝑚𝑜𝑣𝑒

Period length Unit energy consumption of sending Unit energy consumption of receiving Time windows of node 𝑖 Speed of the vehicle Capacity of sensor battery Current energy of sensor battery Capacity of vehicle battery Charging power Path length of a vehicle Charging efficiency% Min energy threshold of sensors Energy consumption of the vehicle

Fig. 1. A wireless rechargeable sensor network.

issue to improve the utilization of charging and network performance. To fill the batteries of nodes, the charging process may consume too much time, such that some nodes might not be replenished in time. Further, to reduce the depletion of energy, the nodes with an urgent need of charging should be arranged at their charging time window as far as possible. Therefore, the charging time and sequence of the recharged nodes need to be optimized to satisfy the nodes to complete the charging as far as possible in the time window. To improve the received energy efficiency, the charging scheduling and the time are considered together in each period. Fig. 2. Charging function(CV = 1000 J, p = 100 s).

3.1. Network model Assume that a wireless sensor network consists of a set of sensors 𝑉 and a fixed base station 𝑆. A wireless sensor network can be denoted as a weighted graph 𝐺 = (𝑉 ∪𝑆, 𝐸, 𝑊 , 𝐵), where 𝐸 represents a set of links between any two nodes and where the distance between two nodes is in the scope of communication. In the network, sensor nodes sense and generate data and transmit sensing data to the sink node or base station through a multi-hop path which is generated by routing algorithm in advance; 𝑊 represents the set of Euclidean distances between any two nodes, and 𝐵 represents the initial energy of sensor nodes. The base station communicates directly with the wireless charging vehicle through long-distance communication (such as 4/5G), and the sensors use short-distance communication (such as Zigbee [30]) to form a selforganizing network. For instance, a network scenario of a rechargeable wireless sensor network is presented in Fig. 1. As shown in the figure, a wireless charging vehicle starts from a depot and traverses to replenish sensors in turn according to the charging requests of the nodes. After charging the sensors, the vehicle will return to the depot to replenish energy for the next scheduling. In sensor networks, sensor nodes are responsible for sensing and sending or forwarding the data to sink nodes. A node consumes energy when collecting, receiving and forwarding data, which is determined by network topology and data rate. Further, the data rate are assumed to be precise estimated through information measurements [31]. Let the received data of node 𝑖 from the others in unit time be denoted as ∑ 𝑗∈𝐷𝑖 𝑟𝑗 , and 𝐷𝑖 represents the set of nodes their data are forwarded by node 𝑖; moreover, the nodes construct a tree rooted in node 𝑖, and the data generated by node 𝑖 in unit time are 𝑟𝑖 . The energy consumption of node 𝑖 is denoted as 𝑒𝑐𝑖 and can be calculated through formula (1): ∑ ∑ 𝑒𝑐𝑖 = 𝑒𝑡𝑖 (𝑟𝑖 + 𝑟𝑗 ) + 𝑒𝑟𝑖 𝑟𝑗 (1) 𝑗∈𝐷𝑖

less than the minimum energy threshold in a period, that is 𝐵 −𝑒𝑐𝑖 ×𝑇 < 𝐵𝑚𝑖𝑛 , they will send recharge requests in that period. Let 𝑠𝑡𝑖 denotes the time at which node 𝑖 sends a request and 𝑑𝑡𝑖 denote the time at which node 𝑖 expires, which indicates that node 𝑖 should be replenished in the time window [𝑠𝑡𝑖 , 𝑑𝑡𝑖 ]. In addition, the labels of parameters used in this paper are presented in Table 1. 3.2. Charging model According to the characteristics of charging for batteries, the charging process varies from fast charging to slow charging [32]. Accord𝑡 ingly, here the charging process is assumed to be ∫0 𝜌(𝑧)𝑑𝑧, where 𝑡 is 1 (−𝑧∕𝑝) , where 𝐶𝑉 represents the the charging duration, and 𝜌(𝑧) = 𝑝 𝐶𝑉 𝑒 maximum capacity of node batteries and 𝑝 is the time constant [33], which is determined by the charging parameters. It is known that the distance is a very important key to harvest energy according to the characteristics of wireless energy transfer. Thus, the wireless charging vehicle runs near the sensor and stops to transfer energy. Since the distance between a vehicle and a sensor is fixed, it is considered that the charging efficiency is a constant [34]. The charging amount obtained by the battery from energy transfer is presented in Fig. 2. Formula (2) formulates a nonlinear function, it can be observed from the figure that charging efficiency has a marginal effect. The received energy is not linear increased with the extension of time. This process reflects that is a process from fast charging to slow. The specific energy obtained by battery is calculated in (2): 𝑡

∫0

𝑗∈𝐷𝑖

−𝑡

𝜌(𝑧)𝑑𝑧 = 𝐶𝑉 [1 − 𝑒 𝑝 ]

(2)

Obviously, if all requesting nodes are fully charged, it will result in a low efficiency since it will need large amount time. A single WCV with maximum battery capacity 𝐶𝑡𝑜𝑡𝑎𝑙 is considered in this paper to replenish nodes. Since the vehicle also needs to consume energy during its run, the sum of energy consumption of all charged

where the energy consumption of sending and receiving unit data are expressed 𝑒𝑡𝑖 and 𝑒𝑟𝑖 . In addition, the energy consumption of calculation is very trivial and is neglected here. Without loss of generality, the scheduling period is assumed to be 𝑇 . When the energy of the nodes is 157

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Computer Communications 152 (2020) 155–170

charging vehicle is determined by the sequence of the charging nodes and the corresponding distance between them. Formula (11) represents the driving time of vehicle movement. The driving time of the vehicle is determined by the driving length and the speed. Spatiotemporal charging scheduling is considered through integrating route vector 𝑠 and charging time vector 𝑡 in Problem 1. In detail, formula (10) is calculated by route scheduling vector 𝑠, while the charging utility in formula (6) is determined by charging time vector 𝑡. This is a complex optimization problem, and its scheduling part is NP-hard problem, while time allocation is a continuous variables; thus, it is difficult to solve directly.

nodes 𝑖 ∈ 𝑁 and energy consumption of the driving vehicle does not exceed the maximum battery capacity, ∑ 𝑡𝑖 × 𝑐 + 𝐸𝑚𝑜𝑣𝑒 × 𝑇𝑡𝑜𝑢𝑟 < 𝐶𝑡𝑜𝑡𝑎𝑙 (3) 𝑖∈𝑁

where 𝑐 represents the unit energy consumption of charging and 𝑡𝑖 is allocated charging time for sensor 𝑖. 𝐸𝑚𝑜𝑣𝑒 denotes the unit energy consumption of the driving vehicle, and 𝑇𝑡𝑜𝑢𝑟 represents the travel length based on charging scheduling. 3.3. Time constraints After the requested nodes are replenished, the mobile vehicle will return to the depot to replenish energy for the next round. Assuming that the time is limited by the predefined charging period 𝑇 during the charging journey, the charging cycle 𝑇 constraint is defined as: ∑ 𝑇𝑡𝑜𝑢𝑟 + 𝑡𝑖 ≤ 𝑇 . (4)

4. Joint optimization of charging scheduling and time allocation To dispatch energy, the working time is divided into multiple continuous work periods. At the beginning of each cycle, all nodes calculate the current energy consumption rate independently, so the energy consumption of nodes can be calculated using formula (1). Therefore, the node can estimate the energy status at the end of this period based on historical information and all nodes whose energy is less then the threshold at the end of this period need to be recharged. By sending requests to the base station, all nodes whose energy status below the threshold in the current cycle will be collected. The requests of the nodes are used to build a recharged list. The charging sequence and time are optimized according to the list to guide a mobile vehicle for energy replenishment. Due to the complexity of the monitoring, the data flow in wireless sensor networks may be heterogeneous. Monitoring tasks may include different types, such as periodic and real-time sensing. Further, periodic sensing data, such as environmental parameters, do not change significantly. Thus, the energy consumption of nodes can be calculated using historical information in each cycle. In addition, other tasks are ondemand scheduling [35], in which the charging demand of some nodes cannot be determined in advance. This paper considers these two cases simultaneously. More concretely, the first case only considers periodic scheduling, which is the common working mode of environmental parameter sensors, whereas the second case considers the dynamic charging request in the network. Since the variables of model P1 need to be optimized with discrete node sequence and continuous charging time, it is a difficult typical mixed planning model. Even the node scheduling problem is also a typical NP problem since it is a TSP problem. Therefore, it is very difficult to solve the refinement problem. It is necessary to design a heuristic algorithm to solve the problem P1.

𝑖∈𝑁

To prevent the node energy falling below the threshold, each node is required to be recharged before the residual energy reaches the minimum threshold. Let 𝑠(𝑘) denotes the node index of the 𝑘th charged node on the scheduling list. 𝑇𝑠𝑡𝑎𝑟𝑡 +

𝑖−1 ∑

𝑡𝑠(𝑘) +

𝑘=1

𝑖−1 ∑ 𝑑𝑖𝑠𝑡(𝑠(𝑘 − 1), 𝑠(𝑘)) < 𝑑𝑡𝑠(𝑖) . 𝑉𝑐 𝑘=1

(5)

where 𝑇𝑠𝑡𝑎𝑟𝑡 indicates the start time of the charging cycle, 𝑡𝑠(𝑘) represents the charging time allocated to node 𝑠(𝑘), 𝑉𝑐 represents the running speed of the charging vehicle, 𝑑𝑖𝑠𝑡(𝑠(𝑘−1),𝑠(𝑘)) represents travel time taken 𝑉𝑐 by the mobile vehicle from the node 𝑠(𝑘 − 1) to node 𝑠(𝑘), and 𝑑𝑖𝑠𝑡(𝑠(𝑘 − 1), 𝑠(𝑘)) denotes the Euclidean distance between nodes 𝑠(𝑘 − 1) and 𝑠(𝑘). Constraint (5) ensures that each node will be charged before it runs out of energy. 3.4. Charging problem statement Note that every sensor need not replenished in every round since the energy consumption rates of different sensors may vary significantly. Thus, the charged sensors are vary in different rounds. In each period, the service cost of a vehicle to find a series of charging scheduling is expected to minimize the total distance 𝑇𝑡𝑜𝑢𝑟 . Accordingly, the energy consumption 𝐸𝑚𝑜𝑣𝑒 × 𝑇𝑡𝑜𝑢𝑟 is minimized. In addition, let 𝑁 denote the set of charged sensors in the period, the received energy of all charged ∑ 𝑡 nodes can be expressed as 𝑖∈𝑁 ∫0 𝑖 𝜌(𝑧)𝑑𝑧. The purpose of scheduling is to maximize the received energy of nodes while minimizing the service cost. The problem can be expressed as follows (P1): ∑ 𝑡𝑖 𝑖∈𝑁 ∫0 𝜌(𝑠)𝑑𝑠 𝑀𝑎𝑥𝐬,𝐭 (6) 𝐸𝑚𝑜𝑣𝑒 × 𝐿𝑇 𝑜𝑢𝑟 ∑ 𝑠.𝑡. 𝑇𝑡𝑜𝑢𝑟 + 𝑡𝑖 ≤ 𝑇 (7) ∑

4.1. Charging scheduling and time allocation for period task Although the base station has collected accurate information about sensors, it is still difficult to solve the problem due to the complexity of the problem. In recent years, heuristic algorithms have become a popular design method in large-scale wireless sensor networks [36]. For this reason, this section designs an improved teaching and learning based optimization algorithm to solve mixed discrete and continuous variables. The new algorithm is used to synchronously optimize the charging sequence and time allocation. Once the mobile vehicle finishes charging, it will return to the depot to replenish its energy and prepare to start a new cycle. It should be noted that the new cycle starts immediately to reduce the waiting time of the vehicle so as to replenish the nodes on time.

𝑖∈𝑁

(8)

𝑡𝑖 𝑐 + 𝐸𝑚𝑜𝑣𝑒 𝑇𝑡𝑜𝑢𝑟 < 𝐶𝑡𝑜𝑡𝑎𝑙

𝑖∈𝑁

𝑇𝑠𝑡𝑎𝑟𝑡 +

𝑗−1 ∑

𝑡𝑘 +

𝑘=1

𝐿𝑡𝑜𝑢𝑟 =

∑

𝑗−1 ∑ 𝑑𝑖𝑠𝑡(𝑣𝑠(𝑘−1) , 𝑣𝑠(𝑘) ) 𝑘=1

𝑉𝑐

< 𝑑𝑡𝑗 , ∀𝑗 ∈ 𝑁

𝑑𝑖𝑠𝑡(𝑣𝑠(𝑖−1) , 𝑣𝑠(𝑖) ) + 𝑑𝑖𝑠𝑡(𝑣𝑠(𝑖) , 𝑣0 )

(9) (10)

𝑖∈𝑁

𝑇𝑡𝑜𝑢𝑟 =

𝐿𝑡𝑜𝑢𝑟 𝑉𝑐

(11) 4.2. Teaching-learning-based optimization

As mentioned above, the scheduling objective is represented by formula (6), and the constraints are listed in formula (7)–(11). Specifically, formula (7) requires the scheduling result to satisfy the period length constraint, formula (8) specifies the scheduling to satisfy the energy constraint; formula (9) denotes that each node should be charged in time window, and formula (10) indicates that the driving length of the

Teaching-learning-based Optimization (TLBO) is a new intelligent optimization algorithm proposed by Rao et al. [37] and has achieved excellent performance in industrial optimization applications [38]. The algorithm simulates teacher education behavior in human society. By simulating the teacher’s teaching behavior and the learning behavior 158

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Computer Communications 152 (2020) 155–170

between students, the students’ academic performance is improved to achieve the global optimal solution. In this algorithm, teaching and learning are two important knowledge-obtaining behaviors. The improvement of the level of students in the class requires the teacher’s teaching to guide study, at the same time, the students need to learn from each other to promote the absorption of knowledge. Teachers and students are equivalent to individuals in algorithms, and teachers are some of the best individuals. In addition, every subject learned by each student is equivalent to a decision variable. Fig. 3. Student individual encoding.

4.3. Relevant definitions of teaching-learning-based optimization For an optimization problem 𝑧 = 𝑚𝑎𝑥𝑋∈𝑆 𝑓 (𝑋), a solution is denoted 𝑋 = (𝑥1 , 𝑥2 , … , 𝑥𝑑 ), where 𝑑 represents the dimension of the solution 𝑈 (the number of decision variables). Note that 𝑥𝐿 𝑖 ≤ 𝑥𝑖 ≤ 𝑥𝑖 , where 𝑈 are the upper and lower bounds of each dimension. Let 𝑥𝐿 and 𝑥 𝑖 𝑖 𝑋 𝑗 = (𝑥𝑗1 , 𝑥𝑗2 , … , 𝑥𝑗𝑑 )(𝑗 = 1, 2, … , 𝑁𝑃 ) be a point in the search space, where 𝑁𝑃 is the number of spatial search points (population size). To facilitate the description of the algorithm, the following definitions are given: (a). Class. In the TLBO algorithm, a collection of all points of the search space is called a class (population). (b). Students. An individual 𝑗 in the class is called a student 𝑋𝑗 . (c). Teacher. The best student 𝑋𝑏𝑒𝑠𝑡 in the class is called a teacher. According to the above three definitions, a class can be expressed as follows: ⎡ 𝑥1 ⎢ 2 ⎢ 𝑥 ⎢ ⋯ ⎢ 𝑥𝑁𝑃 ⎣

𝑓 (𝑥1 ) 𝑓 (𝑥2 ) ⋯ 𝑓 (𝑥𝑁𝑃 )

⎤ ⎡ 𝑥1 ⎥ ⎢ 𝑥12 ⎥=⎢ 1 ⎥ ⎢ ⋯ ⎥ ⎢ 𝑥𝑁𝑃 ⎦ ⎣ 1

𝑥12 𝑥22 ⋯ 𝑥𝑁𝑃 2

𝑥1𝑑 𝑥2𝑑 ⋯ 𝑥𝑁𝑃 𝑑

⋯ ⋯ ⋯ ⋯

𝑓 (𝑋 1 ) 𝑓 (𝑋 2 ) ⋯ 𝑓 (𝑋 𝑁𝑃 )

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

1, 2, … , 𝑁𝑃 ), select a classmate 𝑋 𝑗 = (𝑥𝑗1 , 𝑥𝑗2 , … , 𝑥𝑗𝑑 ), (𝑗 ≠ 𝑖) as the learning object from the excellent students of the class to learn and adjust by analyzing the difference between 𝑋 𝑖 and 𝑋 𝑗 . The process is similar to the differential mutation operator in the difference algorithm. The ‘‘learning’’ behavior can be carried out as follows: { 𝑖 𝑋𝑜𝑙𝑑 + 𝑟𝑖 (𝑋 𝑖 − 𝑋 𝑗 ), 𝑓 (𝑋 𝑗 ) < 𝑓 (𝑋 𝑖 ) 𝑖 𝑋𝑛𝑒𝑤 = (15) 𝑖 𝑋𝑜𝑙𝑑 + 𝑟𝑖 (𝑋 𝑗 − 𝑋 𝑖 ), 𝑓 (𝑋 𝑖 ) < 𝑓 (𝑋 𝑗 ) where 𝑟𝑖 is a random number less than 1 but greater than 0. After the learning process, each student is compared against his/her score after the study and previous score before the study. If his/her new score is better, then the student is updated. Otherwise, the student keeps the previous status. The process of ‘‘teaching’’ and ‘‘learning’’ iteratively updates the population until the maximum number of iterations is reached.

(12)

4.6. Apply ETLBO for charging scheduling and allocation in the wireless sensor network

In addition, the teacher is selected from the class, that is, 𝑋𝑡𝑒𝑎𝑐ℎ𝑒𝑟 = arg 𝑚𝑎𝑥𝑓 (𝑋). 𝑁𝑃 is the number of students, and 𝑑 is the number of subjects.

The basic TLBO optimization algorithm cannot handle the problem since the charging sequence of nodes is a discrete problem rather than a continuous problem. Therefore, this section draws on the idea of TLBO optimization algorithm to improve the evolutionary algorithm to deal with problem P1. To this end, we design an evolutional teaching and learning based algorithm (ETLBO) that combines crossover and mutation operations to solve discrete scheduling and continuous time allocation problems simultaneously.

4.4. Principle of the teaching process At the beginning of the semester, students have very little mastery of new knowledge since they did not learn systematically. Thus, the average grade of the class is low. After many times of teaching by a teacher, the students are getting more and more knowledge, which results in the average score being improved. In the teaching stage, each student learns from the teacher, and the teaching process is achieved as follows: 𝑖 𝑖 𝑋𝑛𝑒𝑤 = 𝑋𝑜𝑙𝑑 + 𝑑𝑖𝑓 𝑓 𝑒𝑟𝑒𝑛𝑐𝑒

(13)

𝑑𝑖𝑓 𝑓 𝑒𝑟𝑒𝑛𝑐𝑒 = 𝑟𝑖 × (𝑋𝑡𝑒𝑎𝑐ℎ𝑒𝑟 − 𝑇 𝐹𝑖 × 𝑚𝑒𝑎𝑛)

(14)

4.6.1. Individual coding scheme The basic TLBO uses continuous coding for function optimization; however, problem P1 includes discrete and continuous variables. For the sake of solving problem P1, we design a new hybrid discrete and continuous variable coding scheme, shown in Fig. 3, where the scheduling vector 𝑥 is a permutation of recharged nodes. For instance, the value of 𝑥1 of student 1 is 3, which means that node 1 is the third recharged in the sequence of student 1. The charging time vector 𝑦 represents the allocated time of nodes; for instance, the time allocated by node 1 of student 1 is 12.6. By jointly encoding the discrete vector 𝑥 and the continuous vector 𝑦, an individual can be used to represent the complete solution. The class denotes the population of solutions shown in Fig. 3.

𝑖 𝑖 Among them, 𝑋𝑛𝑒𝑤 and 𝑋𝑜𝑙𝑑 represent the 𝑖th student’s new and pre∑𝑁𝑃

𝑋𝑖

vious scores, respectively. 𝑚𝑒𝑎𝑛 = 𝑖=1 denotes the average score 𝑁𝑃 for all students. 𝑟𝑖 is the learning step size between 0 and 1, that is, 𝑟𝑖 = 𝑟𝑎𝑛𝑑(0, 1). 𝑇 𝐹𝑖 is the teaching factor, which is calculated as 𝑇 𝐹𝑖 = 𝑟𝑜𝑢𝑛𝑑(1 + 𝑟𝑎𝑛𝑑(0, 1)). 𝑟𝑜𝑢𝑛𝑑() is a decimal function to round up and round down numbers; thus, the value of 𝑇 𝐹𝑖 is 1 or 2. In addition, once the teaching behavior is finished, the students’ grades are updated. Each student is compared using the results after the study and the results before the study. If the score of a student is better, the student is updated. Otherwise, the student keeps the previous status.

4.6.2. Student individual initialization The first step of the TLBO algorithm is to initialize the population. Since student individuals consist two parts, the two parts are initialized separately. To initialize the charge scheduling 𝑥, 𝑁 random numbers are randomly generated in the range of (0, 1) first. Then, the random numbers are sorted, and the values of each node are initialized according to sort results. Assuming that the random number of the corresponding node 1 is 3 in the order of all random numbers, the 𝑥 value corresponding to the node 1 is initialized to 3 accordingly. The second part is to initialize time variables 𝑦; the total time that can be used to charge node should less than 𝑇𝑡𝑜𝑢𝑟 . Thus, every charging time 𝑇 −𝑇 can be set to 𝑟𝑖 × 𝑁𝑡𝑜𝑢𝑟 , where 𝑟𝑖 denotes a random variable between 0 and 1.

4.5. Principle of learning process In addition to learning from a teacher, students may learn from the excellent students in the class. Through learning between students, the average academic performance of the class will also be improved as well. Therefore, in the learning stage, we first set the ratio 𝑟 of excellent students, and then for each student 𝑋 𝑖 = (𝑥𝑖1 , 𝑥𝑖2 , … , 𝑥𝑖𝑑 ), (𝑖 = 159

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may violate the constraints (7)–(9) in the problem P1. Traditionally, the constraints can generally be handled by the penalty function [39]; however, it is more difficult to design an effective penalty function for a multi-constraint problem. In this paper, a combination of the modified method and the penalty function is used to deal with the constraints in problem P1. Specifically, for the constraints (7) and (8), the infeasible solution is directly adjusted to satisfy the two constraints through a repair approach. The repair method randomly selects nodes in the individual solution and then reduce the charging time according to a probability range. The process will repeat until the solution of the student individual can satisfy the constraints of the models (7) and (8) or the maximum number of iteration is reached. The advantage of the direct modification method is to limit the solution search to a feasible space. The repair method is described in Algorithm 1. Note that 𝑒𝑖 denote the minimum charging time of node 𝑖.

4.6.3. Teaching and learning based on evolution operation To deal with student encoding in Fig. 3, the evolutionary operators are designed to enhance the TLBO algorithm for solving hybrid coding. Operators such as selection, crossover and mutation of the evolutionary algorithm can be integrated for the TLBO algorithm for individual updating. The detailed description of the operators is as follow. First, the selection operator refers to the reference fitness function, which randomly selects some individuals from the population (students) to survive according to the individual selection probability, whereas the remaining individuals are eliminated. According to the individual selection probability, the next population can be selected. Obviously, individuals that have good fitness have a higher probability of being selected than worse ones. Second, crossover operators are used to transfer parent genes to newly generated individuals. The crossover is divided into two steps since the encoding in Fig. 3 consists of two type data structure. The first data structure is the charging time, which is a real-valued vector. Without loss of generation, 𝑦 and 𝑦′ are used to, respectively, represent the charging time vectors of the two individuals; then, two new charging scheduling are generated using the formula (16). 𝑦1𝑛𝑒𝑤 = 𝜆𝑦 + (1 − 𝜆)𝑦′ 𝑦2𝑛𝑒𝑤

Algorithm 1 Individual repair algorithm INPUT: a solution validates the constraints (7)–(8) OUTPUT: the repaired solution 1: iteration=0; 2: while less maximum iteration C do 3: Random select a node, and generate 𝑟 ∈ (0, 1); 4: Modify allocated time variable 𝑦𝑖 , set 𝑦𝑖 = 𝑟(𝑦𝑖 − 𝑒𝑖 ). 5: if the constraints (7)–(8) satisfied or lower bound is reached then 6: Break; 7: else 8: Continue; 9: end if 10: iteration++; 11: end while

(16)

′

(17)

= 𝜆𝑦 + (1 − 𝜆)𝑦 𝑥′ ,

For a given charging sequences 𝑥 and two intersections are randomly selected, and the sequence between the two intersections are interchanged. Further, the generated conflict sequence is eliminated to avoid duplicate sensors on a charging path. For example, the following two individuals perform, crossover operation: 𝑥 ∶ {9, 5, 1, 3, 7, 4, 2, 8, 6} and 𝑥′ ∶ {5, 4, 6, 3, 7, 8, 2, 1, 9}. Then the new generated individuals 𝑥1𝑛𝑒𝑤 ∶ {9, 5, 6, 3, 7, 8, 2, 8, 6} and 𝑥2𝑛𝑒𝑤 {5, 4, 1, 3, 7, 4, 2, 1, 9} are obtained. It can be seen that the gene segments exchanged by the two individuals are 6, 3, 7, 8 and 1, 3, 7, 4 respectively. For individual 𝑥1𝑛𝑒𝑤 , the first conflicting gene is 8, and the position of 8 in the exchange segment of 𝑥′ is No. 6. Next, replace the collision gene outside the exchange segment with the gene at the corresponding position in 𝑥, that is, 4 (No. 6 in 𝑥). After conflict elimination, the feasible scheduling sequences 𝑥1𝑛𝑒𝑤 ∶ {9, 5, 6, 3, 7, 8, 4, 1} and 𝑥2𝑛𝑒𝑤 ∶ {5, 4, 1, 3, 7, 4, 2, 6, 9} are obtained. The mutation operator is beneficial to maintain the diversity of a population by disturbing individuals. For the encoding in Fig. 3, the mutation operation is divided into two steps. The first is mutation operation of the charging time 𝑦, which randomly selects a node 𝑖 on the scheduling path and re-initialize the charging time 𝑦𝑖 . The second is the mutation operation of the scheduling vector 𝑥, which randomly select two points on the scheduling sequence and exchanges them. In the process of teaching, each student learns from a teacher. The charging path and allocated time of the student undergo crossover with the ones corresponding to the teacher. Then, new candidate solutions are produced. Moreover, the mutation operation is performed on a student based on probability. After the evolution operation, the student will be compared with the previous scores and updated if the new solution is better than the current solution. Otherwise the student keeps the original solution. In the process of learning, a group of excellent students are selected. Each student randomly selects an excellent student from the group to perform crossover operation. Then new paths and charging time vectors are obtained. Thus, new candidate solutions are produced. Further, the mutation operation is performed on a student based on probability. If the fitness of the new solution is better than its original fitness, the student encoding will be updated. Otherwise, the student keeps the previous encoding.

Theorem 1. The lower bound of the sum of charging time is 𝑇𝑡𝑜𝑢𝑟 + ∑ (𝑅𝐸𝑖 −𝐵𝑚𝑖𝑛 )+𝑒𝑐𝑖 ×𝑇 . 𝑖∈𝑉 𝐸 (𝑇 𝑆 ) 𝑐

𝑖

Proof. The repair process only the adjusts charging time 𝑦. Thus, the driving time of vehicle 𝑇𝑡𝑜𝑢𝑟 is fixed in order for replenished node to work until the next period. The least charging time of node 𝑖, that is (𝑅𝐸𝑖 −𝐵𝑚𝑖𝑛 )+𝑒𝑐𝑖 ×𝑇 , where 𝑇 is period length, 𝑒𝑐𝑖 𝑒𝑖 can be calculated by 𝐸𝑐 (𝑇 𝑆𝑖 ) is energy consumption rate, and 𝐸𝑐(𝑇 𝑆𝑖 ) denotes the harvested energy rate. Thus, the sum of charging times 𝑦 should not be less than the ∑ (𝑅𝐸𝑖 −𝐵𝑚𝑖𝑛 )+𝑒𝑐𝑖 ×𝑇 . [End] lower bound 𝑇𝑡𝑜𝑢𝑟 + 𝑖∈𝑉 𝐸 (𝑇 𝑆 ) 𝑐

𝑖

The repair method is helpful to search for solutions in the feasible space. However, constraint (9) is difficult to be dealt with since it requires all nodes to satisfy a time constraint. For this reason, a penalty function is designed to add to the objective function. Let 𝑓2 (𝑡) = ⎧ 1 ⎪ 𝑅𝐸 (𝑓 𝑐𝑖 − 𝑑𝑡𝑖 ), 𝑖𝑓 (𝑑𝑡𝑖 ≤ 𝑓 𝑐𝑖 ) ∑ 𝑝𝑐 × 𝑝𝑡 , where 𝑝𝑡 = ; 𝑓 𝑐𝑖 denotes 𝑖 ⎨ 𝑖 𝑖 𝑖∈𝑁 ⎪ 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 ⎩ the start charging time of node 𝑖, which is calculated as 𝑓 𝑐𝑖 = 𝑇𝑠𝑡𝑎𝑟𝑡 + ∑𝑖−1 ∑𝑖−1 𝑑𝑖𝑠𝑡(𝑠(𝑘−1),𝑠(𝑘)) ; and 𝑝𝑐 is an adjustable parameter. Thus 𝑘=1 𝑡𝑠(𝑘) + 𝑘=1 𝑉𝑐 the new fitness function is defined as: (18)

𝑓 (𝑡) = 𝑓1 (𝑡) − 𝑓2 (𝑡) ∑

𝑡𝑖 𝑖∈𝑁 ∫0

𝜌(𝑠)𝑑𝑠

where 𝑓1 (𝑡) = 𝐸 ×𝐿 . Combining the repair process and penalty 𝑚𝑜𝑣𝑒 𝑇 𝑜𝑢𝑟 function, constraints (7)–(9) can be overcome in the iteration of ETLBO.

4.6.4. The constraint repair The basic TLBO is used to deal with unconstrained optimization problems. However problem P1 is a multi-constraint optimization problem, so the constraints needs to be processed in the optimization process. When evaluating the objective function, some students solutions

4.6.5. Framework of the evolution TLBO Based on the above description, a flow chart of the proposed evolution TLBO algorithm is presented in Fig. 4. First, the number of nodes 160

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Fig. 4. The framework of the evolution TLBO for wireless sensor network charging scheduling.

5. Dynamic insertion algorithm for real-time charging request

to be charged is determined according to the energy state and consumption of the nodes. The initial students are generated by randomly generating scheduling paths and charging time vectors (i.e., solutions), the fitness of each student is calculated based on fitness function (18). The teachers (i.e., the best student), and the superior students are selected. After initialization, the teachers and students perform the ‘‘teaching’’ process and the ‘‘learning’’ process, respectively, and then calculate the fitness of the new solutions again. If the new fitness is better than the old one, then the student will replace the old solution with the new one, otherwise, the previous solution is maintained. The process will stop when the maximum number of iteration is reached.

In heterogeneous sensor networks, there are multiple types of data streams. Some data streams may change dynamically in real time, such as tracking targets in a sensor network [40]. When the target appears, the sensor energy consumption of the corresponding monitoring area will change significantly, which is difficult to be predicted in advance. Accordingly, the emergence of new tasks will also change the energy consumption [41]. However, the period scheduling method cannot cope with hybrid tasks. Therefore, it is necessary to consider the existence of dynamic data streams and periodic data streams in heterogeneous networks. During the charging process, if a new node generates a new data service, it might send a new charging request. At this time, the new node that sends the charging request may be scheduled to the charging subsequent, and the charging time is allocated simultaneously. The insertion algorithm selects the most appropriate insertion point based on the fitness and inserts the node at the corresponding position. The charging duration of nodes are adjusted because the newly inserted node may result in the scheduling duration and energy constraints being unsatisfied. Let the list denote the set of nodes that the ETLBO

The iteration will continue until the maximum number of iteration is reached. Then, the best student will be output; this result in an optimized scheduling path and charging time. 2

Theorem 2. The computation complexity of ETLBO is 𝑂(𝑀𝑁|𝑉 𝑑 | 𝐶) Proof. The computational complexity of fitness evaluation of an in2 dividual student is 𝑂(1 + 2 + 3 + ⋯ + |𝑉 𝑑 |) = 𝑂(|𝑉 𝑑 | ). In addition the computational complexity of repair process of individual is 𝑂(𝐶), 2 the algorithm needs 𝑂(|𝑉 𝑑 | + 𝐶) to evaluate and repair an individual student of the charging plan. The number of algorithm iteration is 𝑀, and the number of students is 𝑁. Thus, the computation complexity of 2 ETLBO is 𝑂(𝑀𝑁(|𝑉 𝑑 | + 𝐶)).

∑

𝑡 ∫ 𝑖 𝜌(𝑠)𝑑𝑠

𝑖∈𝑙𝑖𝑠𝑡 0 is the algorithm has been scheduled; then, 𝑓 (𝑙𝑖𝑠𝑡) = 𝐸𝑚𝑜𝑣𝑒 ×𝑇𝑡𝑜𝑢𝑟 fitness function of the previous∑solution. Thus, the new fitness can be 𝑡 ∫ 𝑖 𝜌(𝑠)𝑑𝑠

0 expressed as 𝑓 (𝑙𝑖𝑠𝑡 ∪ {𝑚}) = 𝑖∈𝑙𝑖𝑠𝑡∪{𝑚} , where 𝐸𝑚𝑜𝑣𝑒 and 𝑇𝑡𝑜𝑢𝑟 𝐸𝑚𝑜𝑣𝑒 ×𝑇𝑡𝑜𝑢𝑟 are determined by the scheduling solution after the node 𝑚 is added to the list. Therefore, a greedy algorithm can be proposed that selects the

161

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insertion point 𝑚 in the already scheduled list to maximize the fitness ⋃ 𝑓 (𝑙𝑖𝑠𝑡 {𝑚}). The details of the algorithm are presented in Algorithm 2. Algorithm 2 Dynamic Insertion Algorithm INPUT: A planned charging path with ETLBO algorithm: 𝑙𝑖𝑠𝑡 A set of nodes to be inserted: 𝑛𝑜𝑑𝑒𝑙𝑖𝑠𝑡 Time set of the node to be inserted: 𝑑𝑡𝑙𝑖𝑠𝑡 Current time 𝑡′ OUTPUT: A new planned charging 1: Select first candidate position 𝑖 in scheduled list after time 𝑡′ ⋃ 2: Set 𝑓 (𝑙𝑖𝑠𝑡 𝑚)𝑖 = 0 for all candidate position 𝑖 3: while 𝑛𝑜𝑑𝑒𝑙𝑖𝑠𝑡 not empty do 4: Take out one node 𝑚 from 𝑛𝑜𝑑𝑒𝑙𝑖𝑠𝑡 5: Insert node 𝑚 to candidate position 𝑖 of list 6: Repair charging time scheduling of list use Algorithm 1 7: Compute assuming node 𝑚 is insert the position 𝑖; 8: Move to next position 𝑖 = 𝑖 + 1; 9: if candidate position is empty then 10: finished=ture; 11: end if 12: end while ⋃ 13: Select 𝑀𝑎𝑥𝑓 (𝑙𝑖𝑠𝑡 𝑚)𝑖 and insert 𝑚 to the position 𝑖 in scheduled list ⋃ 14: Repair charging time according to selected 𝑓 (𝑙𝑖𝑠𝑡 𝑚)𝑖 15: return new scheduling and charging time

Parameter

Value

Number of sensors Area Period length Speed of vehicle 𝑉𝑐 Capacity of sensor battery CV Capacity of vehicle battery 𝐶𝑡𝑜𝑡𝑎𝑙 Charging power 𝑐 Energy consumption of sensor 𝑒𝑡𝑖 Number of sinks Charging efficiency 𝜌 Min energy threshold 𝐵𝑚𝑖𝑛 Energy consumption of the vehicle

40–200 150 m × 150 m 120 min 50 m/min 103 J 106 J 10 mJ/s 3 μJ/bit 4–6 80% 40% B 50 J/m

will send a charging request when the remaining power in the period is lower than the threshold 𝐵𝑚𝑖𝑛 . In addition, a real-time task generates a new task according to a Poisson distribution, and the load of nodes on the path of the node to the sink is correspondingly increased. The parameters of the ETLBO algorithm are set as follows: the crossover parameter is 0.3, the mutation parameter is 0.1, the superior student ratio parameter is 20%, and the student population size is 200. The adjustable parameter 𝑝𝑐 is set to be 0.1 to adjust the penalty function 𝑓2 (𝑡). 6.2. The performance metrics and compared algorithms To measure the effectiveness of the proposed algorithm, the proposed algorithm is compared with the heuristic offline algorithm [9], EDF algorithm, greedy algorithm [41] and heuristic online algorithm [9]. Moreover, the evaluation metrics include the moving distance of the charging vehicle, total received energy of nodes, charging utility, charging success ratio, number of successfully replenished nodes, starvation ratio and the number of starved nodes. The charging utility indicates the ratio of the total received energy of nodes and the energy consumed by the vehicle in movement. The charging success ratio denotes the proportion of successfully replenished nodes in the set of nodes that request charging in the cycle time. On the other hand, the starvation ratio is defined as the number of starved nodes compared with the number of nodes that request charging in the same period. The heuristic offline algorithm was proposed by Ye et al. [9], which is optimized scheduling based on the 1.5-approximation algorithm. The EDF (earliest deadline first) algorithm is an earliest deadline time priority scheduling algorithm. The charging vehicle will replenish the node according to the sorted charging queue. The greedy (greedy recharging algorithm) [41] algorithm is proposed by Gao et al. The algorithm first selects the node where the vehicle is near as the starting node, and then selects the node with the highest utility as the final node.

The dynamic insertion algorithm (DIA) can be performed independently or integrated after ETLBO. First, it optimizes the scheduling and time variables of periodic tasks via ETLBO; then, after vehicle starts to replenish nodes, the DIA is integrated to respond to real-time charging requests. Theorem 3. The computational complexity of the dynamic insertion 2 algorithm is 𝑂(|𝑉 𝑑 |(|𝑉 𝑑 | + 𝐶)). Proof. The worst of the repair algorithm in step 6 is 𝑂(𝐶). The computational complexity at step 7 to compute is 𝑂(1+2+3+⋯+|𝑉 𝑑 |) = 2 𝑂(|𝑉 𝑑 | ). The maximum number of candidate position is |𝑉 𝑑 |, so the 2 iteration step is 𝑂(|𝑉 𝑑 |). Step 4 through 12 need 𝑂(|𝑉 𝑑 |(|𝑉 𝑑 | + 𝐶)). Step 13 needs to select the best position, and step needs 𝑂(𝐶) to repair new solution. Thus, the computational complexity of the dynamic 2 insertion algorithm is 𝑂(|𝑉 𝑑 |(|𝑉 𝑑 | + 𝐶)). 6. Experiment and results analysis 6.1. Experimental parameters To evaluate the performance of the proposed ETLBO, various network topologies are designed to compare the result of different algorithms. Assuming that a base station is located in the center of the area, sensor nodes are randomly distributed in the area, and each sensor establishes a path to sink nodes in advance. Heterogeneous data service applications are considered, including periodic tasks and non-periodic tasks in the network. Specifically, in periodic tasks such as periodic environmental parameters monitoring, the amount data transmitted by such tasks is basically constant; thus, energy consumption can also be predicted. Non-periodic tasks include burst monitoring tasks that respond to real-time needs, such as intrusion detection. The networks and mobile vehicle are set according to the parameters reported in Table 2. At the center of the scene, there is a charging post that can replace the battery of the vehicle. In each period, the vehicle replenishes the nodes that issues charging requests. Note that nodes

6.3. Periodic task comparison We first consider performance evaluation of periodic tasks. As mentioned above, a node can estimate the residual energy in the current period according to the historical energy consumption, so all nodes that need be charged can be predicted at the beginning of the period. Firstly, we choose the heuristic offline algorithm, EDF algorithm and greedy algorithm as comparison algorithms. Different from that these algorithms, which only optimize the node charging sequence, the proposed ETLBO algorithm optimizes the scheduling order and charging time simultaneously. In the simulation, we construct 9 different network scenarios. 162

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Fig. 5. Network topology of 4 scenarios.

To replenish nodes, the moving vehicle needs to traverse the nodes. Here, the charging path refers to an optimized path obtained by scheduling algorithm to replenish nodes. After optimization, the vehicle starts from the depot to sequentially travel every node that is scheduled to be charged, replenishes the energy for them, and returns to the base station in a period. To clearly demonstrate the result, the depot is denoted as node 0, and the sensors are numbered sequentially. The total length of the paths in different scenarios is counted separately. The result obtained by ETLBO is the average of 10 times results. First, the charging path lengths of the four algorithms in the 9 scenarios are shown in Fig. 6. For the same network topology, the paths planned by different algorithms are different. Obviously, it can be seen from Fig. 6 that the result obtained EDF has the longest length among all algorithms. This is because the EDF algorithm only considers the request order without considering the energy cost associated with the moving distance, so the total path length is the longest. The path length of the ETLBO algorithm is not much different from that of the greedy algorithm. This is because the ETLBO algorithm replenishes every node during the scheduling process. In some scenarios, if the scale is 180 and 200, the total path length is slightly higher than that of the greedy algorithm. In small-scale scenarios, the heuristic offline algorithm and the EDF algorithm have fewer moving paths. The moving paths obtained by the ETLBO algorithm and the greedy algorithm are longer than the others. This is due to the reason that it is easy to find an optimal solution due to the small number of nodes in a small-scale scenario. To further analyze the characteristics of paths obtained by the four algorithms in detail, a small-scale network is first chosen for path comparison. The detailed path obtained by the four algorithms in a network with 40 nodes are shown Fig. 7. It can be observed that only the ETLBO algorithm passes all nodes and replenishes a certain amount of power during the period. However, the heuristic offline algorithm, EDF algorithm and greedy algorithm cannot satisfy the charging request of all nodes due to the limitation of the cycle time length. Although the moving paths obtained by the three compared algorithms are short, three nodes cannot be charged in the period. Since ETLBO satisfies

Fig. 6. Charging path lengths of different algorithms.

6.4. Network topology In each network, a base station and multiple sink nodes are deployed, which are powered by wired power. All nodes whose energies are lower than the threshold are collected to constitute the charging request list, and the vehicle starts from the base station to replenish nodes based on the scheduling result. To demonstrate the simulation results, 4 scenarios are selected and presented in Fig. 5. In the figure, the circle points represent ordinary sensors, the stars represent sink nodes, and the base station resides in the center of the network. In addition, the connection line indicates the route from the sensor to the sink node. 6.4.1. Results comparison for periodic tasks (1) Charging path comparison for different algorithms 163

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Fig. 7. Paths of vehicles obtained by four algorithms over 40 nodes.

the charging requests of all nodes, its moving path is the longest. The heuristic offline has the shortest path, but 3 nodes are not satisfied, which causes these nodes to run out of energy during the cycle. Furthermore, a larger network scenario is selected for comparison. Four algorithms are each run over a network with 30 nodes sending request. The different paths planned by the four algorithms are shown in Fig. 8. In this instance, there are 30 nodes sending charging requests among 200 nodes. Only the ETLBO algorithm can satisfy the charging requests of all sensors; on the contrary, the result of the greedy algorithm has 9 starved nodes, the heuristic offline algorithm has 15 starved nodes, and the EDF algorithm has 17 starved nodes. The ratio of starved nodes of the heuristic offline and the EDF algorithms are greater than 50%, and the greedy algorithm also reaches 30%, whereas ETLBO still guarantees to satisfy all charging requests while the movement path remains short. It can be inferred that optimal charging time enhance the number of successfully replenished nodes. In addition, the moving path obtained by the greedy algorithm is the shortest, while the moving path obtained by EDF is the longest. Further, the path length obtained by ETLBO is better than the result obtained by the heuristic offline and EDF algorithms. Although the heuristic offline algorithm optimizes the path using the 1.5 approximation algorithm, the quality of the approximation algorithm is less than the quality of ETLBO. (2) Charging utility comparison of different algorithms The charging utility is defined in formula (6) and refers to the sum of the energy received by nodes divided by the energy consumed by the vehicle on the path. Since the vehicle is traveling at a constant speed, it means that a longer path corresponding to more energy consumption due to movement, which result in a shorter time for charging nodes. The comparison of the charging utility of the four algorithms is shown in Fig. 9. It can be observed from the figure that the ETLBO and greedy algorithms achieve higher charging utility than the other two algorithms. This is due to the reason that the heuristic offline and EDF algorithms only take the charging demand into account without considering the cost of vehicle movement. In most scenarios, the charging utility obtained by two algorithms is lower than those

of the ETLBO and greedy methods. According to the definition of the charging utility, when the moving distance is longer, more energy is consumed due to the vehicle movement, so the energy supplemented by the node is correspondingly reduced, which results into reduced charging utility. (3) Comparison of total received energy The total of the received energy represents the sum of the harvested energy of nodes from wireless energy transfer carried by the mobile vehicle. The four algorithms are run in different scenarios to calculate total the received energy, and the results are shown in Fig. 10: It can be observed from the figure that the ETLBO algorithm has a lower total received energy than the heuristic offline and EDF algorithm in a small-scale network. This is because the ETLBO algorithm replenishes all nodes, which results in a long path; thus, the vehicle has to consume more energy to drive a long distance, which affects the total amount of charging energy. In the other eight scenarios, the total received energy obtained by ETLBO algorithm is better than the ones obtained by the heuristic offline and EDF algorithms; it is higher than the heuristic offline algorithm by 0.42–16.95% and higher than EDF algorithm about 0.06–25.05%, while exceeding 10% over 6 scenes. Compared with the greedy algorithm, ETLBO has 6 scenes over greedy, and greedy has 3 scenes over ETLBO; this indicates that there is no obvious difference between the two algorithms. As mentioned in formula (2), the harvested energy does not linearly increase with the allocated time. Although the EDF algorithm and heuristic offline algorithms have more energy to replenish nodes, the total received energy in the batteries of nodes is still less than total received energy obtained by ETLBO. This is because ETLBO algorithm finds the optimal solution due to population search, such that the moving path is the shorter; another reason is that ETLBO algorithm does not fill each node, so that all nodes can be replenished with a certain power. Although each node does not get the most energy, all nodes gain energy and eventually achieve the maximum total received energy. 164

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Fig. 8. Paths of vehicle obtained by four algorithms over 200 nodes.

Fig. 9. Charging utility comparison.

Fig. 10. Comparison of total received energy.

(4) Comparison of successful charging ratio and the number of replenished nodes The number of successfully charged nodes refers to the number of nodes that send requests and are replenished during this period. On the other hand, nodes that have issued a request but for which the request is not satisfied during the period become starved nodes. Accordingly, the charging success rate is defined as the number of successfully charged nodes in proportion to the total number of nodes that send charged requests. To ensure the continuous and stable work of the network, it is necessary to reduce the number of starved nodes as much as possible. Therefore, the scheduling algorithm must ensure that as few nodes as possible are starved to ensure that the network runs continuously and effectively. The successful charging ratio and the number of successfully charged nodes in different scenarios are counted and shown in Fig. 11. In 9 scenarios, the ETLBO algorithm achieves a success charging ratio of 100%. The greedy algorithm reaches a

successful charging ratio of 100% in 3 scenes over 9 test scenarios. The heuristic offline algorithm and EDF algorithm obtain 100% successful charging ratios with 2 and 1 scenes respectively. ETLBO achieves a charging success ratio of 100% in all scenarios because the ETLBO optimizes charging time and order at the same time. By allocating the charging time, it can reduce the time of lower charging utility to ensure that more nodes are charged in this period. The comparison algorithms compensate the sensor nodes for 90% battery capacity each time, such that there is no remaining energy and time to replenish more nodes. In particular, the EDF and heuristic offline algorithms consume a great deal of power due to vehicle movement, such that nodes cannot be replenished in the period. The two algorithms charging success ratios of less than 70% over 7 scenes. In addition, greedy algorithm has an average success ratio higher than EDF and heuristic offline due to the reason that the vehicle consumes less driving energy. The average success ratio is higher than those of the two algorithms, and the 165

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Fig. 11. The successful charging ratio and the number of successfully replenished nodes comparison.

the heuristic offline and EDF algorithms are greater than 40% over 4 and 5 scenes, respectively. In detail, the heuristic algorithm, EDF algorithm, and greedy algorithm have 15, 17 and 9 nodes, respectively, that cannot be satisfied in the scenario with 200 nodes in the cycle. On the other hand, no nodes are starved in the schedule obtained by ETLBO. This indicates that cooperative optimization scheduling and charging time allocation greatly prolong the life of the entire wireless sensor network and improve the quality of service for wireless network data transmission. Through periodic service comparison, it can be found that the ETLBO algorithm not only significantly improves the charging success ratio through collaborative optimization of time allocation and scheduling but also enhances the total network charging utility and total received energy.

charging success ratio is greater than 70% over 6 scenes. Specifically, the four algorithms are compared in a scenario in which 200 nodes are randomly distributed. During the cycle, 30 nodes issue charging requests. In particular, ETLBO replenishes 30 nodes, while the heuristic offline algorithm, EDF algorithm, and greedy algorithm replenish 15, 13, and 21 nodes, respectively. Compared with different algorithms, the proposed algorithm has a significant improvement in terms of the charging success ratio. This result demonstrates that allocation of the charging time at the same time can more effectively guarantee the continuous operation of the sensor network. As a result, it will significantly enhance the network transmission quality and ensure data to be transferred to sinks in time. (5) Node starvation ratio and the number of starvation nodes comparison In contrast to the successful charging ratio, this section analyzes the node starvation ratio and the number of starved nodes obtained by different algorithms. A starved node means its energy will be below the threshold and cannot be replenished in time during this period. The node starvation ratio denotes the starved node in proportion to the total number of nodes need to be charged in this cycle. Because the starved nodes significantly affect network performance, the data flow through starved nodes unable be continuously transmitted and received, which affects the network data transmission quality and reduces the network performance. Thus, an excellent charging scheduling algorithm needs to reduce the number of starved nodes as much as possible. The charging results of the nine scenarios are counted separately, and the results are shown in Fig. 12. In 9 different scenarios, the ETLBO algorithm obtains the best starvation ratio, where no nodes are starved to death. This result shows the superiority of the ETLBO algorithm in guaranteeing the energy supply. However, the other comparison algorithms have starved nodes. Concretely, it can be seen from Fig. 12 that the starvation ratio of the EDF algorithm is higher than the ones of the other three algorithms. This is because the EDF algorithm is locally considered from the individual charging requirements of the nodes, which results in that it is difficult to achieve global optimal scheduling. The heuristic offline algorithm is aimed at optimizing the single charging utility. Although it can obtain a better charging effect, those nodes with low utility will be eliminated in this period, which will cause these nodes to not be replenished in time. Although the greedy algorithm optimizes the scheduling path by using the optimization algorithm, it still prefers the node with high utility to insert the charging queue in the period. Since these three algorithms do not optimize the charging time of each node, some nodes cannot be charged in this period due to the reason that mobile vehicle has to consume a long time to replenish nodes. The starvation ratio of the greedy algorithm is higher than 20% over 6 scenes, but five of them are lower than 30%. The starvation ratios of

6.5. Comparison of different algorithms with mixed tasks This section, mixed tasks are considered. The charging vehicle will still perform according to the result of scheduling periodic tasks. The difference is when there is a newly emerged node that cannot support the running in the cycle due to a new real-time tasks, then, the charging vehicle has to respond to the new charging request of the burst node and dynamically modify the scheduling path. To this end, we choose the heuristic online and EDF algorithms as comparison objects. The heuristic online algorithm was proposed by Ye [9], it responds to new nodes in real time. If there are multiple nodes issue new charging requests at the same time, the algorithm will select the node with the best charging utility to respond and add it to the charging queue. The remaining unresponsive nodes will push to the next cycle. Next, the performance differences of different algorithms are compared in the case with mixed tasks. The node size ranges from 20 to 200 nodes. The length of the periodically requested charging queue is 5, 9, 12, 14, 15, 16, 20, 28, 32, and 33, respectively. In addition to the periodic request service, 𝑛 node sending real-time requests (𝑛 < 3) are randomly generated according to the Poisson distribution in a period after the vehicle starts from the depot. 6.5.1. Comparison of moving distance The average result of 10 times for ETLBO+DIA is set as the reference result. The lengths of the running paths for the three algorithms in 10 scenarios are shown in Fig. 13. Obviously, it can be seen from figure that EDF has the longest path length. This is because the EDF algorithm only considers the request order and does not consider the energy cost on moving distance of the vehicle. Therefore, the total path length is the longest, and the total path length difference increases as the network scale expands. The total length of path obtained by ETLBO+DIA is not much different from the heuristic online algorithm. 166

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Fig. 12. The starvation ratio and the number of starved nodes comparison.

Fig. 14. Charging utility comparison for mixed tasks.

Fig. 13. Charging vehicle moving distance.

An explanation is that ETLBO algorithm charges every node during the scheduling process, however heuristic online deletes some nodes with low utility, and the actual number of charged nodes is lower than the number of nodes charged by ETLBO+DIA. For example, in a scenario with 180 nodes, ETLBO+DIA plans for 33 nodes, while heuristic online plans for 26 nodes. ETLBO+DIA replenishes more nodes, so there is no significant superior to the total length obtained by heuristic online one. 6.5.2. Comparison of charging utility The comparison of charging utility results for these three algorithms is shown in Fig. 14. It can be seen from the figure that the charging utility obtained by ETLBO+DIA is significantly higher than that of the comparison algorithm. In 10 scenarios, except for the small-scale scene with 20 nodes, the charging utility obtained by ETLBO+DIA is higher than that one of the heuristic offline algorithm by 23.1–109.8% in the other 9 scenarios, which is higher than the one obtained by EDF algorithm by 45.2–293.1%. Obviously, ETLBO+DIA has achieved excellent charging utility. This is due to two reasons: one is that the ETLBO+DIA replenishes more nodes since the obtained distance is shorter than the path obtained by the other algorithms, and the other reason is due to the battery charging function relationship (Fig. 2), it can obtain more charging utility when same energy is allocated to more nodes. The heuristic offline algorithm and the EDF algorithm charge the node up to 89% to complete the charging, which makes the charging time longer than allocation time by ETLBO+DIA. According to the

Fig. 15. Total received energy comparison for mixed tasks.

marginal characteristics of the battery charging function, ETLBO+DIA makes nodes achieve higher energy utility. Therefore, ETLBO+DIA can achieve excellent charging utility. 6.5.3. Comparison of total received energy The total received energy represents the sum of the harvested energy by the nodes in the network. The three algorithms are run in 167

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Fig. 16. The ratio of successful replenishment and the number of replenished nodes comparison for mixed tasks.

Fig. 17. Comparison of the ratio of starvation and the number of starved nodes for mixed tasks.

different scenarios and the total amounts of energy supplemented by the nodes are counted. The result is shown in Fig. 15: It can be seen from the figure that the small-scale networks with 40 nodes, the total harvested energy obtained of the algorithm node is lower than the one obtained by the heuristic online algorithm node. This is because ETLBO+DIA replenishes all 9 nodes in the scene, while the heuristic online algorithm only charges 4 nodes. Therefore, the algorithm proposed in this paper needs to take a longer path and consume more energy due to vehicle movement. This leads to less energy that can be used to replenish nodes. In the other seven scenarios, the total harvested energy obtained by ETLBO+DIA is better than the one obtained by the heuristic online and EDF algorithms; it is higher than the heuristic online algorithm by 1.45–27.9%, and higher than EDF algorithm by 3.32–31.17%. When the number of nodes is greater, if the number exceeds 160, the difference in the total harvested energy obtained by different algorithms is reduced. One reason is that although the ETLBO algorithm optimization can obtain excellent paths, because the energy is added to the more nodes, all the total path lengths are not significantly smaller than paths obtained by the comparison algorithm. This makes the vehicle consume more energy on the path.

To ensure the continuous and stable operation of the network, the scheduling algorithm must ensure that as few nodes as possible are starved to ensure the continued and effective operation of the network. In 10 scenarios, ETLBO+DIA algorithm achieves 100% successful energy replenishment. The heuristic online algorithm and EDF algorithm only replenish all nodes in a small-scale scenario with 20 nodes. Because ETLBO optimizes scheduling and charging time at the same time, it can reduce the charging time of lower utility to replenish more nodes in this period. Even in the mixed service charging, the path planned charging time is re-allocated by the DIA algorithm, thereby ensuring time allocation or the newly added nodes. The comparison algorithm compensates the sensor node for 90% each time; it consumes a longer time, and consumes more energy, so that there is no remaining power and time to replenish more nodes. In particular, the EDF and heuristic online algorithms consume more power due to movement of vehicle, such that some nodes cannot be replenished in this period. There are 7 scenes in which the ratio of success energy replenishment is less than 70% for the two compared algorithms. 6.5.5. The starvation ratio and the number of starved nodes In contrast to the ratio of success energy replenishment, this section analyzes the ratio of node starvation and the number of starved nodes for different algorithms. Because the starved nodes significantly affect network performance, the data stream flowing through the node will not be continuously transmitted and received, which affects the network data transmission quality and decreases the network performance.

6.5.4. The successful charging ratio and the number of replenished nodes comparison Next, the ratio of success energy replenishment and the number of replenished nodes of different algorithms are compared in the hybrid service, the results in different scenarios are counted and shown in Fig. 16. 168

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References

The results of the nine scenarios are counted separately and shown in Fig. 17. In the 9 different scenarios, ETLBO achieves minimum ratio of node starvation, which is 0. This result shows the superiority of ETLBO algorithm for guaranteeing energy supply. It can be seen from the figure that the number of starved nodes obtained by the EDF algorithm is greater than the ones obtained by the other algorithms. The ratio of starvation is even greater than 50% in 2 scenarios. The heuristic online algorithm is aimed at optimizing the single charging utility; those nodes with low utility will be eliminated in this period, which causes these nodes to not be replenished in time. There are 3 to 8 starved nodes over 8 scenarios. The comparison of the ratio of node starvation and the number of starved nodes demonstrates that ETLBO maintains 100% of the charging request by cooperative optimizing charging sequence and time, which greatly prolongs the life of the entire wireless sensor network and improves quality of service for data transmission in the wireless sensor network.

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7. Conclusion Wireless energy replenishment significantly enhances the network lifetime of wireless sensor networks through feasible scheduling. However, most existing studies only considered how to optimize the charging scheduling without considering charging time allocation. To effectively prolong network working time and improve the efficiency of the sensor network, this paper proposes to jointly optimize the charging scheduling and charging time allocation. The simulation results show that the proposed algorithm can achieve a charging success rate of 100% in periodic and hybrid services, significantly higher than the comparison algorithms. This result demonstrates that cooperative charging scheduling and time allocation are important to reduce starved nodes, while starved nodes significantly reduce the performance of network. However, in large-scale networks, one vehicle is difficult to satisfy all charging request due to a large number of request nodes. Thus, multiple vehicles may be studied to replenish nodes in large-scale networks. Through the K-means method, a largescale network may be separated into multiple subnetwork, and then corresponding multiple vehicles may be scheduled to replenish the nodes. This is an interesting issue to be explored further in future work.

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

CRediT authorship contribution statement Chuaxin Zhao: Conceptualization, Writing - original draft. Hengjing Zhang: Validation, Software. Fulong Chen: Investigation, Supervision. Siguang Chen: Validation. Changzhi Wu: Methodology. Taochun Wang: Writing - review & editing.

Acknowledgment This work was supported in part by the National Natural Science Foundation of China under Grant 61871412, Grant 61971235, Grant 61972438, and Grant 61872194 in part by the Natural Science Foundation of Anhui Province of China under Grant 1908085MF214 and Grant 1808085MF172. 169

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