The Optimization Based Dynamic and Cyclic Working Strategies for Rechargeable Wireless Sensor Networks with Multiple Base Stations and Wireless Energy Transfer Devices

In this paper, the optimal working schemes for wireless sensor networks with multiple base stations and wireless energy transfer devices are proposed. The wireless energy transfer devices also work as data gatherers while charging sensor nodes. The wireless sensor network is firstly divided into sub networks according to the concept of Voronoi diagram. Then, the entire energy replenishing procedure is split into the pre-normal and normal energy replenishing stages. With the objective of maximizing the sojourn time ratio of the wireless energy transfer device, a continuous time optimization problem for the normal energy replenishing cycle is formed according to constraints with which sensor nodes and wireless energy transfer devices should comply. Later on, the continuous time optimization problem is reshaped into a discrete multi-phased optimization problem, which yields the identical optimality. After linearizing it, we obtain a linear programming problem that can be solved efficiently. The working strategies of both sensor nodes and wireless energy transfer devices in the pre-normal replenishing stage are also discussed in this paper. The intensive simulations exhibit the dynamic and cyclic working schemes for the entire energy replenishing procedure. Additionally, a way of eliminating “bottleneck” sensor nodes is also developed in this paper.


Introduction
Nowadays, wireless sensor networks (WSNs) have been widely used in environment investigations, disaster monitoring and resource management, etc., since they provide us with more comprehensive information platforms and more advanced techniques in order to deal with monitoring tasks. Researches have been conducted carefully with regard to locating targets [1,2], data routing [3][4][5], and MAC (Medium Access Control) layer issues [6][7][8] in wireless sensor networks. Methods of localizing senor nodes have also been well studied in recent years. Chen et al. proposed a range-free localizing algorithm to accurately locate regular nodes in WSNs [9]. In their work, a localizing scheme takes advantages of anchor nodes locations and the possible areas of the neighboring and non-neighboring nodes of regular nodes. Compared with existing localizing schemes using anchor nodes information only, the presented approach provides a significant improvement of the localization accuracy. Additionally, the QoS of WSNs is closely related to the continuous maintenance of sensing coverage. Chen et al. introduced a hybrid memetic framework for coverage optimization [10]. They proposed a memetic algorithm, as well as a heuristic recursive algorithm, in their paper. The results from real-world experiments using a WSN test-bed show that the approach proposed is able to maximize the sensing coverage while achieving energy efficiency simultaneously. Wireless sensors are also widely used in tracking physiological index of human body. Mukhopadhyay et al. pointed out the usage of wearable sensors in monitoring activities of human beings continuously [11]. Together with advances in sensing technologies, wireless communication technologies, and embedded systems, it is also possible to develop smart human body WSNs for the purpose of health and well-being. Since sensor nodes may be deployed in some harsh environments, such as volcano areas and underwater areas, it turns out to be an ordeal for sensor nodes performing long-term operation. In addition to this external reason, the very cause of hindering sensor nodes from long-term operation is their limited energy reserves provided by installed batteries.
In order to overcome the drawbacks stemming from the limited energy supply, Murphy developed an energy-efficient cross layer scheme to prolong the network lifetime [12]. Gandelli et al. comprehensively elaborated advances in antenna technologies for conserving energy in large distributed wireless sensor networks for extensive monitoring tasks [13]. Antonio et al. developed an innovative mobile platform for heterogeneous sensor networks. In order to maximize the system performances and the network lifespan, they also worked out a hybrid technique, based on evolutionary algorithms [14]. Some other scientists also developed tactics for sensor nodes to harvest energy from the environment in order to make them function perpetually. Shigeta developed a software control method for maximizing the sensing rate of WSNs, which could harvest energy from the ambient RF power [15]. Yang et al. proposed a distributed networking architecture for WSNs [16], which could acquire energy from solar power. In their paper, they presented the AutoSP-WSNs (Automatic Solar Powered Wireless Sensor Networks), which were sensor networks of distributed fashion and could achieve the optimal end-to-end network performance. Li considered the energy efficient scheduling problem of WSNs with energy harvesting [17]. a drop in energy transfer efficiency. Based on this phenomenon, they developed a touch/proximity/hover sensing system using such property of wireless energy transfer [35]. They also developed a wireless energy transfer system for power and/or high-data rate transmission using sheet-like waveguides [36]. This system is still capable while transferring energy and/or data through thick metal walls.
In this paper, we find that the wireless energy transfer technique proposed by Kurs is a promising way of replenishing sensor nodes since (1) the energy transferring procedure is not vulnerable to climate changes, and can be performed from a long distance; (2) the energy can be transferred without the line of sight, and it can still be delivered successfully even if there are obstacles between the transferor and the transferee; (3) the energy transferred is clean, which means it will not affect the sending and receiving data procedures in WSNs. Therefore, the wireless energy transfer technique, indeed, sheds light on the promising future of WSNs. Inspired by this novel technique, Xie proposed working schemes for WSNs with wireless energy transfer devices [37]. The scheme developed by Xie is also feasible when multiple sensor nodes are charged by one replenishing device at the same time. However, the model they used in that paper is static, which means that the variables in that model are not functions over time.
Inspired by the device invented by Khripkov [38], we could combine the wireless energy transfer and the mobile data collecting together. The aim of this paper is to provide dynamic and cyclic working schemes for both sensor nodes and wireless energy transfer devices based on a series of optimization problems. In order to achieve this goal, we develop a dynamic network model with variables, which are continuous functions over time. With the help of this dynamic model, we can deduce the constraints that sensor nodes and wireless energy transfer devices should comply with at arbitrary time instance t . The optimization problem is firstly established from the dynamic model, and then reshaped into a linear optimization problem, which will yield the identical optimality. The rest of this paper is organized as follows: in Section 2, the working scenario and the preliminary modeling of the prime optimization problem is given. In Section 3, by analyzing the working mechanism of the wireless energy transfer device, the optimization problem established in Section 2 is then transformed into a discrete multi-phased optimization problem. In Section 4, two necessary conditions for the optimality of the optimization problem are proposed. Additionally, the linearization of the discrete optimization problem is carried out in the latter half of this section. In Section 5, issues relating to the pre-normal replenishing stage are studied comprehensively. In Section 6, intensive simulations and numerical analysis are performed. Section 7 concludes this paper.
The main contributions of this paper can be summarized as follows: firstly, we discuss a WSN of multiple base stations in a divide-and-conquer fashion due to the concept of Voronoi diagram. Secondly, the entire energy replenishing procedure is split into the pre-normal and normal replenishing stages. We, then, form a continuous time optimization problem for the normal replenishing cycles in the normal replenishing stage. Later on, the continuous time problem is reformulated into a discrete multi-phased optimization problem with the same optimality by exploiting the working strategies of wireless energy transfer device. In addition, the working schemes for sensor nodes and the wireless energy transfer devices are also obtained for the pre-normal replenishing stage. The intensive simulations exhibit the dynamic and cyclic natures of the working schemes proposed in this paper. In addition to this, a way to eliminate "bottleneck" nodes is also presented in this paper.

Problem Description
In this section, we mainly deal with the modeling issues of the problem, which are mentioned in previous parts of this paper. The goal of modeling this problem, as mentioned in former sections, is to retrieve the dynamic and periodic working strategies for both wireless sensor nodes and wireless energy transfer devices. Before getting fully involved into the discussion of the modeling issues, it would be better to introduce the duties engaged by different components of wireless sensor networks, e.g., sensor nodes, base stations, and wireless energy transfer devices. In the meantime, from the aspect of intelligibility of this article, we table all the symbols used in this paper, which are followed by their corresponding definitions. Please refer to Table 1 for details.
The distance between two successive points along the travelling path The wireless sensor network discussed in the scope of this paper is supposed to be located in a certain two-dimensional area, shown in Figure 1, and the area is denoted as D . This wireless sensor network is equipped with several base stations and couple of sensor nodes. The set of all base stations is denoted as data gathering after deployment. Additionally, certain amount of sensor nodes also play the role of routing nodes which are responsible for rallying the acquired date back to one base station and/or one wireless energy transfer device. Base stations are working as data sinks in which the tasks are to store, process, and/or exchange the received data with other data centers, etc. The wireless energy transfer devices discussed in this paper have to perform two main functions. On the one hand, the device roams around the area D , and charges sensor nodes wirelessly in case that their remaining battery energy falls below certain level. On the other hand, it plays the role of a data mule while replenishing battery energy of sensor nodes, simultaneously. In order to ensure that sensor nodes will not malfunction due to the insufficient energy supply, the working schemes of both sensor nodes and wireless energy transfer devices should be designed and conducted carefully, which is the main problem to be dealt with in this paper.

Subnet Partition Using Voronoi Diagram
Intuitively, discussions of the problem stemming from WSNs with multiple base stations could be carried out in a so-called divide-and-conquer fashion. In this paper, the whole WSN is firstly divided into a number of sub networks, which is shown in Figure 2. Each sensor node is within the "reign" of only one base station, that is, the data gathered by a sensor node will be directed to its own base station. Then, the modeling of the whole WSN could be casted as modeling each sub network one after the other. The subnet partition scheme adopted here is based on the Voronoi diagram. The definition of Voronoi diagram is described as follow.

Definition 1. (Voronoi diagram) Space (
) X X ≠ ∅ is endowed with a binary operation, denoted as : d X X ×   , where  is the set of all real numbers. For arbitrary , x y X ∈ , if the binary operation d satisfies: Then d defines a "distance" on X . Suppose that K is an index set, and ( ) k P k K ∈ is a tuple consisting of several nonempty sets in X . If the set of elements in X , denoted as k R satisfies: , then k R is a Voronoi region of the set X . Voronoi regions generated by all ( ) k P k K ∈ form a Voronoi diagram of the set X .
The space X , specifically in this paper, is the area D (assuming that the location information of sensor nodes and base stations, which can be acquired by several means, such as GPS, is revealed to us beforehand

The Modeling of Normal Replenishing Cycles
Generally speaking, as we put in previous paragraphs, there are B N subnets in the entire WSN.
For subnet k SN , there are one base station, one wireless-energy transfer device, and k N sensor nodes.
Each sensor node samples the environment information every fixed time interval, which produces the data at the rate of i R (1 ) k i N ≤ ≤ bits/s. Every sensor node is equipped with one fully charged wirelessly rechargeable battery, and the initial energy of it is max E . When the remaining energy of the battery falls below certain level, known as min E , the sensor node cannot fulfill its functions properly. The wireless energy transfer device k W roams within D , and charges sensor nodes in order to keep them alive. While charging, the replenishing device can also retrieve data directly from the sensor node being charged concurrently. The time interval that the device spends on charging sensor node i s ( ) After charging all sensor nodes, it returns to service station S and stays there for the time period The total replenishing procedure could be categorized into two distinct stages, i.e., the pre-normal replenishing stage and the normal replenishing stage. The pre-normal replenishing stage corresponds to the first replenishing duty executed by the wireless energy transfer device, and the normal one corresponds to the rest replenishing duties. Furthermore, the normal replenishing stage is composed of a series of normal replenishing cycles of time duration τ . The pre-normal replenishing stage will be carefully examined in Section 5. As is shown in later sections, the normal replenishing cycles are periodic extensions of the first normal replenishing cycle. Therefore, we will take the first normal replenishing cycle as an example while modeling the WSN. The energy-time curve of the sensor node i s is depicted in Figure 4, revealing the pre-normal replenishing stage, the first, and the second normal replenishing cycles. Obviously, the problem we want to attack in this paper involves networking issues together with energy issues of WSNs. Hence, the modeling of WSNs with wireless energy-transfer devices should, of course, exploit both networking and energy properties of each network component. Assume that at the time instance t in the first normal replenishing cycle, the sensor node i s sends data to j s , and the sending data rate is ( ) ij R t bits/s, and i s receives data from k s at the rate of ( ) ki R t bits/s. Nevertheless, i s may transmit data directly to one base station or one wireless-energy transfer device at the rate of ( ) R t bits/s, separately. Hence, at i s , the following constraint must be satisfied: where the ij I , and the power usage at time instance t , known as ( ) i p t , can be calculated as: where ρ denotes the power factor associated with receiving per unit data from other sensor nodes, while ij C , l iB C and l iW C relate to power factors when transmitting per unit data to other sensors, base stations and wireless energy-transfer devices, respectively. As shown in Figure 4, the energy consumption of sensor node i s within any normal replenishing cycle should be equivalent to the energy replenished by the wireless energy transfer device during i τ , which is also the requirement of designing cyclic working schemes for both sensor nodes and replenishing devices. Therefore, the following equation should also be met: Equations (1)-(4) are constraints that a sensor node should satisfy. In other words, they are the very constraints that contribute to the development of the cyclic working strategies for sensor nodes. The rest constraints or equations are related to wireless energy transfer devices which will help us regulate the behaviors of them.
On the behalf of the replenishing device, the entire normal replenishing cycle τ can be split into three main parts, i.e., where P τ represents the time spent on roaming around the area D , and P is the path taken by a replenishing device.
where p D is the total length of the path P , and V is the travelling velocity of a replenishing device. 0 ( ) i i π ≠ stands for the -th i sensor node along the travelling the path P .
( 1)mod( is the distance between two successive spots along the travelling path. Our aim, by introducing wireless energy transfer devices into wireless sensor networks, is to keep sensor nodes from malfunctioning caused by the lack of energy reserves. However, paradoxically, we do not want these devices to be so "hardworking". On the contrary, it is fabulous for these devices staying at service stations as long as possible while keeping all sensor nodes above certain energy level. Therefore, evidentially, we have the following optimization problem OPT-1: OPT-1 max : The objective function of this optimization problem stands for our pursuit that the sojourn time of a replenishing device at service station S should be as long as possible. S τ τ is called the sojourn time ratio in this paper. The optimization variables are ( ) and all the indicator functions. Before carrying on our discussion on the above optimization problem, we need to state two properties in advance.
Proof. Please refer to the Appendix for more details.

Property 2.
If each sensor node is fully charged, i.e., the remaining energy of i s at i i t τ + achieves max E , the optimality of OPT-1 will remain the same, as shown in Figure 5, which indicates that OPT-1 will yield the same optimal objective value no matter if the sensor nodes are fully charged or not. Therefore, the constraint Equation (8) can be substituted with: Consequently, the OPT-1 is then reformulated as: OPT

Modeling the Normal Replenishing Cycle from a Multi-Phased Aspect
In the previous section, we have deduced the optimization problem of wireless sensor networks with wireless energy transfer devices, i.e., OPT-1. With the help of Property 1 and 2 listed above, OPT-1 is then reformulated into OPT-2. At first examination of this optimization problem, it is hard to find an efficient way to obtain the optimal solution in that the variables are continuous functions of time, and the constraints of OPT-2 contain large amount of product and integral terms. In this section, we will firstly form another optimization problem OPT-3 from a multi-phased aspect. Then, by proving the equal optimality of OPT-3 and OPT-2, our focus on solving OPT-2 would be altered to find an optimal solution to OPT-3.

The Analysis of Working Phases of a Wireless Energy Transfer Device
Generally speaking, within one normal replenishing cycle, the working phases of a wireless energy transfer device fall into two categories. During the first phase, the wireless energy transfer device has no interaction with sensor nodes deployed in area D , i.e., it travels along its path or stays at service station S . During the other phase, it charges each sensor node and collects the data from the sensor node being charged.

The Discrete Model with Respect to Multiple Phases
In order to get rid of the integral terms contained in the constraint Equation (4) of OPT-2, certain compromise on the flexibility of variables is made in this paper. Obviously, the optimization variables in OPT-2, such as ( ) R t are all continuous over time t , which is the reason for the existence of the integral terms. We want to make above variables less flexible while this optimization problem still yields the identical optimality. The flexibility of variables is compromised by fixing the variables in each phase m ∈, i.e., these variables are constant during each phase, however, values of them may be exposed to change among different phases. The continuous optimization variables ( ) Combining all the constraints together, we have an optimization problem OPT-3: OPT-3 max : subject to : (10),(7),(9),(11), (12), (13) In the next section, we will focus on the justification of the equal optimality of OPT-3 and OPT-2. Proof. The similar proof can be found in our earlier work [39]. Here, we postponed the detailed proof in the Appendix section in order to save space.

The Analysis of OPT-3 and Its Linearization
In the previous section, by Theorem 1, we come to the conclusion that OPT-3 and OPT-2 yield the same optimality. Thus, the problem of solving OPT-2 is equivalent to that of solving OPT-3. Nevertheless, it is still not easy for us to handle the product terms embedding in the constraints of OPT-3. In this section, we will firstly investigate the necessary conditions for obtaining the optimal solution to OPT-3. Then, efforts will be made to turn OPT-3 into a linear programming problem, which is easy to be solved by some already existing tools, such as Lindo, CPLEX, etc. Proof. Please refer to the Appendix for more details.

The Linearization of OPT-3
It is still not easy to obtain the optimal solution to OPT-3 since the product terms in the constraints, The constraint Equations (7) and (9) can be unified as: The constraint Equations (11) and (12) can be unified as: The constraint Equation (13) is reformulated as: The constraint Equation (6) can be omitted here since the optimal travelling path P , which is the shortest Hamilton cycle, is determined in advance. Now, we have the new optimization problem, denoted as OPT-4.

The Pre-Normal Replenishing Stage
In last section, two necessary conditions for the optimal solution are given. Additionally, by substituting variables in OPT-3, we obtain OPT-4, and this problem can be simplified into a linear programming problem by determining values of indicator functions in advance. In this section, we will complete our discussion by studying the pre-normal replenishing stage.
The above modeling and corresponding optimization problems are only reasonable for normal replenishing cycles. The energy-time curve of sensor node i s in the pre-normal replenishing stage is not the periodic extension of any curve in normal replenishing cycle since the initial battery energy of arbitrary sensor node in pre-normal replenishing stage is max E , which is higher than the initial battery energy, denoted as i E , in any normal replenishing cycle. Nevertheless, as stated in Theorem 4, we can find a solution to the pre-normal replenishing stage by taking advantage of the solution to the normal replenishing cycle, which will make the transition between the pre-normal stage and the first normal cycle smoothly.
Denote the wireless energy replenishing power for sensor node i s in the pre-normal stage as Proof. Please refer to the Appendix for more details.

The Simulation Scenarios
Suppose that there are 150 sensor nodes deployed in an area D . Additionally, in D , we have three base stations. Next to each base station, there is one service station which accommodates one wireless energy transfer device. We make an assumption here that the location information of sensor nodes, base stations and service stations is revealed to us in advance. Assume that the area D is a 1000 m × 1000 m square area. The travelling velocity of any wireless energy transfer device is V which is set to five meters per second here, and it charges sensor nodes with the power of U which is five Watts. The minimum energy required to keep sensor nodes functioning is set to five percent of its maximum energy, i.e., 0.05 In this paper, max E is equal to 10,800 J. The sending and receiving energy factors, such as ij C and ρ , are adopted from [40].

Simulation Results and Numerical Analysis
The locations of sensor nodes are shown in Figure 6. The features of sensor nodes are listed in    (c) The optimal travelling path in 3 SN .
As stated before, values of indicator functions of OPT-4 will be determined after the network partition is done. The next step is about to solve the optimal traveling path P of the wireless energy transfer device of each sub network. As mentioned in Theorem 2, the optimal travelling path P is the shortest Hamilton cycle connecting all sensor nodes and the service station. This path can be obtained by certain existing tools, such as concorde, which is developed by the Math Department of the University of Waterloo [41]. The optimal path for each sub network is calculated and drawn in Figure 8. In the following discussion, we will intensively investigate the simulation results by taking sub network 1 SN for example. for the time duration that the wireless energy transfer device should spend on charging and retrieving data from a specific sensor node. It may not be possible for us to draw routing schemes corresponding to all phases due to the limited space, therefore please allow us to pick up certain phases for future discussing. The data routing schemes of phase 0, 1, 4, and 42 are drawn as shown in Figure 9. The reason why these phases are chosen will be elaborated afterwards. The time which is spent by the wireless energy transfer device on staying at service station is 125,347 s, and the sojourn time ratio is about 87.4%. If we apply the working scheme described in the previous work [39] to 1 SN , the optimal sojourn time ration, which can be obtained from solving the formulated optimization problem in the previous work, is about 58.7%. Therefore, the objective value is increased by nearly 50%. The data routing schemes of phase 0, 1, 4, and 42 are drawn as shown in Figure 10. From Figure 10b-d, we can figure out that in phase 1, 4, and 42, sensor nodes 1, 4, and 42 transmit data to the base station over a much longer distance. Hence, the power usages of these sensor nodes are higher than those with respect to the working scheme proposed in this paper, which leads to more energy consumptions. This is the main reason accounting for the relatively smaller sojourn time ratio in the previous work.   In these figures, the solid arrow drawn from one sensor node to the other represents that the former one sends data to the later one. The hollow arrow pointing toward the wireless energy transfer device means the sensor node is sending data directly to the replenishing device while being charged. The number next to each sensor node represents the visiting order of it, i.e., the number "1" means that the sensor node is the first node along the travelling path.
In Table 3, the forth column contains the time at which the wireless energy transfer device arrives at the -th i node along the path P . The fifth column lists the time durations spent on charging different nodes. The sixth column gives the remaining battery energy of each sensor node when visited by the wireless energy transfer device. The dynamic nature of the data routing schemes: The dynamic nature of the data routing schemes is obvious since 0 . Specifically, in phase 42, the 42nd sensor node receives data from 41st, 40th and 43rd nodes, and then sends data to 1 W , while in phase 0, it sends data to 1 B , and receives data only from 40th and 41st nodes. The cyclic nature of the date routing schemes: OPT-4 is based on the modeling of the first normal replenishing cycle. And the optimal solution to it is also applied to the normal replenishing cycles later on, which embodies the cyclic nature. The cyclic nature of the data routing schemes will be fully exhibited by the energy-time curve of a sensor node lately.
The "bottleneck" sensor nodes: There must exist at least one "bottleneck" sensor node according to Theorem 3. Therefore, we will try to locate these sensor nodes and find a way to eliminate these nodes in the sequel. The sensor node i s which is the -th i sensor node along the path P is the bottleneck node if and only if the following equation holds: After calculating the power used by all sensor nodes in different phases, we find that the 4th and 42nd along P satisfy Equation (9). Therefore, they are both bottleneck nodes. In addition, it is also the reason why we draw the routing schemes in phase 4 and 42. The existence of bottleneck nodes do harm to the stability of WSNs. Therefore, for this kind of sensor nodes, we developed a "cushion" strategy in case of their energy level falling below min E . This strategy is implemented by conducting an extra replenishing assignment during the sojourn time both in pre-normal replenishing stage and normal replenishing cycles only for bottleneck sensor nodes. During the extra replenishing assignment, the wireless energy transfer device visits bottleneck nodes one after another, and offers them with an extra amount of energy, which increases their energy by five percent of their maximum energy. The travelling path is the shortest Hamilton cycle connecting all bottleneck sensor nodes and the service station as shown in Figure 12.
The energy replenishing rate applied in the normal replenish assignment is changed to ' U watts. Hence, the initial energy of each normal replenishing cycle will be The extra replenishing rate of the sensor node 42 in normal replenishing cycles is adjusted to 4.73 W in accordance with Equation (20), and the extra replenishing time duration ex i τ can be calculated as: For the 42nd sensor node, the duration of the extra replenishing operation is about 110 s. The pre-normal replenishing stage: The working strategies for sensor nodes and the wireless energy transfer device in the pre-normal replenishing stage are almost the same as those in normal replenishing cycles. The energy charging rate is adjusted according to Theorem 4, which is approximately 0.37 W for 42nd sensor node. The power usages of the 42nd sensor node in each phase are listed in Table 4.  The energy-time curve combining the pre-normal stage, the first normal cycle and the second normal cycle of the 42nd sensor node with and without extra replenishing assignment are drawn in Figures 13 and 14, respectively. From the energy consumption curve over time in the first and second normal cycle, we can figure out the cyclic nature of working strategies of sensor nodes and the wireless energy transfer device.
The extra replenishing in the pre-normal energy replenishing stage starts at 98,013 s, and ends at 98,123 s. The remaining battery energy at the beginning of each normal cycle is 4.68 kJ, and 5.22 kJ when the extra replenishing ends.
The values of notations in Figures 13 and 14 are listed in Table 5.    Figure 13) 1.78 kJ (with extra replenishing in Figure 14)

Conclusions
In this article, we investigate the working strategies of sensor nodes and wireless energy transfer devices in WSNs with multiple base stations. The entire sensor network is firstly divided into sub networks according to the concept of Voronoi diagram. The whole energy replenishing procedure is firstly categorized into the pre-normal and normal replenishing stages. By enumerating constraints that sensor nodes and wireless energy transfer devices should comply with, we obtain the optimization problem OPT-1 for the normal replenishing cycles. After dividing the cycle into individual phases, we reach the multi-phased optimization problem OPT-3 which retains the same optimality as OPT-2. OPT-3 is linearized through variable substitution, and then we have OPT-4. Later on, issues relating to the pre-normal replenishing stage are also studied in this article. The simulation conducted in this paper offers us the working strategies of different phases, denoted as ( ) ( , ), m m G V E τ , for both sensor nodes and wireless energy transfer devices. After analyzing the simulation results, the dynamic and cyclic natures of the strategies are revealed. In addition, the bottleneck nodes are pointed out with the help of Equation (19), and an extra replenishing assignment is carried out to provide a cushion for these sensor nodes.
It is claimed that φ is a feasible solution to OPT-3, which can be verified by plugging above variables back into constraints of OPT-3 and testing whether they are compatible with them. The validation procedure is listed below: For constraint Equation ( The third equation holds since ( ) For constraint Equation (7), we have: The second equation holds in that φ is a feasible solution to OPT-2.
Therefore, φ is indeed a feasible solution to OPT-3. The objective value achieved by φ can be calculated as: i.e., φ yields the same objective value as φ does.
The proof of the second half of this theorem is straightforward. Since the values of optimization variables are fixed in phase m∈, any solution to OPT-3 is also a feasible solution to OPT-2 with less flexibility. Therefore, the optimal solution to OPT-3 is of course a feasible to OPT-2, which means that the optimal value of OPT-3 will not exceed that of OPT-2. In conclusion, OPT-3 and OPT-2 are of the same optimality.

Proof of Theorem 2. (The Optimal Travelling Path)
Intuitively, if a wireless energy transfer device spends less time on roaming, it will be able to enjoy more time staying at the service station. The formal proof of this assertion is based on contradiction.
Before going through the proof, please allow us to explain some notations used in the proof. Let P denote the travelling path, and P can be represented by { } Suppose that φ  is the optimal solution to OPT-3, and that the travelling path P  is not the shortest Hamilton cycle connecting all sensor nodes and the service station. We can construct a feasible solution φ from φ  , and 3 ( ) The first and third equations hold in that ( ) ( Constraint Equation (11) holds for the same reason. For constraint Equation (7), we have: Therefore, φ is indeed a feasible solution to OPT-3. The objective value yielded by φ can be calculated as follows: The " > " holds since P is the shortest Hamilton cycle along which it is less time consumed.
Therefore, φ yields better objective value of OPT-3 than φ  does, which contradicts the assumption that φ  is the optimal solution to OPT-3. The proof is complete.

Proof of Theorem 3. (The Existence of "Bottleneck" Nodes)
The proof of this theorem is also by contradiction. Suppose that φ  is the optimal solution to we have a candidate solution to OPT-3, denoted as φ . What we should do next is to verify the feasibility of φ . Firstly, we will show that 0 τ is greater than 0.  Therefore, φ is a feasible solution to OPT-3. The objective value achieved by φ can be calculated as follow: The " > " is tenable since 1 γ > . From the inequality, we can draw the conclusion that φ yields better objective value than φ  does, which contradicts the assumption that φ  is an optimal solution. Hence, there must be at least one bottleneck sensor node in the WSN under optimal solution φ  .
Proof of Theorem 4. In conclusion, in the pre-normal replenishing stage, each sensor node and the wireless energy transfer device adopt the same working strategies as those in normal replenishing cycles, and the wireless energy