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Energy-efficient renewable scheme for rechargeable sensor networks

Abstract

Wireless energy transfer (WET) is a promising technology to fundamentally settle energy and lifetime problems in a wireless sensor network (WSN). In this paper, we study the operation of WSN based on WET using a mobile charging vehicle (MCV) and construct a periodic strategy to make the network operational permanently. Our goal is to decrease energy consumption of the entire system while maintaining the network operational forever. Based on the analysis of total energy consumption, we propose an energy-efficient renewable scheme (ERSVC) to achieve energy saving. Compared to previous schemes where the MCV visits and charges all nodes in each cycle, the MCV only needs to visit a portion of nodes in ERSVC. Numerical results show that our scheme can significantly decrease the total energy consumption with no performance loss. It is also validated that ERSVC can maintain the network operational forever with lower complexity than other schemes, making it more practical for real networks.

Introduction

Wireless sensor networks (WSNs) nowadays are primarily powered by batteries. Due to limited energy capacity in a battery at each node, a WSN can only remain operational for limited time. To prolong its lifetime, there have been extensive researches in the last decade [14]. However, lifetime remains a performance bottleneck of a WSN and is one of the main reasons that limit its widespread application.

Recently, wireless energy transfer (WET) based on magnetic resonant coupling [5] was shown to be a promising technology to solve energy and lifetime problems in WSNs fundamentally [69]. Surprisingly, this new technology is immune to the surrounding environment. Compared with other WET technologies such as electro-magnetic radiation [10, 11], magnetic resonant coupling has many significant advantages including higher energy conversion efficiency, further transmission distance, and no requirement of line-of-sight.

Some researchers investigated the problems in WSNs based on this non-radiative energy transfer. In [6], the authors proposed a model to make the WSN operational forever, with the objective of maximizing the ratio of vacation time. In [7], the authors extended the research in [6] to a dense network where several sensors could be charged at the same time. The authors in [6, 7] assumed that the mobile charger must visit all of the sensors in each cycle. However, sensors may have different energy consumption rate in real networks. Charging all nodes in each cycle will increase the traveling distance of MCV and thus degrading the energy efficiency of the entire system. In [8, 9], the authors studied the approach of combining mobile recharging and data gathering to further reduce energy consumption. These schemes assumed that the transmitter was omni-directional and could charge several nodes at the same time. However, the assumed scenarios in [8, 9] are difficult to be deployed in practice. Since magnetic resonant coupling has strict requirement with distance and orientation, which will make an effect on the charging efficiency [12]. Another drawback is that the complexities in [8, 9] are very high, causing them not suitable to be implemented in real networks.

In this paper, we propose an energy-efficient renewable scheme (ERSVC) with low complexity based on this non-radiative energy transfer. Our goal is to decrease energy consumption of the entire system while maintaining the network operational permanently. Due to the potential large coverage of a WSN, we employ a mobile charging vehicle (MCV) to periodically visit each sensor node and charge it wirelessly. Different from previous schemes in [68], the MCV only visits and charges a portion of sensor nodes in ERSVC during each cycle. The set of charged nodes during each cycle is redesigned by taking into account the power consumption of each node. We use the total power consumption and MCV’s vacation time ratio as the performance metric. Morevover, we derive the proof theoretically that ERSVC can make sure the network operational forever. Numerical results have shown that ERSVC achieves the goal of energy saving with lower complexity than other schemes in [69], which suggest it seems to be more practical for real applications.

The main contributions of this paper include several aspects. Firstly, we study the operation of a sensor network and construct a periodic strategy to make the network operational permanently. Secondly, we analyze the total energy consumption and point out that the traveling distance of MCV is the main factor influencing total energy consumption. Therefore, we primarily aim to decrease the traveling distance of MCV as much as possible in proposed ERSVC. Thirdly, we develop a comprehensive design for data flow, charging period and visiting set in ERSVC to achieve energy saving. Furthermore, we also demonstrate that ERSVC can make the network immortal theoretically.

Methods

In this section, we present the model for MCV’s behavior and the control strategy in WSN. Table 1 lists the main abbreviations and notations in this paper.

Table 1 Abbreviations and notations

MCV and travel path

We consider a sensor network \(\mathcal {N}\) deployed over a two-dimensional area, similar to the scenario that was widely adopted in [68]. Each sensor node has a battery capacity of Emax and is fully charged initially. Also, denote Emin as the minimum energy at a sensor node battery (for it to be operational). Each sensor node i generates sensing data with a rate of Ri (in bit/s), \(i \in \mathcal {N}\). Inside the network, there is a fixed base station (B), which is the sink node for all data generated by the sensor nodes. Multi-hop data routing can be employed for forwarding data by the sensor nodes.

To charge the battery at each sensor node, a mobile wireless charging vehicle (MCV) is employed in the network. As shown in Fig. 1, the MCV starts from a service station (denoted as origin O), and the traveling speed of the MCV is V (in m/s). When it arrives at node i, it will spend a time of ti to charge the node’s battery wirelessly via non-radiative energy transfer. Denote U as the energy transfer rate of the MCV. After ti, the MCV leaves node i and travels to the next node.

Fig. 1
figure1

A wireless sensor network with a mobile charging vehicle (MCV). This is the model for MCV’s behavior. During each cycle, the MCV starts from a service station, visits and charges some nodes in the network, and finally returns to the service station

After the MCV visits, all the sensor nodes that need to be charged in this cycle will return to the service station to be serviced (e.g., replacing or recharging its battery) and get ready for the next trip. We call this resting period vacation time, denoted as tvac.

For a given set of nodes S, the MCV starts from O(OS), visits and charges all nodes ni(niS), and finally returns to O. This problem is defined as the traveling salesman problem (TSP) [13]. In TSP, the shortest path which connects all the nodes in S and has the shortest overall length is called the shortest Hamiltonian cycle. It was proved in [6] that the MCV must move along the shortest Hamiltonian cycle in an optimal solution. It is obvious that the shortest Hamiltonian cycle is also the best path to achieve the energy-efficient optimization.

In each cycle, the MCV needs to visit and charge some nodes in the network. Denote Fk as the set of nodes which need to be visited in the kth cycle. In the kth cycle, the MCV travels along the shortest Hamiltonian cycle which connects all nodes in Fk and O. Denote Pk as the traveling path of this shortest Hamiltonian cycle. Denote Dk as the distance of path Pk, and tP=Dk/V as the time spent for traveling over distance Dk.

Denote T as the period for a trip cycle of MCV, and \(t_{\text {vac}}^{k}\) as the vacation time of MCV in the kth cycle. In the kth cycle, the MCV travels from O, visits and charges all nodes in Fk, and finally returns to O for a vacation time \(t_{\text {vac}}^{k}\). Then, the cycle time T can be written as

$$ T = {t_{P}} + t_{\text{vac}}^{k} + \sum\limits_{j \in {F_{k}}} {{t_{j}}} = {D_{k}}/V + t_{\text{vac}}^{k} + \sum\limits_{j \in {F_{k}}} {{t_{j}}} $$
(1)

where \({\sum \nolimits }_{j \in {F_{k}}} {{t_{j}}} \) is the total amount of time that MCV spends charging all nodes in Fk via non-radiative energy transfer.

Control strategy

In this paper, we use periodic strategy to make sure none of the sensor nodes runs out of energy. As mentioned above, a MCV is employed to charge the sensor nodes periodically with a cycle time of T. On the other hand, each node should be charged in time to supplement its energy consumption.

In previous researches [69], the MCV visits and charges all the nodes in each cycle. However, sensors may have different energy consumption rate in real networks. Some nodes which are close to the base station may consume energy several times higher than the remote ones and it is unnecessary to visit all nodes in each cycle. Therefore, we can adopt some strategies by taking into account the energy consumption rate of each node. For example, the node with the highest energy consumption rate will be visited in each cycle. In contrast, the node with lower energy consumption can be visited every 2 cycles or more.

For each node i (\(i \in \mathcal {N}\)), it should be recharged periodically to supplement its energy consumption in time. Denote Ti as the charging period of node i. The charging process for node i is shown in Fig. 2.

Fig. 2
figure2

The charging period of node i in WSN. For each node i, the MCV charges i periodically with a mean period time of Ti. Ti is set to be an integral multiple of T. The blue and white blocks indicate that i will be charged and not be charged in this cycle, respectively. Ti is the visiting and charging interval of i, which is defined and explained detailly in the discussion section

Denote pi as the energy consumption rate of node i. According to the energy conservation principle, the energy consumption of node i should be equal to the energy supplied by MCV. Therefore, we have the following relationship,

$$ {T_{i}} \cdot {p_{i}} = {t_{i}} \cdot U\;\;(i \in \mathcal{N}) $$
(2)

Analysis on total energy consumption

In this section, we analyze the energy consumption of the entire system with MCV. For convenience of comparison, we use a plug-in hybrid vehicle (i.e., PHEV) to carry the mobile battery as the MCV. The total energy consumption consists of two parts as follows:

  • The energy that the MCV needs to charge all the sensor nodes in the network.

  • The energy consumed by the MCV to travel inside the network.

According to the research data given by the Pacific Northwest National Laboratory [14], the energy consumption per mile (i.e., ECPM) of a mid-size PHEV is 0.3 kW, that is, 675 J/m. Suppose the traveling speed of the MCV is V = 5 m/s, then the power consumed by the MCV traveling is about 3 kW. However, the power consumption of a sensor node is about several milliwatts to several hundreds of milliwatts [1]. Apparently, the power consumed by the MCV traveling is the main part of the total energy consumption in the system.

In this paper, we propose an energy-efficient renewable scheme with variable cycle (ERSVC). In ERSVC, we firstly set the value of T properly, then we design the charging period Ti for each node i based on its energy consumption rate pi. The value of Ti is set to be an integral multiple of T. Thus, the MCV can visit a portion of nodes in each cycle, and the traveling distance of MCV can be decreased. We use two major steps to decrease the two parts of total energy consumption respectively. We primarily aim to decrease the traveling distance of MCV as much as possible, since it is the main part of total energy consumption.

We use two parameters to measure the performance of network, which are stated as follows:

  • The total power consumption (denoted as Ptotal), which is the sum of the two parts mentioned above. Denote λ as the energy conversion efficiency of non-radiative energy transfer. Then, Ptotal can be written as follows:

    $$ {P_{\text{total}}} = \frac{1}{\lambda} \cdot \sum\limits_{i,i \in \mathcal{N}} {{p_{i}}} + \frac{{{L_{\text{total}}} \cdot {\text{ECPM}}}}{{{T_{\text{total}}}}} $$
    (3)

    where Ttotal and Ltotal are the total time and traveling distance of MCV over all of the cycles, respectively, and ECPM is defined as the energy consumption per mile in [14].

    Note that all the sensor nodes in WSN are powered by electric energy converted from WET. Therefore, the first part of total power consumption can be written as \(\frac {1}{\lambda } \cdot {\sum \nolimits }_{i,i \in \mathcal {N}} {{p_{i}}}\).

  • The ratio of the MCV’s vacation time (denoted as ηvac), which is the optimization objective in [69]. In this paper, we define ηvac as the mean percentage of time in each cycle that MCV spends on its vacation, and it can be calculated by the following equation:

    $$ {\eta_{\text{vac}}} = \frac{{{\sum\nolimits}_{k} {t_{\text{vac}}^{k}} }}{{{T_{\text{total}}}}},\;\;{\eta_{\text{vac}}} \in [0,1] $$
    (4)

    where \({\sum \nolimits }_{k} {t_{\text {vac}}^{k}}\) is the total amount of time that MCV spends on its vacation over all of the cycles.

In this paper, our goal is to decrease the value of Ptotal while maintaining the network operational forever. In (4), we can see that when ηvac increases, it means that MCV has more time to replace or recharge its battery at the service station, which indicates better performance of the network.

Implementation of proposed ERSVC

In this section, we present the procedure of ERSVC. The implementation of ERSVC includes two major steps. The first step is to optimize the first part of total energy consumption (i.e., \({\sum \nolimits }_{i,i \in \mathcal {N}} {{p_{i}}}\)). In the second step, we make a joint design to decrease the traveling distance of MCV.

Optimization with flowing rate and data routing

Denote xij as the flow rate from node i to node j and xiB as the flow rate from node i to the base station B, respectively. Then, we have the following flow balance constraint [6] at each node i,

$$ \sum\limits_{k \in \mathcal{N}}^{k \ne i} {{x_{ki}}} + {R_{i}} = \sum\limits_{j \in \mathcal{N}}^{j \ne i} {{x_{ij}}} + {x_{iB}}\;\;(i \in \mathcal{N}) $$
(5)

Each sensor node consumes energy for data transmission and reception. In this paper, we use the following energy consumption model [1] which is widely used in previous researches [68].

$$ {p_{i}} = {\beta_{r}} \cdot \sum\limits_{k \in \mathcal{N}}^{k \ne i} {{x_{ki}}} + \sum\limits_{j \in \mathcal{N}}^{j \ne i} {{\beta_{ij}} \cdot {x_{ij}}} + {\beta_{iB}} \cdot {x_{iB}}\;\;(i \in \mathcal{N}) $$
(6)

where βr is the energy consumption rate for receiving a unit of data rate, βij (or βiB) is the energy consumption rate for transmitting a unit of data rate from node i to node j (or the base station B). Furthermore, \({\beta _{ij}} = {C_{1}} + {C_{2}} \cdot d_{ij}^{\alpha } \), where dij is the distance between node i and node j, C1 is a distance independent constant term, C2 is a coefficient of the distance dependent term, and α is the path loss index. In this model, \({\beta _{r}} \cdot {\sum \nolimits }_{k \in \mathcal {N}}^{k \ne i} {{x_{ki}}}\) is the energy consumption rate for reception, and \({\sum \nolimits }_{j \in \mathcal {N}}^{j \ne i} {{\beta _{ij}} \cdot {x_{ij}}} + {\beta _{iB}} \cdot {x_{iB}}\) is the energy consumption rate for transmission.

We assume that the flow rate (xij and xiB) in the network is invariant with time. The optimization objective is the total power consumption of all sensor nodes (i.e., \({\sum \nolimits }_{i,i \in \mathcal {N}} {{p_{i}}} \)) which is the first part of Ptotal in (3). Each node should satisfy the basic flow balance constraint in (5) and energy consumption model in (6). Therefore, the optimal problem can be formulated to a linear programming problem as follows:

$$ \begin{aligned} \text{min} \;\;\;\;\;\; & \sum_{i,i \in \mathcal{N}} p_{i} \\ s.t.\;\;\;\;\;\;& \sum_{k\in \mathcal{N}}^{k\neq i} x_{ki} + R_{i} = \sum_{j \in \mathcal{N}}^{j\neq i} x_{ij} + x_{iB} \\ & p_{i} = \beta_{r}\cdot \sum_{k\in \mathcal{N}}^{k \neq i} x_{ki} + \sum_{j \in \mathcal{N}}^{j\neq i} \beta_{ij} \cdot x_{ij} + \beta_{iB} \cdot x_{iB} \\ & 0 \leq p_{i} \leq \kappa \\ & x_{ij}\geq 0 \end{aligned} $$
(7)

In this problem, xij,xiB and pi are the optimization variables. Ri,βr,βij, and βiB are constants. κ is a constant which is defined in the following Theorem 1.

Define DTSP as the distance of the shortest Hamilton cycle which connects all nodes in \(\mathcal {N}\) and the service station O. Then, the system should satisfy a feasibility condition which is stated in Theorem 1 as follows.

Theorem 1

For any node \(i\;(i \in \mathcal {N})\), the optimization problem has a feasible solution when pi satisfies the following condition.

$$ {\begin{aligned} 0 \leq p_{i} \leq \frac{AU+(2N+1)E_{1}-\sqrt{A^{2}U^{2}+(2N+1)^{2}E_{1}^{2}+2(2N-1)AUE_{1}}}{2A} \end{aligned}} $$
(8)

where N is the number of sensor nodes in the network, A=2DTSP/V and E1=EmaxEmin. The right value in (8) is a constant, which is defined as κ. The proof of Theorem 1 is given in the “Discussion” section.

Note that the constrained condition in Theorem 1 is set up to ensure that the vacation time meets the condition \(t_{\text {vac}}^{k} \ge 0\) in (1), which means the sum of the traveling time of MCV and the charging time for all nodes should not be greater than the total cycle time T.

Procedure of joint design

We can obtain the power consumption pi of each node i after solving the optimization problem in (7). In this subsection, we design the system for combining the charging period Ti, the visiting set Fk in each cycle, and the traveling path of MCV. The steps are as follows.

Step 1

In step 1, we set the value of T for MCV and the number of set needs to be classified.

Firstly, compute the maximum and minimum of pi respectively as follows:

$$ {p_{\max }} = \max \;{p_{i}},\;\;{p_{\min }} = \min \;{p_{i}} $$
(9)

Secondly, set the value of T, and the maximum charging period (denoted as Tmax) corresponding to the node which has the minimum power consumption as follows:

$$ T = \frac{{{E_{\max }} - {E_{\min }}}}{{2 \cdot {p_{\max }}}},\;\;{T_{\max }} = \frac{{{E_{\max }} - {E_{\min }}}}{{{p_{\min }}}} $$
(10)

Denote r as the number of set that needs to be classified, which is set up as follows:

$$ r = \left\lceil {{{\log }_{2}}\left\lfloor {{T_{{\text{max}}}}{/}T} \right\rfloor} \right\rceil $$
(11)

where and is the operating of making a number round up to and down to the nearest integer, respectively.

Step 2

In this step, we set the charging period Ti for each node i and classify the set Sk.

Firstly, for each node \(i\;(i \in \mathcal {N})\), we set its charging period Ti as follows:

$$ {T_{i}} = {2^{a - 1}} \cdot T\;\;\;\;(1 \le a \le r) $$
(12)

where a is the approximate logarithm of the ratio of Ti and T and is calculated as follows:

$$ a = \left\lfloor {{{\log }_{2}}(\frac{{{E_{\max }} - {E_{\min }}}}{{{p_{i}} \cdot T}} - 1)} \right\rfloor + 1 $$
(13)

Define set Sk(1≤kr) and let iSa, then the MCV should visit node i in the (n·2a−1)th trip cycle. Note that we multiply the parameter T by coefficient 1/2 in (10) to guarantee the paramenter a meets the condition a≥1.

Step 3

In this step, we will obtain the visiting set Fj of MCV during the jth cycle and design the traveling path of MCV. We can express j (1≤j≤2r−1) as j=m·2c, where m is an odd number, c is an integer, and c≥0. During the jth cycle, denote Fj as the set of nodes which should be visited and recharged, then Fj can be obtained as follows:

$$ {F_{j}}\ =\ \left\{ \begin{array}{l} {S_{1}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(c = 0)\\ {S_{1}} \cup {S_{2}}... \cup {S_{c + 1}}\;\;\;\;(c \ge 1) \end{array} \right. $$
(14)

Denote Pj as the traveling path of MCV during the jth cycle. It is obvious that Pj should be the shortest Hamilton cycle which connects all nodes in Fj and the service station O, that is,

$$ {P_{j}} = {\text{Hamiltonian}}({F_{j}} \cup O) $$
(15)

During the jth cycle, for each node niFj, the MCV travels to ni and charges its battery to Emax. After 2r−1 cycles, the MCV will recount from the first cycle and keep circulating like this.

The detailed steps of this subsection are summarized in Algorithm 1.

Numerical results

In this section, we present some numerical results to explain how ERSVC works in a real network. In order to evaluate the performance for proposed ERSVC and other schemes in previous researches, we use the network topology and parameter settings similar to those in [69]. The simulations are conducted with MatLab software.

Simulation settings

We consider two randomly generated WSNs consisting of 50 and 100 nodes, respectively. The sensor nodes are deployed randomly in a square area of 1 × 1 km2. Both the base station and the service station are assumed to be located at (500, 500) (in m). The data rate Ri from each node i is randomly generated within [1, 10] kb/s. The parameters in (6) are C1=50 nJ/b,C2=0.0013 pJ/(b·m4),βr=50 nJ/b, and α=4 [68]. The traveling speed of MCV is V=5 m/s, and the energy transfer rate of MCV is U=5 W which is well within feasible range [5]. We assume the charging distance between MCV and each sensor node is about 1 m. Thus, the value of energy conversion efficiency can be set to λ=0.85 [5].

For a sensor node’s battery, we choose a regular NiMH battery and its cell voltage and the quantity of electricity is 1.2 V/2.5 Ah. Let Emax=10.8 KJ, and Emin=0.05·Emax=540 J [6].

Results

50-Node network

We firstly present the results for the 50-node network. Table 2 gives the location of each node and its data rate for a 50-node network.

Table 2 Location and data rate for each node in a 50-node network

Figure 3 illustrates the data routing result after optimizing in the first step of ERSVC. In Fig. 3, an arrow from node i to node j indicates that there is data transmitting from node i to node j (i.e., xij>0).

Fig. 3
figure3

Data routing for proposed ERSVC in the 50-node network. After implementing the first step of ERSVC, we can obtain the dating routing result. Multi-hop data routing is employed for forwarding data by the sensor nodes. In this figure, an arrow from node i to node j indicates that there is data transmitting from node i to node j

After the first step, we can obtain the power consumption pi of node i by (6). Then, the node with the highest power and the node with the lowest power can be calculated, which correspond to the 48th node and the 12th node, respectively. So the number of set needs to be classified is r=12, which can be calculated by (11). The result of the classified set Sk is shown in Table 3.

Table 3 The result of the classified set Sk(k=1,2,…,r) for the 50-node network

In ERSVC, the period of a trip cycle is T=14.4 h. The optimal traveling path of MCV during the (25=32)th cycle is shown in Fig. 4. According to (14), the visiting set of MCV is F32=S1S2S6 in the 32th cycle. It is obvious that the optimal traveling path is the shortest Hamiltonian cycle. We can solve this problem by using the Concorde solver [15].

Fig. 4
figure4

The optimal traveling path of MCV during the 32th cycle. During the 32th cycle, the visiting set of MCV is F32=S1S2S6. Different from other schemes in previous researches, the MCV only needs to charge a portion of nodes during the 32th cycle. Based on previous analysis, the optimal traveling path is the shortest Hamiltonian cycle which connects all nodes in F32 and the service station

As shown in Fig. 4, different from other schemes in previous researches [68], the MCV only needs to charge a portion of nodes during the 32th cycle, which is consistent with our previous analysis. After 2r−1=2048 cycles, the MCV will recount from the first cycle and so on.

The energy cycle behavior of the 48th node which has the highest power consumption is shown in Fig. 5. We can see that the remaining energy of the battery at the 48th node is always higher than Emin during a charging period. Thus, it will remain operational forever due to the periodicity of each node in the network.

Fig. 5
figure5

The energy behavior of the 48th node during two charging periods. The 48th node has the highest power consumption among all nodes in the network. The energy behavior of each node is periodic and can be calculated after the implementation of ERSVC. Similarly, we can also obtain the energy behavior for other nodes. If the remaining energy of a node’s battery is always higher than Emin, it will remain operational forever

After the implementation of ERSVC, we can finally compute the total power consumption Ptotal by (3). Then, the vacation time ratio ηvac can be obtained by bringing (2) into (1). Ptotal and ηvac are two parameters to measure the performance of the network. In traditional scheme (such as [6, 7]), the MCV visits and charges all the nodes in the network with constant cycle time, and we call it the traditional scheme with constant cycle (TSCC). Table 4 illustrates these two parameters for proposed ERSVC, TSCC, and the scheme in [9].

Table 4 Comparison of performance for proposed ERSVC and TSCC in the 50-node network

In Table 4, \(\overline {{L_{\text {total}}}} (m)\) is defined as the average traveling distance of MCV over 2048 cycles. From Table 4, we can see that ERSVC decreases total energy consumption Ptotal by about 48% compared to TSCC, while maintaining the vacation time ratio ηvac nearly equal to that in other schemes. In [9], the authors designed a cluster-based network topology and selected some nodes as cluster heads to reduce the traveling distance and total energy consumption. Nevertheless, the MCV still need to visit all nodes in each cycle in [9]. Therefore, ERSVC outperforms the scheme in [9] in terms of energy consumption.

In ERSVC, the total power consumption of all nodes is \({\sum \nolimits }_{i,i \in \mathcal {N}} {{p_{i}}} {{ = }}0.58\) and it is much less than Ptotal, as shown in Table 4. Therefore, we can deduce that the traveling distance of MCV (i.e., \( \overline {{L_{\text {total}}}} \)) is the main factor influencing Ptotal in (3). In Table 4, \( \overline {{L_{\text {total}}}} \) is significantly decreased by about 75.4% compared to TSCC, since the MCV only needs to charge a portion of nodes in each cycle.

100-Node network

Table 5 gives the location of each node and its data rate for a 100-node network. Figure 6 illustrates the data routing result after optimizing in the first step of ERSVC.

Fig. 6
figure6

Data routing for proposed ERSVC in the 100-node network. Similar to Fig. 3, this figure illustrates the data routing result for the 100-node network

Table 5 Location and data rate for each node in a 100-node network

The result of the classified set Sk is shown in Table 6. Table 7 shows the network performance measured by Ptotal and ηvac for proposed ERSVC and traditional TSCC.

Table 6 The result of the classified set Sk(k=1,2,…,r) for the 100-node network
Table 7 Comparison of performance for proposed ERSVC and TSCC in the 100-node network

In Table 7, we can see that ERSVC decreases the total energy consumption Ptotal by about 51% and 42% compared to TSCC and the scheme in [9] respectively, with no performance loss in terms of the vacation time ratio ηvac. Therefore, ERSVC outperforms TSCC and the scheme in [9] in terms of energy consumption and achieves the goal of energy saving.

Discussion

In this section, we firstly give the proof that ERSVC can make the network immortal. Then, we prove the Theorem 1 which is mentioned in previous implementation section.

Proof of the network’s sustainability with ERSVC

In this subsection, we prove that ERSCV can guarantee the network operational permanently. Before giving the proof, we firstly prove a Lemma which is stated in Lemma 1.

Lemma 1

For any node \(i\;(i \in \mathcal {N})\) in the network, the sufficient condition to make it immortal is that the visiting interval of i satisfies inequation: Ti≤(EmaxEmin)/pi.

Proof

Define Ti the visiting and charging interval of i, which is the time interval between two adjacent arrivals at node i by MCV. Figure 7 is the energy behavior of node i during the nth and (n + 1)th charging period. Different from Ti which is set to be an integral multiple of T, the value of Ti during each charging period is different and its mean value is Ti, as shown in Fig. 7.

Fig. 7
figure7

The energy behavior of node i during the nth and (n + 1)th charging period. Ti is defined as the visiting and charging interval of i, which is the time interval between two adjacent arrivals at i by MCV. In ERSVC, the MCV only visits and charges a portion of nodes in each cycle, and the visiting set in each cycle is different. Therefore, the moment when MCV arrives at i is different every time, and the visiting interval Ti is variable. In Fig. 7, ai is the moment when MCV arrives at node i in the (n + 1)th charging period. After the time of ti, the MCV charges the battery of node i to Emax. Apparently, the slope is (Upi) when the MCV is charging node i and the slope is pi at other times

Denote Elow as the lowest remaining energy of the battery at node i. To prove Lemma 1, we only need to prove that Elow is greater than or equal to Emin. As shown in Fig. 7, denote ai as the moment that MCV arrives at i for the (n + 1)th time, and hi the moment when MCV charges the battery at node i to Emax for the nth time. Assume the remaining energy of the battery at node i is Elow at ai; therefore,

$$ {{}\begin{aligned} {E_{{\text{low}}}} = {E_{\max }} - {p_{i}} ({a_{i}} - {h_{i}}) &\ge {E_{\max }} - {p_{i}} {T_{i}}^{\prime} \ge {E_{\max }} \\&- {p_{i}} ({E_{\max }} - {E_{\min }})/{p_{i}} = {E_{\min }} \end{aligned}} $$
(16)

where the second inequation is based on the sufficient condition in Lemma 1. This completes the proof. □

Theorem 2

For any node \(i\;(i \in \mathcal {N})\) in the network, the proposed scheme ERSVC can make it operational forever.

In ERSVC, the MCV only visits and recharges a portion of nodes in each cycle, and the visiting set in each cycle is different. Therefore, the moment when MCV arrives at i is different every time, and the visiting interval Ti is variable, as shown in Figs. 2 and 7. According to (12), we have Ti=2a−1·T. As shown in Fig. 2, we can obtain the following relationship,

$$ ({2^{a - 1}} - 1) \cdot T < {T_{i}}^{\prime} < ({2^{a - 1}} + 1) \cdot T $$
(17)

Therefore, we can obtain the following result based on (13),

$$\begin{array}{@{}rcl@{}} {T_{i}}^{\prime} & < &({2^{a - 1}} + 1) \cdot T = ({2^{\left\lfloor {{{\log }_{2}}(\frac{{{E_{\max }} - {E_{\min }}}}{{{p_{i}} \cdot T}} - 1)} \right\rfloor }} + 1) \cdot T \\ & \le & (\frac{{{E_{\max }} - {E_{\min }}}}{{{p_{i}} \cdot T}} \,-\, 1 \,+\, 1) \cdot T \,=\, ({E_{\max }} \,-\, {E_{\min }})/{p_{i}} \end{array} $$
(18)

According to Lemma 1, for any node \(i\;(i \in \mathcal {N})\) in the network, it can operate with unlimited time. This completes the proof.

Proof of theorem 1

Recall that the constrained condition in Theorem 1 is set up to ensure that the vacation time meets the condition \(t_{\text {vac}}^{k} \ge 0\) in (1). Therefore, we have the following inequality during the kth cycle:

$$ T \geq D_{k}/V + \sum_{j \in F_{k}} t_{j} $$
(19)

Since T is invarient in (19), we only need to prove that T is not less than the maximum of the right value in (19). The maximum of the right value occurs when the visiting set Fk includes all sensor nodes in the network (namely, \(F_{k}=\mathcal {N}\)), and we have Dk=DTSP at this time. Therefore, we have the following relationship,

$$ \frac{{{D_{k}}}}{V} + \sum\limits_{j \in {F_{k}}} {{t_{j}}} \le \frac{{{D_{TSP}}}}{V} + \sum\limits_{j \in \mathcal{N}} {{t_{j}}} $$
(20)

In Fig. 7, ai is the moment when MCV arrives at node i in the (n + 1)th charging period. After the time of ti, the MCV charges the battery of node i to Emax. Apparently, the slope is (Upi) when the MCV is charging node i at a rate of U. Therefore, we can obtain the following relationship:

$$ {E_{\max }} = {E_{{\text{low}}}} + {t_{i}} \cdot (U - {p_{i}}) \ge {E_{\min }} + {t_{i}} \cdot (U - {p_{i}}) $$
(21)

Then, we can derive the following inequality based on (20) and (21):

$$ {\begin{aligned} \frac{{{D_{k}}}}{V} + \sum\limits_{j \in {F_{k}}} {{t_{j}} \le} \frac{{{D_{\text{TSP}}}}}{V} + \sum\limits_{j \in \mathcal{N}} {t_{j}} &\le \frac{{{D_{\text{TSP}}}}}{V} + \sum\limits_{j \in \mathcal{N}} \frac{{{E_{1}}}}{{U - {p_{i}}}} \\&\quad\le \frac{{{D_{\text{TSP}}}}}{V} + \frac{{N \cdot {E_{1}}}}{{U - {p_{\max }}}} \end{aligned}} $$
(22)

According to (10), (19), and (22), the parameter T should satisfy the following inequality to ensure the optimization problem in (7) has a feasible solution:

$$ T = \frac{{{E_{1}}}}{{2 \cdot {p_{\max }}}} \ge \frac{{{D_{\text{TSP}}}}}{V} + \frac{{N \cdot {E_{1}}}}{{U - {p_{\max }}}} $$
(23)

Then, we can rewrite the inequality in (23) to the following quadratic inequality which is defined as the function f(pmax),

$$ f(p_{\max})=A\cdot p_{\max}^{2}-(AU+B)\cdot p_{\max}+C \geq 0 \\ $$
(24)

where A=2DTSP/V,B=(2N+1)E1 and C=E1·U. Define x1 and x2 (x1x2) as the two roots of the equation f(pmax)=0. Then, the inequality in (23) can be established if pi satisfies the following relationship,

$$ p_{i} \le p_{\max} \le x_{1}=\frac{AU+B-\sqrt{(AU+B)^{2}-4AC}}{2A} $$
(25)

Then, we can obtain the Theorem 1 by bringing B and C into (25). This completes the proof.

Conclusion

In this paper, we studied the operation of a sensor network based on WET, in a scenario where a MCV was employed to charge the sensor nodes wirelessly inside the network. We analyzed the energy consumption of the entire system and pointed out that the traveling distance of MCV is the main factor influencing total energy consumption. Based on energy consumption analysis and periodic strategy, we proposed a scheme ERSVC to decrease total energy consumption while maintaining the network operational forever with low complexity. Compared to previous schemes where the MCV visits and charges all nodes in each cycle, the MCV only needs to visit a portion of nodes in ERSVC by taking into the account the difference of energy consumption rate at each node. Therefore, the traveling distance of MCV was significantly decreased, and the total energy consumption could be saved. In ERSVC, we firstly formulated a practical optimization problem with flow rate to obtain the energy consumption rate, then designed the system for combining the charging period of each node, the visiting set, and traveling path during each cycle. Subsequently, we gave the proof that ERSVC can guarantee the network operational permanently. Simulations were carried out, showing that ERSVC can significantly decrease the total energy consumption with no performance loss in terms of the vacation time ratio.

The limitation of this work is that the flow rate is assumed to be invariant with time in order to simplify the optimization. However, it is time-varying in many real applications. In addition, we multiply the parameter T by coefficient 1/2 due to the variability of Ti which will degrade the energy saving performance. This can be improved by using the heuristic approach, such as genetic algorithm and ant colony algorithm. We will investigate these problems in our future research.

Availability of data and materials

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Acknowledgements

This work is supported by the National Natural Science Foundation of Guangdong Province in China with No. 2016A030310027.

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Qian Zhang designed the system model and the implementation of the proposed scheme. Rong Cheng collected the relative literatures and performed some experiments to obtain the numerical results. Zhihua Zheng was a major contributor in writing and revising the manuscript. All authors read and approved the final manuscript.

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Correspondence to Qian Zhang.

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Zhang, Q., Cheng, R. & Zheng, Z. Energy-efficient renewable scheme for rechargeable sensor networks. J Wireless Com Network 2020, 74 (2020). https://doi.org/10.1186/s13638-020-01687-4

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Keywords

  • Wireless energy transfer
  • Energy-efficient
  • Renewable
  • Wireless sensor network