On-demand Multi-Node Partial Charging Scheme using Multiple Wireless Charging Vehicles in WRSN

Wireless rechargeable sensor networks (WRSNs) have been widely used in various monitoring and surveillance applications, including environmental monitoring, military surveillance, structural health monitoring, etc. [8, 13, 17, 27]. The sensors are powered by the batteries with limited energy capacity and consume their energy to collect information from the surroundings. Therefore, the sensors remain active for a limited duration [5, 12]. To prolong the lifetime of the sensors in WRSN, the use of wireless charging vehicles (WCVs) becomes a promising technique, where the energy-critical sensors are charged wirelessly [12, 15, 29].

Existing charging methods adopted a single-node charging scheme [1, 9, 16, 25], in which a WCV charges only one sensor at a time. Since charging the battery of a sensor to its full capacity takes a long time [22], many sensors may exhaust their energy, i.e., be dead before the WCV arrives to charge them, and it also increases the charging delay. To decrease the number of dead sensors and charging delay of the sensors, some works used a multi-node charging scheme [14, 24, 26], where a WCV can charge multiple sensors within its charging range simultaneously by adjusting the operating frequencies of both the WCV and the sensor coil [10]. In most of the existing multi-node charging schemes [14, 24, 26], a WCV charges the batteries of multiple sensors in its range simultaneously to their full capacity. Since the charging efficiency decreases with the increase in the distance between the WCV and the sensor, a sensor located at a longer distance from the WCV has a slower charge receiving rate. It causes a longer time to fully charge the sensors in the charging range. As a result, many sensors may die in large-scale WRSN, which require urgent energy replenishment. If the sensors remain dead for a long time, some applications of WRSN, like forest fire detection, may be severely affected, as it may cause a delay in detecting the fire in the forest, leading to its uncontrollable spreading.

A more effective and scalable solution is the multi-node partial charging scheme [3, 19, 30], where multiple sensors may be charged partially and simultaneously. Since this scheme reduces the charging time at each charging location, a higher number of sensors may be charged before they die. However, this scheme increases the total distance travelled by a WCV, as the WCV has to revisit the sensors for recharging, which were previously partially charged. Such an increase in the travel distance is acceptable because the continuous working of the sensors is a fundamental need for the majority of WRSN applications. The studies in [19], [30] reduce the number of dead sensors by considering a multi-node partial charging scheme, but they did not consider the charge scheduling of dead sensors. The charge scheduling of dead sensors is essential, otherwise some information may be missed. Also, the new data generated by the other live sensors may not be transmitted to the BS if the dead sensors act as the relay sensors.

Many studies deploy a single WCV to charge the energy-critical sensors [11, 19, 26, 30]. Since a WCV consumes a considerable amount of its energy in its travel than charging the sensors, it results in a high charging cost [23, 29]. A single WCV with limited battery capacity, therefore, may not be able to charge the sensors in a large-scale WRSN [22]. Multiple WCVs are needed to charge many energy-critical sensors, while reducing the number of dead sensors and minimizing their dead durations.

In this paper, we study a multi-node partial charging scheme with multiple WCVs. Given a set of sensors with different energy consumption rates, a set of WCVs, and a fixed base station (BS), the aim is to find the efficient charging tours for the WCVs, which minimize the number of dead sensors and the average dead duration. We refer to the problem as Multi-node Partial Charging with Minimum Dead Sensor (MinDS) problem. The key approach to solve the MinDS problem is to attempt to charge the sensors before they run out of their energy while minimizing the number of times the WCVs perform partial charging. Precisely, our proposed algorithm consists of the following three steps. First, we divide the energy-critical sensors into WCVs so that their charging load is evenly distributed as follows. We create a single tour of the energy-critical sensors and the BS using Christofides algorithm [2]. Then, energy-critical sensors are distributed among the WCVs by considering the energy requirement of the sensors and the energy requirement of the WCVs for movement. Second, a set of charging locations, referred to as stop locations is determined for each WCV based on the deadline of the sensors. Finally, we find the charging tour of each WCV while giving priority to the sensors with near deadlines. If some sensors cannot be charged within the deadline, we use partial charging at some stop locations, where a WCV takes a large charging time. It saves time for the WCV to visit more stop locations in time while charging a higher number of sensors before they die, thereby minimizing the number of dead sensors and their dead duration. The major contributions of this paper are as follows.

The remainder of the paper is organized as follows. The system model is described and the problem is formulated formally in Section 2. The algorithms are discussed in Section 3. Simulation settings are described, and the results are analyzed in Section 4. Finally, Section 5 concludes the paper.

In this section, we first introduce the system model and then formulate the problem formally.

A. Network Model: We consider a WRSN with a set SS = {s₁, s₂, …, s_n} of n static sensors. There is a base station (BS), where a set W = {w₁, w₂, …, w_h} of h wireless charging vehicles (WCVs) are initially parked. Each WCV w_i ∈ W consumes energy in its mechanical movement and in charging the sensors. Let ec (J/s) denotes the energy transmission of each WCV to charge the sensors and em (J/m) denotes the energy consumption of each WCV to move from one location to another location. The charging range of each WCV is denoted by γ. Table 1 summarizes the variables used in the problem formulation.

The sensing information from a sensor is transmitted to the BS either directly or via multi-hop relays [21]. Each sensor s_i ∈ SS has a rechargeable battery of energy capacity b. Let ρ_i denotes the energy consumption of s_i for data sensing, transmitting, and receiving. Let RE_i denote the residual energy and \(RL_i = \frac{RE_i}{\rho _i}\) denote the residual lifetime of s_i. Fig. 1 shows an example of multi-node charging of sensors by two WCVs and their charging tour.

B. Charging Request Model: A fully charged sensor can have its lifetime from several weeks to months depending on its sensing characteristics. It is also seen that a WCV usually takes up to several days to finish charging a few hundred sensors in the network in one round of recharging [16], [1]. Therefore, a WCV should only charge the sensors which have a low residual lifetime. In particular, a sensor s_i will send a charging request REQ_i = ⟨ s_i, t_i, RE_i, ρ_i, ED_i ⟩ to the BS if its residual lifetime RL_i is below a certain threshold (L_thrld), where t_i is the time of raising charging request, RE_i is the residual energy, ρ_i is the energy consumption rate, and ED_i = b − RE_i is the energy demand of s_i. Let SC denotes the set of to-be-charged sensors, i.e., SC = {s_i | s_i ∈ SS, RL_i(t_i) ≤ L_thrld}. The BS divides the charging requests among the WCVs. Let SC_j be the set of to-be-charged sensors which is assigned to a WCV w_j, i.e., \(SC = \bigcup _{j = 1}^{h} SC_j\). The BS constructs a charging tour for each WCV w_j.

C. Multi-node Charging Model: In each charging round, a WCV w_j starts its charging tour from the BS with full battery capacity, stops at a few locations, referred to as stop locations to charge the sensors, and then returns to the BS. At each stop location, single or multiple sensors may be charged simultaneously by w_j. Since there can be an infinite number of stop locations for w_j, the optimal choice of the stop locations for w_j is intractable. In the paper, a set SL_j = {\(sl_j^1, sl_j^2, \ldots , sl_j^q\)} of q stop locations has been determined for w_j, where it can stop and charge the sensors. For a stop location \(sl_j^k\), the neighbouring set \(NS_j^k\) is the set of to-be-charged sensors that are within the charging range γ from \(sl_j^k\), i.e., \(NS_j^k\) = \(\lbrace s_i\;|\; s_i \in SC_j, d(s_i, sl_j^k) \le \gamma \rbrace\), where \(d(s_i, sl_j^k)\) is the Euclidean distance between a sensor s_i and a stop location \(sl_j^k\).

In this paper, a WISP-reader charging model is adopted to charge the sensors [4], [6]. In this model, energy transfer efficiency decreases rapidly with the Euclidean distance. Thus, the distant sensors need a large charging time to full charge their battery. Specifically, the amount of energy received by a sensor s_i from the WCV w_j located at a stop location \(sl_j^k\), is denoted as charge receiving rate (\(P_{r}(s_i, sl_j^k)\)), and is calculated as follows.where α and β are two constant parameters which are defined by the hardware design of the WCV, \(d(s_i,sl_j^k)\) is the distance between s_i and w_j which is at location \(sl_j^k\), ec is the energy transmission of w_j, and γ is the charging range of w_j.

In this paper, omni-directional wireless charging is used, therefore, the maximum charging time at each stop location is equal to the time required to charge the sensors located furthest from the stop location.

D. Multi-node Partial Charging Model: In the previous studies [26], [28] the multi-node full charging scheme was adopted, where the sensors in the charging range of w_j are fully charged at each stop location. In this paper, a multi-node partial charging model is considered, where a sensor s_i can be charged more than once in a single charging tour of w_j. We assume that the total amount of energy replenished to s_i is ED_i = b − RE_i as per the charging request. Therefore, a sensor s_i can be charged for a maximum duration of \(\frac{ED_i}{P_{r}(s_i, sl_j^k)}\).

It may not be practical to transfer a small amount of energy to a sensor multiple times by revisiting its stop location since it unnecessarily increases the energy consumption of w_j due to its travelling to the stop location multiple times. Therefore, we assume at least δ amount of energy must be replenished to all the to-be-charged sensors within the charging range γ from the stop location at each charging. Specially, the amount of energy replenished to a sensor \(s_i \in NS_j^k\) by w_j from the location \(sl_j^k\) ranges in {δ, 2 × δ, …, k_i × δ, ED_i}, where \(k_i = \lfloor \frac{ED_i}{\delta } \rfloor , \forall s_i \in NS_j^k\). In Section 4, the effect of δ has been investigated in Fig. 5, and the proper value of δ is estimated.

E. Problem Definition: In this paper, a sensor can be in two states: alive and dead. A sensor s_i is said to be dead in a charging round if its residual energy RE_i becomes zero; otherwise, it is said to be alive. A sensor s_i is said to be charged in time if s_i is alive till its energy demand ED_i is fulfilled.

Let TD_i = t_i + RL_i be the deadline of a sensor s_i, where t_i is the time at which s_i has raised the charging request. Note that TD_i changes with each partial charging of s_i. Let △_i be the time at which s_i will start receiving charge. Then, the dead duration of s_i is DD_i = △_i − TD_i, which is the time interval for which s_i remains dead due to insufficient energy. Thus, for a dead sensor s_i, DD_i > 0 and for an alive sensor, DD_i is considered to be zero since DD_i is undefined for an alive sensor. Let \(DD_{avg} =\frac{\sum _{i=1}^{|SC|} DD_i}{|SC|}\) denotes the average dead duration of a sensor, and \(DS = \bigcup \limits _{j=1}^{h} DS_j\) be the set of dead sensors, where DS_j denotes the set of dead sensors that are not charged within the deadline by w_j.

The Multi-node Partial Charging with Minimum Dead Sensor (MinDS) Problem: Given h WCVs, and a set SC of to-be-charged sensors in WRSN at time t, the energy consumption rate ρ_i, residual lifetime RL_i, the energy demand ED_i of each s_i ∈ SC, the objective is to find the sequence of stop locations travelled by all the WCVs to charge the sensors in SC, such that the total number of dead sensors |DS| is minimized while minimizing the average dead duration of a sensor (DD_avg) in one charging round, subject to the constraint that the total energy given to a sensor s_i ∈ SC by the WCVs is always equal to its energy demand ED_i.

Note that each sensor s_i ∈ SC∖DS receives the energy ED_i from one of the WCVs. Subsequently, the same WCV w_j is used to charge all the dead sensors in DS_j using multi-node full charging scheme by visiting the same stop locations.

The algorithm proposed to solve the MinDS problem is referred to as MinDS algorithm, which works in three stages.

Stage I: In the first stage, the set SC of to-be-charged sensors (where |SC| = m) has been evenly distributed into h disjoint groups, where each group is assigned a WCV for charging the sensors in this group. To balance the charging load among h WCVs, two factors are taken into account: (i) the total energy consumption of a WCV in its travel, and (ii) the energy demand by the sensors in a group.

The input to the algorithm is the set SC of m to-be-charged sensors, em, h, BS, and the energy demand ED_i of each sensor s_i ∈ SC. The output of the algorithm is h disjoint subsets SC₁, …SC_h, where each SC_i is assigned a WCV w_i.

The Algorithm 1 works as follows. A complete weighted undirected graph G = (SC ∪ {BS}, E) is created. The weight of an edge between two vertex v_i, v_j ∈ SC ∪ {BS} is calculated as w(v_i, v_j) = (ED_i + ED_j)/2 + \(ET_{v_i, v_j}\), where ED_i and ED_j are the energy demand of s_i and s_j respectively and \(ET_{v_i, v_j} = d(v_i, v_j) \times em\) is the energy required by a WCV to travel a distance d(v_i, v_j) from v_i to v_j (Line 1). Note that ED_BS = 0. We then find an approximate shortest tour R of the graph G using Christofides algorithm [2] such that the tour starts from the BS, visit each of the sensors exactly once, and returns to the BS. Say, R = ⟨BS, u₁, …, u_|m|, BS⟩ (Line 2). Let W(R) be the total weight of R. Let E_max denotes the maximum energy required by a WCV to charge a sensor in the tour R, i.e, \(E_{max} = max_{u_i \in SC \lbrace ED(u_i) + 2 \times ET_{BS, u_i}\rbrace }\), where \(ET_{BS, u_i}\) is the energy needed to travel from BS to a sensor u_i, and vice versa (Line 3). Since R contains all the to-be-charged sensors and the cost to visit them, to-be-charged sensors are distributed among h WCVs by splitting R into h subtours R₁, R₂, …, R_h, where each subtour starts from the BS, visits some of the sensors in SC and returns to the BS. The h subtours are determined by finding h − 1 locations \(u_{l_1}, u_{l_2}, \ldots , u_{l_{h-1}}\), where \(u_{l_j}\) (where 1 ≤ j ≤ h − 1) is the last vertex along R such that the total weight of the path from BS to \(u_{l_j}\) along R is not greater than \(\frac{j}{h}(W(R) - E_{max})+ \frac{E_{max}}{2}\), but the total weight of the path from BS to \(u_{l_j + 1}\) is strictly greater than \(\frac{j}{h}(W(R) - E_{max})+ \frac{E_{max}}{2}\), where 1 ≤ j ≤ h − 1 (Lines 4 – 7). Then, the sensors \(u_1, u_2\ldots , u_{l_1}\) along R₁ are assigned to w₁, the sensors \(u_{l_1+1}, \ldots , u_{l_2}\) along R₂ are assigned to w₂, …, the sensors \(u_{l_{h-1}+1}, \ldots , u_m\) along R_h are assigned to w_h (Line 8). The pseudo code of the algorithm is given in Algorithm 1 .

Step 2 determines a tour R of total weight W(R) using Christofides algorithm [2]. Let \(R_{\mathcal {O}}\) be a single tour of the minimum weight \(R_{\mathcal {O}}\) in the optimal solution. Since the edge weight in graph G holds the triangle inequality, \(W(R)\le 1.5\times W(R_{\mathcal {O}})\) [2].

Steps 4 – 7 divide R into h subtours by the edge weight and E_max. Therefore, the weight of the path from the BS to \(u_{l_1}\) and from \(u_{l_{h-1} +1}\) to BS along R are each \(\le \frac{1}{h}\times (W(R)-E_{max}) + \frac{E_{max}}{2}\). Also, for each j, 1 ≤ j ≤ h − 2, the weight of the path from \(u_{l_{j}+1}\) to \(u_{l_{j+1}}\) is \(\le \frac{1}{h}\times (W(R)-E_{max})\). Therefore, the weight of each subtour that starts from the BS and ends at the BS does not exceed \(\frac{1}{h}\times (W(R)-E_{max}) +E_{max}\), i.e., \(W(R_{h}^\prime) \le \frac{W(R)}{h} + (1-\frac{1}{h})\times E_{max}\) or \(W(R_{h}^\prime) \le \frac{1.5\times W(R_{\mathcal {O}})}{h} + (1-\frac{1}{h})\times E_{max}\).

Let \(R_{1-\mathcal {O}}, R_{2-\mathcal {O}}, \dots , R_{h-\mathcal {O}}\) are the h subtours that start and end at the BS of the optimal solution. Let OPT be the subtour with maximum weight in the optimal solution, i.e., \(OPT = Max_{i=1}^{h} W(R_{i-\mathcal {O}})\). Thus, \(W(R_{\mathcal {O}}) \le \sum _{i=1}^{h} W(R_{i-\mathcal {O}})\) or \(W(R_{\mathcal {O}}) \le h\times OPT\). Also, using triangle inequality, we have E_max ≤ OPT. Therefore \(W(R_{h}^\prime) \le 1.5\times OPT + (1-\frac{1}{h})\times OPT\) or \(W(R_{h}^\prime) \le (2.5 - \frac{1}{h})\times OPT\). □

Stage II: In the second stage, the stop locations of a WCV w_j has been determined. Let the set SC_j = {u₁, …, u_p} contains p sensors assigned to w_j for charging. In this stage, we partition p to-be-charged sensors into q (≤ p) clusters based on the deadline of the sensors. The intuition behind the partitioning of the sensors into multiple clusters is two-fold: (i) to minimize the number of stop locations, which allows w_j to charge many sensors before they die, and (ii) to put the sensors with less remaining lifetime into the same cluster so that they can be charged as early as possible, which helps to reduce the number of dead sensors and the dead duration of dead sensors. The sensors in each cluster can be charged simultaneously from the stop location, which is centroid of the cluster.

The input to the algorithm is the set SC_j of p to-be-charged sensors, where the deadline \(TD_{u_i} = t_{u_i}+ RL_{u_i}\) of each u_i in SC_j. The output of the algorithm is the set SL_j of q (≤ p) stop locations.

The Algorithm 2 works as follows. The charging requests in SC_j are sorted in non-decreasing order of their deadlines (Line 1). Let NS_j be the set of clusters, where each cluster is a set of sensors. The cluster set NS_j, and the corresponding stop location set SL_j are initialized to ϕ (Line 2). Next, we select all the sensors in SC_j one by one and insert them either into one of the previously constructed clusters or into a newly created cluster (Lines 3 – 20). For a sensor u_k from SC_j, we try to add it in one of the previously created clusters \(NS_j^i\) (1 ≤ i ≤ |NS_j|). u_k is added to \(NS_j^i\) if, after adding u_k to \(NS_j^i\), all the sensors in \(NS_j^i\) are in the charging range γ from the cluster center \(cc_j^i\) (Lines 6 – 12). Otherwise, a new cluster is created for u_k (Lines 14 -– 19). Note that each sensor in SC_j belongs to exactly one of the clusters \(NS_j^i\) (1 ≤ i ≤ |NS_j|). After that, the cluster center \(sl_j^i\) of each cluster \(NS_j^i\) is determined and appended to the list SL_j (Lines 21 – 24). The pseudo code of the algorithm is given in Algorithm 2 .

Stage III: The input to the algorithm is the set of q clusters NS_j and the set of q stop locations SL_j for WCV w_j, where \(sl_j^k\) is the stop location of cluster \(NS_j^k\). The output of the algorithm is the visiting sequence \(SL_j^{visit}\) of stop locations in SL_j, the set of dead sensors DS_j, and the dead duration DD_i of each sensor s_i in SC_j.

The idea of stage III is as follows. The WCV w_j tries to visit the stop locations in SL_j one by one. Say, after a few iterations, \(SL_j^{visit}\) = {\(sl_j^1\), …, \(sl_j^k\)} (k ≤ q). It indicates that all the sensors {\(NS_j^1\), …, \(NS_j^k\)} in the range γ from the stop locations of {\(sl_j^1\), …, \(sl_j^k\)} can be charged by w_j before they are dead. The next stop location is \(sl_j^{k+1}\) with \(NS_j^{k+1}\) as its neighboring sensors. Three cases are possible. First, all the sensors in \(NS_j^{k+1}\) remain alive after w_j reaches \(sl_j^{k+1}\). In this case, \(sl_j^{k+1}\) is appended in \(SL_j^{visit}\). Second, at least one sensor in \(NS_j^{k+1}\) will be dead before w_j reaches \(sl_j^{k+1}\). In this case, the earlier visiting sequence \(SL_j^{visit}\) = {\(sl_j^1\), …, \(sl_j^k\)} will be altered at most k times by putting \(sl_j^{k+1}\) just before \(sl_j^{i}\), where 1 ≤ i ≤ k. If for some i, 1 ≤ i ≤ k, all the neighboring sensors of {\(NS_j^1\), …, \(NS_j^{k+1}\)} are fully charged, then \(SL_j^{visit}\) will be updated to {\(sl_j^1\), …, \(sl_j^{k+1}\), \(sl_j^{i}\), …, \(sl_j^k\)}. In the next iteration, \(SL_j^{visit}\) will be read as {\(sl_j^1\), …, \(sl_j^{k+1}\)}. Third, If for all 1 ≤ i ≤ k, some of the sensors are not fully charged, we go for partial charging, which is discussed later in the pseudo code.

The Algorithm 3 works as follows. Recall that, for a sensor s_i ∈ SC_j, the energy demand ED_i = b − RE_i. Then, the maximum time to fully charge a sensor s_i by w_j at a stop location \(sl_j^k \in SL_j\) is \(\frac{ED_i}{P_{r}(s_i, sl_j^k)}\). Since w_j can charge multiple sensors simultaneously, the maximum charging time of w_j at location \(sl_j^k\) to fully charge the sensors within the charging range γ is calculated as follows.

Let tt_{j, (i, k)} denotes the time taken by w_j to reach \(sl_j^k\) from \(sl_j^i\), and is calculated as follows.where \(d(sl_j^i, sl_j^k)\) is the Euclidean distance between \(sl_j^i\) and \(sl_j^k\).

Our algorithm gradually determine the visiting sequence \(SL_j^{visit}\) = \(\langle \:(BS,0)\rightarrow (sl_j^{1^\prime }, td_j^{1^\prime })\rightarrow (sl_j^{2^\prime }, td_j^{2^\prime })\rightarrow \ldots \rightarrow (sl_j^{q^\prime }, td_j^{q^\prime })\rightarrow (BS, 0)\: \rangle\) for w_j to charge the sensors in SC_j by inserting \((sl_j^{k}, td_j^k)\) to the partially constructed visiting sequence \(SL_j^{visit}\). \(sl_j^{k}\) is the k^th stop location in \(SL_j^{visit}\) and \(td_j^k\) is the charging time of w_j at \(sl_j^{k}\). The arrival time of w_j at stop location \(sl_j^{k}\) is calculated as follows.For any stop location \(sl_j^k\), if the deadline TD_i of any sensor \(s_i\in NS_j^k\) is less than \(at_j^k\), some sensors cannot get replenished before they run out of energy. We refer to such a location as dropped stop location. To rescue a dropped stop location \(sl_j^k\), we try to rearrange the visiting sequence of stop locations in \(SL_j^{visit}\) by inserting it to a preceding position. Then, we find the proper position to insert \(sl_j^k\) by scanning every stop location from \(sl_j^{1^\prime }\) to \(sl_j^{{k-1}^{\prime }}\) (Lines 10 – 18). We will insert \(sl_j^k\) after location \(sl_j^{f^{\prime }}\), if two conditions are satisfied (i) After charging sensors at \(sl_j^{f^{\prime }}\), the sensors at \(sl_j^k\) are still alive and (ii) Placing \(sl_j^k\) after \(sl_j^{f^{\prime }}\) will not cause other sensors to die. Otherwise, the death of some sensors in \(NS_j^k\) can not be avoided by only rearranging the visiting sequence of stop locations. In that case, we use a partial charging scheme at some stop locations in \(SL_j^{visit}\) by charging the sensors partially to try to save the dropped location \(sl_j^k\) as follows (Lines 19 – 38).

When a stop location \(sl_j^k\) is not visited by w_j before the deadline TD_i of each sensor s_i in \(NS_j^k\), it can not be added to \(SL_j^{visit}\). Then, we try to insert \(sl_j^k\) in \(SL_j^{visit}\) by shortening the charging time of the stop locations in the constructed \(SL_j^{visit}\) (Lines 21 – 34). A stop location \(sl_j^{l^\prime }\) having the longest charging time \(td_j^{l^\prime }\) is selected from \(SL_j^{visit}\) (Line 22). Then, the saved time \(svt_j^{l^\prime }\) at \(sl_j^{l^\prime }\) by shortening the charging time by replenishing δ amount of energy to each \(s_i \in NS_j^{l^\prime }\) is calculated as follows (Lines 23 – 24).where ED_i is the energy demand of s_i, \(P_{r}(s_i, sl_j^{l^\prime })\) is the charge receiving rate of s_i from w_j which is at \(sl_j^{l^\prime }\), and δ is the amount of charging energy unit. Note that when a sensor s_i is partially charged with an amount of δ, the deadline TD_i and energy demand ED_i of s_i is updated to \(TD_i = TD_i + \frac{\delta }{\rho _i}\) and ED_i = ED_i − δ (Lines 25 – 26).

Lets assume that w_j arrives at a dropped location \(sl_j^k\) at time \(at_j^k\). If (\(at_j^k - svt_j^{l^\prime }\)) < TD_i, ∀s_i ∈ \(NS_j^k\), w_j can timely charge all the sensors in \(NS_j^k\) by only reducing the charging time at \(sl_j^{l^\prime }\). Therefore, we cut \(svt_j^{l^\prime }\) at \(sl_j^{l^\prime }\) and insert \(sl_j^k\) at the end of \(SL_j^{visit}\) (Lines 28 – 34). If not, we continuously shorten the charging time at more stop locations in the decreasing order of their charging time and store the total saved time in tsvt_j (Line 27). If after shortening the charging time at all stop locations in \(SL_j^{visit}\) and tsvt_j is not enough for w_j to reach timely at \(sl_j^k\), the location \(sl_j^k\) is inserted into \(SL_{j}^{drop}\) which stores the dropped stop locations (Lines 36).

After processing all the stop locations in SL_j, if \(SL_{j}^{drop}\) is not empty, w_j visits all the stop locations in \(SL_{j}^{drop}\) in the same order as they are inserted in \(SL_{j}^{drop}\) to reduce the dead duration of the sensors (Lines 41 – 52). For each \(sl_j^k\in SL_{j}^{drop}\), the algorithm computes the dead sensors and stores them in DS_j. It also stores the dead duration of a dead sensor s_t in DD_t (Lines 44 – 49). For every \(sl_j^k\), the charging time \(td_j^k\) is also computed and then \(sl_j^k\) is inserted in \(SL_{j}^{visit}\) (Lines 50 – 51). Finally, Algorithm 3 returns the visiting sequence of stop locations along with the charging time, the set of dead sensors, and the dead duration of each sensor, which are assigned to w_j (Line 53). The pseudo code of the algorithm is given in Algorithm 3

Let sl_j(pc) are the set of stop locations, where partial charging is used by w_j and \(sl_j^i(pc)\) is the i-th stop location, where charging time is shortened for w_j or multiple sensors are charged simultaneously and partially. Let \(svt_j^i\) is the time saved for w_j at \(sl_j^i(pc)\). \(Tsvt_j^q\) is the total saved time for w_j at all q stop locations by Algorithm 3 . Let \(Tsvt^q_jopt\) be the total saved time for w_j at all q stop locations using an optimal algorithm. We assume that there are Z partial charging stop locations in \(SL_{j}^{visit}\). We use mathematical induction on Z to prove Theorem 2.

Base Case: Z = 1. There is one partial charging stop location in \(SL_{j}^{visit}\). Let Algorithm 3 does a partial charging at a stop location \(sl_j^1(pc)\). We assume that Algorithm 3 is not optimal. Let there exists an optimal algorithm OPT which chooses another stop location \(sl_j^1opt(pc)\) for partial charging and the saved time \(Tsvt_j^1opt\) at \(sl_j^1opt(pc)\) is greater than \(Tsvt^1_j\). However, \(sl_j^1(pc)\) is the stop location having the longest charging time in \(SL_{j}^{visit}\). Therefore, \(Tsvt_j^1opt\) must be shorter than \(Tsvt^1_j\). Since the hypothesis is that OPT is optimal and \(Tsvt_j^1opt \gt Tsvt^1_j\) is a contradiction. Hence there is no such OPT exists. Therefore, Algorithm 3 is the optimal algorithm.

Induction Step: The inductive hypothesis is as follows. Algorithm 3 is optimal for Z = q − 1.

For Z = q, let Algorithm 3 is not optimal, and OPT is the optimal algorithm. Thus, \(Tsvt_j^qopt \gt Tsvt^q_j\). \(Tsvt^q_j = Tsvt^{q-1}_j + svt_j^q\), where \(svt_j^q\) is the saved time at q-th stop location in Algorithm 3 for w_j. Since, based on the hypothesis, Algorithm 3 is optimal when Z = q − 1, the total saved time using Algorithm 3 by doing partial charging at q − 1 stop locations is maximum. Since OPT is optimal, the saved time \(svt_j^qopt\) must be greater than the saved time \(svt_j^q\) using Algorithm 3 at q-th stop location. However, Algorithm 3 always chooses a stop location having the longest charging time for partial charging. Then it can be concluded that there exists no such \(sl_j^qopt(pc)\) and \(Tsvt_j^qopt\) can not be larger than \(Tsvt^q_j\). Hence, no such OPT exists, therefore, Algorithm 3 is the optimal algorithm. □

In this section, a WRSN with 100 to 1500 rechargeable sensors, distributed randomly in an area of 500m × 500m, has been considered. The base station is positioned at the center of the area. A sensor s_i has a battery capacity (b) of 10.8 kJ [18]. The residual energy RE_i of s_i is randomly chosen between 10% – 90% of its battery capacity. Energy consumption ρ_i of s_i varies between 0.05 – 0.5J/s [19]. In the simulation, a sensor s_i sends the charging request to the BS once its RL_i reaches L_thrld = 5 hours.

The number of WCVs h varies from 1 to 5. The speed of a WCV is v = 5 m/s and it consumes 50 J/m in its travel [18]. The charging range γ of each WCV is 10 m. The value of δ is fixed at \(\frac{b}{4}\), the parameters α and β are fixed at 90 and 10, respectively [19]. The charging rate of a WCV is fixed to 15W [19].

The effect of important parameters such as the number of sensors (n), charging range (γ), and charging energy unit (δ) on the number of dead sensors (DS), average dead duration (DD_avg), maximum dead duration (DD_max) and average travel distance for different number of WCVs have been analyzed. The performance of the proposed algorithm MinDS is compared with three benchmark algorithms. (i) Algorithm SNFC [7] uses a single-node full charging scheme, where a sensor is charged to its full capacity at a time. It determines the charging tour of to-be-charged sensors based on the distance between the to-be-charged sensors and the WCV. (ii) Algorithm MNFC [20] uses a multi-node full charging scheme, where multiple sensors are charged simultaneously to their full capacity at a time. In MNFC, the locations of the sensors are considered as the stop locations. The stop locations are determined based on the remaining energy of the sensors within the charging range from the stop location and the total amount of energy required by the WCV to charge the sensors. Then the Christofides algorithm [2] is used to determine the sequence to visit the stop locations. (iii) Algorithm MNPC [19] uses a multi-node partial charging scheme, where multiple sensors are charged simultaneously. It reduces the number of dead sensors by charging some sensors partially, which requires a long charging time. All results reported in the figures are the average of 100 runs, where in each run, an identical number of sensors has a different random distribution in the WRSN.

A. Effect of the Number of Sensors: We study the effect of the number of sensors on the number of dead sensors (DS) (Fig. 2(a)), average dead duration (DD_avg) (Fig. 2(b)), maximum dead duration (DD_max) (Fig. 2(c)), and the average travel distance (Fig. 2(d)) by increasing the number of sensors from 100 to 500 by deploying a single WCV.

Fig. 2(a) shows that the proposed algorithm MinDS, MNFC, and MNPC gives considerably smaller DS than SNFC. This shows that multi-node charging is quite effective for extending the network’s lifetime. Moreover, MinDS and MNPC performs better than MNFC. This is due to the fact that partial charging of some sensors allows the WCV to visit more sensors and charge them before they run out of their energy.

For a sparse distribution of sensors in WRSN, MNPC performs better than MinDS. As shown in Fig. 2(a), when the number of sensors in the network is small (< 300), MNPC produces less number of DS than MinDS. In MNPC, the WCV moves to the next stop location when it completes the charging of the nearest sensor from the current stop location, while in MinDS, the WCV moves to the next stop location when it completes the charging of the farthest sensor from the current stop location. Therefore, when the number of sensors is small, some sensors may be far away from a stop location. Thus, the WCV stays longer at the stop location in case of MinDS than MNPC, while charging almost the same number of sensors at each location. Due to this, the WCV cannot reach more stop locations in time, leading to more number of DS than MNPC.

However, for a dense distribution of sensors in WRSN, MinDS performs better than MNPC. For example, when the number of sensors in the network is large (≥ 300), MinDS produces less number of DS than MNPC (Fig. 2(a)). If the number of sensors is large with a dense distribution, the average distance between two sensors is short. As a result, the WCV in MinDS stays longer at each stop location than MNPC while charging a higher number of sensors. Therefore, MinDS performs better than MNPC in a dense network in terms of the number of DS.

Fig. 2(b) shows that DD_avg of a sensor by all the algorithms increases as the number of sensors in the network increases. As the number of sensors increases, the number of to-be-charged sensors also increases, and more sensors may not get recharged in time. It is also observed that DD_avg of a sensor produced by algorithm MinDS is shorter than that of SNFC, MNFC, and MNPC. The number of stop locations produced by MinDS is significantly less than that produced by the other algorithms. Due to this, the WCV requires less time to visit and charge all the dead sensors at the stop locations. For example, when there are n = 500 sensors, DD_avg by MinDS is 132 minutes, while DD_avg by SNFC, MNFC, and MNPC are 1816, 1218, and 1189 minutes, respectively, which are significantly longer than by the proposed algorithm MinDS.

Fig. 2(c) illustrates that the DD_max of a sensor produced by algorithm MinDS is shorter than that of SNFC, MNFC, and MNPC for the same reason stated before. For example, when there are n = 500 sensors, DD_max by MinDS is 869 minutes, while DD_max by SNFC, MNFC, and MNPC are 2975, 1834, and 1775 minutes, respectively, which are significantly longer than by the proposed algorithm MinDS. Fig. 2(d) shows that the average travel distance of WCV in MinDS is shorter than that of the other three algorithms as the WCV visits a small number of stop locations to charge all the to-be-charged sensors, which helps the WCV to reach more stop locations before the deadline of the sensors in their charging range.

From Fig. 2(a), it can be observed that when the number of sensors is large, a single WCV cannot charge a large number of sensors in time. Also, with a single WCV, the DD_avg and the DD_max of a sensor are high. Therefore, multiple WCVs are required to charge the sensors in a WRSN.

Fig. 3(a) - 3(d) shows that for a given number of sensors, the DS, DD_avg, DD_max, and the average travel distance decreases significantly when the number of WCVs increases from 1 to 5. As the number of WCVs increases, the number of to-be-charged sensors assigned to a WCV decreases. As a result, it can charge a higher number of sensors in time by travelling a shorter distance, thereby, DS, DD_avg, DD_max, and the average travel distance decrease.

In Fig. 3(a), for a fixed number of WCVs, the DS increases with the increase of the number of sensors in WRSN. The reason is as follows. With the increase in the number of sensors, the number of to-be-charged sensors assigned to a WCV is also increasing. Due to this, a WCV cannot charge more sensors in time. Thus, the DS increases.

B. Effect of the charging Range (γ) of WCVs: In this section, we investigate the effect of charging range (γ) on the DS (Fig. 4(a)), the DD_avg (Fig. 4(b)), the DD_max (Fig. 4(c)), and the average travel distance (Fig. 4(d)) by varying γ from 5 to 15, when the number of sensors is fixed to 1500.

In Fig. 4(a), it is seen that the DS increases as the γ increases. When γ is small, say 5 meters, the number of sensors in the neighbouring set of a stop location is small, and they are close to the stop location. So, the WCVs stay for a shorter duration at each stop location to charge them. Due to this, the WCVs can visit more stop locations in time and charge more sensors. However, when γ is large, say 15 meters, the number of sensors in the neighbouring set of a stop location is large, and some of the sensors are far away from the stop location. Therefore, the WCVs stay for a longer duration at each stop location to charge them. As a result, a higher number of sensors die.

In Fig. 4(b) - 4(d), it is observed that DD_avg, DD_max, and the average travel distance decreases as γ increases. A large value of γ gives fewer stop locations that cover almost all the to-be-charged sensors. Consequently, the WCVs travel a shorter distance while reaching all the stop locations earlier.

C. Effect of the charging energy unit (δ): We study the effect of δ for different algorithms on the DS, and the DD_avg by deploying a single WCV. The number of sensors and the charging range are fixed at 500 and 10, respectively. The value of δ decreases from b to b/7, where b is the battery capacity of a sensor. When δ = b, the full charging scheme is adopted by the algorithms MinDS and MNPC, whereas when the value of δ varies from b/2 to b/7, the partial charging scheme is adopted.

In Fig. 5(a) - 5(b), it is seen that when δ = b, the MinDS produces a higher number of DS which indicates that MinDS does not work well when the battery of the sensors is charged to full capacity. However, with δ = b, the DD_avg is smaller than the other algorithms as the MinDS visits a few stop locations to charge the sensors. In Fig. 5(a) - Fig. 5(b), it is observed that the DS and DD_avg for MinDS decreases when δ decreases from b to b/4. This indicates that the MinDS uses the partial charging scheme to charge the sensors, and the WCV stays for a shorter duration at each stop location, which helps to reduce the DS and DD_avg of a sensor. The DS and DD_avg do not change for the algorithms SNFC and MNFC with the decrease of δ as they use the full charging scheme. When δ varies from b/4 to b/7, the value of DS and DD_avg almost remains unchanged for MinDS, which indicates that further decrease of δ will not affect the performance of MinDS.

In summary, a dense distribution of sensors allows a WCV to charge more sensors before their energy depletes, resulting in a smaller number of dead sensors and a shorter average dead duration for our proposed MinDS. Simulation results show that multiple WCVs help in reducing the number of dead sensors and also the average dead duration when a large number of to-be-charged sensors are densely distributed in a WRSN when γ = 10 meters and δ is fixed at one-fourth of its total battery capacity. In a dense large-scale WRSN, our proposed scheme produces a considerably lower number of dead sensors and smaller average dead duration using multiple WCVs, where a WCV can partially charge multiple sensors in its range simultaneously.

In this paper, a charge scheduling problem using multiple WCVs is formulated to reduce the number of dead sensors and their dead duration, where a WCV can partially charge multiple sensors in its range simultaneously. In the paper, an approximation algorithm with a provable approximation bound has been proposed, which divides the charging requests among the WCVs such that their charging loads are evenly distributed. Next, with the objective of charging the maximum number of sensors before they die, the charging locations for each WCV are determined based on the deadline of the sensors. Finally, a multi-node partial charging scheme is adopted to determine the visiting sequence of the charging locations for each WCV while calculating the appropriate charging time at each location such that the maximum number of charging requests can be served in time. It has also been proven that our scheme uses partial charging at the minimum number of stop locations for a WCV. The simulation results show that the proposed scheme outperforms the existing algorithms in producing a lower number of dead sensors with a shorter average dead duration and a shorter maximum dead duration of a sensor, making our scheme feasible for a large-scale WRSN.