An energy-efficient routing algorithm for dual-energy harvesting-assisted wireless sensor networks based on whale optimization strategy

Abstract

Energy harvesting (EH) wireless sensor networks (WSNs) have wide applications in various fields due to their ability to sense and transmit environmental information, while current routing algorithms on EH-WSN generally rely on a single EH method, such as solar or wind. Due to the uncertainty of EH efficiency, a single-type EH technique may not be able to meet the energy demand of WSNs. In light of this, we propose an energy-efficient dual EH routing algorithm based on whale optimization (WO) strategy (DEHRA-WO). In the modeling process, we firstly propose a dual EH (including solar EH and radio frequency charging (RFC) techniques) switching mechanism. Namely, if the collected solar energy is not enough to maintain the operation of WSNs, RFC is employed to harvest additional energy, in which the collected solar energy is predicted using Kalman filter theory. With this, a residual energy model is constructed to measure the energy status of nodes. Secondly, we provide a communication range model to confirm the transmission area of nodes, where the wireless fading environment is considered. After that, a data link layer model is established to reflect the node blocking state. Based on the above models, we define the evaluation function to indicate the possibility of a node to be selected as the next hop and then propose the overall path selection scheme using WO strategy. Moreover, we also prove the convergence of DEHRA-WO and analyze its complexity. Through extensive simulation experiments, we demonstrate the superior performance of DEHRA-WO in terms of packet loss rate and energy utilization.

Appendices

Appendix 1: DEHRA-WO convergence analysis

Since the convergence of our proposed algorithm is determined by WO strategy, we only need to prove the convergence of WO strategy. In Eq. 27, M decreases linearly throughout the execution of the WO strategy, making $M<1$ constant in the later stages of strategy execution. Consequently, the lead whale does not make the other whales to randomly search prey in later iterations. Thus, the convergence of the WO strategy is determined by the spiral way and the straight line way. Next, we prove the convergence of the spiral way.

Proof

Assume U is the globally optimal solution, and that the ith whale $W_t^i$ in the t-th iteration belongs to the set of solutions. Then, its probability $P(W_t^i \in U)$ can be expressed as:

$$\begin{aligned} \begin{aligned} P(W_t^i \in U)&= P(W_t^i \in U \mid W_{t-1}^i \in U )p(W_{t-1}^i \in U ) \\&\quad + p(W_t^i \in U \mid W_{t-1}^i \notin U)p(W_{t-1}^i \notin U ) \end{aligned} \end{aligned}$$

(37)

As the sequence of optimal solution sets in the WO strategy forms an absorbing Markov chain [60], we can deduce that $P(W_t^i \in U \mid W_{t-1}^i \in U )=1$. We make the assumption:

$$\begin{aligned} \varphi ^{i}_{t-1}=p(W_t^i \in U \mid W_{t-1}^i \notin U) \end{aligned}$$

(38)

therefore, we have:

$$\begin{aligned} \begin{aligned} 1 - P(W_t^i \in U)&= 1 - P(W_{t-1}^i \in U) -\varphi ^{i}_{t-1}P(W^{i}_{t-1}\notin U) \\&=(1-\varphi ^{i}_{t-1})(1-P(W^{i}_{t-1}\in U)) \end{aligned} \end{aligned}$$

(39)

because:

$$\begin{aligned} \begin{aligned} 1 - P(W^{i}_{t-1}\in U)&= (1-\varphi ^{i}_{t-2})(1-P(W^{i}_{t-2}\in U)) \\ 1 - P(W^{i}_{t-2}\in U)&= (1-\varphi ^{i}_{t-3})(1-P(W^{i}_{t-3}\in U)) \\ \cdots \cdots \\ 1 - P(W^{i}_{t}\in U)&= (1-\varphi ^{i}_{t-1})\cdots (1 - P(W^{i}_{0}\in U)) \end{aligned} \end{aligned}$$

(40)

due to $(1-\varphi ^{i}_{t-1}) \in (0,1)$, we take the limit of the above equation:

$$\begin{aligned} \begin{aligned}&\lim _{t \rightarrow \infty } 1-p(W^{i}_{t} \in U) = 0 \\&\quad \Longrightarrow \lim _{t \rightarrow \infty } p(W^{i}_{t} \in U) = 1 \end{aligned} \end{aligned}$$

(41)

Therefore, the spiral way can converge to the optimal solution. The proof procedure for straight lines way is similar to the above and not be demonstrated here. $\square$

Appendix 2: DEHRA-WO complexity analysis

Initially, the computation of the distance between nodes during initialization has a computation complexity of $O(n^2)$. Subsequently, the process of selecting the next jump node using WO strategy goes through m iterations with a computation complexity of O(mn). Finally, each node has to update its information, which has a computation complexity of O(n). Therefore, the overall computation complexity is

$$\begin{aligned} O(n^2) + O(mn) + O(n) = O(n^2) \end{aligned}$$

(42)