## 2 Related Work
### System Modelling
Further, we need to find the weight of RN (\(Weight_{RN}^{CL}\)) considering the particular candidate location (CL) for optimal placement, as represented by Equation (1). \[Weight_{RN}^{CL}=\frac{Energy_{harvest}^{max}-Energy^{CL}}{Energy_{harvest}^{ max}}+1 \tag{1}\]  
Here, \(CL\in RN_{CL}\), \(0\leq Energy^{CL}\leq Energy_{harvest}^{max}\), \(Weight_{RN}^{CL}\in[1,2]\), and \(Energy^{CL}\) represent the energy harvested at the selected candidate location (CL)  
The candidate locations having a higher potential for harvesting the energy should have lower weights and these candidate locations are selected for optimal placement of the RNs. Further, the collector node/sink node or base station is designated for accessing the complete network information through a centralized mechanism, such as the node's position, executing the algorithm for finding the optimal locations of the relay nodes, etc. The deployed EH-WSNs become disjoint after a certain period, and this situation can be announced after analyzing the aggregate data collected at the BS. \(Range_{SN}\), \(Range_{RN}\) are the sensing range of the SN and RN, respectively, with the condition of \(Range_{RN}=2*\)\(Range_{SN}\). Let \(SN_{set}\) denote the set of SNs that are deployed in a two-dimensional area and \(SN_{set}=\{SN_{1},SN_{2},\ldots,\ldots,SN_{c}\}\). Similarly, \(RN_{set}\) represents the set of relay nodes and \(RN_{set}=\{RN_{1},RN_{2},\ldots,RN_{d}\}\) that are required for maintaining effective connectivity in the EH-WSN, with \(c>d\). The Euclidean distance (\(euclidean\_distance\)) between the two sensor nodes (\(SN_{i},\ SN_{j}\)) can be calculated by using the following equation:  
\[euclidean\_distance_{(SN_{i},\ SN_{j})}=\sqrt{\left(SN_{i}-SN_{j}\right)^{2}+ \left(SN_{i}-SN_{j}\right)^{2}} \tag{2}\]  
Furthermore, it is considered that the sensor nodes \(SN_{i}\)_and_\(SN_{j}\) are connected if  
\[euclidean\_distance_{(SN_{i},\ SN_{j})}\leq Range_{SN} \tag{3}\]  
\(Range_{SN}\) represents the range of sensor node. Similarly, it is considered that sensor node \(SN_{i}\) is connected to relay node \(RN_{j}\) and \(d<c\), if  
\[euclidean\_distance_{(SN_{i},\ SN_{j})}\leq Range_{RN} \tag{4}\]  
\(Range_{RN}\  
with the BS through intermediate SNs, due to the dead conditions, then the RNs come to the picture, which are deployed by adopting a control random approach with a non-uniform nature to maintain the proper flow of data by maintaining the optimal connectivity. In the first phase of the proposed scheme, we have used two metaheuristic optimization algorithms, namely, the grey wolf optimizer and sine cosine optimizer, for designing the hybrid framework, named the hybrid adapted grey wolf sine cosine optimizer (HA-GWSCO), for exploring the maximum node-disjoint routes from all SNs to BS in the EH-WSN; this hybrid framework also discovers \(k\)-node-disjoint routes from each SN to BS, with \(k\geq 2\), which enhances the fault-tolerance capability in EH-WSN. This hybrid framework utilizes the fitness function as described by Equation (6) below. In EH-WSN, the maximum approximate number of node-disjoint routes \(max\{A(|R_{SN_{net}}|)\}\) that exists between all the SNs \((|SN_{set}|)\) and BS utilizing the least number of RNs can be determined by Equation (6). \[max\{A(|R_{SN_{net}}|)\}\geq min\left\{A(RN_{(BS)}),\sum_{i=1}^{|SN_{set}|}A \left(RN_{(SN_{i})}\right)\right\} \tag{6}\]  
subject to  
\[\left\{A\left(RN_{(SN_{i})}\right)\geq k\,if\,all\,the\,SN_{s}\,are\,having\, same\,number\,of\,\,\,k-node\,disjoint\, routes\right. \tag{7}\]  
Equation (6) is assumed as the fitness function for exploring all possible maximum approximate number of node-disjoint routes \(max\{A(|R_{SN_{set}}|)\}\) that exists between all the SNs \((|SN_{set}|)\) and BS utilizing the least number of RNs. Now, we will illustrate the role of the adapted GWO and adapted SCO metaheuristic optimization algorithms.