## 4 Improved GWO-Based Load-Balanced Clustering
### Illustration of the Search for Prey (Exploration)  
This section presents the exploration of solution space in GWO. Gray wolves generally search to follow the **Alpha**, **Beta** and **Delta** wolves. The solution may exist in the global search space. Therefore, A's value is used to diverge from the solution if the value of A is greater than 1 or less than \(-1\). Due to this, it can explore the global search space. In this phase, the less loaded CH is selected to update; otherwise, it is discarded. It diverges from prey to hopefully find the fittest prey, as shown in Figure 6, considering that a better solution may exist in the global search space. Therefore, this phase tends to go for it by avoiding local optima. Figure 5: Exploitation. Figure 6: Exploration. **Remark 4:**_The convergence of the proposed exploration approach is faster compared to the traditional GWO exploration phase. In the traditional GWO exploration phase, the convergence becomes sluggish due to random selection. Higher loaded CH from the global search space also has the chance of replacement with less loaded in the solution vector. This may result in divergence. Therefore, the proposed approach for exploration will always converge toward the optimum value of the fitness function and result in faster convergence._  
**Lemma 3:**_The new solution vector produced in the above exploration process is valid._  
**Proof:** At the time of exploration, a less loaded CH is selected from global search space to update; otherwise, it is discarded in this phase. Therefore, it always replaces a less loaded CH from the global search space with a maximum loaded CH in the solution vector. Hence, the exploration process does not hamper the validity of the solution vector. \(\square\)  
Primary considerations of GWO for energy-efficient load-balanced clustering are described as under the following:  
* The social hierarchy of _Alpha_, _Beta_, _Delta_ and _Omega_ wolves is proposed for load-balanced clustering to determine the optimal solution. * The proposed encircle behavior minimizes a load of maximum loaded CH. It is considered that the best search agents _Alpha_, _Beta_, and _Delta_ maintain the knowledge of the desired solution. * The proposed hunting mechanism of the load-balanced clustering algorithm reinforces the best search agents to find the best solution. * The values of \(a\) and \(A\) help with exploring the solution in the global search space and prevent stagnation from local optima. * These values of \(a\) and \(A\) smoothly balance the load of CHs by transitioning in between exploration (searching) and exploitation (attacking). * Therefore, this algorithm has crucial parameter settings that need to be adjusted. **5. Test System, Assumptions and Result Analysis**  
In a sustainable cities environment, the proposed algorithm is demonstrated for a large network that consists of 500 sensors and 50 CHs deployed in an area of 500 \(\times\) 500 m\({}^{2}\). The various parameters for performance analysis of the WSN parameters are depicted in Table 1. The simulation is performed in the MATLAB environment, and the results are obtained using IGWO, traditional GWO, GALBCA and HSCA. The GWO-based approaches are considered as traditional GWO and improved-GWO (IGWO) based load balanced clustering. Furthermore, the performance of IGWO and traditional GWO is evaluated. The results obtained using the proposed approach are compared with several other recently developed \begin{table}
\begin{tabular}{c l c} \hline \hline
**S.No.** & **Parameters** & **Values** \\ \hline
1 & Round & 2000 \\
2 & Sink location & 500 \(\times\) 250 \\
3 & \(d_{0}\) & 60 \\
4 & CH communication range & 130 m \\
5 & \(d_{max}\) & 65 \\
6 & Sensor initial energy & 2.0 J \\
7 & CH initial energy & 12 J \\
8 & Packet size & 4000 bits \\
9 & Message Size & 500 bits \\
10 & Area (m\({}^{2}\)) & 500 \(\times\) 500 \\
11 & Sensor & 500 \\
12 & CH & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of the parameters used.