## 1 Introduction  
Over the past few years, Wireless Sensor Networks (WSNs) have played significant role in areas such as health monitoring, military operations, environmental monitoring, and autonomous vehicular network and drone-related technologies due to their extensive capability of applications. The WSN consists of a set of multiple sensor nodes that continuously monitor and collect data, transmitting it from their surroundings to a central base station (BS) through other intermediate nodes. A key advantage of WSNs is their capability to function in remote or inaccessible environments, making them essential for applications such as disaster management and safety surveillance. WSNs face various challenges, such as energy consumption, sensor node deployment, routing algorithms, energy efficiency, cluster-head (CH) selection, and robustness. Sensor nodes in the WSN [1, 2] are often deployed in large quantities, particularly in locations that are difficult for humans to access. Consequently, it is crucial to use the limited energy of these sensor nodes efficiently since it is typically impractical to replace or recharge their batteries. Therefore, one of the primary objectives in WSN design is to minimize energy consumption at each node to enhance network energy efficiency. If a wireless sensor node exhausts its energy, it becomes inoperable; moreover, if a certain threshold (e.g., 50% or 80%) of nodes in the network exhaust their energy, the network itself may become nonfunctional. Moreover, WSNs face several challenges, including energy consumption, sensor node deployment and its CH selection, and their routing algorithms. To address these challenges, researchers have developed various routing protocols and proposed optimization techniques to establish the most efficient path between transmitting and receiving nodes. Particularly, to reduce node energy consumption and extend network lifetime, research work in [3; 4; 5] have focused on addressing these challenging issues. Among these previously studied protocols, the Low Energy Adaptive Clustering Hierarchy (LEACH) is one of the most widely adopted clustering-based algorithms, which adopts a hierarchical method and aims at enhancing energy efficiency [6]. In [7], a comprehensive survey presents a comparative analysis of various protocols from LEACH, highlighting that LEACH had been used as the prototypical model of a clustering algorithm. LEACH is widely adopted across various applications due to its cluster-based structure, which not only reduces the power consumption but also optimizes energy and data transmission efficiency. LEACH organizes nodes into clusters and employs a probabilistic threshold to elect a CH within each cluster. Although LEACH is somewhat effective, it does not always ensure optimal cluster configurations. Actual clustering in the LEACH algorithm can sometimes result in clusters with too few or too many nodes due to its probabilistic selection of cluster heads, leading to inefficiencies and data transfer failures, such as the first node dead (FND). Additionally, LEACH does not consider the spatial distribution or residual energy levels of nodes when selecting cluster heads, which can further exacerbate energy consumption and reduce network lifetime. To address the challenging issue of the LEACH algorithm, various LEACH-based protocols have been proposed to address these limitations by improving CH selection criteria, considering factors like residual energy, mobility, energy efficiency, centralized formation, and balanced cluster formation [7]. Among these protocols, energy-aware versions of the LEACH protocol (LEACH-B in [8], I-LEACH in [9]) adjust the CH election probability based on the residual energy of nodes to reduce the likelihood of nodes with low energy becoming CHs. Specifically, the energy-aware LEACH protocol modifies the CH election threshold by factoring in the residual energy so that nodes with lower remaining energy have a reduced probability of being selected as CHs.