## 1 Introduction
Computational intelligence has the ability to deal with imprecise information and find approximate yet good-enough solutions to these problems [27]. Computational intelligence methods are often inspired by biological processes. As one of the technologies in computational intelligence, evolutionary computation [28] imitates the process of natural evolution to provide a near-optimal solution for an  
Figure 1: Simplified diagram of sensor node clustering: (**a**) grid-based clustering method, (**b**) low-energy adaptive clustering hierarchy (LEACH)-based clustering method, and (**c**) adaptive sensor node clustering method for determining the number of clusters and grouping the number of sensor nodes into clusters evenly. optimization problem. The advantages of evolutionary computation are self-organization, self-learning, and parallel search for the global optimal solution. Among many evolutionary algorithms, the particle swarm optimization algorithm (PSO) can find the optimal solution to a problem at a higher velocity by considering previous global and local best experiences of the entire population. This study proposes an energy-balanced cluster-routing protocol (EBCRP) using PSO with five mutation operators for WSN that groups sensor nodes into clusters evenly. In the step of EBCRP, the number of clusters and their centroids is optimized (Figure 1c), in which some sensor nodes are selected as the centroids of clusters and other sensor nodes are assigned to the nearest centroid. The average distance between sensor nodes belonging to one cluster and the centroids of this cluster, as well as balancing the number of sensor nodes between clusters are used as optimization objectives. Five mutation operators are specially designed to improve the performance of PSO in optimizing the clustering of sensor nodes. According to the clustering results of sensor nodes, in each round, the sensor node with the highest residual energy in each cluster has the priority to be selected as a CH to aggregate data from other sensor nodes in the same cluster. Then, these CHs are responsible for sending aggregated data to the sink. The proposed schemes for balanced clustering scheme and rotation CH selection based on the highest residual energy are helpful in balancing the energy consumption of sensor nodes and thus prolonging the network lifetime. The main contributions of this work are the following:  
* An energy-balanced cluster-routing protocol for WSNs is proposed to balance the energy consumption of sensor nodes and prolong the network lifetime. * An adaptive sensor node clustering scheme based on PSO is proposed to determine the number of clusters and group sensor nodes into clusters evenly. * Five mutation operators are specially proposed to improve the performance of PSO in optimizing the clustering of sensor nodes. The rest of the paper is organized as follows. Section 2 introduces related works on clustering sensor nodes and prolonging the network lifetime. The considered network model is described in Section 3. Section 4 outlines the proposed EBCRP method. Simulations and their results are represented in Section 5. Section 6 concludes the study and suggests directions for future work.