## 4 Multi-Agent Particle Swarm Optimization (MAPSO) Approach
### Mapso  
MAPSO is a new algorithm combining PSO and MAS [63; 64]. First of all, determining the structure suitable for MAS so as to confirm the domain of an agent is carried out. Each agent not only competes and cooperates with its neighbors, but also takes advantage of the evolution mechanisms of PSO, which speeds up the information transfer process [65; 66]. Thus, MAPSO converges faster and more accurately to global optimal solution. 1. Meaning and purpose of each agent  
In MAPSO, each agent \(\alpha\) is equivalent to a particle in PSO. The particle has an optimized objective function value which is called the fitness value. The fitness function with the best EE and maximum EUE as the optimization goal is represented by Equations (13) and (14). In other words, the fitness values of agent \(\alpha_{1}\) and agent \(\alpha_{2}\) are expressed as Equations (13) and (14). \[f(\alpha_{1}\ )=C \tag{13}\]  
\[f(\alpha_{2}\ )=E \tag{14}\]  
The purpose of agent \(\alpha_{1}\) is to minimize its fitness value as long as the constraints are met. On the contrary, the goal of agent \(\alpha_{2}\) is to maximize the fitness value. 1. Global environment of MAPSO  
For the definition of global environment, this paper adopts a simple lattice-like structure, as shown in Figure 2. In Figure 2, each circle represents an agent which is the same as a particle in PSO. The coordinates \((a,b)\) in this circle represent the specific location of an agent in the global environment, and the coordinates have a one-to-one correspondence with each agent. In addition, every agent consists of two pieces of data, namely the speed and location of the particle in PSO. Furthermore, \(a\times b\) is the total grid number of this global environment, and the relationship between \(a,b\) and the number of particles for PSO \(z\) is \(z=a\times b\). (3) Local environment of MAPSO  
In order to realize communication among particles and obtain the next behavioral strategy, the local environment needs to be defined. Then, each agent will communicate and interact with its neighboring particles in the predefined local environment. Assume \(A_{m,n}\) represents the agent with position \((m,n)\), and \(m=1,2,3,\ldots,a;n=1,2,3,\ldots,b\), \((a,b\) are the global environment boundaries of MAPSO). The local environment \(N_{m,n}\) of \(A_{m,n}\) is defined as Equation (15). \[N_{m,n}\ =\ \{A_{m,n1},A_{m,n2},A_{m1,n},A_{m1,n1},A_{m1,n2},\ A_{m2,n},A_{m2,n1},\ A_{m2,n2}\} \tag{15}\]  
where \(m_{1}\ =\ \ \left\{ \begin{array}{c}m-1,\ m\
eq 1\\ a,\ m=1\end{array} \right.\); \(n_{1}\ =\ \ \left\{ \begin{array}{c}n-1,\ n\
eq 1\\ b,\ n=1\end{array} \right.\); \(m_{2}\ =\ \ \left\{ \begin{array}{c}m+1,\ m\
eq a\\ 1,\ m=a\end{array} \right.\); \(n_{2}\ =\ \left\{ \begin{array}{c}n+1,\ n\
eq b\\ 1,\ n=b\end{array} \right.\). According to the above definition, as shown in Figure 2, there are several local environments in the global environment. The shade of Figure 2 is the local environment of an agent with coordinates (2,2) and the eight particles around it are the neighbors used to obtain environmental information. (4) Behavioral strategy of each agent  
The agent competes and cooperates with its neighbors. Take the EE of ESC as an example, assuming the position of agent \(A_{m,n}\) in the solution space is \(A_{m,n}=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\) and \(M_{m,n}\) is the agent with the minimum fitness value among eight neighbors of \(A_{m,n}\). If \(A_{m,n}\) satisfies Equation (16), the position of agent \(A_{m,n}\) remains unchanged, otherwise changing according to Equation (17) [48]. \[f(A_{m,n}\ )\leq f(M_{m,n}) \tag{16}\]  
\[A^{\prime}_{m,n}\ =M_{m,n}+rand(-1,1)\cdot(A_{m,n}-M_{m,n}) \tag{17}\]  
where \(rand(-1,1)\) is a random number in the range [\(-1\),\(1\)]. As can be seen from Equation (16), \(A_{m,n}\) not only retains its original useful information, but also fully absorbs the information of neighbor \(M_{m,n}\) and further reduces its fitness value.