## 3 Particle Swarm Optimization  
### Standard PSO Algorithm  
PSO is an optimization method inspired by swarm behavior observed in nature, where a population of particles represents the optimization parameters [53]. These particles collectively search for optimal solutions within a multi-dimensional search space. The objective of the algorithm is to converge toward the best-possible values for each parameter [54]. The fitness of each particle is evaluated using a fitness function, which quantifies the quality of the particle's solution estimate [5; 55]. Each particle maintains two state variables: its position (\(x(i)\)) and velocity (\(v(i)\)), where \(i\) represents the iteration index. The fitness of a particle is determined by evaluating a cost function associated with its solution estimate. Through information sharing, each particle combines its own best solution with the best solution found by the entire swarm, adjusting its search pattern accordingly. This iterative process continues until an optimal solution is reached or a termination criterion is met. The equation of the standard PSO algorithm is given as  
\[v_{k}^{d}(i+1)=v_{k}^{d}(i)+c_{1}.r_{1},k(i).(p_{k}^{d}-x_{k}^{d}(i))+c_{2}.r_{ 2},k(i).(g^{d}-x_{k}^{d}(i)) \tag{1}\]  
\[x_{k}^{d}(i+1)=x_{k}^{d}(i)+v_{k}^{d}(i+1) \tag{2}\]  
The equation represents the velocity and position update mechanism in PSO. The velocity is updated by combining the particle's previous velocity, the cognitive component based on its personal best solution, and the social component based on the global best solution found by the swarm. This combination allows the particle to maintain its momentum, explore its individual best solution, and be influenced by the overall best solution. The updated velocity is then used to update the particle's position, determining its next location in the search space. The updated version of PSO having an inertia term is given below:  
\[v_{k}^{d}(i+1)=v_{k}^{d}(i).r_{1},k(i).(p_{k}^{d}-x_{k}^{d}(i))+c_{2}.r_{2},k(i ).(g^{d}-x_{k}^{d}(i)) \tag{3}\]  
where \(c_{1}\) represents the cognitive term, \(c_{2}\) represents the social term, \(d\) is the dimension of the particles, \(P_{k}\) is the local best, \(g\) is the global best of the particle, and \(r_{1}\) and \(r_{2}\) are the random variables, their range lying between 0 and 1 [56], while the momentum of a particle is controlled by inertia, represented by \(w\).  
When the inertia of the particle is zero, the model will only explore and become independent of past values. The convergence rate of the PSO algorithm refers to the speed at which the algorithm converges toward an optimal solution [57]. The convergence rate of PSO can be influenced by various factors, including problem complexity, population size, inertia weight, acceleration coefficients, and termination conditions [58]. The flow chart of the standard PSO algorithm mentioned in [59] is shown in Figure 1.  
PSO has the potential for fast convergence due to its ability to share information among particles in the swarm [60]. Collective knowledge sharing enables particles to converge towards promising regions of the search space [61]. However, the convergence rate of standard PSO can be affected by the balance between exploration and exploitation. If the exploration is too dominant, the algorithm may take longer to converge. On the other hand, if the exploitation is too dominant, the algorithm may converge prematurely to local optima [62]. To enhance the convergence rate, several strategies can be employed. One approach is to adaptively adjust the parameters of the algorithm during the optimization process. This includes modifying the inertia weight and acceleration coefficients to balance exploration and exploitation at different stages of the optimization [63]. Different variants of PSO algorithms exist in the literature, shown in Figure 2, to achieve better complexity and faster convergence.