## 3 Result Discussion and Assessment
### Reliability Enhancement via PSO Method  
[PERSON] and [PERSON] [62] originally introduced the particle swarm optimization (PSO) technique, in which a solution is represented by a particle that encodes six parameters of the PI control strategy: \(K_{P_{1}},K_{I_{1}},K_{P_{2}},K_{I_{2}},K_{P_{3}}\), and \(K_{I_{3}}\).  
Each particle's fitness value in the optimization algorithm is calculated by assessing its performance through the application of a specifically designed fitness function. This function quantitatively evaluates how well a particle meets the criteria of the optimization problem, thereby determining its suitability within the search space. It is important to note that if the total runtime \(T_{0}\) is still early in the system's lifecycle, the reliability of the LFC system may be close to 1. This suggests that the component wear and tear is minimal at this stage [16].  
In this situation, the PI controller must ensure that reliability remains at one while also optimizing the quality parameters (PT, PO, and ST) of the LFC process as much as possible. Consequently, two scenarios need to be considered:  
* If the reliability is one, the fitness function of each particle will focus on optimizing the quality parameters of the LFC process.
* Conversely, the fitness function combines the reliability measure with a penalty for deviating from a reliability of one.  
Based on the preceding discussions, the objective function for PSO is composed of two main components: the reliability-related component and the quality parameters-related component. These components are balanced using an equal penalty. Consequently, the fitness function for each particle is given by the following equation:  
\[f(T_{o})=(1-\Phi(R^{*}(T_{o})))\cdot(1-R^{*}(T_{o})+Q)+\Phi(R^{*}(T_{o}))\cdot W \tag{12}\]  
where  
\[\Phi(x)=\left\{\begin{array}{c}0,ifx\
eq 1\\ 1,\textit{else}\end{array}\right. \tag{13}\]\[Q=a_{1}\sum_{i=1}^{\text{N}_{T}}\text{PT}_{\max_{i}}+a_{2}\sum_{i=1}^{\text{N}} \text{PO}_{\max_{i}}+a_{3}\sum_{i=1}^{\text{N}}\text{ST}_{\max_{i}} \tag{14}\]  
\[W=\sum_{m=1}^{M}\left(\frac{\sum_{i=1}^{\text{N}_{T}}\left(b_{1}\cdot\sum_{i=1}^ {\text{N}}\text{PT}_{i}+b_{2}\cdot\sum_{i=1}^{\text{N}}\text{PO}_{i}+b_{3} \cdot\sum_{i=1}^{\text{N}}\text{ST}_{i}\right)}{N_{T}\cdot M}\right) \tag{15}\]  
where \(a_{1}\), \(a_{2}\), and \(a_{3}\) are weights to determine the value of the quality parameters-related part, and are set to 4, 10, and 0.3 based on a sufficient number of realistic simulations. Similarly, \(b_{1}\), \(b_{2}\), and \(3_{3}\) are weights to determine the penalty function, and their values are set 2, 2, and 1.  
Each particle in the swarm moves towards its best-known position, referred to as the local best, \(p_{best}\), and towards the best position found by the entire swarm during the current iteration, known as the global best, \(g_{best}t\). For a particle \(i\) with \(j\) dimensions, its local best-known position is denoted as \(p_{best_{i}}=(p_{i1},\ldots,p_{ij})\). The particle updates its velocity \(v_{ij}^{k+1}\), and the position \(x_{ij}^{k+1}\) using the following equations:  
\[v_{ij}^{k+1}=w_{1}\cdot v_{ij}^{k}+c_{1}\cdot\rho_{1}\cdot\left(p_{ij}-x_{ij}^ {k}\right)+c_{2}\cdot\rho_{2}\cdot\left(p_{g}-x_{ij}^{k}\right) \tag{16}\]  
\[x_{ij}^{k+1}=x_{ij}^{k}+v_{ij}^{k} \tag{17}\]  
where \(k\in 1,\ldots,K;i\in 1,\ldots,I;j\in 1,\ldots,J\). \(w_{1}\) indicates the inertia weight, \(c_{1}\) and \(c_{2}\) stand for the cognition learning factor and the social learning factor, respectively, and \(\rho_{1}\) and \(\rho_{2}\) are random factors limited to \([0,1]\).  
The procedural steps of the proposed MCS-PSO methodology are illustrated in Figure 4.