## 4 Proposed Algorithms for the Unit Sizing Problem
### EESAs  
The research on hybrid algorithms has dramatically grown during recent years [36]. [PERSON] et al. [37] proposed a hybrid technique (HT) by combining the EDE and TLBO schemes for energy management. HT outperforms other schemes in terms of achieving better cost values along with other decision variables. In [38], the authors proposed a hybrid approach by merging tabu search (TS) and simulated annealing for unit sizing of a PV-WT-FC-battery system along with diesel and bio-generators. The optimal results with convergence speed were achieved by the hybrid algorithm. [PERSON] et al. [39] proposed a hybrid iterative-Pareto-fuzzy approach to obtain the optimal number of components of the PV-WT-battery system for a region, located in Malaysia. The objective was to minimize the user's TAC and also achieve maximum reliability. In another study, multi-objective GA along with multi-criteria decision making was proposed for the optimal combination of the PV-WT system in a grid connected environment [40]. Motivated by these studies, this paper presents two EESAs (TLBO + EDE and TLBO + SSA) composed of hybrid algorithms. The flowchart of EESA (TLBO + EDE) is depicted in Figure 2. Here, the TLBO process starts by initializing the parameters, like population, termination criteria, etc. At first, the process initiates by executing the teacher phase, which is then followed by the student phase. The updated population of TLBO is further explored using trial vectors of EDE. In another case of hybrid EESA (TLBO + SSA), the steps of TLBO remain the same until the new population (\(X_{new}\)) is generated. The SSA steps used to update the population are shown in Figure 3. Here, the best fitness value obtained by the solution is assigned to the variable \(F^{d}\), and then, the coefficient \(C_{1}\) is updated using Equation (32). An iteration of a size equal to the population size is set to update the position of the leader and follower salps. The leader's position is updated using Equation (31), while the follower's position is updated via Equation (33). The newer solution should also satisfy the constraints. The salps' new population (\(G_{new}\)) is compared with the old population \(X_{new}\), and the best solutions are accepted based on the fitness criteria. The mapping steps of EESA to the unit sizing problem for the PV-WT-battery hybrid system are given below. 1. The first step includes initialization of parameters: hourly input solar irradiation, ambient temperature, the speed of the wind, and user's load profile data. 2. Here, the power generated by single PV and WT is calculated using Equations (1) and (3), respectively. 3. In the third step, a solution space of two decision variables is randomly generated within the upper and lower bounds as given below. \[Gen^{i}= \begin{bmatrix}s_{1}^{1}&s_{2}^{1}\\ s_{1}^{2}&s_{2}^{2}\\ \vdots&\vdots\\ s_{1}^{j}&s_{2}^{j}\end{bmatrix} \] (34) In Equation (34), the first and second columns are associated with the number of PVs and WTs, respectively. 4. In this step, based on the RESs' generation and user's load, the total number of batteries required for each \(j\) solution is calculated using Equation (8). Thus, the cluster of configurations showing the solution space is depicted as: \[Gen^{i}= \begin{bmatrix}s_{1}^{1}&s_{2}^{1}&s_{3}^{1}\\ s_{1}^{2}&s_{2}^{2}&s_{3}^{2}\\ \vdots&\vdots&\vdots\\ s_{1}^{j}&s_{2}^{j}&s_{3}^{j}\end{bmatrix} = \begin{bmatrix}S^{1}\\ S^{2}\\ \vdots\\ S^{j}\end{bmatrix} .\] (35) Here, the third column shows the total number of batteries \(N^{b}\) in the battery bank. In Equation (35), for the \(i^{\text{th}}\) population generation, \(S^{i}\) represents \(j\) possible configurations. Each configuration represents a possible solution competing to fulfill the EESA objective. 5. In the fifth step, the LPSP of all \(j\) possible configurations is calculated via applying Equation (9). The population is updated, and only those solutions are considered that satisfy the \(LPSP^{max}\) constraint as given in Equation (18). 6.