## 3 Proposed Improved Hybrid SSA with an EDO (ISSAEDO) Algorithm
### Computational Complexity of the ISSAEDO Algorithm  
The overall proposed ISSAEDO algorithm's computational complexity (\(O(ISSAEDO)\)) can be determined by computing the complexity of each step individually, which can be summarized in five essential steps, namely population initialization, position upgrading, random position amendment appreciation of fitness function, and DE strategy. We let \(D\) be the dimensional space of the problem, \(N\) be the total number of candidate positions (population size), we let \(T\) equal of maximum time, and we let population of \(M\) equal the population of \(N\) in all allowed iterations. Let us point out the complications as follows:  
1. \(O(\text{Population initialization})=O(N)\). 2. \(O(\text{Position upgrading})=O(N\times T\times D)\). 3. \(O(\text{Random Position Amendment})=O(N\times T\times D)\). 4. \(O(\text{Appreciation of fitness function})=O(N\times T)\). 5. \(O(\text{DE strategy})=O(N\times T\times D)\). By combining these complexities to get the overall computational complexity:  
Figure 3: Flowchart of the ISSAEDO algorithm. \[O(ISSAEDO) =O(\text{Population initialization})+O(\text{Position Upgrading})\] \[+O(\text{Random Position Amendment})+O(\text{DE strategy})+\] \[+O(\text{Appreciation of Fitness Function})\] \[=O(N)+O(N\times T\times D)+O(N\times T\times D)+O(N\times T)+O(N \times T\times D).\]  
Grouping similar terms, we obtain  
\[O(ISSAEDO)=O(N+N\times T+N\times T\times D).\]  
Therefore, the overall computational complexity of the ISSAEDO algorithm is  
\[O(N+N\times T+N\times T\times D).\]  
According to the flowchart and assumption within each algorithm, the following can be obtained as follows:  
1. \(M\) is the population size. 2. \(G(Max)\) is the maximum number of iterations. 3. \(O(E)\) is the time complexity for the single solution evaluation. 4. \(O(S)\) assumed that the SSA algorithm per iteration is a standard which is this one. 5. \(O(D)\) assumed the same for the EDO algorithm. 6. \(O(DE)\) assumed the same for the EDO algorithm per iteration as well. The Overall Time Complexity would be:  
\[O(G(Max)*(ME+S+D+DE))\]  
Since \(M\) is the population size, each one will be evaluated once per iteration, if we Suppose the evaluation takes \(O(E)\), so this entire step takes \(O(ME)\). **4. PV Models Results and Experimental Analysis**  
Obtaining the parameters for the two PV models (SDM, DDM), this section presents computational experiments comparing the ISSAEDO algorithm to several competing metaheuristic algorithms. The PV models are described in Table 1. The details of the I-V for SDM and DDM are given in [35]. In addition, Table 2 displays the search range for all undiscovered parameters. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **SDM** & **DDM** \\ \hline Type & Poly-crystalline & Poly-crystalline \\ \hline Number of cells in zeros \((N_{\text{s}})\) & \(1\) & \(1\) \\ \hline Number of cells in parallel \((N_{p})\) & \(1\) & \(1\) \\ \hline Size & \(57\) mm diameter & \(57\) mm diameter \\ \hline Test temperature (T) & \(33\)\({}^{\circ}\)C & \(33\)\({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of different PV models. Research was conducted using the Python programming language in a computing environment equipped with a Dual Xeon Gold 5115 2.4 GHz CPU and 128 GB of RAM. The operating system used was Microsoft Windows Server 2019. It should be noted that the execution of each method was repeated a total of thirty times, and the resulting data included the minimum, maximum, mean, and standard deviation (STD) values. In order to maintain a consistent and equitable evaluation, all algorithms under consideration were executed until reaching the specified stopping condition (\(G_{max}\)) as outlined in Table 3 for the different PV models employed. The variable values of the competing approaches were obtained from the articles associated with each strategy.