**4. Discussion**

To evaluate the reliability of the WOAPSO, the proposed hybrid algorithm is compared with six well established metaheuristics algorithms, i.e., GSA [56], SCA [57], GWO [58], PSO [59], WOA [37], PSOGSA [60] as well as other algorithms existing in the literature. It is observed that the estimated parameters based on the optimization process are highly consistent with the experimental data for SDM, DDM, and SS2018P PV module.

For SDM, the hybrid WOAPSO algorithm generates the lowest RMSE values (7.1700 × 10 −4 ) compared to the GSA, SCA, GWO, PSO, and WOA, PSOGSA algorithms (Table 2). The RMSE of the proposed WOAPSO algorithm is also compared with previously studied algorithms (Table S2). It is noted that the hybrid WOAPSO algorithm provides the lowest RMSE values than that of others. Table S3 represents the absolute IAE for SDM analysis. The magnitude of IAE for different observations is less than 0.0018 (Table S3), which indicates that the parameters optimized by the WOAPSO are very precise.

7.1700 × 10 −4 In the case of DDM analysis, the MLBSA, EHHO, IJAYA, and GOTLBO algorithms produce the best value of RMSE (Table S4). However, WOAPSO generates the third-best value of RMSE (9.8412 × 10 −4 ), which is very close to MLBSA (9.8249 × 10 −4 ), EHHO (9.8360 × 10 −4 ), IJAYA (9.8293 × 10 −4 ), and GOTLBO (9.8317 × 10 −4 ). However, the computational cost in terms of function evaluation is 1/3 of MLBSA, EHHO, IJAYA, and GOTLBO. Moreover, WOAPSO shows superiority over other algorithms in terms of RMSE (Table 3). For DDM, the magnitude of IAE for different observations is depicted in Table S5. It is noticed that the IAE values are less than 0.0097, which demonstrates the accuracy of optimized parameters produced by WOAPSO.

9.8412 × 10 −4 9.8249 × 10 −4 For the SS2018P PV module, the hybrid WOAPSO algorithm produces the lowest RMSE values compared to the GSA, SCA, GWO, PSO, WOA, and PSOGSA algorithms. The IAE magnitudes for different observations (at 1000 W/m<sup>2</sup> ) are less than 0.0018 (Table S6). More importantly, the computational time for WOAPSO is less than other algorithms (Table 4). The average execution time of each algorithm on the three PV models is calculated and illustrated in Figure 9. The WOAPSO algorithm requires less time (about 26.1 s)

−4 9.8317 × 10

−4

−4 9.8293 × 10

9.8360 × 10

than GWO, PSO, SCA, WOA, and PSOGSA, while GSA has the worst execution time of approximately 52 s.

**Figure 9.** Comparison of the execution time.

Furthermore, the Friedman ranking test is also performed for all algorithms and depicted in Table 6. Table 6 shows that the proposed WOAPSO algorithm significantly outperforms the GSA, SCA, GWO, PSO, WOA, PSOGSA algorithms for all three models, i.e., single-diode, double-diode, and PV module models.
