*3.3. Biased Initialization EHO (BIEHO)*

The main idea of the biased initialization algorithm is that the algorithm did not start evolving while the population's average fitness did not exceed a certain threshold. Therefore, the clan should be satisfied with its population's quality and ensure high-quality elephants. Start the generation with a population with functional fitness. The next step of evolution will not begin until the quality of the first generation reaches a suitable predetermined threshold. Biased algorithms are used in the initialization step by adding a rule or a limit [54]. Forcing the first generation of the population to have a good candidate solution may lead to another good production.

### **4. Computer Results and Simulations**

EHO variants were tested using 57 mm diameter commercial silicon solar cells from the RTC Company of France to verify their performance against single- and double-diode models. The experiment is carried out under 1 sun (1000 W/m2) at 33 ◦C [8,42,55]. A multi-crystalline PV solar module CS6P-240P is used to represent the three-diode model. CS6P-240P experimental data based on [56,57] are established for four irradiance levels (109.2, 246.65, 347.8, and 580.3 W/m2) at temperatures (37.32, 40.05, 347.8, and 51.91 ◦C), respectively. Table 2 shows the manufacture specification for CS6P-240P under standard test conditions (STD). The basic EHO and its three variants are compared with the results of two algorithms from [42] called Artificial Bee Swarm Optimization algorithm (ABSO) and Harmony Search (HS) algorithm. The few adjustable parameters for EHO can be set as *α* = 0.9, β = 0.1, number of clans = 4, population size = 32, and maximum iteration = 5000.

**Table 2.** Manufacture specification under standard test condition.


Tables 3 and 4 present the optimal solar cell parameters and RMSE by EHO algorithms, Artificial Bee Swarm Optimization algorithm (ABSO), and Harmony Search (HS) for single- and double-diode modes. The single-diode model is considered the simplest model among all models with only five parameters. Table 3 shows that the four EHO algorithms obtained the same result due to the model's simplicity, but all four algorithms outperformed ABSO and HS. Table 4 shows the results for the double-diode model with seven parameters, showing differences between the extracted parameters and the RMSE. Compared to other algorithms, CEHO achieved the lowest RMSE. Figure 7 shows the convergence of the four EHO algorithms for the single-diode and double-diode model at the first 250 generations, respectively. In addition, it showed the fast convergence of the proposed EHO algorithms for obtaining good results.

**Table 3.** Comparison between EHO algorithms, ABS, and HS for single-diode solar cells.



**Table 4.** Comparison between EHO algorithms, ABS, and HS for double-diode solar cells.

**Figure 7.** Convergence rates of EHO and its variants. (**a**) single diode. (**b**) double diode.

As demonstrated by Table 5, the measured current is very close to the calculated current. In addition, cultural-based EHO leads to outperformed results compared with other EHO variants.

Figures 8 and 9 show the power and current of the calculated and measured current from cultural-based EHO. Again, the measured and calculated curves are almost identical, while the relative error for the double-diode model for cultural-based EHO is presented in Table 6.

The previous results were for the PV panels at standard temperature and radiation. The four EHO algorithms were tested against three other algorithms at different irradiance levels and temperatures for more testing. Table 7 shows the extracted parameters for the seven algorithms at different irradiance levels and temperatures. Finally, the three-diode model is tested against three algorithms from [43] (Moth-Flame Optimizer (MFO), FPA, and Hybrid Evolutionary algorithm (DEIM)). The RMSEs for each algorithm at varying irradiance levels are listed in Table 8. Again, at low radiation with 109.2 W/m2, CEHO outperforms EHO with a slightly small difference but a big difference compared to other algorithms. CEHO outperformed other algorithms at other radiations, and BIEHO's results were slightly different from CEHO's. The superiority of the CEHO algorithm is proven as the best compared with the other three variants and the other three algorithms for all irradiance levels. Figure 10 shows that calculated data fit the I-V curve of measured data for CEHO.


**Table 5.** The relative error for 26 measurements (single diode) with CEHO.

**Figure 8.** Measured power vs. calculated by CEHO. (**a**) single diode. (**b**) double diode.

**Figure 9.** Measured current vs. calculated by CEHO. (**a**) single diode. (**b**) double diode.


**Table 6.** The relative error for 26 measurements (double diode) with CEHO.


**Table 7.** Comparison between different EHO algorithms among irradiance levels.

**Table 8.** Comparison between EHO algorithms, MFO, FBA, and DEIM for three-diode solar cells.


**Figure 10.** Measured current vs. calculated for three-diode model by CEHO.

### **5. Conclusions**

This paper presents a new optimization algorithm based on elephant herding behavior called Elephant Herding Optimization (EHO) and three improved variants called αEHO, CEHO, and BIEHO. The EHO and its three variants are developed to estimate single, double, and three-diode solar cell models. The 57 mm diameter RTC Company of France commercial silicon solar cell with 26 points of measured data was chosen to present single and double models' problem under one irradiance level (25 ◦C and 1000 W/m2). The EHO variants results are compared with two good algorithms (ABSO, HS). For presenting the three-diode model multi-crystalline PV solar module CS6P-240P under four irradiance levels (109.2, 246.65, 347.8, and 580.3 W/m2) at temperature (37.32, 40.05, 347.8, and 51.91 ◦C) respectively. The EHO algorithms are compared with another three algorithms (MFO, FBA, and DEIM). The superiority of the four EHO algorithms is proven in the results. Cultural-based algorithms outperformed all algorithms used in the double- and three-diode models and ABSO, HS, and Biased in the single-diode model. Finally, it can be concluded from the results that EHO algorithms are very suitable for solving parameters extraction of solar cell problems for variant models.

Among the drawbacks of conventional EHO is its scale factor alpha being a constant value. Additionally, the behavior of EHO requires more attention to the solutions. Therefore, it would be helpful to employ more hybrid solutions, as this study recommends. Moreover, due to the practical nature of elephant herding, there are more processes involved than clan updating and separating. Thus, more models should be developed and incorporated into the EHO method that models elephant behavior. Finally, the main EHO was designed for solving continuous problems, so it must be validated for continuous and discrete problems [58].

Future work will include extracting parameters for more complex models for more accurate parameter extraction. In addition, the adaptive scaling factor is more promising than being a constant value in the range [0, 1]. Moreover, due to the superiority of the CEHO algorithm, we can do more enhancements to the CEHO algorithm to get more accurate results for more complex optimization problems. In addition, more behavior characteristics are recommended to investigate an advanced version of EHO accomplished with new hybrid algorithms.

**Author Contributions:** Conceptualization, Y.I.R. and M.A.E., R.A.E.-S.; methodology, A.M., A.A.M.; software, A.M., A.A.M.; validation, A.M., A.A.M.; formal analysis, M.A.E., R.A.E.-S.; investigation, A.M., A.A.M.; resources, A.M., A.A.M.; data curation, Y.I.R.; writing—original draft preparation, Y.I.R.; writing—review and editing, M.A.E., R.A.E.-S. and A.M.; visualization, A.M., A.A.M.; supervision, M.A.E., R.A.E.-S.; project administration, A.M., A.A.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data available upon request.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Nomenclature**

