**4. Hybrid Grey Wolf Optimizer-Sine Cosine Algorithm (HGWOSCA)**

SCA and grey wolf optimiser (GWO) are meta-heuristic optimisation algorithms recently developed by Mirjalili et al. [44,77]. Both SCA and GWO approaches show high performance compared with other well-known meta-heuristic algorithms [44,77]. The hybrid GWO-SCA technique was introduced by N. Singh et al. [43] for combining the advantages of both approaches. In the GWO-SCA hybrid approach, GWO presents the main part, whereas the implementation of SCA assists in the optimisation of GWO. An improvement in the position, speed, and convergence of the best grey wolf individual alpha (α) by using the original equation expressed in [77], is achieved by applying the position updating equations of the SCA approach, as illustrated in [44].

The position of the current space agent is updated on the basis of the following equation:

$$
\overrightarrow{\dot{\bf x}}\_2 = \overrightarrow{\dot{\bf x}}\_\rho - \overrightarrow{\dot{a}}\_2 \cdot (\overrightarrow{\dot{d}}\_\rho), \overrightarrow{\dot{x}}\_3 = \overrightarrow{\dot{x}}\_\delta - \overrightarrow{\dot{a}}\_3 \cdot (\overrightarrow{\dot{d}}\_\delta) \tag{52}
$$

where <sup>→</sup> *a* is random value in the gap [−2a, 2a]. The position of <sup>→</sup> *X*β, → *<sup>X</sup>*δ. and <sup>→</sup> *X*α. is updated using the following equation:

$$\frac{\overrightarrow{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{\cdots}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}$$
}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} $}$ }} $}$ }} $}$ }}} $}$ }}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} $(\sharp$ 

Details and description of the HGWOSCA approach can be found in reference [43]. Furthermore, the computational procedure of the HGWOSCA approach is illustrated in Figure 8.


**Figure 8.** Pseudo-code of the hybrid grey wolf optimiser-sine cosine algorithm (HGWOSCA).

Although the GWO and SCA are able to expose an efficient accuracy in comparison with other well-known swarm intelligence optimisation techniques, it is not fitting for highly complex functions and may still face the difficulty of getting trapped in local optima [43]. Thus, a new hybrid variant based on GWO and SCA is used to solve recent real-life problems.

#### **5. Results and Discussion**

The proposed methodology has been implemented in MATLAB software and applied to the development of the optimal design of a PV plant connected to the electric grid. Solar irradiance, ambient temperature, and wind speed data for 1 year from the installation field are required. The effect of minimum LCOE and maximum annual energy objective functions on the PV plant design was determined. The HGWOSCA optimisation technique and a single SCA algorithm were applied with 400 search agents and 30 iterations to solve the design problem.

According to the results presented in Table 7, the PV plant optimal design variables depend on the selected objective function. The minimum LCOE and maximum annual energy result in two completely different optimal PV plant structures. PV power plant results are presented in Table 8. The optimisation process applying HGWOSCA outperforms the single SCA for minimum and maximum objective functions.


**Table 7.** Optimal design variables using semi-hourly measurement data.

**Table 8.** Results of optimal design algorithms using semi-hourly measurement data.


For both objectives using HGWOSCA, the optimisation process has selected mono-crystalline PV module type 3 (PV3) from the list of candidates. This module uses 295 W, and inverter type 3 (INV3) was selected from a list of three inverters. This inverter uses 500 kW and presents the central topology of the PV power plant. With the objective of maximum annual energy, the suggested number of PV modules is 1394 and 2 inverter to have 786.5035 (MWh). In this case, the LCOE was 32.1174 (\$/MWh), and the PV plant total cost was the highest at 0.6315 (M\$). With the objective of minimum LCOE, the number of PV modules is 1376 and only 1 inverter is required to have 28.6283 (\$/MWh) of LCOE. In this case, the annual energy generation is equal to 786.5035 (MWh), and the total cost was reduced to 0.5557 (M\$) compared with the first case. The use of LCOE's objective function to optimise the design of PV plants can reduce the financial risks, as proven in this case study. The total cost of using minimum LCOE decreased by 12% with a benefit of 71,800 (\$) in terms of installation cost, maintenance and operation costs. Figure 9 illustrates the maintenance and operational costs and the installation cost throughout the life of the PV plant for minimum LCOE and maximum annual energy generation.

The area occupied by the PV power plant can be calculated based on the summation of the occupied area by all PV rows, according to the length of each row and the inter-row area of all adjacent rows. The total available area of the installation field is equal to 3131 m<sup>2</sup> and the installed PV modules occupied 3094 m<sup>2</sup> of the installation site, which is nearly the same as the total area of the field. Therefore, the percentage of the occupied area by PV modules in the two cases presents 99% of the available area. The arrangement of PV modules in rows within the installation area is illustrated in Figure 10 using the LCOE objective function. The length of each row changed from one row to another according to the shape of the PV plant. Furthermore, this configuration has been designed in terms of the shape of the installation area, reflecting the actual situation. The difference obtained on the energy production

using LCOE and maximum energy objective functions is due to the configuration and the arrangement of the PV modules within the available installation area. On the one hand, the optimal design of the PV plant under the maximum annual energy resulted in the minimum number of lines *Nr* installed in each row, which is equal to 1. Additionally, this arrangement allowed the PV modules to capture more reflected radiation from the ground. Furthermore, at *Nr* = 1, a small distance between two adjacent rows in terms of shading effect is required, thereby increasing the total number of rows in the installation area to *Drow* = 36 with one PV module line in each row and increasing the reflected radiation on PV modules. Moreover, the total number of PV modules for maximum energy is equal to 1394 and distributed among two central inverters. However, the number of PV modules for LCOE is less, leads to 1379 and arranged among only one inverter. PV modules are installed in multiple lines in case of LCOE objective function. In this configuration, the number of lines *Nr* for each row is equal to 4 and leads only to 14 rows. Moreover, this configuration decreases the reflected radiation from the ground to be captured by PV modules and cannot be absorbed by the rest of the lines (*Nr* > 1). The PV modules are installed horizontally for minimum LCOE (*PVorien* = 1) and vertically (*PVorien* = 2) for maximum annual energy.

**Figure 9.** Throughout the life of the PV plant optimised by HGWOSCA.

**Figure 10.** Rows arrangement for minimum LCOE using HGWOSCA.

Figure 11 illustrates the monthly energy generation by the PV power plant for the LCOE objective function. The PV plant energy generation remained high over the year, with an energy average of 65 (MWh) per month. The highest value of the energy generated by the PV power plant is obtained in March, because this condition is due to the high solar irradiance in this period.

**Figure 11.** PV plant energy generation (MWh).

For comparison, the semi-hourly average time was compared with the hourly average time meteorological data to examine the step time effect on the PV plant performance. The peaks of the meteorological data can influence the design solution. Therefore, the usage of annual semi-hourly average time rather than monthly, daily and hourly is recommended, as semi-hourly data contain the troughs and peaks of solar irradiation, ambient temperature, and wind speed. According to the results presented in Tables 9 and 10, the step time data can affect the objective functions. The LCOE for semi-hourly average time is 28.6283 (\$/MWh), and that obtained for hourly average time is higher and equal to 28.637 (\$/MWh). The use of semi-hourly average time meteorological data in designing the PV plant can increase the financial benefits.


**Table 9.** Optimal design variables using hourly measurement data.

**Table 10.** Results of optimal design algorithms using hourly measurement data.


In all resulting cases, the proposed HGWOSCA optimisation approach was applied successfully and showed higher efficiency than that of a single SCA technique, with high performance in determining the optimal solution and solving the PV plant complex design problem. The convergence optimisation of annual energy and LCOE is illustrated in Figures 12 and 13.

**Figure 12.** Convergence of the optimisation of annual energy using HGWOSCA algorithm for semi-hourly data.

**Figure 13.** The convergence of the optimisation of LCOE using HGWOSCA algorithm for semi-hourly data.

#### *E*ff*ect of PV Module Reduction Coe*ffi*cient*

A sensitivity analysis was applied to evaluate the PV power plant performance. Accordingly, the variations in the PV module annual reduction coefficient were investigated. The optimisation results were obtained for different annual reduction coefficient values, from 0.3% to 0.7% per year. The annual reduction coefficient used in this study was 0.5%, as mentioned in Equation (44).

The optimum results for five different values for the annual reduction coefficient of the PV module are presented in Figures 14 and 15. According to the results, by increasing the PV module reduction coefficient, the PV plant energy production is reduced throughout its lifetime period. The LCOE of the PV plant increases by increasing the PV module reduction coefficient. By contrast, the total cost of the PV power plant is not affected and has the same value for all reduction coefficient values.

**Figure 14.** Total energy for reduction coefficient variations.

**Figure 15.** LCOE for reduction coefficient variations.

In economic terms, an improved PV module annual reduction coefficient leads to the recovery of capital investment of the PV plant within a smaller time period, making the PV plant economically profitable. Moreover, the sensitivity of the PV power plant improved by the decrement of the PV module annual reduction coefficient and vice versa.

## **6. Conclusions**

The proposed methodology was executed using semi-hourly time-resolution (i.e., 30 min-average) values of meteorological input data, including solar irradiance, ambient temperature, and wind speed. The procedure considers PV modules and inverter specifications, including a list of different

commercially available PV modules and inverter technologies as candidates. The optimisation process selects only one PV module and inverter from a list of several alternatives, presenting the optimum combination. The proposed PV plant area model considers the shape and size of the installation field to properly arrange all the existing components.

The minimum LCOE and maximum annual energy objective functions were used to design the PV power plant. On the basis of the optimal results, the total cost of using the minimum LCOE objective function decreased by 12% with a benefit of 71,800 (\$), including installation cost and maintenance and operation costs compared with the maximum annual energy. In this methodology, the HGWOSCA optimisation technique and a single SCA algorithm were applied. The optimum design solution shows that the proposed HGWOSCA is more efficient. Additionally, the PV plant optimal design variables depend on the selected objective function. The minimum LCOE and maximum annual energy result in two different optimal PV plant structures. LCOE improved with the use of semi-hourly average time meteorological data for designing the PV plant and can increase the financial benefits. Moreover, the sensitivity analysis shows that the PV power plant can be improved by the decrement of the PV module annual reduction coefficient and makes the PV plant economically more profitable.

**Author Contributions:** T.E.K.Z. contributed theoretical approaches, simulation, and preparing the article; M.R.A., M.F.N.T., S.M.Z. and A.D. contributed to supervision; M.R.A., A.D. and S.M. contributed to article editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the School of Electrical System Engineering Research Fund (SESERF), UniMAP.

**Acknowledgments:** We gratefully acknowledge the support of the Algerian company of electricity SONELGAZ, for providing the measurement data.

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