**1. Introduction**

Renewable energy sources (RESs) like wind and solar should be considered in order to mitigate the effects of climate change and rising temperatures, as well as to protect the planet from the pollution and destruction produced by traditional fossil energy [1]. The process of ecological transition involves identifying consumption and sustainable community models to reduce harmful emissions and to create reliance on power generation from renewable sources [2]. One of the aims of the sustainable development goals (SDGs), especially the seventh goal, is to obtain modern energy which is sustainable and highly reliable at the lowest cost [3]. There is a great deal of interest in RESs due to the enormous financial and environmental problems associated with traditional energy sources like fossil

**Citation:** Ali, H.H.; Ebeed, M.; Fathy, A.; Jurado, F.; Babu, T.S.; A. Mahmoud, A. A New Hybrid Multi-Population GTO-BWO Approach for Parameter Estimation of Photovoltaic Cells and Modules. *Sustainability* **2023**, *15*, 11089. https://doi.org/10.3390/su151411089

Academic Editor: Idiano D'Adamo

Received: 29 June 2023 Revised: 13 July 2023 Accepted: 14 July 2023 Published: 16 July 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

fuels. It is essential to transform the solar energy into different forms that may be utilized in daily life with the assistance of an appropriate device to exploit it [4,5]. Even though solar energy is abundant, its expansion is hampered by problems like fractional shadow, high construction cost, weather variation, and the need for costly storage. As a result, photovoltaic (PV) modeling is necessary to estimate the performance of a PV system before installation. Furthermore, the prediction of PV panel operating attributes is critical in solar PV system design, evaluation, simulation analysis, and control. Also, modeling aids in comprehending the functioning precept and attributes of the solar PV system under variable meteorological situations. The PV solar system is useful for capturing the solar energy and converting it into electrical power [5–7]; it has penetrated into many applications [5]. Moreover, the economic implications of the decreased lifetime and its causes are presented in [8]. One of the scientists' priorities is to improve the efficiency and dependability of these technologies. Understanding the mechanisms of power absorption and conversion in solar cells, as well as correct modelling, can help in forecasting and designing them properly. One of the most critical challenges that researchers are facing is how to build a reliable model of the solar panel [9–11].

Changes in temperature and sun irradiance have significant impacts on the performance of PV systems [12]. Therefore, to maximize the performance of these systems, adequate mathematical models are required that precisely replicate the PV system behavior under several operational scenarios. Three of the most common PV system models, the single, double, and triple diode models (SDM, DDM, and TDM), are used [13,14].

The parameters of the SDM are simple to estimate as it only has five parameters, but its performance suffers from minimal irradiance scales and as a consequence of temperature changes. The DDM includes seven unknown parameters; it employs a second diode to achieve current reunification and to deal with other non-idealities [15]. However, the DDM suffers from some defects in recombining the current and other non-idealities. The final model is TDM, with nine ungiven parameters; it was introduced in [16]. Unfortunately, the nine parameters should be calculated as the manufacturers do not directly give them. To decrease the difference between the measured assessed power–voltage (*P*-*V*) and current– voltage (*I*-*V*) curves, the issue is converted into an optimization problem with a nonlinear objective function and a significant number of local minima.

Researchers are interested in employing metaheuristic algorithms to estimate the PV model parameters due to their notable success in handling various real-world optimization problems [5–7]. A hybrid seagull optimization algorithm architecture (HSOA) has been described for assessing the PV model parameters and developing a nonlinear control factor, which is dependent on the cosine function, to stabilize exploitation and exploration capabilities [1]. A springy whale optimization algorithm is described as an enhanced optimization technique to determine the parameters of PV cell/panel models [9]. Changes have been made to the way that the whales move in order to improve the algorithm performance. This helped the algorithm avoid the local solution, and the algorithm convergence speed was enhanced. In [13], an improved cuckoo search optimizer (ICSO) and a modified cuckoo search optimizer (MCSO) are implemented to solve the parameter evaluation issue of a PV system. Solar cell parameters have been evaluated through a genetic neural network (GNN) strategy [14]. The PV module characteristics have been identified with the aid of the tabu search optimizer (TSO) [15]; moreover, the lightning search algorithm, pattern search (PS), gravity search algorithm (GSA), genetic algorithm (GA), and PSO have been applied and compared to the presented approach [16].

In order to define the values of the ungiven parameters, the sooty tern optimization (STO) approach was developed for parameter evaluation of the PV cells/modules [17]. The hybrid particle swarm optimization (PSO) and rat search algorithm have been presented and combined as a hybrid approach for extracting the parameters of hybrid systems, including those of fuel cells and solar PVs [18]. The presented approach in that work reduced the likelihood of a local minimum and increased the algorithm accuracy. In [19], the animals migration optimizer (AMO) was introduced to construct the SDM of a PV

system. The approach capacity for producing prompt, dependable, and consistent outcomes has been considered. In [20], a chaotic WOA for estimating the solar cell parameters was introduced; the key benefit of this method is that its parameters are automatically computed and adjusted using chaotic maps. In [21], a mathematical model for PV solar cells was created using the equilibrium optimizer (EO). The results using the EO have been compared with Harries hawk optimization (HHO), the teaching learn-based optimizer (TLBO), and PSO. In [22], the many approaches employed in constructing the SDM, DDM, and TDM of PV systems were reviewed and compared in terms of pros and cons.

The fractional-order Darwinian PSO methodology was used in [23] to enhance the conventional PSO method in evaluating the electrical parameters of PV cells/modules. To assign the solar cell parameters, the authors in [24] presented a hybrid honey badger algorithm and GTO [25]. These algorithms reduced the root mean square error (RMSE) between the simulated and measured results. In [26–28], a marine predatory animal (MPA) algorithm is described for computing the parameters of PV cells/panels in constant and varying weather situations. An improved stochastic fractal search algorithm has been used to solve the parameter appreciation of SDM solar cells and PV panels [29]. The authors in [30] presented the computational optimization method for extracting the parameters of solar cells/panels using an enhanced arithmetic optimization algorithm. In order to study the DDM-based circuit of a PV panel, practical tests to obtain the measured *I*-*V* and *P*-*V* characteristics have been conducted while considering various statistical analyses to determine the average, maximum, minimum, and standard deviations. A quick and efficient method for collecting the solar cell/panel parameters from the datasheet is provided in [31]. A niche PSO using a parallel computing technique was presented in [32] to identify the PV panel parameters. A multi-agent system (MAS) has been combined with CSO to estimate the parameters of various PV cells [33]. The circuits of SDM, DDM, and TDM for PV cells have been analyzed using the atomic orbital search to determine the ungiven parameters [34]. The tree seed algorithm has been used to calculate the parameters of the STM6-40/36 PV panel with different maximum fitness evaluations [35]. Moreover, a heterogeneous mechanism for the differential evolution algorithm (DE) [36], population diversity controlled DE [37], the artificial parameter-less optimization algorithm [38], random reselection PSO [39], the arithmetic operation algorithm based on the Newton–Raphson and Lambert W approaches [40], and adaptive slime mold [41] have been utilized to construct different equivalent circuits of PV cells/panels. A mayfly algorithm [42], northern goshawk optimization [43], and Newton–Raphson (NR) with an enhancement of a tuna swarm optimizer by a chaotic tent map [44] have been presented to evaluate the parameters of a TDM circuit. The parameters of a PV equivalent circuit were resolved by a chimp optimization algorithm with a robust niching approach [45], hybrid PSO with a gravitational search algorithm [46], chaos game optimization [47], an improved gradient-based optimizer based on sine cosine [48], DE enhanced by a chaotic map [49], and the predict output-based backpropagation neural network with EO [50]. Furthermore, the forensic-based investigation algorithm [51], the supply–demand optimizer [52], the enhanced hunger games search via the Laplacian Nelder–Mead approach [53], the Rao-1 optimization-based chaotic sequence [54], the arithmetic optimization algorithm-based guaranteed convergence and modified third-order NR [55], and the hybridized wind-driven optimization with fruit fly optimization [56] have been used to compute the parameters of various types of PV models.

Most of the reported studies have limitations, such as the falling into local optima, the requirement for numerous controlling parameters, and the complexity in implementation, in addition to the use of absolute algorithms without fundamental changes or modifications. The motivation of this study is to introduce a novel hybrid multi-population gorilla troops optimizer and beluga whale optimization (HGTO-BWO) to determine the PV cell/panel parameters such that all the gaps in the previous works are covered.

GTO is characterized by its ability to solve real-world problems with limited and unknown search space. On the other hand, the BWO has better stability, good convergence accuracy, stronger search ability, and a faster convergence rate. Therefore, hybridization between GTO and BWO results in a strong optimizer which is able to solve the handled problem with good efficiency. Table 1 provides a comparison of the recent work published in 2023 with regard to parameter estimations of PVs. The multi-population technique is applied to enhance the algorithm performance and avoid early convergence through dividing the entire population into many subgroups to preserve population variety. Different subgroups can be discovered throughout the whole search area and can reach the optimal solution efficiently by searching in different locations inside the search area at one time. Moreover, the optimization techniques can be easily and efficiently incorporated into multi-population methods [57,58]. The following are the major contributions of this article:



**Table 1.** A comparison of recent work published in 2023.

The rest of this article is as follows. Section 2 describes the mathematical model of solar PVs. Section 3 illustrates the problem expression, while the proposed hybrid multi-population GTO and BWO algorithm is presented in Section 4. The testing of the benchmark functions is presented in Section 5, and the application of the PV parameter estimation is given in Section 6. The conclusions are clarified in Section 7.

#### **2. Modeling of Solar Photovoltaic (PV)**

A solar PV cell is typically described through an electrical analogous circuit that includes current source, resistors, and a diode. Numerous PV cell modeling systems have evolved due to nonlinearity. The models of a PV cell are divided into three categories: single, double, and triple diode models. The prediction accuracy of the *I*-*V* curve is defined by the number of diodes in the model. Also, adding another diode, from one to three, enhances the model performance and precision at minimal irradiance levels. Similarly, the growth of modeling results in the development of the TDM. The model of the analogous circuit, its

equations, and the specifications of the ungiven parameters are shown in Figures 1 and 2. As the number of diodes grows, the number of model parameters to be evaluated grows and then the complexity of the problem is increased [64,65].

**Figure 1.** DDM equivalent circuit.

**Figure 2.** TDM equivalent circuit.
