Estimation of solar cell parameters through utilization of adaptive sine–cosine particle swarm optimization algorithm

Based on recent research, it is projected that by 2050, 85% of the global population will live in urban areas, creating a significant need for specialized services to be available around the clock. [1]. Approximately 75% of the world's energy production is consumed by cities, and they are responsible for generating 80% of greenhouse gas emissions [1, 2]. To address the issue of pollution, it is necessary to develop new environmentally friendly energy sources [2]. In recent years, there has been a growing global concern for environmental protection policies that aim to promote the development of clean fuel technology [3]. Solar energy is one of the most lucrative renewable sources being explored today, as it can help meet the increasing demand for energy supplies [4]. Solar energy is recognized for its quiet operation, ease of deployment, and lack of pollutants [5].

Photovoltaic modules (PVs), consisting of solar cells (SCs), can convert solar energy into electricity without the need for any additional intervention. This means that both SC and PV modules must be capable of functioning in conditions that are influenced by weather [6], in addition, the maintenance of SC and PV modules should be cost-effective [7]. Meeting these objectives requires rigorous design of both SC and PV modules. Before a PV system is installed, its efficiency must be optimized, and its capacity must be evaluated using a reliable and effective simulator [8].

The design of SCs involves the use of mathematical models to estimate the parameters that define the cell. These models are utilized to simulate the internal variables that govern the behavior of the SC and establish the relationship between current and voltage (I–V). Two models are commonly employed for this task: the single-diode (SD) model and the double-diode (DD) model. In [9], the essential concepts of single- and double-diode modeling of PV solar cells are described.

The single-diode (SD) and double-diode (DD) models are electronic circuits used to depict the nonlinear behavior of SC. In the SD and DD models, various factors are considered, including the photo-generated current, diode saturation current, series resistance, and diode ideality factor. The configuration of these elements precisely reflects the performance of the SC or PV modules. The SD model employs five parameters to describe the SC, while the DD model uses seven. To achieve a precise balance between current and voltage (I–V) and obtain current estimates that closely match the measured values, it is essential to accurately estimate these parameters.

Several approaches have been proposed in recent years to optimize the estimation of SC model parameters [9, 10]. The methods for estimating SC model parameters can be categorized into three types: numerical techniques, analytical techniques, and soft-computing techniques. Numerical techniques employ nonlinear optimization approaches, such as the Newton–Raphson technique [11], the Levenberg–Marquardt algorithm [12], and the conductivity method for estimating the parameters of SC [13]. One of the main disadvantages of numerical techniques is their sensitivity to the initialization of parameters. This can result in becoming trapped in local minima, as the nature of the SC parameter estimation problem is multimodal [14].

Several attempts have been made to estimate parameters using analytical methods, such as employing elementary functions [15] and Lambert W-function [16], and there is a review of the common methods used in [9] that shows the advantages and drawbacks of each method. The main issue with analytical methods is that they require more approximations due to the large number of parameters to be estimated (five for a single diode and seven for a double diode), which can lead to increased computation time for solving the system [9]. That is clear in [17] when the number of parameters was reduced from seven to four to be solved analytically. Another drawback of analytical methods is that they require additional coefficients, and their values may not be readily available in the datasheets [9]. As a result, analytical methods may provide less accurate parameter estimates for PV solar cells due to the approximations required and the lack of available data [18].

Soft computing is a third approach that aims to overcome the disadvantages of numerical and analytical methods. Soft-computing techniques typically utilize meta-heuristic algorithms (MAs), which search for optimal global solutions based on a search strategy that imitates natural behavior. Meta-heuristic algorithms employ different metaphors, such as the genetic algorithm (GA), to achieve this goal [19] and differential evolution (DE) [20], which is based on the evolutionary theory. Meanwhile, physics-based algorithms include methods such as the sine–cosine algorithm (SCA) [21] and the gravitational search algorithm (GSA) [22]. There also exists another group of methods based on animals and insects like particle swarm optimization (PSO) [23], artificial bee colony (ABC) [24, 25], or moth-flame optimization (MFO) [26]. MA succeeded in optimizing many applications such as the optimization of radiative transfer function [27], induction motor design [28], Handwritten Arabic Manuscript Image Binarization [29], feature selection [30], PID controller tuning parameters [31], and multiple sequence alignment [32].

Meta-heuristic algorithms can explore complex and multimodal search spaces using various operators to find the optimal solution. In the context of estimating SC parameters using MA, the root-mean-square error (RMSE) is commonly used as an objective function. The RMSE compares the values obtained from the dataset with the parameters calculated by the diode models. There are several different types of MA discussed in the literature. For example, in [33], the GA is used to increase the accuracy of the parameters estimated by the DD. The PSO is applied to estimate the parameters of solar cells using SD and DD models [34‐38] and in another development of PSO using chaos theory to increase exploration [39]. Besides, PSO was used for solar fabrication with the aid of neural networks [40]. PSO also succeeded in estimating the parameters of solar cells but in a more complex model of the circuit where it modeled as three diodes [41].

Another interesting approach uses simulated annealing (SA) to compute the values of the SD and DD [42], according to the authors, the results obtained using simulated annealing (SA) were superior to those obtained using other approaches that were compared. More recently, a new method called cat swarm optimization (CSO) has been proposed for determining the optimal parameters of SC using both the SD and DD models [43]. Moreover, the CSO has also been compared with different methods to verify the quality of the solutions. In Rajasekar [44], the bacterial foraging algorithm (BFA) is introduced as an alternative to accurately model the characteristics of an SC using a new equation proposed by the authors. In the same context, in Askarzadeh and Rezazadeh [45], various versions of the harmony search (HS) algorithm have been proposed for identifying unknown parameters in both single- and double-diode models of solar cells. While these methods are generally efficient, they may still suffer from accuracy issues. In real-world scenarios, such as PV or SC model identification, it is crucial to have accurate outputs to minimize the costs associated with energy systems [46]. Other related works are the pattern search (PS) [47] and the bird-mating optimizer [48].

In Das et al. [49], a hybrid scheme between PSO and DE [20]. This technique involves integrating particle swarm optimization (PSO) with differential evolution (DE) for the optimization of digital filter circuits. The primary advantage of this integration is that it enhances the exploration scheme of PSO, thereby reducing the likelihood of becoming trapped in local optima. This is achieved by adding a new term in the estimation of particle velocity, which is the percentage difference in distances between two random particles. This promotes greater exploration of the particles. Additionally, particles are moved to new positions if they provide better fitness than their previous positions. However, the main drawback of this technique is its slow convergence, and it may provide low accuracy for optimizing mathematical benchmark functions.

The no-free-lunch (NFL) theorem states that there is no single optimization technique that is universally suitable for all optimization problems. In other words, different optimization techniques may perform better or worse depending on the specific problem being addressed. It emphasizes the importance of selecting an appropriate optimization technique based on the characteristics of the problem at hand [50]. Given the NFL theorem, this paper proposes the use of a hybrid optimization algorithm that combines the best features of two different methods. Specifically, it introduces the use of a recently proposed method called the sine–cosine algorithm, which is known for its simplicity and efficiency. By combining the sine–cosine algorithm with other optimization techniques, the proposed hybrid algorithm aims to achieve better performance than either method alone [21] with the PSO [23]. This method is called an adaptive sine–cosine optimization algorithm integrated with particle swarm optimization (ASCA-PSO), and it has two optimization layers [51]. The proposed hybrid optimization algorithm consists of two layers. The first layer uses particle swarm optimization (PSO) to exploit the most prominent regions of the search space. Meanwhile, the second layer employs the sine–cosine algorithm (SCA) to explore different sections of the same space using its unique operators. By combining these two methods, the proposed ASCA-PSO algorithm aims to achieve a better balance between exploration and exploitation of the search space, thereby improving the overall performance compared to either method alone. The ASCA-PSO algorithm has demonstrated its capabilities for solving complex optimization problems such as local sequence alignment.

The primary advantage of ASCA-PSO is its ability to perform both exploitation and exploration processes in parallel, which speeds up the convergence of the best solution and improves the quality of the solutions by combining the benefits of SCA for exploring the search space with the accurate tuning provided by PSO for exploitation. This makes it an ideal algorithm for enhancing the estimation of parameters for PV solar cells, which typically involve numerous parameters that must be accurately tuned. By efficiently exploring the search space, ASCA-PSO can find optimal solutions in multidimensional search spaces while achieving a high degree of convergence.

ASCA-PSO is executed in two layers, with SCA in the bottom layer and PSO in the upper layer, which helps to increase the accuracy of the search process. This is achieved through the combination of operators and the co-evolutionary learning scheme, which guides the algorithm toward optimal solutions. However, due to the nonlinearity of the parameter estimation problem for solar cells, SCA may produce poor results when used alone, in comparison to PSO. This is because PSO can exploit the search space to find the best solution, while SCA explores the search space. The objective function used in ASCA-PSO is the root-mean-square error (RMSE), which measures the differences between a dataset and the values estimated using the optimal diode models (SD and DD).

Thus, this paper's primary contributions are outlined as follows: firstly, introducing a reliable and precise tool for identifying PV solar cell parameters through a hybrid method of PSO and SCA. Secondly, it showcases the ability of ASCA-PSO to simultaneously conduct exploration and exploitation processes, thereby improving the accuracy of SC parameters. Thirdly, applying ASCO-PSO to estimate parameters for two SC models, namely the SD and DD circuit models. Finally, the paper compares the performance of ASCA-PSO with other algorithms used for estimating PV solar cell parameters.

The organization of this paper considers the following sections: Sect. 2 presents the diode models used for SC and how the parameter estimation can be formulated as an optimization problem. In Sect. 3, the preliminaries of ASCA-PSO are presented. Section 4 describes the experimental methodology and presents the results. Meanwhile, Sect. 5 includes some conclusions and future work.

In photovoltaic (PV) systems, diodes are used to manage the flow of electrical current. There are two common types of diodes used in PV systems: single diodes and double diodes. A single diode, also known as a bypass diode or blocking diode, is a basic diode component used in PV modules. Its primary function is to prevent reverse current flow through a specific cell or group of cells in a PV module when they are shaded or operating under low-light conditions. When a solar cell is shaded, it can act as a resistor and impede the flow of current, potentially reducing the overall performance of the module. The single diode is connected in parallel to the shaded cell or group of cells. When the voltage across the shaded cell(s) becomes negative (due to shading), the diode becomes forward biased and allows the current to flow through it, by passing the shaded cells.

A double-diode configuration is a more advanced setup used in some high-efficiency PV modules. It includes both a forward-biased diode (like the single diode) and a reverse-biased diode. The forward-biased diode functions as described for the single diode, allowing current to bypass shaded cells. However, the additional reverse-biased diode offers extra protection and performance benefits. The reverse-biased diode is connected in series with the PV cell or string of cells and helps reduce losses caused by voltage drop and thermal effects. It prevents current from flowing in the reverse direction, which can improve the overall efficiency and reliability of the PV module.

The choice between single- and double-diode configurations depends on the specific design and requirements of the PV system. Double-diode configurations are typically found in high-end, high-efficiency solar modules, while single diodes are more common in standard PV modules. The use of diodes in PV systems is crucial to maximizing energy harvest and protecting the cells from damage under various operating conditions.

In this section, the basic concepts of the general two solar cell models, the single diode (SD) and the double diode (DD) are discussed.

This section is divided into two parts, the first subsection introduces the basics of particle swarm optimization (PSO) and the sine–cosine algorithm (SCA). The second part explains all the steps of the adaptive sine–cosine and particle swarm optimization algorithm (ASCA-PSO).

This section presents the implementation of the ASCA-PSO for parameter estimation of PV cells. The algorithm and problem could be adapted to accurately estimate the best solutions. The optimization problem is defined \(as\,\,minimizing\,\,the\,\,RMSE\left(X\right)\) that depends on the variables of each diode model. In the ASCA-PSO, the population X contains the candidate solutions defined as \(X=\left[{x}_{1},{x}_{2},...,{x}_{N}\right]\) and each element is constructed as \({x}_{{\varvec{i}}}=\left[{x}_{i,1},{ x}_{i,2},...,{ x}_{i,d}\right]\) where \(d\in \left[\mathrm{5,7}\right].\) The variable d corresponds to the dimension of the problem and depends on the parameters to estimate using the diode models for that reason its value could be five or seven. The set of solutions is randomly initialized and then evaluated in the objective functions defined by the root-mean-square error (RMSE). The ASCA-PSO then starts the iterative search process. Here is important to mention that to compute the RMSE first is necessary to test the set of parameters (candidate solution) on the model of the solar cells to compute the output current of the circuit. Algorithm 1 describes the details for computing the parameters of a diode model.

This section discusses the verification of the performance and efficiency of the ASCA-PSO approach for estimating the parameters of the PV solar cell design. The evaluation criteria were used for testing the performance of the optimization technique as follows:

Two commercial solar cell modules are used for verification which are the R.T.C module and the STM6-40/36 module with 36 mono-crystalline cells. In the experimental testing of ASCA-PSO in comparison with standard SCA and PSO. The following parameters are used which are chosen within the allowed range specified by the authors of algorithms and by several runs each parameter is tuned individually to deliver the best results:

All experimental tests were performed using Matlab software on a machine with a 64-bit Intel Dual Core processor with 2.0 GHz and 4GB RAM.

The performance of the proposed algorithm is evaluated by analyzing the complexity and measuring the processing time. The complexity of the ASCA-PSO algorithm is \(O(T\times M\times N\times ({{\text{C}}}_{{\text{SCA}}}+{{\text{C}}}_{{\text{PSO}}} ))\) where \(M\) and \(N\) are the size of search agents in the top and bottom layer in order and C_SCA and C_PSO is the time cost of updating all one search agent per one iteration for SCA and PSO in order, and T is the number of iterations. The time complexity of SCA and PSO respectively is O( \(T\times N\times \) C_SCA) and O(\(T\times N\times \) C_PSO). According to the previous setting parameters values the program coded in ASCA-PSO, SCA, and PSO for estimating the parameters is run 30 times. The average time for estimating the parameters consumes 16.4 s by ASCA-PSO, while SCA consumes 9.3 s and PSO executes 10.5 s.

This paper examines the effectiveness of ASCA-PSO, a recently developed optimization algorithm, in estimating the parameters of photovoltaic cells for both single- and double-circuit models. The proposed algorithm employs a two-layer structure, with the top layer consisting of search agents controlled by PSO, each representing the global solution found by the agents in the corresponding bottom layer. The bottom layer comprises groups of search agents that update their movements based on SCA. This combination allows for simultaneous exploration and exploitation of the search space in the same iteration, thus enhancing the convergence rate and solution quality. The performance of ASCA-PSO is evaluated using two commercial photovoltaic modules, the R.T.C module and the STM6-40/36 module, each with 36 mono-crystalline cells and employing both single- and double-circuit models. The experimental results are compared to other related works and demonstrate the ability of ASCA-PSO to find global solutions for multimodal and complex objective functions with greater precision and stability, even in the presence of noise. The proposed model has the potential for application in more types of solar cells and more complex problems in the future.