**1. Introduction**

Fossil fuels' total reserves are limited, and their overuse has threatened human health and the ecological environment. Thus, developing renewable energy sources is an extremely urgent concern [1–5]. Renewable energy, including the energy sources of solar, hydro, wind, geothermal, and biomass energy [6–8], is inexhaustible or short-term renewable. Solar energy is a form of energy that contains a tremendous amount of energy and has the potential to meet all the energy requirements of current human activities [9]. As a result, solar energy has been employed in varied applications such as desalination, heating plants, and photovoltaic (PV) power generation [10,11]. Due to the clean and widespread availability of electrical energy in various fields, PV power generation is an important project for developing renewable energy sources [12].

Accurate modeling is essential for the assessment, efficiency improvement, fault analysis, and simulation of PV systems [13–15]. A PV system consists of an aggregation of PV cells, and they are typically modeled with equivalent circuits, mainly including single diode (SDM), double diode (DDM), and triple diode (TDM) models [16–18]. These equivalent circuits can simulate PV cells' electrical characteristics. They have five, seven, and nine parameters to be extracted, respectively. As the number of diodes increases, more parameters to be extracted are involved, which results in more computational difficulty. The challenges faced by the issue include not only the multiplication of solution complexity due to multiple unknown parameters but also the coupling between electrical quantities, leading to a highly implicit function [4,19–21]. Moreover, the non-linear characteristics are challenging to solve due to the exponential functions in the characteristic equations. These challenges render determining accurate PV models a puzzle.

Extracting proper parameters of PV models is a thorny issue, and it is primarily solved by three types of methods: point-specific-based methods, traditional numerical optimization methods, and metaheuristic methods [19]. The first category, also referred to as analytical methods, relies heavily on the analytical treatment of the models to reduce the parameters and on specific points to deduce the model parameters [13,22,23]. They generally have low accuracy, especially when there is noise on specific data points. The second category is also known as the deterministic methods, which extensively use the idea of gradients. They are highly exploitable and computationally fast but are sensitive to initialization settings, and the accuracy of the solutions can be insufficient [24–26]. That dilemma is because the PV model's mathematical formulation is implicit, has exponential functions, and requires extraction of multiple parameters. As a result, the mentioned issue has multi-peaked, non-linear, and strongly coupled characteristics, which pose a significant challenge to solving the issue using deterministic methods. Unlike the above two categories, natural phenomena inspire the third class of methods: metaheuristics. They do not rely on gradients and detailed data, are conceptually simple and computationally convenient, and can solve complex optimization issues with high accuracy [27–31]. Therefore, scholars have identified the merits of metaheuristics and applied them to many problems.

Nowadays, the metaheuristics for this paper's problems have evolved considerably, and it is necessary to review the current developments in parameter extraction techniques. Recently, several reviews have partially covered the application of metaheuristics in this area. Abbassi et al. [19] comprehensively described and summarized different indicators and cases and briefly assessed the results. However, the authors were biased towards a broad overview of different methods and ignored details about the metaheuristics' application mechanisms. They merely measured the indicators' presence, without specific results to give the methods' effectiveness. Oliva et al. [32] undertook a dedicated review, tabulated each indicator's results, and described the details of some metaheuristics. Nevertheless, the work mainly focused on PV cells, with insufficient attention to PV modules, and ignored a review of the TDM and algorithmic settings. Venkateswari et al. [33] summarized the indicators and case names, described improved concepts, and compared some metaheuristics. However, they just summarized the minimum root mean square error (RMSE) results and lacked data on other indicators. Li et al. [20] overviewed the environmental factors' presence and surveyed the results of various approaches. However, the review mainly focused on the SDM and DDM and lacked the algorithmic settings of metaheuristics. Overall, the available reviews mainly highlighted the statistics of the RMSE values for SDM and DDM. Specific data on other indicators, i.e., the total number of fitness evaluations (TNFES), the sum of individual absolute errors (SIAE), and the mean, maximum, and standard deviation (STD) of RMSE, were unavailable for judging different methods' performance in computational resources, accuracy, reliability, and robustness. We also note the following shortcomings in past reviews: (a) a lack of holistic evaluation of metaheuristics in recent years for cells and modules, (b) no discussion or literature screening of the situation when the temperature changes, and (c) omission of a presentation of data changes when partial shade is applied.

A holistic view of this type of research takes time to establish for researchers unfamiliar with this area. Meanwhile, the available reviews should include the results of the last several years of study. However, although some reviews comprehensively summarize all solutions to the problem, they mention too few metaheuristics and need more numerical details. Others focus on PV cells and modules, but omit the analysis of metaheuristics. These shortcomings make their conclusions rather one-sided and make it difficult for the reader to understand the research results from multiple dimensions. Therefore, a persuasive article that considers the model's various aspects, the parameter settings, and the evaluation metrics and integrates the results of a large number of applications of metaheuristics to the problem is needed to present the recent research results. This paper provides a comprehensive and detailed summary and analysis of the application of metaheuristics to model PV accurately in recent years. Specifically, the metaheuristics are categorized and

their rationale is outlined. The algorithmic settings are summarized, and the results are compared and ranked in various indicators. The variation of the parameters in different environments is studied, and a brief description of the relevant literature in recent years is given. Some cell models that are temporarily not in widespread use today but are of high research value are analyzed. Then, their advantages and disadvantages are analyzed, and the remaining challenges are analyzed. Eventually, future directions for research are summarized in solution approaches and application scenarios.

This work's main contributions are as follows:


The remainder is briefly sketched as follows. The PV cell's mathematical model and the evaluation indicators are explained in Section 2. Section 3 illustrates different metaheuristics. Section 4 provides an overall analysis of different methods, existing challenges, and possible research directions. Finally, Section 5 gives the conclusion.
