## 1 Introduction  
Due to the pressing necessity of safeguarding the environment against pollution, researchers have been diligently engaged in constructing a diverse range of precise models aimed at generating effective renewable energy (RE) alternatives to conventional fossil fuel-derived energy [1, 2]. The utilization of traditional energy sources has led to the emergence of environmental contamination, which stands as a prominent concern within the realm of environmental challenges. The utilization of alternate renewable energy sources has garnered significant attention recently as a potential solution to this problem. The most economically viable sources of renewable energy include wind, solar, tidal, wave, and nuclear power, among others. The aforementioned sources possess a diverse range of applications in practical domains such as photovoltaic (PV) systems, wireless sensors, hot water delivery, and agriculture. Solar power has garnered significant recognition from institutions, governments, and researchers in recent times related to its enduring advantages in generating electricity via PV energy, while avoiding environmental harm, in contrast in the context of the effects of fossil fuels [1, 2, 3]. PV systems are indispensable to facilitate the worldwide advancement of renewable energy by effectively transforming solar energy into electrical power [4, 5]. However,the presence of external elements, such as dust accumulation in the PV grid's extensive systems, poses challenges to the efficient energy supply of PV systems. Particles in the air, as well as uncertain weather and changing temperatures, have a negative effect on the performance of solar cells, which makes them less efficient overall. To solve these problems, one possible answer is to find accurate PV models that make solar cell energy generation as efficient as possible. To utilize a PV system effectively, at first, current-voltage (I-V) characteristics of the system should be well determined both theoretically and practically, and this requires good mathematical modeling for the system. In the most prevalent iterations, solar PV systems may simply be represented by using a current source paralleled with a diode and two resistors of which one is shunt and the other is connected in series [6]. In the context of PV parallel circuits, the creation of a single-diode model (SDM) involves the combination of a single diode with two resistors as shown in Figure 1. The augmentation of diodes in the PV models leads to an amplification of unidentified parameters, hence resulting in a heightened level of complexity inside the model. In a PV equivalent circuit, SDM encompasses five unidentified characteristics related to PV behavior. Conversely, The double-diode model (DDM) includes seven unidentified parameters, and subsequent models follow a similar pattern, DDM is formed by pairing two diodes, as shown in Figure 2. Diode models integrate a number of variables, including saturation current, series/parallel resistance, shunt-resistant current, and shunt resistor. These parameters are not known and need to be computed and obtained from the data on the I-V characteristic curve. The comprehensive consideration of these components is essential to the effectiveness of solar cell process models, as imprecise estimations of these parameters can lead to substantial discrepancies in the manufacturing process. Consequently, the estimation of these features is a crucial endeavor that enhances the optimization of the PV system. Mathematical models for PV systems frequently use implicit, non-linear, multi-variable, and multi-modal equations [7]. According to the researchers, the resolution of PV models continues to pose a challenging issue. In order to address this, they put forward a number of approaches aimed at extracting, assessing, and simulating PV parameters with the objectives of timeliness, accuracy, and reliability [4; 5; 8].