1. Introduction
Sustainable electricity that is regenerated over time is provided by renewable energy sources like wind, rain, sunshine, tides, and waves. Being able to utilize renewable energy without creating harmful byproducts is one of its strongest justifications. Among the renewable energy sources, solar power extraction using PV panels as shown in Figure 1 has attracted the most attention and study [1]. Solar power is widely used as a renewable energy source due to its year-round availability, low maintenance requirements, and high energy purity [2]. Solar energy must overcome many obstacles, such as high installation costs and low efficiency. The PV module also has a nonlinear maximum power point (MPP) and fixed features. Operating the PV system at its MPP, where it generates the most electricity, is only possible with the help of an appropriate MPP tracking (MPPT) control approach [3]. MPPTs must be verified by computer simulation before being physically implemented. Before installing a PV MPPT system, it is crucial to have a dependable and efficient PV model. Therefore, it is important to have a reliable model of the PV panel for use in analysis and simulation [4,5].
By utilizing a photovoltaic cell, also known as a solar cell, the concept of the photovoltaic effect can be put into practice to convert the energy from light into usable electrical energy. When electrons begin to flow in one direction rather than another, a current is created. The generation of this flow occurs if a photon’s energy is higher than the energy required to cross the band gap [6]. The fundamental characteristic of a photovoltaic cell is its persistent forward bias. The creation of a current or voltage signal in a photodiode occurs because of the interaction between light rays and the N-channel of a semiconductor junction. It is common knowledge that factors such as temperature, sun irradiation, and weather have a significant impact on the amount of energy that can be generated by photovoltaic (PV) power systems. In addition, these systems will invariably suffer from degradation, which may or may not be accompanied by the incidence of electrical problems. An accurate PV model should be built so that the reliable I–V and P–V output characteristics under normal operation can be predicted, which is essential for both the design and evaluation of a PV system’s performance. Several characteristics must be accurately extracted to design, simulate, and evaluate the performance of a PV system [7].
Effective PV cell modeling is essential to regulate and forecast the performance of solar systems under varied operating conditions. One of the most significant tasks is to model and assess the parameters of PV cells. It can be challenging to determine cell restrictions due to the presence of nonlinear dimensions and intermittent climatological static. The physical process and associated factors of PV cells were used as the basis for the development of several different models. The behavior of PV systems has been successfully modeled using, for instance, single-diode, double-diode, and triple-diode models. Because of its simplicity and widespread acceptance, the single-diode model (SDM) is frequently used to estimate similar circuit parameters [8,9]. Also, mathematical modeling is required to analyze the impact that shifting parameter values have on the output voltage-current and power-voltage characteristics of a configuration consisting of a single diode. This is performed with the intention of improving the accuracy of the analysis. As part of our investigation, we take a closer look at how temperature and irradiance levels affect the output properties. With this information, we can better understand how various parameters affect our system’s output and how to obtain the most out of our system. Using this method, we may achieve the maximum possible power point [10,11].
Analytical, numerical, and meta-heuristic methods are the three primary types of methodologies that have been utilized throughout the history of parameter estimation [12]. However, analytical and numerical methods cannot extract the accurate parameter value for every condition. To solve these problems, most researchers have used meta-heuristic methods. Genetic algorithm (GA) [13], artificial bee colony optimization (ABC) [14], grey wolf optimization (GWO) [15], particle swarm optimization (PSO) [16], whale optimization algorithm (WOA) [17], coyote optimization algorithm (COA) [18], war strategy optimization (WSO) [19], and Cuckoo Search (CS) [20] are some examples of meta-heuristic algorithms.
In this paper, the mathematical modeling of the PV model manufactured by Kyocera (KC200GT) is modeled in MATLAB, and after that, the five unknown parameters (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, n, ${\mathrm{R}}_{\mathrm{s}}$ and ${\mathrm{R}}_{\mathrm{sh}}$) are extracted using GA and PSO. After, we derive the I–V equation of a single-diode configuration PV in two variables (${\mathrm{R}}_{\mathrm{S}}{\mathrm{and}\mathrm{V}}_{\mathrm{t}}$). A unique two-step optimization method for precisely calculating the parameters of single-diode photovoltaic modules is presented in this study. The method’s accuracy and convergence speed are better than those of classic single-stage optimization techniques such as PSO, GA, GWO [15], Villalva’s Method [21], Accarino’s Method [22], Iterative Method [23], and Silva’s Method [24], as confirmed by the Kyocera (KC200GT) module validation. The two-step method drastically lowers the number of iterations needed and the fluctuating parameter values. Improving PV model accuracy—crucial for designing and optimizing PV systems—is achieved by bringing parameter values closer to the MPP. Improving the technique’s practical usefulness for real-time online parameter detection will be the focus of future efforts.
2. Characteristics of PV Cell
The effect that photovoltaics has on PN junction diodes is crucial to the operation of PV arrays. Without this effect, the arrays would not be able to generate electricity [8]. When light is shone on a photovoltaic (PV) cell, a direct current (DC) is produced. The amount of this current that is generated is linearly proportional to the amount of solar irradiance as well as the temperature. A model of an equivalent circuit that is based on a single diode is presented in Figure 2. To describe it and to develop the mathematical equation, we will first employ the more sophisticated physics of the PV module [25].
2.1. Mathematical Equations
The mathematical representation of a PV model, specifically when employing a single-diode configuration, can be expressed as follows:
$$\mathrm{I}={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_{\mathrm{D}}-{\mathrm{I}}_{\mathrm{sh}}$$
where
$$\{{\mathrm{I}}_{\mathrm{D}}={\mathrm{I}}_{0}\left[{\mathrm{e}}^{\left({\displaystyle \frac{\mathrm{V}+{\mathrm{IR}}_{\mathrm{s}}}{{\mathrm{nV}}_{\mathrm{T}}}}\right)}-1\right],{\mathrm{I}}_{\mathrm{sh}}={\displaystyle \frac{\left(\mathrm{V}+{\mathrm{IR}}_{\mathrm{s}}\right)}{{\mathrm{R}}_{\mathrm{sh}}}},{\mathrm{V}}_{\mathrm{T}}={\displaystyle \frac{{\mathrm{KT}}_{\mathrm{C}}{\mathrm{N}}_{\mathrm{S}}}{\mathrm{q}}}\}$$
The following equation describes the relationship between the output current (I) and the output voltage (V) in a photovoltaic cell:
$$\mathrm{I}={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_{0}\left[{\mathrm{e}}^{\left({\displaystyle \frac{\mathrm{V}+{\mathrm{IR}}_{\mathrm{S}}}{{\mathrm{nV}}_{\mathrm{T}}}}\right)}-1\right]-{\displaystyle \frac{\left(\mathrm{V}+{\mathrm{IR}}_{\mathrm{s}}\right)}{{\mathrm{R}}_{\mathrm{sh}}}}$$
where ${\mathrm{I}}_{\mathrm{D}}$, ${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{\mathrm{sh}}$, and ${\mathrm{I}}_{0}$ stand, respectively, for diode current, the current that is generated by the light hitting the solar cell, the current that is lost because of shunt resistances, and saturation current. Both shunt resistance and series resistance are represented by the symbols ${\mathrm{R}}_{\mathrm{sh}}$ and ${\mathrm{R}}_{\mathrm{s}}$, respectively. The symbol for thermal voltage is denoted by ${V}_{T}$, the value of Boltzmann’s constant is (1.381 × ${10}^{-23}$$\raisebox{1ex}{$\mathrm{J}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{K}$}\right.}$), and denoted by K, and the value for elementary charge, q, is equal to $1.602\times {10}^{-19}\complement $. The ideality factor of a diode is represented by n, where G is the solar irradiation in watts per square meter ($\raisebox{1ex}{$\mathrm{W}$}\!\left/ \!\raisebox{-1ex}{${\mathrm{m}}^{2}$}\right.}$), T is the operating temperature, ${\mathrm{I}}_{\mathrm{ph}}$ is the photon current, and the open circuit voltage and the short circuit current are represented by ${\mathrm{V}}_{\mathrm{OC}}$ and ${\mathrm{I}}_{\mathrm{sc}}$, respectively [26]. It is possible for the PV’s load terminal to be shorted, resulting in a short circuit, and the associated short-circuit current (${\mathrm{I}}_{\mathrm{sc}}$) will be (substituting V = 0 in Equation (2)). If the solar PV system’s load terminal is an open circuit, take the following steps: (Place I = 0 in (Equation (2)) at this point).
$${\mathrm{I}}_{\mathrm{Ph}}-{\mathrm{I}}_{0}\left[\mathrm{exp}\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{OC}}}{{\mathrm{V}}_{\mathrm{t}}}}\right)-1\right]-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{OC}}}{{\mathrm{R}}_{\mathrm{S}}}}=0$$
The short-circuit current (${\mathrm{I}}_{\mathrm{sc}}$) is
$${\mathrm{I}}_{\mathrm{SC}}={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_{0}\left[\mathrm{exp}\left({\displaystyle \frac{{\mathrm{I}}_{\mathrm{SC}}{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{nV}}_{\mathrm{t}}}}\right)-1\right]-{\displaystyle \frac{{\mathrm{I}}_{\mathrm{SC}}{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{Sh}}}}$$
If we substitute ${\mathrm{V}}_{\mathrm{MP}}$ and Imp into Equation (2), then we obtain the following:
$${\mathrm{I}}_{\mathrm{MP}}={\mathrm{I}}_{\mathrm{ph}}-{\mathrm{I}}_{0}\left[\mathrm{exp}\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{MP}}+{\mathrm{I}}_{\mathrm{MP}}{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{nV}}_{\mathrm{t}}}}\right)-1\right]-{\displaystyle \frac{{\mathrm{V}}_{\mathrm{MP}}+{\mathrm{I}}_{\mathrm{MP}}{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{Sh}}}}$$
Since the PV characteristic curve is aligned in the same direction as the voltage axis at MPP, at MPP: $\frac{\mathrm{dP}}{\mathrm{dV}}}=0$.
When we solve for that condition, we obtain the following:
$$\frac{\mathrm{dI}}{\mathrm{dV}}}=-{\displaystyle \frac{{\mathrm{I}}_{\mathrm{MP}}}{{\mathrm{V}}_{\mathrm{MP}}}$$
The result of solving Equations (2) and (6) is as follows:
$${\mathrm{I}}_{\mathrm{MP}}=\left({\mathrm{V}}_{\mathrm{MP}}-{\mathrm{I}}_{\mathrm{MP}}{\mathrm{R}}_{\mathrm{S}}\right)\times \left({\displaystyle \frac{{\mathrm{I}}_{0}}{{\mathrm{nV}}_{\mathrm{t}}}}\left[\mathrm{exp}\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{MP}}+{\mathrm{I}}_{\mathrm{MP}}{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{V}}_{\mathrm{t}}}}\right)\right]+{\displaystyle \frac{1}{{\mathrm{R}}_{\mathrm{Sh}}}}\right)$$
In short-circuit conditions, the I–V slope in PV is as follows:
$$\frac{\mathrm{dI}}{\mathrm{dV}}}{|}_{\mathrm{I}={\mathrm{I}}_{\mathrm{SC}}}=-{\displaystyle \frac{1}{{\mathrm{R}}_{\mathrm{SC}}}$$
As a result of solving this equation, we obtain the following:
$$\frac{{\mathrm{I}}_{0}}{{\mathrm{V}}_{\mathrm{t}}}}\left[\mathrm{exp}\left({\displaystyle \frac{{\mathrm{I}}_{\mathrm{SC}}{\mathrm{R}}_{\mathrm{S}}}{n{\mathrm{V}}_{\mathrm{t}}}}\right)\right]\left[{\mathrm{R}}_{\mathrm{Sh}}-{\mathrm{R}}_{\mathrm{S}}\right]={\displaystyle \frac{{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{Sh}}}$$
2.2. Optimized the Characteristic Equation of PV
First, we derive the characteristics equation of solar cells in terms of ${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{\mathrm{O}}$, ${\mathrm{V}}_{\mathrm{T}}$, ${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{R}}_{\mathrm{sh}}$, in Equation (2). In this section, we derive the characteristic equation in terms of (${\mathrm{R}}_{\mathrm{s}},{\mathrm{V}}_{\mathrm{T}}$) or (${\mathrm{R}}_{\mathrm{s}}$, n). So, we then determine all the variables for the PV cell’s single-diode setup using these variables in [27,28].
From the value of ${\mathrm{I}}_{\mathrm{Ph}}$ using Equation (3), we obtain the following:
$${\mathrm{I}}_{\mathrm{Ph}}={\mathrm{I}}_{0}\left[\mathrm{exp}\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{OC}}}{{\mathrm{nV}}_{\mathrm{t}}}}\right)-1\right]+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{OC}}}{{\mathrm{R}}_{\mathrm{S}}}}$$
The value of ${\mathrm{I}}_{\mathrm{Ph}}$ is substituting in Equations (4) and (5), we obtain the following:
$${\mathrm{I}}_{\mathrm{SC}}=\left(\mathsf{\beta}-\mathsf{\alpha}\right){\mathrm{I}}_{0}+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{OC}}-{\mathrm{I}}_{\mathrm{SC}}{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{sh}}}}$$
$${\mathrm{I}}_{\mathrm{mp}}=\left(\mathsf{\beta}-\mathsf{\gamma}\right){\mathrm{I}}_{0}+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{OC}}-{\mathrm{V}}_{\mathrm{mp}}-{\mathrm{I}}_{\mathrm{mp}}{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{sh}}}}$$
From Equation (8), we obtain the value of ${\mathrm{I}}_{0}$ as follows:
$${\mathrm{I}}_{0}={\displaystyle \frac{n{\mathrm{V}}_{\mathrm{t}}{\mathrm{R}}_{\mathrm{S}}}{\left(1+\mathsf{\alpha}\right){\mathrm{R}}_{\mathrm{Sh}}\left({\mathrm{R}}_{\mathrm{Sh}}-{\mathrm{R}}_{\mathrm{S}}\right)}}$$
Placing the Value of ${\mathrm{I}}_{0}$ in Equation (7), we obtain the following:
$${\mathrm{I}}_{\mathrm{MP}}=\left({\mathrm{V}}_{\mathrm{MP}}-{\mathrm{I}}_{\mathrm{MP}}{\mathrm{R}}_{\mathrm{S}}\right)\times \left({\displaystyle \frac{{\mathrm{R}}_{\mathrm{S}}\left(1+\mathsf{\gamma}\right)}{{\mathrm{R}}_{\mathrm{Sh}}\left(1+\mathsf{\alpha}\right)\left({\mathrm{R}}_{\mathrm{Sh}}-{\mathrm{R}}_{\mathrm{S}}\right)}}+{\displaystyle \frac{1}{{\mathrm{R}}_{\mathrm{Sh}}}}\right)$$
Again, we place the value of ${\mathrm{I}}_{0}$ from Equation (12) in Equations (10) and (11) we obtain the following:
$${\mathrm{I}}_{\mathrm{SC}}={\displaystyle \frac{\left(\mathsf{\beta}-\mathsf{\alpha}\right){\mathrm{nV}}_{\mathrm{t}}{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{Sh}}\left(1+\mathsf{\alpha}\right)\left({\mathrm{R}}_{\mathrm{Sh}}-{\mathrm{R}}_{\mathrm{S}}\right)}}+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{OC}}-{\mathrm{I}}_{\mathrm{SC}}{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{sh}}}}$$
$${\mathrm{I}}_{\mathrm{mp}}={\displaystyle \frac{\left(\mathsf{\beta}-\mathsf{\gamma}\right){\mathrm{nV}}_{\mathrm{t}}{\mathrm{R}}_{\mathrm{S}}}{\left(1+\mathsf{\alpha}\right){\mathrm{R}}_{\mathrm{Sh}}\left({\mathrm{R}}_{\mathrm{Sh}}-{\mathrm{R}}_{\mathrm{S}}\right)}}+{\displaystyle \frac{{\mathrm{V}}_{\mathrm{OC}}-{\mathrm{V}}_{\mathrm{mp}}-{\mathrm{I}}_{\mathrm{mp}}{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{sh}}}}$$
Comparing the value ${\mathrm{I}}_{\mathrm{mp}}$ using Equations (13) and (15) we obtain the following:
$${\mathrm{R}}_{\mathrm{sh}}={\mathrm{R}}_{\mathrm{S}}\left\{1+{\displaystyle \frac{({\mathrm{nV}}_{\mathrm{t}}\left(\mathsf{\beta}-\mathsf{\gamma}\right)-\left(1+\mathsf{\gamma}\right)\left({\mathrm{V}}_{\mathrm{mp}}-{\mathrm{I}}_{\mathrm{mp}}{\mathrm{R}}_{\mathrm{S}}\right)}{\left(2{\mathrm{V}}_{\mathrm{mp}}-{\mathrm{V}}_{\mathrm{OC}}\right)\left(1+\mathsf{\alpha}\right)}}\right\}$$
Again, we solve the Equations (10) and (11) we obtain the following:
$${\mathrm{R}}_{\mathrm{sh}}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{OC}}\left(\mathsf{\alpha}-\mathsf{\gamma}\right)+{\mathrm{I}}_{\mathrm{SC}}{\mathrm{R}}_{\mathrm{S}}\left(\mathsf{\gamma}-\mathsf{\beta}\right)+\left({\mathrm{V}}_{\mathrm{MP}}-{\mathrm{I}}_{\mathrm{MP}}{\mathrm{R}}_{\mathrm{S}}\right)\left(\mathsf{\beta}-\mathsf{\alpha}\right)}{{\mathrm{I}}_{\mathrm{SC}}\left(\mathsf{\beta}-\mathsf{\gamma}\right){\mathrm{I}}_{\mathrm{MP}}\left(\mathsf{\alpha}-\mathsf{\beta}\right)}}$$
where [$\mathsf{\alpha}=\left\{\mathrm{exp}\left({\displaystyle \frac{{\mathrm{I}}_{\mathrm{SC}}{\mathrm{R}}_{\mathrm{S}}}{n{\mathrm{V}}_{\mathrm{t}}}}\right)-1\right\}$, $\mathsf{\beta}=\left\{\mathrm{exp}\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{OC}}}{n{\mathrm{V}}_{\mathrm{t}}}}\right)-1\right\}$, and $\mathsf{\gamma}=\left\{\mathrm{exp}\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{mp}}+{\mathrm{I}}_{\mathrm{mp}}{\mathrm{R}}_{\mathrm{S}}}{{\mathrm{nV}}_{\mathrm{t}}}}\right)-1\right\}$].
After the analysis of Equations (16) and (17), we obtain the optimized characteristics equation in terms of (${\mathrm{V}}_{\mathrm{T}}$, ${\mathrm{R}}_{\mathrm{s}}$):
$$\mathrm{F}\left({\mathrm{R}}_{\mathrm{S}},{\mathrm{V}}_{\mathrm{t}}\right)=\left\{{\mathrm{R}}_{\mathrm{S}}\left(1+{\displaystyle \frac{{\mathrm{nV}}_{\mathrm{T}}\left(\mathsf{\beta}-\mathsf{\gamma}\right)-\left(1+\mathsf{\gamma}\right)\left({\mathrm{V}}_{\mathrm{mp}}+{\mathrm{I}}_{\mathrm{mp}}{\mathrm{R}}_{\mathrm{S}}\right)}{\left(2{\mathrm{V}}_{\mathrm{mp}}-{\mathrm{V}}_{\mathrm{OC}}\right)\left(1+\mathsf{\alpha}\right)}}\right)-\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{OC}}\left(\mathsf{\alpha}-\mathsf{\gamma}\right)+{\mathrm{I}}_{\mathrm{SC}}{\mathrm{R}}_{\mathrm{S}}\left(\mathsf{\gamma}-\mathsf{\beta}\right)+\left({\mathrm{V}}_{\mathrm{mp}}+{\mathrm{I}}_{\mathrm{mp}}{\mathrm{R}}_{\mathrm{S}}\right)\left(\mathsf{\beta}-\mathsf{\gamma}\right)}{{\mathrm{I}}_{\mathrm{SC}}\left(\mathsf{\beta}-\mathsf{\gamma}\right){\mathrm{I}}_{\mathrm{mp}}\left(\mathsf{\alpha}-\mathsf{\beta}\right)}}\right)\right\}$$
Equation (18) shows the optimized characteristics equation of solar PVs in terms of (${\mathrm{R}}_{\mathrm{S}},{\mathrm{V}}_{\mathrm{t}}$).
2.3. Effect of Irradiance and Temperature on PV Cell
One way to measure the amount of energy that the sun reaches a specific spot-on Earth is by looking at its irradiance, which is measured in $\mathrm{W}/{\mathrm{m}}^{2}$. Irradiance is the collective amount of energy from incoming sunlight per square meter. It is crucial to remember that irradiation is the term used to assess the energy content of sunshine. It is worth mentioning that the photocurrent produced is directly linked to the intensity of solar radiation, commonly known as irradiance. As irradiance increases, ${\mathrm{I}}_{\mathrm{sc}}$ also increases linearly. Open-circuit voltage $\left({\mathrm{V}}_{\mathrm{oc}}\right)$ is also proportional to the photocurrent, so if there is an increment in irradiance, the ${\mathrm{V}}_{\mathrm{oc}}$ increases logarithmically. The open-circuit voltage $\left({\mathrm{V}}_{\mathrm{oc}}\right)$ of the solar cell is most strongly influenced by temperature changes. This results from the parameter ${\mathrm{I}}_{0}$, which is temperature-dependent and reduces the open-circuit voltage as the temperature rises.
$${\mathrm{V}}_{\mathrm{oc}}={\mathrm{nV}}_{\mathrm{T}}\mathrm{ln}\left\{{\displaystyle \frac{{\mathrm{I}}_{\mathrm{ph}}+{\mathrm{I}}_{0}}{{\mathrm{I}}_{0}}}\right\}$$
As ${\mathrm{I}}_{\mathrm{ph}}\gg {\mathrm{I}}_{0}$ so $\left({\mathrm{I}}_{\mathrm{ph}}+{\mathrm{I}}_{0}\right)\cong {\mathrm{I}}_{\mathrm{ph}}$ substituting this value in the above Equation (19) we obtain the following:
$${\mathrm{V}}_{\mathrm{oc}}={\mathrm{nV}}_{\mathrm{T}}\mathrm{ln}\left\{{\displaystyle \frac{{\mathrm{I}}_{\mathrm{ph}}}{{\mathrm{I}}_{0}}}\right\}$$
where ${\mathrm{I}}_{0}{\mathsf{\alpha}\mathrm{T}}^{\mathrm{m}}{\mathrm{e}}^{{\displaystyle \frac{-{\mathrm{V}}_{{\mathrm{G}}_{0}}}{{\mathrm{nV}}_{\mathrm{T}}}}}$, and the following:
$${\mathrm{I}}_{0}={\mathrm{KT}}^{\mathrm{m}}{\mathrm{e}}^{{\displaystyle \frac{-{\mathrm{V}}_{{\mathrm{G}}_{0}}}{{\mathrm{nV}}_{\mathrm{T}}}}}$$
Taking ‘log’ on both sides of the above Equation (21) we obtain the following:
$$\frac{\mathrm{d}\mathrm{ln}\left({\mathrm{I}}_{0}\right)}{\mathrm{dT}}}={\displaystyle \frac{\mathrm{M}}{\mathrm{T}}}+{\displaystyle \frac{{\mathrm{V}}_{{\mathrm{G}}_{0}}}{{\mathrm{nTV}}_{\mathrm{T}}}$$
Now, differentiating Equation (20), we can write the following:
$$\frac{\mathrm{d}\left(\frac{{\mathrm{V}}_{\mathrm{oc}}}{{\mathrm{nV}}_{\mathrm{T}}}\right)}{\mathrm{dT}}}={\displaystyle \frac{\mathrm{d}\mathrm{ln}\left({\mathrm{I}}_{\mathrm{Ph}}\right)}{\mathrm{dT}}}-{\displaystyle \frac{\mathrm{d}\mathrm{ln}({\mathrm{I}}_{0})}{\mathrm{dT}}$$
Substituting Equation (23) with Equation (22), we obtain the following:
$$\frac{\mathrm{d}({\mathrm{V}}_{\mathrm{oc}})}{\mathrm{dT}}}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{oc}}-\left({\mathrm{V}}_{{\mathrm{G}}_{0}}+{\mathrm{mnV}}_{\mathrm{T}}\right)}{\mathrm{T}}$$
According to Equation (24), as the temperature (T) increases, the open-circuit voltage (${\mathrm{V}}_{\mathrm{oc}}$) decreases, resulting in a reduction in PV cell output.
2.4. Standard Equation for Mathematical Modeling
The utilization of mathematical modeling to examine the behavior of a photovoltaic (PV) cell array using a single-diode design has grown significantly in significance. The complicated impact of various parameters on the effectiveness of solar cells is elucidated in part because of this methodology. They carefully assess the effects of these characteristics on the performance of solar cells using mathematical modeling techniques. The main goal in carrying out this extensive investigation is to enhance the operational effectiveness of both separate solar cells as well as integrated modules. The mathematical equation required for the modeling of a single-diode configuration of a PV cell is as follows:
Reverse Saturation Current:
$${\mathrm{I}}_{\mathrm{RS}}={\mathrm{e}}^{\left({\displaystyle \frac{\frac{{\mathrm{I}}_{\mathrm{SC}}}{\mathrm{q}\times {\mathrm{V}}_{\mathrm{OC}}}}{\mathrm{n}\times {\mathrm{N}}_{\mathrm{s}}\times \mathrm{K}\times \mathrm{T}}}\right)}-1$$
Saturation Current:
$${\mathrm{I}}_{\mathrm{O}}={\mathrm{I}}_{\mathrm{RS}}{({\displaystyle \frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{n}}}})}^{3}\times {\mathrm{e}}^{\left[{\displaystyle \raisebox{1ex}{$\mathrm{q}\times {\mathrm{E}}_{\mathrm{go}}\times \left(\frac{1}{{\mathrm{T}}_{\mathrm{n}}}-\frac{1}{\mathrm{T}}\right)$}\!\left/ \!\raisebox{-1ex}{$\mathrm{n}\times \mathrm{K}$}\right.}\right]}$$
Photon Current:
$${\mathrm{I}}_{\mathrm{ph}}=\left[{\mathrm{I}}_{\mathrm{sc}}+{\mathrm{K}}_{\mathrm{i}}\left(\mathrm{T}-298\right)\right]\times {\displaystyle \frac{\mathrm{G}}{1000}}$$
Current Throw Shunt Resistance:
$${\mathrm{I}}_{\mathrm{sh}}=\left({\displaystyle \frac{\mathrm{V}+{\mathrm{IR}}_{\mathrm{S}}}{{\mathrm{R}}_{\mathrm{sh}}}}\right)$$
The given equation is a commonly used standard for the mathematical modeling of solar systems.
3. Optimization Techniques
The ideal, single-diode and double-diode photovoltaic (PV) models are all well-documented in the scientific literature. Because of its ease of use and great reliability, the single-diode variant has a high degree of application. The exact model parameters of a PV module are required for accurate modeling. However, the manufacturer does not provide all the parameters required for modeling. The datasheet of a PV module usually lists some STC parameters, including ${\mathrm{V}}_{\mathrm{oc}}$, ${\mathrm{I}}_{\mathrm{sc}}$, ${\mathrm{I}}_{\mathrm{MPP}}$, and ${\mathrm{V}}_{\mathrm{MPP}}$. Therefore, one of the parameter estimation methods must be used to estimate the values of the unknown parameters of the single-diode model, such as ${\mathrm{R}}_{\mathrm{S}}$, ${\mathrm{R}}_{\mathrm{Sh}}$, diode ideality factor, ${\mathrm{I}}_{\mathrm{Ph}}$, and ${\mathrm{I}}_{\mathrm{Sh}}$. The properties of the model are highly dependent on these variables, so their accurate evaluation is critical. The literature has released multiple methods for estimating parameters. The literature contains extensive research on various parameter estimation strategies. However, there are still several obstacles to determining the precise parameters of real-world models. Researchers turn to optimization techniques to solve the problem by using various techniques to find the unknown parameters of PVs. Optimization in science involves finding the best solution from a set of options. One or more personalized criteria and conditions influence the optimality of solutions. Thus, the user or problem usually limits the set of feasible solutions. Feasible solutions satisfy all limitations, so, “viable solution” was introduced. Determining the best solution from a group of choices is a global optimization issue. Sometimes, non-feasibility is possible. Local optimization refers to situations where poor solutions are accepted because they are better than the ideal alternative. These challenges necessitated the development of modern meta-heuristic optimization strategies for precisely solving fundamental nonlinear problems. Using optimization algorithms that take their cues from biology, we can solve nonlinear transcendental equations without resorting to elaborate calculus. As a result, many different optimization strategies have been applied recently for PV module parameter estimation. The unknown parameters of solar PVs are discussed here, along with genetic algorithms, particle swarm optimization, and purposed two-stage optimization (GPSO).
3.1. Genetic Algorithm
Genetic algorithms (GAs) are a type of search and optimization procedure inspired by the tenets of evolution and natural selection. Artificially constructed search algorithms that are both robust and require minimum problem information are created by borrowing certain fundamental notions from genetics. It is important to note that the GA operates on a fundamentally different concept than traditional optimization methods.
The GA estimation process commences by generating a population of individual chromosomes as shown in Figure 3. The individual serves as the candidate’s proposed resolution to the specified problem. The fitness value of each population is determined based on the specified criterion, also known as the objective function. The optimal individual is chosen through the selection process to create a fresh population. Next, the parental individuals are chosen for the purpose of reproduction and subjected to the processes of crossover and mutation to generate novel solutions. The iterative procedure will continue until the problem satisfies the predetermined termination conditions. To find the unknown parameter of the single-diode configuration of solar cells using genetic algorithms, first, we derive the cost function in terms of unknown parameters (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, ${\mathrm{V}}_{\mathrm{t}}$, ${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{R}}_{\mathrm{sh}}$) of the single-diode configuration of PV cells. Once the function’s population has been established, the characteristic equation’s cost function is assessed. Then, we start the selection process, and after, we perform the crossover and mutation. Then, the convergence criteria of the algorithm is checked. If convergence criteria are met, the iteration is stopped, and the optimal parameters value is obtained; otherwise, the next step commences. This whole process will continue till the convergence conditions are not met.
3.2. Particle Swarm Optimization
In 1995, Eberhart and Kennedy introduced the Particle Swarm Optimization (PSO) technique for the purpose of solving nonlinear functions. The PSO algorithm is one form of population-based search; it mimics the coordinated activity of a flock of birds. The aim of this strategy is to quickly identify the most effective solution to a problem. The idea of a particle swarm was developed to visualize the fluid and unpredictable behavior of a flock of birds. The goal of this study was to better understand how flocks of birds can fly in unison and quickly change course while re-forming themselves in the most efficient formation. This primary goal prompted the creation of a simple yet powerful optimization method, which was then put into practice. PSO compares people to particles and leads them through complex hyperdimensional search areas. These particle placements inside the search domain are adaptively altered, indicating people’s social-psychological propensity to strive for the same levels of success as those seen among their peers. Because of this, a particle’s evolution within the swarm will be affected by the experiences and knowledge of its neighbors. Therefore, the search behaviors of a particle are influenced by the search behaviors of other particles in the swarm (hence PSO’s classification as a type of symbiotic cooperative algorithm). As a direct result of modeling these social behaviors, the search method has been built in such a way that particles will sporadically return toward previously successful regions in the search space. This is performed to facilitate the process. This is something that happens as a direct result of the social behaviors being represented by the particles. A set number of “particles,” which act as searching agents, are initially chosen by the algorithm. Every particle participated in the search according to two basic rules: first, it moved in the same general direction as the best particle, and second, it moved in the same general direction as the best location attained by the best particle. Each particle in the search process modifies its position in accordance with Equations (29) and (30).
$${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{n}+1\right)=\mathrm{w}\left(\mathrm{n}\right){\mathrm{V}}_{\mathrm{i}}\left(\mathrm{n}\right)+{\mathrm{c}}_{1}{\mathrm{r}}_{1}\left({\mathrm{P}}_{\mathrm{best},\mathrm{i}}-{\mathrm{X}}_{\mathrm{i}}\left(\mathrm{n}\right)\right)+{\mathrm{c}}_{2}{\mathrm{r}}_{2}\left({\mathrm{G}}_{\mathrm{best}}-{\mathrm{X}}_{\mathrm{i}}\left(\mathrm{n}\right)\right)$$
$${\mathrm{X}}_{\mathrm{i}}\left(\mathrm{n}+1\right)={\mathrm{X}}_{\mathrm{i}}\left(\mathrm{n}\right)+{\mathrm{V}}_{\mathrm{i}}\left(\mathrm{n}+1\right)$$
where ${\mathrm{X}}_{\mathrm{i}}$ represents the location of the ith particle, ${\mathrm{V}}_{\mathrm{i}}$ is its velocity, and ‘n’ is the number of iterations. Coefficients ${\mathrm{c}}_{1}$ and ${\mathrm{c}}_{2}$ stand for the corresponding learning factors, while ‘w’ is the corresponding inertia weight. The inertia weight strikes a balance between in-depth exploration and broad surveying. Both ${\mathrm{r}}_{1}$ and ${\mathrm{r}}_{1}$ are arbitrary values in the range [0, 1]. ${\mathrm{P}}_{\mathrm{best},\mathrm{i}}$ is the best location for the ith particle and ${\mathrm{G}}_{\mathrm{best}}$ is the best position of all the particles found so far in this. The inertia weight is chosen so that it is higher at the outset for good exploration and lowers gradually for a precise solution, resulting in faster convergence.
Different search and optimization methods are used to find out the parameters of the single-diode models for the solar cell. The characteristics of the model must be reproduced to implement an optimization or search algorithm. Cell and module parameters can be determined in a variety of ways, including using datasheet parameters or I/V curve measurements. Single-diode model specifications are not directly provided by PV module makers. Here are some of the standard test conditions (STC) values: open circuit voltage (${\mathrm{V}}_{\mathrm{OC}}$), short circuit current (${\mathrm{I}}_{\mathrm{SC}}$), voltage at MPP (${\mathrm{V}}_{\mathrm{MP}}$), current at MPP $\left({\mathrm{I}}_{\mathrm{MP}}\right)$, maximum power $\left({\mathrm{P}}_{\mathrm{max}}\right)$, and temperature coefficient of ${\mathrm{V}}_{\mathrm{OC}}$ and ${\mathrm{I}}_{\mathrm{SC}}$. The electrical model’s unknown parameters can be connected to the datasheet information by analyzing the electrical model at its three I–V curve operation points (${\mathrm{I}}_{\mathrm{SC}}$, ${\mathrm{V}}_{\mathrm{OC}}$, and ${\mathrm{P}}_{\mathrm{max}}$). All parameter estimate methods follow the same procedure: using I–V or P–V curves from manufacturer datasheets, the unknown parameters are estimated at the STC, followed by simulations at other conditions. Each I–V or P–V curve is then given a metric, which is then computed. When all five estimated parameters have been evaluated under various STC circ*mstances, the electrical model is ready. I–V curves can be used to properly determine the amount of power extracted from the electrical model.
The absolute inaccuracy in power for a specific voltage point is determined as follows:
$$\mathrm{Error}={\mathrm{P}}_{\mathrm{Actual}}-{\mathrm{P}}_{\mathrm{experimental}}$$
To determine the mean absolute error in power (MAEP), the following is used:
$$\mathrm{MAEP}={\displaystyle \frac{\u2a0a\mathrm{Error}}{{\mathrm{N}}_{\mathrm{curve}}}}$$
where, ${\mathrm{N}}_{\mathrm{curve}}$ is points taken from the datasheet I–V curve. In this study, the PV cell’s single-diode configuration’s unknown parameters were determined using the PSO. In this case, we will assume the five parameters of the PV cell (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, ${\mathrm{V}}_{\mathrm{t}}$, ${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{R}}_{\mathrm{sh}}$) are unknown and use PSO in MATLAB to try to determine the actual value or best values of these parameters. By using mathematical modeling, we first collect voltage and current data at discrete places and use these values as input parameters for the characteristic equation of the single-diode configuration in PV systems. This procedure yields the previously unknown parameters. The PSO method’s flow chart, which is used to extract the PV cell parameter, is shown in Figure 4. Additionally, Table 1 displays the PSO algorithm’s parameter values, which can be initialized here.
3.3. Proposed Two-Step-Based Optimization Techniques
Genetic Particle Swarm Optimization (GPSO) is a meta-heuristic optimization technique that is inspired by the collective behavior observed in social insects. It combines the principles of PSO and GA to optimize complex problems. GPSO shares characteristics with both PSOs and GAs, including the ability to search for solutions in a high-dimensional space, a focus on exploration and exploitation, and the ability to handle multiple objectives. GPSO is most used to solve non-convex optimization problems, such as multi-objective optimization, global optimization, and combinatorial optimization. The technique can also be used to solve a variety of engineering design problems, including design optimization, parameter estimation, and control optimization. The proposed two-step optimizing technique is used for parameter estimation of the PV cell/model. In these techniques, first, we extract the parameters using a genetic algorithm and then choose the optimal parameters solution after performing all the steps involving the genetic algorithm. Then, we estimate these optimal parameter values using article swarm optimization and obtain the estimated parameter value.
Flow chart of Proposed Algorithm:
Step 1: Genetic Algorithm (GA) for (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, ${\mathrm{V}}_{\mathrm{t}}$, ${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{R}}_{\mathrm{sh}}$) estimation using Equation (2).
- ❖
Initialize GA population with random values for (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, ${\mathrm{V}}_{\mathrm{t}}$, ${\mathrm{R}}_{\mathrm{s}}$ And ${\mathrm{R}}_{\mathrm{sh}}$);
- ❖
Evaluate the fitness of each individual using the cost function derived from the PV model equations;
- ❖
Perform selection, crossover, and mutation operations;
- ❖
Obtain optimal values of (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, ${\mathrm{V}}_{\mathrm{t}}$, ${\mathrm{R}}_{\mathrm{s}}$ And ${\mathrm{R}}_{\mathrm{sh}}$) from the best individual
Step 2: Particle Swarm Optimization (PSO) for (${\mathrm{R}}_{\mathrm{s}}$ and ${\mathrm{V}}_{\mathrm{t}}$) estimation using Equation (18).
- ❖
Initialize PSO parameters (swarm size, inertia weight, acceleration coefficients, etc.);
- ❖
Initialize particle positions with random values for (${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{V}}_{\mathrm{t}}$), using optimal (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, and ${\mathrm{R}}_{\mathrm{sh}}$) from Step 1;
- ❖
Evaluate the fitness of each particle using the modified cost function with (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, and ${\mathrm{R}}_{\mathrm{sh}}$) fixed;
- ❖
Set personal best (P-best) and global best (G-best) position;
- ❖
For each particle, perform the following:
- ❖
Update particle velocity and position using PSO equations;
- ❖
Evaluate particle fitness using the modified cost function
- ❖
Update P-best and G-best;
- ❖
Obtain optimal values of (${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{V}}_{\mathrm{t}}$) from G-best
- ❖
Check Convergence Criteria
- ❖
If not met, then perform the following:
- ❖
Return to step 1 with the obtained values of (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, ${\mathrm{V}}_{\mathrm{t}}$, ${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{R}}_{\mathrm{sh}}$).
- ❖
If Convergence Criteria are met:
- ❖
End
For parameter estimation of solar cells using GPSO, first, we instilled the parameter for the genetic algorithm and then initialized the population. For the cost function for genetic algorithm-based parameter extraction, we derive Equation (2) in terms of five parameters of the PV cell (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, ${\mathrm{V}}_{\mathrm{t}}$, ${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{R}}_{\mathrm{sh}}$) which are unknown. After, we used the roulette wheel method for the selection process, and then in the next step, we performed the crossover and mutation. Then, we optioned the optimal value of the unknown parameters. After, we jumped to the next steps and initialized the parameter value and cost function for the PSO method. For this, we used the modified cost function derived in (18) for a single-diode configuration of PVs. For this, if only ${\mathrm{R}}_{\mathrm{s}}$ and ${\mathrm{V}}_{\mathrm{t}}$ are variables, and the remaining parameter is taken as a constant value optioned by GA techniques. For the finding of ${\mathrm{R}}_{\mathrm{s}}$ and ${\mathrm{V}}_{\mathrm{t}}$, First, we find the fitness value of each particle and then find the best individual fitness values and best global fitness values. Next, we evaluate all the particles and update the particle’s velocity and position. Achieving the objective function to reach the specified minimum value or to reach the maximum number of iterations is considered a stop criterion. In this study, it is accepted as a criterion to reach the maximum number of iterations to limit the calculation time. If not, we then progressed to the next iteration. Otherwise, we stopped and obtained the best-estimated value of a PV cell (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, ${\mathrm{V}}_{\mathrm{t}}$, ${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{R}}_{\mathrm{sh}}$).
4. Results and Discussion
First, we extracted the parameters (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, n, ${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{R}}_{\mathrm{sh}}$) of the PV cell single-diode configuration using GA, PSO, and GPSO. The initial parameters for the GPSO algorithm are shown in Table 1. The extracted parameter values using these three algorithms are shown in Table 2. Details requiring a set of points representing input voltage and predicted current can be found in reference [11]. Additionally, the value of the extracted parameter is compared to the one found in [11].
The I–V and P–V characteristic curves of a single-diode PV cell are shown in Figure 5a,b, along with a comparison to the estimation parameters derived from the use of the suggested two-step GPSO methodology. Table 3 shows the compression between % relative change in a PV cell parameter value extracted using the proposed techniques and extracted parameter values in reference [11].
After extracting the PV cell parameters of single-diode configurations, we extracted the parameters of the PV model Kyocera (KC200GT) using GA, PSO, and the proposed two-step optimization techniques. Table 4 shows the parameters and rating of KC200GT PV modules provided by the manufacturers, and Table 5 shows the extracted parameters (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, ${\mathrm{V}}_{\mathrm{t}}$, ${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{R}}_{\mathrm{sh}}$) values using GA, PSO, and the proposed techniques.
Figure 6 shows the estimated parameter (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, ${\mathrm{V}}_{\mathrm{t}}$, ${\mathrm{R}}_{\mathrm{s}}$. and ${\mathrm{R}}_{\mathrm{sh}}$) values using PSO and GPSO for each iteration. A total of 101 values show the mean values of the parameter values obtained in 100 iterations. The best parameter values for PSO showed more variety when we analyzed all iterations. As a result, using PSO to establish the ideal parameter value can be difficult. To overcome this problem, we presented two-stage optimization methods in this paper. The parameter value extracted using the proposed two-step optimization technique was compared to the GA and PSO approaches; the fluctuation in the optimal parameter values in each iteration was less.
Table 6 presents the comparative chart between the optimization techniques that have been used to obtain the unknown parameters of the Kyocera (KC200GT) module. In this section, the extracted parameters of previously established methods, such as GWO [15], Villalva’s Method [21], Accarino’s Method [22], Iterative Method [23], and Silva’s Mehod [24], are directly taken from the references and compared with the GA, PSO, and proposed GPSO-based extraction techniques.
Figure 7a,b shows the output PV curve of the KC200GT PV module using PSO, GA, and GPSO. The performance of the proposed GPSO-based optimization techniques was better compared to other optimization techniques.
The relative error for power was calculated for GA, PSO, and GPSO methods to determine the efficacy of the suggested model. Comparisons were made between GA, PSO, and GPSO approaches and other methods in the literature about the relative error in computed power. Relative error for power (${\mathrm{RE}}_{\mathrm{P}}$) is computed using Equation (33) as follows:
$${\mathrm{RE}}_{\mathrm{P}}={\displaystyle \frac{{\mathrm{P}}_{\mathrm{A}}-{\mathrm{P}}_{\mathrm{C}}}{{\mathrm{P}}_{\mathrm{A}}}}$$
where ${\mathrm{P}}_{\mathrm{A}}$ and ${\mathrm{P}}_{\mathrm{C}}$ are the experimental and computed power, respectively. Table 7 analyzes the actual power and calculated power based on different approaches. In addition to this, we evaluated the relative power inaccuracy using various approaches. This illustration shows that using the proposed technique has lesser relative output power error compared to other approaches; it produces significantly better outcomes.
The fitness value curve is illustrated in Figure 8. The GPSO method proved to be highly effective in determining the unknown parameters of a PV cell/model, exhibiting a significantly faster convergence compared to other optimization techniques.
Equation (34) is utilized to compute the absolute error for current (${\mathrm{AE}}_{\mathrm{C}}$) as follows:
$${\mathrm{AE}}_{\mathrm{C}}=\left|{\mathrm{I}}_{\mathrm{A}}-{\mathrm{I}}_{\mathrm{C}}\right|$$
where ${\mathrm{I}}_{\mathrm{A}}$ and ${\mathrm{I}}_{\mathrm{C}}$ are the actual and calculated currents respectively. As most applications require a precise model to extract exact power, the suggested method utilizes P–V characteristics for error calculation; whereas, in most prior research, only the I–V curve is employed [15]. Power errors are also computed, such as current errors. The absolute power error (AEP) is determined by plugging in a specific voltage level into Equation (35).
$${\mathrm{AE}}_{\mathrm{P}}=\left|{\mathrm{P}}_{\mathrm{A}}-{\mathrm{P}}_{\mathrm{C}}\right|$$
Figure 9 and Figure 10 depict the computed absolute error graph for the current and power of the KC200GT PV panel at STC. Because the values of the current do not fluctuate very much in the current source region, the error obtained is typically smaller. The accuracy of the estimating approach is mostly determined by the magnitude of the error obtained in the voltage source region. The suggested GPSO approach yields lower inaccuracy in the current source region and the voltage source region compared to alternative methods.
Table 8 summarizes that the suggested GPSO technique is advantageous compared to traditional parameter estimation techniques. Firstly, it is more efficient since it does not require large computational resources. Secondly, it is more accurate since it allows for a more precise estimation of the parameters. Lastly, the technique is also faster since it can quickly identify the best set of parameters in a short period of time.
5. Conclusions
The development of the characteristic equation governing the single-diode configuration within a photovoltaic (PV) cell is the initial emphasis of this study, which is then followed by its evaluation under a range of other scenarios. It is also necessary to calculate the saturation current, the reverse saturation current, the photon current, and the current throw shunt resistance numerically in addition to the PV cell modeling. Each point’s voltage and current can be determined using mathematical modelling. The present study introduces a novel parameter estimate strategy that precisely determines the unknown parameters by applying the GPSO method. For the implementation of the proposed optimizing techniques, we drove the single-diode configuration PV equation in terms of ${\mathrm{R}}_{\mathrm{s}}$ and ${\mathrm{V}}_{\mathrm{t}}$. Then, the parameters of PV cells (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, n, ${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{R}}_{\mathrm{sh}}$) were estimated using the GA, PSO, and the proposed techniques in MATLAB. Additionally, we compared the estimated parameter value to a specific reference publication and displayed the relative change in parameter values and errors. Then, using the GA, PSO, and the proposed optimization techniques, we extracted an unknown parameter (${\mathrm{I}}_{\mathrm{ph}}$, ${\mathrm{I}}_{0}$, ${\mathrm{V}}_{\mathrm{t}}$, ${\mathrm{R}}_{\mathrm{s}}$, and ${\mathrm{R}}_{\mathrm{sh}}$) of a PV model from Kyocera (KC200GT), and it was discovered that the GPSO technique estimated the module parameters more accurately than other techniques. Table 5 illustrates the extracted parameter values for the Kyocera (KC200GT) PV model. Additionally, we compared the extracted parameter value for the Kyocera (KC200GT) PV model obtained using GPSO approaches with that obtained for the same PV model using other techniques developed by various authors. Moreover, the derived model parameters were verified by checking that the P–V characteristics agreed with the experimental curves. The results showed that the proposed GPSO method was preferable and the best one for estimating the PV module’s parameters.
Author Contributions
Conceptualization, M.K.; Methodology, M.K. and K.P.P.; Software, K.P.P.; Validation, K.P.P.; Investigation, M.K., R.T.N., R.T. and G.P.; Resources, R.T.N. and R.T.; Data curation, R.T.; Writing—original draft, M.K.; Writing—review & editing, G.P.; Visualization, R.T.N.; Supervision, G.P. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Data are contained within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
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Figure 1. Electricity generated using photovoltaic cell.
Figure 1. Electricity generated using photovoltaic cell.
Figure 2. Photovoltaic cell with a single-diode arrangement.
Figure 2. Photovoltaic cell with a single-diode arrangement.
Figure 3. Flow chart of genetic algorithm-based optimization.
Figure 3. Flow chart of genetic algorithm-based optimization.
Figure 4. Flow Chart of PSO technique.
Figure 4. Flow Chart of PSO technique.
Figure 5. I–V and P–V characteristic curves of a single-diode PV cell.
Figure 5. I–V and P–V characteristic curves of a single-diode PV cell.
Figure 6. Estimated parameters value of PV module KC200GT using PSO and GPSO techniques.
Figure 6. Estimated parameters value of PV module KC200GT using PSO and GPSO techniques.
Figure 7. PV curve of the KC200GT PV module.
Figure 7. PV curve of the KC200GT PV module.
Figure 8. Fitness curve parameter extraction using different methods.
Figure 8. Fitness curve parameter extraction using different methods.
Figure 9. Absolute error in the current of PV module KC200GT at STC.
Figure 9. Absolute error in the current of PV module KC200GT at STC.
Figure 10. Absolute error in the power of PV module KC200GT at STC.
Figure 10. Absolute error in the power of PV module KC200GT at STC.
Table 1. Parameters for optimization techniques.
Table 1. Parameters for optimization techniques.
Description | PSO | GPSO |
---|---|---|
Population Size | 20 | 20 |
Maximum Iteration | 100 | 100 |
Crossover probability | -- | 0.5 |
Mutation probability | -- | 0.01 |
C1 | 0.5 | 0.5 |
C2 | 2 | 2 |
Tolerance | 1 × 10^{−5} | 1 × 10^{−10} |
Table 2. Estimated PV cell parameter (single-diode configuration).
Table 2. Estimated PV cell parameter (single-diode configuration).
Parameter | [11] Reference Paper Parameters | Estimated Parameters Using GA | Estimated Parameters Using PSO | Estimated Parameters Using GPSO |
---|---|---|---|---|
${\mathrm{I}}_{\mathrm{ph}}$ | 0.7619 | 0.76973 | 0.7653 | 0.76078 |
${\mathrm{I}}_{0}$ | 8.085 × ${10}^{-7}$ | 4.82 × ${10}^{-7}$ | 7.164 × ${10}^{-7}$ | 3.242 × ${10}^{-7}$ |
n | 1.5751 | 1.5799 | 1.5619 | 1.5462 |
${\mathrm{R}}_{\mathrm{s}}$ | 0.03 | 0.0540 | 0.034 | 0.0363 |
${\mathrm{R}}_{\mathrm{sh}}$ | 42.3728 | 48.54369 | 43.628 | 53.742 |
Table 3. Relative change in PV cell parameters (single-diode configuration).
Table 3. Relative change in PV cell parameters (single-diode configuration).
Parameter | [11] Reference Paper Parameters | Relative Change in Parameters Using GA | Relative Change in Parameters Using PSO | Relative Change in Parameters Using GPSO |
---|---|---|---|---|
${\mathrm{I}}_{\mathrm{ph}}$ | 0.7619 | 0.28% | 0.446% | 0.147% |
${\mathrm{I}}_{0}$ | 0.8085 × ${10}^{-6}$ | 2.3% | 11.391% | 59.982% |
n | 1.5751 | 0.3% | 0.838% | 1.866% |
${\mathrm{R}}_{\mathrm{s}}$ | 0.03 | 44.4% | 13.34% | 21.07% |
${\mathrm{R}}_{\mathrm{sh}}$ | 42.3728 | 12.7% | 2.961% | 26.83% |
Table 4. Kyocera’s (KC200GT) PV module’s parameters.
Table 4. Kyocera’s (KC200GT) PV module’s parameters.
Parameters for Kyocera (KC200GT) | ${\mathbf{V}}_{\mathbf{m}\mathbf{p}}$ | ${\mathbf{I}}_{\mathbf{m}\mathbf{p}}$ | ${\mathbf{V}}_{\mathbf{o}\mathbf{c}}$ | ${\mathbf{I}}_{\mathbf{s}\mathbf{c}}$ | ${\mathbf{N}}_{\mathbf{s}}$ |
---|---|---|---|---|---|
Value | 26.3 | 7.61 | 32.9 | 8.21 | 54 |
Table 5. Kyocera’s (KC200GT) PV module’s extracted parameter using GA, PSO, and two-step optimization.
Table 5. Kyocera’s (KC200GT) PV module’s extracted parameter using GA, PSO, and two-step optimization.
Solar PV Module | Parameters to Be Extracted | Parameters Extraction Using Genetic Algorithm | Parameters Extraction Using Particle Swarm Optimization | Proposed (GPSO) Two-Step Optimization Techniques | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Lower Value | Upper Value | Estimated Parameter | Lower Value | Upper Value | Estimated Parameter | Lower Value | Upper Value | Estimated Parameter | ||
Kyocera KC200GT Modules | ${\mathrm{I}}_{\mathrm{ph}}$ | 8.1 | 8.3 | 8.2056 | 8.1 | 8.3 | 8.1958 | 8.1 | 8.3 | 8.21615 |
${\mathrm{I}}_{0}$ | 1 × 10^{−8} | 1 × 10^{−5} | 4.7278 × 10^{−7} | 1 × 10^{−8} | 1 × 10^{−5} | 5.38 × 10^{−7} | 1 × 10^{−8} | 1 × 10^{−5} | 4.315 × 10^{−7} | |
${\mathrm{V}}_{t}$ | 1.2 | 1.9 | 1.5842 | 1.2 | 1.9 | 1.5437 | 1.2 | 1.9 | 1.6819 | |
${\mathrm{R}}_{\mathrm{s}}$ | 0 | 0.4 | 0.1930 | 0 | 0.4 | 0.2175 | 0 | 0.4 | 0.20868 | |
${\mathrm{R}}_{\mathrm{sh}}$ | 100 | 600 | 397.1536 | 100 | 600 | 421.2257 | 100 | 600 | 487.8684 |
Table 6. Extracting parameter comparison chart for the PV module Kyocera (KC200GT).
Table 6. Extracting parameter comparison chart for the PV module Kyocera (KC200GT).
Parameters Extraction Techniques | Parameters | ||||
---|---|---|---|---|---|
${\mathbf{I}}_{\mathbf{p}\mathbf{h}}$ | ${\mathbf{I}}_{0}$ | Vt | ${\mathbf{R}}_{\mathbf{s}}$ | ${\mathbf{R}}_{\mathbf{s}\mathbf{h}}$ | |
GWO [15] | ---- | ---- | n = 1.37 | 0.2072 | 1691.75 |
Villalva’s Method [21] | 8.214 | 9.825 × 10^{−8} | 1.8036 | 0.221 | 415.405 |
Accarino’s Method [22] | 8.21 | 2.154 × 10^{−9} | 1.4922 | 0.2844 | 157.536 |
Iterative Method [23] | 8.2233 | 2.152 × 10^{−9} | 1.4926 | 0.308 | 193.0493 |
Silva’s Method [24] | 8.193 | 0.30 × 10^{−9} | 1.3874 | 0.271 | 171.2 |
GA | 8.2056 | 4.7278 × 10^{−7} | 1.5842 | 0.1930 | 397.154 |
PSO | 8.1958 | 5.1107 × 10^{−7} | 1.5437 | 0.2175 | 421.226 |
GPSO | 8.2161 | 4.13 × 10^{−7} | 1.6819 | 0.2086 | 487.4175 |
Table 7. Relative power error comparisons chart.
Table 7. Relative power error comparisons chart.
Parameters Extraction Techniques Used for KC200GT PV Module | Actual Maximum Power (W) | Maximum Power Using Extracted Parameters | Relative Error |
---|---|---|---|
GPSO | 200.143 | 200.1157 | $1.364\times {10}^{-2}$ |
Practical Swam Optimization (PSO) | 200.143 | 199.782 | $1.803\times {10}^{-1}$ |
Genetic Algorithm | 200.143 | 199.926 | $1.084\times {10}^{-1}$ |
GWO [15] | 200.143 | 200.04 | |
Villalva’s Method [21] | 200.143 | 199.883 | $1.29\times {10}^{-1}$ |
Accarino’s Method [22] | 200.143 | 199.972 | $8.51\times {10}^{-2}$ |
Iterative Method [23] | 200.143 | 199.866 | $1.38\times {10}^{-1}$ |
Silva’s Method [24] | 200.143 | 204.7336 | $-2.293\times {10}^{0}$ |
Table 8. Comparisons between GA, PSO, and GPSO.
Table 8. Comparisons between GA, PSO, and GPSO.
Parameter | GA | PSO | Two-Step Optimization (GPSO) |
---|---|---|---|
Convergence Speed | Moderate | Fast | Very Fast |
Accuracy of Parameters | Moderate | High | Very High |
Runtime per Iteration | Low | Moderate | High |
Computational Efficiency | High | Moderate | Low |
Stability | Moderate | High | Very High |
Complexity | Low | Moderate | High |
Scalability | High | High | Moderate |
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