Novel Fractional Order and Stochastic Formulations for the Precise Prediction of Commercial Photovoltaic Curves

Abstract

To effectively represent photovoltaic (PV) modules while considering their dependency on changing environmental conditions, three novel mathematical and empirical formulations are proposed in this study to model PV curves with minimum effort and short timing. The three approaches rely on distinct mathematical techniques and definitions to formulate PV curves using function representations. We develop our models through fractional derivatives and stochastic white noise. The first empirical model is proposed using a fractional regression tool driven by the Liouville-Caputo fractional derivative and then implemented by the Mittag-Leffler function representation. Further, the fractional-order stochastic ordinary differential equation (ODE) tool is employed to generate two effective generic models. In this work, multiple commercial PV modules are modeled using the proposed fractional and stochastic formulations. Using the experimental data of the studied PV panels at different climatic conditions, we evaluate the proposed models’ accuracy using two effective statistical indices: the root mean squares error (RMSE) and the determination coefficient (R²). Finally, the proposed approaches are compared to several integer-order models in the literature where the proposed models’ precisely follow the real PV curves with a higher R² and lower RMSE values at different irradiance levels lower than 800 w/m², and module temperature levels higher than 50 °C.

1. Introduction

The requirement for renewable energy is growing in significance to support transmission systems by reducing the amount of electricity that can be delivered from traditional sources [1] and reducing greenhouse gas emissions such as CO₂ [2]. As a result, the utilization of renewable energy sources such as biomass energy [3,4], solar energy [5,6], wind energy [6,7], wave energy [7], and others [8] has become vital and widespread worldwide. The planning of renewable energy sources utilizing various approaches [9] and technologies [10] has been the subject of numerous research studies. Power electronic devices such as soft open points (SOPs) [11], static voltage compensators [12], smart inverters [13], and others are among these technologies. To store excess power during off-peak hours and improve reliability, energy storage systems (ESS) [14] have been investigated in several electrical fields.

Therefore, accurate modeling of these energy resources is required for energy harvesting. Many research articles [15,16,17,18] have used mathematical frameworks to model energy resources. A scenario reduction technique was used to quantify the wind speed, solar irradiance, and load profiles in a set of wind speeds with certain probabilities [16]. In [18], the authors also mixed four well-known probability density functions (PDFs), including Weibull, log-normal, gamma, and Rayleigh, to create a probabilistic model. In five places in Egypt, they demonstrated their capacity to mimic the probabilistic nature of wind speeds. The cumulative distribution functions of the Weibull, log-normal, gamma, and composite Weibull-Gamma were used by the authors of [19] to describe wind fluctuations. Then, they used the invasive weed optimization technique to optimize their parameters. Further, solar PV irradiance was modeled via beta PDF to optimally fit its uncertainty [20]. Also, multiple PDFs, including Weibull, Gamma, log-normal, and Rayleigh, were employed [21] to estimate the power supplied to various residential load types. In addition, various mixture PDFs were employed to model the wind turbine (WT) and photovoltaic (PV) fluctuations.

A variety of methods, including equivalent circuit-based models (diodes number models), trigonometric function-based models, power forecast models, and empirical PV models, were used in earlier studies to handle the mathematical modeling of PV panel I-V and P-V curves. The single-diode, two-diode, or three-diode circuits are the basis for the corresponding equivalent circuits [22,23,24]. Higher precision is attained when more diodes are incorporated into the model, but this comes at a cost in terms of complexity and computing effort, because more parameters need to be estimated. Numerous mathematical techniques exist for estimating unknown PV parameters, including analytical, numerical, AI-based, evolutionary algorithm-based, and hybrid methods [25,26,27,28,29].

A different PV cell mathematical representation based on trigonometric functions (a sine and cosine function-based model) is described in [30] in addition to the analogous circuit-based PV modeling methodologies. It depends on detecting the relationship between changes in PV open-circuit voltage and short-circuit current as a function of temperature and irradiance, and then converting that relationship into a trigonometric property. This trigonometric function, however, contains seven constants the values of which must be determined using PV experimental features, which increases the computational work and reduces the practicality of the function.

Accurate forecasting is necessary for PV models based on short- and long-term PV power forecasts utilizing artificial neural networks (ANN) and machine learning approaches. This requires a challenging implementation, a large amount of data samples, and ongoing training and fitting results [31,32]. The generic PV model, on the other hand, is demonstrated, and it includes an empirical mathematical equation based on the function representation of figures extracted from the datasheet. Based on any practical PV datasheet under standard testing circumstances (STC), this model can generate characteristic curves for any PV device [33].

The Lambert W function is used in [34] to determine the PV output current expression using a combination of numerical and analytical techniques. The Newton-Raphson method is used to quantitatively and concurrently calculate the PV voltage. To calculate PV current-voltage (I-V) and power-voltage (P-V) curves, the Lambert W function and an artificial neural network were used [35]. The authors of [33] suggested a novel mathematical formulation to replicate the data sheet curves of various PV modules, including the characteristics of (I-V) and (P-V) curves. They were able to establish rough I-V curves for many commercial PV modules.

Further, fractional order derivatives and stochastic modeling have been crucial in many research articles in different disciplines [36,37,38,39]. This is due to a greater degree of flexibility in the model, which provides an excellent instrument for describing the properties of various practical processes and dynamical systems. Therefore, we use these tools to enhance our study’s modeling.

In this paper, novel mathematical formulations of the I-V curves of commercial PV modules are presented using fractional data fitting, fractional order derivatives, and stochastic operators. The aim is to gain more precise insight compared to the extracted I-V curves from modules’ datasheets under various weather conditions. The proposed formulas use the module’s main nameplate data (short circuit current, maximum power point current, maximum power point voltage, open circuit voltage, and voltage sensitivity to temperature) as input without needing any complex data collection or forecasting.

The remaining sections of this paper are presented as follows: Section 2 illustrates the essential mathematical preliminaries and definitions required to introduce the modeling process; Section 3 discusses the basic integer-order models and the detailed description of the generated fractional and stochastic models; Section 4 provides the obtained results with complete analysis and comparisons; and Section 5 presents the conclusions.

2. Preliminaries

Engineering, science, and physics have all paid considerable attention to stochastic differential equations and fractional-order differential operators. Fractional derivatives provide the advantage of having more modeling freedom for real-world processes and dynamical systems. By maintaining allowable random process terms in the deterministic modeling equations, stochastic differential equations bring more realism to the modeling process. Using fractional derivatives and stochastic white noise, the following preliminaries provide the necessary lemmas, definitions, and theories to handle the fundamental mathematical concepts.

Definition 1.

Suppose ϕ > 0, t > a, α, a, t $\in$ $\mathbb{R}$. The fractional operator (${\mathit{D}}_{\mathit{t}}^{\mathit{\varphi }}$), defined as [40]:

$${\mathit{D}}_{\mathit{t}}^{\mathit{\varphi }}\mathit{f}\left(\mathit{t}\right)=\left\{\begin{array}{c}\frac{\mathbf{1}}{\mathit{\Gamma }\left(\mathit{n}-\mathit{\varphi }\right)}\left[\underset{\mathbf{0}}{\overset{\mathit{t}}{\int }}{\left(\mathit{t}-\mathit{\zeta }\right)}^{\mathit{n}-\mathit{\alpha }-\mathbf{1}}{\mathit{f}}^{\left(\mathit{n}\right)}\left(\mathit{\zeta }\right)\mathit{d}\mathit{\zeta }\right], \mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e} \mathit{t}>\mathbf{0}, \mathbf{0} \le \mathit{n}-\mathbf{1}<\varphi <\mathit{n}\in \mathbb{N}. \hfill \\ \frac{{\mathit{d}}^{\mathit{n}}}{\mathit{d}{\mathit{t}}^{\mathit{n}}}\mathit{f}\left(\mathit{t}\right), \mathit{\varphi }=\mathit{n}\in \mathbb{N}. \hfill \end{array}\right.$$

(1)

 is called the Liouville-Caputo fractional derivative or of order ϕ and $\mathbb{N}$ is the set of positive integer numbers.

Definition 2.

The Liouville-Caputo fractional derivative of order ϕ of the function f(t) has a Laplace transform that takes the form [41],

$$\mathit{Ŀ}\left({\mathit{D}}_{\mathit{t}}^{\mathit{\varphi }}\mathit{f}\left(\mathit{t}\right)\right)={\mathit{s}}^{\mathit{\varphi }}\mathit{F}\left(\mathit{s}\right)-\sum _{\mathit{m}=\mathbf{0}}^{\mathit{n}-\mathbf{1}}\frac{\mathbf{1}}{\mathit{s}}{\mathit{s}}^{\mathit{\varphi }-\mathit{m}}{\mathit{f}}^{\mathit{m}}\left(\mathbf{0}\right),\mathit{n}\in \mathbb{N},\mathit{n}-\mathbf{1}<\mathit{\varphi }<\mathit{n}.$$

(2)

Definition 3.

In series form, a Mittag-Leffler function with two parameters ${\mathit{E}}_{\mathit{\varphi },\mathit{\beta }}\left(\mathit{z}\right)$ for $\mathit{z}\in \mathbb{C}$ is [41],

$${\mathit{E}}_{\mathit{\varphi },\mathit{\beta }}\left(\mathit{z}\right)=\sum _{\mathit{j}=\mathbf{0}}^{\mathit{\infty }}\frac{{\mathit{z}}^{\mathit{j}}}{\mathit{\Gamma }\left(\mathit{\varphi }\mathit{j}+\mathit{\beta }\right)},\mathit{\varphi },\mathit{\beta }<\mathbf{0}.$$

(3)

$\mathbb{C}$ is the set of all real and complex numbers.

Definition 4.

A standard one-dimensional Brownian motion is a stochastic process $\left\{\mathit{B}\right(\mathit{t}),\mathit{t}>{\mathit{t}}_{\mathbf{0}}\}$ indexed by nonnegative real numbers t with the following properties [42]:

(1) 

$\mathit{B}\left({\mathit{t}}_{\mathbf{0}}\right)=\mathbf{0}$ with probability 1.

(2) 

The function $\mathit{t}\to \mathit{B}\left(\mathit{t}\right)$  is continuous on t.

(3) 

If ${\mathit{t}}_{\mathbf{1}}\ne {\mathit{t}}_{\mathbf{2}}$  then $\mathit{B}\left({\mathit{t}}_{\mathbf{1}}\right)$  and $\mathit{B}\left({\mathit{t}}_{\mathbf{2}}\right)$  are independent.

(4) 

For Ɐ${\mathit{t}}_{\mathit{i}}\ge {\mathit{t}}_{\mathbf{0}}$, all increments $∆{\mathit{B}}_{\mathit{i}}={\mathit{B}}_{\mathit{i}+\mathbf{1}}-{\mathit{B}}_{\mathit{i}}$, are normally distributed with mean 0 and variance $\mathit{h}={\mathit{t}}_{\mathit{i}+\mathbf{1}}-{\mathit{t}}_{\mathit{i}}$; i.e., $∆{\mathit{B}}_{\mathit{i}}\sim \mathit{Ɲ}(\mathbf{0},\mathit{h})$.

Definition 5.

;The fractional regression model that can be used to fit bivariate data with a function $\mathit{f}\left(\mathit{t}\right)$ is defined as [41],

$$\mathit{f}\left(\mathit{t}\right)={\mathit{f}}_{\mathit{I}}\left(\mathit{t}\right)-{\mathit{c}}_{\mathit{p}}{\mathit{D}}_{\mathit{t}}^{-\mathit{\varphi }}\mathit{f}\left(\mathit{t}\right),\mathbf{0}<\mathit{\varphi }.$$

(4)

where ${\mathit{f}}_{\mathit{I}}\left(\mathit{t}\right)$ is the integer order fitting model, ${\mathit{D}}_{\mathit{t}}^{-\mathit{\varphi }}$ is the Liouville-Caputo fractional integral of order $\mathit{\varphi }$ [39], and ${\mathit{c}}_{\mathit{p}},\mathit{\varphi }$ represent the parameters that must be calculated for least square regression criteria satisfaction.

Theorem 1.

Using Definition 5, Equation (4) can be transformed into a differential equation as,

$${\mathit{D}}_{\mathit{t}}^{\mathit{\varphi }}\mathit{z}\left(\mathit{t}\right)=-{\mathit{c}}_{\mathit{p}}\mathit{z}\left(\mathit{t}\right)-{\mathit{c}}_{\mathit{p}}{\mathit{f}}_{\mathit{I}}\left(\mathit{t}\right),\mathbf{0}<\mathit{\varphi },\mathit{z}\left(\mathbf{0}\right)=\mathbf{0}.$$

(5)

where $\mathit{z}\left(\mathit{t}\right)$  is an auxiliary function defined by,

$$\mathit{z}\left(\mathit{t}\right)=\mathit{f}\left(\mathit{t}\right)-{\mathit{f}}_{\mathit{I}}\left(\mathit{t}\right).$$

(6)

Proof. 

Through rearranging (4) and using (6) for substitution, then Equation (5) can be easily obtained. □

There are generally two steps in using the fractional regression model (4). Firstly, the differential Equation (5) is solved, substituting in (6). Secondly, the parameters ${\mathit{c}}_{\mathit{p}},\mathit{\varphi }$ are calculated using the least squares techniques.

Lemma 1.

Let us assume the following fractional order differential equation driven by Liouville-Caputo fractional derivative and order ϕ ∈ (0,1]:

$${\mathit{D}}_{\mathit{t}}^{\mathit{\varphi }}\mathit{x}\left(\mathit{t}\right)=\mathit{f}\left(\mathit{t},\mathit{x}\right),{\mathit{t}}_{\mathbf{0}}>\mathbf{0}, \mathit{a}\mathit{n}\mathit{d} \mathit{x}\left({\mathit{t}}_{\mathbf{0}}\right)={\mathit{x}}_{\mathbf{0}}.$$

(7)

where f: $\left[{\mathit{t}}_{\mathbf{0}},\mathit{\infty }\right)\times \mathcal{R}\to {\mathbb{R}}^{\mathit{n}},\mathcal{R}\complement {\mathbb{R}}^{\mathit{n}}$ is a function, if the Lipschitz condition is satisfied by $\mathit{f}(\mathit{t},\mathit{x})$ with respect to x(t), then the system (7) has a unique solution on the interval $\left[{\mathit{t}}_{\mathbf{0}},\mathit{\infty }\right)\times \mathcal{R}$ [43].

Definition 6.

A fractional-order stochastic differential equation (SDE) driven by the Liouville-Caputo fractional derivative and one-dimensional Brownian motion for a function $\mathit{x}\left(\mathit{t}\right)$ is defined as

$${\mathit{D}}_{\mathit{t}}^{\mathit{\varphi }}\mathit{x}\left(\mathit{t}\right)=\mathit{\mu }\left(\mathit{t}, \mathit{x}\left(\mathit{t}\right)\right)\mathit{d}\mathit{t}+\mathit{\sigma }\left(\mathit{t},\mathit{x}\left(\mathit{t}\right)\right)\frac{\mathit{d}\mathit{B}\left(\mathit{t}\right)}{\mathit{d}\mathit{t}},{\mathit{x}}_{\mathbf{0}}=\mathit{x}\left({\mathit{t}}_{\mathbf{0}}\right).$$

(8)

where $\mathit{t}\ge {\mathit{t}}_{\mathbf{0}}$ represents nonnegative real numbers, the functions $\mathit{\mu }(\mathit{t},\mathit{x})$ and $\mathit{\sigma }(\mathit{t},\mathit{x})$ are the drift and diffusion terms, respectively, of the SDE, and $\mathit{B}\left(\mathit{t}\right)$ represents the stochastic Brownian motion.

The indicated dynamic model in definition 6 can be numerically solved using the following fractional-order Euler Maruyama scheme indicated in Equation (9) [37], $\mathit{t}\in ({\mathit{t}}_{\mathbf{0}},\mathit{T}]$,

$$\begin{array}{cc}\hfill {\mathit{x}}^{\left(\mathit{n}\right)}\left(\mathit{t}\right)={\mathit{x}}_{\mathbf{0}}& +\frac{\mathbf{1}}{Г\left(\mathit{\varphi }\right)}{\int }_{{\mathit{t}}_{\mathbf{0}}}^{\mathit{t}}\frac{\mathit{\mu }\left(\right({\mathit{\tau }}_{\mathit{n}}\left(\mathit{s}\right),{\mathit{x}}^{\left(\mathit{n}\right)}\left({\mathit{\tau }}_{\mathit{n}}\left(\mathit{s}\right)\right))}{{\left(\mathit{t}-\mathit{s}\right)}^{\mathbf{1}-\mathit{\varphi }}}\mathit{d}\mathit{s}\hfill \\ & +\frac{\mathbf{1}}{Г\left(\mathit{\varphi }\right)}{\int }_{{\mathit{t}}_{\mathbf{0}}}^{\mathit{t}}\frac{\mathit{\sigma }\left(\right({\mathit{\tau }}_{\mathit{n}}\left(\mathit{s}\right),{\mathit{x}}^{\left(\mathit{n}\right)}\left({\mathit{\tau }}_{\mathit{n}}\left(\mathit{s}\right)\right))}{{\left({\mathit{\rho }}_{\mathit{n}}\left(\mathit{t}\right)-{\mathit{\tau }}_{\mathit{n}}\left(\mathit{s}\right)\right)}^{\mathbf{1}-\mathit{\varphi }}}\mathit{d}\mathit{B}\left(\mathit{t}\right),{\mathit{\tau }}_{\mathit{n}}\left(\mathit{s}\right)=\frac{\mathit{k}\mathit{T}}{\mathit{n}}, \mathit{s}\hfill \\ & \in \left(\frac{\mathit{k}\mathit{T}}{\mathit{n}},\frac{\left(\mathit{k}+\mathbf{1}\right)\mathit{T}}{\mathit{n}}\right]{\to \mathit{\rho }}_{\mathit{n}}\left(\mathit{s}\right)=\frac{\left(\mathit{k}+\mathbf{1}\right)\mathit{T}}{\mathit{n}},\mathit{k}=\mathbf{0},\mathbf{1}\dots ,\mathit{n}-\mathbf{1}. \hfill \end{array}$$

(9)

where n is the total time discretization and belongs to the set of positive integer numbers. The previous iterative equation can be solved step by step at each interval $(\frac{\mathit{k}\mathit{T}}{\mathit{n}},\frac{\left(\mathit{k}+\mathbf{1}\right)\mathit{T}}{\mathit{n}}]$. The convergency of the used fractional Euler Maruyama scheme is described in detail in [44].

3. Mathematical Modeling

In this section, the three formulations are developed based on previous basic integer order empirical formulas from the literature, so we start by introducing these integer models. Then, a detailed illustration of each proposed generic formula is made. All of the empirical formulas under study seek the same objective: a precise description of the commercial PV module output under variable weather conditions with available data on most of the modules’ data sheets. These models surpass the single diode and double models as there is no need to seriously measure the series and shunt resistances of modules, which is mandatory in all physical models. Also, we need to focus on commercial panels, which have near-but-not-ideal characteristics.

3.1. Integer Order Models

3.1.1. Integer Model One

This empirical formula is one of the most commonly used formulas in modeling the PV module curves, which derives that the PV module current $\mathit{I}\left(\mathit{v}\right)$ function of module voltage ($\mathit{v})$ can take the form

$$\mathit{I}\left(\mathit{v}\right)={\mathit{I}}_{\mathit{s}\mathit{c}}-{\mathit{k}}_{\mathbf{1}}{\mathit{e}}^{-\frac{{\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{k}}_{\mathbf{2}}}}\left({\mathit{e}}^{\frac{\mathit{v}}{{\mathit{k}}_{\mathbf{2}}}}-\mathbf{1}\right).$$

(10)

where ${\mathit{I}}_{\mathit{s}\mathit{c}}$ is the module short circuit current, ${\mathit{v}}_{\mathit{o}\mathit{c}}$ is the module open circuit voltage, and ${\mathit{k}}_{\mathbf{1}},{\mathit{k}}_{\mathbf{2}}$ can be approximated though two operating conditions:

-

Open circuit condition ($\mathit{I}=\mathbf{0},\mathit{v}={\mathit{v}}_{\mathit{o}\mathit{c}})$:

$${\mathit{k}}_{\mathbf{1}}=\frac{{\mathit{I}}_{\mathit{s}\mathit{c}}}{\mathbf{1}-{\mathit{e}}^{-\frac{{\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{k}}_{\mathbf{2}}}}},$$

(11)

for $\frac{{\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{k}}_{\mathbf{2}}}\gg \mathbf{1}$: ${\mathit{k}}_{\mathbf{1}}\cong {\mathit{I}}_{\mathit{s}\mathit{c}}$.

-

Maximum power point condition ($\mathit{I}={\mathit{I}}_{\mathit{m}\mathit{p}},\mathit{v}={\mathit{v}}_{\mathit{m}\mathit{p}})$:

$${\mathit{k}}_{\mathbf{1}}=\frac{{\mathit{I}}_{\mathit{s}\mathit{c}}-{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{e}}^{-\frac{{\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{k}}_{\mathbf{2}}}}\left({\mathit{e}}^{\frac{{\mathit{v}}_{\mathit{m}\mathit{p}}}{{\mathit{k}}_{\mathbf{2}}}}-\mathbf{1}\right)}.$$

(12)

where ${\mathit{I}}_{\mathit{m}\mathit{p}}$ and ${\mathit{v}}_{\mathit{m}\mathit{p}}$ are the module maximum power point current and voltage, respectively.

$$\mathrm{for} \frac{{\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{k}}_{\mathbf{2}}}\gg \mathbf{1}\text{:} {\mathit{k}}_{\mathbf{1}}\cong \frac{{\mathit{I}}_{\mathit{s}\mathit{c}}-{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{e}}^{\frac{{\mathit{v}}_{\mathit{m}\mathit{p}}-{\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{k}}_{\mathbf{2}}}}}.$$

Substituting the approximated value of ${\mathit{k}}_{\mathbf{1}}$ in (11) into (12),

$${\mathit{k}}_{\mathbf{2}}=\frac{{{\mathit{v}}_{\mathit{m}\mathit{p}}-\mathit{v}}_{\mathit{o}\mathit{c}}}{\mathbf{l}\mathbf{n}(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}})}.$$

(13)

Now, substituting the approximated values of ${\mathit{k}}_{\mathbf{1}}$ and ${\mathit{k}}_{\mathbf{2}}$ from (11) and (13), respectively, in Equation (10) yields

$$\mathit{I}\left(\mathit{v}\right)={\mathit{I}}_{\mathit{s}\mathit{c}}\left[\mathbf{1}+{\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}}\right)}^{-\frac{{\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}-{\mathit{v}}_{\mathit{o}\mathit{c}}}}-{\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}}\right)}^{\frac{{\mathit{v}-\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}-{\mathit{v}}_{\mathit{o}\mathit{c}}}}\right].$$

(14)

The main pitfall of this formula is the approximations made on ${\mathit{k}}_{\mathbf{1}}$ and ${\mathit{k}}_{\mathbf{2}}$, which actually affect the accuracy of the modeling process as well as how the accuracy of the formula is affected under varying weather conditions.

3.1.2. Integer Model Two

This model is a recent empirical formula developed by the authors of [33] and tested for its accuracy by applying it to different commercial PV modules using valid data sheet curves for checking under a range of variable irradiances and temperatures. This model equation describes the relation between the module current and voltage and takes the form

$$\mathit{I}\left(\mathit{v}\right)=\left[{\mathit{I}}_{\mathit{s}\mathit{c}}+\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}-{\mathit{I}}_{\mathit{s}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}}\mathit{v}-{\mathit{e}}^{-\frac{{\mathit{v}}_{\mathit{m}\mathit{p}}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}}\left(\mathbf{1}-{\mathit{e}}^{\frac{\mathit{v}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}}\right)\left(\mathbf{1}-{\mathit{e}}^{\mathit{A}(\mathit{v}-{\mathit{v}}_{\mathit{m}\mathit{p}}}\right)\right].$$

(15)

$$\mathit{A}=\left(\mathbf{1}+\frac{{\mathit{v}}_{\mathit{o}\mathit{c}}-\mathit{v}}{{\mathit{v}}_{\mathit{m}\mathit{p}}}\right)\left(\frac{\mathbf{1}}{{\mathit{v}}_{\mathit{o}\mathit{c}}-{\mathit{v}}_{\mathit{m}\mathit{p}}}\right)\mathbf{ln}\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{s}\mathit{c}}+\left(\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}-{\mathit{I}}_{\mathit{s}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}}\right){\mathit{v}}_{\mathit{o}\mathit{c}}}{\left(\mathbf{1}-\mathit{e}\right){\mathit{e}}^{-\frac{{\mathit{v}}_{\mathit{m}\mathit{p}}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}}}\right).$$

(16)

However, this empirical model shows fine modeling accuracy, as described in [33]. Still, with increased accuracy, it will be developed by adding stochastic white noise to its definition and giving it more realistic curve shapes.

3.2. Fractional Order Proposed Models

3.2.1. Fractional Model One

Starting with the integer model 1 in (14) and using the fractional curve fitting definition in (4), the PV module current function of module voltage yields

$$\mathit{I}\left(\mathit{v}\right)={\mathit{I}}_{\mathit{I}}\left(\mathit{v}\right)-{\mathit{c}}_{\mathit{p}}{\mathit{D}}_{\mathit{v}}^{-\mathit{\varphi }}\mathit{I}\left(\mathit{v}\right).$$

(17)

where ${\mathit{I}}_{\mathit{I}}\left(\mathit{v}\right)$ is the PV module current as defined in (14).

Let the difference between the integer order model ${\mathit{I}}_{\mathit{I}}\left(\mathit{v}\right)$ and the fractional order model $\mathit{I}\left(\mathit{v}\right)$ be defined by

$$\mathit{z}\left(\mathit{v}\right)=\mathit{I}\left(\mathit{v}\right)-{\mathit{I}}_{\mathit{I}}\left(\mathit{v}\right)=\mathit{I}\left(\mathit{v}\right)-{\mathit{c}}_{\mathbf{0}}+{\mathit{c}}_{\mathbf{1}}{\mathit{c}}_{\mathbf{2}}^{\mathit{v}}.$$

(18)

$$\begin{gathered} {\mathit{c}}_{\mathbf{0}}={\mathit{I}}_{\mathit{s}\mathit{c}}+{\mathit{I}}_{\mathit{s}\mathit{c}}{\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}}\right)}^{-\frac{{\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}-{\mathit{v}}_{\mathit{o}\mathit{c}}}},{\mathit{c}}_{\mathbf{1}}={{\mathit{I}}_{\mathit{s}\mathit{c}}\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}}\right)}^{-\frac{{\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}-{\mathit{v}}_{\mathit{o}\mathit{c}}}}, \\ {\mathit{c}}_{\mathbf{2}}={{\mathit{I}}_{\mathit{s}\mathit{c}}\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}}\right)}^{\frac{\mathbf{1}}{{\mathit{v}}_{\mathit{m}\mathit{p}}-{\mathit{v}}_{\mathit{o}\mathit{c}}}}. \end{gathered}$$

(19)

Now, transforming (17) into a fractional differential equation using the assumptions in (19),

$${\mathit{D}}_{\mathit{v}}^{\mathit{\varphi }}\mathit{z}\left(\mathit{v}\right)=-{\mathit{c}}_{\mathit{p}}\mathit{z}\left(\mathit{v}\right)-{\mathit{c}}_{\mathit{p}}{\mathit{c}}_{\mathbf{0}}+{\mathit{c}}_{\mathit{p}}{\mathit{c}}_{\mathbf{1}}{\mathit{e}}^{\mathit{v}\mathbf{ln}\left({\mathit{c}}_{\mathbf{2}}\right)}.$$

(20)

Taking the Laplace transform to (20)

$${\mathit{s}}^{\mathit{\varphi }}\mathit{Z}=-{\mathit{c}}_{\mathit{p}}\mathit{Z}-\frac{{{\mathit{c}}_{\mathit{p}}\mathit{c}}_{\mathbf{0}}}{\mathit{s}}+{\mathit{c}}_{\mathit{p}}{\mathit{c}}_{\mathbf{1}}\left(\frac{\mathbf{1}}{\mathit{s}}+\frac{\mathbf{ln}\left({\mathit{c}}_{\mathbf{2}}\right)}{{\mathit{s}}^{\mathbf{2}}}+\frac{(\mathbf{ln}{\left({\mathit{c}}_{\mathbf{2}}\right))}^{\mathbf{2}}}{\mathbf{2}!{\mathit{s}}^{\mathbf{3}}}+\dots +\frac{(\mathbf{ln}{\left({\mathit{c}}_{\mathbf{2}}\right))}^{\mathit{n}}}{\mathit{n}!{\mathit{s}}^{\mathit{n}+\mathbf{1}}}\right).$$

(21)

then,

$$\mathit{Z}=-\frac{{\mathit{c}}_{\mathit{p}}{\mathit{c}}_{\mathbf{0}}}{\mathit{s}\left({\mathit{s}}^{\mathit{\varphi }}+{\mathit{c}}_{\mathit{p}}\right)}+{\mathit{c}}_{\mathit{p}}{\mathit{c}}_{\mathbf{1}}\left(\left(\frac{\mathbf{1}}{\mathit{s}\left({\mathit{s}}^{\mathit{\varphi }}+{\mathit{c}}_{\mathit{p}}\right)}+\frac{\mathbf{ln}\left({\mathit{c}}_{\mathbf{2}}\right)}{{\mathbf{1}!\mathit{s}}^{\mathbf{2}}\left({\mathit{s}}^{\mathit{\varphi }}+{\mathit{c}}_{\mathit{p}}\right)}+\frac{(\mathbf{ln}{\left({\mathit{c}}_{\mathbf{2}}\right))}^{\mathbf{2}}}{\mathbf{2}!{\mathit{s}}^{\mathbf{3}}\left({\mathit{s}}^{\mathit{\varphi }}+{\mathit{c}}_{\mathit{p}}\right)}+\dots +\frac{(\mathbf{ln}{\left({\mathit{c}}_{\mathbf{2}}\right))}^{\mathit{n}}}{\mathit{n}!{\mathit{s}}^{\mathit{n}+\mathbf{1}}\left({\mathit{s}}^{\mathit{\varphi }}+{\mathit{c}}_{\mathit{p}}\right)}\right)\right).$$

(22)

Taking the inverse Laplace transform to (22)

$$\begin{array}{cc}\hfill \mathit{z}\left(\mathit{v}\right)=-{\mathit{c}}_{\mathbf{0}}+{\mathit{c}}_{\mathbf{0}}{\mathit{E}}_{\mathit{\varphi },\mathbf{1}}& \left(-{\mathit{c}}_{\mathit{p}}{\mathit{v}}^{\mathit{\varphi }}\right)+{\mathit{c}}_{\mathbf{1}}\left(\mathbf{1}+\mathit{v}\mathbf{ln}\left({\mathit{c}}_{\mathbf{2}}\right)+\frac{{\mathit{v}}^{\mathbf{2}}{\mathbf{ln}\left({\mathit{c}}_{\mathbf{2}}\right)}^{\mathbf{2}}}{\mathbf{2}!}+\dots +{\mathit{v}}^{\mathit{n}}\frac{{\mathbf{ln}\left({\mathit{c}}_{\mathbf{2}}\right)}^{\mathit{n}}}{\mathit{n}!}\right)\hfill \\ & -{\mathit{c}}_{\mathbf{1}}\left({\mathit{E}}_{\mathit{\varphi },\mathbf{1}}\left(-{\mathit{c}}_{\mathit{p}}{\mathit{v}}^{\mathit{\varphi }}\right)+{\frac{\mathit{v}\mathit{l}\mathit{n}\left({\mathit{c}}_{\mathbf{2}}\right)}{\mathbf{1}!}\mathit{E}}_{\mathit{\varphi },\mathbf{2}}\left(-{\mathit{c}}_{\mathit{p}}{\mathit{v}}^{\mathit{\varphi }}\right)+\dots +\frac{{\left(\mathit{v}\mathit{l}\mathit{n}\left({\mathit{c}}_{\mathbf{2}}\right)\right)}^{\mathit{n}-\mathbf{1}}}{\mathbf{1}!}{\mathit{E}}_{\mathit{\varphi },\mathit{n}}\left(-{\mathit{c}}_{\mathit{p}}{\mathit{v}}^{\mathit{\varphi }}\right)\right).\hfill \end{array}$$

(23)

Making mathematical simplifications yields

$$\mathit{z}\left(\mathit{v}\right)=-{\mathit{c}}_{\mathbf{0}}+{\mathit{c}}_{\mathbf{1}}{\mathit{c}}_{\mathbf{2}}^{\mathit{v}}+{\mathit{c}}_{\mathbf{0}}{\mathit{E}}_{\mathit{\varphi },\mathbf{1}}\left(-{\mathit{c}}_{\mathit{p}}{\mathit{v}}^{\mathit{\varphi }}\right)-{\mathit{c}}_{\mathbf{1}}\sum _{\mathbf{0}}^{\mathit{n}}\frac{{\mathit{v}}^{\mathit{n}}\mathbf{ln}\left({\mathit{c}}_{\mathbf{2}}\right)}{\mathit{n}!}{\mathit{E}}_{\mathit{\varphi },\mathit{n}+\mathbf{1}}\left(-{\mathit{c}}_{\mathit{p}}{\mathit{v}}^{\mathit{\varphi }}\right).$$

(24)

Using Equations (18), (19), and (24), then

$$\begin{array}{cc}\hfill \mathit{I}\left(\mathit{v}\right)={[\mathit{I}}_{\mathit{s}\mathit{c}}+{\mathit{I}}_{\mathit{s}\mathit{c}}& {\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}}\right)}^{-\frac{{\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}-{\mathit{v}}_{\mathit{o}\mathit{c}}}}]{\mathit{E}}_{\mathit{\varphi },\mathbf{1}}\left(-{\mathit{c}}_{\mathit{p}}{\mathit{v}}^{\mathit{\varphi }}\right)\hfill \\ & -{{\mathit{I}}_{\mathit{s}\mathit{c}}\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}}\right)}^{-\frac{{\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}-{\mathit{v}}_{\mathit{o}\mathit{c}}}}{\displaystyle \sum _{\mathbf{0}}^{\mathit{n}}}\frac{{\mathit{v}}^{\mathit{n}}\mathbf{ln}\left({{\mathit{I}}_{\mathit{s}\mathit{c}}\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}}\right)}^{\frac{\mathbf{1}}{{\mathit{v}}_{\mathit{m}\mathit{p}}-{\mathit{v}}_{\mathit{o}\mathit{c}}}}\right)}{\mathit{n}!}{\mathit{E}}_{\mathit{\varphi },\mathit{n}+\mathbf{1}}\left(-{\mathit{c}}_{\mathit{p}}{\mathit{v}}^{\mathit{\varphi }}\right).\hfill \end{array}$$

(25)

where ${\mathit{E}}_{\mathit{\varphi },\mathbf{1}}\left(-{\mathit{c}}_{\mathit{p}}{\mathit{v}}^{\mathit{\varphi }}\right)$ and ${\mathit{E}}_{\mathit{\varphi },\mathit{n}+\mathbf{1}}\left(-{\mathit{c}}_{\mathit{p}}{\mathit{v}}^{\mathit{\varphi }}\right)$ are Mittag-Leffler functions with parameters ($\mathit{\varphi },\mathbf{1}$) and ($\mathit{\varphi },\mathit{n}+\mathbf{1}$), respectively.

The previously obtained formula is a generalized form of the integer model 1 with two unknown fitting parameters ($\mathit{\varphi },{\mathit{c}}_{\mathit{p}})$. All PV modules share the following two generated empirical formulas for estimating $\mathit{\varphi }$ and ${\mathit{c}}_{\mathit{p}}$ at different irradiances (G) and temperatures (t),

$$\mathit{\varphi }={\mathit{\varphi }}_{\mathbf{0}}+\left\{\begin{array}{c}\mathbf{2}\left(\mathbf{0.005}-\mathit{d}\mathit{I}\right)\left(\mathit{t}-\mathbf{25}\right), \mathit{d}\mathit{I}>\mathbf{0.005} \\ \mathbf{3}\left(\mathbf{0.005}-\mathit{d}\mathit{I}\right)\left(\mathit{t}-\mathbf{25}\right), \mathit{d}\mathit{I}<\mathbf{0.005} \end{array}\right..$$

(26)

$${\mathit{c}}_{\mathit{p}}=-\mathbf{0.5}-{\left(\frac{{\mathit{I}}_{\mathit{s}\mathit{c}}}{{\mathit{I}}_{\mathit{m}\mathit{p}}}\times \frac{{\mathit{v}}_{\mathit{o}\mathit{c}}-{\mathit{v}}_{\mathit{m}\mathit{p}}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}\right)}^{\mathit{k}}\left[\left(\mathbf{1}-\frac{\mathit{G}}{\mathbf{1000}}\right)+\mathbf{0.2}\mathit{k}{\left(\mathbf{1}-\frac{\mathit{G}}{\mathbf{1000}}\right)}^{\mathbf{2}}\right],$$

(27)

$$\mathit{k}=\mathbf{100}\left(\mathit{d}{\mathit{V}}_{\mathit{G}}-\mathbf{0.1}\right),$$

(28)

$$\mathit{d}{\mathit{V}}_{\mathit{G}}=\mathbf{0.24}-\mathit{d}{\mathit{V}}_{\mathit{t}}.$$

(29)

where ${\mathit{\varphi }}_{\mathbf{0}}$ is the model fractional order value in standard test conditions (STC) and it can be easily estimated within the tested range [0.5, 0.8], $\mathit{d}\mathit{I}$ is the PV module current sensitivity in Ampere (A) per degree Celsius, $\mathit{d}{\mathit{V}}_{\mathit{G}}$ is the module voltage sensitivity in Volt (V) per unit irradiance (rad), and $\mathit{d}{\mathit{V}}_{\mathit{t}}$ is the module voltage absolute sensitivity in Volt (V) per degree Celsius decrease.

3.2.2. Fractional Stochastic Model Two

Starting with Equation (14) of integer model 1, we can change the equation form into a fractional stochastic form defined in (8) through fractional differentiation using the Liouville-Caputo derivative and by adding a drift white noise function $\mathit{\sigma }(\mathit{v},\mathit{I}\left(\mathit{v}\right))$, so it yields

$${\mathit{D}}_{\mathit{v}}^{\mathit{\lambda }}\mathit{I}\left(\mathit{v}\right)={\mathit{I}}_{\mathit{s}\mathit{c}}{\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}}\right)}^{-\frac{{\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}-{\mathit{v}}_{\mathit{o}\mathit{c}}}}\mathit{l}\mathit{n}\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}}\right)\left[{\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}}\right)}^{\frac{{\mathit{v}-\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}-{\mathit{v}}_{\mathit{o}\mathit{c}}}}+\mathit{\sigma }\left(\mathit{v},\mathit{I}\left(\mathit{v}\right)\right)\frac{\mathit{d}\mathit{B}\left(\mathit{v}\right)}{\mathit{d}\mathit{v}}\right],$$

(30)

$$\mathit{\sigma }\left(\mathit{v},\mathit{I}\left(\mathit{v}\right)\right)={\mathit{\sigma }}_{\mathit{o}}{\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}}{{\mathit{I}}_{\mathit{s}\mathit{c}}}\right)}^{\frac{{\mathit{v}-\mathit{v}}_{\mathit{o}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}-{\mathit{v}}_{\mathit{o}\mathit{c}}}}.$$

(31)

where $\mathit{\lambda }$ is the fractional order parameter and ${\mathit{\sigma }}_{\mathit{o}}$ is the drift term coefficient.

This modified stochastic equation to the integer model 1 aims to adhere to more accuracy and stochasticity in modeling the commercial PV characteristics under variable irradiances and temperatures. We use the following two empirical formulas to present the values of $\mathit{\lambda }$ and ${\mathit{\sigma }}_{\mathit{o}}$ under varying weather conditions for the commercial PV modules under study.

$$\mathit{\lambda }=\mathbf{0.96}-\mathbf{0.035}\left(\frac{\mathit{t}-\mathbf{25}}{\mathbf{25}}\right)+\mathbf{0.01}{\left(\frac{\mathit{t}-\mathbf{25}}{\mathbf{25}}\right)}^{\mathbf{2}}.$$

(32)

$${\mathit{\sigma }}_{\mathit{o}}=\mathbf{0.15}-\mathbf{0.05}\left(\frac{\mathit{t}-\mathbf{25}}{\mathbf{25}}\right)-\mathbf{0.1}\left(\mathbf{1}-\frac{\mathit{G}}{\mathbf{1000}}\right).$$

(33)

3.2.3. Fractional Stochastic Model Three

Using integer model 2 equations in (15) and (16), and the fractional stochastic form defined in (8), we developed the following fractional stochastic differential equation describing the relation between the PV module current and voltage.

$${\mathit{D}}_{\mathit{v}}^{\mathit{\alpha }}\mathit{I}\left(\mathit{v}\right)=\left[\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}-{\mathit{I}}_{\mathit{s}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}}+{\mathit{e}}^{-\frac{{\mathit{v}}_{\mathit{m}\mathit{p}}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}}\left[\left(\mathbf{1}+{\mathit{e}}^{\frac{\mathit{v}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}}\right){\mathit{D}}_{\mathit{v}}^{}\left(\mathit{A}\right)+{\frac{\mathbf{1}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}\mathit{e}}^{\frac{\mathit{v}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}}\right){\mathit{e}}^{\mathit{A}\left(\mathit{v}-{\mathit{v}}_{\mathit{m}\mathit{p}}\right)}+\frac{\mathbf{1}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}{\mathit{e}}^{\frac{\mathit{v}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}}\right]+\mathit{\rho }\left(\mathit{v},\mathit{I}\left(\mathit{v}\right)\right)\frac{\mathit{d}\mathit{B}\left(\mathit{v}\right)}{\mathit{d}\mathit{v}},$$

(34)

$$\mathit{\rho }\left(\mathit{v},\mathit{I}\left(\mathit{v}\right)\right)={\mathit{\rho }}_{\mathit{o}}\left[\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}-{\mathit{I}}_{\mathit{s}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}}+{\mathit{e}}^{-\frac{{\mathit{v}}_{\mathit{m}\mathit{p}}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}}\left[\left(\mathbf{1}+{\mathit{e}}^{\frac{\mathit{v}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}}\right){\mathit{D}}_{\mathit{v}}^{}(\mathit{A})+{\frac{\mathbf{1}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}\mathit{e}}^{\frac{\mathit{v}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}}\right){\mathit{e}}^{\mathit{A}\left(\mathit{v}-{\mathit{v}}_{\mathit{m}\mathit{p}}\right)}+\frac{\mathbf{1}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}{\mathit{e}}^{\frac{\mathit{v}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}}\right],$$

(35)

$${\mathit{D}}_{\mathit{v}}^{}\left(\mathit{A}\right)=\left(\frac{\mathbf{1}}{{\mathit{v}}_{\mathit{o}\mathit{c}}-{\mathit{v}}_{\mathit{m}\mathit{p}}}\right)\mathbf{ln}\left(\mathbf{1}-\frac{{\mathit{I}}_{\mathit{s}\mathit{c}}+\left(\frac{{\mathit{I}}_{\mathit{m}\mathit{p}}-{\mathit{I}}_{\mathit{s}\mathit{c}}}{{\mathit{v}}_{\mathit{m}\mathit{p}}}\right){\mathit{v}}_{\mathit{o}\mathit{c}}}{\left(\mathbf{1}-\mathit{e}\right){\mathit{e}}^{-\frac{{\mathit{v}}_{\mathit{m}\mathit{p}}}{{\mathit{v}}_{\mathit{o}\mathit{c}}}}}\right)\left(\mathbf{1}+\frac{\mathbf{1}}{{\mathit{v}}_{\mathit{m}\mathit{p}}}\right)\left({\mathit{v}}_{\mathit{m}\mathit{p}}+{\mathit{v}}_{\mathit{o}\mathit{c}}-\mathbf{2}\mathit{v}\right).$$

(36)

where $\mathit{\rho }\left(\mathit{v},\mathit{I}\left(\mathit{v}\right)\right)$ is the drift term presenting the stochastic part, and ${\mathit{\rho }}_{\mathit{o}}$ is the model fractional order parameter and the drift term coefficient, respectively. In this empirical formula, all commercial modules have the same $\mathit{\alpha }$ value equal to 0.98 and constant drift coefficient value ${\mathit{\rho }}_{\mathit{o}}$ equal to 0.22. These selected values are tested using trial and error estimations.

The three generated models are tested under different irradiance and temperature levels from different manufacturers of three different PV commercial modules. The assumed equations for ${\mathit{I}}_{\mathit{s}\mathit{c}},{\mathit{I}}_{\mathit{m}\mathit{p}},{\mathit{v}}_{\mathit{o}\mathit{c}}$, and ${\mathit{v}}_{\mathit{m}\mathit{p}}$, as functions of both irradiance $\mathit{G}$ and temperature $\mathit{t}$, used as inputs for all models, are:

$$\begin{array}{cc}\hfill {\mathit{I}}_{\mathit{s}\mathit{c}}=\frac{\mathit{G}}{{\mathit{G}}_{@\mathit{S}\mathit{T}\mathit{C}}}{\mathit{I}}_{\mathit{s}{\mathit{c}}_{@\mathit{S}\mathit{T}\mathit{C}}}+\mathit{d}\mathit{I}& \left(\mathit{t}-{\mathit{t}}_{@\mathit{S}\mathit{T}\mathit{C}}\right), {\mathit{I}}_{\mathit{m}\mathit{p}}=\frac{\mathit{G}}{{\mathit{G}}_{@\mathit{S}\mathit{T}\mathit{C}}}{{\mathit{I}}_{\mathit{m}\mathit{p}}}_{@\mathit{S}\mathit{T}\mathit{C}},{\mathit{v}}_{\mathit{m}\mathit{p}}={\mathit{v}}_{\mathit{m}{\mathit{p}}_{@\mathit{S}\mathit{T}\mathit{C}}}-\mathit{d}{\mathit{V}}_{\mathit{t}}\left(\mathit{t}-{\mathit{t}}_{@\mathit{S}\mathit{T}\mathit{C}}\right),\hfill \\ & {\mathit{v}}_{\mathit{o}\mathit{c}}={{\mathit{v}}_{\mathit{o}\mathit{c}}}_{@\mathit{S}\mathit{T}\mathit{C}}(\mathbf{1}-\mathbf{0.1}\left[\left(\mathbf{1}-\frac{\mathit{G}}{{\mathit{G}}_{@\mathit{S}\mathit{T}\mathit{C}}}\right)-\mathbf{0.2}{\left(\mathbf{1}-\frac{\mathit{G}}{{\mathit{G}}_{@\mathit{S}\mathit{T}\mathit{C}}}\right)}^{\mathbf{2}}\right]-\mathit{d}{\mathit{V}}_{\mathit{t}}(\mathit{t}-{\mathit{t}}_{@\mathit{S}\mathit{T}\mathit{C}}).\hfill \end{array}$$

(37)

where ${\mathit{G}}_{@\mathit{S}\mathit{T}\mathit{C}},{\mathit{t}}_{@\mathit{S}\mathit{T}\mathit{C}},{\mathit{I}}_{\mathit{s}{\mathit{c}}_{@\mathit{S}\mathit{T}\mathit{C}}},{{\mathit{I}}_{\mathit{m}\mathit{p}}}_{@\mathit{S}\mathit{T}\mathit{C}},{\mathit{v}}_{\mathit{m}{\mathit{p}}_{@\mathit{S}\mathit{T}\mathit{C}}},$ and ${{\mathit{v}}_{\mathit{o}\mathit{c}}}_{@\mathit{S}\mathit{T}\mathit{C}}$ are the irradiance, temperature, short circuit current, maximum power point current, maximum power point voltage, and open circuit voltage, respectively, at STC.

4. Results and Discussion

The three established fractional models are applied to three commercial PV modules from different manufacturers to assess the generated fractional models’ validation. The objective is to show whether the proposed models give high accuracy in modeling the I-V curves of commercial modules compared to the basic integer models. The commercial PV modules under study at STC are indicated in

Table 1. Date of commercial PV modules at STC.

Module No.	Commercial Name	${\mathit{I}}_{\mathit{s}\mathit{c}}$ (A)	${\mathit{v}}_{\mathit{o}\mathit{c}}$  (V)	${\mathit{I}}_{\mathit{m}\mathit{p}}$ (A)	${\mathit{v}}_{\mathit{m}\mathit{p}}$ (V)	$\mathit{d}\mathit{I}$ (A/°C)	$\mathit{d}{\mathit{V}}_{\mathit{t}}$ (V/°C)
Module 1	KYOCERA (KK280P-3CD3CG) [45]	9.53	38.9	8.89	31.5	0.00559	0.138
Module 2	KFSolar (KF245-280P-200) [46]	9.17	37.9	8.49	31.2	0.00459	0.1213
Module 3	Amerisolar (AS-6P30-275P) [47]	9.2	38.5	8.79	31.3	0.0046	0.1194

. Also, the commercial modules’ [45,46,47] actual I-V curves from datasheets at different irradiance and temperature levels are extracted in order to be compared with the generated fractional models and basic integer ones. The PV modules’ actual datasheet curves were extracted using a well-known extraction online software, named “WebPlotDigitizer”, version 4.6 [48] in which the graph of the I-V curves can be transformed into a set of data points.

While applying the fractional model 1, the fractional order value (${\mathit{\varphi }}_{\mathbf{0}}$) is estimated for the different commercial modules using trial-and-error estimations. The stochastic models 2 and 3, presented in stochastic differential equation form, are numerically solved when applied to the three commercial PV modules using the previously defined fractional Euler-Maruyama scheme. MATLAB version 2021a is used to apply the numerical scheme with a number of paths equal to 10 and a numerical step equal to 0.1. The average path solution is compared to the other models when applied to the three PV modules. The accuracy of each model compared to the actual data extracted from the datasheets is assessed using the determination coefficient (R²) and the root mean square error (RMSE) and they are expressed as follows:

$${\mathit{R}}^{\mathbf{2}}=\mathbf{1}-\frac{\sum _{\mathit{i}=\mathbf{1}}^{\mathit{n}}{\left({\mathit{y}}^{\mathit{P}\mathit{V}}-{\mathit{y}}^{\mathit{e}\mathit{s}\mathit{t}}\right)}^{\mathbf{2}}}{\sum _{\mathit{i}=\mathbf{1}}^{\mathit{n}}{\left({\mathit{y}}^{\mathit{P}\mathit{V}}-\overline{{\mathit{y}}^{\mathit{e}\mathit{s}\mathit{t}}}\right)}^{\mathbf{2}}}.$$

(38)

$$\mathit{R}\mathit{M}\mathit{S}\mathit{E}={\left[\frac{\mathbf{1}}{\mathit{n}}{\sum }_{\mathit{i}=\mathbf{1}}^{\mathit{n}}{\left({\mathit{y}}^{\mathit{P}\mathit{V}}-{\mathit{y}}^{\mathit{e}\mathit{s}\mathit{t}}\right)}^{\mathbf{2}}\right]}^{\mathbf{1}/\mathbf{2}}.$$

(39)

where ${\mathit{y}}^{\mathit{P}\mathit{V}}$ and ${\mathit{y}}^{\mathit{e}\mathit{s}\mathit{t}}$ are the extracted data from the graph of the PV module in its datasheet and the estimated data using the proposed estimation models, respectively. $\mathit{n}$ is the number of data points extracted from the actual datasheet curves.

We begin by presenting and analyzing the results for all the appropriate PV panels using the desired models by fixing the irradiance and changing the temperature, and then setting the temperature and varying the irradiance to see how the accuracy of the models is affected by varying these factors.

When applying the different models to the first commercial PV module and changing the temperature while fixing the irradiance at 1000 W/m², we arrived at the results shown in

Table 2. Accuracy of modeling for the first PV module at constant irradiance (1000 W/m²).

Module	Condition	Integer Model 1  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Integer Model 2  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Fractional Model 1  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Stochastic Model 2  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Stochastic Model 3  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$
KYOCERA (KK280P-3CD3CG) [45]	t = 25	(1.07, 0.9783)	(0.5564, 0.9951)	(0.9143, 0.9786)	(0.7508, 0.9786)	(0.2624, 0.9982)
	t = 50	(1.6171, 0.9848)	(0.3771, 0.9942)	(1.7486, 0.9747)	(1.507, 0.989)	(0.576, 0.992)
	t = 75	(3.2308, 0.9457)	(1.2676, 0.9632)	(2.8876, 0.9562)	(3.0308, 0.9497)	(0.6033, 0.9915)

, which describe how the efficiency of the models is affected by changing the temperature. Through analyzing Table 2, at STC, each proposed model compared to its governing one gave very close accuracy. But, at 50 °C, the proposed fractional stochastic models gave slightly higher accuracy than their corresponding integer models. Increasing the temperature to 75 °C showed the superiority of the proposed models compared to the integer models. The fractional model 1 gave an R² value equal to 0.9562, while its corresponding integer model 1 gave 0.9457. In the case of stochastic model 3, it provided the best value of R², which equaled 0.9915, while its corresponding integer model 2 gave 0.9632. Figure 1 shows the I-V curves of module one at different temperatures for both the proposed models and the integer models compared with the real curves extracted from the module datasheet.

Table 3. Accuracy of modeling for the first PV module at constant temperature (25 °C).

Module	Condition	Integer Model 1  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Integer Model 2  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Fractional Model 1  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Stochastic Model 2  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Stochastic Model 3  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$
KYOCERA (KK280P-3CD3CG) [45]	G = 1000	(1.07, 0.9783)	(0.5564, 0.9951)	(0.9143, 0.9786)	(0.7508, 0.9786)	(0.2624, 0.9982)
	G = 800	(0.765, 0.9816)	(0.326, 0.9992)	(0.432, 0.9892)	(0.6011, 0.9949)	(1.4669, 0.9896)
	G = 600	(0.7622, 0.9793)	(0.4933, 0.9893)	(0.4622, 0.9936)	(0.4431, 0.9943)	(0.8622, 0.9723)
	G = 400	(0.6708, 0.9743)	(0.3032, 0.9881)	(0.3708, 0.9923)	(0.2079, 0.9946)	(0.7302, 0.9683)
	G = 200	(0.4814, 0.9716)	(0.1696, 0.982)	(0.2814, 0.9962)	(0.1278, 0.9967)	(0.391, 0.9625)

shows the accuracy of the proposed and integer models compared to actual I-V curves while varying the irradiance at a fixed temperature (25 °C). Decreasing the irradiance has a substantial impact on the modeling accuracy. From the results in Table 3, the accuracy of the integer models 1 and 2 decreases with decreasing irradiance levels. The accuracy of the stochastic model 3 also decreases with decreasing irradiance. The accuracies of the developed fractional model 1 and stochastic model 1 increase by reducing the irradiance levels. Figure 2 shows the I-V curves of module 1 at different irradiance for both the proposed models and the integer models compared with the real curves extracted from the module datasheet.

In commercial PV module 2, when the temperature changed from 25 to 70 °C at fixed irradiance, the obtained models’ accuracy is described in

Table 4. Accuracy of modeling for the second PV module at constant irradiance (1000 W/m²).

Module	Condition	Integer Model 1  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Integer Model 2  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Fractional Model 1  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Stochastic Model 2  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Stochastic Model 3  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$
KFSolar (KF245-280P-200) [46]	t = 25	(0.171, 0.9981)	(0.2772, 0.9991)	(1.9789, 0.9823)	(0.564, 0.9941)	(1.9189, 0.9873)
	t = 55	(1.7908, 0.989)	(0.5433, 0.9974)	(0.398, 0.9977)	(1.339, 0.9903)	(0.318, 0.9992)
	t = 70	(2.708, 0.9843)	(2.0279, 0.9869)	(0.9922, 0.996)	(0.8619, 0.9943)	(1.878, 0.9892)

. At 25 °C, integer model 1 gives an R² equal to 0.9981, which is higher than fractional model 1 with an R² equal to 0.9823, and slightly higher than stochastic model 2, with an R² equal to 0.9941. While integer model 2 accuracy gives an R² equal to 0.9991, it is higher than its corresponding developed model, stochastic model 3, with an R² equal to 0.9873. At higher temperature levels (55 and 70 °C), the developed models’ accuracy exceeds the corresponding integer models’ accuracy with higher R² values and a lower RMSE. The I-V curves of module two at fixed irradiance and variable temperature compared to the real extracted curves from the module datasheet are represented in Figure 3.

By varying the irradiance from 1000 to 200 W/m² at a fixed temperature of 25 °C, the third commercial PV module data is extracted and compared with the developed models and their integer ones. The results are shown in Figure 4, with RMSE and R² as indicated in

Table 5. Accuracy of modeling for the second PV module at constant temperature (25 °C).

Module	Condition	Integer Model 1  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Integer Model 2  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Fractional Model 1 $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Stochastic Model 2  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Stochastic Model 3  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$
KFSolar (KF245-280P-200) [46]	G = 1000	(0.171, 0.9981)	(0.2772, 0.9991)	(1.9789, 0.9823)	(0.564, 0.9941)	(1.9189, 0.9873)
	G = 800	(0.7819, 0.9982)	(0.5617, 0.9996)	(2.1789, 0.9721)	(1.8302, 0.9898)	(1.8302, 0.9898)
	G = 600	(1.2939, 0.9834)	(0.428, 0.987)	(1.1677, 0.9902)	(0.3178, 0.9979)	(0.8199, 0.9859)
	G = 400	(0.8478, 0.9831)	(0.3429, 0.979)	(0.8199, 0.9859)	(0.1977, 0.9973)	(0.6781, 0.969)
	G = 200	(0.4007, 0.9803)	(0.2432, 0.973)	(0.1996, 0.9934)	(0.1137, 0.995)	(0.276, 0.9672)

. For an irradiance level of 1000, integer model 1 gives an accuracy higher than the fractional model 1 and slightly higher accuracy than the stochastic model 2. While integer model 2 provides higher accuracy than stochastic model 3. At irradiance levels of 800 W/m², the integer models have better accuracy than the developed models. At irradiance levels of 600, 400, and 200 W/m², stochastic model 2 is the best model with the highest R² values, followed by fractional model 1 and integer model 1. Integer model 2 and the stochastic model 3 have the lowest accuracy at these irradiance levels.

By applying the different models to module 3 and changing the temperature while fixing the irradiance at 1000 W/m², we arrived at the results in

Table 6. Accuracy of modeling for the third PV module at constant irradiance (1000 W/m²).

Module	Condition	Integer Model 1  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Integer Model 2  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Fractional Model 1 $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Stochastic Model 2  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Stochastic Model 3  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$
Amerisolar (AS-6P30-275P) [47]	t = 25	(3.0905, 0.9626)	(1.6674, 0.9876)	(2.7928, 0.9694)	(2.4498, 0.9698)	(1.398, 0.9824)
	t = 50	(3.571, 0.9849)	(0.7343, 0.9896)	(0.654, 0.9945)	(0.871, 0.9896)	(0.4436, 0.9977)
	t = 75	(0.7068, 0.9841)	(0.415, 0.9935)	(2.2835, 0.9861)	(0.5068, 0.9941)	(0.305, 0.9966)

, which describe how the efficiency of the models is affected by changing the temperature. Analyzing Table 6, at STC, each proposed model is compared to its basic integer one with near accuracy. However, at 50 °C, the proposed fractional stochastic models have higher accuracy than their corresponding integer models. Fractional model 1 gives an accuracy equal to 0.9945, while its integer model 1 gives 0.9849. Fractional stochastic model 3 gives the best accuracy with an R² value equal to 0.9977. Increasing the temperature to 75 °C shows that the proposed stochastic model 3 provides the best accuracy, better than stochastic model 2, integer model 2, and fractional model 1. While integer model 1 gives the lowest accuracy, Figure 5 shows the I-V curves of module 3 at different temperatures for both the proposed and integer models compared with the real curves extracted from the module datasheet.

To check the modeling accuracy of the generated I-V curves under varying irradiance at a fixed temperature (25 °C), the obtained results are presented in

Table 7. Accuracy of modeling for the third PV module at constant temperature (25 °C).

Module	Condition	Integer Model 1  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Integer Model 2  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Fractional Model 1 $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Stochastic Model 2  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$	Stochastic Model 3  $(\mathit{R}\mathit{M}\mathit{S}\mathit{E},{\mathit{R}}^{\mathbf{2}})$
Amerisolar (AS-6P30-275P) [47]	G = 1000	(3.0905, 0.9626)	(0.6674, 0.9876)	(2.7928, 0.9694)	(2.4498, 0.9698)	(1.398, 0.9824)
	G = 800	(0.5089, 0.9767)	(0.398, 0.9912)	(0.8189, 0.9891)	(1.0132, 0.9826)	(0.7065, 0.989)
	G = 600	(0.5012, 0.9824)	(0.4369, 0.9855)	(0.6432, 0.9836)	(0.476, 0.9847)	(1.3481, 0.9824)
	G = 400	(0.531, 0.9844)	(0.4819, 0.9832)	(0.434, 0.9875)	(0.4893, 0.9848)	(0.9126, 0.9825)
	G = 200	(0.1018, 0.9889)	(0.1864, 0.9807)	(0.1821, 0.9932)	(0.2330, 0.9921)	(0.3133, 0.9781)

and visualized as shown in Figure 6. From the obtained results in Table 7, at an irradiance of 1000 W/m², the best model is the integer model 2, and then the stochastic model 3. At 800 W/m², the best model is integer model 2, and then fractional model 1. In the range from 600 to 200 W/m², fractional model 1 is the best, followed by fractional stochastic model 2 and, after that, integer model 2.

5. Conclusions

This paper introduces three novel empirical formulas to model commercial PV modules’ I-V curves. The article targets modeling the actual modules’ curves extracted from the modules’ datasheets. The proposed formulas use modules’ commercial nameplate data (short circuit current, maximum power point current, maximum power point voltage, open circuit voltage, and module current and voltage sensitivity to temperature) as inputs without complex data collection or forecasting. The targeted objective is to generate realistic I-V curves with minimum root mean square errors (RMSE) and maximum determination coefficients (R²). According to the results, the developed models provide more accurate predictions than the integer models at irradiance levels of less than 800 W/m² and PV module temperatures greater than 50 °C. Therefore, these developed models are recommended for modeling commercial PV modules’ I-V curves within these specified operating condition ranges. Finally, adding fractional operators and stochastic white noise results in more realistic I-V curves tangible to the actual datasheet curves. Future works will introduce the application of the proposed PV modeling in power systems [49,50,51,52,53].