A Critical Review on the Estimation Techniques of the Solar PV Cell’s Unknown Parameters

Abstract

To meet the exponentially growing demand for clean and green energy, the solar photovoltaic (PV) system’s importance is increasing day by day, for which PV modeling is considered to be one of the most important work in the current state-of-the-art methods. To effectively model a PV system, accurate PV parameter estimation is of the utmost importance. In line with this, although the values of some of the parameters are provided in the manufacturer’s datasheet, the values of unknown parameters, such as shunt resistance, series resistance, the diode ideality factor, photo-generated current and diode saturation current, are not provided. To estimate these values a lot of algorithms are already reported in the literature. After careful observation of all the reported algorithms, a few best-reported algorithms are identified and their performances are compared with respect to accuracy, convergence issues, computational complexity and thermal stability. All kind of algorithms, such as numerical, analytical and evolutionary algorithms, are considered in this study, and only the best reported algorithms are considered for the comparison.

1. Introduction

Solar photovoltaic (PV) technology is one of the leading renewable energy technologies. It has the potential to meet the global energy demand without harming the environment. According to [1], the global solar PV installed capacity is increasing exponentially, and the total installed capacity had reached 629 GW by 2019. Unlike conventional energy sources, such as coal, oil, petrol, etc., solar PV has fewer harmful effects on the environment during operation and maintenance. Again due to the reduction in its cost, solar PV is gaining a new horizon in the energy industry. As a result, PV is considered one of the highest power-producing technologies across the globe. Although the solar PV has manifold advantages compared to all other energy sources, it has a number of demerits too. The power generation from a PV plant depends on the solar irradiance, which is not reliable. Moreover, if a part of the PV module or array is shaded (partial shading condition), the performance of a PV plant decreases drastically. When a solar cell is shaded, it acts as a reverse biased diode, and it affects the performance of the PV module. In such a condition, the short-circuit current ($I_{sc}$) and fill factor ($FF$) change [2], which in turn affects the output power of the PV system. Again, to analyse the behaviour of current, voltage and $FF$ in different atmospheric conditions, it is integral to know the different parameters of the PV cell/module.

To optimise the performance, efficiency, size and cost of the PV systems extensive studies have been carried out which help the researchers in efficiently modelling a PV module. For the accurate modelling of a PV cell/module, accurately estimating the PV system’s parameters is very important. In line with that, although several parameters are available in the manufacturer’s datasheet, a set of parameters are normally not available. It creates difficulty in accurately modelling a PV module. Accordingly, the accurate measurement of modelling parameters, such as diode saturation current ($I_0$), series resistance ($R_s$), diode ideality factor (n), shunt resistance ($R_{sh}$) and photo-genated current ($I_{ph}$), are essential. To estimate those unknown parameters, several algorithms are proposed in the literature, which have been reviewed and analysed carefully in this paper, along with critical reasoning for all the best-reported models. The existing algorithms can be broadly divided into three categories: analytical algorithms, non-analytical algorithms and meta-heuristic approaches. Analytical algorithms comparatively take less time for computation, and there is no convergence failure problem, whereas numerical or non-analytical methods consume more time for computation and suffer from convergence issues. However, to solve a non-linear PV system equation, an analytical algorithm cannot be used singly. Therefore, in many algorithms, a combination of both analytical and numerical algorithms is used to optimise the computational time and convergence issues. In the current state of art, meta-heuristic approaches are also gaining importance among the PV researchers because of their flexibility in computation time and convergence. However, sometimes, the meta-heuristic algorithms get stuck at local maxima or local minima, and as a result, they do not converse [3,4]. In the case of evolutionary algorithms, if the fitness function is not derived properly, there is a high chance of obtaining the wrong output.

Many reviews on PV parameter estimation have already been published in the literature. However, after carefully observing the last 5 years of publications, it is noticed that some critical points are yet to be highlighted precisely. A few observations are given below:

• The combination of both analytical and numerical techniques (combined technique) are observed to be less time-consuming compared to other techniques that are used for parameter estimation of solar PV systems;
• The combined technique has the highest accuracy compared to others in the existing literature;
• In analytical algorithms, the number of non-linear exponential terms can be reduced by four times compared to the existing literature. Hence, the combined technique is computationally efficient;
• The shunt resistance ($R_{sh}$) should be considered as one of the iteration parameters, which makes the approach more realistic;
• The application of the nominal operating cell temperature (NOCT) value in the PV parameter estimation strengthens the accuracy in varying temperatures and irradiance conditions. It is noticed that, on average, a 10% performance degradation (PD) is present in the MPP obtained at $T_{cell}$ compared to $T_{amb}$.

Solar PV modelling is discussed in the second part of the paper, in which a single-diode model, double-diode model and three-diode model are discussed with suitable diagrams and relevant mathematical equations. The algorithms reported so far for the estimation of PV parameters are discussed in the third part, in which analytical algorithms, non-analytical algorithms and meta-heuristic algorithms are discussed with suitable examples. The performance analysis of the reported algorithms are discussed in part four of the paper. Here, the effect of temperature and radiance on the PV system are discussed with neat diagrams provided. Finally, the conclusion is written in part five of this paper.

2. Solar PV Modelling

The performance of a PV module depends on the value of irradiance (G) and temperature (T) at which the module operates. To simulate a solar PV module, accurately modelling the solar PV cell is very important. Based on the semiconductor PN junction physics, the equivalent circuit of a solar PV cell is designed, in which there may be one cell, two cells or more than two cells. According to the number of diodes present, the nomenclature of the models are given. In the equivalent circuit, if one diode is present, it is called a single-diode model (SDM). Accordingly, based on the presence of diodes, they are called a double-diode model (DDM), three-diode model and so on.

2.1. Single-Diode Model

The equivalent circuit diagram of a single-diode model solar PV cell is shown in Figure 1. The terminal current (I) of the PV cell is calculated with Equation (1).

$$I=I_P-I_D-\frac{V+I{R}_S}{R_{SH}}$$

(1)

where $I_P$ and $I_D$ are the photo-generated current and diode current, respectively; V is the terminal voltage; $R_S$ and $R_{SH}$ are the series and shunt resistances, respectively. $I_D$ can be expressed as given in Equation (2).

$$I_D=I_0\left\{e^{\frac{V+I{R}_S}{V_T}}-1\right\}$$

(2)

where $V_T$ ($=\frac{nkT}{q}$) is the junction thermal voltage. Here $n,k,T$ and q are the diode ideality factor, Boltzmann’s constant, cell temperature in $°$C and electron charge in coulomb, respectively. Combining Equations (1) and (2), we can write Equation (3).

$$I=I_P-I_0\left\{e^{\frac{V+I{R}_S}{V_T}}-1\right\}-\frac{V+I{R}_S}{R_{SH}}$$

(3)

For the calculation of the terminal current, it is of the utmost importance to estimate all the parameters involved in Equations (1) and (2). Although the information related to the open-circuit voltage, short-circuit current and maximum power point is provided in the manufacturer’s datasheet, the value of the five parameters, namely $I_P$, $I_0$, n, $R_S$ and $R_{SH}$, are normally not provided in the manufacturer’s datasheet. Therefore, several algorithms are developed in the literature to estimate those unknown parameters. A comparative review was presented in [5] and included almost all the then best-reported algorithms. After this, a lot of algorithms were developed to estimate the unknown parameters of a PV cell, which are comparatively better in terms of accuracy, computational complexity, convergence issues and many other aspects. By analysing all the best-reported algorithms, a critical review is provided, and the same is reported in this paper.

2.2. Double-Diode Model

The diode ideality factor ‘n’ mainly depends on the voltage between the two terminals of the device. The value of n approaches 1 when the recombination is dominated by the bulk region and the surfaces. However, if recombination is the dominant parameter, n approaches 2, which happens at lower voltages. To introduce this phenomenon, a second diode is connected in parallel with the diode present in the single-diode model’s equivalent circuit, and this model is named the double-diode model (DDM) of the PV system. The equivalent circuit of the DDM PV system is shown in Figure 2, and the expression for the generated current using DDM of a PV system is given in Equation (4).

$$I=I_P-I_{0_1}\left\{e^{\frac{(V+I{R}_S)q}{n_{1}kT}}-1\right\}-I_{0_2}\left\{e^{\frac{(V+I{R}_S)q}{n_{2}kT}}-1\right\}-\frac{V+I{R}_S}{R_{SH}}$$

(4)

From Equation (4), it is seen that there are a total of seven unknown parameters whose values are not provided in the manufacturer’s datasheet. These seven parameters are $I_P$, $I_{0_1}$,$I_{0_2}$, $n_1$, $n_2$, $R_S$ and $R_{SH}$. The equations during open-circuit and short-circuit conditions in the case of DDM are given in Equations (5) and (6):

$$I=I_P-I_{0_1}\left\{e^{\frac{\left(I{R}_S\right)q}{n_{1}kT}}-1\right\}-I_{0_2}\left\{e^{\frac{\left(I{R}_S\right)q}{n_{2}kT}}-1\right\}-\frac{I{R}_S}{R_{SH}}$$

(5)

$$0=I_P-I_{0_1}\left\{e^{\frac{Vq}{n_{1}kT}}-1\right\}-I_{0_2}\left\{e^{\frac{\left(V\right)q}{n_{2}kT}}-1\right\}-\frac{V}{R_{SH}}$$

(6)

2.3. Three-Diode Model

To address the effects of leakage current and grain boundaries, a three-diode model of the PV system is introduced. The equivalent diagram of the same is shown in Figure 3, and the expression for generated current using DDM of a PV system is given in Equation (7).

$$\begin{array}{c}\hfill I=I_P-I_{0_1}\left\{e^{\frac{(V+I{R}_S)q}{n_{1}kT}}-1\right\}-I_{0_2}\left\{e^{\frac{(V+I{R}_S)q}{n_{2}kT}}-1\right\}-\\ \hfill I_{0_3}\left\{e^{\frac{(V+I{R}_S)q}{n_{3}kT}}-1\right\}-\frac{V+I{R}_S}{R_{SH}}\end{array}$$

(7)

From Equation (7), it is seen that there are a total of nine unknown parameters whose values are not provided in the manufactures datasheet. These nine parameters are $I_P$, $I_{0_1}$,$I_{0_2}$, $I_{0_3}$, $n_1$, $n_2$, $n_3$, $R_S$ and $R_{SH}$.

3. Algorithms Reported So Far

3.1. Analytical Algorithms

Whenever the number of diodes increases in PV modelling, the unknown parameters also increase accordingly. With the increase in every diode, at least two unknown parameters increase. Hence, in the already reported literature, it is observed that the researchers mainly opt for SDM and DDM for parameter estimation work. Again, in SDM, five unknown parameters need to be calculated, whereas, in DDM, seven unknown parameters are present; as a result, the DDM algorithm suffers from more computational complexities [6,7,8]. In some reported algorithms such as [9,10], DDM is used for parameter estimation, where the computation is carried out for seven unknown parameters without much improvement in accuracy. Therefore, the SDM reported model is analysed critically, and the same is reported below.

To estimate the unknown parameters, several analytical algorithms (AAs) have been reported. In almost all of the algorithms, the mathematical modelling is conducted with the help of three strategic points of the I–V characteristics. These points are the open-circuit point ($V_{OC}$,0), short-circuit point (0,$I_{SC}$) and maximum power point ($V_{MP}$,$I_{MP}$), as shown in Figure 4.

As the value of voltage in short-circuit point is 0 (i.e., $V=0$), the current is therefore calculated as Equation (8). In Equation (8), the short-circuit current is denoted by $I_{SC}$.

$$I_{SC}=I_P-I_0\left\{e^{\frac{I_{SC}{R}_S}{N{V}_T}}-1\right\}-\frac{I_{SC}{R}_S}{R_{SH}}$$

(8)

Similarly, the generated current at the open-circuit point is 0 (i.e., $I=0$) expressions is written as Equation (9).

$$0=I_P-I_0\left\{e^{\frac{I_{SC}{R}_S}{N{V}_T}}-1\right\}-\frac{V_{OC}}{R_{SH}}$$

(9)

Simplifying Equation (9) for $I_P$, we obtain

$$I_P=I_0\left\{e^{\frac{I_{SC}{R}_S}{N{V}_T}}-1\right\}+\frac{V_{OC}}{R_{SH}}$$

(10)

At ($V_{MP},I_{MP}$), by putting $I=I_{MP}$ and $V=V_{MP}$ in Equation (2), we obtain Equation (11).

$$I_{MP}=I_P-I_0\left\{e^{\frac{V_{MP}+I_{MP}{R}_S}{N{V}_T}}-1\right\}-\frac{V_{MP}+I_{MP}{R}_S}{R_{SH}}$$

(11)

After simplifying Equation (8) with the help of Equation (10), we obtain

$$I_{SC}=I_0\left\{e^{\frac{I_{SC}{R}_S}{N{V}_T}}-1\right\}-I_0{e}^{\frac{I_{SC}{R}_S}{N{V}_T}}-\frac{I_{SC}{R}_S}{R_{SH}}$$

(12)

Further simplifying Equation (12) for $I_0$, we obtain Equation (13),

$$I_0=\frac{I_{SC}-\left\{\frac{V_{OC}-I_{SC}{R}_S}{R_{SH}}\right\}}{e^{\frac{V_{OC}}{N{V}_T}}-e^{\frac{I_{SC}{R}_S}{N{V}_T}}}$$

(13)

Simplifying $I_P$ and $I_0$ in Equation (11) using the corresponding expression from Equations (10) and (13), respectively, we obtain Equation (14):

$$I_{MP}=I_0\left\{e^{\frac{I_{SC}{R}_S}{N{V}_T}}-1\right\}+\frac{V_{OC}}{R_{SH}}-\frac{I_{SC}-\left\{\frac{V_{OC}-I_{SC}{R}_S}{R_{SH}}\right\}}{e^{\frac{V_{OC}}{N{V}_T}}-e^{\frac{I_{SC}{R}_S}{N{V}_T}}}\left\{e^{\frac{V_{MP}+I_{MP}{R}_S}{N{V}_T}}-1\right\}-\frac{V_{MP}+I_{MP}{R}_S}{R_{SH}}$$

(14)

Equation (14) is used for the estimation of unknown parameters in many of the existing algorithms [11]. However, Equation (14) comprises four numbers of exponential terms, and accordingly, it increases non-linearly. Therefore, Equation (14) is not used in the reported models, such as [12]. Therefore, by neglecting the less significant terms, a different approach is introduced, which is explained below.

In the case of the PV systems that are made of silicon, the value of $N{V}_T$ is smaller than $(V+I{R}_S)$. Therefore, $e^{\frac{V+I{R}_S}{N{V}_T}}\gg 1$. Hence, for simplicity, ‘−1’ is neglected and rewritten as Equations (3), (8) and (10), as given in Equations (15)–(17), respectively.

$$I=I_P-I_0{e}^{\frac{V+I{R}_S}{V_T}}-\frac{V+I{R}_S}{R_{SH}}$$

(15)

$$I_{SC}=I_P-I_0{e}^{\frac{I_{SC}{R}_S}{N{V}_T}}-\frac{I_{SC}{R}_S}{R_{SH}}$$

(16)

$$I_P=I_0{e}^{\frac{I_{SC}{R}_S}{N{V}_T}}+\frac{V_{OC}}{R_{SH}}$$

(17)

Incorporating Equation (17) into Equation (16) and eliminating the term $e^{\frac{I_{SC}{R}_S}{N{V}_T}}$ (as $e^{\frac{V_{OC}}{N{V}_T}}\gg e^{\frac{I_{SC}{R}_S}{N{V}_T}})$, we obtain Equation (18).

$$I_{SC}=I_0{e}^{\frac{I_{SC}{R}_S}{N{V}_T}}+\frac{V_{OC}-I_{SC}{R}_S}{R_{SH}}$$

(18)

Further resolving Equation (18), we obtain an expression for $I_0$ as given in Equation (19).

$$I_0=\left\{I_{SC}-\frac{V_{OC}-I_{SC}{R}_S}{R_{SH}}\right\}{e}^{-\frac{V_{OC}}{N{V}_T}}$$

(19)

In Equation (11), $I_P$ and $I_0$ are replaced by the corresponding expressions of Equations (10) and (19), respectively, which results in Equation (20).

$$\begin{array}{c}\hfill I_{SC}=I_{MP}+\frac{V_{MP}+I_{MP}{R}_S-I_{SC}{R}_S}{R_{SH}}+\\ \hfill \left\{I_{SC}-\frac{V_{OC}-I_{SC}{R}_S}{R_{SH}}\right\}{e}^{\frac{V_{MP}+I_{MP}{R}_S-V_{OC}}{N{V}_T}}\end{array}$$

(20)

For estimating the unknown parameters, Equation (20) is used, in which only one non-linear term is present. Therefore, it decreases the issues related to non-linearity and accordingly minimises the computational complexity.

Except for $R_S$, $R_{SH}$ and $V_T$, the values of all the other parameters in Equation (20) are provided in the manufacturer’s datasheet. The estimation of these three unknown parameters helps in obtaining the values of $I_0$ and $I_P$ using Equations (19) and (10), respectively. As reported by [13,14], the initial values of shunt ($R_{SH0}$) and series resistance ($R_{S0}$) can be measured from the I–V curve using Equations (21) and (22).

$$R_{S0}=-\frac{dV}{dI}{|}_{V=V_{OC}}$$

(21)

$$R_{SH0}=-\frac{dV}{dI}{|}_{I=I_{SC}}$$

(22)

Following the same procedure described in [15], the analytical expressions for $V_T$, $R_S$ and $R_{SH}$ are derived, which are mentioned in Equations (23) and (24).

$$\begin{array}{c}\hfill V_T=\frac{V_{MP}+R_{S0}{I}_{MP}-V_{OC}}{ln\left\{I_{SC}-\frac{V_{MP}}{R_{SH0}}-I_{mpp\left(s\right)}\right\}-ln\left\{I_{SC}-\frac{V_{OC}}{R_{SH}}\right\}+\frac{I_{MP}}{I_{SC}-\left\{\frac{V_{OC}}{R_{SH0}}\right\}}}\end{array}$$

(23)

$$\left\{\begin{array}{c}{R}_S=R_{S0}-\frac{V_T}{I_0}{e}^{\frac{-V_{OC}}{V_T}}\hfill \\ R_{SH}={\left\{\frac{1}{R_{SH0}-R_S}-\frac{I_0}{V_T}{e}^{\frac{I_{SC}{R}_S}{V_T}}\right\}}^{-1}\hfill \end{array}\right.$$

(24)

By solving Equations (20), (23) and (24), all three unknown parameters—$R_S$, $R_{SH}$ and $V_T$—can be obtained. To solve these equations, several algorithms are adopted in the existing literature, such as non-linear optimisation algorithms or numerical algorithms (NAs), evolutionary algorithms (EAs), etc.

3.2. Non-Analytical Algorithms

For parameter estimation, several non-linear optimisation algorithms or EAs are developed in the existing literature, out of which some of the best-reported algorithms are highlighted in [5]. Here, it is observed that the result of some of the reported algorithms, such as [16,17,18,19], are very significant. After these, a number of algorithms such as [20,21,22,23,24,25,26,27,28,29,30] are reported. From the critical observation of all the reported algorithms, it is noticed that, although a number of reported algorithms are found to be efficient, they show poor performance in some cases. Again, a comparison of these algorithms is also a tedious job. In the literature, it is seen that, for the purposes of comparison, the authors have validated the algorithms with a common PV system. To check the cell level [31] parameter estimation, they have considered a RTC France silicon solar cell (57 mm diameter) at 1000 W/m${}^2$ irradiance and a temperature of 33 $°$C (Case Study 1), and for the module level, they have considered a solar module comprised of 36 series connected cells (Photowatt-PWP 201) at 1000 W/m${}^2$ irradiance and a temperature of 45 $°$C (Case Study 2). In [32]. Finally, 26 data points are considered from the experimentally obtained I–V characteristics. The same is implemented in the developed algorithms, and the corresponding values of the estimated current ($I_{{estd}_j}$; here, j changes from 1 to 26) are calculated. To check the accuracy, the root mean square error (RMSE), mean absolute error (MAE), deviation, normalised sum of squared error (NSSE) and mean absolute error in power (MAEP) are calculated, as given in Equations (25)–(29), respectively.

$$RMSE=\sqrt{\frac{{\sum }_{j=1}^p{(I_{{estd}_j}-I_j)}^2}{p}}$$

(25)

$$\mathrm{MAE}=\frac{{\sum }_{j=1}^p(I_{{estd}_j}-I_j)}{p}$$

(26)

$$\mathrm{Deviation}(\%)=\left(\frac{I_{{estd}_j}}{I_j}-1\right)\times 100$$

(27)

$$\mathrm{NSSE}(\%)=\frac{{\sum }_{j=1}^p{(I_j-I_{{estd}_j})}^2}{{\sum }_{j=1}^p{\left(I_I\right)}^2}$$

(28)

$$\mathrm{MAEP}=\frac{{\sum }_{j=1}^p|P_j-P_{{estd}_j}|}{p}$$

(29)

Parameter estimation is carried out for both case studies (Case Study 1 and Case Study 2), and the reported results are given in

Table 1. Comparison of the parameters obtained in Case Study 1.

Literature	Year	Algorithm	${\mathit{I}}_{\mathit{p}\mathit{h}}$ (A)	${\mathit{I}}_0$($\mathsf{\mu }$A)	${\mathit{R}}_{\mathit{s}\mathit{h}}$($\mathbf{\Omega }$)	${\mathit{R}}_{\mathit{s}}$($\mathbf{\Omega }$)	n	No. of Steps	RMSE
[42]	2019	EA	0.76080000	0.32230000	53.76340000	0.036400000	1.4837000000	-	1.0072 × 10${}^{-2}$
[43]	2018	NA	0.76074014	0.31285196	55.90738000	0.036615485	1.477729500	-	7.7301 × 10${}^{-4}$
[44]	2020	EA	0.76038466	0.23082625	53.67788300	0.037991668	1.447929015	-	9.7505 × 10${}^{-4}$
[45]	2019	EA	0.76080000	0.32300000	53.71850000	0.036400000	1.481200000	-	9.8602 × 10${}^{-4}$
[46]	2017	EA	0.76080000	0.32280000	53.75950000	0.036400000	1.481100000	-	9.8603 × 10${}^{-4}$
[47]	2020	EA	0.76077600	0.32302100	53.71852000	0.036377000	1.481184000	-	9.8602 × 10${}^{-4}$
[48]	2019	NA + EA	0.76078797	0.31068450	52.88979426	0.036546950	1.477267780	-	7.7301 × 10${}^{-4}$
[49]	2019	EA	0.76077552	0.32302000	53.71852000	0.036370000	1.481108170	-	9.8602 × 10${}^{-4}$
[50]	2019	EA	0.76079000	0.31062000	52.88500000	0.036548000	1.477100000	-	7.7300 × 10${}^{-4}$
[51]	2019	EA	0.76080000	0.32300000	53.71850000	0.036400000	1.481200000	-	9.8602 × 10${}^{-4}$
[52]	2019	EA	0.76077562	0.32301700	53.71821748	0.036377160	1.481182200	-	9.8602 × 10${}^{-4}$
[53]	2019	EA	0.76078000	0.32302000	53.71852000	0.036380000	1.481180000	-	9.8602 × 10${}^{-4}$
[54]	2019	EA	0.76080000	0.32300000	53.71850000	0.036400000	1.481200000	-	9.8602 × 10${}^{-4}$
[55]	2019	EA	0.76077500	0.32302100	53.71867900	0.036377000	1.481108000	-	9.8602 × 10${}^{-4}$
[56]	2019	EA	0.76077450	0.32300180	53.73000000	0.036377500	1.481177400	-	9.8602 × 10${}^{-4}$
[57]	2019	EA	0.76078000	0.32302000	53.71852000	0.036380000	1.481180000	-	9.8602 × 10${}^{-4}$
[58]	2020	EA	0.76076000	0.32314000	53.71489000	0.036370000	1.481140000	-	9.8482 × 10${}^{-4}$
[59]	2018	EA	0.76077000	0.32320000	53.68360000	0.036300000	1.520800000	-	9.8600 × 10${}^{-5}$
[60]	2018	EA	0.76078700	0.31068300	52.88971000	0.036546000	1.475262000	-	7.7301 × 10${}^{-4}$
[61]	2018	EA	0.76077700	0.32262200	53.67840000	0.036381900	1.481060000	-	9.8602 × 10${}^{-4}$
[62]	2018	EA	0.76077553	0.32302083	53.71852771	0.036377090	1.481183600	-	9.8602 × 10${}^{-4}$
[63]	2018	EA	0.76069712	0.43244110	53.40180803	0.033410590	1.452456660	-	5.1382 × 10${}^{-4}$
[64]	2018	EA	0.76078000	0.32302000	53.71636000	0.036380000	1.481180000	-	9.8602 × 10${}^{-4}$
[65]	2020	NA	0.76870000	9.9414E-07	100.0000000	0.030966000	1.602000000	26	2.7756 × 10${}^{-17}$
[66]	2017	NA	0.76072000	0.31911000	54.19241000	0.036290000	1.479860000	-	8.1291 × 10${}^{-4}$
[67]	2018	EA	0.76077600	0.32302100	53.71852400	0.036377000	1.481718000	-	9.8602 × 10${}^{-4}$
[68]	2019	EA	0.76078000	0.33971000	54.43370000	0.036160000	1.486290000	-	9.9185 × 10${}^{-4}$

and

Table 2. Comparison of the parameters obtained in Case Study 2.

Literature	Algorithm	${\mathit{I}}_{\mathit{p}\mathit{h}}$ (A)	${\mathit{I}}_0$($\mathsf{\mu }$A)	${\mathit{R}}_{\mathit{s}\mathit{h}}$($\mathbf{\Omega }$)	${\mathit{R}}_{\mathit{s}}$($\mathbf{\Omega }$)	n	No. of Steps	RMSE	MAE
[69]	AA + NA	1.032377	2.517957	745.7122	1.239060	1.3173635	27	2.0465456 × 10${}^{-3}$	1.6925284 × 10${}^{-3}$
[43]	AA + NA	1.0323823	2.5129059	744.71302	1.3001512	1.3171591	6	2.0465347 × 10${}^{-3}$	1.6923215 × 10${}^{-3}$
[70]	AA + NA	1	2.3	830	1.3	1.3056	-	3.26 × 10${}^{-2}$	-
[6]	AA + NA	1.033285	1.82	850.7068	1.357607	1.2857	-	5.181 × 10${}^{-3}$	-
[15]	AA + NA	1.0323729	2.5129158	744.713061	1.2456174	1.3248753	4	2.046479 × 10${}^{-3}$	3.423077 × 10${}^{-4}$
[44]	EA	1.0263	9.5710	6842.2	0.0298	1.5255	-	3.819492 × 10${}^{-3}$	-
[65]	NA	1.0285	4.9614 × 10${}^{-6}$	1632.5	1.1638	-	-	2.6174×10${}^{-3}$	-

, respectively, as they appeared in their articles. Recently,a set of algorithms, such as differential algorithm (DE) [33], success-history based adaptive DE with linear population size reduction (LSHADE) [34], iLSHADE [35], LSHADE-EpSin [36], LSHADE-SPACMA [37], jSO [38], Gaussian EDA (GEDA) [39], covariance matrix adaptation evolution strategy (CMA-ES) [40], nuclear reaction optimisation (NRO) [41], etc., have been developed. According to [37], LSHADE is quite competitive compared to state-of-the-art EA.

3.3. Meta-Heuristic Approach

In order to attain the accurate modelling of the advance photovoltaic solar cell and module, it is pertinent to identify the optimal value of the corresponding parameters. Among the various methods to estimate the parameters of the photovoltaic cell model, meta-heuristic algorithms appear to be a sensible deterministic approach. In order to find the required parameters to enhance the performance of a solar cell, a range of meta-heuristics techniques are considered, such as particle swarm optimisation, the gravitational search algorithm, the flower pollination algorithm, wind-driven optimisation, whale optimisation, the artificial bee colony algorithm, differential evolution, genetic algorithms and cat swarm optimisation. Based on the literature of meta-heuristic optimisation techniques, there are of four types, namely, evolutionary algorithms wherein the genetic algorithm, differential evolution and shuffled complex evolution are involved; physics-based algorithms wherein wind-driven optimisation, the flower pollination algorithm and the gravitational search algorithm are involved; swarm-based algorithms wherein artificial bee colony and particle swarm optimisation is involved; and human-based algorithms wherein harmony search is involved. The documented reports on gradient-based methods [71,72] suggest that the meta-heuristic algorithms have an edge over parameters, such as accuracy, computational time and the avoidance of local minima trap. The optimisation function of the PV array using the meta-heuristic methods is adopted because it uses simple mathematical derivatives. In the evaluation of the state-of-the-art PV cell design, the non-linearity of the current–voltage (IV) curves is an evident consideration, which can be satisfied with meta-heuristic algorithms. The Deterministic methods, including analytical and iterative methods [73], do not support non-linear equations as the complexity increases with the increase in the number of unknown parameters in the proposed model. Moreover, in the iterative method, an erroneous choice during the trial-and-error approach may cause trapping in local optima. A meta-heuristic method [74] was adopted based on the way a group of hawks would approach a prey. Here, Harris hawks are considered for their remarkable behaviour of hunting cooperatively in packs. There exist various stages of chase where the hawks will adopt different strategies based on the dynamic location and credible escape route of the prey. The Harris hawks optimisation (HHO) algorithm emulates the chasing strategies of the Harris hawks using a dynamic pattern. However, it has been observed by Chen, H. (2020 [75]) that in order to obtain the global optimum solution, the HHO algorithm gets stuck in local minima.

In order to estimate the model parameter in the single-diode model (SDM), double-diode model (DDM) and three-diode model (TDM), in 2021, S. Maryam et al. [76] proposed an efficient optimisation algorithm, namely whippy Harris hawks optimisation (WHHO), which is an improved version of the HHO algorithm. Whippy Harris hawks optimisation (WHHO) has higher global search capability, convergence speed and robustness over the original algorithm.

The inability to derive accurate parameters under the non-linear condition [77] is the cause for the inaccurate modelling of the solar photovoltaic cell and module. Based on the documented literature [78], in general, there exist three types of algorithms under non-linear conditions, namely analytical approaches, numerical methods and evolutionary algorithms. The analytical approach is not preferred for non-linear conditions as the complexity increases as the parameters of the advanced model of the PV solar cell and module design increase. Under the various diode-based models [79], to achieve the characteristics of a PV cell, it has been observed that the unknown parameters increase with the increase in the numbers of diodes in the model. This results in a compromise between model accuracy and model simplicity [80]. Another advanced meta-heuristic algorithm [81] adopts the tunicate swarm optimisation (TSA) to accurately identify the model parameter of a PV cell and modules. This algorithm is based on the iterative process where the tunicate is randomly selected within the search area and locates the best tunicate position. The approach helps in improving the exploring capability and averts premature convergence. Analytical methods cannot solve non-linear equations as multiple unknowns result in unacceptable assumptions and produce erroneous results. To devise the unknown parameters in analytical approaches, it is necessary to manipulate model equations mathematically [82].

Due to existence of various characteristic types ranging from multi-modality, non-linearity and multi-variability in the voltage current curve of the photovoltaic cell, meta-heuristic algorithms are the finest approach. Sofiane et al. [83] came up with a hummingbird’s optimisation method termed as artificial hummingbird algorithm (AHA). AHA is a meta-heuristic bio-inspired optimisation algorithm in which populations of n humming birds are randomly initialised and placed on n food tables, as in Equation (30).

$$x_i=l+r(u-l)$$

(30)

where i = 1, 2, …, n;

• $x_i$ is the position of the $ith$ foodTable;
• l is the lower boundary;
• u is the upper boundary;
• r is the random vector in (0, 1).

The initialisation of the visit table of food sources is given by Equation (31).

$$V{t}_{i,j}=\left\{\begin{array}{c}0;ifi\ne j\hfill \\ null;ifi=j\hfill \end{array}\right.$$

(31)

There exist two conditions: if $i\ne j$, then $ith$ humming bird obtains the $jth$ food table, whereas if $i=j$, then the humming bird obtains its corresponding food table. A guided foraging behaviour is adopted to obtain direction control in d-dimensional space where the food source is given by Equation (32).

$$v_i(t+1)=x_{i,tar}\left(t\right)+\alpha D(x_i\left(t\right)-x_{i,tar}\left(t\right));\alpha \to N(0,1)$$

(32)

where $\alpha$ is the guided factor to reach the desired location, and $x_{i,tar}\left(t\right)$ is the target location of the food table where the $ith$ humming bird is supposed to visit. The cluster probability of $\alpha$ ranging from 0 to 1 is represented by $\alpha \to N(0,1)$.

The new position of the $ith$ food source is given by Equation (33).

$$x_i(t+1)=\left\{\begin{array}{c}{x}_i\left(t\right);f\left(x_i\left(t\right)\right)\le f\left(v_i(t+1)\right)\hfill \\ v_i(t+1);f\left(x_i\left(t\right)\right)>f\left(v_i(t+1)\right)\hfill \end{array}\right.$$

(33)

In territorial foraging, only specific food sources will be available for the humming birds in the local search corresponding to Equation (34).

$$v(t+1)=x_i\left(t\right)+bD\left(x_i\left(t\right)\right);b\to N(0,1)$$

(34)

where $‘b’$ represents the territorial factor corresponding to the normal distribution N (0,1).

The migration foraging of a humming bird from the source corresponding to the worst nectar-refilling rate to a randomly produced new one can be expressed by Equation (35).

$$x_{wor}(t+1)=L+r(U-L)$$

(35)

where $x_{wor}$ represents the food source corresponding to the worst nectar-refilling rate.

Various meta-heuristic techniques, including CPMPSO [47], EHHO [84], EJADE [85], ELBA [86], NMSOLMFO [87], GBO [88], RUN [89], GSK [90], RLGBO [91], DSCSE [92], IMPA [93], CCNMHHO [94], SEDE [95], WLCSODGM [96], SGDE [97], EABOA [98], MTLBO [99] and WHHO [76], were compared in [83] based on squared statistical error (SSE), standard deviation (StD) and Root mean square error (RMSE). Based on the following performance indicator, the proposed AHA stands out as an effective parameter-extraction algorithm. The AHA results in a satisfactory RMSE under limited iterations compared to other meta-heuristic techniques, with close matches regarding the experimental dataset.

Another improved algorithm based on flower pollination by [100] has been proposed for determining the unknown parameters of PV cells and module models. Yang [101] proposed the flower pollination algorithm (FPA) wherein the concept of cross-pollination and self-pollination is adopted for pollen transfer and abiotic and biotic pollination for carriers of pollens. In the conventional PFA, the probability factor was the only controlling parameter to determine the local or global process of pollination. Under the improved variant, double exponential based dynamic switch probability is adopted to maintain equilibrium between the local and global searches, and a dynamic step size function is used to avoid premature convergence and local optima stagnation by tuning the search speed.

To study the efficacy of the improved algorithm, an objective function has to be deduced as the root mean square error (RMSE) in Equation (36).

$$RMSE\left(X\right)=\sqrt{\frac{1}{M}{\displaystyle \sum _{d=1}^M}\left(errorfunction\right)}$$

(36)

where ‘X’ is the unknown parameter of the model, and ‘M’ is the number of measured I–V data. The error function of $V_L$ and $I_L$ denote the measured I–V data acquired from the PV cell.

In order to attain the global pollination process, the pollinators should fly with biotic pollination and cross-pollination mechanisms obeying levy flight. Moreover, the rule based on the probability of reproduction on the percentage of correspondence between the involved flowers is stated as Equation (37).

$${X_i}^{t+1}={X_i}^t+L({X_i}^t-g^{*})$$

(37)

where ${X_i}^t$ represents the $X_i$ solution at iteration number t, L mimics the Levy flight distribution feature, $g^{*}$ represents the best solution from the present population and S represents the step size of a Levy flight. Using the Mantegna algorithm, the value of L is identified as Equation (38).

$$L=\frac{\lambda \Gamma \left(\lambda \right)sin(\Pi \lambda /2)}{\Pi }\frac{1}{S^{1+\lambda }};(S>>S_0>0)$$

(38)

To attain the local pollination process, the abiotic and self-pollination mechanism is obeyed. Here, the rule based on the probability of reproduction on the percentage of correspondence between the involved flowers is stated as Equation (39).

$${X_i}^{t+1}={X_i}^t+ϵ({X_j}^t-{X_k}^t)$$

(39)

where ${X_j}^t$ and ${X_k}^t$ are two random chosen solutions from a given set of solutions, and $ϵ$ is a random number varies from 0 to 1.

The conventional FPA does not support the searching agents to gather around the optimal value as the value of probability ‘p’ will decide the value of search agents towards the local and global pollination equation. The higher values of ‘p’, i.e., towards 1, will update the search agents using the global pollination equation, which in turn is affected by the Levy fight distribution mechanism. Furthermore, the search agents will be updated by local pollination if the value of ‘p’ is towards 0, which traps the solution in the local optimum. This detrimental effect is mitigated by creating a balance between the local and global pollination models with the advancement of the convergence rate using the dynamic step function search capability.

In general, the estimation of the model parameter for a photovoltaic cells and modules do not embrace degradation due to environmental effects. The actual working conditions should include the changes in the PV cell’s characteristics due to ageing, faults, maintenance and degradation [102]. With the support of string currents, voltages, irradiance and temperature data, the model parameters are estimated using an adaptive module string model. In [103], it is stated that this method supports the estimation of complex module string parameters. In the estimation algorithm, the four standard test condition parameters are reset to the initial values based on the module datasheet. The previous value error will be set to infinity before performing the following steps for each cell and subsequently the substring: minimise the error by searching the STC parameters and set the minimum value as $E_{min}$. The algorithm is sufficed if the condition $E_{min}<ϵ$ holds, where $ϵ$ is 0.1. Under this condition, if $max((1-\delta )E_{min,err},E_{min,err}-\Delta )\le E_{min}$ is not sufficient to achieve minimum value of error, it is the end of the algorithm. The present parameters are considered as the estimation result with $E_{min,err}=E_{min}$.

There are a few methods, such as Kohno’s [104], Harrou’s [105] and Mansouri’s [106] methods, to detect the presence of shading conditions. Ref. [103] has proposed a method that will exhibit higher detecting accuracy for areas covered by cells larger than four in number using the true-positive rate (TPR) and the false-positive rate (FPR) [105] conditions. However, Mansouri’s method is preferred for a module to estimate partial shade detection accuracy.

The existence of non-linear voltage–current characteristics is evidence for considering the organic solar photovoltaic cell modelling. This would lead to the estimation of unknown parameters under a dynamic irradiation profile pattern. Ref. [107] proposed an adaptive wind-driven optimisation (WDO) algorithm for a three-diode electrical equivalent model of an organic solar photovoltaic cell that could emulate the kink effect [108].

In order to test the quality of the solution in the WDO algorithm, the root mean square error via curve fitting is considered. As per the initial procedure of testing, the absolute error (actual data – simulated data) of individual datum is formulated as the function of individual absolute error using Equation (40).

$$f_i(V_m,I_m,x)=abs(I_{act}-I_{sim})$$

(40)

where $I_{act}$ is experimental data, and $I_{sim}$ is estimated values from simulation. x is the function of all nine dimensions (two parallel resistance + three diode reverse saturation currents + three diode ideality factors + initial photon current) of the three-diode PV model. The sum of the squared error is written as Equation (41).

$$Sumofsquarederror=\sum _{n=1}^N{\left(Individualabsoluteerror\right)}^2$$

(41)

where N is the individual data in the I–V curve. Finally, the root mean square error (RMSE) is given by Equation (42).

$$RMSE=\sqrt{\frac{1}{M}\left(Sumofsquarederror\right)}$$

(42)

The WDO algorithm has four velocity update processes that prevent convergence to local optima. Hence, the parameter estimation process is more precise. The WDO algorithm is based on the horizontal imbalance of air pressure in our atmosphere due to various factors, such as topography, temperature variation and suspended air particles. Under the Lagrangian function, the mathematical depiction of force acting on the air parcel is represented as Equation (43).

$$\rho \overrightarrow{\alpha }=\sum {\overrightarrow{F}}_t$$

(43)

where ${\overrightarrow{F}}_t$ is the algebraic sum of applied force (which includes pressure gradient force, frictional force, gravitational force and Coriolis force), $\overrightarrow{\alpha }$ is the acceleration in air parcel and $\rho$ is the air density. It is difficult to devise an accurate model of an organic solar photovoltaic cell as the manufacturer uses various active materials and inter-layers that are reactive to changes in the environment. The adaptive wind-driven optimisation algorithm is a meta-heuristic optimisation technique that utilises four various velocity update processes for a single particle [109] for fast computation. This provides control tuning of the model parameter to obtain the global optimal region using adaptive velocity generation strategy.

4. Performance Analysis of the Reported Algorithms

Although analytical algorithms are very much powerful for estimating any unknown parameter, it has limited use for non-linear equations. Again, numerical methods have the potential to solve non-linear equations, but their precision is less, and they suffer from convergence failure. Compared to the analytical and numerical algorithms, evolutionary algorithms are observed to be more precise. Keeping that in view, a number of evolutionary algorithms are developed and reported in the existing literature. However, from our careful observation, it is noticed that the reported algorithms show poor performance in some of the vital points, despite showing better performance in some other criteria. Less than ±5% error is observed in [110], where the genetic algorithm (GA) is used. The artificial immune system (AIS) algorithm is used in [5] to estimate the parameter of the DDM PV module and observed that AIS performs better compared to the GA and particle swarm optimisation (PSO) techniques in terms of convergence speed. In line with the short convergence time, the pattern search (PS) algorithm is also very useful, which is reported in [27,61]. The bacterial foraging algorithm (BFA) [5] is useful for high precision, faster convergence speed and reliable output compared to GA and AIS. However, the computation of BFA is a bit challenging. DDM PV module parameter estimation is carried out in [111] using the differential evaluation (DE) technique, which is also used for thin-film technologies. However, finding the control parameter in DE is a difficult job. The simulated annealing (SA) [65,112] technique gives better accuracy compared to other optimisation algorithms, but accommodating the temperature effect is tough in SA. To overcome the problem of premature outcome, the fireworks algorithm (FA) shows good performance, as reported in [113]. However, the computational time is high in the case of FA. For less convergence time, the flower pollination algorithm (FPA) [114] is a useful algorithm, but accurately determining the fitness function is a difficult job in FPA. The hybrid flower pollination algorithm (HFPA) [115,116] shows good performance in convergence speed and reliability, but it has less precision. The harmony search (HS) [117,118] algorithm shows better accuracy compared to the PS and SA techniques. HS has better convergence speed compared to FA. The artificial bee colony (ABC) [64,119,120] algorithm shows better accuracy and convergence speed compared to HS, GA, BFA and PSO. However, it shows convergence failure in the case of repeated progression. On the other hand, for fast computation, PSO [121,122,123] shows better performance compared to the other EAs. However, the selection of an initial value of the parameters in PSO is a difficult job.

Effect of Temperature and Irradiance

By applying the estimated values of the unknown parameters, PV modelling is conducted in [15]. Now, to check the performance of the newly modelled PV module, the same is tested at different temperatures. In much of the literature, it is observed that, although the reported algorithms work well within a certain ambient temperature range, the performance deteriorates in some other range of ambient temperatures. In line with this, applying a set of best-reported algorithm simulations are performed for a PV module (datasheet is given in

Table 3. Datasheet values of a 315 Wp PV module.

Parameter	Symbol	Value
Maximum power	$P_{MP}$	315 W
Short-circuit current	$I_{SC}$	8.95 A
Open-circuit voltage	$V_{OC}$	45.6 V
Current at Maximum power	$I_{MP}$	8.45 A
Voltage at Maximum power	$V_{MP}$	37.3 V
Co-efficient of current	$K_I$	0.05%/$°$C
Co-efficient of voltage	$K_V$	−0.35%/$°$C
Co-efficient of power	$K_P$	−0.40%/$°$C
Nominal operating cell temperature	NOCT	45 ± 2 $°$C

). The simulated voltage–power (V–P) characteristics of the reported models are shown in Figure 5 at temperatures of 25 $°$C and 45 $°$C. For the simulation, irradiance is considered as 888 W/m${}^2$ at air-mass 1.5 gobal (AM1.5g). From Figure 5, it is observed that, although the performance of [124] (mentioned as EA-1 in Figure 5) is near the tolerance limit at 25 $°$C ambient temperature, it deteriorates at 45 $°$C. On the other hand, the performance of [58] (mentioned as EA-2 in Figure 5) is good at 45 $°$C, but it degrades at 25 $°$C. Similarly, the performance of the algorithms reported by Toledo [43], Laudani [125], Changmai [15] and Cardenas [69] are shown in Figure 5, from which it can be stated that the performance pattern is not linear with the change in temperature. Based on our careful observation, it is noticed that the nominal operating cell temperature (NOCT) is a vital component that affects the performance of a PV module. The effect of NOCT in the PV cell’s temperature is given in Equation (44). As reported by [15], there may be around a 10% performance error if the NOCT is not considered for comparison purposes. The specifications of 315 Wp PV modules is presented in Table 3.

$$T_{cell}=T_{amb}+G·\frac{\mathrm{NOCT}-{20}^{\circ }}{800{\mathrm{W}/\mathrm{m}}^2}$$

(44)

When the irradiance increases, the current generation in the PV module also increases. Moreover, from the voltage and power relationship of a PV module at different irradiance levels, it is seen that as irradiance increases, the module is able to generate more power represented by higher peaks on the V–P curve. The effect of temperature and irradiance (G) in the PV cell are shown in Figure 6a,b, respectively.

Temperature is a vital factor in deciding the efficiency of a photovoltaic cell. Moreover, the rate of degradation is prominent on the PV panel under higher temperatures. Various cooling techniques are adopted to improve the efficiency of the solar panel. The force air stream technique [126,127] and the adoption of phase change materials (PCM) [128] are some of the latest methods known to cool off the PV module. With the advancement of composite PCM material embedded with nanoparticles, it has been observed that the efficiency of the photovoltaic module increases by about 13% [129]. Some of the recent active cooling systems adopted are based on the surface water cooling and aluminium heat sink [130]; M.S chips and thermal grease [131,132]; saturated zeolite with water [133]; and the use of palm wax [134] to regulate the temperature of the PV panel to the desired value. An approach for cooling the PV module on both the front and rear surface using a cotton wick mesh has been developed [135,136,137]. Based on the capillary action, water is allowed to spread along the cotton wick mesh throughout the rear surface of the panel. A perforated aluminium sheet is designed along the back of the PV panel to trap the evading vapour. An overall improvement of 10.89% in voltage is observed in the PV module considering the mentioned arrangement. This results in an enhancement of 11.9% in the efficiency of the suggested cooled panel. Agyekum et al. [138] assessed the viability of combining aluminium fins and paraffin wax to cool a PV module. A combination of both active and passive cooling was adopted [138] to cool a PV system wherein an ultrasonic humidifier was considered for producing a humid environment to cool off along with the aluminium fins.

5. Conclusions

Parameter estimation of a solar photovoltaic system is a vital in PV modelling. A number of algorithms have been reported to estimate the unknown parameters of the PV cell/module across the globe in the last decade. Based on the observation of all the reported algorithms, a few recent best-reported algorithms were selected for this review in terms of accuracy, computational complexity and convergence issues. To strengthen the review, all types of algorithms, including analytical, numerical and evolutionary algorithms, were considered, and the outcome was reported in a tabular manner. Simulations were performed for a few of the best-reported models, and the performance was checked at various temperatures. In this study, it is observed that the combination of both analytical and numerical techniques (combined technique) are less time-consuming compared to other techniques that are used for parameter estimation of solar PV systems. The combined technique has the highest accuracy compared to the other existing literature. In analytical algorithms, the number of non-linear exponential terms can be reduced by 4 times compared to the existing literature. Hence, the combined technique is computationally efficient. The shunt resistance ($R_{sh}$) should be considered as one of the iteration parameters, which makes the approach more realistic. The application of the NOCT value in the PV parameter estimation strengthens the accuracy in varying temperatures and irradiance conditions. Based on our careful observations, it is noticed that, on average, 10% performance degradation is present in the MPP obtained at $T_{cell}$ compared to $T_{amb}$.