Parameter Estimation Techniques for Photovoltaic System Modeling

Abstract

In improving PV system performance, the parameters associated with electrical photovoltaic equivalent models play a pivotal role. However, due to the increased mathematical complexities and non-linear traits of PV cells, the precise prediction of these parameters is a challenging task. To estimate the parameters associated with PV models, a reliable, robust, and accurate optimization technique is needed. This paper introduces a new algorithm, Rat Swarm Optimizer (RSO), for obtaining the optimum PV cell and module parameters. The proposed method maintains an adequate balance between the exploration and exploitation phases to overcome premature particle issues. The results obtained using RSO are compared with those of other algorithms, i.e., Particle Swarm Optimization (PSO), Ant Lion Optimizer (ALO), Salp Swarm Algorithm (SSA), Harris Hawks Optimization (HHO), and Grasshopper Optimization (GOA), in this work. The modified one-diode model (MODM) and modified two-diode model (MTDM) are used to analyze the parameters of the mono-crystalline PV cell using the suggested RSO. The obtained findings imply that the parameters estimated by the suggested RSO are more accurate than those calculated by the other algorithms taken into consideration in the paper. The statistical results are compared, and it is clear that RSO is a very accurate, fast, and dependable approach for the parameter estimation of PV cells.

1. Introduction

Due to massive demographic growth and economic development, the demand for electrical energy is rapidly growing. Electricity is primarily generated by burning coal in counties like India. It is available in limited amounts in nature, and even other fossil fuels are not capable of meeting the demand; therefore, there is a need to shift the focus toward renewable energy sources [1,2,3]. Moreover, fossil fuels are not green, i.e., they emit greenhouse gases (GHG) into the air and result in pollution. Solar energy is available in abundance and seems to be the most feasible and optimal solution, as it is omnipresent and has the potential to address the most critical energy crisis issues [4,5,6]. Many models of solar photovoltaic (PV) modules [7,8] have been reported in the literature like one-diode [9], two-diode [10], and three-diode [11] models.

The analysis of a substantial mechanism requires knowledge of superior characteristics, measurements of intensity, and the optimization of a one-diode solar cell model [12,13]. Various methods have been suggested for optimizing the model’s characteristics under different constraints [14,15]. It was observed in previous works that the implementation of a PV system involves a one-diode solar PV cell model [16]. For the evaluation and study of electrical characteristics, the one-diode model is commonly used [17,18].

Photovoltaic (PV) models are categorized based on the number of diodes they incorporate, namely single-, double-, three-, and four-diode models. The inclusion of multiple diodes in a PV system enhances its ability to provide more accurate and realistic output values. As a result, the design optimization of three- and four-diode models assumes greater significance than single-diode and double-diode studies do [19]. These advanced models offer improved accuracy and are better suited for complex real-world applications.

Accurate parameter estimation is very important for many reasons, i.e., it offers improved efficiency, an enhanced performance, and reliable operation. Accurate parameter estimation enables engineers and researchers to fine-tune the PV system’s configuration, ensuring that it operates at its peak efficiency. With precise parameters, the system can more effectively match the incoming solar irradiance, leading to higher energy yields and reduced losses. This, in turn, maximizes the utilization of the available solar resource and enhances the economic viability of PV installations. The performance of a PV system is heavily influenced by various factors, such as temperature, shading, module mismatch, and aging effects [2]. Accurate parameter estimation allows a better understanding of these influencing factors, leading to the improved modeling and prediction of system behavior under different environmental conditions. This knowledge empowers operators to optimize the system’s performance and adapt to varying weather patterns, ultimately resulting in a more reliable energy output. PV systems often experience degradation over time due to environmental stress and material deterioration. Accurate parameter estimation facilitates better monitoring and the early detection of system faults or anomalies [20]. By promptly identifying and addressing such issues, the system downtime can be minimized, and the overall reliability and availability of the PV system can be increased.

For each PV model, the number of parameters to be optimized remains constant. However, as the number of diodes increases, so does the complexity of the model, resulting in a larger set of parameters to be optimized. Specifically, the optimization process for the single-diode model typically involves optimizing five parameters. In contrast, the three-diode model requires the optimization of seven parameters, the four-diode model necessitates nine parameters, and the complexity reaches its peak with the optimization of eleven parameters for the five-diode model.

The optimization of these parameters plays a crucial role in maximizing the efficiency and performance of PV systems. By fine-tuning the values of these parameters, such as the diode currents, resistances, and photocurrents, it becomes possible to achieve the desired performance characteristics and accurately model the behavior of the PV system. The optimization process ensures that the PV system operates optimally under various external conditions, including varying light intensities, load characteristics, and temperature fluctuations [21].

For the electrical parameter estimation of the PV module/cell, most researchers have favored the SDM or TDM in the literature, and the Triple-Diode Model (TDM) is not normally preferred because it has a complicated design. The researchers have attempted to estimate the unknown parameters of PV modules using iterative, meta-heuristic, and analytical methods alone or in comparison with iterative and analytical methods. The nominal operating cell temperature (NOCT) and standard operating temperature (STC) datasheet values are used in some of the analytical methods [22]. However, empirical approaches use rough solutions due to the presumption of Rsh and Rse constant values. Certain methods, for example, the bond graph and Lambert function, neglect Rsh and Rse values. For parameter estimation problems, there are a few iterative approaches, including the Gauss–Seidel, Newton–Raphson, and least squares methods. However, analytical and iterative methods are not preferable due to their limitations, such as them missing a few parameters, assuming constant values, and having an inaccurate initial value range. Meta-heuristic algorithms are now used to approximate both modified one-diode model (MODM) and modified two-diode model (MTDM) parameters by decreasing the objective function and errors due to their high accuracy, fast convergence rate, and cost effectiveness. There are many optimization strategies that are mainly divided into groups based on the type of inspiration; the most common ones are swarm-based algorithms. Swarm-based optimizers are meta-heuristic algorithms inspired by the collective behavior of social organisms such as bees or ants. They work by simulating the movement and interaction of a group (swarm) of solution candidates to find optimal solutions in complex search spaces. The most commonly known swarm-based algorithms are the Particle Swarm Optimizer (PSO), Ant Lion Optimizer (ALO), Harris Hawk Optimizer (HHO), etc. They are effective at handling high-dimensional, non-linear problems and can often find global optima. The drawback of swarm-based optimizers is that they do not have a good memory. The root-mean-square error (RMSE) is known to be an objective function in most of the literature.

The main contributions of this manuscript are as follows:

• This paper introduces a new optimization algorithm called the Rat Swarm Optimizer (RSO) for accurately predicting the parameters associated with electrical photovoltaic (PV) equivalent models.
• RSO is designed to strike a balance between the exploration and exploitation phases, addressing premature particle issues that commonly arise in optimization tasks.
• The performance of RSO is compared with those of other algorithms, such as Particle Swarm Optimization (PSO), Ant Lion Optimizer (ALO), Salp Swarm Algorithm (SSA), Harris Hawks Optimization (HHO), and Grasshopper Optimization (GOA).
• The modified one-diode model (MODM) and modified two-diode model (MTDM) are utilized to analyze the parameters of mono-crystalline PV cells using RSO.
• The results indicate that the suggested RSO provides more accurate and reliable parameter estimations for PV cells than the other algorithms tested in this study do, making it a fast and dependable approach for PV system performance improvement.

2. Materials and Methods

2.1. Diode Modeling

One-Diode Model (ODM):

A one-diode model is employed to produce the I-V characteristics of the solar cell, as illustrated in Figure 1. The PV effect, which has to do with a possible differential in exposure to radiation other than visible light at the P-N junction, serves as the foundation for the operational theory of photovoltaic cells. When a photovoltaic cell is exposed to light, semiconductor materials absorb photons; as a result, charged carriers are produced. The solar cell is powered by sunlight, as photons are absorbed by the semiconductor material, and the generation of charge carriers occurs.

The potential difference between the external circuit and current implies that the carrier is mostly separated by the P-N barrier in the confined electric field and converges on the electrodes. The charged carriers produced are recognizable in the form of an electrical current, notably, a photovoltaic current. If the photovoltaic current has no effect, light parameters are not defined, so that the photovoltaic cell diode behaves like an ordinary diode. In general, the diode drift current is defined as the Shockley diode equation, as shown in Equation (1).

$$I=I_{ph}-I_d\left(e^{\frac{V+{IR}_{se}}{N{V}_T}}-1\right)-\frac{V+{IR}_{se}}{R_{sh}}$$

(1)

Modified One-Diode Model (MODM):

The one-diode model (ODM) is improved to become the modified one-diode model (MODM). In the MODM, supplementary resistance is included in a series with the basic ODM, which reflects losses in the quasi-neutral area. The equivalent circuit of the MODM is represented in Figure 2.

The mathematical equation of the MODM is described below in Equation (2):

$$I=I_{ph}-I_d\left(e^{\frac{V+{IR}_{se-I_d{R}_s}}{N{V}_T}}-1\right)-\frac{V+{IR}_{se}}{R_{sh}}$$

(2)

Two-Diode Model (TDM):

A further examination is conducted in order to identify more sophisticated models, despite the one-diode model’s uncertain precision. By taking into account that two diodes are linked in parallel to the current source, as shown in Figure 3, a two-diode model is used to address this issue. The better version of the one-diode model, which incorporates the impact of recombination by including a second parallel diode, is modeled after the two-diode model. There are two more parameters in this model that are related to unknown diode quality variables. This resulted in more equations, which make the computations more difficult. So, this model shows greater precision at a low insolation level. The modeling of the two-diode model is represented in Equation (3).

$$I=I_{ph}-I_d\left(e^{\frac{V+{IR}_{se}}{N{V}_T}}-1\right)-I_{d1}\left(e^{\frac{V+{IR}_{se}}{N{V}_T}}-1\right)-\frac{V+{IR}_{se}}{R_{sh}}$$

(3)

Modified Two-Diode Model:

The modified two-diode model (MTDM) is the modified version of the two-diode model (TDM). An MTDM is formed by adding additional resistance to a basic TDM, which indicates that grain boundaries have greater resistance than crystallite boundaries do. The equivalent circuit of MTDM is represented in Figure 4.

The mathematical equation of MTDM is described below in Equation (4):

$$I=I_{ph}-I_d\left(e^{\frac{V+{IR}_{se}}{N{V}_T}}-1\right)-I_{d1}\left(e^{\frac{V+{IR}_{se}-I_{d1}{R}_s}{N{V}_T}}-1\right)-\frac{V+{IR}_{se}}{R_{sh}}$$

(4)

In Equation (1), I_d represents the saturation current of the diodes, V is known as the diode voltage, V_T is termed as the equivalent of thermal voltage, N is the number of cells in series, I is the output current, I_ph is the photo-current, the series resistance is denoted as R_se, and Shunt resistance is denoted as R_sh.

2.2. Problem Formulation

The objective function is used to determine the parameters of the PV cell. Based on the following error for the MSDM and MTDM, the optimization problem can be formulated mathematically.

$$SSE={\sum }_{i=1}^{n}f{\left(V,I,M\right)}^2$$

(5)

$$AE={\sum }_{i=1}^{n}f\left(V,I,M\right)$$

(6)

$$MAE=\frac{1}{n}{\sum }_{i=1}^{n}f\left(V,I,M\right)$$

(7)

$$MSE=\frac{1}{n}{\sum }_{i=1}^{n}f{\left(V,I,M\right)}^2$$

(8)

$$RMSE=\sqrt{\frac{1}{n}{\sum }_{i=1}^{n}f{\left(V,I,M\right)}^2}$$

(9)

where SSE represents the Sum of Square Error, AE represents the Absolute Error, MAE represents the Mean Absolute Error, MSE represents the Mean Squared Error, and RMSE represents the root-mean-square Error. The solution vector is represented by M, the number of experiments is represented by n, and the voltage and current are represented by V and I. In addition to the statistical analysis presented in Equations (5)–(9), we also performed analysis based on the amount of error, the Absolute Error, the average error, the mean square error, and the root-mean-square error.

2.3. Proposed Algorithm (RSO)

The average rat is a medium-sized rodent with a long tail, although rats can vary in size and weight. In terms of species, there are two types of rat: black rats and brown rats. Male and female rats of the rat family are called bucks and does, respectively. Rats are socially intelligent animals. In addition to grooming each other, they tumble, hop, box, and chase each other. A rat community consists of both males and females, and they are territorial. Rats can be extremely violent, which can lead to the death of some animals. The main inspiration for this research comes from predators chasing and competing with prey. Mathematical models have been developed to analyze the chasing and battle actions of rats [23,24].

Mathematical Modeling:

Step 1: Chasing: Rats are usually social animals who engage in a social, agonistic activity to pursue prey in groups. To mathematically characterize this action, it is safe to conclude that the best search agent is aware of the position of the prey. The other search agents will revise their positions in comparison to the best search agent obtained thus far. With this in mind, the following Equation (10) is introduced.

$$\overrightarrow{Q}=B.\overrightarrow{Q_j}\left(z\right)+D.\left(\overrightarrow{Q_s}\left(z\right)-\overrightarrow{Q_j}\left(z\right)\right)$$

(10)

In this equation, the rat position is represented by $\overrightarrow{Q_j}\left(z\right)$, and the best optimal solution is represented by $\overrightarrow{Q_s}\left(z\right)$. However, the parameters B and D are shown in Equations (11) and (12).

$$B =S-z\times \left(\frac{S}{{Max}_{iter}}\right) \mathrm{where}, \mathrm{z}=0, 1, 2, \dots , {Max}_{iter}.$$

(11)

D = 2. Rand ()

(12)

where the random number is represented by S and D, and the range of the random number is between [1, 5] and [0, 2]. For better exploration and exploitation over the course of several iterations, the parameters B and D are used.

Step 2: Fighting: The following Equation (13) has been developed to mathematically describe the how rats battle with prey.

$$\overrightarrow{Q_j}\left(z+1\right)=\left|\overrightarrow{Q_s}\left(z\right)-\overrightarrow{Q}\right|$$

(13)

where the next position of the rat is updated as $\overrightarrow{Q_j}\left(z+1\right)$. It preserves the optimal solution and changes other search agents’ locations in relation to the optimal search agent. As a result, the modified values of parameters B and D ensure exploration and exploitation. With the fewest operators, the suggested rat algorithm gives the desired solution. The RSO algorithm’s flow chart and pseudo code are shown in Figure 5 and Algorithm 1, respectively.

Algorithm 1: RSO Algorithm

Initialize the population of rats

Initialize the parameters B, S and D

The fitness value of the search agent is calculated

While (Z < Max iteration) do

for each search agent do

the position if the current search agent is updated by Equation (13)

end for

Update the parameters

Adjust the parameter if required

Calculate the fitness of each search agent

Update the fitness function accordingly

z ← z + 1

end while

return Qj

Stop

3. Results

The parameter estimation datasheet for a monocrystalline PV cell is shown in

Table 1. Parameter estimation datasheet.

Company	H&T GmbH
Model	TS265D60
Cell Type	Mono-Crystalline
Vm [V]	30.90
Im [A]	8.58
Voc [V]	38.10
Isc [A]	9.19
Ns [Cells]	60
T [°C]	25

. This section uses modified one-diode and modified two-diode models to estimate parameters. Using the compared algorithms, the parameters are determined, followed by the numerous errors listed above. The RSO algorithm and all the algorithms are set with their standard parameters except the no. of search agents and maximum no. of iterations, i.e., 30 and 1000, respectively. Compared with the other algorithms analyzed in this manuscript, such as PSO [25,26,27], ALO [28], SSA [29], HHO [30], and GOA [31], the evolved RSO algorithm leads to better outcomes.

Table 2. Algorithm parameters.

Algorithm	Parameters	Values
RSO	Search agents	30
	Maximum iteration	1000
	S	[1, 5]
	D	[0, 2]
PSO	Search agents	30
	Maximum iteration	1000
	Vmax	4
	Vmin	−4
	C1 & C2	0.5
	r1 & r1	0.5
	wg	0.25
ALO	Search agents	30
	Maximum iteration	1000
SSA	Search agents	30
	Maximum iteration	1000
	C1	2*exp(−4/Max_iteration)2
	C2	rand
	C3	rand
HHO	Search agents	30
	Maximum iteration	1000
	E1	2*(1 − (t/T))
	E0	2*rand − 1 (range lies between −1 to 1)
GOA	Search agents	30
	Maximum iteration	1000
	CMax	1
	CMin	0.00004

represents the algorithm parameters. The Friedman ranking is clearly superior to that of the compared algorithm based on the results of the test. RSO is clearly a better algorithm than the others based on this test. In this section, two cases are discussed.

Case 1: MODM

According to

Table 3. Parameter estimation and computation time of MODM under STC.

Parameter/Algorithms	ALO	PSO	SSA	GOA	HHO	RSO
Ipv	9.2483	9.3851	9.2127	9.2029	9.2430	9.2016
Alpha1	1.4523	1.3438	1.4371	1.5128	1.4839	1.5261
Rse	0.1061	0.0185	0.0240	0.0357	0.0901	0.0283
Rsh	260.250	110.949	283.493	324.592	294.329	305.099
Io1	3.59 × 10−7	2.49 × 10−7	4.04 × 10−7	6.48 × 10−7	5.01 × 10−7	7.75 × 10−7
Rs	0.1021	0.0152	0.0140	0.0241	0.0758	0.0385
Computation time	1.1038	1.1864	1.0597	1.1520	1.5247	1.0276

, different optimization methods were used to estimate the parameters. As shown in

Table 4. Statistical results of MODM of PV cell.

Algorithms	Errors	SSE	AE	MAE	MSE	RMSE
ALO	Minimum	1.02 × 10−3	3.27 × 10−2	1.67 × 10−3	5.35 × 10−5	7.31 × 10−3
	Average	5.24 × 10−3	7.21 × 10−2	3.65 × 10−3	2.65 × 10−4	1.61 × 10−2
	Maximum	9.07 × 10−3	9.53 × 10−2	4.74 × 10−3	4.51 × 10−4	2.13 × 10−2
	Mean	5.24 × 10−3	7.21 × 10−2	3.65 × 10−3	2.65 × 10−4	1.61 × 10−2
	S.D	2.33 × 10−3	4.89 × 10−2	2.48 × 10−3	1.19 × 10−4	1.09 × 10−2
PSO	Minimum	5.37 × 10−2	2.32 × 10−1	1.15 × 10−2	2.67 × 10−3	5.18 × 10−2
	Average	2.84 × 10−1	5.33 × 10−1	2.66 × 10−2	1.42 × 10−2	1.19 × 10−1
	Maximum	6.60 × 10−1	8.13 × 10−1	4.06 × 10−2	3.30 × 10−2	1.82 × 10−1
	Mean	2.84 × 10−1	5.33 × 10−1	2.66 × 10−2	1.42 × 10−2	1.19 × 10−1
	S.D	1.86 × 10−1	4.32 × 10−1	2.16 × 10−2	9.34 × 10−3	9.66 × 10−1
SSA	Minimum	1.05 × 10−2	1.03 × 10−1	5.11 × 10−2	5.38 × 10−3	2.30 × 10−2
	Average	3.23 × 10−2	1.79 × 10−1	8.95 × 10−2	1.67 × 10−2	4.02 × 10−2
	Maximum	7.34 × 10−2	2.71 × 10−1	1.35 × 10−1	3.67 × 10−2	6.06 × 10−2
	Mean	3.23 × 10−2	1.79 × 10−1	8.95 × 10−1	1.67 × 10−2	4.02 × 10−2
	S.D	1.67 × 10−2	1.29 × 10−1	6.43 × 10−2	8.64 × 10−3	2.89 × 10−2
GOA	Minimum	2.82 × 10−2	5.29 × 10−2	2.64 × 10−2	1.44 × 10−4	1.18 × 10−2
	Average	7.08 × 10−2	8.39 × 10−2	4.29 × 10−2	3.52 × 10−4	1.87 × 10−2
	Maximum	9.91 × 10−2	9.94 × 10−2	4.93 × 10−2	4.97 × 10−4	2.22 × 10−2
	Mean	7.08 × 10−2	8.39 × 10−2	4.29 × 10−2	3.52 × 10−4	1.87 × 10−2
	S.D	2.32 × 10−2	4.83 × 10−2	2.47 × 10−2	1.18 × 10−4	1.08 × 10−2
HHO	Minimum	1.05 × 10−3	3.26 × 10−2	1.62 × 10−3	5.35 × 10−5	7.38 × 10−3
	Average	4.03 × 10−3	6.37 × 10−2	3.19 × 10−3	2.07 × 10−4	1.42 × 10−2
	Maximum	8.41 × 10−3	9.18 × 10−2	4.50 × 10−3	4.21 × 10−4	2.05 × 10−2
	Mean	4.03 × 10−3	6.37 × 10−2	3.19 × 10−3	2.07 × 10−4	1.42 × 10−2
	S.D	2.27 × 10−3	4.73 × 10−2	2.38 × 10−3	1.16 × 10−4	1.05 × 10−2
RSO	Minimum	1.11 × 10−4	1.09 × 10−2	5.47 × 10−4	5.95 × 10−6	2.49 × 10−3
	Average	4.35 × 10−4	2.07 × 10−2	1.05 × 10−3	2.15 × 10−5	4.64 × 10−3
	Maximum	9.91 × 10−4	3.14 × 10−2	1.59 × 10−3	4.95 × 10−5	7.02 × 10−3
	Mean	4.35 × 10−4	2.07 × 10−2	1.05 × 10−3	2.15 × 10−5	4.64 × 10−3
	S.D	2.62 × 10−4	1.62 × 10−2	8.04 × 10−4	1.32 × 10−5	3.60 × 10−3

, the MODM mono-crystalline PV cell statistical results are presented. Figure 6 shows the convergence graph for MODM SSE, which indicates that the method created is superior to the comparable algorithms. From Table 4, it is concluded that the developed algorithm (RSO) minimum values are 1.11 × 10⁻⁴, 1.09 × 10⁻², 5.47 × 10⁻⁴¹, 5.95 × 10⁻⁶, and 2.49 × 10⁻³; the maximum values are 9.91 × 10⁻⁴, 3.14 × 10⁻², 1.59 × 10⁻³, 4.95 × 10⁻⁵, and 7.02 × 10⁻³; the average values or mean values are 4.35 × 10⁻⁴, 2.07 × 10⁻², 1.05 × 10⁻², 2.15 × 10⁻⁵, and 4.64 × 10⁻³; and the standard deviation values are 2.62 × 10⁻⁴, 1.62 × 10⁻², 8.04 × 10⁻⁴, 1.32 × 10⁻⁵, and 3.60 × 10⁻³ for the SSE, AE, MAE, MSE, and RMSE, respectively. A modified one-diode model is represented in Figure 7 as an I-V curve, and a modified one-diode model is represented as a P-V curve in Figure 8. The results indicate that the parameter estimation using the designed RSO is superior to the other methods compared.

The Friedman ranking test [32,33] is illustrated in

Table 5. Statistical results based on Friedman ranking test for mono-crystalline PV cell.

Algorithms	Friedman Ranking
ALO	4
PSO	6
SSA	5
HHO	2
GOA	3
RSO	1

. In this test, the newly developed algorithm clearly outperformed the other compared algorithms. RSO is clearly the best algorithm when compared to the other algorithms based on the Friedman ranking test results.

Case 2: MTDM

Several optimization techniques were employed to estimate the parameters under standard temperature conditions (STC), as shown in Table 6. The statistical findings of the MTDM for the mono-crystalline PV cell are shown in Table 7. Figure 9 shows the convergence graph of the MTDM for the mono-crystalline PV cell of SSE, which clearly shows that the proposed hybrid method outperforms the other techniques evaluated. Figure 10 represents the I-V curve of the MTDM under STC, and Figure 11 represents the P-V curve of the MTDM under STC. From Table 7, it is concluded that for the developed algorithm (RSO):

(a)

The minimum values are 1.17 × 10⁻⁴, 1.08 × 10⁻², 2.16 × 10⁻³, 2.34 × 10⁻⁵, and 4.83 × 10⁻³.

(b)

The maximum values are 6.87 × 10⁻⁴, 2.62 × 10⁻², 5.24 × 10⁻³, 1.37 × 10⁻⁴, and 1.17 × 10⁻².

(c)

The average values or mean values are 4.81 × 10⁻⁴, 2.19 × 10⁻², 4.39 × 10⁻³, 9.62 × 10⁻⁵, and 9.81 × 10⁻³.

(d)

The standard deviation values are 2.23 × 10⁻⁴, 1.49 × 10⁻², 2.99 × 10⁻³, 4.47 × 10⁻⁵, and 6.68 × 10⁻³.

These results are presented for the SSE, AE, MAE, MSE, and RMSE, respectively. Based on the results, the parameter estimation utilizing the designed RSO is superior to those of the other methods examined.

Table 6. Parameter estimation and computation time of MTDM under STC.

Parameter/Algorithms	ALO	PSO	SSA	GOA	HHO	RSO
Ipv	9.1281	9.2096	9.2837	9.2137	9.2342	9.1246
Alpha1	1.9466	1.6706	1.4002	1.9432	1.5777	1.0485
Alpha2	1.5675	1.0237	1.0300	1.4761	1.2238	1.7029
Rse	0.0153	0.0332	0.0557	0.0043	0.0437	0.0119
Rsh	270.759	153.932	80.733	93.221	212.921	176.179
Io1	2.66 × 10−7	6.22 × 10−7	1.41 × 10−7	8.98 × 10−7	3.90 × 10−7	1.95 × 10−7
Io2	3.29 × 10−7	2.00 × 10−7	4.67 × 10−7	3.65 × 10−7	2.97 × 10−7	6.04 × 10−7
Rs	0.0148	0.0233	0.0685	0.0021	0.0521	0.0018
Computation time	1.2511	1.1257	1.1028	1.1942	1.2864	1.1103

Table 6. Parameter estimation and computation time of MTDM under STC.

Parameter/Algorithms	ALO	PSO	SSA	GOA	HHO	RSO
Ipv	9.1281	9.2096	9.2837	9.2137	9.2342	9.1246
Alpha1	1.9466	1.6706	1.4002	1.9432	1.5777	1.0485
Alpha2	1.5675	1.0237	1.0300	1.4761	1.2238	1.7029
Rse	0.0153	0.0332	0.0557	0.0043	0.0437	0.0119
Rsh	270.759	153.932	80.733	93.221	212.921	176.179
Io1	2.66 × 10−7	6.22 × 10−7	1.41 × 10−7	8.98 × 10−7	3.90 × 10−7	1.95 × 10−7
Io2	3.29 × 10−7	2.00 × 10−7	4.67 × 10−7	3.65 × 10−7	2.97 × 10−7	6.04 × 10−7
Rs	0.0148	0.0233	0.0685	0.0021	0.0521	0.0018
Computation time	1.2511	1.1257	1.1028	1.1942	1.2864	1.1103

Figure 9. Convergence graph of MTDM for SSE.

Figure 9. Convergence graph of MTDM for SSE.

Figure 10. I-V curve of MTDM.

Figure 10. I-V curve of MTDM.

Figure 11. P-V curve of MTDM.

Figure 11. P-V curve of MTDM.

Table 7. Statistical results of MTDM under STC.

Algorithms	Errors	SSE	AE	MAE	MSE	RMSE
ALO	Minimum	2.40 × 10−3	4.89 × 10−2	9.79 × 10−3	4.79 × 10−4	2.19 × 10−2
	Average	6.01 × 10−3	7.75 × 10−2	1.55 × 10−2	1.20 × 10−3	3.47 × 10−2
	Maximum	9.00 × 10−3	9.49 × 10−2	1.90 × 10−2	1.80 × 10−3	4.24 × 10−2
	Mean	6.01 × 10−3	7.75 × 10−2	1.55 × 10−2	1.20 × 10−3	3.47 × 10−2
	S.D	2.72 × 10−3	5.21 × 10−2	1.04 × 10−2	5.43 × 10−4	2.33 × 10−2
PSO	Minimum	1.75 × 10−1	4.18 × 10−1	8.37 × 10−2	3.50 × 10−2	1.87 × 10−1
	Average	3.63 × 10−1	6.03 × 10−1	1.21 × 10−1	7.27 × 10−2	2.70 × 10−1
	Maximum	8.56 × 10−1	9.25 × 10−1	1.85 × 10−1	1.71 × 10−1	4.14 × 10−1
	Mean	3.63 × 10−1	6.03 × 10−1	1.21 × 10−1	7.27 × 10−2	2.70 × 10−1
	S.D	2.84 × 10−1	5.33 × 10−1	1.07 × 10−1	5.68 × 10−2	2.38 × 10−1
SSA	Minimum	1.20 × 10−2	1.09 × 10−1	2.19 × 10−2	2.39 × 10−3	4.89 × 10−2
	Average	3.84 × 10−2	1.96 × 10−1	3.92 × 10−2	7.69 × 10−3	8.77 × 10−2
	Maximum	8.75 × 10−2	2.96 × 10−1	5.92 × 10−2	1.75 × 10−2	1.32 × 10−1
	Mean	3.84 × 10−2	1.96 × 10−1	3.92 × 10−2	7.69 × 10−3	8.77 × 10−2
	S.D	3.17 × 10−2	1.78 × 10−1	3.56 × 10−2	6.33 × 10−3	7.96 × 10−2
GOA	Minimum	1.98 × 10−2	1.41 × 10−1	2.81 × 10−2	5.35 × 10−5	6.29 × 10−2
	Average	3.47 × 10−2	1.86 × 10−1	3.73 × 10−2	6.94 × 10−3	8.33 × 10−2
	Maximum	6.43 × 10−2	2.54 × 10−1	5.07 × 10−2	1.29 × 10−2	1.13 × 10−1
	Mean	3.47 × 10−2	1.86 × 10−1	3.73 × 10−2	6.94 × 10−3	8.33 × 10−2
	S.D	1.82 × 10−2	1.35 × 10−1	2.70 × 10−2	3.64 × 10−3	6.04 × 10−2
HHO	Minimum	1.20 × 10−3	3.46 × 10−2	6.92 × 10−3	2.39 × 10−4	1.55 × 10−2
	Average	3.75 × 10−3	6.12 × 10−2	1.22 × 10−2	7.50 × 10−4	2.74 × 10−2
	Maximum	6.60 × 10−3	8.12 × 10−3	1.62 × 10−2	1.32 × 10−3	3.63 × 10−2
	Mean	3.75 × 10−3	6.12 × 10−2	1.22 × 10−2	7.50 × 10−4	2.74 × 10−2
	S.D	2.28 × 10−3	4.78 × 10−3	9.56 × 10−3	4.57 × 10−4	2.14 × 10−2
RSO	Minimum	1.17 × 10−4	1.08 × 10−2	2.16 × 10−3	2.34 × 10−5	4.83 × 10−3
	Average	4.81 × 10−4	2.19 × 10−2	4.39 × 10−3	9.62 × 10−5	9.81 × 10−3
	Maximum	6.87 × 10−4	2.62 × 10−2	5.24 × 10−3	1.37 × 10−4	1.17 × 10−2
	Mean	4.81 × 10−4	2.19 × 10−2	4.39 × 10−3	9.62 × 10−5	9.81 × 10−3
	S.D	2.23 × 10−4	1.49 × 10−2	2.99 × 10−3	4.47 × 10−5	6.68 × 10−3

Table 7. Statistical results of MTDM under STC.

Algorithms	Errors	SSE	AE	MAE	MSE	RMSE
ALO	Minimum	2.40 × 10−3	4.89 × 10−2	9.79 × 10−3	4.79 × 10−4	2.19 × 10−2
	Average	6.01 × 10−3	7.75 × 10−2	1.55 × 10−2	1.20 × 10−3	3.47 × 10−2
	Maximum	9.00 × 10−3	9.49 × 10−2	1.90 × 10−2	1.80 × 10−3	4.24 × 10−2
	Mean	6.01 × 10−3	7.75 × 10−2	1.55 × 10−2	1.20 × 10−3	3.47 × 10−2
	S.D	2.72 × 10−3	5.21 × 10−2	1.04 × 10−2	5.43 × 10−4	2.33 × 10−2
PSO	Minimum	1.75 × 10−1	4.18 × 10−1	8.37 × 10−2	3.50 × 10−2	1.87 × 10−1
	Average	3.63 × 10−1	6.03 × 10−1	1.21 × 10−1	7.27 × 10−2	2.70 × 10−1
	Maximum	8.56 × 10−1	9.25 × 10−1	1.85 × 10−1	1.71 × 10−1	4.14 × 10−1
	Mean	3.63 × 10−1	6.03 × 10−1	1.21 × 10−1	7.27 × 10−2	2.70 × 10−1
	S.D	2.84 × 10−1	5.33 × 10−1	1.07 × 10−1	5.68 × 10−2	2.38 × 10−1
SSA	Minimum	1.20 × 10−2	1.09 × 10−1	2.19 × 10−2	2.39 × 10−3	4.89 × 10−2
	Average	3.84 × 10−2	1.96 × 10−1	3.92 × 10−2	7.69 × 10−3	8.77 × 10−2
	Maximum	8.75 × 10−2	2.96 × 10−1	5.92 × 10−2	1.75 × 10−2	1.32 × 10−1
	Mean	3.84 × 10−2	1.96 × 10−1	3.92 × 10−2	7.69 × 10−3	8.77 × 10−2
	S.D	3.17 × 10−2	1.78 × 10−1	3.56 × 10−2	6.33 × 10−3	7.96 × 10−2
GOA	Minimum	1.98 × 10−2	1.41 × 10−1	2.81 × 10−2	5.35 × 10−5	6.29 × 10−2
	Average	3.47 × 10−2	1.86 × 10−1	3.73 × 10−2	6.94 × 10−3	8.33 × 10−2
	Maximum	6.43 × 10−2	2.54 × 10−1	5.07 × 10−2	1.29 × 10−2	1.13 × 10−1
	Mean	3.47 × 10−2	1.86 × 10−1	3.73 × 10−2	6.94 × 10−3	8.33 × 10−2
	S.D	1.82 × 10−2	1.35 × 10−1	2.70 × 10−2	3.64 × 10−3	6.04 × 10−2
HHO	Minimum	1.20 × 10−3	3.46 × 10−2	6.92 × 10−3	2.39 × 10−4	1.55 × 10−2
	Average	3.75 × 10−3	6.12 × 10−2	1.22 × 10−2	7.50 × 10−4	2.74 × 10−2
	Maximum	6.60 × 10−3	8.12 × 10−3	1.62 × 10−2	1.32 × 10−3	3.63 × 10−2
	Mean	3.75 × 10−3	6.12 × 10−2	1.22 × 10−2	7.50 × 10−4	2.74 × 10−2
	S.D	2.28 × 10−3	4.78 × 10−3	9.56 × 10−3	4.57 × 10−4	2.14 × 10−2
RSO	Minimum	1.17 × 10−4	1.08 × 10−2	2.16 × 10−3	2.34 × 10−5	4.83 × 10−3
	Average	4.81 × 10−4	2.19 × 10−2	4.39 × 10−3	9.62 × 10−5	9.81 × 10−3
	Maximum	6.87 × 10−4	2.62 × 10−2	5.24 × 10−3	1.37 × 10−4	1.17 × 10−2
	Mean	4.81 × 10−4	2.19 × 10−2	4.39 × 10−3	9.62 × 10−5	9.81 × 10−3
	S.D	2.23 × 10−4	1.49 × 10−2	2.99 × 10−3	4.47 × 10−5	6.68 × 10−3

Table 8. Statistical results based on Friedman ranking test under STC of MTDM.

Algorithms	Friedman Ranking
ALO	3
PSO	6
SSA	5
GOA	4
HHO	2
RSO	1

illustrates the Friedman ranking test. According to this test, the newly created algorithm performs better than the other compared algorithms. When compared with other algorithms, RSO achieved the highest ranking according to the Friedman ranking test, proving that it is a superior algorithm.

4. Conclusions

To obtain the exact solar cell parameter values of an MODM and MTDM equivalent, a new algorithm (RSO) is proposed in this manuscript. The proposed methodology provides many advantages, such as an increased convergence speed, a precise solution, and equilibrium between exploration and exploitation. In this method, the parameter values of MODM and MTDM solar cells are extracted by minimizing the error between the theoretical and experimental results. The main objective of the work presented in this manuscript is to minimize the error function. From the statistical analysis carried out based on the error analysis and Friedman ranking, it is clear that the RSO technique is more accurate in estimating the PV parameters than other algorithms like ALO, PSO, SSA, HHO, and GOA are. The main drawback of swarm-based algorithms is that they can get stuck in the local minima, but their advantage is that they have a fast iteration rate, which is why researchers modify swarm-based algorithms to eliminate this problem. The RSO algorithm is also a swarm-based algorithm, but it has obtained the best results, which proves that, for this particular application, RSO does not get stuck in the local minima. The scope of the application of the RSO technique can also be extended to the parameter estimation of solar PV cells using other models reported in the literature. In the end, it can be safely concluded that RSO has the immense potential to estimate the performance parameters of a PV cell.

The limitation of the RSO algorithm is that it can be used for the optimization of various other applications, like the parameter estimation of PEMFC, the estimation of MPP of solar, etc. It can be improved by increasing its memory using different methods such as like hybridization with a genetic algorithm.