Modeling Based on Daily Data of PV Power Plants

Abstract

Modeling performance characteristics is essential for the design and optimal operation of solar power plants. However, due to the influence of various factors on the performance of solar panels and data changes over time, determining an exact relationship between output power and weather conditions is still challenging. In this chapter, a hybrid method based on genetic programming will be presented for accurate modeling of solar power plant characteristics, which includes two steps. First, three points of open-circuit voltage, maximum power point, and short-circuit current are modeled as functions of atmospheric conditions. For this purpose, by using the modeling process based on genetic programming, relationships with high fit will be obtained for these three points in terms of cell temperature and radiation. Then, with the help of these equations, the voltage–current characteristics are modeled based on the circuit analysis methods and without the need for factory data. To evaluate the modeling for a 3 kW solar power plant, and based on the results, the effectiveness of the proposed method will be shown.

1. Introduction

Solar power plants based on photovoltaic systems are considered as one of the solutions to the energy crisis [1]. These power plants have been widely integrated into power systems due to their easy installation and utilization, low cost, and high lifetime [2]. This extensive integration, on a large scale, has brought challenges in consumption management and load-frequency control of microgrids because the production power of these power plants is intermittent and unplannable [3, 4]. In general, an accurate and reliable prediction of output power is of vital importance for the design and optimal operation of these power plants, which will lead to the improvement of grid stability and energy consumption efficiency.

Also, it is necessary to have an accurate model of the functional characteristics of solar power plants to detect faults, plan the time of washing the surface of the panels, identify loose electrical connections, and evaluate the performance of electronic power converters [5, 6]. For example, in fault diagnosis, after determining the voltage–current characteristic curve, if the working point of the power plant is at the point of maximum power, then the fault has not occurred and the control and power electronics, connections, and panels work correctly, but if the point If the power plant is not working at maximum power, an error has occurred, in this case, if the operating point is on the voltage–current characteristic, the error is related to sensors, electrical connections, or voltage converters, but if the power plant’s operating point is on the voltage–current characteristic If there is no current, the fault is related to the solar panels.

So far, different methods have been presented for modeling solar panels [7]. Many PV power forecasting models have been developed in previous studies based on different strategies, such as data-driven modeling methods and equivalent circuit modeling. Different methods can be divided into three categories: 1) methods based on artificial intelligence, 2) methods based on equivalent circuits, and 3) combined methods.

In methods based on artificial intelligence, neural networks are usually used to model or identify the solar panel. Neural networks, although in topics such as fault detection [8], production energy prediction [9], estimation of some parameters such as solar cell temperature [10], amount of radiation on cloudy days [11], and losses due to dirt [12] perform well, the error of modeling the characteristics of solar panels is significant compared to other methods [13, 14]. In some research [14, 15], to improve accuracy, in addition to atmospheric conditions, voltage is also included as input and the output is only current or power. However, there was not much improvement in modeling accuracy. So, the modeling of solar panels based on these methods received less attention.

In methods based on the equivalent circuit, the solar panel is modeled by a non-linear circuit [16]. The main goal of these methods is to extract the unknown parameters of this non-linear circuit, which are dependent on atmospheric conditions. The algorithm of these methods is based on the methods of circuit analysis and the work data of the manufacturer [17, 18]. Complex circuit calculations are one of the challenges of these methods. In some of the proposed algorithms [19] some unknown parameters are assumed to be constant to reduce the calculation burden. Nonetheless, in practice, these parameters are variable, leading to a decrease in modeling accuracy [20, 21]. Another challenge of these methods is the dependence of calculations on the manufacturer’s data (datasheet) [22]. These data change over time and therefore the accuracy of modeling in this method decreases. These cases are more frequent when dealing with the accuracy of the modeling of the solar power plant, which consists of interconnected solar panels.

The third category is the combined methods that do not suffer from the disadvantages of the previously mentioned methods. In combined methods, both equivalent circuits and neural networks are used to model the voltage–current characteristics of solar panels. In these methods [21, 23, 24], first, the neural network determines the unknown parameters according to atmospheric conditions. Then, the parameters are placed in the main equation of the non-linear equivalent circuit and the voltage–current characteristic can be provided. The main challenge of these methods is to form a database for training neural networks. A neural network needs a set of corresponding input and output data for training, but direct measurement of unknown parameters is practically not possible.

The importance of modeling accuracy is evident in the utilization process of a solar power plant that faces various challenges every day. Different methods presented for modeling the solar power plant are not free from challenges. In this regard, using the advantages of other modeling methods can be a solution to this problem.

Genetic programming (GP) is known as one of the most widely used tools in the field of modeling. GP can be considered a completed version of the genetic optimization algorithm as it is capable of extracting diverse and complex equations from databases and presenting them in the form of intuitive formulae [25]. Mohaghin used GP to solve various problems, including transportation energy demand prediction [26], analysis and optimization of thermocline solar energy storage [27], design and development of circuits and antennas [28], and identification of nonlinear systems [29]. Yet, so far, this powerful tool has not been adopted for modeling solar panels.

In this regard, in this chapter, a new method based on genetic programming for solar power plant modeling is presented, which is both accurate and independent of catalog data, so that it is possible to model old-installed solar power plants. The proposed method consists of two parts. In the first part, three main points of the voltage–current characteristic curve, i.e., open-circuit voltage, short-circuit current, and maximum power, are modeled according to weather conditions using the GP. In the second part, which is based on the equivalent circuit of a single diode, five unknown parameters of the equivalent circuit are calculated using the results of the first part and with the help of circuit analysis equations. In summary, the main contributions of this chapter are:

• A new hybrid method for modeling solar power plants based on daily data.

• Using genetic programming to model the solar power plant.

• Independent solar power plant modeling from datasheet.

The next sections of this chapter are organized as follows: In Section 2, the statement of the problem is discussed. The proposed method including two parts of GP and circuit analysis will be presented in Section 3. In Section 4, by simulating the proposed algorithm for a solar power plant, its performance is evaluated and the effective factors in the simulation are discussed. Finally, in Section 5, the general conclusions of the present study are summarized.

2. Statement of the problem

The solar cell, which consists of various materials such as silicon semiconductors, produces electricity from sunlight. A solar panel consists of several solar cells connected in series and parallel. Figure 1 shows the equivalent circuit of a solar panel.

Figure 1.

Equivalent circuit of a single-diode solar panel.

Considering the parameters of Figure 1 and circuit analysis, the characteristics of the solar panel are given here [30]:

$$I=I_{\mathit{PV}}-I_O\left[exp\left(\frac{V+R_{S}I}{{\mathit{aV}}_t}\right)-1\right]-\frac{V+R_{S}I}{R_P}$$E1

I_PV and I_O are related to the irradiance and temperature changes as follows [30]:

$$I_{\mathit{PV}}=\left(I_{S,n}+K_I\mathrm{\Delta }T\right)\frac{G}{G_n}$$E2

$$I_O=\frac{I_{\mathit{SC},n}+K_I\mathrm{\Delta }T}{exp\left(V_{\mathit{OC},n}+K_V\mathrm{\Delta }T\right)/{\mathit{aV}}_t-1}$$E3

Additionally, Open-circuit voltage and short-circuit current are important points on the I-V characteristic curve of a solar panel. These points vary with changes in weather conditions. Using Eqs. (4) and (5), which are derived from the model equations, it is possible to calculate the short-circuit current and open-circuit voltage under different weather conditions [30].

$$I_{\mathit{SC}}=\left(I_{\mathit{SC},n}+K_I\mathrm{\Delta }T\right)\frac{G}{G_n}$$E4

$$V_{\mathit{OC}}=V_{\mathit{OC},n}+K_V\mathrm{\Delta }T$$E5

In the manufacturer’s datasheet of a PV module, information such as open-circuit voltage (V_OC), short-circuit current (I_SC), maximum power value (Pm), current and voltage values corresponding to maximum power, Im and Vm, and temperature coefficients of current and voltage (K_I and K_V) are listed. However, the catalogs do not provide information about solar cell performance such as optical current (I_PV), saturation current (I_O), diode ideality factor (a), series resistance (R_S), and shunt resistance (R_P). These unknown parameters are essential for modeling a PV module.

So far, various methods have been presented for extracting unknown parameters. In these methods, parameter extraction is based on catalog data, such as K_I, K_V, I_SC,n, and V_OC,n, but the data for solar panels change over time and the modeling error increases. The challenge of combined methods in modeling solar panels is collecting data for neural network training, while the values of unknown parameters cannot be measured and must be extracted or calculated. In other words, it is not possible to model an old-installed solar power plant using analytical methods and combined methods.

In the next section, the proposed method for modeling the solar power plant is presented. The algorithm of the proposed method is independent of the factory datasheet, so old solar power plants can be properly modeled.

3. The proposed method

The suggested method for modeling the solar power plant includes two steps: 1) applying the GP, and 2) using circuit analysis. In the first step, three important points of voltage–current characteristics are modeled as formulae using the GP and according to cell temperature conditions and solar panel surface radiation. In the second step, the unknown parameters of the equivalent circuit are calculated based on the circuit analysis equations and using the formulae of the previous step. Each of the steps is described below.

3.1 Genetic programming

GP is a variant of genetic algorithm (GA) and was proposed (Koza [25]) to automatically code computer programs to perform predefined tasks. The GP method [25] is founded on the “survival of the fittest” and genetic propagation of characteristics principles followed by biologically evolving species. Although GA and GP employ the same principles of Darwinian evolution, there is a significant difference between their application domains. That is, while the GA performs function maximization/minimization, GP implements symbolic regression (SR). Given an example input–output data set, SR obtains an appropriate linear or a nonlinear function and all of its parameters that best fit the data.

The general form of the model obtained by GP-based SR for modeling is given as follows.

$$y=f\left(\alpha ,\beta \right)$$E6

where f is a linear/non-linear function and $\alpha$ and $\beta$ are the input values of the function. In this research, these values are the cell temperature and irradiance, the value of y, the output of the function, and one of the three values of open-circuit voltage, short-circuit current, and maximum power.

The stages of GP include four parts, which are performed by GeneXprotools.5 software in this research. The first part is the initialization, where the problem and the general structure of the solution are explained. The other three parts, namely, evaluation and selection of fit, intersection, and mutation are done by the software.

A summary of the GP procedure consisting of four major steps is given below.

1. Initialization: Creates a random initial population of candidate (probable) solutions to the given data-fitting problem using tree structures.

2. Fitness evaluation and selection: Evaluate the fitness of each candidate solution in the current population using a fitness function and selects fitter solutions to form a pool of parent candidates (see Figure 1b) to undergo crossover.

3. Crossover: Forms a new generation of candidate solutions (offspring); a typical crossover operation executed on a randomly selected pair from the parent pool is shown in Figure 2a.

4. Mutation: Applies small changes to offspring candidate solutions (see Figure 2b).

Figure 2.

Schematic of GP: (a) random selection of branches for reproduction, (b) crossover operation.

Among these, steps (ii)–(iv) are performed iteratively until the best data-fitting candidate is achieved. Finally, statistical indicators are used to evaluate predictive models and compare modeling accuracy.

3.2 Circuit analysis

To plot the V-I characteristics of the solar panel, it is necessary to calculate or derive five parameters: a, I_O, I_PV, R_S, and R_P. Therefore, in this section, these parameters are calculated using the results of the previous step and based on circuit analysis equations.

The coefficient a has a value between 1 and 2. Initially, its value is close to 1, and over time, and with the decrease in efficiency of the installed panels, its value approaches 2 [31]. Accordingly, the value of this coefficient is taken into account in the form of Eq. (16), which is the inverse of the fill factor,

$$a=\frac{I_{\mathit{SC}}^{\mathit{GP}}\times V_{\mathit{OC}}^{\mathit{GP}}}{P_m^{\mathit{GP}}}$$E7

By comparing the values of $I_{\mathit{sc}}^{\mathit{GP}}$and $V_{\mathit{oc}}^{\mathit{GP}}$with Eqs. (4) and (5) and substituting them in Eq. (3), we have,

$$I_O=\frac{I_{\mathit{SC}}^{\mathit{GP}}}{exp\left(V_{\mathit{OC}}^{\mathit{GP}}\right)/{\mathit{aV}}_t-1}$$E8

Considering that voltage and current values at the maximum power point are approximately seven-tenths of the open-circuit voltage and nine-tenths of the short-circuit current, Vm ≈ 0.7 × V_OC and Im ≈ 0.9 × I_SC, [32] the initial values of resistances Rs and Rp can be found by the following equations,

$$R_{P0}=\frac{V_m}{I_{\mathit{SC}}-I_m}\approx 7\times \frac{V_{\mathit{OC}}}{I_{\mathit{SC}}}$$E9

$$R_{S0}=\frac{V_{\mathit{OC}}-V_m}{I_m}\approx 0.34\times \frac{V_{\mathit{OC}}}{I_{\mathit{SC}}}$$E10

Also, using the ratio $\frac{I_{\mathit{PV},n}}{I_{\mathit{SC},n}}=\frac{R_P+R_S}{R_P}$ [31], the following equation will replace Eq. (2),

$$I_{\mathit{PV}}=I_{\mathit{SC}}^{\mathit{GP}}\times \frac{R_P+R_S}{R_P}$$E11

To calculate the two R_S and R_P parameters, open-circuit voltage and short-circuit current values are inserted in Eq. (1),

$$0=I_{\mathit{PV}}-I_O\left[exp\left(\frac{V_{\mathit{OC}}}{{\mathit{aV}}_t}\right)-1\right]-\frac{V_{\mathit{OC}}}{R_P}$$E12

$$I_{\mathit{SC}}=I_{\mathit{PV}}-I_O\left[exp\left(\frac{R_S{I}_{\mathit{SC}}}{{\mathit{aV}}_t}\right)-1\right]-\frac{R_S{I}_{\mathit{SC}}}{R_P}$$E13

Using Eqs. (7) to (13), the voltage–current characteristics curve is extracted based on the following algorithm.

1. Start

2. Calculate V_OC, I_SC, and Pm.

3. Calculate a using Eq. (7).

4. Calculate I_O using Eq. (8).

5. Set the initial values of R_P and R_S using Eqs. (9) and (10).

6. Calculate I_PV using Eq. (11).

7. Calculate R_P using Eq. (12).

8. Calculate Rs using Eq. (13).

9. Repeat steps 6–8 until the parameters converge.

Figure 3 shows the flowchart of the proposed method, which includes the above two steps to calculate the voltage–current characteristic curve.

Figure 3.

The flowchart of the proposed method includes three steps: GP, circuit analysis, and V-I curve.

4. Simulation and discussion

In this section, the performance of the proposed algorithm is evaluated for modeling a 3 kW power plant. For this purpose, first, by using GP and the data of a sunny day, open-circuit voltage, short-circuit current, and maximum power are determined in terms of solar cell temperature and radiation. Next, based on these formulae and using the circuit analysis, unknown parameters are extracted to model the voltage–current characteristic.

4.1 Genetic programming

In this section, short-circuit current, open-circuit voltage, and maximum power of a 3 kW power plant are modeled by GP according to cell temperature and solar panel surface radiation. The output of this section includes three formulae that can be used to calculate these three quantities for all cell temperatures and radiations. The initial structure of the formula of these three quantities is considered according to Eqs. (14)–(16).

$$I_{\mathit{sc}}=f_{{\mathit{pv}}_1}\left(T_C,G\right)$$E14

$$V_{\mathit{oc}}=f_{{\mathit{pv}}_2}\left(T_C,G\right)$$E15

$$P_m=f_{{\mathit{pv}}_3}\left(T_C,G\right)$$E16

After the implementation of GP, with the initial population size of 500, the number of generations of 1000, the crossover rate of 0.8, and the mutation probability of 0.12, the three suggested formulae for estimating the values of ISC, VOC, and Pm are:

$$I_{\mathit{sc}}^{\mathit{GP}}=-0.253+0.0047T_C+0.0303T_C^{0.022}\times G$$E17

$$V_{\mathit{oc}}^{\mathit{GP}}=-222.07-0.57T_C+308.52{\mathrm{G}}^{0.017}+0.044T_C\times {\mathrm{G}}^{0.17}$$E18

$$\begin{array}{ll}{P}_m^{\mathit{GP}}=& \left(-0.294+0.0294T_C^{0.0047}\times G\right)(66.43-0.237{\mathrm{G}}^{0.67}-0.536T{\mathrm{G}}^{-0.044}\\ & +6.144T_C^{0.011}\times {\mathrm{G}}^{0.325})\end{array}$$E19

Two statistical indices of root mean square error (RMSE) and coefficient of determination, R2, according to the following equations are used to evaluate the accuracy of the proposed formulae.

$$\mathit{\text{RMSE}}=\sqrt{\frac{{\sum }_{i=1}^N{\left(y_o-y_p\right)}^2}{N_{\mathit{sa}}}}$$E20

$$R^2=1-\frac{{\sum }_{i=1}^N{\left(y_o-y_p\right)}^2}{{\sum }_{i=1}^N{\left(y_o-\overline{y}\right)}^2}$$E21

Statistical indices including RMSE and R2 are reported in Table 1, Appendix A.

4.2 Circuit analysis

In this section, according to the proposed algorithm, Figure 3, having the formulae of open-circuit voltage, short-circuit current, and maximum power, the values of five unknown parameters can be calculated. Table 2 shows the simulation results for the test data, Appendix B.

To check the performance of the proposed method, the modeling for other data should also be evaluated. For this purpose, the modeling of the solar power plant for a sunny day with cell temperature and radiation intensity shown in Figure 4 has been done, Figure 5 shows the performance of the proposed method.

Figure 4.

Atmospheric conditions. a) Temperature, b) irradiance.

Figure 5.

Solar power plant modeling a) open-circuit voltage, b) short-circuit current, c) maximum power.

Figure 6 shows the voltage–current characteristics. Also, the values of RMSE and R2 are presented in Table 3, Appendix C. As can be seen, the results match the initialization data very accurately, which is due to the accurate modeling of five parameters based on the proposed equations. In this case, the parameters change depending on atmospheric conditions, such as the parameters a, Rp, and Rs, which are usually included in fixed circuit analysis methods. Here, based on the proposed formulae, the parameters change proportionally.

Figure 6.

Voltage–current characteristic of a 3 kW power plant for actual and modeled values.

The proposed method can be compared with the hybrid method presented in the article [23]. This comparison can be made in the two fields of modeling process and modeling results. In the modeling process, databases play an effective role. In the proposed method, the databases are small and the required data can be measured, but the required data of the [23] method are five unknown parameters that cannot be measured directly, and these data are extracted with calculations and optimization methods. Also, to train the neural network in the method, the databases should be wider.

Another area where two methods can be compared is comparing their results. In this regard, the method of the [23] is implemented by the data of Table 2, so that part of the data is used for training and part as a test. The results of the simulations for the test data are shown in Table 4, Appendix D. The main reason for the low accuracy of the modeling done is the small amount of training data.

5. Discussion

In the utilization of the solar power plant, modeling the instantaneous values of three parameters: maximum power, open-circuit voltage, and short-circuit current are important. These values are effective both in the process of controlling and extracting power and in evaluating the performance of the controller. In the following, the effective factors in modeling and determining these three points are discussed in different ways.

5.1 The effect of time

The parameters and coefficients of solar panels change over time. These changes are the main challenge of the analytical methods because, in these methods, the parameters are extracted based on the factory data. The passage of time does not play a role in optimization methods because the parameter extraction algorithm is re-executed when the weather conditions change. In the modeling with the proposed algorithm method, the passage of time is taken into account in such a way that after each periodical maintenance service, once or twice a year, older data is deleted, newer data is added, and the formula is renewed. On other days of the year, modeling is performed based on the calculated formulae.

5.2 The effect of data type

In the analytical methods, the data required for modeling are diverse and include factory data (solar panel catalog), three important characteristic points of voltage–current, and atmospheric conditions. After modeling, the input data is the atmospheric condition and the output data is the voltage–current characteristic. With this feature, it is possible to extract three important points. In modeling by optimization method, the required data include different voltage–current points in certain atmospheric conditions. The output of this method is the five unknown parameters of Eq. (1). In other words, the voltage–current characteristic in specific atmospheric conditions it is possible to determine three important points. In the proposed method, the required data includes a set of three important points and weather conditions for two or three days of the year. After modeling the input data, atmospheric conditions and output data are three important points. Also, based on the proposed formulae, it is possible to extract five unknown parameters by analytical method without the need for a factory datasheet. Simply put, the proposed method is a combination of two analytical and optimization methods.

5.3 Fault analysis

Among these three methods, analytical methods provide less accuracy. In analytical methods, three out of five unknown parameters usually have a fixed value for different weather conditions. In other words, these three constant parameters are determined for STC; therefore, in other atmospheric conditions for low irradiance, modeling is associated with errors. In large power plants, including series and parallel panels, this error is more visible. In the optimization methods, the error has been reported as negligible because the optimization algorithm is implemented for each weather condition. Due to the continuous implementation of algorithms in practice, there is a possibility of measurement errors and hardware failure. In these methods, there is a compromise between the speed and accuracy of the response and the hardware used. The error of the proposed method depends on the quality of the collected data. If the data includes both sunny and cloudy days, the modeling error is negligible. Also, due to the functional changes of solar panels over time, older data should be excluded from the process of modeling and calculation of formulae.

6. Conclusions

Accurate modeling of the solar power plant is important both from the point of view of the power plant to diagnose faults, plan maintenance, and improve performance and from the smart networks to manage production and consumption and control frequency and stability. In this chapter, using the GP technique, three important points of voltage–current characteristics are modeled as formulae of atmospheric conditions. To calculate the proposed formulae, the data from one day at a solar power plant including atmospheric conditions are used as variables and three important points are adopted as outputs. It was also shown in this chapter that with the help of the calculated formulae, it is possible to extract five unknown parameters of the single-diode model with high accuracy and without the need for factory data. Finally, the proposed method was simulated for a 3 kW power plant. The simulation results were effective and the proposed method showed three important points of voltage–current characteristics while extracting the unknown parameters of the single-diode model.

Highlights

• A hybrid method for modeling the solar power plant will be investigated, which is simple and efficient.

• The proposed method is independent of the data sheet information, as it enables the modeling of panels that have been installed for several years.

• The method reviewed in this chapter performs modeling with high accuracy based on the daily data of the solar power plant, the required data are small and measurable.

Appendix A

See Table 1

Index	VOC	ISC	Pm
RMSE	0.0154	0.0162	0.0236
R2	0.9998	0.9997	0.9988

Table 1.

Statistical indices corresponding to each proposed equation.

Appendix B

See Table 2

GP input		GP output			Circuit analysis results
T (W/m^2)	Tc (Co)	VOC (V)	ISC (A)	Pm (kW)	a	IPV (A)	IO (μA)	Rs (Ω)	Rp (Ω)
302.34	16.8	110.28	9.5739	0.7533	1.40	9.82	0.030	3.57	139.2
307.91	17.2	110.21	9.7605	0.7671	1.40	10.01	0.032	3.57	139.0
309.25	18.1	109.83	9.8192	0.7674	1.41	10.07	0.039	3.56	138.3
311.73	18.8	109.56	9.9109	0.7715	1.41	10.17	0.044	3.56	137.8
409.36	20.9	110.30	13.1071	1.0270	1.41	13.44	0.060	3.55	137.7
427.01	22.0	110.07	13.6997	1.0692	1.41	14.05	0.073	3.54	137.1
544.91	26.2	109.75	17.6112	1.3616	1.42	18.07	0.149	3.52	135.8
598.33	27.4	109.82	19.3753	1.4957	1.42	19.88	0.182	3.51	135.6
618.37	27.9	109.81	20.0387	1.5453	1.42	20.56	0.199	3.50	135.5
687.09	29.5	109.80	22.3141	1.7147	1.43	22.89	0.263	3.49	135.7
815.44	34.9	108.58	26.6278	2.0007	1.45	27.32	0.669	3.46	133.4
831.48	34.1	109.06	27.1356	2.0499	1.44	27.84	0.588	3.46	133.6
882.79	35.2	108.98	28.8412	2.1707	1.45	29.59	0.715	3.45	133.4
867.38	35.9	108.56	28.3518	2.1239	1.45	29.09	0.799	3.45	132.8
894.87	36.2	108.64	29.2598	2.1910	1.45	30.02	0.846	3.44	132.9
923.39	36.9	108.56	30.2110	2.2560	1.45	30.99	0.957	3.43	132.7
948.13	37.4	108.52	31.0342	2.3133	1.46	31.84	1.047	3.43	132.6
972.29	38.2	108.35	31.8455	2.3652	1.46	32.67	1.201	3.42	132.2

Table 2.

Simulation results for test data.

Appendix C

See Table 3

Index	VOC	ISC	Pm
RMSE	0.0213	0.0197	0.0251
R^2	0.9989	0.9991	0.9983

Table 3.

Statistical indices corresponding to each proposed equation.

Appendix D

See Table 4

Index	a	IPV (A)	IO (μA)	RS (Ω)	RP (Ω)
RMSE	0.1316	0.3548	0.3827	0.2161	0.1925
R^2	0.8485	0.6971	0.7283	0.7935	0.8218

Table 4.

Statistical indices corresponding to each proposed equation.

Appendices and nomenclature

STC	the standard test condition (Tn = 25°C and Gn = 1000 W/m^2)
Tn	the solar cell temperature at standard test conditions (25°C)
TC	the cell temperature (°C)
ΔT	the cell temperature difference from STC
Gn	the solar irradiance at standard test conditions (1000 W/m^2)
G	the solar irradiance (W/m^2)
IPV,n	the photovoltaic current at standard test conditions (A)
IPV	the photovoltaic current (A)
VOC,n	the open circuit voltage at STC (V)
VOC	the open circuit voltage (V)
ISC,n	the short circuit current at STC (A)
ISC	the short circuit current (A)
IO	the reverse saturation current
$V_t$	the thermal voltage which is equal to $N_S{\mathit{KT}}_C/q$(V)
NS	the number of cells connected in a series
K	the Boltzmann constant
q	the electron charge
a	the diode ideality factor
RS	the series resistances
RP	the parallel resistances
KI	the ratio of short-circuit current to temperature
KV	the temperature coefficient of open-circuit voltage
Nsa	the number of samples
yO	the data observed from practical experiments
yP	the corresponding data obtained using the GP
$\overline{y}$	the mean of the data