Evaluation and Validation of Equivalent Five-Parameter Model Performance for Photovoltaic Panels Using Only Reference Data

This paper presents the modeling of electrical I-V verification of photovoltaic modules using fiveparameter models based on the minimum usage of input data, which are usually provided by manufacturer’s datasheet. However, we vary them with a step of 10−4, the ideality factor γ between 0.0 and 4 for each iteration in order to choose the γ value, which gives a minimal relative error of the maximum power point. Moreover, when γ is known, the other four parameters (i.e., Rs, I0, Iph and Rsh) are known. Finally, the effectiveness of this approach is then validated through comparison of the experimental results data under outdoor weather conditions.


Introduction
Due to the fossil fuels consumption, the increase of energy demand and the negative effect of pollution on the environment, the photovoltaic solar cell has become one of the promising green and renewable energy solutions.Ge-nerally, the three import points (i.e., maximum power points, short circuit current and open-circuit voltage) are provided on manufacturer datasheets under STC, i.e., 1000 W/m 2 , 25˚C and NOCT conditions, i.e., 800 W/m 2 , 47˚C.
For the modeling of (the) photovoltaic module, the use of equivalent circuits can be seen from the literature [1]- [4].Reference [5] proposed an equivalent circuit called three-diode model.The third diode is to present the influence of the recombination of carrier and the leakage current.This model is more effective for the poly-silicon cells.The double exponential model [6]- [8] is the most accurate model, which contains seven unknown parameters.In general, this model is more accurate for polycrystalline silicon cells [9].
To make the model easier to be solved, the equivalent circuit can be reduced to a circuit with a one-diode [10]- [14].In addition, one diode model can be divided into two categories.The first is called the four parameters model [15] [16], where the shunt resistance is neglected and considered as infinite.The second is called the five parameters model [17]- [19], where the shunt resistance is considered as finite.A comparison of five different mathematical one-diode models was studied in detail in [20].Reference [21] developed an empirical model to produce the current and voltage curve using five located points at the current-voltage curve.However, the method needs empirically coefficients parameters which are typically not available from the manufacturers datasheet.The coefficients are provided by the Sandia National Laboratory [22].
A modified electrical five-parameter-model is presented in the present work.The multi-crystalline PV module is studied by the model under real meteorological conditions in the desert climate.

The Equivalent Circuit of a Solar Cell
The five-parameter model is one of the famous photovoltaic models in the scientific world.Figure 1 shows the one diode equivalent circuit which contains of a current source ( ) ph I , a one diode (D), and series/parallel resistances ( s R and sh R , respectively).The characteristic equation model is given by Equation ( 1).

(
) where ph I is the photocurrent (A), 0 I is the reverse saturation current (A), q is the electron's charge (1.602 × 10 −19 ˚C), γ the ideality factor, The five unknown parameters necessities to estimate before to solve Equation (1) are: γ, I 0 , I ph , R s , and R sh .The ideality factor value γ is indicative of the recombination mechanism, it has been observed in the c -Si cells.The increasing of solar radiation from 1.6 to 1.1 includes a decrease of the ideality factor in even range [23].Moreover, at the three important manufacturer points at STC (i.e., short circuit, open circuit tension and maximum power points), Equation (1) can be written as: Short-circuit current: Open circuit tension: Maximum power point: In [24], the Newton-Raphson method was chosen to calculate the series resistance value and also its convergence time for various initial value of series resistance 0 s R .In this work, the authors developed this method to find the values of the four unknown parameters ph I , 0 I , sh R and s R at various ideality factor values γ .To simplify the calculations, several authors take an ideality factor fixed value.In this paper we vary with a step of 10 −2 , the ideality factor γ between 0 and 3, for each iteration, in order to choose its value which gives a mi- nimal relative error of the maximum power.Reference [24] used the following relationship, Equations ( 5)-( 7) relating to the photo-current ph I , the saturation current 0 I , and shunt resistance sh R .
( ) ( ) ( ) where: ( ) ) ( ) The derivative of the maximum power point as expressed in Equation ( 12): ( ) The Equation (1) derivative compared to the tension can be expressed by: ( ) We introduce Equation ( 12) in ( 13), and then we define a s R f function given by: As 0 I and sh R are depending on s R , the s R f function behaves so.For the series resistance s R , we suppose that the initial value of the crystalline PV cells is given by the approximate function [25] with the constants: 1:1: 4 i i α = .
We note that Equation ( 16) is independent of material properties of the solar cell [26].

Effect of Outdoor Condition on the Parameters
The unknown electrical parameters (i.e., ph I , 0 I , γ , s R and sh R ), was measured under standard test condi- tion (1000 W/m 2 and 25˚C) were rarely found in outdoor climate [27].Moreover, the dependence of the parameters on the real conditions is described in Equations ( 17)-( 21) [27].
( ) ,ref ,ref where g E is the band gap energy.

The Cell Temperature
To predict the performance of PV modules, it is necessary to know the PV cell temperature.In the literature, the modeling of PV cell temperature is obtained by several correlations mathematic models as a function of solar irradiance, ambient temperature and wind speed, etc., [28] [29].Equation ( 22) is used to calculate the PV-cell temperature [30].
NOCT is the Nominal Operating Cell Temperature which is estimated at 20˚C, 0.8 kW•m −2 and 1 m•s −1 .

The Maximum Power Point Current and Voltage
At any operating condition the maximum power point current and voltage in Equations ( 23) and ( 24) can be obtained by iterative method, we used the instrument fsolve in MatLab [31].

Short Circuit Current and Open Circuit Voltage
Short circuit current and open circuit voltage of the panel at a cell temperature and irradiance other than the ref-erence values, can be obtained by simultaneously solving Equations ( 2) and (3), using fsolve in MatLab/Simulink [22].

Outdoor Tested Photovoltaic Module
To validate the five parameter model we used an experimental data extracted from specific desert climate environment.The outdoor exposure tests were started on March 21-23, 2013 within the Research Unit in Renewable Energies in the Saharan Medium, Adrar, Algeria.A mono-crystalline silicon solar cell, i.e., S-Energy SM-XXXMH1 Series PV module was experimentally selected to perform the characterization and modeling method, Table 1 shows the electrical specifications in the module reference conditions.Table 2 shows the electric parameters calculated by the proposed model.The measurements of the I-V photovoltaic characteristics were performed with a data acquisition MP-160 I-V curve tracer and a CM11 type Kipp & Zonenpyranometer (last calibrate February 2011) was used to collect the measurements of meteorological radiation data.
Also, the MP-160 I-V curve tracer was used for determining the important points (i.e., sc I , oc V , p I and p V ) and for tracing the IV curves.The panel temperature was calculated using Equation ( 22).The maximum power point for a PV module and the relative error of maximum power point described by the model of this paper according to ideality factor, are illustrated in Figure 2  we give a maximum power almost equal to that of the manufacturer.
It is noted that the maximum power point of the PV modules studied in this article is calculated according to the manufacturer datasheet by producing the maximum power point current and voltage, i.e., p p P I V = × .This appears clearly in Figure 3, where the relative error is very weak for a value γ which corresponds to 1.58.
The manufacturer gives under the test standard conditions (1000 W/m 2 with 25˚C) that the maximum power point equal to 250.2 W, and the simulation gives us 250.151W p P = .However, the value of the maximum power relative error is 5.63%.To verify also the model of this paper, it is necessary to observe the curves resulting from the five parameters under varied atmospheric conditions.However, Figure 6 and Figure 7 show the I-V and P-V characteristics of S-energy modules in outdoor conditions.Scattered points are the experimental results and solid line is the model results.It can be seen that the I-V and PV curves extracted with the method of this paper is on a good agreement at the current zone with experimental dataset.But at the zone of voltage and maximum power there is a difference between experimental and calculated values.The differences between the experimental data and the calculated values was occurred as a result of limitations in the cell model itself, as well as in the methods used to calculate cell temperature.
For the verification of the model at the three important points (i.e., sc I , oc V and p P ), Figure 8 to Figure 10 show the comparison between the simulated and experimental values of these points during the test days.R of maximum power point are respectively 11.0121 W and 0.9743.The correlation coefficient is greater than 0.97, a good agreement between the simulated and measured values is observed.However, the values of root mean square error RMSE and the correlation coefficient 2  R of short circuit current are respectively 0.6802 A and 0.9743.A good agreement between the simulated and measured values is observed.
In Figure 10 we observe a remarkable deviation between simulated and experimental values.However, the values of root mean square error RMSE and the correlation coefficient 2  R of open circuit voltage are respec- tively 2.1746 V and 0.0677.By far, the open-circuit voltage point oc V is more temperature dependent than the short circuit current sc I [28].From this, the disparity was also supposed to be mainly caused by the error esti- mate by the module temperature, which was theoretically measured.

Conclusions
In this paper, the modeling of the photovoltaic five-parameter-model was proposed.This approach is based on datasheet parameters with a minimum of input information to find and calculate the parameters γ , 0 I , ph I , s R and sh R at STC (25˚C, 1000 W/m 2 ).The model accuracy was also analyzed through comparison between experimental data and simulation results using mono-crystalline PV-panel in outdoor climate.The plot of I-V and P-V curves extracted by the method of this paper is in a good agreement at the current zone with experimental data, but in the voltage zone and the maximum power zone there is a difference between experimental data and the calculated value.To present the model accuracy more clearly, the maximum power point, short circuit current and open circuit voltage during the three test days were described and evaluated.The agreement between the theoretical estimates and the experi- s constant and T is the temperature of the p n junction (K), s R the series resistance ( ) Ω and sh R the shunt resistance ( ) Ω .

Figure 1 .
Figure 1.One-diode equivalent circuit for a PV panel.

and Figure 3 .
The graph represented in Figure 2, reveals that the value of 1.58 γ =

Figure 2 .
Figure 2. Maximum power point P mp for different accepted values of ideality factor at SRC.

Figure 3 .
Figure 3. Relative Error for P mp for different accepted values of ideality factor γ at SRC.

Figure 4 and
Figure 4 and Figure 5 indicate the current-voltage and power-voltage of all acceptable values of S-energy PV panel at SRC.It is observed that the majority of the simulated values results are in very good agreement with the three important points: short circuit current, maximum power point and open circuit voltage.To verify also the model of this paper, it is necessary to observe the curves resulting from the five parameters under varied atmospheric conditions.However, Figure6and Figure7show the I-V and P-V characteristics of S-energy modules in outdoor conditions.Scattered points are the experimental results and solid line is the model results.It can be seen that the I-V and PV curves extracted with the method of this paper is on a good agreement at the current zone with experimental dataset.But at the zone of voltage and maximum power there is a difference between experimental and calculated values.The differences between the experimental data and the calculated values was occurred as a result of limitations in the cell model itself, as well as in the methods used to calculate cell temperature.For the verification of the model at the three important points (i.e., sc I , oc V and p P ), Figure8to Figure10show the comparison between the simulated and experimental values of these points during the test days.

Figure 4 .
Figure 4. Current-Voltage curves at SRC at different values of ideality factor γ.

Figure 5 . 2 R
Figure 5. Power-Voltage curves at SRC for different values of ideality factor γ.To compare the estimate results with the experimental measurements, the statistical parameters i.e., the root mean square error (RMSE) and the correlation coefficient ( ) 2 R are used.The values of root mean square error RMSE and the correlation coefficient 2R of maximum power point are respectively 11.0121 W and 0.9743.The correlation coefficient is greater than 0.97, a good agreement between the simulated and measured values is observed.However, the values of root mean square error RMSE and the correlation coefficient2  R of short circuit current are respectively 0.6802 A and 0.9743.A good agreement between the simulated and measured values is observed.In Figure10we observe a remarkable deviation between simulated and experimental values.However, the values of root mean square error RMSE and the correlation coefficient2  R of open circuit voltage are respec- tively 2.1746 V and 0.0677.By far, the open-circuit voltage point oc V is more temperature dependent than the short circuit current sc I[28].From this, the disparity was also supposed to be mainly caused by the error esti- mate by the module temperature, which was theoretically measured.

Figure 8 .
Figure 8.The simulated and experimental values of maximum power point during the three test days.

Figure 9 .
Figure 9. Simulated and experimental values of short-circuit current during the three test days.

Figure 10 .
Figure 10.The simulated and experimental values of open circuit voltage during the three test days.

2 R
P mental data of the maximum power point and short circuit current is very good in the whole days.The root mean square error RMSE and the correlation coefficient ( ) of the maximum power point and short circuit current respectively is 0.9743 and 0.9743, which is very satisfactory.By far, we observe a remarkable deviation between theoretical estimate values and experimental data of the open-circuit voltage point Voc.However, the values of root mean square error RMSE and the correlation coefficient R 2 of open circuit voltage are 2.1746 V and 0.0677, respectively. .

Table 1 .
The modules nominal electrical specifications.

Table 2 .
Electrical parameters model for the selected PV module.