Analytical Modeling and Determination of the Characteristic Parameters of the Different Commercial Technologies of Photovoltaic Modules

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1. Introduction

At present, we are witnessing: Rapid decline in fossil fuel reserves due to increased use of thermal power plants; Increased air pollution correlated with the burning of fossil fuels, which generates greenhouse gases.

Therefore, in the current scenario, there is an urgent need to accelerate research and development of renewable energy technology, especially solar energy, to meet global energy demand. Solar energy applications have been progressively increasing worldwide. This is due to the decrease in the cost of photovoltaic panels with the increasing demand, and the increase in the duration of use (lifetime). Photovoltaic is very competitive in areas far away from the conventional grid [1] . However, its exploitation requires a well-optimized design and dimensioning. The performance and economic profitability of this type of technology depends on different parameters that characterize its mathematical model. A precise knowledge of these parameters makes it possible to predict the performance of photovoltaic solar cells [2] . The five parameters of interest in the equivalent circuit are the photocurrent ( ${I}_{pv}$ ), the series resistance ( ${R}_{s}$ ), the diode saturation current ( ${I}_{o}$ ), the parallel resistance ( ${R}_{sh}$ ) and the ideality factor (n). The voltage-current relationship of a solar cell is described by a mathematical equation [3] . These various parameters make it possible to describe the behavior of the module and predict its performance. The models and methods used to evaluate these parameters have been the subject of several studies. Soto et al. (2006) [4] studied a five-parameter model using only manufacturer-supplied data with semi-empirical equations to predict the I-V and P-V curve of the cell for any operating condition and compared with experimental data from a Building integrated photovoltaic system for four different cellular technologies (mono-crystalline, multi-crystalline, silicon thin film and triple junction amorphous). They showed that the five-parameter model can be a precise tool for predicting energy production for single-junction cell types. Ould Mohamed Yahya et al. (2008) [5] also studied a five-parameter simulation model to predict the performance of a photovoltaic (PV) system operating in the meteorological conditions of the installation site and validated the simulation model from the experimental data of an individual 1.2 kWp system installed in Nouakchott, Mauritania. R. Merahi et al. (2010) [6] used the four-parameter model to simulate the operation of the PV module (PW500 of PHOTOWATT) for different conditions of sunshine and temperature. They found that the increase in series resistance and the quality factor is due to the degradation of I(V) curve at the elbow. Dongue et al. (2012) [7] investigated the performance of the four- and five-parameter models used to predict the electrical response of multi-crystalline Shell SP75 and GES-P70 mono-crystalline PV modules for different operating conditions. They concluded that the four- and five-parameter models accurately adjust the experimental data of the two PV modules used for various operating conditions analyzed. Zerdoudi et al. (2015) [8] modeled a photovoltaic generator using the four-parameter model to simulate the operation of the PV model (SPR315 E) for different conditions of sunshine and temperature. The aim of this work is to characterize and modelisize four different commercial photovoltaic modules technologies and to study the efficiency of the method compared to different photovoltaic module technologies by using the average absolute relative error between the values simulated by the model and those given by the manufacturers.

2. Models and Methods

2.1. Models

There are several commercial models of photovoltaic technologies that have certain performances depending on their location. Among these different technologies, the most exploited in Mauritania [9] are listed in Table 1. The data in Table 1 were obtained under standard test conditions (G = 1000 W/m^{2} and T = 25˚C) according to the manufacturers of each technology.

2.2. Method

In the literature, there are two main models of photovoltaic electric generators; namely one and two diode models, with three or more parameters. In this work, a one diode photovoltaic module with five parameters whose equivalent diagram is presented in Figure 1 is studied. The five parameters here are: ${I}_{ph}$ , ${R}_{s}$ , ${R}_{sh}$ , ${I}_{0}$ and n [10] [11] .

The current produced by the generator is obtained from Kirchhoff’s laws as follows:

$I={I}_{ph}-{I}_{D}-{I}_{sh}$ (1)

The diode current can be obtained through Shockley equation as follows [12] :

${I}_{D}={I}_{0}\cdot \left({\text{e}}^{\left(\frac{q\left(V+IRs\right)}{nNk{T}_{c}}\right)}-1\right)$ (2)

While the shunt current is given by the relation:

Table 1. Electrical characteristics of photovoltaic modules used.

Figure 1. Electrical model of a PV generator with five parameters [4] .

${I}_{sh}=\frac{\left(V+I{R}_{s}\right)}{{R}_{sh}}$ (3)

Replacing (2) and (3) into (1) give the photovoltaic current as:

$I={I}_{ph}-{I}_{0}\cdot \left({\text{e}}^{\left(\frac{q\left(V+IRs\right)}{nNk{T}_{c}}\right)}-1\right)-\frac{\left(V+I{R}_{s}\right)}{{R}_{sh}}$ (4)

If we assume that the parallel resistance ${R}_{sh}$ is very large (case of crystalline silicon) [13] .

For this purpose Equation (4) becomes as follows:

$I={I}_{ph}-{I}_{0}\cdot \left({\text{e}}^{\left(\frac{q\left(V+IRs\right)}{nNk{T}_{c}}\right)}-1\right)$ (5)

Determination of the PV Generator Parameters

1) Evaluation of I_{ph}

The light current ${I}_{ph}$ depends on both irradiance and temperature. It is given by [12] [14] :

${I}_{ph}=\left[{I}_{sc}+\alpha \left({T}_{c}-{T}_{r}\right)\right]\cdot \left(\frac{G}{{G}_{r}}\right)$ (6)

2) Evaluation of I_{0}

The reverse saturation current depending of cells temperature is given as follows [15] [16] :

${I}_{0}={I}_{on}\cdot {\left(\frac{{T}_{c}}{{T}_{r}}\right)}^{3}\cdot {\text{e}}^{\left[\frac{q{E}_{g}\left(\frac{1}{{T}_{r}}-\frac{1}{{T}_{c}}\right)}{nk}\right]}$ (7)

At the open circuit voltage $I=0,V={V}_{oc}$ and $I=0,V={V}_{oc}$

$0={I}_{sc}-{I}_{0n}\cdot \left({\text{e}}^{\left(\frac{q{V}_{oc}}{nNk{T}_{c}}\right)}-1\right)$ (8)

So the nominal saturation current is obtained through:

${I}_{0n}=\frac{{I}_{sc}}{{\text{e}}^{\left(\frac{q{V}_{oc}}{nkN{T}_{c}}\right)}-1}$ (9)

By replacing the Equation (9) into Equation (7) one gets:

${I}_{0}=\frac{{I}_{sc}}{{\text{e}}^{\left(\frac{q{V}_{oc}}{nkN{T}_{c}}\right)}-1}\cdot {\left(\frac{{T}_{c}}{{T}_{r}}\right)}^{3}\cdot {\text{e}}^{\left[\frac{q{E}_{g}\left(\frac{1}{{T}_{r}}-\frac{1}{{T}_{c}}\right)}{nk}\right]}$ (10)

3) Evaluation of R_{s}

Various techniques have been used to determine the series resistance ${R}_{s}$ [3] [17] . In this work the series resistance is evaluated as follows:

${R}_{s}=\left(-{\frac{\text{d}V}{\text{d}I}|}_{V={V}_{oc}}\right)$ (11)

Considering the asymptotic behavior of the I-V curve under short-circuit and open-circuit conditions ${R}_{s}$ can be calculated as [18] :

${R}_{s}=\frac{\frac{{N}_{s}nk{T}_{c}}{q}\cdot \mathrm{ln}\left(1-\frac{{I}_{mp}}{{I}_{sc}}\right)+{V}_{oc}-{V}_{mp}}{{I}_{mp}}$ (12)

4) Evaluation of n:

At the short circuit point, $I={I}_{sc}$ , $V=0$ :

${I}_{sc}={I}_{ph,ref}-{I}_{0,ref}\cdot \left({\text{e}}^{\left(\frac{q{R}_{s}{I}_{sc}}{nNk{T}_{c}}\right)}-1\right)$ (13)

At the maximum power point, $I={I}_{mp}$ , $V={V}_{mp}$ :

${I}_{mp}={I}_{ph,ref}-{I}_{0,ref}\cdot \left({\text{e}}^{\left(\frac{q\left({V}_{mp}+{I}_{mp}{R}_{s}\right)}{nNk{T}_{c}}\right)}-1\right)$ (14)

The reverse saturation current Io for any diode is a very small quantity, on the order of 10^{−}^{5} or 10^{−6} A [19] . This minimizes the impact of the exponential term in Equation (13), so it is safe to assume that the photocurrent equals the short-circuit current [20] . Another simplification [21] can be made regarding the first term in Equations (8) and (14). In both cases, regardless of the system size, the exponential term is much greater than the first term. For this reason the first term can be neglected. Then, the equation system becomes:

${I}_{sc}\approx {I}_{ph,ref}$ (15)

$0={I}_{sc}-{I}_{0,ref}\cdot \left({\text{e}}^{\left(\frac{q{V}_{oc}}{nNk{T}_{r}}\right)}-1\right)$ (16)

${I}_{mp}={I}_{ph,ref}-{I}_{0,ref}\cdot {\text{e}}^{\left(\frac{q\left({V}_{mp}+{I}_{mp}{R}_{s}\right)}{nNk{T}_{r}}\right)}$ (17)

Combining (16) and (17), the ideality factor is evaluated as follows:

$n=\frac{q\left(2{V}_{mp}-{V}_{oc}\right)}{Nk{T}_{r}\left(\frac{{I}_{mp}}{{I}_{sc}-{I}_{mp}}+\mathrm{ln}\left(\frac{{I}_{sc}-{I}_{mp}}{{I}_{sc}}\right)\right)}$ (18)

2.3. Maximum Power Point Determination

By using the expression of the PV current defined by Equation (5), the voltage supplied by the generator is:

$V=\frac{nNK{T}_{c}}{q}\mathrm{ln}\left(1+\frac{{I}_{ph}-I}{{I}_{o}}\right)\cdot {R}_{s}I$ (19)

The electric power produced by the generator is given by:

$P=V\cdot I$ (20)

By replacing (19) into (20) one obtain,

$P=\frac{nNK{T}_{c}}{q}\mathrm{ln}\left(1+\frac{{I}_{ph}-I}{{I}_{o}}\right)\cdot {R}_{s}{I}^{2}$ (21)

From the function $P=f\left(I\right)$ , the extremum is obtained by the resolution of the equation

$\frac{\text{d}P}{\text{d}I}=0$

$\begin{array}{l}2{R}_{s}{I}^{2}-\left[C\mathrm{ln}\left(1+\frac{{I}_{ph}-I}{{I}_{o}}\right)+C+2{R}_{s}\left({I}_{o}+{I}_{ph}\right)\right]\cdot I\\ \text{\hspace{0.05em}}+C\mathrm{ln}\left(1+\frac{{I}_{ph}-I}{{I}_{o}}\right)\left({I}_{o}+{I}_{ph}\right)=0\end{array}$ (22)

In the above Equation (22),

$C=\frac{nNk{T}_{c}}{q}$ (23)

The limit development near of $I=0$ , in one order for $\mathrm{ln}\left(1+\frac{{I}_{ph}-I}{{I}_{o}}\right)$ is

given by:

$\mathrm{ln}\left(1+\frac{{I}_{ph}-I}{{I}_{o}}\right)=\mathrm{ln}\left(1+\frac{{I}_{ph}}{{I}_{o}}\right)-\frac{I}{{I}_{ph}+{I}_{o}}$ (24)

By replacing Equation (24) into Equation (22), one can have:

$\begin{array}{l}2{R}_{s}{I}^{2}-\left[C\mathrm{ln}\left(1+\frac{{I}_{ph}}{{I}_{o}}\right)-\frac{C{I}_{o}}{{I}_{ph}+{I}_{o}}+C+2{R}_{s}\left({I}_{o}+{I}_{ph}\right)\right]\cdot I\\ \text{\hspace{0.05em}}+C\mathrm{ln}\left(1+\frac{{I}_{ph}}{{I}_{o}}\right)\left({I}_{o}+{I}_{ph}\right)-\frac{CI\left({I}_{o}+{I}_{ph}\right)}{{I}_{ph}+{I}_{o}}=0\end{array}$ (25)

By rearranging Equation (25), one can obtain the equation of the second degree (26) below:

$\begin{array}{l}\left(C+2{R}_{s}\left({I}_{o}+{I}_{ph}\right)\right){I}^{2}-\left({I}_{o}+{I}_{ph}\right)\left[C\mathrm{ln}\left(1+\frac{{I}_{ph}}{{I}_{o}}\right)+2C+2{R}_{s}\left({I}_{o}+{I}_{ph}\right)\right]I\\ \text{\hspace{0.05em}}+C\mathrm{ln}\left(1+\frac{{I}_{ph}}{{I}_{o}}\right){\left({I}_{o}+{I}_{ph}\right)}^{2}\end{array}$ (26)

To solve this equation of the second degree let’s put:

${X}_{1}=\left(C+2{R}_{s}\left({I}_{o}+{I}_{ph}\right)\right)$ (27)

${X}_{2}=\left({I}_{o}+{I}_{ph}\right)\left[C\mathrm{ln}\left(1+\frac{{I}_{ph}}{{I}_{o}}\right)+2C+2{R}_{s}\left({I}_{o}+{I}_{ph}\right)\right]$ (28)

${X}_{3}=C\mathrm{ln}\left(1+\frac{{I}_{ph}}{{I}_{o}}\right){\left({I}_{o}+{I}_{ph}\right)}^{2}$ (29)

The Equation (26) could be rewritten as follows:

${X}_{1}{I}^{2}+{X}_{2}I+{X}_{3}=0$ (30)

The resolution of Equation (30) permits to obtain the following solutions:

${I}_{\mathrm{max}}=\frac{-{X}_{2}\pm \sqrt{{X}_{2}^{2}-4{X}_{1}{X}_{3}}}{2{X}_{1}}$ (31)

${V}_{\mathrm{max}}=\frac{nNK{T}_{c}}{q}\mathrm{ln}\left(1+\frac{{I}_{ph}-{I}_{\mathrm{max}}}{{I}_{o}}\right)\cdot {R}_{s}{I}_{\mathrm{max}}$ (32)

${P}_{\mathrm{max}}={V}_{\mathrm{max}}\cdot {I}_{\mathrm{max}}$ (33)

${I}_{\mathrm{max}}$ , ${V}_{\mathrm{max}}$ and ${P}_{\mathrm{max}}$ are respectively the maximum current, the maximum voltage and the maximum power.

3. Results and Discussion

The accuracy of the modeling methods described in this work is validated by experimental data published by the manufacturers of the selected PV modules. Four modules of different technologies are used for the verification. These include: the BYD model (BYD 320P6C-36), the Atersa Grupo model (A-320P GSE), the SunPower model (E19-320) and the model operated in the 50 MW Nouakchott power station (15.983˚W, 18.1553˚N) in Mauritania (JKM320PP-72-V) of JinkoSolar. The experimental data (I, V) are extracted from the data sheets [22] [23] [24] [25] for the different technologies studied. Table 2 presents the results of these unknown parameters of these different commercial technologies under standard conditions (T = 25˚C and G = 1000 W/m^{2}).

The values of the parameters calculated using the method proposed in this work are compatible with the literature [4] [5] [6] [7] [8] .

The determination of these parameters in parallel with the exploitation of the

Table 2. Unknown parameters of STC modules (T = 25˚C and G = 1000 W/m^{2}).

proposed method made it possible to obtain the optimal parameters of various technologies which are: the open circuit voltage, the maximum power, the short-circuit current. The results of these parameters are shown in Table 3.

Figures 2-9 show the I-V and P-V curves for the different photovoltaic module technologies used under the standard test conditions (T = 25˚C and G = 1000 W/m^{2}).

Table 3. Optimal parameters of STC modules (T = 25˚C and G = 1000 W/m^{2}).

Figure 2. Characteristic I(V) for BYD 320P6C-36 at T = 25˚C and G = 1000 W/m^{2}. A comparison between the experimental values and the calculated values.

Figure 3. Characteristic P(V) for BYD 320P6C-36 at T = 25˚C and G = 1000 W/m^{2}. A comparison between the experimental values and the calculated values.

Figure 4. Characteristic I(V) for A-320P GSE at T = 25˚C and G = 1000 W/m^{2}. A comparison between the experimental values and the calculated values.

Figure 5. Characteristic P(V) for A-320P GSE at T = 25˚C and G = 1000 W/m^{2}. A comparison between the experimental values and the calculated values.

Figure 6. Characteristic I(V) for E19-320 at T = 25˚C and G = 1000 W/m^{2}. A comparison between the experimental values and the calculated values.

Figure 7. Characteristic P(V) for E19-320 at T = 25˚C and G = 1000 W/m^{2}. A comparison between the experimental values and the calculated values.

Figure 8. Characteristic I(V) for JKM320PP-72-V at T = 25˚C and G = 1000 W/m^{2}. A comparison between the experimental values and the calculated values.

Figure 9. Characteristic P(V) for JKM320PP-72-V at T = 25˚C and G = 1000 W/m^{2}. A comparison between the experimental values and the calculated values.

These figures show a consistency between the experimental results and the expected results. We note that the calculated values are in good agreement with the experimental values provided by the manufacturers.

In order to quantify the quality of the modeling procedure for the I-V characteristics of different PV module technologies, the performance parameter is used to compare the values simulated by the method and the values given by the manufacturers of different technologies. This parameter is the average absolute relative error. It is defined as following:

${E}_{x}=\frac{{x}_{i}-{x}_{mi}}{{x}_{mi}}\cdot 100$ (36)

${x}_{i}$ and ${x}_{mi}$ are the theoretical value given by the method and the measured value given by the manufacturer, respectively.

The results of comparisons between the simulations and the manufacturer's data using four photovoltaic module technologies under standard test conditions (T = 25˚C and G = 1000 W/m^{2}) are shown in Table 4.

The results obtained prove the precision of the modeling method with an average absolute relative error between the estimated power and the measured power is less than 0.035% and that BYD 320P6C-36 technology is the most efficient among the four different PV module technologies studied with the average absolute relative error for the maximum point current is 0.46%, 0.51% for the maximum point voltage and 0.021% for the maximum point power.

Table 4. Mean absolute relative error between simulated values and values provided by the manufacturers.

4. Conclusion

This paper focuses on the characterization and modeling of various commercial solar photovoltaic module technologies most used in Mauritania through an analytical modeling method to describe its behavior under conditions of use in the Sahel. The modeling data of these different technologies were taken from the data sheets of different manufacturers. Four types of technologies, namely: BYD 320P6C-36, A-320P GSE, E19-320 and JKM320PP-72-V were studied and compared according to the maximum power current, the maximum power voltage and the power maximum. The results obtained prove the precision of the modeling method with an average absolute relative error between the estimated power and the measured power is less than 0.035%. The comparison results of these different technologies show that BYD 320P6C-36 technology is the most efficient among the four different PV module technologies studied with the average absolute relative error for maximum point current is 0.46%, 0, 51% for the maximum point voltage and 0.021% for the maximum point power.

Acknowledgements

The authors express their gratitude to the African Center of Excellence in Mathematics, Computer Science and ICT (CEA-MITIC) for their financial grant.

Nomenclature

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