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This work presents a method of optimization of the photovoltaic generator (PV) based on the electrical model with a diode. The method consists of solving a second degree equation representing the derivative of the power function. The current and the maximum voltage being determined, the maximum power is deduced. Four popular types of photovoltaic panels from different manufacturers were considered for the study: BYD Model (BYD 320P6C-36), Atersa Grupo Model (A-320P GSE), SunPower Model (E19-320) and Model operated in the 50 MW power plant of Nouakchott-Mauritania (JKM320PP-72-V) of JinkoSolar. A comparative study is carried out between the simulated results and the data of the manufacturer of different technologies. The results obtained prove the effectiveness of the proposed method and that the BYD 320P6C-36 model is the most efficient among the four different technologies studied.

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 [

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 [^{2} and T = 25˚C) according to the manufacturers of each technology.

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

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

I = I p h − I D − I s h (1)

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

I D = I 0 ⋅ ( e ( q ( V + I R s ) n N k T c ) − 1 ) (2)

While the shunt current is given by the relation:

Manufacturer | BYD [ | Atersa Grupo [ | SunPower [ | JinkoSolar [ |
---|---|---|---|---|

Model | BYD 320P6C-36 | A-320P GSE | E19-320 | JKM320PP-72-V |

Peak power P m (Wc) | 320 | 320 | 320 | 320 |

Power tolerance (%) | 0 - 5% | ±1.5% | +5/−0% | ±3% |

Maximum power voltage V m p (V) | 36.78 | 37 | 54.7 | 37.4 |

Maximum power current I m p (A) | 8.7 | 8.65 | 5.86 | 8.56 |

Open circuit voltage V c o (V) | 46.39 | 45.5 | 64.8 | 46.4 |

Short circuit current I s c (A) | 9.15 | 9.17 | 6.24 | 9.05 |

Module efficiency (%) | 16.5 | 16.49 | 19.8 | 16.49 |

Temperature coefficients of V c o μ V o c (%/˚C) | −0.31 | −0.33 | −0.176 | −0.30 |

Temperature coefficients of P m μ P m (%/˚C) | −0.39 | −0.43 | −0.38 | −0.40 |

Temperature coefficients of I s c μ I s c (%/˚C) | 0.07 | 0.05 | 0.035 | 0.06 |

NOCT (Nominal operating cell temperature) (˚C) | 45 ± 2 | 45 ± 2 | 45 ± 2 | 45 ± 2 |

Number of cells | 72 | 72 | 96 | 72 |

Area (m^{2}) | 1.94 | 1.94 | 1.63 | 1.94 |

I s h = ( V + I R s ) R s h (3)

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

I = I p h − I 0 ⋅ ( e ( q ( V + I R s ) n N k T c ) − 1 ) − ( V + I R s ) R s h (4)

If we assume that the parallel resistance R s h is very large (case of crystalline silicon) [

For this purpose Equation (4) becomes as follows:

I = I p h − I 0 ⋅ ( e ( q ( V + I R s ) n N k T c ) − 1 ) (5)

Determination of the PV Generator Parameters1) Evaluation of I_{ph}

The light current I p h depends on both irradiance and temperature. It is given by [

I p h = [ I s c + α ( T c − T r ) ] ⋅ ( G G r ) (6)

2) Evaluation of I_{0}

The reverse saturation current depending of cells temperature is given as follows [

I 0 = I o n ⋅ ( T c T r ) 3 ⋅ e [ q E g ( 1 T r − 1 T c ) n k ] (7)

At the open circuit voltage I = 0 , V = V o c and I = 0 , V = V o c

0 = I s c − I 0 n ⋅ ( e ( q V o c n N k T c ) − 1 ) (8)

So the nominal saturation current is obtained through:

I 0 n = I s c e ( q V o c n k N T c ) − 1 (9)

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

I 0 = I s c e ( q V o c n k N T c ) − 1 ⋅ ( T c T r ) 3 ⋅ e [ q E g ( 1 T r − 1 T c ) n k ] (10)

3) Evaluation of R_{s}

Various techniques have been used to determine the series resistance R s [

R s = ( − d V d I | V = V o c ) (11)

Considering the asymptotic behavior of the I-V curve under short-circuit and open-circuit conditions R s can be calculated as [

R s = N s n k T c q ⋅ ln ( 1 − I m p I s c ) + V o c − V m p I m p (12)

4) Evaluation of n:

At the short circuit point, I = I s c , V = 0 :

I s c = I p h , r e f − I 0 , r e f ⋅ ( e ( q R s I s c n N k T c ) − 1 ) (13)

At the maximum power point, I = I m p , V = V m p :

I m p = I p h , r e f − I 0 , r e f ⋅ ( e ( q ( V m p + I m p R s ) n N k T c ) − 1 ) (14)

The reverse saturation current Io for any diode is a very small quantity, on the order of 10^{−}^{5} or 10^{−6} A [

I s c ≈ I p h , r e f (15)

0 = I s c − I 0 , r e f ⋅ ( e ( q V o c n N k T r ) − 1 ) (16)

I m p = I p h , r e f − I 0 , r e f ⋅ e ( q ( V m p + I m p R s ) n N k T r ) (17)

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

n = q ( 2 V m p − V o c ) N k T r ( I m p I s c − I m p + ln ( I s c − I m p I s c ) ) (18)

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

V = n N K T c q ln ( 1 + I p h − I I o ) ⋅ R s I (19)

The electric power produced by the generator is given by:

P = V ⋅ I (20)

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

P = n N K T c q ln ( 1 + I p h − I I o ) ⋅ R s I 2 (21)

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

d P d I = 0

2 R s I 2 − [ C ln ( 1 + I p h − I I o ) + C + 2 R s ( I o + I p h ) ] ⋅ I + C ln ( 1 + I p h − I I o ) ( I o + I p h ) = 0 (22)

In the above Equation (22),

C = n N k T c q (23)

The limit development near of I = 0 , in one order for ln ( 1 + I p h − I I o ) is

given by:

ln ( 1 + I p h − I I o ) = ln ( 1 + I p h I o ) − I I p h + I o (24)

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

2 R s I 2 − [ C ln ( 1 + I p h I o ) − C I o I p h + I o + C + 2 R s ( I o + I p h ) ] ⋅ I + C ln ( 1 + I p h I o ) ( I o + I p h ) − C I ( I o + I p h ) I p h + I o = 0 (25)

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

( C + 2 R s ( I o + I p h ) ) I 2 − ( I o + I p h ) [ C ln ( 1 + I p h I o ) + 2 C + 2 R s ( I o + I p h ) ] I + C ln ( 1 + I p h I o ) ( I o + I p h ) 2 (26)

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

X 1 = ( C + 2 R s ( I o + I p h ) ) (27)

X 2 = ( I o + I p h ) [ C ln ( 1 + I p h I o ) + 2 C + 2 R s ( I o + I p h ) ] (28)

X 3 = C ln ( 1 + I p h I o ) ( I o + I p h ) 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 max = − X 2 ± X 2 2 − 4 X 1 X 3 2 X 1 (31)

V max = n N K T c q ln ( 1 + I p h − I max I o ) ⋅ R s I max (32)

P max = V max ⋅ I max (33)

I max , V max and P max are respectively the maximum current, the maximum voltage and the maximum power.

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 [^{2}).

The values of the parameters calculated using the method proposed in this work are compatible with the literature [

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

Parameters | BYD 320P6C-36 | A-320P GSE | E19-320 | JKM320PP-72-V |
---|---|---|---|---|

I p h ( A ) | 9.15 | 9.17 | 6.24 | 9.05 |

I o ( A ) | 7.2298 × 10 − 12 | 2.6219 × 10 − 9 | 6.7684 × 10 − 8 | 4.2692 × 10 − 10 |

R s ( Ω ) | 0.5282 | 0.2957 | 0.0361 | 0.3866 |

R s h ( Ω ) | +∞ | +∞ | +∞ | +∞ |

n | 0.9003 | 1.1197 | 1.4331 | 1.0553 |

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

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}).

Parameters | BYD 320P6C-36 | A-320P GSE | E19-320 | JKM320PP-72-V |
---|---|---|---|---|

I s c ( A ) | 9.15 | 9.17 | 6.24 | 9.05 |

V o c ( V ) | 46.39 | 45.5 | 64.8 | 46.4 |

P max ( W ) | 319.9150 | 319.9472 | 320.4132 | 320.0529 |

V max ( V ) | 36.5999 | 36.7724 | 54.3209 | 37.1864 |

I max ( A ) | 8.7409 | 8.7007 | 5.8985 | 8.6067 |

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 = x i − x m i x m i ⋅ 100 (36)

x i and x m i 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

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.

Parameters | I s c ( A ) | V o c ( V ) | I max ( A ) | V max ( V ) | P max ( W ) |
---|---|---|---|---|---|

BYD 320P6C-36 | |||||

Measured values | 9.15 | 46.39 | 8.7 | 36.78 | 319.98 |

Calculated values | 9.15 | 46.39 | 8.74 | 36.59 | 319.91 |

E x | 0 | 0 | 0.46 | 0.51 | 0.021 |

A-320P GSE | |||||

Measured values | 9.17 | 45.5 | 8.65 | 37 | 320.05 |

Calculated values | 9.17 | 45.5 | 8.7 | 36.77 | 319.94 |

E x | 0 | 0 | 0.57 | 0.62 | 0.034 |

E19-320 | |||||

Measured values | 6.24 | 64.8 | 5.86 | 54.7 | 320.524 |

Calculated values | 6.24 | 64.8 | 5.8985 | 54.3209 | 320.4132 |

E x | 0 | 0 | 0.6570 | 0.6931 | 0.0346 |

JKM320PP-72-V | |||||

Measured values | 9.05 | 46.4 | 8.56 | 37.4 | 320.144 |

Calculated values | 9.05 | 46.4 | 8.6067 | 37.1864 | 320.0529 |

E x | 0 | 0 | 0.5456 | 0.5711 | 0.0285 |

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.

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

Sidibba, A., Ndiaye, D., El Bah, M. and Bouhamady, S. (2018) Analytical Modeling and Determination of the Characteristic Parameters of the Different Commercial Technologies of Photovoltaic Modules. Journal of Power and Energy Engineering, 6, 14-27. https://doi.org/10.4236/jpee.2018.63002