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Nowadays in laboratories and in industries, distribution of solar cells sizes could be very large, hence; for the first time it is rather difficult to compare photovoltaic parameters which are size dependent (current, serial resistance, shunt resistance...) and performances of these cells. Also, it will be useful for scale effect to extrapolate performances calculated on a unit size solar cell to commercial products, especially in the case of heterogeneous wafers used for the device.

In this work, we have focused our model on standard multicrystalline p-type silicon solar cells; the standard solar cell was produced by the most industries in 2011. Because of better radiations hardness and the primary spatial domain applications, p-type bulk silicon is the first studied and produced solar cell [

We have both used PC1D [

Single diode model for illuminated solar cell could be used here [_{sh}), the lumped resistance (r_{s}).

In this model, the output (I-V) characteristic is given by the Equation (1)

I = I p h − I D − I R s h = I p h − I o [ exp ( q ( V + R s ⋅ I ) a k T ) − 1 ] − V + R s ⋅ I R s h (1)

Equation (1): (I-V) analytical expression for one exponential model solar cell

The difficulty in equation 1is that two kinds of parameters coexists, intensive parameter such as voltage (V), temperature (T),… and extensive ones such as current (I), resistances (R_{s} and R_{sh}_{)}. If we want a quantitative comparison between solar cells with different surface areas (S), it is necessary to give the current density, J, versus voltage, V, equation. This expression is simply derived from (1) with the assumption:

J = I S (2)

J = J p h − J D − J R s h = J p h − J o [ exp ( q ( V + R s ⋅ S ⋅ J ) a k T ) − 1 ] − V + R s ⋅ S ⋅ J R s h ⋅ S (3)

We can now introduce a normalized area solar cell (S = 1 cm^{2} for instance) with resistance specifications given in Ω∙cm^{2} units.

r s = R s ⋅ S (4)

and

r s h = R s h ⋅ S (5)

J = J p h − J D − J R s h = J p h − J o [ exp ( q ( V + r s ⋅ J ) a k T ) − 1 ] − V + r s ⋅ J r s h (6)

It is now obvious that for a given solar cell (material and process) the serial and shunt resistance decrease if the area increases. To highlight this, for two solar cells with S and S' area respectively and if S' > S, Equations (4)-(6) imply that:

r s = R s ⋅ S = R ′ s ⋅ S ′ (7)

and

r s h = R s h ⋅ S = R ′ s h ⋅ S ′ (8)

For instance, if S ′ = 2 ⋅ S ,

R ′ s = R s 2 (9)

and

R ′ s h = R s h 2 (10)

Best values for parameters used in Equation (6)

To estimate J_{o}, J_{ph}, a, r_{s} and r_{sh} values, we have made some statistical measurements on elaborated solar cells characterized under standard one sun test condition (AM1, 5G) (

Thanks to this model, it is relatively easy to extrapolate photovoltaic parameters computed for a unit size solar cell to another size, just by calculating R_{s} and R_{sh} for the size S. The formula used in this case is:

For size S:

R s = r s S (11)

and

R s h = r s h S (12)

In order to verify this assumption, PC1D and SPICE codes are used, we have modelized a unit solar cell with parameters generally used in conventional p- type silicon solar cells.

Concentrated photovoltaic is very attractive due to the fact that for the same solar cell size, the photovoltaic properties are increased with X, when the device is illuminated under X sun.

Obscurity current density (mA/cm^{2}) | Diode ideality factor | Serial resistance (Ω) | Shunt resistance (Ω) | Photocurrent density (mA/cm^{2}) (one sun) |
---|---|---|---|---|

10^{−}^{7} | 1.2 | 1.5 | 1500 | 33 |

Area (cm^{2}) | V_{oc} (mV) | J_{sc} (mA/cm^{2}) | I_{sc}(mA) | Efficiency (%) | R_{s} (ohm) | R_{sh} (ohm) |
---|---|---|---|---|---|---|

1 | 602.2 | 32.8 | 32.8 | 14.80 | 1.5 | 1500 |

10 | 602.2 | 32.8 | 328 | 14.78 | 0.15 | 150 |

100 | 602.2 | 32.8 | 3280 | 14.77 | 0.015 | 15 |

A current question is whether conventional solar cells can be used for low concentration applications, typically X < 10.

With our model, it is evident that it is not possible, because of the high serial resistance value.

In fact, _{s} = 1.5 Ω∙cm^{2}), Jsc increases linearly if X < 10 and saturates after this value, but and most importantly, there is a decrease in conversion efficiency.

If you want to optimise this possible utilisation for conventional solar cells, the r_{s} value must be divided by a factor of ten (r_{s} = 0.15 Ω∙cm^{2}), which is not easy with today’s technology.

In a solar cell, global performances are largely relative to local ones, specifically in heterogeneous material such as multi-crystalline silicon.

To evaluate local properties it is useful to divide a large size into N unit solar cells, in

Application 1: Homogeneous solar cell

For this case each unit solar cells are identical (

Evidently, the performances for the final solar cell are conserved, but this ideal case could be attributed to an homogeneous solar cell, in terms of materiel, for instance monocrystalline wafer and uniform processes.

Generally, especially for multicrystalline ingots, heterogeneous properties govern final solar cell performances. In a rapid approach we can evaluate the effects of local defects at the scale of individual USC on global performance, such as local shunt (r_{sh}), local bad electrical contact (r_{s}) and local bad photon collection or conversion (J_{ph})

Now, to evaluate the impact of unit solar cells on the final device, we have chosen to see the influence of r_{s}, r_{sh} and J_{ph} after a 50% variation of their initial value.

Application 2: Serial resistance discrepancy

In the electrical model

In this case, illustrated by Figures 5(a)-(c), we have just increased the rs value for one cell, this modified value is equal to 2.25 ohms, which represents also 50% variation. We can see in

Application 3: Shunt resistance discrepancy

For this case, we have taken into account a 50% decrease of the shunt resistance in the unit solar cell (Figures 6(a)-(c)). Especially in multicristalline wafers, shunt leakage must be attributed to crystallographic defects decorated with metallic impurities crossing the junction. During the solar cell fabrication, shunt resistance degradation must appear at the aperture of the junction process.

We can see that for this local variation, the total properties of the cell are not greatly affected, but evidently if the shunt leakage increases the impact will be more important.

Application 4: Local photocurrent generation discrepancy

This case must be attributed to an accidental partial shadowing of the cell, or more problematically to a local area with poor electronic quality due to the presence of electrical recombining impurities for instance.

We have also kept with the same total variation (50%) of the unit J_{sc} value, 17.5 mA/cm^{2} instead of 35 mA/cm^{2}, and the results are described in Figures 7(a)-(c).

It is obvious that this situation is the worst case, because of the degradation of all photovoltaic parameters (V_{oc}, J_{sc} and η).

We have proposed a method seeking to explain properties of solar cell by local analysis thanks to a normalized area solar cell concept. We studied here the influence of three macroscopic parameters (r_{s}, r_{sh} and J_{sc}); these parameters depend at the same time on implemented materials and on the solar cell technology. A principal result is that the contribution of local photo-current density (J_{sc}) is the most important.

Most of actual solar cell technologies are based on multicrystalline materials (Si, CIGS, CdTe,) and for each category as we have done for silicon, parameters for each unit solar cell must be defined.

As we have described in this article, several applications could be concerned with this analysis and for further studies, it will be such interesting to implement and compare this model with local photoelectrical characterizations as Light Beam Induced Current (LBIC), Photoluminescence and Electroluminescence Imaging.

Pasquinelli, M., Simon, J.-J., Le Rouzo, J., Flory, F. and Escoubas, L. (2017) Normalized Area Solar Cell and Potential Applications. Journal of Applied Mathematics and Physics, 5, 1106- 1112. https://doi.org/10.4236/jamp.2017.55097