# Estimating Solar Irradiation Absorbed by Photovoltaic Panels with Low Concentration Located in Craiova, Romania

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Modeling and Simulation of Solar Irradiation

#### 2.1. Extraterrestrial Irradiation

- S is the solar constant;
- n is the days’ number of the year;
- φ is the latitude of the considered location;
- δ represents the declination of the Earth;
- and ω
_{s}is the solar angle.

#### Influence of atmosphere

_{z}is equal to 60°, then the air mass m = 2, i.e., the solar ray will travel a path through atmosphere 2 times higher than for θ

_{z}= 0°. In the second situation the solar ray will be more attenuated and will transport less energy. This way could be explained the decrease of the solar radiation intensity in the north and south hemispheres, respectively, in comparison with the equatorial zone.

#### 2.2. Terrestrial Irradiation

**Figure 2.**Parameters describing the sun position in sky roof, according to [14].

_{s}= 0 is imposed, and one could calculate the horary angles of sunrise and sunset, respectively, by using the relationship:

_{s}might also be determined.

**Figure 3.**Components of solar irradiation according to [13].

- Adnot model, which models global solar irradiation under conditions of a clear sky, by using the relationship [9]:$${G}_{g}=951.39{(\mathrm{sin}{\text{\alpha}}_{s})}^{1.15}\left[{\text{W/m}}^{2}\right]$$

- Haurwitz model [9].$${G}_{g}=1098\cdot {e}^{\frac{0.057}{\mathrm{sin}{\text{\alpha}}_{s}}}\cdot \mathrm{sin}{\text{\alpha}}_{s}\left[{\text{W/m}}^{2}\right]$$
- Kasten model [16]$${G}_{g}=910\cdot \mathrm{sin}{\text{\alpha}}_{s}-30\left[{\text{W/m}}^{2}\right]$$
- Empirical model (EIM) [17]

**Figure 4.**Chart of global solar irradiation, in conditions of a clear sky, on June 21 for the location Craiova.

**Figure 5.**Chart of global solar irradiation, in conditions of a clear sky, on December 21 for the location Craiova.

#### 2.3. Solar Irradiation Absorbed by a Photovoltaic Panel without Concentration

_{z}= β, and from the relationship (4) we obtain:

_{D}, on the horizontal plane (a), and G

_{Dβ}, on a plane inclined to the horizontal with the angle β, (b) according to [14].

_{Dn}, in order to determine the ratio between G

_{D}and G

_{D}

_{β}. Hence, the ratio between the direct radiation on an inclined plane and on a horizontal plane is denoted by ${R}_{G}={G}_{D\text{\beta}}/{G}_{D}$.

_{G}:

_{z}= θ (see Figure 7a).

_{D}is the direct radiation on a horizontal plane calculated on basis of one of the previous patterns.

- ρ is the reflection coefficient of the Earth surface; and
- Gg is the global radiation on a horizontal surface.

**Figure 8.**Solar irradiation absorbed by a PV panel located in Craiova city on 21 June nm (

**a**) simulation graphs’ results; (

**b**) literature graphs according to [18].

**Figure 9.**Solar irradiation absorbed by a PV panel located in Craiova city on December 21 (

**a**) simulation graphs’ results; (

**b**) literature graphs according to [18].

Hour | June | December | ||||||
---|---|---|---|---|---|---|---|---|

${G}_{g\text{\beta}}$ Calculation (W/m^{2}) | ${G}_{g\text{\beta}}$ Literature (W/m^{2}) | Relative Error (%) | Medium Error (%) | ${G}_{g\text{\beta}}$ Calculation (W/m^{2}) | ${G}_{g\text{\beta}}$ Literature (W/m^{2}) | Relative Error (%) | Medium Error (%) | |

5 | 0 | 0 | 0 | 3.84 | 0 | 0 | 0 | 3.16 |

6 | 73.5 | 100 | −26.5 | 0 | 0 | 0 | ||

7 | 218.2 | 250 | −12.72 | 0 | 0 | 0 | ||

8 | 377.5 | 400 | −5.625 | 40.5 | 55 | −26.3636 | ||

9 | 535 | 540 | −0.92593 | 138 | 158 | −12.6582 | ||

10 | 675 | 660 | 2.272727 | 230 | 220 | 4.545455 | ||

11 | 785.5 | 730 | 7.60274 | 290 | 260 | 11.53846 | ||

12 | 856 | 750 | 14.13333 | 311 | 280 | 11.07143 | ||

13 | 785.5 | 730 | 7.60274 | 290 | 260 | 11.53846 | ||

14 | 675 | 660 | 2.272727 | 230 | 220 | 4.545455 | ||

15 | 535 | 540 | −0.92593 | 138 | 158 | −12.6582 | ||

16 | 377.5 | 400 | −5.625 | 40.5 | 55 | −26.3636 | ||

17 | 218.2 | 250 | −12.72 | 0 | 0 | 0 | ||

18 | 73.5 | 100 | −26.5 | 0 | 0 | 0 | ||

19 | 0 | 0 | 0 | 0 | 0 | 0 |

## 3. Mathematical Model of Solar Irradiation Absorbed by Photovoltaic Panel with Low Concentration

**Figure 10.**WS Heliots system, with DoubleSun technology, according to [22].

- -
- x is the angle between the mirror and the photovoltaic module, and is a constant parameter;
- -
- h is the maximum incidence angle created by the solar ray with the normal to the photovoltaic arrays;
- -
- k represents the ratio between the mirrors width (Lm) and the photovoltaic module width (Lp);
- -
- c
_{1}, c_{2}are the width coefficients of PV module brushed by the rays reflected by the mirror; - -
- k
_{l}is the longitudinal deviation coefficient, defined as the ratio between the additional length of the mirror (necessary for the compensation of the deviation of solar rays reflected, caused by the elevation deviations of the PV module from the solar elevation) and the photovoltaic module width.

_{M1,2}is the irradiation reflected by the two mirrors

_{R1,2}is the incidence angle formed between the ray reflected by each mirror and the normal to the photovoltaic panel.

_{1}= c

_{2}= 1.35; k

_{l}= 0.2.

## 4. Experimental Results

**Figure 15.**Experimental equipment: (1) LCPV panel; (2) PV panel without concentration, (3) pyranometer; (4) PC; (5) solar lamp; (6) data logger.

^{2}.

^{2}.

_{1}= c

_{2}= 1.35; k

_{l}= 0.2.

^{2}, while that measured on the plane of the photovoltaic panel with low concentration for mirrors tilt x = 55° (see Figure 17) has values roughly of 1280 W/m

^{2}. For the mirrors’ tilt at x = 65° the solar irradiation has the value 1400 W/m

^{2}(see Figure 18).

**Figure 16.**Chart of solar irradiation measured on the plane of the photovoltaic panel without concentration.

**Figure 17.**Chart of solar irradiation measured on the plane of the photovoltaic panel with low concentration for mirrors tilt x = 55°.

**Figure 18.**Chart of solar irradiation measured on the plane of the photovoltaic panel with low concentration for mirrors tilt x = 65°.

## 5. Conclusions

- -
- the models Adnot and Empiric for the global irradiation assessment can be adapted in the best way for the conditions of the location chosen for this study;
- -
- the main advantage of these models is their simplicity (these models have as input parameters only the location and time marks), which makes them easy to use in practice;
- -
- the values of solar irradiations obtained through the simulations are close enough to those indicated on the maps elaborated by professional institutes for the site took into study;
- -
- during travel through the atmosphere the solar rays decrease intensity as they are approaching the Earth;
- -
- the values of the solar irradiation on the PV panel plane are roughly two times higher than those on the PV panel plane without a concentration system.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Alboteanu, I.L.; Bulucea, C.A.; Degeratu, S.
Estimating Solar Irradiation Absorbed by Photovoltaic Panels with Low Concentration Located in Craiova, Romania. *Sustainability* **2015**, *7*, 2644-2661.
https://doi.org/10.3390/su7032644

**AMA Style**

Alboteanu IL, Bulucea CA, Degeratu S.
Estimating Solar Irradiation Absorbed by Photovoltaic Panels with Low Concentration Located in Craiova, Romania. *Sustainability*. 2015; 7(3):2644-2661.
https://doi.org/10.3390/su7032644

**Chicago/Turabian Style**

Alboteanu, Ionel L., Cornelia A. Bulucea, and Sonia Degeratu.
2015. "Estimating Solar Irradiation Absorbed by Photovoltaic Panels with Low Concentration Located in Craiova, Romania" *Sustainability* 7, no. 3: 2644-2661.
https://doi.org/10.3390/su7032644