The desert aerosols are soil particles suspended in the atmosphere in regions with easily degradable arid soils, sparse vegetation and strong winds   . These aerosols represent one of the most important features in terms of mass and optical depth and can have a significant impact on solar radiation during high dust cloud events or even in the monthly and annual average   . Burkina Faso, as a Sahelian country, is swept each year by sheets of migratory dust with an average optical depth of 0.8 µm, coming from the foothills of the Saharan mountains. Therefore, visible solar radiation is disturbed when passing through the Earth’s atmosphere and in contact with a PV field. The composition of the atmosphere and the state of the central solar PV field can be considered as time and scale variables. Rigorous modeling of the radiance received by a solar field would require knowing a priori all the parameters τa and X related to the state of the atmosphere and to the surface of the PV field. Many experimental and numerical works had been carried out on the subject. It is known, for example, that dust can cause attenuation of solar PV radiation of around 90%  . In contrast, little is known about the impact of desert aerosols on solar radiation received by solar PV systems. The purpose in this research is therefore to propose in this article the ideal model to study the impact of desert aerosols on the global solar radiation of a mini central solar PV of 40 kWp in Ouagadougou. We consider this mini central solar photovoltaic with the desert aerosol deposits and the atmosphere laden with desert aerosols as being a multilayer to model  .
2. Description of the Physical Model
The physical model studied is represented in Figure 1. The mini central solar PV is assimilated to a multilayer.
The physical model studied includes the optical thickness of desert aerosols in the upper atmosphere (τa), the thickness of the various aerosol deposits (X) and finally a glass surface representing the solar PV field (S = 234 m2). The principle consists in observing τa and X on the electrical production of the mini central solar PV. Thus, the optical properties for each layer are shown in Figure 2.
The solar radiation which encounters aerosol particles undergoes transformations either by absorption, diffusion and emission  - . For this, we consider the graph of Figure 2 of the optical model studied, which characterizes the dusty atmosphere (layer 1) in contact with dust deposits (layer 2) on the glazing (layer 3) of the PV solar field. The layer 3 is doesn’t take account in this numerical studies.
Thus, Figure 2 of the optical model shows that the solar intensity I0 passes through layer 1 at an incident angle θ, of optical air mass ma and optical depth of aerosols τa. Layer 1 and 2 respectively represent the suspension of desert aerosols in the atmosphere    and the deposition of desert dust whose deposition
Figure 1. Studied physical model.
Figure 2. Optical poperties of the studied physical model.
thickness on the PV solar field is X. Layer 2 is crossed by a solar intensity I0 at an incident angle θ.
3. Mathematical Formulation of the Studied Physical Model and Numerical Method Resolution
We have established the mathematical equations that govern our studied physical model. Thus, the clear sky radiation reaching the surface of the Earth (normal to the rays) is given by a commonly used model by treating the attenuation as an exponential decay function. The transmittance due to this multilayer is given as follows:
Knowing the transmisttance (Equation (1)) caused by the suspension and deposits of desert aerosols, direct radiation can be written:
The final equation for direct solar radiation in a multilayer is:
with the following coefficients:
The expression of diffuse solar radiation is:
The expression of reflected solar radiation is:
Finally, the global solar radiation is:
The Equations (3)-(6) are solved using the Finish Differences Method (FDM). The discretization of the equations is:
Direct solar radiation:
Diffuse solar radiation:
Reflected solar radiation:
Global solar radiation:
4. Results and Discussions
We present the numerical results of the code developed on MATLAB  . Thus, the colored bars representing the colored curves in Figures 3-5 shows the
Figure 3. Variation of the solar radiation incident.
Figure 4. Variation of solar radiation: (a) direct radiation; (b) diffuse radiation and (c) reflected radiation.
Figure 5. Variation of global solar radiation.
evolution of solar radiation (incident, direct, diffuse reflected and global) on the solar PV field. When the solar radiation is about 1400 W/m2 on the solar field, we contact the color of the curves pulls dark red and when it is weak it tends towards 0, the color of the curves pulls towards dark blue.
4.1. Solar Incident Radiation
The curves of Figure 3 represent the variation of the incident radiation. We find that the daily extraterrestrial solar radiation (I0) is a function of the incident solar radiation and the angle of incidence (θ). For , the incident solar radiation is equal to the extraterrestrial solar radiation. The incident solar radiation decreases as the angle of incidence increases and at the incident radiation is 0.
4.2. Direct, Diffuse and Reflected Solar Radiation
We present the variations of direct, diffuse and reflected solar radiation on the solar PV field. The curves of Figures 4(a)-(c) show that they depend on the parameters τa and X. We find that the solar radiation decreases when the main parameters increase by attenuating the extraterrestrial solar radiation. Indeed, if and this corresponds to a clear blue sky, the maximum direct solar radiation is 1000 W/m2. When the parameters and , the direct solar radiation (IBC) decreases and tends towards 0 (Figure 4(a)). The diffuse solar radiation is estimated at a maximum value of 120 W/m2 when and . If and we contact that the diffuse solar radiation (IDC) decreases to 0 (Figure 4(b)). Concerning the solar radiation reflected from the solar PV field (IRC), if , , we notice that the reflected solar radiation varies and reaches a value of 140 W/m2. The reflected solar radiation tends towards 0 if and (Figure 4(c)).
4.3. Global Solar Radiation
Figure 5 curves show the evolution over time of global solar radiation on the surface of the solar field. For the following parameters: if and , with a cloudless sky; it can be seen that the maximum global solar radiation of the field surface is 1260 W/m2.
4.4. Influence of τa and X Thickness Constants Parameters on Global Solar Radiation and Field Solar PV
The curves in Figures 6(a)-(d) represent the variations of the global solar radiation on the solstices and equinoxes as well as the days 21 of the months of the year, under the influence of the parameters τa and X. By varying τa and X with a thickness of: ( , ; , ; , and , ), we note that the global solar radiation (IC) drops considerably from an average value of 935 W/m2: for , (Figure 6(a)) to an average value 4 W/m2 at , (Figure 6(d)). Also, we contact that τa and X have a negative impact on the global solar radiation of the mini central solar PV.
In summary, for the two parameters (τa and X) of our study, we give the results of the different solar radiations which cross the surface of the solar PV field. Based on the geographic data of the mini central solar PV site. We have an average daily solar time of 12.25 hours or an annual solar time of 4476 hours. The Table 1 shows the values of solar radiation received by the mini central solar PV at .
Table 1. Maximum solar radiations of the central solar PV.
Figure 6. Influence of τa and X on global solar radiation of mini central solar PV.
Finally, the quantitative study shows that out of 100% (1400 W/m2) of extraterrestrial solar radiation in the direction of the PV solar field, only 71% of this radiation is converted into direct solar radiation (IBC), 10% into solar radiation reflected (IRC), 9% in diffuse solar radiation (IDC) and 10% of solar radiation are considered lost. Thus, global solar radiation is 90% of extraterrestrial solar radiation. According to a variable angle of incidence (θ) on the solstices and equinoxes of the year, the average global solar radiation on the solar PV field is: if and , ; if and , ; if and , and if and then
At the end of our study on the influence of desert aerosols on the mini central solar PV, we described the mathematical equations which govern the system of the studied physical model. Then, these equations are discretized using a Finite Difference Method and solved by the Gaussian algorithm. The numerical results show that if and , with a cloudless sky; we have a maximum global solar radiation at the surface of the PV solar field of 1260 W/m2. This value of global solar radiation indicates a very significant improvement in the productivity of the mini central solar. And if and , the global solar radiation tends towards very low values (<10 W/m2). This shows a low productivity of the mini central solar PV. Finally, the parameters and X considerably influence the operation of the mini central solar PV. We showed that our numerical results on the desert aerosols can be to influence the central solar PV productivity in West Africa countries. From a theoretical point of view, this modeling research has enabled us to establish a new model for evaluating the impacts of desert aerosols on the radiation received by a solar PV system. It is a true model that solves all cases of issues that deal with mineral dust deposits and suspensions on solar PV systems. Our numerical method consisted of the desert aerosols on global solar radiation, another digital method could have consisted of showing the impact of gaseous aerosols on solar radiation.
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