In this work, we collect samples directly from PV surfaces in Dakar (14˚44'N latitude and 17˚27'28''W longitude). These samples are collected in vials as shown in Figure 1(b). The samples are then analyzed by X-ray fluorescence (EDXRF) to determine the different chemical elements. The instrument used to determine these elementary compositions is a Horiba Jobin Yvon XGT-5000. It is equipped with a tungsten X-ray tube, a beryllium window and a high-purity Si detector; it operates at 50 kV-max/1mA-max. Next, thin layers of dust are made on glass plates (see Figure 1(c)) with a spin coater according to the spin coating technique using an isopropanol solution as described by Lawrence et al. [26]. A digital microscope (Keyence, VHX1000) with a magnification of 100 - 1000 nm or 500 - 5000 nm is used to determine the thickness of dust layer. In addition, a SOPRA GES-5 type ellipsometer with a spectral ranging from 190 to 2500 nm is used to obtain real and imaginary dust indices (Pristinski et al. [27]). These optical indices obtained from the ellipsometer are used as input parameters for the proposed model. To ensure the reliability of the ellipsometer results, refractions index are compared with those measured by the AERONET sunphotometer stationed in Dakar (Holben et al. [28]). Indeed, the same particles measured by the sunphotometer were deposited on PV modules surface in Dakar. And finally, UV-Visible spectrophotometer is used to validate the model.

The approach consists of simulating an electromagnetic wave with oblique or normal incidence on dust layer deposited on glass as shown in Figure 2. Electromagnetic wave passes through a multilayer optical system (Air-Dust-Glass). This wave can be reflected, transmitted or partially absorbed at each interface by the different constituents of the layers. Air and glass are characterized by their refractive index: n₀ and n₂ respectively. The dust is characterized by its optical index (n₁) which varies as a function of wavelength according to Equations (1) and (2) described by Bashara and Azzam [29].

n 1 ( λ ) = n ( λ ) − i k ( λ ) (1)

k = λ 4 π α ( λ ) (2)

n, k, α and λ respectively represent the refractive index (real index) of dust, the dust extinction coefficient (imaginary index), the dust absorption coefficient and the solar radiation wavelength. Next, we calculate the Fresnel reflection and transmission coefficients at the layer interfaces for p and s polarization by applying electric E and magnetic B field conservation and Descartes’ formulas as described by Mahdjoub et al. [30], Berthier and Lafait [31]. We use the equations of Macleod [32], Born and Wolf [33] and McCrackin et al. [34]:

For the air-dust interface:

n 0 sin Θ 0 = n 1 sin Θ 1 (3)

And for the dust-glass interface the equation becomes:

n 1 sin Θ 1 = n 2 sin Θ 2 (4)

β = 2 π ⋅ n 1 cos Θ 1 ( d 1 λ ) (5)

where β is the phase difference introduced by the reflection and d₁ is the dust layer thickness. The overall reflection (r_p, r_s) and transmission (t_s, t_p) coefficients of the system for a p or s polarization are shown in the equations below:

r p = r 01 p + r 12 p exp ( − 2 i β ) 1 + r 01 p r 12 p exp ( − 2 i β ) (6)

r s = r 01 s + r 12 s exp ( − 2 i β ) 1 + r 01 s r 12 s exp ( − 2 i β ) (7)

t s = t 01 s t 12 s exp ( − i β ) 1 + r 01 s r 12 s exp ( − 2 i β ) (8)

t p = t 01 p t 12 p exp ( − i β ) 1 + r 01 p r 12 p exp ( − 2 i β ) (9)

T p = | t p t p | n 2 cos Θ 2 n 0 cos Θ 0 (10)

T s = | t s t s | n 2 cos Θ 2 n 0 cos Θ 0 (11)

T_s, T_p are Transmittances for s and p polarization.

R p = | r p r p | (12)

R s = | r s r s | (13)

R_s and R_p are reflectances for s and p polarization.

T = T s + T p 2 (14)

R = R s + R p 2 (15)

A = 100 − ( R + T ) (16)

T, R and A are the mean reflectance, mean transmittance and absorbance of the system, respectively.

Figure 3 shows the dust samples elemental analysis by X-ray fluorescence (XRF). It indicates that the dust is dominated by elements such as: Si, Fe, Ca, K, Ti, P, S, Zn, Mn, Zr. Some elements such as Cr, Cd, Br, Pb, Cu and Hg are minority or trace elements.

Figure 4 shows the detected element mass concentration in Dakar dust. It determines the mass proportions of each chemical element. The analysis reveals that this powder is mainly composed of Silicon (Si) with a rate of more than 50.1% of the total mass. The remainder consists of 16.8% of Fe, 10.6% of Ca, 1.5% of K, 1.4% of Ti, 0.98% of P and 0.5% of S. Trace elements such as Cu, Zn, Sr, Zr were also detected. However, X-ray fluorescence diffraction (XRF) does not detect very light elements such as aluminum.

These two materials are used as reference and validation elements to determine the concentrations of sand elements. We find the same chemical elements for these three samples, which proves that the dust collected in Dakar is of sandy nature. However, main elements such as Si, Ca, Fe, S and P are more important in mass in Saharan dust rather than the other samples (Figure 5).

In this section, we perform the model sensitivity studies and then assess the impact of dust on radiation components.

The first step is to simulate the influence of dust layers on solar radiation transmission (transmittance) through an air-dust-glass system. For this purpose, the model is executed with different layer sizes. Figure 8 shows the transmittance for an air-dust-glass system for different dust layer sizes. These values are

30 µm (corresponding to a dust deposit of 1.4 g/m²), 45 µm (corresponding to a dust deposit of 2.1 g/m²), 55 µm (corresponding to a dust deposit of 2.6 g/m²) and 70 µm (corresponding to a dust deposit of 3.3 g/m²) respectively.

The correspondence between layer size in µm and dust deposition in g/m² is made by considering a dust density about 2.6 g/cm³ according to Ansmann et al. [22]. The cyan curve in Figure 8 corresponds to the transmittance for clean glass (without dust), which is considered as absolute reference. Overall, a clean glass allows all the radiation to pass through with a transmittance of around 95%.

Next, Figure 8 shows that a dust deposit of 1.4 g/m² (corresponding to a layer size of 30 µm) causes 18% loss of transmittance. Then, we note that the transmittance decreases inversely with the layer thickness. Indeed, for thickness of 45, 55 and 70 µm, the mean transmittance is of 70%, 64.7% and 58% respectively. Overall, the results show that the transmittances decrease in short wavelengths. This is due to the Mie scattering of these particles (Xie et al. [35]). These results are in good agreement with those of Yun-Yun and Zhang [36]. However, these losses are greater than those found by Mastekbayeva and Kumar [37]. They showed that for a period of 30 days and for 3.72 g/m² of dust deposition, transmission losses were about 24.2%. These differences are due to the nature of the dusts considered. In summary, half of the radiation is lost when the dust layer thickness reaches 70 µm. This radiation is blocked by dust, as shown by Hegazy et al. [20].

Figure 9 shows the reflectance losses (of the air-dust-glass system) for different dust layers considered in the model. Indeed, this figure represents the reflectance as a function of wavelength for four dust layer thicknesses (30, 45, 55 and 70 µm). Overall, we note that the average reflectance increases from 26% to 53% if the layer thickness increases from 30 to 70 µm.

Reflectance is more important for short wavelengths between 400 and 600 nm. According to Kaufman et al. [38], the high reflectance at wavelengths 462 - 500 nm is due to the presence of iron atoms in the dust. Other studies such as that of Müller et al. [39] confirm these radiative properties of Saharan dust.

Figure 10 shows the transmission losses as a function of different dust layers with thicknesses ranging from 0 to 70 μm. We note that the transmittance decreases as the thickness of the dust layer increases. These results are in good agreement

with those of Said and Walwil [40], Al-Hasan [41], Ebert and Bhushan [42], Duell et al. [43], Chaouki et al. [44] who showed the negative impact of dust layers on transmittances. Thus, this figure shows that the transmittance of solar radiation of a clean glass compared to an unclean glass could decrease by up to 58% for a layer of 70 μm.