# Simulation of a Low Concentrator Photovoltaic System Using COMSOL
## 2 Materials and Methods  
### The Model  
The V-trough model is a simple design that consists of one PV panel with two attached flat mirrors that are inclined by an angle (\(\theta\)) as illustrated in Figure 1. To simulate a V-trough, the parameters that affect the model must be determined. These parameters can be divided into two categories: the first category is related to the geographical location where the amount of direct solar radiation falling on the V-trough will be determined. The second category is related to the geometrical design parameters of the V-trough [13; 16; 24]. Firstly, the geographical location parameters include the latitude angle (\(\varphi\)), the declination angle (\(\delta\)), the hour angle (\(\omega\)), and the solar altitude angle (\(\alpha\)). The latter is also known as the elevation angle, and it refers to the angle between an imaginary line in the middle of the sun in the sky and the horizontal. The elevation angle can be calculated from the following equation [25]:  
\[\alpha=\sin^{-1}\left(\cos\,\varphi\,\cos\,\delta\,\cos\,\omega+\sin\,\varphi\, \sin\,\delta\right) \tag{1}\]  
where the hour angle is given by \(\omega=15^{\circ}\) (\(12-T_{\mathrm{S}}\)); here, \(T_{\mathrm{S}}\) is the solar time. From these parameters, the direct solar radiation (\(B_{\mathrm{S}}\)) can be calculated from the relation [26]  
\[B_{\mathrm{S}}=B_{0}\,\exp\,\left(-T_{\mathrm{R}}/(0.9+9.4\,\sin\,\alpha)\right) \tag{2}\]  
where \(T_{\mathrm{R}}\) is the turbidity factor that describes the condition of the sky. It is classified into ten classifications, each of which describes how the air contains particles of dust and water vapor that cause solar beam attenuation [27]. \(T_{\mathrm{R}}\) plays a critical role in solar-system efficiency, as it measures the clearness of the sky. It has values greater or equal to one. For example, for polluted air \(T_{\mathrm{R}}=6\), while for clean warm air \(T_{\mathrm{R}}=3\), and it equals 2 for very clean cold air [16]. That is, a high value reflects an elevated content of atmospheric water vapor and dust particles [27]. The values of \(T_{\mathrm{R}}\) for Riyadh, Jeddah, Dhahran, Aljawf, and Sharurah are listed in Table 1. Since \(T_{\mathrm{R}}\) gives an indication of the optical thickness of the atmosphere, its value affects the amount of solar radiation reaching the surface of the Earth, hence, affecting the performance of LCPV systems. \(B_{0}\) is the extra-terrestrial radiation given by [13; 25]  
\[B_{0}=1367[1+0.0334\mathrm{cos}(0.9856\times(N-2.27))] \tag{3}\]  
Here, N is the day number of the year. Secondly, the parameters due to the geometrical design describe the design of the V-trough, which includes the width of both mirrors (\(L_{2}\)), the width of the PV panel (\(L_{1}\)) (which is constant), and the inclination angle for each mirror (\(\theta\)). [PERSON] et al. [16] concluded that the tracking system had no effect on the amount of total radiation that falls on the PV. Therefore, in this work, the system was placed horizontally with no tilt and without a tracking system to track the sun over the day. The PV is assumed stationary, while the attached mirrors can be tilted at different angles. The width of the mirrors was fixed at two different lengths to enable us to investigate two cases: the first when the mirror length, \(L_{2}\), is twice the width of the PV panel, \(L_{1}\), to ensure that the reflected solar radiation covers the whole PV surface as shown in Figure 2a (this case is called double coverage). The second case, when \(L_{2}\) is the same length of \(L_{1}\) so the reflected solar radiation from each mirror will partially cover the PV surface, is illustrated in Figure 2b (this case is called partial coverage). To simulate the two cases of V-trough design, the geographical locations where the system could be assembled were selected to be Dhahran, Jeddah, Aljawf, Sharurah, and Riyadh. These cities represent different locations in Saudi Arabia, in the east, west, north, south, and center, respectively.