## 2 Materials and Methods  
### Astronomical Bases and Irradiance Model  
Knowing the concrete position of the sun at each moment of the day and on each day of the year is fundamental for determining the direction of solar beams. It is possible to determine the position of the sun accurately by using astronomical geometry. For this purpose, a solar vector was defined, expressed with respect to a terrestrial reference system (Figure 2) and formed by the axes Ox, towards the west; Oy, towards the south; and Oz in the zenithal direction. The position was determined by Equations (1) and (2):  
\[\overset{\rightarrow}{s}=s_{x}\overset{\rightarrow}{i}+s_{y}\overset{ \rightarrow}{j}+s_{z}\overset{\rightarrow}{k} \tag{1}\]  
\[\overset{\rightarrow}{s}=sin\Omega t\text{-}cos\overset{\rightarrow}{i}+( cos\Omega t\text{-}cos\overset{\rightarrow}{\delta}\text{-}sin\varphi-sin\delta\text{-} cos\varphi)\overset{\rightarrow}{j}+ \tag{2}\]  
\[(cos\Omega t\text{-}cos\overset{\rightarrow}{\delta}\text{-}cos\varphi+sin \delta\text{-}sin\varphi)\overset{\rightarrow}{k}\]  
where \(\delta\) represents the solar declination, \(\varphi\) the latitude and \(\Omega t\) the hour angle. This angle was calculated as the Earth's rotation speed, 2\(\pi\)/24 rad/h, for the hours since solar noon [30].  
In PV technology, all irradiance components (direct, diffuse and reflected irradiance) are usable. Traditionally, the astronomical model has been used to determine the position of the sun. Applying the astronomical model to solar tracking means that the angle formed between the direct solar rays and the normal angle to the collector's surface \(\theta\) must be as low as possible. With astronomical tracking, the value of the direct irradiance component is maximised, which is appropriate for applications focused on this component (such as concentration technologies). Therefore, this type of tracking is not the most suitable for the purposes of this study. For instance, on cloudy days, when the solar disk is not visible and direct radiation does not reach the collectors, those located on a fixed horizontal position collect more energy than those using astronomical tracking. Therefore, to study the influence of diffuse and reflected components on solar tracking, several sky models have been proposed.  
The [PERSON] method [31] was used for the calculations in this work because it adequately describes the anisotropy of radiation [32] in addition to being notable for its simplicity, as opposed to other more complex models such as those of [PERSON] [33] and [PERSON] [34], and for obtaining high quality results [35; 36; 37]. This model establishes that a determined fraction of diffuse irradiance is directed according to the direction of the solar disk. Models such as that of [PERSON] [31] describe the irradiance affecting a solar panel by considering its three components, direct, diffuse and reflected irradiance, but take into account the fact that the diffuse component has a preferential direction in which the diffuse radiation is greater. According to [PERSON] [31], the mathematical expression of the model is given by Equation (3), where the first term indicates the direct irradiance \(I_{B}\), the second term refers to the diffuse irradiance \(I_{D}\) and the third term corresponds to the reflected component, taking into account the visible soil fraction and the albedo, \(\rho\). \(I_{B}\) and \(I_{D}\) are determined by the Collares-Pereira model [38].  
\[I=\frac{cos\theta}{cos\theta_{z}}I_{B}+\left[\left(\frac{cos\theta}{cos\theta _{z}}\right)\frac{I_{B}}{I_{OH}}+\left(1-\frac{I_{B}}{I_{OH}}\right)\frac{1+ cos\xi}{2}\right]I_{D}+\rho\frac{1-cos\xi}{2}(I_{B}+I_{D}) \tag{3}\]  
where \(\xi\) is the inclination angle of the collector and \(\theta_{Z}\) is the solar zenith angle, that is, the angle between the solar vector and the zenith axis.