## 2 Materials and Methods
### Numerical Model for Azimuth-Based SAST  
To calculate the solar azimuth angle, which is the angular distance between the south and the ground projection of the line of sight to the sun as seen in Figure 2, Equation (1) can be used. Additionally, the position east of south is signified by a positive solar azimuth angle, while a negative azimuth angle indicates a position west of south. \[\gamma_{s}=\sec(\alpha_{s})\cos(\delta)\sin(\omega) \tag{1}\]  
where \(\gamma_{s}\) is the solar azimuth angle, \(\alpha_{s}\) is the altitude angle, \(\delta\) is the declination angle, and \(\omega\) is the hour angle. The altitude angle shows the height of the sun in the sky. The angle is calculated by measuring the distance between an imaginary line between the observer and the sun and the horizontal plane on which the observer stands. When the sun sets below the horizon, the altitude angle becomes negative. The altitude angle can be calculated using Equation (2) below,  
\[\alpha_{s}=90-\theta_{z} \tag{2}\]  
where \(\alpha_{s}\) is the altitude angle, and \(\theta_{z}\) the zenith angle. The azimuth angle is the compass direction from which the sunlight is coming. At solar noon, the sun is always directly south in the Northern Hemisphere and directly north in the Southern Hemisphere. At  
Figure 1: CAD model for (**a**) sensor-based SAST and (**b**) azimuth-based SAST. the equinoxes, the sun rises directly east and sets directly west regardless of the latitude, thus making the azimuth angles \(90^{\circ}\) at sunrise and \(270^{\circ}\) at sunset. The zenith angle can be calculated using Equation (3). \[\cos(\theta_{z})=\cos(\phi)\cos(\delta)\cos(\omega)+\sin(\phi)\sin(\delta) \tag{3}\]  
where \(\theta_{z}\) is the zenith angle, \(\phi\) is the latitude, \(\delta\) is the declination angle, and \(\omega\) is the hour angle. The latitude is the angle formed by the radial line connecting the place to the center of the earth with the projection of the line on the equatorial plane as a point or location. At \(90^{\circ}\) latitude (North Pole) and \(-90^{\circ}\) latitude (South Pole), the earth's axis of rotation contacts the earth's surface (South Pole). Therefore, the intersection of a longitude and a latitude angle may be used to designate any point on the earth's surface. The declination angle is the angular distance between the North and South Poles to the earth's equator, and it can be calculated using Equation (4),  
\[\delta=23.45\sin(360\frac{284+n}{365}) \tag{4}\]  
where \(\delta\) is the declination angle, and \(n\) is the number of days in the year. The hour angle is the angular displacement of the sun east or west of the local meridian due to the rotation of the earth on its axis at \(15^{\circ}\) per hour considering Solar Time, where morning times are negative and afternoon times are positive. Because the earth's axis is inclined by \(23.34^{\circ}\) from the plane of the earth's orbit around the sun, the declination angle varies from \(23.45^{\circ}\) north on 21 December in winter to \(23.45^{\circ}\) south on 21 June in Summer [32]. The Solar Time parameter is highly significant since it is the actual time affected by the sun's location in the sky. It can be calculated using the Equations (5)-(7) shown below. The fundamental basis of this time is the day order in the year, which begins on 1 January and ends with 365 on 31 December [13]. \[Solar\ time-Standard\ time=4(L_{st}-L_{loc})+E \tag{5}\]  
\[E=229.2\ (0.000075+0.001868\ \cos B-0.032077\ \sin B\ -0.014615\cos 2B-0.0 4089\sin 2B\ ) \tag{6}\]  
Figure 2: The direction of beam radiation. \[B=\frac{(n-1)\times 360}{365} \tag{7}\]  
where \(L_{st}\) is the standard meridian time zone, \(L_{loc}\) is the location's longitude, B is the displacement of time from the standard time, and n is the number of days in the year. Additionally, to measure the standard meridian time zone, the time difference between local standard actual clock time and Greenwich Meantime should be multiplied by \(15^{\circ}\) (e.g., the \(L_{loc}\) for KSA is \(45^{\circ}\)).