## 2 Materials and Methods
The refinement of this physical model and further modeling consists in determining the angle of inclination as the angle between the Earth's inclined axis and its projection onto a vertical plane perpendicular to the line connecting the centers of the Earth and the Sun. This model allows us to determine the average annual efficiency of the photovoltaic module's installation depending on the angle of its installation to the horizon for different values of geographical latitude. The correlation coefficient of the set of experimental data of the deviation of the installation angle of photovoltaic modules from the value of geographical latitude and the corresponding set of calculated data according to the modeling data was 0.83 [18]. The results of mathematical modeling are confirmed by numerical experimental data obtained in different cities of the world at different values of geographical latitude of the place of installation, and are given in the study [18]. A general view of the installation and orientation of the solar PV-modules is shown in Figure 1. One module was stationary, with a vertically installed double-sided solar panel with an \"East-West\" orientation. The other module was equipped with a solar panel with a \"South\" orientation. This research was carried out on the geographical latitude of 50\({}^{\circ}\). The angle of inclination of the PV-module panel with \"South\" orientation in the winter period is 54.1\({}^{\circ}\), and for the summer period it is \(-34.9^{\circ}\)[28]. The rest of the data, necessary for conducting experimental work on PV-module panels with a \"South\" orientation, are given in other works [17; 18]. The angle of inclination of the double-sided panels of the vertical PV-module to the horizon is equal to zero. To achieve our research goal, the modeling algorithm was as follows. First, the angular length of daylight was determined using the following expression [17]:  
\[\delta\prec 0\rightarrow\alpha=2 arctg\sqrt{\frac{ctg^{2}\phi}{tg^{2}\delta}- 1;}\ \delta\geq 0\rightarrow\alpha=2\pi-2 arctg\sqrt{\frac{ctg^{2}\phi}{tg^{2} \delta}-1} \tag{2}\]  
Figure 1: Examples of photovoltaic modules installation: (**a**) vertically mounted double-sided solar panel with “East-West” orientation; (**b**) inclined solar panel with “South” orientation; 1—foundation slab; 2—frame structure; 3—solar panel. where \(\alpha\) is the angular length of daylight; \(\delta\) is the declination angle (angular position of the Sun at noon relative to the plane of the equator); \(\theta\) is the latitude at which solar panels of PV modules are installed on the surface of the Earth. Next, the angle of incidence of the Sun's rays with respect to the y-axis, which is located in the equatorial plane and runs parallel to the plane of the PV-module panel, was determined. The next step was to determine the angle of incidence of the Sun's rays relative to the x-axis. The indicated axis is located in the meridional plane of the solar panel. The mathematical expression (3) is used to calculate the angle of incidence of the Sun's rays relative to the z-axis. The z-axis is perpendicular to the surface of the solar panel. Therefore, according to the geometry of three-dimensional space, the specified angle will be [17]:  
\[\cos\theta_{Z}=\sqrt{1-\cos^{2}\theta_{H}-\cos^{2}\theta_{V}}=\sqrt{\sin^{2} \theta_{H}-\cos^{2}\theta_{V}} \tag{3}\]  
where \(\theta_{H},\theta_{V}\) and \(\theta_{Z}\) are the angles of incidence of the Sun's rays relative to the \(y\), \(x\) and \(z\) axes, accordingly. Later, based on the values of the angular length of daylight and the cosine of the angle of incidence of the Sun's rays relative to the z-axis, the weighted average daily value of the cosine of the angle of the Sun's rays on the plane of the panel was determined [17],  
\[\cos\theta_{Zi}^{d}=\frac{\sum_{j=0}^{n_{j}}a_{j}\cos\theta_{j}}{\sum_{j=0}^{ n_{j}}a_{j}} \tag{4}\]  
where \(\alpha_{i}\)--angular length of the ith day, degrees; \(\cos\theta_{Zi}^{d}\)--daily installation efficiency of PV-module installation.