## 3 Results
\begin{array}{l}E(\beta)=\left(7245.99\cdot e^{-7.53823\cdot 10^{-9}\beta} \right)\\ +\left(e^{-0.01322\beta}\cdot(-24.2152\cdot\sin(0.0841679\beta)+268.311\cdot\cos (0.0841679\beta))\right)\end{array}  \tag{7}\]  
From the global extreme analysis of the model polynomial function described by Equation (6) and analytical model function--Equation (7), the global maxima of the functions were identified for the tilt angle 34,405\({}^{\circ}\). The influence of the tilt angle on the PV system electricity production during the year was also modelled by three-dimensional graphical dependence shown in Figure 9. On the presented 3D dependence was applied polynomial approximation of second degree, which was used for the \(x\)-axis and the \(y\)-axis. The \(x\)-axis shows the time represented by calendar months; the \(y\)-axis shows the individual positions of PV modules which correspond to the tilt angles. The \(z\)-axis shows the average values of the electricity produced by the PV system. The final three-dimensional model dependency is shown in Figure 10. It can be described by the characteristics which are presented in Table 1 and for mathematical description of three-dimensional relation was obtained Equation (8). The relation can be used for prediction of PV system energy balance as a function of tilt angle and month. The importance of mathematical modelling dependencies that enable the calculation of the PV system energy production was also declared by the authors [18; 47; 62]. \[E_{m}(t,\ \beta)=-1438+56.33\cdot t+4218\cdot\beta-0.8993\cdot t^{2}+0.8507\cdot t \cdot\beta-340.9\cdot\beta^{2} \tag{8}\]  
where \(E_{m}(t,\ \beta)\) is the energy produced by the PV system per month (kWh), \(\beta\) is the tilt angle of PV system (\({}^{\circ}\)), and \(t\) is the time (month). Figure 9: Three-dimensional dependence before polynomial approximation—the influence of tilt angle on the electric energy production of the PV system. The author [2] in the publication presents that any change in the tilt angle of the PV panel is the cause of electric energy production decreasing up to 10%. Measurements on the model PV module showed the significant influence of the tilt angle, where the perpendicular placement of the PV module (90\({}^{\circ}\)) reduces the electricity production from 13,200 kWh to 6520 kWh, which represents 49.39%. If the tilt angle of the PV module is in the range from 0\({}^{\circ}\) to 30\({}^{\circ}\) during the months from April to August, the difference of electric energy production by PV system is minimal. Especially, the energy range was from 12,700 kWh to 13,100 kWh, which means the difference is 3.05%. However, if we compare the arithmetic average representing the energy production by the PV system in different months for different tilt angles, it is obvious that they differ from the total monthly average of 7668.571 kWh. This is especially in the range 5800-8560 kWh, which represents 24.37-11.62%. From the photovoltaic practice point of view, the average influence of the tilt angle on the energy production is approximately 18%. A considerable influence of the tilt angle is evident also from the graphical dependencies presented in Figures 6, 7, 9 and 10. The general mathematical description of the tilt angle influence is represented by Equation (4). This equation allows the simple calculation of electric energy production by the PV system after entering the time value and the position which corresponds to the tilt angle \(\beta\) of the PV panel. These parameters are easy to identify in practice. Based on the model equation, it is possible to