## 3 Results
After analyses of the energy production during the winter months in a similar way, the amount of produced energy in January was 1580 kWh for the azimuth angle \(+90^{\circ}\) to 2520 kWh at the azimuth angle in range (0-\(5^{\circ}\)). It means change of 37.30% from the maximum energy production in January. The influence of the azimuth angle changes in winter is on average 36.55%. It can be compensated for example by optimizing of the PV module tilt angle. From the spring months typical March is the model month, where the amount of produced energy was (6850-9610) kWh; the change in the amount of produced energy in March was affected by an azimuth angle of 28.72%. Minimum energy production was reached in March with azimuth orientation of \(+90^{\circ}\) and maximum energy production with azimuth orientation of \(0^{\circ}\). Numerically, the effect of the azimuth orientation changes on energy production is during the spring 19.38%. Similarly, model autumn month October was analysed. The minimum energy production during October was 4090 kWh at \(+90^{\circ}\) and the maximum energy production was 5950 kWh for azimuth orientation \(0^{\circ}\). The difference in October was 31.26%, and the average change of energy balance was 31.51% in the autumn season. The presented results show that in terms of the energy balance, the maximal production was achieved for azimuth angle from \(-5^{\circ}\) to \(-10^{\circ}\) Southeast when the average energy production was 8590 kWh. More generally, the assessment of the effect of the PV module azimuth angle on the amount of produced energy is shown in Figure 13. The final observed dependence can be described by the regression Equation (9). It represents the polynomial function of the second degree with the coefficient of determination \(R^{2}\) = 0.998. \[E_{m}(\gamma)=-0.1943\cdot\gamma^{2}-2.1746\cdot\gamma+8568.8 \tag{9}\]  
where \(E_{m}(\gamma)\) is the energy produced by the PV system per month (kWh) and \(\gamma\) is the azimuth angle (\({}^{\circ}\)). Figure 13: Relation between the PV system energy production and the azimuth angle. In the case of Figure 13, a purely parabolic trend of functional dependence was detected. Within the analytical model, the first order differential equation with constant coefficients--Equation (10)--was compiled. As in the previous analytical solution, a transfer function in a complex plain described by Equation (11) was used and subsequently a general solution of the differential Equation (10) was found. The coefficient of determination for the transfer function--Equation (11)--was 96.25%,  
\[\frac{dE(\gamma)}{d\gamma}+490.3\cdot 10^{-6}\cdot E(\gamma)=-0.3879\cdot\gamma \tag{10}\]  
\[G(s)=\frac{-0.3879}{s+0.0004903} \tag{11}\]  
where \(\gamma\) is the azimuth angle (\({}^{\circ}\)). From the mathematical description of the parabolic course by polynomial function of the second degree, it is possible to determine the position of the stationary point, which corresponds to an azimuth angle of \(-5.60^{\circ}\). Due to the information known from PV theory, a calculation was performed for the azimuth angle \(\gamma\) = 0\({}^{\circ}\). The general solution of the differential Equation (10) for the mentioned azimuth angle \(\gamma\) can be expressed in the form of Equation (12):  
\[E\left(\gamma\right)=8568.8\cdot e^{-0.0004903}\ \gamma \tag{12}\]  
A comparison of the analytical model results with functional dependence is shown on Figure 14.  
Matlab(r) version R 2015a was used for the mathematical model creation. Three-dimensional dependencies were created by mathematical software. The suitable polynomial approximation was applied to the relation of data files. For the three-dimensional relation, selected statistical parameters and regression coefficients of Equation (13) were calculated, which are summarized in Table 3.  
Figure 14.0  
where \(E_{m}(t,\ \gamma)\) is the energy produced by the PV system per month (kWh), \(\gamma\) is azimuth angle (\({}^{\circ}\)), and \(t\) is the time (month).