## 2 Material and Methods
### Weather Data  
In this work, the provided weather data from Greek Technical Chamber [29] are used in order to estimate the daily solar energy on the tilted surface (H\({}_{\rm T}\)). More specifically, the global irradiation on the horizontal surface (H) and the diffuse irradiation on the horizontal surface (H\({}_{\rm d}\)) are properly combined in order to find the (H\({}_{\rm T}\)). These data regard the mean monthly day for four different Greek cities. Every city is a representative one for every different climate zone. Table 3 gives the data about the (H) and Table 4 the data about the (H\({}_{\rm d}\)) for the studied locations. Moreover, the yearly solar irradiation values are summarized in the previous tables. The data have been taken from Ref. [29]. Figure 3 depicts the examined cities and the respective climate zone separation in Greece. More specifically, in the work Zone A is examined with the city of Heraklion (35\({}^{\circ}\)20\({}^{\prime}\) N, 25\({}^{\circ}\)8\({}^{\prime}\) E), Zone B with the city of Athens (37\({}^{\circ}\)59\({}^{\prime}\) N, 23\({}^{\circ}\)43\({}^{\prime}\) E), Zone C with the city of Thessaloniki (40\({}^{\circ}\)37\({}^{\prime}\) N, 22\({}^{\circ}\)56\({}^{\prime}\) E) and Zone D with the city of Kastoria (40\({}^{\circ}\)31\({}^{\prime}\) N, 21\({}^{\circ}\)16\({}^{\prime}\) E). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Cities** & **JAN** & **FEB** & **MAR** & **APR** & **MAY** & **JUN** & **JUL** & **AUG** & **SEP** & **OCT** & **NOV** & **DEC** & **Year** \\ \hline Heraklion & 65.6 & 81.6 & 125 & 166.5 & 207.3 & 222.4 & 227.1 & 207.0 & 163.0 & 117.3 & 78.6 & 61.2 & 1722.6 \\ Athens & 63.3 & 77.7 & 118.9 & 152.7 & 190.4 & 207.4 & 214.5 & 198.6 & 156.0 & 111.1 & 68.1 & 54.4 & 1613.1 \\ Thessaloniki & 52.6 & 67.5 & 103.2 & 140.7 & 179.1 & 198.6 & 209.5 & 184.7 & 136.7 & 91.4 & 56.6 & 45.5 & 1466.1 \\ Kastoria & 57.6 & 71.3 & 111.2 & 141.1 & 173.6 & 201.8 & 206.3 & 185.5 & 138.5 & 97.0 & 60.0 & 47.7 & 1491.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Data for the daily global solar irradiation on the horizontal surface (H) in [kWh/m\({}^{2}\)]. The next step in this work is the calculation of the solar energy that receives the tilted surface (\(\rm H_{T}\)). This calculation is possible by following the methodology of the isotropic atmosphere proposed by [PERSON] and [PERSON] [33] and extended by [PERSON] [34]. \[\rm H=R_{b}\cdot(H-H_{d})+\left(\frac{1+cos(\beta)}{2}\right)\cdot H_{d}+\rho_{g} \cdot\left(\frac{1-cos(\beta)}{2}\right)\cdot H \tag{16}\]  
where (\(\rho_{g}\)) is the reflectance of the ground equal to 0.2 [27], (\(\beta\)) the collector tile angle, and (\(\rm R_{b}\)) the ratio of the beam irradiation on the tilted surface to the horizontal surface on the monthly base. The parameter (\(\rm R_{b}\)) is calculated according to the suggested methodology in Ref. [27]. For surfaces in the northern hemisphere with zero azimuth (facing the south), it can be written:  
\[\rm R_{b}=\frac{cos(\varphi-\beta)\cdot cos(\delta)\cdot sin(\omega_{s}^{ \prime})+\left(\frac{\pi}{180}\right)\cdot\omega_{s}^{\prime}\cdot sin(\varphi- \beta)\cdot sin(\delta)}{cos(\varphi)\cdot cos(\delta)\cdot sin(\omega_{s})+ \left(\frac{\pi}{180}\right)\cdot\omega_{s}\cdot sin(\varphi)\cdot sin(\delta)} \tag{17}\]  
The parameter (\(\omega_{s}^{\prime}\)) is the sunset hour angle for the collectors and it is calculated as below: