## 4 Conclusions and Discussion  
A mathematical formulation for the determination of the upper and lower diffuse-fraction limits was developed for the first time worldwide. The methodology was based on existing or prepared TMY for the site into consideration. Such a TMY can be downloaded free-of-charge from the PV-GIS website. The TMY must include hourly values of the global horizontal, diffuse horizontal, and direct-normal solar irradiances. Hourly values of the diffuse fraction can then be derived. The method consists of the following analytical steps. Step 1. Prepare a scatter plot of the hourly direct-normal solar radiation vs. diffuse-fraction values that are included in the TMY for the site into consideration. Step 2. Estimate the best-fit second-order polynomial to the data points of step 1. Step 3. Find the upper diffuse-fraction limit, \(\mathrm{k_{du}}\), at the intersection between the line \(\mathrm{y}=120\) Wm\({}^{-2}\) and the best-fit curve, i.e., point C (\(\mathrm{k_{du}}\),120). Step 4. Find the equation of the straight line, which is tangent to point C.  
Step 5. Find point B where the straight line of step 4 crosses the y axis. Step 6. Observe the orthogonal triangle formed by the vertexes A (0,0), B, C (\(\mathrm{k_{du}}\),120). Step 7. Draw the medians from the vertexes A and B, crossing each other at M.  
Step 8. Draw a straight line from M normal to AC, crossing AC at G.  
Step 9. Find the lower diffuse-fraction limit, \(\mathrm{k_{dl}}\), on the x axis, or estimate it as \(\mathrm{k_{dl}}=\mathrm{k_{du}}/3\). Fourteen sites were selected worldwide under the criteria of (i) existing TMY at the PV-GIS platform, and (ii) spread of the locations across the continents. The above procedure was applied to the 14 sites and their \(\mathrm{k_{du}}\), \(\mathrm{k_{dl}}\) values were calculated. To evaluate the methodology derived, 1-min solar radiation data were downloaded from BSRN for 11 sites and for 1 year for each location. Data for the ATH, KIS, and LIN sites were provided free-of-charge. Hourly values of (\(\mathrm{k_{d}}\), SSD) pairs were estimated and statistics about the number of SSD under clear (\(0\leq\mathrm{k_{d}}\leq\mathrm{k_{dl}}\)), intermediate (\(\mathrm{k_{dl}}<\mathrm{k_{d}}\leq\mathrm{k_{du}}\)), and overcast (\(\mathrm{k_{du}}<\mathrm{k_{d}}\leq\mathrm{1}\)) skies were prepared. The analysis showed that, in most sites (GAN and ILO were not taken into account), the anticipated decreasing trend of SSD with increasing values of \(\mathrm{k_{d}}\) was confirmed. Moreover, the number of the SSD values varied from site-to-site according to local climate and sky-conditions considered. The analysis \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Site (Year)** & \(\mathbf{k_{du}<k_{d}\leq 1}\)** & \(\mathbf{k_{dl}<k_{d}\leq k_{du}}\) & \(\mathbf{0\leq k_{d}\leq k_{dl}}\) & \(\mathbf{0\leq k_{d}\leq 1}\) \\
**Overcast Skies** & **Intermediate Skies** & **Clear Skies** & **All Skies** \\ \hline ATH (2000) & 919 (20.9\%) & 1879 (42.7\%) & 1602 (36.4\%) & 4400 (100\%) \\ BOU (1999) & 1477 (31.7\%) & 1429 (30.6\%) & 1758 (37.7\%) & 4664 (100\%) \\ CAR (2018) & 1799 (37.1\%) & 1556 (32.1\%) & 1496 (30.8\%) & 4851 (100\%) \\ DAA (2017) & 733 (15.5\%) & 1288 (27.1\%) & 2715 (57.4\%) & 4736 (100\%) \\ GAN (2015) & - & - & - & - \\ OHY (2018) & 908 (24.3\%) & 957 (25.5\%) & 1881 (50.2\%) & 3746 (100\%) \\ ILO (1999) & - & - & - & - \\ KIS (2020) & 1524 (36.2\%) & 1525 (36.2\%) & 1161 (27.6\%) & 4210 (100\%) \\ LER (2003) & 3476 (68.8\%) & 1225 (24.2\%) & 354 (7.0\%) & 5055 (100\%) \\ LIN (2018) & 1652 (41.5\%) & 1515 (38.1\%) & 811 (20.4\%) & 3978 (100\%) \\ PAY (2013) & 2100 (43.5\%) & 1743 (36.1\%) & 986 (20.4\%) & 4829 (100\%) \\ REG (2003) & 2049 (37.6\%) & 1932 (35.5\%) & 1467 (26.9\%) & 5448 (100\%) \\ SOV (2002) & 872 (18.3\%) & 2199 (46.0\%) & 1704 (35.7\%) & 4775 (100\%) \\ SON (2018) & 2624 (60.2\%) & 952 (21.8\%) & 781 (18.0\%) & 4357 (100\%) \\ \hline \hline \end{tabular}
\end{table}