## 2 Materials and Methods
This straight line is the BC in Figure 2. A rectangle triangle ABC is then formed. The third step is to draw the two medians from the vertexes A and B of the triangle ABC to their opposite sides, i.e., the lines AD and BE, respectively. The two lines are crossed at M; if a straight line parallel to the \(y\)-axis is drawn through M, i.e., the blue line MG, this crosses the x-axis at a value, which is the lower limit of k\({}_{\rm d}\). From the Euclidian geometry, it is known that AG = AC/3. This means that if the upper k\({}_{\rm d}\) value at the point C  
Figure 2: Mathematical determination of the upper (k\({}_{\rm du}\), red solid line) and lower (k\({}_{\rm d}\), blue solid line) limits of k\({}_{\rm d}\) that divide the skies into clear, intermediate, and overcast. The bold black solid line is the best-fit curve to the H\({}_{\rm bn}\)-k\({}_{\rm d}\) data points that correspond to their hourly values; in this case the typical meteorological month of January, which belongs to the PV-GIS TMY-BOU, has been considered. The line BC is tangent to the best-fit line at C (0.76,120). M is the crossing point of the medians AD and BE. The MG line is normal to AC. From the Euclidian geometry AG = AC/3 in the rectangle triangle ABC; therefore, k\({}_{\rm d}\)l = k\({}_{\rm du}\)/3 = 0.76/3 \(\approx\) 0.25.  
is obtained (\(\mathrm{k_{du}}\)), then very easily the lower \(\mathrm{k_{d}}\) value at the point G (\(\mathrm{k_{dl}}\)) can be calculated: \(\mathrm{k_{dl}}=\mathrm{k_{du}}/3\). This way, the estimated \(\mathrm{k_{du}}\) and \(\mathrm{k_{dl}}\) values divide the sky conditions in clear skies, intermediate skies, and overcast skies at a site as shown in Figure 2.  
The above methodology was applied to 14 sites around the world. The selection has been based on the following criteria: (i) different environmental characteristics, (ii) spread across the continents, and (iii) availability of TMY. Table 1 deploys the selected sites in alphabetical order together with their geographical coordinates and environmental description. Sites that do not meet criterion (iii) may not be appropriate for applying this methodology. Figure 3 gives the distribution of the 14 sites over the world. For the sake of the present analysis, TMYs for the selected sites have been downloaded from the PV-GIS platform [17; 18]. These typical meteorological years have been derived from a combination of satellite and re-analysis data in the period 2005-2014. Each TMY contains the following parameters: date, time (h UTC), dry-bulb temperature (in degC), relative humidity (in %), global horizontal solar irradiance (\(\mathrm{H_{g}}\), in \(\mathrm{Wm^{-2}}\)), direct-normal solar irradiance (\(\mathrm{H_{bn}}\), in \(\mathrm{Wm^{-2}}\)), diffuse horizontal solar irradiance (\(\mathrm{H_{d}}\), in \(\mathrm{Wm^{-2}}\)), infrared horizontal solar irradiance (\(\mathrm{in}\)\(\mathrm{Wm^{-2}}\)), wind speed (in \(\mathrm{ms^{-1}}\)), wind direction (in deg), barometric pressure (\(\mathrm{P_{z}}\), in Pa) at the site's altitude (z, in m), and a control parameter. The meteorological and solar radiation values are hourly averages. Therefore, from the 14 downloaded TMYs new files were created to contain the necessary parameters: date, time (h UTC), \(\mathrm{H_{g}}\), \(\mathrm{H_{bn}}\), and \(\mathrm{H_{d}}\). Conversion of time from UTC (universal time constant) into LST (local standard time) at each site was applied to all files. Then, an own-developed