## 3 Determination of VSC
The VSC was obtained from the experimental data of horizontal global, diffuse, and beam illuminance, as shown in Equation (6):  
\[\mathrm{L_{dv}}=\mathrm{L_{gy}}-\frac{\rho}{2}\mathrm{L_{gh}}-\left(\frac{ \mathrm{L_{bh}}}{\sin\alpha_{\mathrm{s}}}\right)\ \cos\alpha_{\mathrm{s}}\cdot\cos(\phi_{\mathrm{s}}-\phi_{ \mathrm{Nr}}). \tag{5}\]  
\[\mathrm{VSC}=\frac{\mathrm{L_{gy}}-\frac{\rho}{2}\mathrm{L_{gh}}-\left(\frac{ \mathrm{L_{bh}}}{\sin\alpha_{\mathrm{s}}}\right)\ \cos\alpha_{\mathrm{s}}\cdot\cos(\phi_{\mathrm{s}}-\phi_{ \mathrm{Nr}})}{\mathrm{L_{dh}}}. \tag{6}\]  
If the luminance distribution in the sky dome is known, one alternative to determine the VSC is through integration. With a well-defined luminance distribution, the daylight on any surface can be estimated by integrating the luminance distribution of the sky dome over each surface. Traditionally, sky luminance distribution has been measured by sky-scanner instruments. These instruments divide the sky hemisphere into a limited number of sky elements (patches) of finite angular size and continuously scan the luminance data corresponding to each patch, \(\mathrm{L_{P}}\). Each sky patch will, it is expected, be treated as point sources with negligible error, but integration over all sky luminance scanning patches is only an approximation of total illuminance. Figure 2 shows a typical sky hemisphere sectorization into 145 patches and the geometric magnitudes that characterize each patch. When sky scanner measurements are available, the horizontal and vertical diffuse illuminance can be obtained from Equations (7) and (8), respectively [37; 41; 42]:  
\[\mathrm{L_{dh}}=\sum_{\mathrm{P=1}}^{145}\mathrm{L_{P}}\sin\alpha_{\mathrm{P }}\cdot\cos\alpha_{\mathrm{P}}\cdot\delta\alpha_{\mathrm{P}}\cdot\delta\phi_{ \mathrm{P}}, \tag{7}\]  
Figure 2: Angles defining the position of a sky element and the pattern of the 145 sky patches. \[\mathrm{L_{dv}}=\sum\
olimits_{\mathrm{p=1}}^{145}\Delta\mathrm{E_{LvD_{P}}}. \tag{8}\]  
In Equation (7), \(\mathrm{L_{P}}\) is the luminance of patch \(\mathrm{p}\), \(\mathrm{\alpha_{P}}\) is the angle of elevation of a patch above the horizon, \(\delta\mathrm{\alpha_{P}}\) is the altitude length of patch \(\mathrm{p}\) (\(\frac{\pi}{2.8}\) for all patches), and \(\delta\mathrm{\Phi_{P}}\) is the azimuth length of patch \(\mathrm{p}\). In Equation (8), \(\Delta\mathrm{E_{LvD_{P}}}\) represents the contribution of each patch to the vertical diffuse illuminance, calculated with Equation (9):  
\[\Delta\mathrm{E_{LvD_{P}}}=\mathrm{D_{P}}\mathrm{\cdot L_{P}}\mathrm{\cdot} \mathrm{\cos\,\alpha_{P}}\mathrm{\cdot}\delta\mathrm{\alpha_{P}}\mathrm{\cdot }\delta\mathrm{\Phi_{P}}. \tag{9}\]  
\(\mathrm{D_{P}}\) is a geometrical factor which depends on the patch position in the sky dome, defined by Equation (10) [32]:  
\[\mathrm{D_{P}}=\left\{ \begin{array}{cl}\mathrm{\cos\,\alpha_{P}\mathrm{\cdot }\mathrm{\cos}}\mathrm{\left(\mathrm{\phi_{P}}-\mathrm{\phi_{Nr}}\right)}&if \ 0\leq\ \left|\mathrm{\phi_{P}}\right|\leq 90^{\mathrm{\circ}}\\ 0&otherwise\end{array} \right., \tag{10}\]  
where \(\mathrm{\phi_{P}}\) is the azimuth of the patch \(\mathrm{p}\) and \(\mathrm{\phi_{Nr}}\) is the azimuth of the vertical surface. Knowledge of the sky luminance distribution can be obtained through the CIE standard sky classification, which establishes a biunivocal relationship between the sky type and the sky luminance pattern.