# Numerical Validation of the Radiative Model for the Solar Cadaster Developed for Greater Geneva
## 2 Materials and Methods
### Tools Considered
#### 2.2.1 Solar Cadaster  
As summarized by [PERSON] et al. [10] and [PERSON] et al. [13], the solar cadaster tool is a geographic information system (GIS) tool. Such tools enable one to process large amounts of data and spatial analyses. These tools also provide automatic or systematic environmental analyses of urban areas, such as solar radiation calculations. These are different from tools that are classified in the category of computer-aided design (CAD). The latter process more accurate spatial and weather data in terms of spatial and time scales, but they require much more computing time and, thus, address the local scale (limited sets of buildings).  
The irradiance received from the direct component \(I_{b}\) on an element of a surface at location \(x\) at a given time \(t\) is given by:  
\[I_{b}(x,t)=BNI(t)\times r_{b}(x,t)\times S_{b}(x,t) \tag{3}\]  
in which \(BNI\) corresponds to the direct normal component of the irradiance (also called the beam), \(r_{b}\) is the transposition factor, and \(S_{b}\) corresponds to the shadow cast from a neighborhood building (\(S_{b}(x,t)=0\) if, at time \(t\), the surface located at \(x\) is shaded by another building, or \(S_{b}(x,t)=1\) otherwise). The transposition factor, \(r_{b}\), depends on the solar elevation \(h\) and the slope of the considered surface \(\beta\) (\(\beta=0\) corresponds to a horizontal surface and \(\beta=90^{\circ}\) corresponds to a vertical surface; see Figure 3). It is calculated as \(r_{b}(x,t)=sin(h(t))/cos(\beta(x))\). The irradiance over the element of the surface is then considered as the average of the calculated irradiances at its edges. The calculation differences between the three tools, as described below, are illustrated in Figure 4.  
In order to model the contribution of the diffuse component to the received irradiance, the Hay model is used. This model considers two components: a circumsolar (anisotropic) and an isotropic component. Similarly to the direct component, the circumsolar component is calculated at each time step by considering the sun's position and the shadowing. For the isotropic component, the sky-view factor is computed. In the solar cadaster, a sky model of 580 light sources is used.  
The diffuse component, \(I_{d}\), of the solar radiation is then calculated as follows:  
\[I_{d}(x,t)=DHI(t)\times\bigg{(}\frac{GHI(t)-DHI(t)}{I_{0}}r_{b}(x,t)+SVF(x) \bigg{(}\frac{GHI(t)-DHI(t)}{I_{0}}\bigg{)}\bigg{)} \tag{4}\]  
Figure 4: Differences in terms of irradiance calculations for a mesh.  
Figure 3: Shadow casting of the solar cadaster, adapted with permission from [10; 14].  
where \(GHI\) is the global irradiation on a horizontal surface, \(DHI\) is the diffuse irradiation on a horizontal surface, \(I_{0}\) is the hourly extraterrestrial irradiation, \(r_{b}\) is the transposition factor as defined in (3), and \(SVF\) is the sky-view factor.  
Finally, the reflected component is simply considered at the current stage as isotropic and is estimated based on [PERSON] (1983) [15] as follows:  
\[I_{r}(x,t)=0.5\times GHI(t)\times\rho(1-\cos\beta(x)) \tag{5}\]  
where \(GHI\) is as defined in (4), \(\rho\) is the coefficient of reflection of the surface, as defined in (1), and \(\beta\) is the slope of the surface.  
More details about the calculation method and the computing tools used are available in [10; 16].