## 4 A Proposal for Minute-Scale GHI Separation Models  
In this section, two innovative separation models are introduced. The models were built on the basis of the D_FIT dataset. Most of the existing separation models are linear, polynomial, logistic, and/or piecewise defined [4; 5; 6]. Differently, the proposed models are defined by rational equations. Basically, the two models estimate the diffuse fraction based on common predictors included in the equations of the most performant separation models (see, e.g., Appendix A). The first proposed model, further denoted M1, is defined by an elaborate equation calling on eight predictors. M1 is the result of intensive research on the effect of various predictors on the diffuse fraction. It gathers together almost all the predictors successfully accounted for by the previous separation models: (1) the clearness index \(k_{t}\), (2) the deviation of the clearness index from its estimated value under clear skies \(\Delta k_{tc}\), (defined by Equation (A3)), (3) the part of diffuse fraction \(k_{d}\) that is attributable to cloud enhancement \(k_{de}\), (defined by Equation (A4)), (4) daily average of the clearness index \(k_{dny}\), (defined by Equation (A1)), (5) hourly average of the clearness index \(k_{hour}\), (6) the persistence factor \(\psi\), defined as the average of a lag and a lead of the clearness index values, (7) the clear sky global solar irradiance \(G_{cs}\) in MJ/(h\(\cdot\)m\({}^{2}\)), and (8) the ratio of measured \(GHI\) and the estimated global clear sky irradiance \(K_{csi}\). M1 is defined by the empirical equation (\(R^{2}=0.929\) and \(nMBE=0.011\)):  
\[k_{d}=\frac{1+7.217K_{csi}k_{t}(\Delta k_{tc})^{2}-0.656K_{csi}^{0.5}k_{t}^{2} +7.947k_{de}+0.187G_{cs}-2.747k_{t}^{2}k_{hour}^{3}-15.379k_{t}\psi(\Delta k_{ tc})^{3}}{1+1.655K_{csi}(\Delta k_{tc})^{3}+14.008\Big{(}k_{day}k_{t}\Big{)}^{3}+17.4 77k_{de}+0.16G_{cs}} \tag{5}\]  
The second proposed model, further denoted M2, is definitely simpler and more accessible than M1. It uses only three predictors, all related to the clearness index: (1) the clearness index itself \(k_{t}\), the daily average of the clearness index \(k_{dny}\), and the hourly average of the clearness index \(k_{hour}\). M2 is defined by the equation: (\(R^{2}=0.917\) and \(nMBE=0.007\)). \[k_{d}=\frac{1-2.525k_{t}+1.834k_{t}^{2}-0.204(k_{t}k_{hour})^{2}}{1-2.578k_{t}+ 1.902k_{t}^{2}+1.054\Big{(}k_{t}k_{dny}\Big{)}^{2}} \tag{6}\]  
The simplicity of M2 does not reduce its ability to capture the essential behavior of the diffuse fraction with respect to the clearness index. It is well illustrated by the sensitivity analysis presented in Figure 1, which displays \(k_{d}\) with respect to \(k_{t}\) for the boundary values of the hourly average of the clearness index: overcast during the whole hour \(k_{hour}=0.3\) (Figure 1a) and clear sky \(k_{hour}=1.0\) (Figure 1b). The curve parameter is \(k_{day}\). In a broad sense, \(k_{day}\) and \(k_{hour}\) measures the cloud cover at two different time scales, hourly and daily, respectively. Figure 1 shows the two parameters acting as a locator for the geometrical place of the \(k_{d}=k_{d}(k_{t})\) curve in the plane \(k_{t}-k_{d}\). As the daily/hourly average of cloudiness increases (measured by a decreasing in \(k_{day}\) and \(k_{hour}\)) the share of \(DHI\) in \(GHI\) increases as well. In other words, as the day and the hourly interval within the diffuse fraction are estimated to become cloudier, Equation (6) increases the likelihood for the diffuse fraction to take higher values. This discussion is valid for the atmospheric column content as well (e.g., atmospheric aerosol loading). Generally, \(k_{day}\) and \(k_{hour}\) capture the whole atmospheric transmittance (atmospheric column content and clouds), the cloud cover is the most influential parameter. Looking again at Figure 2a, it can be seen that the two parameters, \(k_{day}\) and \(k_{hour}\), ascribe to M2 the ability to cover the geometrical place occupied by the measurements in the plane \(k_{t}-k_{d}\).