## 4 A Proposal for Minute-Scale GHI Separation Models
This perspective will be discussed next in more detail. Figure 2 shows how the estimates of the seven models cover the measurements in the plane \(k_{t}-k_{d}\). Figure 2a, already referred to, displays the measured diffuse fraction vs. the clearness index for D_FIT. This is a typical picture whose merit is to illustrate the wide dispersion of \(k_{d}\) compared to the classical predictor \(k_{t}\). The estimates issued by a simple equation \(k_{d}=k_{d}(k_{t})\), irrespective of their complexity, will always be accompanied by a substantial amount of uncertainty. Figure 2b-f displays the estimates issued by the two proposed models (M1 (Equation (5)) and (M2 (Equation (6)) and the reference models  
Figure 1: Diffuse fraction \(k_{d}\) estimated with M2 (Equation (6)) with respect to the clearness index \(k_{t}\) for two values of the hourly average of the clearness index \(k_{hour}\) (**a**) 0.3 and (**b**) 1.0. The curve parameter is the daily average of the clearness index \(k_{day}\). Figure 2: Diffuse fraction \(k_{d}\) vs. the clearness index \(k_{t}\) at the station in Payerme, Switzerland, in 2020. Measured (in gray) and estimated (in red) values, with the tested separation models, are displayed. PB (Equation (A2)), E2 (Equation (A5)), S1 (Equation (A6)), S3 (Equation (A7)), and Y4 (Equation (A8)) superimposed over the measurements. This allows a visual intercomparison of the models' flexibility with the variation of the atmospheric conditions captured by the input parameters. PB (Equation (A2)) appears to overlap the data reasonably well (Figure 2b) but is experiencing a relative underestimation of the diffuse fraction. The overlay of E2 over the measurements is lower, but the appearance and position of the estimates cluster indicate an accurate capture of the data mean (Figure 2c). S1 captures almost all the areas of data (Figure 2d). S3 best covers the space occupied by the measured data (Figure 2e). Y4 also appears to overlap the data reasonably well while displaying a lower dispersion than S3 (Figure 2f). As expected, the proposed models M1 and M2 cover the measurements to a large extent (Figure 2g,h), since the two models are fitted on D_FIT. The flexibilities of M1 and S3 are comparable, explained by their relative high complexity: M1 operates with eight predictors, while S3 operates with seven predictors on two branches. Looking at Figure 2 as a whole, it can be concluded that the proposed M1 and M2 models achieve with rational polynomial equations a similar behavior as the reference logistic models. Figure 3 displays the density plot of \(k_{d}\) vs. \(k_{t}\) at the station in Payerne. It highlights the area in plane \(k_{t}-k_{d}\) where the measurements (Figure 3a) and the estimates issued by each model (Figure 3b-h) are clustered. Thus, Figure 3 gives us a better perspective on how the models locate the estimated pairs \((k_{t},k_{d})\) compared to the measured ones. In general, the models demonstrate the ability to agglomerate the estimates in the regions of the \(k_{d}-k_{t}\) space where the measurements show the highest density. It is useful to remember that the reference models are evaluated as high performing by independent studies [5]. A visual comparison between Figures 3a and 3g indicates that the M1 estimates meet almost the same distribution as the measured joint probability \(k_{t}-k_{d}\), especially in areas with high probability density. Visible differences appear in the area with intermediate values of the clearness index, but fortunately the probability density is much lower here.