# Evaluation and Comparison of Spatial Clustering for Solar Irradiance Time Series
## 2 Solar Irradiation Time-Series Analysis
### Clear Sky Model  
The solar irradiation time-series GHI (for global horizontal irradiation or irradiance) exhibits a seasonality and a periodicity (a yearly cycle and a diurnal one). Most of the machine learning and AI methods can only be used with stationary time series [33] (the \"weak\" assumption is sufficient). To \"delete\" this periodicity, it is common to compute a ratio called clear-sky index and defined as GHI on the solar irradiance in clear-sky conditions (denoted GHIcs), kt = GHI/GHIcs. This one results in a normalized quantity theoretically comprised between 0 and 1. A clear-sky model estimates the solar irradiance that reaches the ground surface when the sky is clear without clouds. Such models were developed and differ from each other mainly in the inputs needed by each model [34]; they generally used meteorological variables (such as ozone layer thickness, precipitable water, aerosol optical depth, etc.) and solar geometry (solar elevation, declination, etc.) [35]. The most widely used clear-sky models are the Solis model developed by [PERSON] et al. [36] and simplified by [PERSON] [37], the European Solar Radiation Atlas (ESRA) model [38], the Reference Evaluation on Solar Transmittance 2 (REST2) model [39], and the McClear Model [40; 41; 42] (this list is not exhaustive). The New Heliosat-4 method [40] is an operational tool for solar irradiance monitoring in the framework of the projects \"Monitoring Atmospheric Composition and Climate (MACC)\" [43] and the Copernicus program. This method processes Meteosat images to create the CAMs radiation service (Copernicus Atmosphere Monitoring Service) used to calculate the clear-sky solar radiation, to estimate the cloud effect (coupled with APOLLO/SEV from DLR), and the ground albedo (derived from MODIS).  
It allows us to have an efficient model with a fast implementation because it is available via a web service that gives time series of horizontal global, horizontal diffuse, and normal beam irradiances for a given point and a given period, from 2004 up to current day \(-\) 2 (minus two days), with a time step from 1 min to 1 day. For this paper, only the data of clear-sky solar irradiance were uploaded via [[http://www.soda-pro.com](http://www.soda-pro.com)]([http://www.soda-pro.com](http://www.soda-pro.com)) (accessed on 10 March 2021). More information about the McClear Model and the upload platform can  
Figure 1: Positions of the weather stations considered in this study with solar data available from 1 January 2017 to 31 December 2020. In red, stations are not used because the data are not considered reliable. In yellow, the stations used. The geographical coordinates of the weather stations can be found in Reference [31].  
be found in references [40,41]. Figure 2 shows, as an example, the superposition of the series generated with this model (clear sky) and SIAR measurements (in real meteorological conditions) on the same locations (here, stations V29 and C01; see Figure 1).  
It is clear that the clear-sky irradiance estimated by CAMs can be quasi-perfectly superimposed onto the data measured on the ground level for days with clear skies. Thus, the quality of the CAM model is shown. It is possible to assess, more globally, the adequacy between measurement and model over a significant period (annual) for the same stations (see Figure 3).  
Using such a methodology, all the data, for each station, were carefully checked, and only stations having passed the tests correctly were selected in this study. As an intermediate outcome, the McClear model provides a good estimate of the clear-sky irradiance, but as shown in the next section, the results can still be improved.  
Figure 3: Presentation of GHI and GHIcs for 2 meteorological stations (V29 and C01) in 2020.  
Figure 2: Presentation of GHI and GHIcs for 2 meteorological stations (V29 and C01) and 2 periods of 5 days in 2020 (winter and spring).