Semi-parametric modelling of sun position dependent solar gain using Bsplines in grey-box models

Christoffer Rasmussen, Linde Frölke, Peder Bacher, Henrik Madsen, Carsten Rode

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Modelling the effects of solar irradiation plays an important role in various applications. This paper describes a semi-parametric (combined grey-box and spline-based), data-driven technique that can be used to model systems in which the solar gain depends on the sun position. The solar gain factor is introduced, i.e. the absorbed fraction of measured solar irradiation, and estimated as a continuous non-parametric function of the sun position. The implementation of the spline-based solar gain factor in a grey-box model framework is described. The method is tested in two case studies—in a model of the internal temperature of a dwelling in Aalborg, Denmark, and a model of the return temperature of a solar collector field in Solrød, Denmark. It is shown that the solar gain factor as a function of sun position is able to account for structural variations in solar gain that may occur due to factors such as shading obstacles and window or absorber orientation. In both test cases, the spline-based solar gain function improved the model accuracy significantly, and largely reduced structural errors in prediction residuals. In addition, the shape of the estimated function provided insight into the dynamics of the system and the local solar input characteristics. Accurate representation of such site characteristics was not possible with any data-driven method found in the literature. Besides the grey-box models used in this study, the solar gain factor can be used in a variety of data-driven models, for example in linear regression models.
Original languageEnglish
JournalSolar Energy
Pages (from-to)249-258
Publication statusPublished - 2020


  • Solar gain modelling
  • Grey-box modelling
  • Splines
  • Thermal dynamics
  • Building energy
  • Solar heat collectors


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