3D isotropic approximation for solar diffuse irradiance on tilted surfaces

A well known 2D approach of Liu and Jordan allows computing isotropic solar diffuse irradiance on a tilted surface. It is a 2D theory as the position of a sky element is characterized by a single (zenith) angle. A more realistic 3D model (that uses both zenith and azimuth angles to describe sky element's position) is developed in this paper for both isotropic diffuse irradiance and ground reflected irradiance incident on an arbitrary oriented surface. The 3D formula predicts a lower diffuse irradiance than the 2D relationship while the ground reflected irradiance is higher in case of the 3D model than in case of the 2D approach. In case of a small tilt angle, the 2D and 3D approximations predict comparable values, higher than the mean of the results obtained with a (reference) non-isotropic model. However, the 3D model is slightly more precise. When a larger tilt angle is considered, the 3D model predicts a few percent larger value than the mean of the values estimated by the reference model while the 2D model gives a significantly higher value.