The Hay transposition model
The Hay transposition model applies differently to the different components of the irradiance.
The Beam component results of a pure geometrical transformation (no model - no intrinsic error):
BeamInc = BeamHor * sin Hsoli / sin Hsol
The Diffuse component is supposed to be mainly constituted of an isotropic distribution, and a circum-solar contribution proportional to Kb
DiffInc = DiffHor * [ (1-Kb) * (1 + cos i) / 2 + Kb * sin HsolI / sin Hsol ]
The Albedo component is the irradiance reflected by the ground "seen" by the plane :
AlbInc = ρ * GlobHor * (1 - cos i) / 2
where
| i | = Plane tilt |
| Hsol | = Sun height on horizontal plane |
| Hsoli | = Sun height on the plane (= 90° - incidence angle) |
| Kb | = Clearness index of beam = BeamHor / (Io * Sin Hsol) |
| Io | = Solar constant (depends on the day of year) |
| ρ | = Albedo coefficient (usual value 0.2) |
The expression (1 + cos i) / 2 is the mathematical result of the spherical integral of a constant irradiance, coming from all directions "seen" by the plane (i.e. the orange slice between the plane and the horizontal).
NB: All transposition models are highly dependent on the diffuse component. The higher diffuse, the lower transposed irradiance in monthly or annual values.
This is usually evaluated from another model (Liu-Jordan or Erbs) and represents the main uncertainty in the transposition result.