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The Hay transposition model

Model

The Hay or Hay-Davies transposition model1 applies different transformations to the different irradiance components.

The Beam component follows a pure geometrical transformation (no model - no intrinsic error):

\[ \mathsf{BeamInc} = \mathsf{BeamHor}\, \frac{\cos \mathsf{AngInc} }{\sin \mathsf{HSol}}\,. \]

The Diffuse component is supposed to be decomposed in an isotropic distribution, and a circumsolar contribution proportional to \(K_b\)

\[ \mathsf{DiffInc} = \mathsf{DiffHor}\,(1-K_b) (1 + \cos \theta) / 2 \,,\\ \mathsf{CircInc} = \mathsf{DiffHor}\,K_b\,\frac{\cos \mathsf{AngInc}} {\sin \mathsf{HSol}}\,. \]

The Albedo component is the irradiance reflected by the ground "seen" by the plane :

\[ \mathsf{AlbInc} = \rho\; \mathsf{GlobHor}\,(1 - \cos \theta) / 2\,. \]

where
\(\theta\) = Plane tilt
\(\mathsf{HSol}\) = Sun height on horizontal plane
\(\mathsf{AngInc}\) = Incidence angle
\(K_b\) = Clearness index of beam \(= \mathsf{BeamHor} / (I_o \sin \mathsf{HSol})\)
\(I_o\) = Solar constant (depends on the day of year)
\(\rho\) = Albedo coefficient (usual value 0.2)

The expression \((1 + \cos \theta ) / 2\) is the mathematical result of the spherical integral of a constant irradiance, coming from all directions "seen" by the plane (i.e. the spherical wedge between the plane and the horizontal).

Warning

With the old transposition methodology (replaced by a new default since PVsyst 7),

\[ \mathsf{DiffInc} = \mathsf{DiffHor}\left[(1-K_b) (1 + \cos \theta) / 2 + \,K_b\,\frac{\cos \mathsf{AngInc}} {\sin \mathsf{HSol}}\right] \,,\enspace\mathsf{CircInc} = 0\,. \]

This difference had consequences in particular for shadings and IAM losses.

Info

All transposition models are highly dependent on the diffuse component. The higher diffuse, the lower transposed irradiance in monthly or annual values.

The diffuse component is usually evaluated from another model (Liu-Jordan or Erbs) and represents the main uncertainty in the transposition result.

Cutoff for low sun heights

PVsyst applies a cutoff for low sun heights \(\mathsf{HSol} < 2°\). In such a case:

\[ \mathsf{BeamInc} = 0\,,\enspace\mathsf{CircInc} = 0\,. \]

  1. John E. Hay and J. A. Davies. Calculations of the solar radiation incident on an inclined surface. In John E. Hay and K. Won Thorne, editors, Proceedings: First Canadian Solar Radiation Data Workshop. Ottawa, Canada : Supply and Services Canada, 1980. URL: https://archive.org/details/proceedingsfirst00cana/mode/2up. ↩