Skip to main content

​
Summary

The Hay-Davies transposition model separates sky diffuse irradiance into two components: a component concentrated around the solar disk and an background component distributed uniformly across the sky dome. The model uses an based on atmospheric transmittance to weight the circumsolar fraction, derived from the ratio of to extraterrestrial DNI. Unlike the Perez model, Hay-Davies does not include a horizon brightening component.

​
Inputs

NameSymbolUnitsDescription
Global Horizontal IrradianceGHIGHIW/m²Total irradiance on horizontal surface
Direct Normal IrradianceDNIDNIW/m²Direct beam irradiance perpendicular to sun
Diffuse Horizontal IrradianceDHIDHIW/m²Diffuse irradiance on horizontal surface
Extraterrestrial DNIDNIextraDNI_{extra}W/m²Direct normal irradiance at top of atmosphere
Solar Zenith Angleθz\theta_zdegreesAngle between sun and local vertical
Angle of Incidenceθ\thetadegreesAngle between sun and surface normal
Surface Tilt Angleβ\betadegreesTilt angle of surface from horizontal
Albedoρ\rho—Ground reflectance (0–1)

​
Outputs

NameSymbolUnitsDescription
POA BeamGbeamG_{beam}W/m²Direct beam irradiance on tilted surface
POA Sky DiffuseGskyG_{sky}W/m²Sky diffuse irradiance on tilted surface
POA Ground DiffuseGgroundG_{ground}W/m²Ground-reflected irradiance on tilted surface

​
Detailed Description

​
Anisotropy Index

The Hay-Davies model uses an anisotropy index AiA_i to quantify the fraction of diffuse irradiance exhibiting directional characteristics. The anisotropy index represents the ratio of beam transmittance through the atmosphere: Ai=DNIDNIextraA_i = \frac{DNI}{DNI_{extra}} At high zenith angles where air mass calculations become unreliable (θz≥87.9°\theta_z \geq 87.9°), the anisotropy index is set to zero (Ai=0A_i = 0). The anisotropy index ranges from 0 (overcast conditions with fully isotropic diffuse) to 1 (clear sky with high circumsolar fraction), though in practice AiA_i rarely exceeds ~0.8 due to atmospheric scattering even under clear skies.

​
Circumsolar Component

The circumsolar component represents diffuse irradiance concentrated around the solar disk. This formulation assumes circumsolar diffuse follows the same geometric projection as beam irradiance: Gcircumsolar=DHI⋅Ai⋅cos⁡(θ)cos⁡(θz)G_{circumsolar} = DHI \cdot A_i \cdot \frac{\cos(\theta)}{\cos(\theta_z)} The ratio cos⁡(θ)/cos⁡(θz)\cos(\theta)/\cos(\theta_z) is the projection ratio that converts horizontal circumsolar irradiance to the tilted plane. When the sun is behind the module (cos⁡(θ)<0\cos(\theta) < 0), the circumsolar component is set to zero. To avoid numerical instabilities near the horizon, a minimum threshold is applied: cos⁡(θz)≥0.01745\cos(\theta_z) \geq 0.01745 (corresponding to θz≤89°\theta_z \leq 89°).

​
Isotropic Component

The isotropic component represents uniform diffuse irradiance distributed across the sky dome: Gisotropic=DHI⋅(1−Ai)⋅1+cos⁡(β)2G_{isotropic} = DHI \cdot (1 - A_i) \cdot \frac{1 + \cos(\beta)}{2} The term (1+cos⁡(β))/2(1 + \cos(\beta))/2 is the from the tilted surface to the sky dome.

​
Circumsolar Allocation

The circumsolar component can be allocated to beam or diffuse sky POA irradiance based on user selection: Circumsolar Allocation to Direct Beam: Gsky=GisotropicG_{sky} = G_{isotropic} Gbeam=DNI⋅cos⁡(θ)+GcircumsolarG_{beam} = DNI \cdot \cos(\theta) + G_{circumsolar} Circumsolar Allocation to Sky Diffuse: Gsky=Gisotropic+GcircumsolarG_{sky} = G_{isotropic} + G_{circumsolar} Gbeam=DNI⋅cos⁡(θ)G_{beam} = DNI \cdot \cos(\theta) Circumsolar irradiance originates from near the solar disk and is blocked by obstructions the same way direct beam is. Allocating it to beam ensures that row-to-row shading calculations apply appropriate losses to circumsolar. Allocating to diffuse treats circumsolar as unaffected by direct shading, which may be appropriate for unshaded systems or when shading is negligible.

​
Ground-Reflected Component

The ground-reflected component accounts for irradiance reflected from the ground onto the tilted surface: Gground=12⋅ρ⋅GHI⋅(1−cos⁡(β))G_{ground} = \frac{1}{2} \cdot \rho \cdot GHI \cdot (1 - \cos(\beta)) The term (1−cos⁡(β))/2(1 - \cos(\beta))/2 is the view factor from the tilted surface to the ground.

​
Quality Control

Physical constraints are enforced by clamping values:
  • If Gisotropic<0G_{isotropic} < 0 → Gisotropic=0G_{isotropic} = 0
  • If Gcircumsolar<0G_{circumsolar} < 0 → Gcircumsolar=0G_{circumsolar} = 0
  • If Gbeam<0G_{beam} < 0 → Gbeam=0G_{beam} = 0
  • If Gsky>800G_{sky} > 800 W/m² → Gsky=800G_{sky} = 800 W/m²

​
References

  • Hay, J. E., & Davies, J. A. (1980). Calculation of the solar radiation incident on an inclined surface. Proceedings of First Canadian Solar Radiation Data Workshop, Toronto, p. 59.