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Summary

The Hay-Davies transposition model separates sky diffuse irradiance into two components: a circumsolar component concentrated around the solar disk and an isotropic background component distributed uniformly across the sky dome. The model uses an anisotropy index based on atmospheric transmittance to weight the circumsolar fraction. This simple yet effective approach provides a balance between computational efficiency and physical accuracy.

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Inputs

NameSymbolUnitsDescription
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
Circumsolar Treatment——Choice of Direct or Diffuse allocation

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Outputs

NameSymbolUnitsDescription
Circumsolar ComponentEcircumsolarE_{circumsolar}W/m²Diffuse irradiance concentrated near solar disk
Isotropic ComponentEisotropicE_{isotropic}W/m²Uniform diffuse irradiance from sky dome
Sky Diffuse on TiltEskyE_{sky}W/m²Total sky diffuse irradiance on tilted surface

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Detailed Description

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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}} If the solar exceeds the zenith limit (θz≥86.5°\theta_z \geq 86.5°), then Ai=0A_i = 0. The anisotropy index ranges from 0 (overcast, isotropic diffuse) to 1 (clear sky, high circumsolar fraction).

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Projection Ratio

The projection ratio RbR_b relates the beam irradiance on the tilted surface to that on a horizontal surface: Rb=cos⁡(θ)cos⁡(θz)R_b = \frac{\cos(\theta)}{\cos(\theta_z)} where cos⁡(θ)\cos(\theta) is the projection of the beam onto the tilted surface normal and cos⁡(θz)\cos(\theta_z) is the projection onto the horizontal. To avoid numerical instabilities near sunrise/sunset, a minimum threshold is applied: cos⁡(θz)≥0.01745\cos(\theta_z) \geq 0.01745 (corresponding to θz≤89°\theta_z \leq 89°).

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Circumsolar Component

The circumsolar component of sky diffuse irradiance is proportional to the anisotropy index and projection ratio: Ecircumsolar=DHI⋅Ai⋅RbE_{circumsolar} = DHI \cdot A_i \cdot R_b This formulation assumes the circumsolar diffuse is concentrated around the sun and follows the same geometric projection as beam irradiance.

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Isotropic Component

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

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Circumsolar Allocation

The circumsolar and isotropic components are allocated to beam or diffuse POA irradiance based on user selection: Direct Allocation: Esky=EisotropicE_{sky} = E_{isotropic} Ebeam=DNI⋅cos⁡(θ)+EcircumsolarE_{beam} = DNI \cdot \cos(\theta) + E_{circumsolar} Diffuse Allocation: Esky=Eisotropic+EcircumsolarE_{sky} = E_{isotropic} + E_{circumsolar} Ebeam=DNI⋅cos⁡(θ)E_{beam} = DNI \cdot \cos(\theta) Direct allocation is typically used when circumsolar diffuse is expected to experience shading and soiling losses similar to beam irradiance.

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Quality Control

Physical constraints are applied to prevent negative or unrealistic values:
  • Ecircumsolar≥0E_{circumsolar} \geq 0
  • Eisotropic≥0E_{isotropic} \geq 0
  • cos⁡(θ)≥0\cos(\theta) \geq 0 (if negative, Ebeam=0E_{beam} = 0)

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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.
  • Reindl, D. T., Beckman, W. A., & Duffie, J. A. (1990). Evaluation of hourly tilted surface radiation models. Solar Energy, 45(1), 9–17.
  • Holmgren, W. F., Hansen, C. W., & Mikofski, M. A. (2018). pvlib python: A python package for modeling solar energy systems. Journal of Open Source Software, 3(29), 884.