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# Hay-Davies Model

export const ViewFactor = () => <Tooltip tip="Fraction of radiation leaving one surface that reaches another; determined by geometry.">
    view factor
  </Tooltip>;

export const AnisotropyIndex = () => <Tooltip tip="Ratio of DNI to extraterrestrial DNI; quantifies the directional fraction of diffuse irradiance.">
    anisotropy index
  </Tooltip>;

export const Isotropic = () => <Tooltip tip="Uniform in all directions; describes diffuse irradiance distributed evenly across the sky dome.">
    isotropic
  </Tooltip>;

export const Circumsolar = () => <Tooltip tip="Diffuse irradiance concentrated in a bright ring around the solar disk.">
    circumsolar
  </Tooltip>;

export const Albedo = () => <Tooltip tip="Ground reflectance (0–1); fraction of incident light reflected by the surface.">
    albedo
  </Tooltip>;

export const GHI = () => <Tooltip tip="Global Horizontal Irradiance: total solar radiation on a horizontal surface (direct + diffuse).">
    GHI
  </Tooltip>;

export const DHI = () => <Tooltip tip="Diffuse Horizontal Irradiance: solar radiation from the sky dome (excluding direct beam) on a horizontal surface.">
    DHI
  </Tooltip>;

export const DNI = () => <Tooltip tip="Direct Normal Irradiance: solar radiation received perpendicular to the sun's rays (direct beam only).">
    DNI
  </Tooltip>;

export const ZenithAngle = () => <Tooltip tip="Angle between the local vertical and the sun (0° = overhead, 90° = horizon).">
    zenith angle
  </Tooltip>;

## 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 <AnisotropyIndex /> based on atmospheric transmittance to weight the circumsolar fraction, derived from the ratio of <DNI /> to extraterrestrial DNI. Unlike the Perez model, Hay-Davies does not include a horizon brightening component.

## Inputs

| Name                                          | Symbol         | Units   | Description                                   |
| --------------------------------------------- | -------------- | ------- | --------------------------------------------- |
| **Global Horizontal Irradiance**              | $GHI$          | W/m²    | Total irradiance on horizontal surface        |
| **Direct Normal Irradiance**                  | $DNI$          | W/m²    | Direct beam irradiance perpendicular to sun   |
| **Diffuse Horizontal Irradiance**             | $DHI$          | W/m²    | Diffuse irradiance on horizontal surface      |
| **Extraterrestrial Direct Normal Irradiance** | $DNI_{extra}$  | W/m²    | Direct normal irradiance at top of atmosphere |
| **Solar Zenith Angle**                        | $\theta_z$     | degrees | Angle between sun and local vertical          |
| **Angle of Incidence**                        | $\theta_{AOI}$ | degrees | Angle between sun and surface normal          |
| **Surface Tilt Angle**                        | $\beta_m$      | degrees | Tilt angle of surface from horizontal         |
| **Albedo**                                    | $\rho$         | —       | Ground reflectance (0–1)                      |

***

## Outputs

| Name                   | Symbol       | Units | Description                                   |
| ---------------------- | ------------ | ----- | --------------------------------------------- |
| **POA Beam**           | $G_{beam}$   | W/m²  | Direct beam irradiance on tilted surface      |
| **POA Sky Diffuse**    | $G_{sky}$    | W/m²  | Sky diffuse irradiance on tilted surface      |
| **POA Ground Diffuse** | $G_{ground}$ | W/m²  | Ground-reflected irradiance on tilted surface |

***

## Detailed Description

### Anisotropy Index

The Hay-Davies model uses an anisotropy index $A_i$ to quantify the fraction of diffuse irradiance exhibiting directional characteristics. The anisotropy index represents the ratio of beam transmittance through the atmosphere:

$$
A_i = \frac{DNI}{DNI_{extra}}
$$

At high zenith angles where air mass calculations become unreliable ($\theta_z \geq 87.9°$), the anisotropy index is set to zero ($A_i = 0$). The anisotropy index ranges from 0 (overcast conditions with fully isotropic diffuse) to 1 (clear sky with high circumsolar fraction), though in practice $A_i$ 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:

$$
G_{circumsolar} = DHI \cdot A_i \cdot \frac{\cos(\theta_{AOI})}{\cos(\theta_z)}
$$

The ratio $\cos(\theta_{AOI})/\cos(\theta_z)$ is the projection ratio that converts horizontal circumsolar irradiance to the tilted plane. When the sun is behind the module ($\cos(\theta_{AOI}) < 0$), the circumsolar component is set to zero. To avoid numerical instabilities near the horizon, a minimum threshold is applied: $\cos(\theta_z) \geq 0.01745$ (corresponding to $\theta_z \leq 89°$).

### Isotropic Component

The isotropic component represents uniform diffuse irradiance distributed across the sky dome:

$$
G_{isotropic} = DHI \cdot (1 - A_i) \cdot \frac{1 + \cos(\beta_m)}{2}
$$

The term $(1 + \cos(\beta_m))/2$ is the <ViewFactor /> 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 (default: Diffuse):

**Circumsolar Allocation to Direct Beam:**

$$
G_{sky} = G_{isotropic}
$$

$$
G_{beam} = DNI \cdot \cos(\theta_{AOI}) + G_{circumsolar}
$$

**Circumsolar Allocation to Sky Diffuse:**

$$
G_{sky} = G_{isotropic} + G_{circumsolar}
$$

$$
G_{beam} = DNI \cdot \cos(\theta_{AOI})
$$

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:

$$
G_{ground} = \frac{1}{2} \cdot \rho \cdot GHI \cdot (1 - \cos(\beta_m))
$$

The term $(1 - \cos(\beta_m))/2$ is the view factor from the tilted surface to the ground.

### Quality Control

Physical constraints are enforced by clamping values:

* If $G_{isotropic} < 0$ → $G_{isotropic} = 0$
* If $G_{circumsolar} < 0$ → $G_{circumsolar} = 0$
* If $G_{beam} < 0$ → $G_{beam} = 0$
* If $G_{sky} > 800$ W/m² → $G_{sky} = 800$ 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.