Perez 1990

Method

The Perez model splits the diffuse sky light into three parts: a uniform dome, a bright ring around the sun (the circumsolar region) and a bright band along the horizon. The circumsolar coefficient $F_1$ moves a share of the diffuse light from the dome to the ring, and the horizon-band coefficient $F_2$ sets the strength of the band. Both coefficients come from a published table: the sky clearness $\varepsilon$ selects the row, and within the row they vary with the sky brightness $\Delta$ and the zenith angle of the sun. On a clear day most of the diffuse light comes from near the sun; under an overcast sky it comes evenly from the dome. The direct sunlight converts to the plane by the cosine of the incidence angle, and the ground-reflected light by the albedo and the tilt. The Perez settings step fixes the solar constant $S$ and its eccentricity factor $e_t$, the source of the direct normal irradiance, the air-mass formula, the altitude factor $f_z$ and the limits $c_\varepsilon$, $F_{1,min}$ and $\alpha_c$.

The math

1. Extraterrestrial irradiance
2. Direct sunlight
3. Sky clearness
4. Sky brightness
5. Circumsolar and horizon-band coefficients
6. Uniform part
7. Circumsolar part
8. Horizon-band part
9. Diffuse sky light on the plane

$$E_0 = S\,e_t \tag{1}$$ $$B = \text{DNI}_{used}\,\max(0,\ \cos\theta) \tag{2}$$ $$\varepsilon = \frac{(D + \text{DNI}_\varepsilon)/D + 1.041\,\theta_z^3}{1 + 1.041\,\theta_z^3},\qquad \text{DNI}_\varepsilon = \min\!\left(\text{DNI}_{used},\ c_\varepsilon\,E_0\right) \tag{3}$$ $$\Delta = \frac{D\,\text{AM}\,f_z}{E_0} \tag{4}$$ $$F_1 = \max\!\left(F_{1,min},\ f_{11} + f_{12}\,\Delta + f_{13}\,\theta_z\right),\qquad F_2 = f_{21} + f_{22}\,\Delta + f_{23}\,\theta_z \tag{5}$$ $$D_{iso} = D\,(1 - F_1)\,\frac{1 + \cos\beta}{2} \tag{6}$$ $$D_{circ} = \begin{cases} D\,F_1\,\dfrac{\max(0,\ \cos\theta)}{\max(\sin 5°,\ \sin\alpha)} & \alpha \ge \alpha_c \\ 0 & \alpha < \alpha_c \end{cases} \tag{7}$$ $$D_{hb} = D\,F_2\sin\beta \tag{8}$$ $$D_{plane} = D_{iso} + D_{circ} + D_{hb} \tag{9}$$

where

Symbol	Meaning	Unit
$E_0$	the extraterrestrial normal irradiance	W/m²
$S$	the solar constant, set by the Perez settings	W/m²
$e_t$	the correction for the Earth's distance from the sun at the time of the timestep, set by the Perez settings	—
$\text{DNI}_{used}$	the direct normal irradiance the model uses, set by the Perez settings	W/m²
$B$	the direct sunlight on the module plane	W/m²
$\theta$	the angle between the sun and the perpendicular to the module plane	degrees
$\varepsilon$	the sky clearness, which selects the row of the coefficient table	—
$D$	the diffuse horizontal irradiance of the weather file	W/m²
$\text{DNI}_\varepsilon$	the direct normal irradiance that enters the sky clearness	W/m²
$\theta_z$	the zenith angle of the sun, 90° minus $\alpha$	radians
$c_\varepsilon$	the cap on the direct part of the clearness, as a fraction of $E_0$, set by the Perez settings	—
$\Delta$	the sky brightness	—
$\text{AM}$	the relative air mass, from the formula the Perez settings select	—
$f_z$	the altitude factor of the sky brightness, set by the Perez settings	—
$F_1$	the circumsolar coefficient	—
$F_2$	the horizon-band coefficient	—
$f_{11} \ldots f_{23}$	the six coefficients of the clearness row, from the published table	—
$F_{1,min}$	the floor of the circumsolar coefficient, set by the Perez settings	—
$D_{iso}$	the uniform part of the diffuse light on the plane	W/m²
$\beta$	the tilt of the module plane from the horizontal	degrees
$D_{circ}$	the circumsolar part of the diffuse light on the plane	W/m²
$\alpha$	the elevation of the sun above the horizon	degrees
$\alpha_c$	the sun elevation below which the circumsolar part is zero, set by the Perez settings	degrees
$D_{hb}$	the horizon-band part of the diffuse light on the plane	W/m²
$D_{plane}$	the diffuse sky light on the module plane	W/m²

References

1. Perez, Ineichen, Seals, Michalsky & Stewart 1990, Modeling daylight availability and irradiance components from direct and global irradiance, Solar Energy 44(5) 271–289

API

array.transposition_model = perez

Symbols follow the symbol registry.