Crossing fraction + diffuse integrals

Method

The sun's path through the sunlit part of each timestep is sampled 24 times, at the middles of 24 equal parts. At each sample the sun height is compared with the skyline height in the sun's direction, and the direct and circumsolar light are scaled by the mean of the sample factors (1). A band 5° wide around the skyline is treated as partly transparent to smooth the transition: the factor is 0 with the sun 2.4° below the skyline, ½ at 0.1° above it and 1 at 2.6° above it. The sky diffuse light is reduced by the share of the sky in front of the plane that lies above the skyline, weighted by the cosine of the incidence angle (2). The ground-reflected light is reduced by a factor that falls linearly from 1 for an open horizon to 0 for a skyline 20° high, averaged over the ground the plane sees and multiplied by the fraction of albedo taken into account (Site tab) (3). Both factors are computed once per plane orientation and held constant over the year; a tracker uses the factors of its orientation at each timestep.

The math

1. Direct and circumsolar light
2. Sky diffuse light
3. Ground-reflected light

$$f_{b,i} = \frac{1}{24}\sum_{k=1}^{24} \text{clamp}\!\left(\tfrac{1}{2} + \frac{\alpha_k - \alpha_{hrz}(\gamma_{s,k}) - 0.1°}{5°},\ 0,\ 1\right) \tag{1}$$ $$f_{sky} = \frac{\int_{e > \alpha_{hrz}(\gamma)} \cos\theta\,d\Omega}{\int \cos\theta\,d\Omega} \tag{2}$$ $$f_{gnd} = a_{hrz}\,\frac{\int \max\!\left(0,\ 1 - \alpha_{hrz}(\gamma)/20°\right) w(\gamma)\,d\gamma}{\int w(\gamma)\,d\gamma} \tag{3}$$

where

Symbol	Meaning	Unit
$f_{b,i}$	the fraction of the direct and circumsolar light kept at timestep $i$	—
$\alpha_k$	the sun elevation at sample $k$	degrees
$\alpha_{hrz}(\gamma)$	the skyline elevation in the direction $\gamma$, clockwise from north	degrees
$\gamma_{s,k}$	the sun's azimuth at sample $k$, clockwise from north	degrees
$\text{clamp}(x, 0, 1)$	$x$ limited to the range 0 to 1	—
$f_{sky}$	the fraction of the sky diffuse light kept	—
$\theta$	the angle between the direction $\Omega$ and the perpendicular to the module plane	degrees
$d\Omega$	an element of solid angle of the sky in front of the plane, at elevation $e$ and azimuth $\gamma$	sr
$f_{gnd}$	the fraction of the ground-reflected light kept	—
$a_{hrz}$	the fraction of albedo taken into account, Site tab	—
$w(\gamma)$	the view the plane has of the ground in the direction $\gamma$: the integral of the cosine of the incidence angle over the ground below the horizon in that direction	rad

References

1. Quaschning & Hanitsch 1998, Irradiance calculation on shaded surfaces, Solar Energy 62(5) 369–375
2. Analysis of PVsyst 8.1.5 outputs shows this convention in its results.

API

options.far_shading_model = crossing_fraction, site.use_horizon = true

Symbols follow the symbol registry.