19-bin histogram

Method

The bins are 5° wide and start at 0°. The sky is taken as uniform, and each direction of the sky in front of the plane goes into a bin by its incidence angle on the plane (1). There are 19 bins, and the 19th, from 90° to 95°, stays empty because no direction that reaches the plane is 90° or more from its normal. The weight of a bin is the light on the plane that arrives through its directions, as a share of the light from the whole visible sky (3). The light of a direction is proportional to its solid angle times the cosine of its incidence angle.

The ground-reflected light is placed in the bins as follows. The ground reflects uniformly, and its directions below the horizon in front of the plane get their weights the same way (4). Each ground direction goes into the bin of the horizontal direction with the same azimuth, whose incidence angle is given by (2). On a plane tilted 35°, the ground light therefore falls in the bins from 55° up. A fixed plane tilted less than 2° sees no ground, and its ground factor is 1.

The grid has 40 heights by 90 azimuths, and each weight is counted at the centre of a cell. The sky grid runs from the horizon to the zenith in steps of 2.25° and all round in steps of 4°. The ground grid runs down from the horizon in 40 rows, over a half turn of azimuth in steps of 2°. Each row is the tilt divided by 40 deep, and at least 0.5°. From 20° of tilt up, the rows end at the tilt angle. Below 20° they reach 20° under the horizon, and the rows deeper than the tilt see no ground in front of the plane and add nothing. The plane is symmetric about its azimuth, so a half turn is enough.

The distribution gives two factors: the angular-response curve is read at the centre of each bin (5), and each factor is the weighted sum of these readings (6). The weights depend only on the tilt, so the rows in front and the horizon do not enter. Fixed rows use one set of weights. Trackers use a table of factors every 5° of the plane's tilt, read at the tilt of each timestep and interpolated linearly. The ground factor's table starts at 2°, the smallest tilt that sees the ground, and a tracker tilted less than 2° keeps the ground factor at 2°.

Settings

Setting	Value
Bin width	5°
Bins	19 from 0°; the 19th, from 90° to 95°, stays empty
Grid	40 heights × 90 azimuths, read at the cell centres
Sky grid step	2.25° in height, 4° in azimuth
Ground grid step	the tilt divided by 40 in height, at least 0.5°; 2° in azimuth
Smallest tilt that sees the ground	2°
Tracker table	every 5° of the plane's tilt; the ground factor from 2°

The math

1. Incidence angle of a direction
2. Incidence angle given to a ground direction
3. Sky weights
4. Ground weights
5. Bin centres
6. Factors

$$\cos\theta = \cos\gamma_r\,\cos h\,\sin\beta + \sin h\,\cos\beta \tag{1}$$ $$\theta_0 = \arccos\left(\cos\gamma_r\,\sin\beta\right) \tag{2}$$ $$w_{sky,k} = \frac{\sum_{\mathcal{S}_k} \cos\theta\,\Delta\Omega}{\sum_{\mathcal{S}} \cos\theta\,\Delta\Omega},\qquad \Delta\Omega = \cos h\,\Delta h\,\Delta\gamma_r \tag{3}$$ $$w_{gnd,k} = \frac{\sum_{\mathcal{G}_k} \cos\theta\,\Delta\Omega}{\sum_{\mathcal{G}} \cos\theta\,\Delta\Omega} \tag{4}$$ $$\theta_k = 5\,(k + 0.5) \tag{5}$$ $$F_{IAM,sky} = \sum_{k=0}^{18} w_{sky,k}\,F(\theta_k),\qquad F_{IAM,gnd} = \sum_{k=0}^{18} w_{gnd,k}\,F(\theta_k) \tag{6}$$

where

Symbol	Meaning	Unit
$\theta$	the incidence angle of a direction on the plane	degrees
$\gamma_r$	the azimuth of the direction, measured from the azimuth the plane faces	degrees
$h$	the height of the direction above the horizon, negative below it	degrees
$\beta$	the tilt of the plane	degrees
$\theta_0$	the incidence angle of the horizontal direction with azimuth $\gamma_r$	degrees
$w_{sky,k}$	the share of the sky light on the plane that falls in bin $k$	—
$k$	the bin number, 0 to 18; bin $k$ holds the angles from $5k$ to $5k + 5$°	—
$\mathcal{S}$	the grid cells of the sky in front of the plane, where $\cos\theta > 0$	—
$\mathcal{S}_k$	the cells of $\mathcal{S}$ whose incidence angle $\theta$ falls in bin $k$	—
$\Delta\Omega$	the solid angle of a grid cell	sr
$\Delta h,\ \Delta\gamma_r$	the height and azimuth steps of the grid	radians
$w_{gnd,k}$	the share of the ground-reflected light on the plane that falls in bin $k$; 0 in every bin when the tilt is below 2°	—
$\mathcal{G}$	the grid cells of the ground in front of the plane, where $\cos\theta > 0$	—
$\mathcal{G}_k$	the cells of $\mathcal{G}$ whose angle $\theta_0$ falls in bin $k$	—
$\theta_k$	the centre of bin $k$	degrees
$F_{IAM,sky}$	the angular-response factor for the sky light	—
$F_{IAM,gnd}$	the angular-response factor for the ground-reflected light; 1 when a fixed plane sees no ground	—
$F(\theta)$	the angular-response curve: the accepted fraction at incidence angle $\theta$	—

References

1. PVsyst 6.8.8, the program's computation of the incidence-angle distribution of the isotropic diffuse and of the albedo, shown in its Detailed Study of IAM dialog (Effect on Diffuse and Albedo tab): the source of this method's bins, grid and ground rule, read from the program
2. PVsyst 6.8 help, Array incidence loss (IAM)

API

iam.diffuse_integration = histogram

Symbols follow the symbol registry.