Incidence Angle on a Module

The incidence angle $\theta$ on any plane is the angle between the normal to a plane's surface and the incident radiation vector. Modules in different racks may have differing incidence angles, so the incidence angle is calculated separately for each rack in the layout using the effective in this case. For a rack with effective tilt angle $\beta$ and azimuth $\gamma$, the incidence angle $\theta$ of direct radiation on the plane of modules mounted on the rack (the "Plane of the Array", POA) is given by [9] as:

$$\begin{align} \cos \left( \theta \right) &= \sin \left( \delta \right) \sin \left( \phi \right) \cos \left( \beta \right) \\ &- \sin \left( \delta \right) \cos \left( \phi \right) \sin \left( \beta \right) \cos \left( \gamma \right) \\ &+ \cos \left( \delta \right) \cos \left( \phi \right) \cos \left( \beta \right) \cos \left( \omega \right) \\ &+ \cos \left( \delta \right) \sin \left( \phi \right) \sin \left( \beta \right) \cos \left( \gamma \right) \cos \left( \omega \right) \\ &+ \cos \left( \delta \right) \sin \left( \beta \right) \sin \left( \gamma \right) \sin \left( \omega \right) \end{align}$$

Where:

$\delta$ - Solar declination, the angle between a plane through the equator and a vector between the Earth's centre and the sun. (North positive; -23.45° ≤ $\delta$ ≤ 23.45)
$\phi$ - local latitude
$\beta$ - Effective tilt angle of module
$\gamma$ - Effective azimuth angle of module
$\omega$ - Hour angle

Hour angle is calculated using the solar time at the site location -- starting with an hour angle of 0°, add/subtract 15° per hour difference from noon in the afternoon/morning respectively.

From [9]:

$$\mathit{solar\ time} - \mathit{standard\ time} = 4\left( L_{\text{st}} - L_{\text{loc}} \right) + E$$

Where $L_{\text{st}}$ is the standard meridian for the local time zone, and $L_{\text{Loc}}$ is the longitude of the site location.

The Equation of Time $E$ is:

$$E = 229.2 \left( 0.000075 + 0.001868 \cos{B - 0.032077 \sin B - 0.014615 \cos{2B} - 0.04089 \sin{2B}} \right)$$

Where $B = \left( n - 1 \right) \frac{360}{365}$ and $n$ is the day number of the year.

Alternatively, where the sun's position is known in terms of zenith angle $\theta_{\text{z}}$ (angle between the vertical and a vector pointing to the sun at the plant location) and solar azimuth angle $\gamma_{\text{s}}$ (the clockwise angle between a line running to true north and the projection of a line from the observer to the sun onto the horizontal plane passing through the nominal position of the plant), the incidence angle can be calculated using (again from [9]):

$$\cos{ \left( \theta \right) } = \cos{ \left( \theta_{z} \right) } \cos{ \left( \beta \right) } + \sin{ \left( \theta_{z} \right) } \sin{ \left( \beta \right) } \cos { \left( \gamma_{s} - \gamma \right) }$$