AOI_PROJECTION

This function computes the cosine of the angle of incidence.

It returns the projection of the sun’s unit vector onto the surface’s normal unit vector. Negative values indicate the sun is behind the surface.

Excel Usage

=AOI_PROJECTION(surface_tilt, surface_azimuth, solar_zenith, solar_azimuth)

surface_tilt (list[list], required): Panel tilt from horizontal (degrees).
surface_azimuth (list[list], required): Panel azimuth direction (degrees).
solar_zenith (list[list], required): Solar zenith angle (degrees).
solar_azimuth (list[list], required): Solar azimuth angle (degrees).

Returns (list[list]): 2D list of AOI projections (unitless), or an error string.

Example 1: Projection for sun directly normal to surface

Inputs:

surface_tilt	surface_azimuth	solar_zenith	solar_azimuth
30	180	30	180

Excel formula:

=AOI_PROJECTION({30}, {180}, {30}, {180})

Expected output:

1

Example 2: Projection negative when sun is behind surface

Inputs:

surface_tilt	surface_azimuth	solar_zenith	solar_azimuth
30	180	30	0

Excel formula:

=AOI_PROJECTION({30}, {180}, {30}, {0})

Expected output:

0.5

Example 3: Vectorized projection over two sun positions

Inputs:

surface_tilt		surface_azimuth		solar_zenith		solar_azimuth
20	20	180	180	20	60	180	210

Excel formula:

=AOI_PROJECTION({20,20}, {180,180}, {20,60}, {180,210})

Expected output:

Result
1
0.726361

Example 4: Scalar projection inputs

Inputs:

surface_tilt	surface_azimuth	solar_zenith	solar_azimuth
20	180	20	180

Excel formula:

=AOI_PROJECTION(20, 180, 20, 180)

Expected output:

1

Python Code

Show Code

import pandas as pd
import numpy as np
from pvlib.irradiance import aoi_projection as result_func

def aoi_projection(surface_tilt, surface_azimuth, solar_zenith, solar_azimuth):
    """
    Calculate the dot product of the sun position and surface normal (cosine of AOI).

    See: https://pvlib-python.readthedocs.io/en/stable/reference/generated/pvlib.irradiance.aoi_projection.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        surface_tilt (list[list]): Panel tilt from horizontal (degrees).
        surface_azimuth (list[list]): Panel azimuth direction (degrees).
        solar_zenith (list[list]): Solar zenith angle (degrees).
        solar_azimuth (list[list]): Solar azimuth angle (degrees).

    Returns:
        list[list]: 2D list of AOI projections (unitless), or an error string.
    """
    try:
        def flatten_num(data):
            if not isinstance(data, list): return [float(data)]
            flat = []
            for row in data:
                row = row if isinstance(row, list) else [row]
                for val in row:
                    if val == "": flat.append(float('nan'))
                    else: flat.append(float(val))
            return flat

        t_list = flatten_num(surface_tilt)
        a_list = flatten_num(surface_azimuth)
        z_list = flatten_num(solar_zenith)
        az_list = flatten_num(solar_azimuth)

        n = len(t_list)
        if n == 0 or len(a_list) != n or len(z_list) != n or len(az_list) != n:
            return "Error: All input arrays must have the same non-zero length"

        res = result_func(
            surface_tilt=np.array(t_list),
            surface_azimuth=np.array(a_list),
            solar_zenith=np.array(z_list),
            solar_azimuth=np.array(az_list)
        )

        return [[float(v) if not pd.isna(v) else ""] for v in res]
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

surface_tilt *

Panel tilt from horizontal (degrees).

surface_azimuth *

Panel azimuth direction (degrees).

solar_zenith *

Solar zenith angle (degrees).

solar_azimuth *

Solar azimuth angle (degrees).