PROJ_SOLAR_ZENITH

This function projects the solar zenith angle onto a plane defined by an axis tilt and azimuth.

This projection is critical for modeling single-axis trackers and for simplifying 3D shading problems into a 2D plane perpendicular to the collector rows.

Excel Usage

=PROJ_SOLAR_ZENITH(solar_zenith, solar_azimuth, axis_tilt, axis_azimuth)

solar_zenith (list[list], required): Sun’s apparent zenith angle (degrees).
solar_azimuth (list[list], required): Sun’s azimuth angle (degrees).
axis_tilt (float, optional, default: 0): Tilt angle of the reference axis (degrees).
axis_azimuth (float, optional, default: 0): Azimuth angle of the reference axis (degrees).

Returns (list[list]): 2D list of projected solar zenith angles (degrees), or an error string.

Example 1: Projection on a north-south axis at noon

Inputs:

solar_zenith	solar_azimuth	axis_tilt	axis_azimuth
30	180	0	180

Excel formula:

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

Expected output:

0

Example 2: Projection on east-west axis

Inputs:

solar_zenith	solar_azimuth	axis_tilt	axis_azimuth
30	180	0	90

Excel formula:

=PROJ_SOLAR_ZENITH({30}, {180}, 0, 90)

Expected output:

30

Example 3: Projection with tilted axis

Inputs:

solar_zenith	solar_azimuth	axis_tilt	axis_azimuth
50	200	10	180

Excel formula:

=PROJ_SOLAR_ZENITH({50}, {200}, 10, 180)

Expected output:

19.0672

Example 4: Vectorized projected zenith angles

Inputs:

solar_zenith			solar_azimuth			axis_tilt	axis_azimuth
20	40	60	120	180	240	0	180

Excel formula:

=PROJ_SOLAR_ZENITH({20,40,60}, {120,180,240}, 0, 180)

Expected output:

Result
-17.4952
0
56.3099

Python Code

Show Code

import pandas as pd
import numpy as np
from pvlib.shading import projected_solar_zenith_angle as result_func

def proj_solar_zenith(solar_zenith, solar_azimuth, axis_tilt=0, axis_azimuth=0):
    """
    Calculate the projected solar zenith angle in the tracker reference plane.

    See: https://pvlib-python.readthedocs.io/en/stable/reference/generated/pvlib.shading.projected_solar_zenith_angle.html

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

    Args:
        solar_zenith (list[list]): Sun's apparent zenith angle (degrees).
        solar_azimuth (list[list]): Sun's azimuth angle (degrees).
        axis_tilt (float, optional): Tilt angle of the reference axis (degrees). Default is 0.
        axis_azimuth (float, optional): Azimuth angle of the reference axis (degrees). Default is 0.

    Returns:
        list[list]: 2D list of projected solar zenith angles (degrees), 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

        z_list = flatten_num(solar_zenith)
        az_list = flatten_num(solar_azimuth)

        n = len(z_list)
        if n == 0 or len(az_list) != n:
            return "Error: solar_zenith and solar_azimuth arrays must have the same non-zero length"

        at = float(axis_tilt) if axis_tilt is not None else 0.0
        aa = float(axis_azimuth) if axis_azimuth is not None else 0.0

        res = result_func(
            solar_zenith=np.array(z_list),
            solar_azimuth=np.array(az_list),
            axis_tilt=at,
            axis_azimuth=aa
        )

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

Online Calculator

solar_zenith *

Sun's apparent zenith angle (degrees).

solar_azimuth *

Sun's azimuth angle (degrees).

axis_tilt

Tilt angle of the reference axis (degrees).

axis_azimuth

Azimuth angle of the reference axis (degrees).