Shaded fraction of a horizontal single-axis tracker#

This example illustrates how to calculate the 1D shaded fraction of three rows in a North-South horizontal single axis tracker (HSAT) configuration.

pvlib.shading.shaded_fraction1d() exposes a useful method for the calculation of the shaded fraction of the width of a solar collector. Here, the width is defined as the dimension perpendicular to the axis of rotation. This method for calculating the shaded fraction only requires minor modifications to be applicable for fixed-tilt systems.

It is highly recommended to read pvlib.shading.shaded_fraction1d() documentation to understand the input parameters and the method’s capabilities. For more in-depth information, please see the journal paper 10.1063/5.0202220 describing the methodology.

Let’s start by obtaining the true-tracking angles for each of the rows and limiting the angles to the range of -50 to 50 degrees. This decision is done to create an example scenario with significant shading.

Key functions used in this example are:

  1. pvlib.tracking.singleaxis() to calculate the tracking angles.

  2. pvlib.shading.projected_solar_zenith_angle() to calculate the projected solar zenith angle.

  3. pvlib.shading.shaded_fraction1d() to calculate the shaded fractions.

import pvlib

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from matplotlib.dates import DateFormatter

# Define the solar system parameters
latitude, longitude = 28.51, -13.89
altitude = pvlib.location.lookup_altitude(latitude, longitude)

axis_tilt = 3  # degrees, positive is upwards in the axis_azimuth direction
axis_azimuth = 180  # degrees, N-S tracking axis
collector_width = 3.2  # m
pitch = 4.15  # m
gcr = collector_width / pitch
cross_axis_slope = -5  # degrees
surface_to_axis_offset = 0.07  # m

# Generate a time range for the simulation
times = pd.date_range(
    start="2024-01-01T05",
    end="2024-01-01T21",
    freq="5min",
    tz="Atlantic/Canary",
)

# Calculate the solar position
solar_position = pvlib.solarposition.get_solarposition(
    times, latitude, longitude, altitude
)
solar_azimuth = solar_position["azimuth"]
solar_zenith = solar_position["apparent_zenith"]

# Calculate the tracking angles
rotation_angle = pvlib.tracking.singleaxis(
    solar_zenith,
    solar_azimuth,
    axis_tilt,
    axis_azimuth,
    max_angle=(-50, 50),  # (min, max) degrees
    backtrack=False,
    gcr=gcr,
    cross_axis_tilt=cross_axis_slope,
)["tracker_theta"]

Once the tracker angles have been obtained, the next step is to calculate the shaded fraction. Special care must be taken to ensure that the shaded or shading tracker roles are correctly assigned depending on the solar position. This means we will have a result for each row, eastmost_shaded_fraction, middle_shaded_fraction, and westmost_shaded_fraction. Switching the parameters will be based on the sign of pvlib.shading.projected_solar_zenith_angle().

The following code is verbose to make it easier to understand the process, but with some effort you may be able to simplify it. This verbosity also allows to change the premises easily per case, e.g., in case of a tracker failure or with a different system configuration.

psza = pvlib.shading.projected_solar_zenith_angle(
    solar_zenith, solar_azimuth, axis_tilt, axis_azimuth
)

# Calculate the shaded fraction for the eastmost row
eastmost_shaded_fraction = np.where(
    psza < 0,
    0,  # no shaded fraction in the morning
    # shaded fraction in the evening
    pvlib.shading.shaded_fraction1d(
        solar_zenith,
        solar_azimuth,
        axis_azimuth,
        shaded_row_rotation=rotation_angle,
        axis_tilt=axis_tilt,
        collector_width=collector_width,
        pitch=pitch,
        surface_to_axis_offset=surface_to_axis_offset,
        cross_axis_slope=cross_axis_slope,
        shading_row_rotation=rotation_angle,
    ),
)

# Calculate the shaded fraction for the middle row
middle_shaded_fraction = np.where(
    psza < 0,
    # shaded fraction in the morning
    pvlib.shading.shaded_fraction1d(
        solar_zenith,
        solar_azimuth,
        axis_azimuth,
        shaded_row_rotation=rotation_angle,
        axis_tilt=axis_tilt,
        collector_width=collector_width,
        pitch=pitch,
        surface_to_axis_offset=surface_to_axis_offset,
        cross_axis_slope=cross_axis_slope,
        shading_row_rotation=rotation_angle,
    ),
    # shaded fraction in the evening
    pvlib.shading.shaded_fraction1d(
        solar_zenith,
        solar_azimuth,
        axis_azimuth,
        shaded_row_rotation=rotation_angle,
        axis_tilt=axis_tilt,
        collector_width=collector_width,
        pitch=pitch,
        surface_to_axis_offset=surface_to_axis_offset,
        cross_axis_slope=cross_axis_slope,
        shading_row_rotation=rotation_angle,
    ),
)

# Calculate the shaded fraction for the westmost row
westmost_shaded_fraction = np.where(
    psza < 0,
    # shaded fraction in the morning
    pvlib.shading.shaded_fraction1d(
        solar_zenith,
        solar_azimuth,
        axis_azimuth,
        shaded_row_rotation=rotation_angle,
        axis_tilt=axis_tilt,
        collector_width=collector_width,
        pitch=pitch,
        surface_to_axis_offset=surface_to_axis_offset,
        cross_axis_slope=cross_axis_slope,
        shading_row_rotation=rotation_angle,
    ),
    0,  # no shaded fraction in the evening
)

Plot the shaded fraction result for each row:

plt.plot(times, eastmost_shaded_fraction, label="East-most", color="blue")
plt.plot(times, middle_shaded_fraction, label="Middle", color="green",
         linewidth=3, linestyle="--")  # fmt: skip
plt.plot(times, westmost_shaded_fraction, label="West-most", color="red")
plt.title(r"$shaded\_fraction1d$ of each row vs time")
plt.xlabel("Time")
plt.gca().xaxis.set_major_formatter(DateFormatter("%H:%M"))
plt.ylabel("Shaded Fraction")
plt.legend()
plt.show()
$shaded\_fraction1d$ of each row vs time

Total running time of the script: (0 minutes 0.211 seconds)

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