plot_martinez_shade_loss.py

"""
Modelling shading losses in modules with bypass diodes
======================================================
"""

# %%
# This example illustrates how to use the loss model proposed by Martinez et
# al. [1]_. The model proposes a power output losses factor by adjusting
# the incident direct and circumsolar beam irradiance fraction of a PV module
# based on the number of shaded *blocks*. A *block* is defined as a group of
# cells protected by a bypass diode. More information on *blocks* can be found
# in the original paper [1]_ and in the
# :py:func:`pvlib.shading.direct_martinez` documentation.
#
# The following key functions are used in this example:
#
# 1. :py:func:`pvlib.shading.direct_martinez` to calculate the power output
#    losses fraction due to shading.
# 2. :py:func:`pvlib.shading.shaded_fraction1d` to calculate the fraction of
#    shaded surface and consequently the number of shaded *blocks* due to
#    row-to-row shading.
# 3. :py:func:`pvlib.tracking.singleaxis` to calculate the rotation angle of
#    the trackers.
#
# .. sectionauthor:: Echedey Luis <echelual (at) gmail.com>
#
# Problem description
# -------------------
# Let's consider a PV system with the following characteristics:
#
# - Two north-south single-axis trackers, each one having 6 modules.
# - The rows have the same true-tracking tilt angles. True tracking
#   is chosen in this example, so shading is significant.
# - Terrain slope is 7 degrees downward to the east.
# - Row axes are horizontal.
# - The modules are comprised of multiple cells. We will compare these cases:
#    - modules with one bypass diode
#    - modules with three bypass diodes
#    - half-cut cell modules with three bypass diodes in portrait and landscape
#
# Setting up the system
# ----------------------
# Let's start by defining the system characteristics, location and the time
# range for the analysis.

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

pitch = 4  # meters
width = 1.5  # meters
gcr = width / pitch  # ground coverage ratio
N_modules_per_row = 6
axis_azimuth = 180  # N-S axis
axis_slope = 0  # flat because the axis is perpendicular to the slope
cross_axis_slope = -7  # 7 degrees downward to the east

latitude, longitude = 40.2712, -3.7277
locus = pvlib.location.Location(
    latitude,
    longitude,
    tz="Europe/Madrid",
    altitude=pvlib.location.lookup_altitude(latitude, longitude),
)

times = pd.date_range("2001-04-11T04", "2001-04-11T20", freq="10min")

# %%
# True-tracking algorithm and shaded fraction
# -------------------------------------------
# Since this model is about row-to-row shading, we will use the true-tracking
# algorithm to calculate the trackers rotation. Back-tracking eliminates
# shading between rows, and since this example is about shading, we will not
# use it.
#
# Then, the next step is to calculate the fraction of shaded surface. This is
# done using :py:func:`pvlib.shading.shaded_fraction1d`. Using this function is
# straightforward with the variables we already have defined.
# Then, we can calculate the number of shaded blocks by rounding up the shaded
# fraction by the number of blocks along the shaded length.

# Calculate solar position to get single-axis tracker rotation and irradiance
solar_pos = locus.get_solarposition(times)
solar_apparent_zenith, solar_azimuth = (
    solar_pos["apparent_zenith"],
    solar_pos["azimuth"],
)  # unpack for better readability

tracking_result = pvlib.tracking.singleaxis(
    apparent_zenith=solar_apparent_zenith,
    solar_azimuth=solar_azimuth,
    axis_slope=axis_slope,
    axis_azimuth=axis_azimuth,
    max_angle=(-90 + cross_axis_slope, 90 + cross_axis_slope),  # (min, max)
    backtrack=False,
    gcr=gcr,
    cross_axis_slope=cross_axis_slope,
)

tracker_theta, aoi, surface_tilt, surface_azimuth = (
    tracking_result["tracker_theta"],
    tracking_result["aoi"],
    tracking_result["surface_tilt"],
    tracking_result["surface_azimuth"],
)  # unpack for better readability

# Calculate the shade fraction
shaded_fraction = pvlib.shading.shaded_fraction1d(
    solar_apparent_zenith,
    solar_azimuth,
    axis_azimuth,
    axis_slope=axis_slope,
    shaded_row_rotation=tracker_theta,
    shading_row_rotation=tracker_theta,
    collector_width=width,
    pitch=pitch,
    cross_axis_slope=cross_axis_slope,
)

# %%
# Number of shaded blocks
# -----------------------
# The number of shaded blocks depends on the module configuration and number
# of bypass diodes. For example,
# modules with one bypass diode will behave like one block.
# On the other hand, modules with three bypass diodes will have three blocks,
# except for the half-cut cell modules, which will have six blocks; 2x3 blocks
# where the two rows are along the longest side of the module.
# We can argue that the dimensions of the system change when you switch from
# portrait to landscape, but for this example, we will consider it the same.
#
# The number of shaded blocks is calculated by rounding up the shaded fraction
# by the number of blocks along the shaded length. So let's define the number
# of blocks for each module configuration:
#
# - 1 bypass diode: 1 block
# - 3 bypass diodes: 3 blocks in landscape; 1 in portrait
# - 3 bypass diodes half-cut cells:
#   - 2 blocks in portrait
#   - 3 blocks in landscape
#
# .. figure:: ../../_images/PV_module_layout_cesardd.jpg
#    :align: center
#    :width: 75%
#    :alt: Normal and half-cut cells module layouts
#
#    Left: common module layout. Right: half-cut cells module layout.
#    Each module has three bypass diodes. On the left, they connect cell
#    columns 1-2, 2-3 & 3-4. On the right, they connect cell columns 1-2, 3-4 &
#    5-6.
#    *Source: César Domínguez. CC BY-SA 4.0, Wikimedia Commons*
#
# In the image above, each orange U-shaped string section is a block.
# By symmetry, the yellow inverted-U's of the subcircuit are also blocks.
# For this reason, the half-cut cell modules have 6 blocks in total: two along
# the longest side and three along the shortest side.

blocks_per_module = {
    "1 bypass diode": 1,
    "3 bypass diodes": 3,
    "3 bypass diodes half-cut, portrait": 2,
    "3 bypass diodes half-cut, landscape": 3,
}

# Calculate the number of shaded blocks during the day
shaded_blocks_per_module = {
    k: np.ceil(blocks_N * shaded_fraction)
    for k, blocks_N in blocks_per_module.items()
}

# %%
# Plane of array irradiance example data
# --------------------------------------
# To calculate the power output losses due to shading, we need the plane of
# array irradiance. For this example, we will use synthetic data:

clearsky = locus.get_clearsky(
    times, solar_position=solar_pos, model="ineichen"
)
dni_extra = pvlib.irradiance.get_extra_radiation(times)
airmass = pvlib.atmosphere.get_relative_airmass(solar_apparent_zenith)
sky_diffuse = pvlib.irradiance.perez_driesse(
    surface_tilt, surface_azimuth, clearsky["dhi"], clearsky["dni"],
    solar_apparent_zenith, solar_azimuth, dni_extra, airmass,
)  # fmt: skip
poa_components = pvlib.irradiance.poa_components(
    aoi, clearsky["dni"], sky_diffuse, poa_ground_diffuse=0
)  # ignore ground diffuse for brevity
poa_global, poa_direct = (
    poa_components["poa_global"],
    poa_components["poa_direct"],
)

# %%
# Results
# -------
# Now that we have the number of shaded blocks for each module configuration,
# we can apply the model and estimate the power loss due to shading.
#
# Note that this model is not linear with the shaded blocks ratio, so there is
# a difference between applying it to just a module or a whole row.

shade_losses_per_module = {
    k: pvlib.shading.direct_martinez(
        poa_global=poa_global,
        poa_direct=poa_direct,
        shaded_fraction=shaded_fraction,
        shaded_blocks=module_shaded_blocks,
        total_blocks=blocks_per_module[k],
    )
    for k, module_shaded_blocks in shaded_blocks_per_module.items()
}

shade_losses_per_row = {
    k: pvlib.shading.direct_martinez(
        poa_global=poa_global,
        poa_direct=poa_direct,
        shaded_fraction=shaded_fraction,
        shaded_blocks=module_shaded_blocks * N_modules_per_row,
        total_blocks=blocks_per_module[k] * N_modules_per_row,
    )
    for k, module_shaded_blocks in shaded_blocks_per_module.items()
}

# %%
# Plotting the results
# ^^^^^^^^^^^^^^^^^^^^

fig, (ax1, ax2) = plt.subplots(2, 1, sharex=True)
fig.suptitle("Martinez power losses due to shading")
for k, shade_losses in shade_losses_per_module.items():
    linestyle = "--" if k == "3 bypass diodes half-cut, landscape" else "-"
    ax1.plot(times, shade_losses, label=k, linestyle=linestyle)
ax1.legend(loc="upper center")
ax1.grid()
ax1.set_ylabel(r"$P_{out}$ losses")
ax1.set_title("Per module")

for k, shade_losses in shade_losses_per_row.items():
    linestyle = "--" if k == "3 bypass diodes half-cut, landscape" else "-"
    ax2.plot(times, shade_losses, label=k, linestyle=linestyle)
ax2.legend(loc="upper center")
ax2.grid()
ax2.set_xlabel("Time")
ax2.xaxis.set_major_formatter(DateFormatter("%H:%M", tz="Europe/Madrid"))
ax2.set_ylabel(r"$P_{out}$ losses")
ax2.set_title("Per row")
fig.tight_layout()
fig.show()

# %%
# Note how the half-cut cell module in portrait performs better than the
# normal module with three bypass diodes. This is because the number of shaded
# blocks is less along the shaded length is higher in the half-cut module.
# This is the reason why half-cut cell modules are preferred in portrait
# orientation.

# %%
# References
# ----------
# .. [1] F. Martínez-Moreno, J. Muñoz, and E. Lorenzo, 'Experimental model
#    to estimate shading losses on PV arrays', Solar Energy Materials and
#    Solar Cells, vol. 94, no. 12, pp. 2298-2303, Dec. 2010,
#    :doi:`10.1016/j.solmat.2010.07.029`.