NREL non-uniform irradiance mismatch loss for bifacial modules

docs/examples/bifacial/plot_irradiance_nonuniformity_loss.py

@@ -0,0 +1,265 @@
+"""
+Plot Irradiance Non-uniformity Loss
+===================================
+
+Calculate the incident irradiance lost to non-uniformity in a bifacial PV array
+"""
+
+# %%
+# The incident irradiance on the backside of a bifacial PV module is
+# not uniform due to neighboring rows, the ground albedo and site conditions.
+# Cells with different irradiance levels produce less power that the sum of
+# the power produced by each cell individually. This is known as irradiance
+# non-uniformity loss.
+#
+# This example demonstrates how to model the irradiance non-uniformity loss
+# due to different global irradiance levels on a bifacial PV module through
+# two different approaches:
+#
+# - Given the irradiance levels of each cell in an instant,
+#   in the first section.
+# - Modelling the irradiance non-uniformity RMAD through the day thanks to a
+#   mock-up horizontal axis tracker system, in the second section.
+#   See [2]_ and [3]_ for more information.
+#
+# The function :py:func:`pvlib.pvsystem.nonuniform_irradiance_loss` will be
+# used to transform the Relative Mean Absolute Deviation (RMAD) of the
+# irradiance into a power loss percentage. Down below you will find a
+# numpy-based implementation of the RMAD function.
+#
+# References
+# ----------
+# .. [1] C. Deline, S. Ayala Pelaez, S. MacAlpine, and C. Olalla, 'Estimating
+#    and parameterizing mismatch power loss in bifacial photovoltaic
+#    systems', Progress in Photovoltaics: Research and Applications, vol. 28,
+#    no. 7, pp. 691-703, 2020, :doi:`10.1002/pip.3259`.
+# .. [2] C. Domínguez, J. Marcos, S. Ures, S. Askins, and I. Antón, 'A
+#    Horizontal Single-Axis Tracker Mock-Up to Quickly Assess the Influence
+#    of Geometrical Factors on Bifacial Energy Gain', in 2023 IEEE 50th
+#    Photovoltaic Specialists Conference (PVSC), Jun. 2023, pp. 1–3.
+#    :doi:`10.1109/PVSC48320.2023.10359580`.
+# .. [3] C. Domínguez, J. Marcos, S. Ures, S. Askins, I. Antón, A Horizontal
+#    Single-Axis Tracker Mock-Up to Quickly Assess the Influence of Geometrical
+#    Factors on Bifacial Energy Gain. Zenodo, 2023.
+#    :doi:`10.5281/zenodo.11125039`.
+#
+# .. sectionauthor:: Echedey Luis <echelual (at) gmail.com>
+
+import numpy as np
+import pandas as pd
+import scipy.stats
+import matplotlib.pyplot as plt
+from matplotlib.cm import ScalarMappable
+from matplotlib.colors import Normalize
+
+from pvlib.pvsystem import nonuniform_irradiance_loss
+
+# %%
+# Theoretical and straightforward problem
+# ---------------------------------------
+# Let's set a fixed irradiance to each cell row of the PV array with the values
+# described in Figure 1 (a), [1]_. We will cover this case for educational
+# purposes.
+# Here we set and plot the global irradiance levels of each cell.
+
+x = np.arange(6, 0, -1)
+y = np.arange(6, 0, -1)
+cells_irrad = np.repeat([1059, 976, 967, 986, 1034, 1128], 6).reshape(6, 6)
+
+color_map = "gray"
+color_norm = Normalize(930, 1150)
+
+fig, ax = plt.subplots()
+fig.suptitle("Global Irradiance Levels of Each Cell")
+fig.colorbar(
+    ScalarMappable(cmap=color_map, norm=color_norm),
+    ax=ax,
+    orientation="vertical",
+    label="$[W/m^2]$",
+)
+ax.set_aspect("equal")
+ax.pcolormesh(
+    x,
+    y,
+    cells_irrad,
+    shading="nearest",
+    edgecolors="black",
+    cmap=color_map,
+    norm=color_norm,
+)
+
+# %%
+# Relative Mean Absolute Deviation
+# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+# Calculate the Relative Mean Absolute Deviation (RMAD) of the cells'
+# irradiance with the following function, Eq. (4) of [1]_:
+#
+# .. math::
+#
+#    \Delta \left[ \% \right] = \frac{1}{n^2 \bar{G}_{total}}
+#    \sum_{i=1}^{n} \sum_{j=1}^{n} \lvert G_{total,i} - G_{total,j} \rvert
+#
+
+
+def rmad(data, axis=None):
+    """
+    Relative Mean Absolute Deviation.
+    https://stackoverflow.com/a/19472336/19371110
+    """
+    mad = np.mean(np.absolute(data - np.mean(data, axis)), axis)
+    return mad / np.mean(data, axis)
+
+
+rmad_cells = rmad(cells_irrad)
+
+# this is the same as a column's RMAD!
+print(rmad_cells == rmad(cells_irrad[:, 0]))
+
+# %%
+# Mismatch Loss
+# ^^^^^^^^^^^^^
+# Calculate the power loss percentage due to the irradiance non-uniformity
+# with the function :py:func:`pvlib.pvsystem.nonuniform_irradiance_loss`.
+
+mismatch_loss = nonuniform_irradiance_loss(rmad_cells)
+
+# %%
+# Total power incident on the module cells
+# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+# It is the sum of the irradiance of each cell
+
+total_irrad = np.sum(cells_irrad)
+
+# %%
+# Mismatch-corrected irradiance
+# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+# The power incident on the module cells is multiplied by the mismatch loss
+# as follows:
+
+mismatch_corrected_irrad = total_irrad * (1 - mismatch_loss)
+
+# %%
+# Results
+# ^^^^^^^
+
+print(f"Total power incident on the module cells: {total_irrad}")
+print(f"RMAD of the cells' irradiance: {rmad_cells}")
+print(f"Power loss % due to the irradiance non-uniformity: {mismatch_loss}")
+print(f"Effective power after mismatch correction: {mismatch_corrected_irrad}")
+
+# %%
+# A practical approach
+# --------------------
+# In practice, simulating each cell irradiance is computationally expensive,
+# and measuring each of the cells irradiance of a real system can also be
+# challenging. Nevertheless, a mock-up system can be used to estimate the
+# non-uniformity through the day.
+#
+# Here we will base our calculations on the work of Domínguez et al. [2]_.
+# The following image in [3]_ shows the backside irradiance non-uniformity of a
+# HSAT mock-up system:
+#
+# .. figure:: ../../_images/Dominguez_et_al_PVSC2023.png
+#    :alt: Plot of backside reference cells of an HSAT mock-up and their RMAD.
+#    :align: center
+#
+#    Blue dots represent the backside irradiance non-uniformity.
+#    *BE* stands for *backside east*.
+
+
+def hsat_backside_rmad_model_through_day(hour):
+    """Model of the blue dots in the image above."""
+    # For demonstration purposes only. Model roughly fit to show an example of
+    # the RMAD variation through the day without including all the data.
+    # fmt: off
+    morning_polynom = [6.71787833e-02, -4.50442998e+00, 1.18114757e+02,
+                       -1.51725679e+03, 9.56439547e+03, -2.36835920e+04]
+    afternoon_polynom = [7.14947943e-01, -6.02541075e+01, 2.02789031e+03,
+                         -3.40677727e+04, 2.85671091e+05, -9.56469320e+05]
+    # fmt: on
+    day_rmad = np.where(
+        hour < 14.75,
+        np.polyval(morning_polynom, hour),
+        np.polyval(afternoon_polynom, hour),
+    )
+    return day_rmad / 100  # RMAD is a percentage
+
+
+# %%
+# Calculating Global RMAD from Backside RMAD
+# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+#
+# .. note::
+#    The global irradiance RMAD is different from the backside irradiance RMAD.
+#
+# The global irradiance is the sum of the front irradiance and the backside
+# irradiance by the bifaciality factor, see equation (2) in [1]_.
+#
+# .. math::
+#
+#    G_{total,i} = G_{front,i} + \phi_{Bifi} G_{rear,i}
+#
+# where :math:`\phi_{Bifi}` is the bifaciality factor.
+#
+# Here we will model front and backside irradiances with normal distributions
+# for simplicity, then calculate the global RMAD and plot the results.
+#
+# The backside irradiance is the one that presents the most significant
+# non-uniformity. The front irradiance is way more uniform, so it will be
+# neglected in this example.
+#
+# Let's calculate the **global RMAD** through the day - it's **different** from
+# the backside RMAD since
+#
+# .. math::
+#
+#    RMAD(k \cdot X + c) = RMAD(X) \cdot k \frac{k \bar{X}}{k \bar{X} + c}
+#    = RMAD(X) \cdot k \frac{1}{1 + \frac{c}{k \bar{X}}}
+#
+# where :math:`X` is a random variable and :math:`k>0, c \neq \bar{X}` are
+# constants (`source
+# <https://en.wikipedia.org/wiki/Mean_absolute_difference#Properties>`_).
+
+times = pd.date_range("2023-06-06T09:30", "2023-06-06T18:30", freq="30min")
+hours = times.hour + times.minute / 60
+bifaciality = 0.65
+front_irrad = scipy.stats.norm.pdf(hours, loc=12.5, scale=1600)
+backside_irrad = scipy.stats.norm.pdf(hours, loc=12.5, scale=180)
+
+global_irrad = front_irrad + bifaciality * backside_irrad
+# See RMAD properties above
+# Here we calculate RMAD(global_irrad)
+# backside_irrad := X, bifaciality := k, front_irrad := c
+backside_rmad = hsat_backside_rmad_model_through_day(hours)
+global_rmad = (
+    backside_rmad
+    * bifaciality
+    / (1 + front_irrad / backside_irrad / bifaciality)
+)
+
+# Get the mismatch loss
+mismatch_loss = nonuniform_irradiance_loss(global_rmad)
+
+# Plot results
+fig, ax1 = plt.subplots()
+fig.suptitle("Irradiance RMAD and Mismatch Losses")
+
+ax1.plot(hours, global_rmad, label="RMAD: global", color="k")
+ax1.plot(
+    hours, backside_rmad, label="RMAD: backside", color="b", linestyle="--"
+)
+ax1.set_xlabel("Hour of the day")
+ax1.set_ylabel("RMAD")
+ax1.legend(loc="upper left")
+
+ax2 = ax1.twinx()
+ax2.plot(
+    hours, mismatch_loss, label="Mismatch loss", color="red", linestyle=":"
+)
+ax2.grid()
+ax2.legend(loc="upper right")
+ax2.set_ylabel("Mismatch loss")
+
+fig.show()
+
+# %%

docs/sphinx/source/reference/effects_on_pv_system_output/loss-models.rst

@@ -8,3 +8,4 @@ Loss models
 
    pvsystem.combine_loss_factors
    pvsystem.dc_ohms_from_percent
+   pvsystem.nonuniform_irradiance_loss

docs/sphinx/source/whatsnew/v0.11.0.rst

@@ -15,6 +15,9 @@ Deprecations
 
 Enhancements
 ~~~~~~~~~~~~
+* Add new losses function that accounts for non-uniform irradiance of bifacial
+  modules, :py:func:`pvlib.pvsystem.nonuniform_irradiance_loss`.
+  (:issue:`2045`, :pr:`2046`)
 
 
 Bug fixes

pvlib/pvsystem.py

@@ -3030,3 +3030,63 @@ def combine_loss_factors(index, *losses, fill_method='ffill'):
         combined_factor *= (1 - loss)
 
     return 1 - combined_factor
+
+
+def nonuniform_irradiance_loss(rmad):
+    r"""
+    Calculate the incident irradiance loss due to irradiance non-uniformity.
+
+    This model is described for bifacial modules in [1]_, where the backside
+    irradiance is less uniform due to mounting and site conditions.
+
+    .. versionadded:: 0.11.0
+
+    Parameters
+    ----------
+    rmad : numeric
+        The Relative Mean Absolute Difference of the irradiance.
+        See the definition in [2]_ and the example
+        :ref:`sphx_glr_gallery_bifacial_plot_irradiance_nonuniformity_loss.py`.
+
+    Returns
+    -------
+    loss : numeric
+        The irradiance loss.
+
+    Notes
+    -----
+    The function this model implements is equation (7) of [1]_:
+
+    .. math::
+
+       M[%] = 0.12 \Delta[%] + 2.77 \Delta[%]^2
+
+    where :math:`\Delta[%]` is the Relative Mean Absolute Difference of the
+    global irradiance, Eq. (4) of [1]_ and [2]_.
+
+    The losses definition is done in Eq. (1) of [1]_:
+
+    .. math::
+
+        M[%] = 1 - \frac{P_{Array}}{\sum P_{Cells}}
+
+    It is recommended to see the example
+    :ref:`sphx_glr_gallery_bifacial_plot_irradiance_nonuniformity_loss.py`
+    for a complete use case and the RMAD function implementation.
+
+    See Also
+    --------
+    pvlib.pvsystem.combine_loss_factors
+
+    References
+    ----------
+    .. [1] C. Deline, S. Ayala Pelaez, S. MacAlpine, and C. Olalla, 'Estimating
+       and parameterizing mismatch power loss in bifacial photovoltaic
+       systems', Progress in Photovoltaics: Research and Applications, vol. 28,
+       no. 7, pp. 691-703, 2020, :doi:`10.1002/pip.3259`.
+    .. [2] “Mean absolute difference,” Wikipedia, Sep. 05, 2023.
+       https://en.wikipedia.org/wiki/Mean_absolute_difference#Relative_mean_absolute_difference
+       (accessed 2024-04-14).
+    """  # noqa: E501
+    # Eq. (7) of [1]
+    return rmad * (0.12 + 2.77 * rmad)

pvlib/tests/test_pvsystem.py

@@ -2521,3 +2521,15 @@ def test_Array_temperature_missing_parameters(model, keys):
         array.temperature_model_parameters = params
         with pytest.raises(KeyError, match=match):
             array.get_cell_temperature(irrads, temps, winds, model)
+
+
+def test_nonuniform_irradiance_loss():
+    """tests pvsystem.nonuniform_irradiance_loss"""
+    premise_rmads = np.array([0.0, 0.05, 0.1, 0.15, 0.2, 0.25])
+    expected_mms = np.array([0, 0.012925, 0.0397, 0.080325, 0.1348, 0.203125])
+    result_mms = pvsystem.nonuniform_irradiance_loss(premise_rmads)
+    assert_allclose(result_mms, expected_mms, atol=1e-5)
+
+    # test datatypes Series IO
+    result_mms = pvsystem.nonuniform_irradiance_loss(pd.Series(premise_rmads))
+    assert isinstance(result_mms, pd.Series)