"""
The ``mismatch`` module in the ``spectrum`` package provides functions for
spectral mismatch calculations. Spectral mismatch models quantify the effect on
a device's photocurrent (or its short-circuit current) of changes in the solar
spectrum due to the atmosphere.
"""
import pvlib
import numpy as np
import pandas as pd
from scipy.integrate import trapezoid
from warnings import warn
[docs]
def calc_spectral_mismatch_field(sr, e_sun, e_ref=None):
"""
Calculate spectral mismatch between a test device and broadband reference
device under specified solar spectral irradiance conditions.
Parameters
----------
sr: pandas.Series
The relative spectral response of one (photovoltaic) test device.
The index of the Series must contain wavelength values in nm. [-]
e_sun: pandas.DataFrame or pandas.Series
One or more measured solar irradiance spectra in a pandas.DataFrame
having wavelength in nm as column index. A single spectrum may be
be given as a pandas.Series having wavelength in nm as index.
[(W/m^2)/nm]
e_ref: pandas.Series, optional
The reference spectrum to use for the mismatch calculation.
The index of the Series must contain wavelength values in nm.
The default is the ASTM G173-03 global tilted spectrum. [(W/m^2)/nm]
Returns
-------
smm: pandas.Series or float if a single measured spectrum is provided. [-]
Notes
-----
Measured solar spectral irradiance usually covers a wavelength range
that is smaller than the range considered as broadband irradiance.
The infrared limit for the former typically lies around 1100 or 1600 nm,
whereas the latter extends to around 2800 or 4000 nm. To avoid imbalance
between the magnitudes of the integrated spectra (the broadband values)
this function truncates the reference spectrum to the same range as the
measured (or simulated) field spectra. The assumption implicit in this
truncation is that the energy in the unmeasured wavelength range
is the same fraction of the broadband energy for both the measured
spectra and the reference spectrum.
If the default reference spectrum is used it is linearly interpolated
to the wavelengths of the measured spectrum, but if a reference spectrum
is provided via the parameter ``e_ref`` it is used without change. This
makes it possible to avoid interpolation, or to use a different method of
interpolation, or to avoid truncation.
The spectral response is linearly interpolated to the wavelengths of each
spectrum with which is it multiplied internally (``e_sun`` and ``e_ref``).
If the wavelengths of the spectral response already match one or both
of these spectra interpolation has no effect; therefore, another type of
interpolation could be used to process ``sr`` before calling this function.
The standards describing mismatch calculations focus on indoor laboratory
applications, but are applicable to outdoor performance as well.
The 2016 version of ASTM E973 [1]_ is somewhat more difficult to
read than the 2010 version [2]_ because it includes adjustments for
the temperature dependency of spectral response, which led to a
formulation using quantum efficiency (QE).
IEC 60904-7 is clearer and also discusses the use of a broadband
reference device. [3]_
References
----------
.. [1] ASTM "E973-16 Standard Test Method for Determination of the
Spectral Mismatch Parameter Between a Photovoltaic Device and a
Photovoltaic Reference Cell" :doi:`10.1520/E0973-16R20`
.. [2] ASTM "E973-10 Standard Test Method for Determination of the
Spectral Mismatch Parameter Between a Photovoltaic Device and a
Photovoltaic Reference Cell" :doi:`10.1520/E0973-10`
.. [3] IEC 60904-7 "Computation of the spectral mismatch correction
for measurements of photovoltaic devices"
"""
# Contributed by Anton Driesse (@adriesse), PV Performance Labs. Aug. 2022
# get the reference spectrum at wavelengths matching the measured spectra
if e_ref is None:
e_ref = pvlib.spectrum.get_reference_spectra(
wavelengths=e_sun.T.index)["global"]
# interpolate the sr at the wavelengths of the spectra
# reference spectrum wavelengths may differ if e_ref is from caller
sr_sun = np.interp(e_sun.T.index, sr.index, sr, left=0.0, right=0.0)
sr_ref = np.interp(e_ref.T.index, sr.index, sr, left=0.0, right=0.0)
# a helper function to make usable fraction calculations more readable
def integrate(e):
return trapezoid(e, x=e.T.index, axis=-1)
# calculate usable fractions
uf_sun = integrate(e_sun * sr_sun) / integrate(e_sun)
uf_ref = integrate(e_ref * sr_ref) / integrate(e_ref)
# mismatch is the ratio or quotient of the usable fractions
smm = uf_sun / uf_ref
if isinstance(e_sun, pd.DataFrame):
smm = pd.Series(smm, index=e_sun.index)
return smm
[docs]
def spectral_factor_firstsolar(precipitable_water, airmass_absolute,
module_type=None, coefficients=None,
min_precipitable_water=0.1,
max_precipitable_water=8,
min_airmass_absolute=0.58,
max_airmass_absolute=10):
r"""
Spectral mismatch modifier based on precipitable water and absolute
(pressure-adjusted) air mass.
Estimates the spectral mismatch modifier, :math:`M`, representing the
effect of variation in the spectral irradiance on the module short circuit
current :math:`M` is estimated from absolute (pressure-corrected) air
mass, :math:`AM_a`, and precipitable water, :math:`Pw`.
Default coefficients are determined for several cell types with
known quantum efficiency curves, by using the Simple Model of the
Atmospheric Radiative Transfer of Sunshine (SMARTS) [1]_. Using
SMARTS, spectrums are simulated with all combinations of AMa and
Pw where:
* :math:`0.5 \textrm{cm} <= Pw <= 5 \textrm{cm}`
* :math:`1.0 <= AM_a <= 5.0`
* Spectral range is limited to that of CMP11 (280 nm to 2800 nm)
* Spectrum simulated on an equatorial facing surface with 37° tilt
* All other parameters fixed at G173 standard
From these simulated spectra, :math:`M` is calculated using the known
quantum efficiency curves. Multiple linear regression is then
applied to fit Eq. 1 to determine the coefficients for each module. More
details on the model can be found in [2]_.
Parameters
----------
precipitable_water : numeric
atmospheric precipitable water. [cm]
airmass_absolute : numeric
absolute (pressure-adjusted) air mass. [unitless]
module_type : str, optional
a string specifying a cell type. Values of 'cdte', 'monosi', 'xsi',
'multisi', and 'polysi' (can be lower or upper case). If provided,
module_type selects default coefficients for the following modules:
* ``'cdte'`` - First Solar Series 4-2 CdTe module.
* ``'monosi'``, ``'xsi'`` - First Solar TetraSun module.
* ``'multisi'``, ``'polysi'`` - anonymous multi-crystalline silicon
module.
* ``'cigs'`` - anonymous copper indium gallium selenide module.
* ``'asi'`` - anonymous amorphous silicon module.
The module used to calculate the spectral correction
coefficients corresponds to the Multi-crystalline silicon
Manufacturer 2 Model C from [3]_. The spectral response (SR) of CIGS
and a-Si modules used to derive coefficients can be found in [4]_
coefficients : array-like, optional
Allows for entry of user-defined spectral correction
coefficients. Coefficients must be of length 6. Derivation of
coefficients requires use of SMARTS and PV module quantum
efficiency curve. Useful for modeling PV module types which are
not included as defaults, or to fine tune the spectral
correction to a particular PV module. Note that the parameters for
modules with very similar quantum efficiency should be similar,
in most cases limiting the need for module specific coefficients.
min_precipitable_water : float, default 0.1
minimum atmospheric precipitable water. Any ``precipitable_water``
value lower than ``min_precipitable_water``
is set to ``min_precipitable_water``. [cm]
max_precipitable_water : float, default 8
maximum atmospheric precipitable water. Any ``precipitable_water``
value greater than ``max_precipitable_water``
is set to ``np.nan``. [cm]
min_airmass_absolute : float, default 0.58
minimum absolute airmass. Any ``airmass_absolute`` value lower than
``min_airmass_absolute`` is set to ``min_airmass_absolute``. [unitless]
max_airmass_absolute : float, default 10
minimum absolute airmass. Any ``airmass_absolute`` value greater than
``max_airmass_absolute`` is set to ``max_airmass_absolute``. [unitless]
Returns
-------
modifier: array-like
spectral mismatch factor (unitless) which can be multiplied
with broadband irradiance reaching a module's cells to estimate
effective irradiance, i.e., the irradiance that is converted to
electrical current.
Notes
----
The ``spectral_factor_firstsolar`` model takes the following form:
.. math::
M = c_1 + c_2 AM_a + c_3 Pw + c_4 AM_a^{0.5}
+ c_5 Pw^{0.5} + c_6 \frac{AM_a} {Pw^{0.5}}.
The default values for the limits applied to :math:`AM_a` and :math:`Pw`
via the ``min_precipitable_water``, ``max_precipitable_water``,
``min_airmass_absolute``, and ``max_airmass_absolute`` are set to prevent
divergence of the model presented above. These default values were
determined by the publication authors in the original pvlib-python
implementation (:pull:`208`).
References
----------
.. [1] Gueymard, Christian. SMARTS2: a simple model of the atmospheric
radiative transfer of sunshine: algorithms and performance
assessment. Cocoa, FL: Florida Solar Energy Center, 1995.
.. [2] Lee, Mitchell, and Panchula, Alex. "Spectral Correction for
Photovoltaic Module Performance Based on Air Mass and Precipitable
Water." IEEE Photovoltaic Specialists Conference, Portland, 2016
.. [3] Marion, William F., et al. User's Manual for Data for Validating
Models for PV Module Performance. National Renewable Energy
Laboratory, 2014. http://www.nrel.gov/docs/fy14osti/61610.pdf
.. [4] Schweiger, M. and Hermann, W, Influence of Spectral Effects
on Energy Yield of Different PV Modules: Comparison of Pwat and
MMF Approach, TUV Rheinland Energy GmbH report 21237296.003,
January 2017
"""
pw = np.atleast_1d(precipitable_water)
pw = pw.astype('float64')
if np.min(pw) < min_precipitable_water:
pw = np.maximum(pw, min_precipitable_water)
warn('Low precipitable water values replaced with '
f'{min_precipitable_water} cm in the calculation of spectral '
'mismatch.')
if np.max(pw) > max_precipitable_water:
pw[pw > max_precipitable_water] = np.nan
warn('High precipitable water values replaced with np.nan in '
'the calculation of spectral mismatch.')
airmass_absolute = np.minimum(airmass_absolute, max_airmass_absolute)
if np.min(airmass_absolute) < min_airmass_absolute:
airmass_absolute = np.maximum(airmass_absolute, min_airmass_absolute)
warn('Low airmass values replaced with 'f'{min_airmass_absolute} in '
'the calculation of spectral mismatch.')
# pvlib.atmosphere.get_absolute_airmass(1,
# pvlib.atmosphere.alt2pres(4340)) = 0.58 Elevation of
# Mina Pirquita, Argentian = 4340 m. Highest elevation city with
# population over 50,000.
_coefficients = {}
_coefficients['cdte'] = (
0.86273, -0.038948, -0.012506, 0.098871, 0.084658, -0.0042948)
_coefficients['monosi'] = (
0.85914, -0.020880, -0.0058853, 0.12029, 0.026814, -0.0017810)
_coefficients['xsi'] = _coefficients['monosi']
_coefficients['polysi'] = (
0.84090, -0.027539, -0.0079224, 0.13570, 0.038024, -0.0021218)
_coefficients['multisi'] = _coefficients['polysi']
_coefficients['cigs'] = (
0.85252, -0.022314, -0.0047216, 0.13666, 0.013342, -0.0008945)
_coefficients['asi'] = (
1.12094, -0.047620, -0.0083627, -0.10443, 0.098382, -0.0033818)
if module_type is not None and coefficients is None:
coefficients = _coefficients[module_type.lower()]
elif module_type is None and coefficients is not None:
pass
elif module_type is None and coefficients is None:
raise TypeError('No valid input provided, both module_type and ' +
'coefficients are None')
else:
raise TypeError('Cannot resolve input, must supply only one of ' +
'module_type and coefficients')
coeff = coefficients
ama = airmass_absolute
modifier = (
coeff[0] + coeff[1]*ama + coeff[2]*pw + coeff[3]*np.sqrt(ama) +
coeff[4]*np.sqrt(pw) + coeff[5]*ama/np.sqrt(pw))
return modifier
[docs]
def spectral_factor_sapm(airmass_absolute, module):
"""
Calculates the spectral mismatch factor, :math:`f_1`,
using the Sandia Array Performance Model approach.
The SAPM spectral factor function is part of the broader Sandia Array
Performance Model, which defines five points on an IV curve using empirical
module-specific coefficients. Module coefficients for the SAPM are
available in the SAPM database and can be retrieved for use in the
``module`` parameter through
:py:func:`pvlib.pvsystem.retrieve_sam()`. More details on the
SAPM can be found in [1]_, while a full description of the procedure to
determine the empirical model coefficients, including those for the SAPM
spectral correction, can be found in [2]_.
Parameters
----------
airmass_absolute : numeric
Absolute airmass [unitless]
Note: ``np.nan`` airmass values will result in 0 output.
module : dict-like
A dict, Series, or DataFrame defining the SAPM parameters.
Must contain keys `'A0'` through `'A4'`.
See the :py:func:`pvlib.pvsystem.sapm` notes section for more details.
Returns
-------
f1 : numeric
The spectral mismatch factor. [unitless]
Notes
-----
The SAPM spectral correction functions parameterises :math:`f_1` as a
fourth order polynomial function of absolute air mass:
.. math::
f_1 = a_0 + a_1 AM_a + a_2 AM_a^2 + a_3 AM_a^3 + a_4 AM_a^4,
where :math:`f_1` is the spectral mismatch factor, :math:`a_{0-4}` are
the module-specific coefficients, and :math:`AM_a` is the absolute airmass,
which is calculated by applying a pressure correction to the relative
airmass. More detail on how this spectral correction function was developed
can be found in [3]_.
References
----------
.. [1] King, D., Kratochvil, J., and Boyson W. (2004), "Sandia
Photovoltaic Array Performance Model", (No. SAND2004-3535), Sandia
National Laboratories, Albuquerque, NM (United States).
:doi:`10.2172/919131`
.. [2] King, B., Hansen, C., Riley, D., Robinson, C., and Pratt, L.
(2016). Procedure to determine coefficients for the Sandia Array
Performance Model (SAPM) (No. SAND2016-5284). Sandia National
Laboratories, Albuquerque, NM (United States).
:doi:`10.2172/1256510`
.. [3] King, D., Kratochvil, J., and Boyson, W. "Measuring solar spectral
and angle-of-incidence effects on photovoltaic modules and solar
irradiance sensors." Conference Record of the 26th IEEE Potovoltaic
Specialists Conference (PVSC). IEEE, 1997.
:doi:`10.1109/PVSC.1997.654283`
"""
am_coeff = [module['A4'], module['A3'], module['A2'], module['A1'],
module['A0']]
spectral_loss = np.polyval(am_coeff, airmass_absolute)
spectral_loss = np.where(np.isnan(spectral_loss), 0, spectral_loss)
spectral_loss = np.maximum(0, spectral_loss)
if isinstance(airmass_absolute, pd.Series):
spectral_loss = pd.Series(spectral_loss, airmass_absolute.index)
return spectral_loss
[docs]
def spectral_factor_caballero(precipitable_water, airmass_absolute, aod500,
module_type=None, coefficients=None):
r"""
Estimate a technology-specific spectral mismatch modifier from
airmass, aerosol optical depth, and atmospheric precipitable water,
using the Caballero model.
The model structure was motivated by examining the effect of these three
atmospheric parameters on simulated irradiance spectra and spectral
modifiers. However, the coefficient values reported in [1]_ and
available here via the ``module_type`` parameter were determined
by fitting the model equations to spectral factors calculated from
global tilted spectral irradiance measurements taken in the city of
Jaén, Spain. See [1]_ for details.
Parameters
----------
precipitable_water : numeric
atmospheric precipitable water. [cm]
airmass_absolute : numeric
absolute (pressure-adjusted) airmass. [unitless]
aod500 : numeric
atmospheric aerosol optical depth at 500 nm. [unitless]
module_type : str, optional
One of the following PV technology strings from [1]_:
* ``'cdte'`` - anonymous CdTe module.
* ``'monosi'`` - anonymous sc-si module.
* ``'multisi'`` - anonymous mc-si- module.
* ``'cigs'`` - anonymous copper indium gallium selenide module.
* ``'asi'`` - anonymous amorphous silicon module.
* ``'perovskite'`` - anonymous pervoskite module.
coefficients : array-like, optional
user-defined coefficients, if not using one of the default coefficient
sets via the ``module_type`` parameter.
Returns
-------
modifier: numeric
spectral mismatch factor (unitless) which is multiplied
with broadband irradiance reaching a module's cells to estimate
effective irradiance, i.e., the irradiance that is converted to
electrical current.
References
----------
.. [1] Caballero, J.A., Fernández, E., Theristis, M.,
Almonacid, F., and Nofuentes, G. "Spectral Corrections Based on
Air Mass, Aerosol Optical Depth and Precipitable Water
for PV Performance Modeling."
IEEE Journal of Photovoltaics 2018, 8(2), 552-558.
:doi:`10.1109/jphotov.2017.2787019`
"""
if module_type is None and coefficients is None:
raise ValueError('Must provide either `module_type` or `coefficients`')
if module_type is not None and coefficients is not None:
raise ValueError('Only one of `module_type` and `coefficients` should '
'be provided')
# Experimental coefficients from [1]_.
# The extra 0/1 coefficients at the end are used to enable/disable
# terms to match the different equation forms in Table 1.
_coefficients = {}
_coefficients['cdte'] = (
1.0044, 0.0095, -0.0037, 0.0002, 0.0000, -0.0046,
-0.0182, 0, 0.0095, 0.0068, 0, 1)
_coefficients['monosi'] = (
0.9706, 0.0377, -0.0123, 0.0025, -0.0002, 0.0159,
-0.0165, 0, -0.0016, -0.0027, 1, 0)
_coefficients['multisi'] = (
0.9836, 0.0254, -0.0085, 0.0016, -0.0001, 0.0094,
-0.0132, 0, -0.0002, -0.0011, 1, 0)
_coefficients['cigs'] = (
0.9801, 0.0283, -0.0092, 0.0019, -0.0001, 0.0117,
-0.0126, 0, -0.0011, -0.0019, 1, 0)
_coefficients['asi'] = (
1.1060, -0.0848, 0.0302, -0.0076, 0.0006, -0.1283,
0.0986, -0.0254, 0.0156, 0.0146, 1, 0)
_coefficients['perovskite'] = (
1.0637, -0.0491, 0.0180, -0.0047, 0.0004, -0.0773,
0.0583, -0.0159, 0.01251, 0.0109, 1, 0)
if module_type is not None:
coeff = _coefficients[module_type]
else:
coeff = coefficients
# Evaluate spectral correction factor
ama = airmass_absolute
aod500_ref = 0.084
pw_ref = 1.4164
f_AM = (
coeff[0]
+ coeff[1] * ama
+ coeff[2] * ama**2
+ coeff[3] * ama**3
+ coeff[4] * ama**4
)
# Eq 6, with Table 1
f_AOD = (aod500 - aod500_ref) * (
coeff[5]
+ coeff[10] * coeff[6] * ama
+ coeff[11] * coeff[6] * np.log(ama)
+ coeff[7] * ama**2
)
# Eq 7, with Table 1
f_PW = (precipitable_water - pw_ref) * (
coeff[8]
+ coeff[9] * np.log(ama)
)
modifier = f_AM + f_AOD + f_PW # Eq 5
return modifier
[docs]
def spectral_factor_pvspec(airmass_absolute, clearsky_index,
module_type=None, coefficients=None):
r"""
Estimate a technology-specific spectral mismatch modifier from absolute
airmass and clear sky index using the PVSPEC model.
The PVSPEC spectral mismatch model includes the effects of cloud cover on
the irradiance spectrum. Model coefficients are derived using spectral
irradiance and other meteorological data from eight locations. Coefficients
for six module types are available via the ``module_type`` parameter.
More details on the model can be found in [1]_.
Parameters
----------
airmass_absolute : numeric
absolute (pressure-adjusted) airmass. [unitless]
clearsky_index: numeric
clear sky index. [unitless]
module_type : str, optional
One of the following PV technology strings from [1]_:
* ``'fs4-1'`` - First Solar series 4-1 and earlier CdTe module.
* ``'fs4-2'`` - First Solar series 4-2 and later CdTe module.
* ``'monosi'`` - anonymous monocrystalline Si module.
* ``'multisi'`` - anonymous multicrystalline Si module.
* ``'cigs'`` - anonymous copper indium gallium selenide module.
* ``'asi'`` - anonymous amorphous silicon module.
coefficients : array-like, optional
user-defined coefficients, if not using one of the default coefficient
sets via the ``module_type`` parameter.
Returns
-------
mismatch: numeric
spectral mismatch factor (unitless) which is multiplied
with broadband irradiance reaching a module's cells to estimate
effective irradiance, i.e., the irradiance that is converted to
electrical current.
Notes
-----
The PVSPEC model parameterises the spectral mismatch factor as a function
of absolute air mass and the clear sky index as follows:
.. math::
M = a_1 k_c^{a_2} AM_a^{a_3},
where :math:`M` is the spectral mismatch factor, :math:`k_c` is the clear
sky index, :math:`AM_a` is the absolute air mass, and :math:`a_1, a_2, a_3`
are module-specific coefficients. In the PVSPEC model publication, absolute
air mass (denoted as :math:`AM`) is estimated starting from the Kasten and
Young relative air mass [2]_. The clear sky index, which is the ratio of
GHI to clear sky GHI, uses the ESRA model [3]_ to estimate the clear sky
GHI with monthly Linke turbidity values from [4]_ as inputs.
References
----------
.. [1] Pelland, S., Beswick, C., Thevenard, D., Côté, A., Pai, A. and
Poissant, Y., 2020. Development and testing of the PVSPEC model of
photovoltaic spectral mismatch factor. In 2020 47th IEEE Photovoltaic
Specialists Conference (PVSC) (pp. 1258-1264). IEEE.
:doi:`10.1109/PVSC45281.2020.9300932`
.. [2] Kasten, F. and Young, A.T., 1989. Revised optical air mass tables
and approximation formula. Applied Optics, 28(22), pp.4735-4738.
:doi:`10.1364/AO.28.004735`
.. [3] Rigollier, C., Bauer, O. and Wald, L., 2000. On the clear sky model
of the ESRA—European Solar Radiation Atlas—with respect to the Heliosat
method. Solar energy, 68(1), pp.33-48.
:doi:`10.1016/S0038-092X(99)00055-9`
.. [4] SoDa website monthly Linke turbidity values:
http://www.soda-pro.com/
"""
_coefficients = {}
_coefficients['multisi'] = (0.9847, -0.05237, 0.03034)
_coefficients['monosi'] = (0.9845, -0.05169, 0.03034)
_coefficients['fs4-2'] = (1.002, -0.07108, 0.02465)
_coefficients['fs4-1'] = (0.9981, -0.05776, 0.02336)
_coefficients['cigs'] = (0.9791, -0.03904, 0.03096)
_coefficients['asi'] = (1.051, -0.1033, 0.009838)
if module_type is not None and coefficients is None:
coefficients = _coefficients[module_type.lower()]
elif module_type is None and coefficients is not None:
pass
elif module_type is None and coefficients is None:
raise ValueError('No valid input provided, both module_type and ' +
'coefficients are None. module_type can be one of ' +
", ".join(_coefficients.keys()))
else:
raise ValueError('Cannot resolve input, must supply only one of ' +
'module_type and coefficients. module_type can be ' +
'one of' ", ".join(_coefficients.keys()))
coeff = coefficients
ama = airmass_absolute
kc = clearsky_index
mismatch = coeff[0]*np.power(kc, coeff[1])*np.power(ama, coeff[2])
return mismatch
[docs]
def spectral_factor_jrc(airmass, clearsky_index, module_type=None,
coefficients=None):
r"""
Estimate a technology-specific spectral mismatch modifier from
airmass and clear sky index using the JRC model.
The JRC spectral mismatch model includes the effects of cloud cover on
the irradiance spectrum. Model coefficients are derived using measurements
of irradiance and module performance at the Joint Research Centre (JRC) in
Ispra, Italy (45.80N, 8.62E). Coefficients for two module types are
available via the ``module_type`` parameter. More details on the model can
be found in [1]_.
Parameters
----------
airmass : numeric
relative airmass. [unitless]
clearsky_index: numeric
clear sky index. [unitless]
module_type : str, optional
One of the following PV technology strings from [1]_:
* ``'cdte'`` - anonymous CdTe module.
* ``'multisi'`` - anonymous multicrystalline Si module.
coefficients : array-like, optional
user-defined coefficients, if not using one of the default coefficient
sets via the ``module_type`` parameter.
Returns
-------
mismatch: numeric
spectral mismatch factor (unitless) which is multiplied
with broadband irradiance reaching a module's cells to estimate
effective irradiance, i.e., the irradiance that is converted to
electrical current.
Notes
-----
The JRC model parameterises the spectral mismatch factor as a function
of air mass and the clear sky index as follows:
.. math::
M = 1 + a_1(e^{-k_c}-e^{-1}) + a_2(k_c-1)+a_3(AM-1.5),
where :math:`M` is the spectral mismatch factor, :math:`k_c` is the clear
sky index, :math:`AM` is the air mass, :math:`e` is Euler's number, and
:math:`a_1, a_2, a_3` are module-specific coefficients. The :math:`a_n`
coefficients available via the ``coefficients`` parameter differ from the
:math:`k_n` coefficients documented in [1]_ in that they are normalised by
the specific short-circuit current value, :math:`I_{sc0}^*`, which is the
expected short-circuit current at standard test conditions indoors. The
model used to estimate the air mass (denoted as :math:`AM`) is not stated
in the original publication. The authors of [1]_ used the ESRA model [2]_
to estimate the clear sky GHI for the clear sky index, which is the ratio
of GHI to clear sky GHI. Also, prior to the calculation of :math:`k_c`, the
irradiance measurements were corrected for angle of incidence using the
Martin and Ruiz model [3]_.
References
----------
.. [1] Huld, T., Sample, T., and Dunlop, E., 2009. A simple model
for estimating the influence of spectrum variations on PV performance.
In Proceedings of the 24th European Photovoltaic Solar Energy
Conference, Hamburg, Germany pp. 3385-3389. 2009. Accessed at:
https://www.researchgate.net/publication/256080247
.. [2] Rigollier, C., Bauer, O., and Wald, L., 2000. On the clear sky model
of the ESRA—European Solar Radiation Atlas—with respect to the Heliosat
method. Solar energy, 68(1), pp.33-48.
:doi:`10.1016/S0038-092X(99)00055-9`
.. [3] Martin, N. and Ruiz, J. M., 2001. Calculation of the PV modules
angular losses under field conditions by means of an analytical model.
Solar Energy Materials and Solar Cells, 70(1), 25-38.
:doi:`10.1016/S0927-0248(00)00408-6`
"""
_coefficients = {}
_coefficients['multisi'] = (0.00172, 0.000508, 0.00000357)
_coefficients['cdte'] = (0.000643, 0.000130, 0.0000108)
# normalise coefficients by I*sc0, see [1]
_coefficients = {
'multisi': tuple(x / 0.00348 for x in _coefficients['multisi']),
'cdte': tuple(x / 0.001150 for x in _coefficients['cdte'])
}
if module_type is not None and coefficients is None:
coefficients = _coefficients[module_type.lower()]
elif module_type is None and coefficients is not None:
pass
elif module_type is None and coefficients is None:
raise ValueError('No valid input provided, both module_type and ' +
'coefficients are None. module_type can be one of ' +
", ".join(_coefficients.keys()))
else:
raise ValueError('Cannot resolve input, must supply only one of ' +
'module_type and coefficients. module_type can be ' +
'one of' ", ".join(_coefficients.keys()))
coeff = coefficients
mismatch = (
1
+ coeff[0] * (np.exp(-clearsky_index) - np.exp(-1))
+ coeff[1] * (clearsky_index - 1)
+ coeff[2] * (airmass - 1.5)
)
return mismatch
