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irradiance.py
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"""
The ``irradiance`` module contains functions for modeling global
horizontal irradiance, direct normal irradiance, diffuse horizontal
irradiance, and total irradiance under various conditions.
"""
import datetime
from collections import OrderedDict
from functools import partial
import numpy as np
import pandas as pd
from scipy.interpolate import splev
from scipy.optimize import bisect
from pvlib import atmosphere, solarposition, tools
import pvlib # used to avoid dni name collision in complete_irradiance
from pvlib._deprecation import pvlibDeprecationWarning
import warnings
# Deprecation warning based on https://peps.python.org/pep-0562/
def __getattr__(attr):
if attr == 'SURFACE_ALBEDOS':
warnings.warn(f"{attr} has been moved to the albedo module as of "
"v0.11.0. Please use pvlib.albedo.SURFACE_ALBEDOS.",
pvlibDeprecationWarning)
return pvlib.albedo.SURFACE_ALBEDOS
raise AttributeError(f"module {__name__!r} has no attribute {attr!r}")
def get_extra_radiation(datetime_or_doy, solar_constant=1366.1,
method='spencer', epoch_year=2014, **kwargs):
"""
Determine extraterrestrial radiation from day of year.
Parameters
----------
datetime_or_doy : numeric, array, date, datetime, Timestamp, DatetimeIndex
Day of year, array of days of year, or datetime-like object
solar_constant : float, default 1366.1
The solar constant.
method : string, default 'spencer'
The method by which the ET radiation should be calculated.
Options include ``'pyephem', 'spencer', 'asce', 'nrel'``.
epoch_year : int, default 2014
The year in which a day of year input will be calculated. Only
applies to day of year input used with the pyephem or nrel
methods.
kwargs :
Passed to solarposition.nrel_earthsun_distance
Returns
-------
dni_extra : float, array, or Series
The extraterrestrial radiation present in watts per square meter
on a surface which is normal to the sun. Pandas Timestamp and
DatetimeIndex inputs will yield a Pandas TimeSeries. All other
inputs will yield a float or an array of floats.
References
----------
.. [1] M. Reno, C. Hansen, and J. Stein, "Global Horizontal Irradiance
Clear Sky Models: Implementation and Analysis", Sandia National
Laboratories, SAND2012-2389, 2012.
.. [2] http://solardata.uoregon.edu/SolarRadiationBasics.html, Eqs.
SR1 and SR2
.. [3] Partridge, G. W. and Platt, C. M. R. 1976. Radiative Processes
in Meteorology and Climatology.
.. [4] Duffie, J. A. and Beckman, W. A. 1991. Solar Engineering of
Thermal Processes, 2nd edn. J. Wiley and Sons, New York.
.. [5] ASCE, 2005. The ASCE Standardized Reference Evapotranspiration
Equation, Environmental and Water Resources Institute of the American
Civil Engineers, Ed. R. G. Allen et al.
"""
to_doy, to_datetimeindex, to_output = \
_handle_extra_radiation_types(datetime_or_doy, epoch_year)
# consider putting asce and spencer methods in their own functions
method = method.lower()
if method == 'asce':
B = solarposition._calculate_simple_day_angle(to_doy(datetime_or_doy),
offset=0)
RoverR0sqrd = 1 + 0.033 * np.cos(B)
elif method == 'spencer':
B = solarposition._calculate_simple_day_angle(to_doy(datetime_or_doy))
RoverR0sqrd = (1.00011 + 0.034221 * np.cos(B) + 0.00128 * np.sin(B) +
0.000719 * np.cos(2 * B) + 7.7e-05 * np.sin(2 * B))
elif method == 'pyephem':
times = to_datetimeindex(datetime_or_doy)
RoverR0sqrd = solarposition.pyephem_earthsun_distance(times) ** (-2)
elif method == 'nrel':
times = to_datetimeindex(datetime_or_doy)
RoverR0sqrd = \
solarposition.nrel_earthsun_distance(times, **kwargs) ** (-2)
else:
raise ValueError('Invalid method: %s', method)
Ea = solar_constant * RoverR0sqrd
Ea = to_output(Ea)
return Ea
def _handle_extra_radiation_types(datetime_or_doy, epoch_year):
# This block will set the functions that can be used to convert the
# inputs to either day of year or pandas DatetimeIndex, and the
# functions that will yield the appropriate output type. It's
# complicated because there are many day-of-year-like input types,
# and the different algorithms need different types. Maybe you have
# a better way to do it.
if isinstance(datetime_or_doy, pd.DatetimeIndex):
to_doy = tools._pandas_to_doy # won't be evaluated unless necessary
def to_datetimeindex(x): return x # noqa: E306
to_output = partial(pd.Series, index=datetime_or_doy)
elif isinstance(datetime_or_doy, pd.Timestamp):
to_doy = tools._pandas_to_doy
to_datetimeindex = \
tools._datetimelike_scalar_to_datetimeindex
to_output = tools._scalar_out
elif isinstance(datetime_or_doy,
(datetime.date, datetime.datetime, np.datetime64)):
to_doy = tools._datetimelike_scalar_to_doy
to_datetimeindex = \
tools._datetimelike_scalar_to_datetimeindex
to_output = tools._scalar_out
elif np.isscalar(datetime_or_doy): # ints and floats of various types
def to_doy(x): return x # noqa: E306
to_datetimeindex = partial(tools._doy_to_datetimeindex,
epoch_year=epoch_year)
to_output = tools._scalar_out
else: # assume that we have an array-like object of doy
def to_doy(x): return x # noqa: E306
to_datetimeindex = partial(tools._doy_to_datetimeindex,
epoch_year=epoch_year)
to_output = tools._array_out
return to_doy, to_datetimeindex, to_output
def aoi_projection(surface_tilt, surface_azimuth, solar_zenith, solar_azimuth):
"""
Calculates the dot product of the sun position unit vector and the surface
normal unit vector; in other words, the cosine of the angle of incidence.
Usage note: When the sun is behind the surface the value returned is
negative. For many uses negative values must be set to zero.
Input all angles in degrees.
Parameters
----------
surface_tilt : numeric
Panel tilt from horizontal.
surface_azimuth : numeric
Panel azimuth from north.
solar_zenith : numeric
Solar zenith angle.
solar_azimuth : numeric
Solar azimuth angle.
Returns
-------
projection : numeric
Dot product of panel normal and solar angle.
"""
projection = (
tools.cosd(surface_tilt) * tools.cosd(solar_zenith) +
tools.sind(surface_tilt) * tools.sind(solar_zenith) *
tools.cosd(solar_azimuth - surface_azimuth))
# GH 1185
projection = np.clip(projection, -1, 1)
try:
projection.name = 'aoi_projection'
except AttributeError:
pass
return projection
def aoi(surface_tilt, surface_azimuth, solar_zenith, solar_azimuth):
"""
Calculates the angle of incidence of the solar vector on a surface.
This is the angle between the solar vector and the surface normal.
Input all angles in degrees.
Parameters
----------
surface_tilt : numeric
Panel tilt from horizontal.
surface_azimuth : numeric
Panel azimuth from north.
solar_zenith : numeric
Solar zenith angle.
solar_azimuth : numeric
Solar azimuth angle.
Returns
-------
aoi : numeric
Angle of incidence in degrees.
"""
projection = aoi_projection(surface_tilt, surface_azimuth,
solar_zenith, solar_azimuth)
aoi_value = np.rad2deg(np.arccos(projection))
try:
aoi_value.name = 'aoi'
except AttributeError:
pass
return aoi_value
def beam_component(surface_tilt, surface_azimuth, solar_zenith, solar_azimuth,
dni):
"""
Calculates the beam component of the plane of array irradiance.
Parameters
----------
surface_tilt : numeric
Panel tilt from horizontal.
surface_azimuth : numeric
Panel azimuth from north.
solar_zenith : numeric
Solar zenith angle.
solar_azimuth : numeric
Solar azimuth angle.
dni : numeric
Direct normal irradiance, see :term:`dni`. [Wm⁻²]
Returns
-------
beam : numeric
Beam component
"""
beam = dni * aoi_projection(surface_tilt, surface_azimuth,
solar_zenith, solar_azimuth)
beam = np.maximum(beam, 0)
return beam
def get_total_irradiance(surface_tilt, surface_azimuth,
solar_zenith, solar_azimuth,
dni, ghi, dhi, dni_extra=None, airmass=None,
albedo=0.25, surface_type=None,
model='isotropic',
model_perez='allsitescomposite1990'):
r"""
Determine total in-plane irradiance and its beam, sky diffuse and ground
reflected components, using the specified sky diffuse irradiance model.
.. math::
I_{tot} = I_{beam} + I_{sky diffuse} + I_{ground}
Sky diffuse models include:
* isotropic (default)
* klucher
* haydavies
* reindl
* king
* perez
* perez-driesse
Parameters
----------
surface_tilt : numeric
Panel tilt from horizontal. [degree]
surface_azimuth : numeric
Panel azimuth from north. [degree]
solar_zenith : numeric
Solar zenith angle. [degree]
solar_azimuth : numeric
Solar azimuth angle. [degree]
dni : numeric
Direct Normal Irradiance. [W/m2]
ghi : numeric
Global horizontal irradiance. [W/m2]
dhi : numeric
Diffuse horizontal irradiance. [W/m2]
dni_extra : numeric, optional
Extraterrestrial direct normal irradiance. [W/m2]
airmass : numeric, optional
Relative airmass (not adjusted for pressure). [unitless]
albedo : numeric, default 0.25
Ground surface albedo. [unitless]
surface_type : str, optional
Surface type. See :py:func:`~pvlib.irradiance.get_ground_diffuse` for
the list of accepted values.
model : str, default 'isotropic'
Irradiance model. Can be one of ``'isotropic'``, ``'klucher'``,
``'haydavies'``, ``'reindl'``, ``'king'``, ``'perez'``,
``'perez-driesse'``.
model_perez : str, default 'allsitescomposite1990'
Used only if ``model='perez'``. See :py:func:`~pvlib.irradiance.perez`.
Returns
-------
total_irrad : OrderedDict or DataFrame
Contains keys/columns ``'poa_global', 'poa_direct', 'poa_diffuse',
'poa_sky_diffuse', 'poa_ground_diffuse'``.
Notes
-----
Models ``'haydavies'``, ``'reindl'``, ``'perez'`` and ``'perez-driesse'``
require ``'dni_extra'``. Values can be calculated using
:py:func:`~pvlib.irradiance.get_extra_radiation`.
The ``'perez'`` and ``'perez-driesse'`` models require relative airmass
(``airmass``) as input. If ``airmass`` is not provided, it is calculated
using the defaults in :py:func:`~pvlib.atmosphere.get_relative_airmass`.
"""
poa_sky_diffuse = get_sky_diffuse(
surface_tilt, surface_azimuth, solar_zenith, solar_azimuth,
dni, ghi, dhi, dni_extra=dni_extra, airmass=airmass, model=model,
model_perez=model_perez)
poa_ground_diffuse = get_ground_diffuse(surface_tilt, ghi, albedo,
surface_type)
aoi_ = aoi(surface_tilt, surface_azimuth, solar_zenith, solar_azimuth)
irrads = poa_components(aoi_, dni, poa_sky_diffuse, poa_ground_diffuse)
return irrads
def get_sky_diffuse(surface_tilt, surface_azimuth,
solar_zenith, solar_azimuth,
dni, ghi, dhi, dni_extra=None, airmass=None,
model='isotropic',
model_perez='allsitescomposite1990'):
r"""
Determine in-plane sky diffuse irradiance component
using the specified sky diffuse irradiance model.
Sky diffuse models include:
* isotropic (default)
* klucher
* haydavies
* reindl
* king
* perez
* perez-driesse
Parameters
----------
surface_tilt : numeric
Panel tilt from horizontal. [degree]
surface_azimuth : numeric
Panel azimuth from north. [degree]
solar_zenith : numeric
Solar zenith angle. [degree]
solar_azimuth : numeric
Solar azimuth angle. [degree]
dni : numeric
Direct Normal Irradiance. [W/m2]
ghi : numeric
Global horizontal irradiance. [W/m2]
dhi : numeric
Diffuse horizontal irradiance. [W/m2]
dni_extra : numeric, optional
Extraterrestrial direct normal irradiance. [W/m2]
airmass : numeric, optional
Relative airmass (not adjusted for pressure). [unitless]
model : str, default 'isotropic'
Irradiance model. Can be one of ``'isotropic'``, ``'klucher'``,
``'haydavies'``, ``'reindl'``, ``'king'``, ``'perez'``,
``'perez-driesse'``.
model_perez : str, default 'allsitescomposite1990'
Used only if ``model='perez'``. See :py:func:`~pvlib.irradiance.perez`.
Returns
-------
poa_sky_diffuse : numeric
Sky diffuse irradiance in the plane of array. [W/m2]
Raises
------
ValueError
If model is one of ``'haydavies'``, ``'reindl'``, ``'perez'``, or
``'perez_driesse'`` and ``dni_extra`` is not specified.
Notes
-----
Models ``'haydavies'``, ``'reindl'``, ``'perez'`` and ``'perez-driesse'``
require ``'dni_extra'``. Values can be calculated using
:py:func:`~pvlib.irradiance.get_extra_radiation`.
The ``'Perez'`` transposition model features discontinuities in the
predicted tilted diffuse irradiance due to relying on discrete input
values. For applications that benefit from continuous output, consider
using :py:func:`~pvlib.irradiance.perez_driesse`.
The ``'perez'`` and ``'perez-driesse'`` models require relative airmass
(``airmass``) as input. If ``airmass`` is not provided, it is calculated
using the defaults in :py:func:`~pvlib.atmosphere.get_relative_airmass`.
"""
model = model.lower()
if dni_extra is None and model in {'haydavies', 'reindl',
'perez', 'perez-driesse'}:
raise ValueError(f'dni_extra is required for model {model}')
if model == 'isotropic':
sky = isotropic(surface_tilt, dhi)
elif model == 'klucher':
sky = klucher(surface_tilt, surface_azimuth, dhi, ghi,
solar_zenith, solar_azimuth)
elif model == 'haydavies':
sky = haydavies(surface_tilt, surface_azimuth, dhi, dni, dni_extra,
solar_zenith, solar_azimuth)
elif model == 'reindl':
sky = reindl(surface_tilt, surface_azimuth, dhi, dni, ghi, dni_extra,
solar_zenith, solar_azimuth)
elif model == 'king':
sky = king(surface_tilt, dhi, ghi, solar_zenith)
elif model == 'perez':
if airmass is None:
airmass = atmosphere.get_relative_airmass(solar_zenith)
sky = perez(surface_tilt, surface_azimuth, dhi, dni, dni_extra,
solar_zenith, solar_azimuth, airmass,
model=model_perez)
elif model == 'perez-driesse':
# perez_driesse will calculate its own airmass if needed
sky = perez_driesse(surface_tilt, surface_azimuth, dhi, dni, dni_extra,
solar_zenith, solar_azimuth, airmass)
else:
raise ValueError(f'invalid model selection {model}')
return sky
def poa_components(aoi, dni, poa_sky_diffuse, poa_ground_diffuse):
r'''
Determine in-plane irradiance components.
Combines DNI with sky diffuse and ground-reflected irradiance to calculate
total, direct and diffuse irradiance components in the plane of array.
Parameters
----------
aoi : numeric
Angle of incidence of solar rays with respect to the module
surface, from :func:`aoi`.
dni : numeric
Direct normal irradiance (Wm⁻²), as measured from a TMY file or
calculated with a clearsky model.
poa_sky_diffuse : numeric
Diffuse irradiance (Wm⁻²) in the plane of the modules, as
calculated by a diffuse irradiance translation function
poa_ground_diffuse : numeric
Ground reflected irradiance (Wm⁻²) in the plane of the modules,
as calculated by an albedo model (eg. :func:`grounddiffuse`)
Returns
-------
irrads : OrderedDict or DataFrame
Contains the following keys:
* ``poa_global`` : Total in-plane irradiance (Wm⁻²)
* ``poa_direct`` : Total in-plane beam irradiance (Wm⁻²)
* ``poa_diffuse`` : Total in-plane diffuse irradiance (Wm⁻²)
* ``poa_sky_diffuse`` : In-plane diffuse irradiance from sky (Wm⁻²)
* ``poa_ground_diffuse`` : In-plane diffuse irradiance from ground
(Wm⁻²)
Notes
------
Negative beam irradiation due to aoi :math:`> 90^{\circ}` or AOI
:math:`< 0^{\circ}` is set to zero.
'''
poa_direct = np.maximum(dni * np.cos(np.radians(aoi)), 0)
poa_diffuse = poa_sky_diffuse + poa_ground_diffuse
poa_global = poa_direct + poa_diffuse
irrads = OrderedDict()
irrads['poa_global'] = poa_global
irrads['poa_direct'] = poa_direct
irrads['poa_diffuse'] = poa_diffuse
irrads['poa_sky_diffuse'] = poa_sky_diffuse
irrads['poa_ground_diffuse'] = poa_ground_diffuse
if isinstance(poa_direct, pd.Series):
irrads = pd.DataFrame(irrads)
return irrads
def get_ground_diffuse(surface_tilt, ghi, albedo=.25, surface_type=None):
r'''
Estimate diffuse irradiance on a tilted surface from ground reflections.
Ground diffuse irradiance is calculated as
.. math::
G_{ground} = GHI \times \rho \times \frac{1 - \cos\beta}{2}
where :math:`\rho` is ``albedo`` and :math:`\beta` is ``surface_tilt``.
Parameters
----------
surface_tilt : numeric
Surface tilt angles in decimal degrees. Tilt must be >=0 and
<=180. The tilt angle is defined as degrees from horizontal
(e.g. surface facing up = 0, surface facing horizon = 90).
ghi : numeric
Global horizontal irradiance. [Wm⁻²]
albedo : numeric, default 0.25
Ground reflectance, typically 0.1-0.4 for surfaces on Earth
(land), may increase over snow, ice, etc. May also be known as
the reflection coefficient. Must be >=0 and <=1. Will be
overridden if ``surface_type`` is supplied.
surface_type : string, optional
If supplied, overrides ``albedo``. ``surface_type`` can be one of
'urban', 'grass', 'fresh grass', 'snow', 'fresh snow', 'asphalt',
'concrete', 'aluminum', 'copper', 'fresh steel', 'dirty steel',
'sea'.
Returns
-------
grounddiffuse : numeric
Ground reflected irradiance. [Wm⁻²]
Notes
-----
Table of albedo values by ``surface_type`` are from [2]_, [3]_, [4]_;
see :py:const:`~pvlib.albedo.SURFACE_ALBEDOS`.
References
----------
.. [1] Loutzenhiser P.G. et. al. "Empirical validation of models to compute
solar irradiance on inclined surfaces for building energy simulation"
2007, Solar Energy vol. 81. pp. 254-267.
.. [2] https://www.pvsyst.com/help/albedo.htm Accessed January, 2024.
.. [3] http://en.wikipedia.org/wiki/Albedo Accessed January, 2024.
.. [4] Payne, R. E. "Albedo of the Sea Surface". J. Atmos. Sci., 29,
pp. 959–970, 1972.
:doi:`10.1175/1520-0469(1972)029<0959:AOTSS>2.0.CO;2`
'''
if surface_type is not None:
albedo = pvlib.albedo.SURFACE_ALBEDOS[surface_type]
diffuse_irrad = ghi * albedo * (1 - np.cos(np.radians(surface_tilt))) * 0.5
try:
diffuse_irrad.name = 'diffuse_ground'
except AttributeError:
pass
return diffuse_irrad
def isotropic(surface_tilt, dhi):
r'''
Determine diffuse irradiance from the sky on a tilted surface using
the isotropic sky model.
.. math::
I_{d} = DHI \frac{1 + \cos\beta}{2}
Hottel and Woertz's model treats the sky as a uniform source of
diffuse irradiance. Thus, the diffuse irradiance from the sky (ground
reflected irradiance is not included in this algorithm) on a tilted
surface can be found from the diffuse horizontal irradiance and the
tilt angle of the surface. A discussion of the origin of the
isotropic model can be found in [2]_.
Parameters
----------
surface_tilt : numeric
Surface tilt angle in decimal degrees. Tilt must be >=0 and
<=180. The tilt angle is defined as degrees from horizontal
(e.g. surface facing up = 0, surface facing horizon = 90)
dhi : numeric
Diffuse horizontal irradiance. [Wm⁻²] DHI must be >=0.
Returns
-------
diffuse : numeric
The sky diffuse component of the solar radiation.
References
----------
.. [1] Loutzenhiser P.G. et al. "Empirical validation of models to
compute solar irradiance on inclined surfaces for building energy
simulation" 2007, Solar Energy vol. 81. pp. 254-267
:doi:`10.1016/j.solener.2006.03.009`
.. [2] Kamphuis, N.R. et al. "Perspectives on the origin, derivation,
meaning, and significance of the isotropic sky model" 2020, Solar
Energy vol. 201. pp. 8-12
:doi:`10.1016/j.solener.2020.02.067`
'''
sky_diffuse = dhi * (1 + tools.cosd(surface_tilt)) * 0.5
return sky_diffuse
def klucher(surface_tilt, surface_azimuth, dhi, ghi, solar_zenith,
solar_azimuth):
r'''
Determine diffuse irradiance from the sky on a tilted surface
using the Klucher (1979) model.
Parameters
----------
surface_tilt : numeric
Surface tilt angles in decimal degrees. ``surface_tilt`` must be >=0
and <=180. The tilt angle is defined as degrees from horizontal
(e.g. surface facing up = 0, surface facing horizon = 90)
surface_azimuth : numeric
Surface azimuth angles in decimal degrees. ``surface_azimuth`` must
be >=0 and <=360. The Azimuth convention is defined as degrees
east of north (e.g. North = 0, South=180 East = 90, West = 270).
dhi : numeric
Diffuse horizontal irradiance, must be >=0. [Wm⁻²]
ghi : numeric
Global horizontal irradiance, must be >=0. [Wm⁻²]
solar_zenith : numeric
Apparent (refraction-corrected) zenith angles in decimal
degrees. ``solar_zenith`` must be >=0 and <=180.
solar_azimuth : numeric
Sun azimuth angles in decimal degrees. ``solar_azimuth`` must be >=0
and <=360. The Azimuth convention is defined as degrees east of
north (e.g. North = 0, East = 90, West = 270).
Returns
-------
diffuse : numeric
The sky diffuse component of the solar radiation. [Wm⁻²]
Notes
-----
The Klucher (1979) model [1]_ [2]_ determines the diffuse irradiance
from the sky (ground reflected irradiance is not included in this
algorithm) on a tilted surface using the surface tilt angle, surface
azimuth angle, diffuse horizontal irradiance, global horizontal irradiance,
sun zenith angle, and sun azimuth angle. The expression for the sky diffuse
irradiance, :math:`I_d`, is as follows:
.. math::
I_{d} = DHI \frac{1 + \cos\beta}{2} (1 + F' \sin^3(\beta/2))
(1 + F' \cos^2\theta\sin^3\theta_z).
DHI is the diffuse horizontal irradiance, :math:`\beta` is the surface
tilt angle, :math:`\theta_z` is the solar zenith angle, and :math:`\theta`
is the angle of incidence. :math:`F'` is a modulating function to account
for when the sky changes from clear to overcast, and is defined as follows:
.. math::
F' = 1 - (DHI / GHI)^2,
where GHI is the global horiztonal irradiance.
References
----------
.. [1] Klucher, T. M. "Evaluation of models to predict insolation on
tilted surfaces" 1979, Solar Energy vol. 23 (2), pp. 111-114
:doi:`10.1016/0038-092X(79)90110-5`
.. [2] Loutzenhiser P. G. et al. "Empirical validation of models to
compute solar irradiance on inclined surfaces for building energy
simulation" 2007, Solar Energy vol. 81. pp. 254-267
:doi:`10.1016/j.solener.2006.03.009`
'''
# zenith angle with respect to panel normal.
cos_tt = aoi_projection(surface_tilt, surface_azimuth,
solar_zenith, solar_azimuth)
cos_tt = np.maximum(cos_tt, 0) # GH 526
# silence warning from 0 / 0
with np.errstate(invalid='ignore'):
F = 1 - ((dhi / ghi) ** 2)
try:
# fails with single point input
F.fillna(0, inplace=True)
except AttributeError:
F = np.where(np.isnan(F), 0, F)
term1 = 0.5 * (1 + tools.cosd(surface_tilt))
term2 = 1 + F * (tools.sind(0.5 * surface_tilt) ** 3)
term3 = 1 + F * (cos_tt ** 2) * (tools.sind(solar_zenith) ** 3)
sky_diffuse = dhi * term1 * term2 * term3
return sky_diffuse
def haydavies(surface_tilt, surface_azimuth, dhi, dni, dni_extra,
solar_zenith=None, solar_azimuth=None, projection_ratio=None,
return_components=False):
r'''
Determine diffuse irradiance from the sky on a tilted surface using the
Hay and Davies (1980) model.
The Hay and Davies model determines the diffuse irradiance from
the sky (ground reflected irradiance is not included in this
algorithm) on a tilted surface using the surface tilt angle, surface
azimuth angle, diffuse horizontal irradiance, direct normal irradiance,
extraterrestrial irradiance, sun zenith angle, and sun azimuth angle.
Parameters
----------
surface_tilt : numeric
Panel tilt from the horizontal, in decimal degrees, see
:term:`surface_tilt`.
surface_azimuth : numeric
Surface azimuth angles in decimal degrees. The azimuth
convention is defined as degrees east of north (e.g. North=0,
South=180, East=90, West=270).
dhi : numeric
Diffuse horizontal irradiance. [Wm⁻²]
dni : numeric
Direct normal irradiance, see :term:`dni`. [Wm⁻²]
dni_extra : numeric
Extraterrestrial normal irradiance. [Wm⁻²]
solar_zenith : numeric, optional
Solar apparent (refraction-corrected) zenith angles in decimal
degrees. Must supply ``solar_zenith`` and ``solar_azimuth`` or
supply ``projection_ratio``.
solar_azimuth : numeric, optional
Solar azimuth angles in decimal degrees. Must supply
``solar_zenith`` and ``solar_azimuth`` or supply
``projection_ratio``.
projection_ratio : numeric, optional
Ratio of angle of incidence projection to solar zenith angle
projection. Must supply ``solar_zenith`` and ``solar_azimuth``
or supply ``projection_ratio``.
return_components : bool, default `False`
If `False`, ``sky_diffuse`` is returned.
If `True`, ``diffuse_components`` is returned.
Returns
--------
numeric, OrderedDict, or DataFrame
Return type controlled by ``return_components`` argument.
If `False`, ``sky_diffuse`` is returned.
If `True`, ``diffuse_components`` is returned.
sky_diffuse : numeric
The sky diffuse component of the solar radiation on a tilted
surface.
diffuse_components : OrderedDict (array input) or DataFrame (Series input)
Keys/columns are:
* sky_diffuse: Total sky diffuse
* isotropic
* circumsolar
* horizon
Notes
------
In the Hay and Davies (1980) model [1]_ [2]_, sky diffuse irradiance,
:math:`I_d`, is as follows:
.. math::
I_{d} = DHI \left( A\cdot R_b + (1 - A) \left(\frac{
1 + \cos\beta}{2}\right ) \right ).
:math:`DHI` is the diffuse horizontal irradiance,
:math:`A` is the anisotropy index, which is the ratio of the direct normal
irradiance to the extraterrestrial irradiation,
:math:`R_b` is the projection ratio, which is defined as the ratio of the
cosine of the angle of incidence (AOI) to the cosine of the zenith angle,
and finally :math:`\beta` is the tilt angle of the array.
If supplying ``projection_ratio``, consider constraining its values
when the zenith angle approaches 90 degrees or the AOI projection is
negative. See code for details.
References
-----------
.. [1] Hay, J. E., Davies, J. A., 1980. Calculations of the solar
radiation incident on an inclined surface. In: Hay, J.E., Won, T.K.
(Eds.), Proc. of First Canadian Solar Radiation Data Workshop, 59.
Ministry of Supply and Services, Canada. Available from:
https://archive.org/details/proceedingsfirst00cana/mode/2up
(Last accessed: 17/09/24)
.. [2] Loutzenhiser P. G. et al. "Empirical validation of models to
compute solar irradiance on inclined surfaces for building energy
simulation" 2007, Solar Energy vol. 81. pp. 254-267
:doi:`10.1016/j.solener.2006.03.009`
'''
# if necessary, calculate ratio of titled and horizontal beam irradiance
if projection_ratio is None:
cos_tt = aoi_projection(surface_tilt, surface_azimuth,
solar_zenith, solar_azimuth)
cos_tt = np.maximum(cos_tt, 0) # GH 526
cos_solar_zenith = tools.cosd(solar_zenith)
Rb = cos_tt / np.maximum(cos_solar_zenith, 0.01745) # GH 432
else:
Rb = projection_ratio
# Anisotropy Index
AI = dni / dni_extra
# these are the () and [] sub-terms of the second term of eqn 7
term1 = 1 - AI
term2 = 0.5 * (1 + tools.cosd(surface_tilt))
poa_isotropic = np.maximum(dhi * term1 * term2, 0)
poa_circumsolar = np.maximum(dhi * (AI * Rb), 0)
sky_diffuse = poa_isotropic + poa_circumsolar
if return_components:
diffuse_components = OrderedDict()
diffuse_components['sky_diffuse'] = sky_diffuse
# Calculate the individual components
diffuse_components['isotropic'] = poa_isotropic
diffuse_components['circumsolar'] = poa_circumsolar
diffuse_components['horizon'] = np.where(
np.isnan(diffuse_components['isotropic']), np.nan, 0.)
if isinstance(sky_diffuse, pd.Series):
diffuse_components = pd.DataFrame(diffuse_components)
return diffuse_components
else:
return sky_diffuse
def reindl(surface_tilt, surface_azimuth, dhi, dni, ghi, dni_extra,
solar_zenith, solar_azimuth):
r'''
Determine the diffuse irradiance from the sky on a tilted surface using
the Reindl (1990) model.
The Reindl (1990) model [1]_ [2]_ determines the diffuse irradiance from
the sky on
a tilted surface using the surface tilt angle, surface azimuth angle,
diffuse horizontal irradiance, direct normal irradiance, global
horizontal irradiance, extraterrestrial normal irradiance, sun zenith
angle, and sun azimuth angle.
Parameters
----------
surface_tilt : numeric
Surface tilt angles in decimal degrees. The tilt angle is
defined as degrees from horizontal (e.g. surface facing up = 0,
surface facing horizon = 90)
surface_azimuth : numeric
Surface azimuth angles in decimal degrees. The azimuth
convention is defined as degrees east of north (e.g. North = 0,
South=180 East = 90, West = 270).
dhi : numeric
diffuse horizontal irradiance. [Wm⁻²]
dni : numeric
direct normal irradiance. [Wm⁻²]
ghi: numeric
Global horizontal irradiance. [Wm⁻²]
dni_extra : numeric
Extraterrestrial normal irradiance. [Wm⁻²]
solar_zenith : numeric
Apparent (refraction-corrected) zenith angles in decimal degrees.
solar_azimuth : numeric
Sun azimuth angles in decimal degrees. The azimuth convention is
defined as degrees east of north (e.g. North = 0, East = 90,
West = 270).
Returns
-------
poa_sky_diffuse : numeric
The sky diffuse component of the solar radiation.
Notes
-----
The Reindl (1990) model for the sky diffuse irradiance,
:math:`I_d`, is as follows:
.. math::
I_{d} = DHI \left(A \cdot R_b + (1 - A)
\left(\frac{1 + \cos\beta}{2}\right)
\left(1 + \sqrt{\frac{BHI}{GHI}} \sin^3(\beta/2)\right) \right).
:math:`DHI`, :math:`BHI`, and :math:`GHI` are the diffuse horizontal, beam
(direct) horizontal and global horizontal irradiances, respectively.
:math:`A` is the anisotropy index, which is the ratio of the direct normal
irradiance to the direct extraterrestrial irradiation, :math:`R_b` is the
projection ratio, which is defined as the ratio of the cosine of the angle
of incidence (AOI) to the cosine of the zenith angle, and :math:`\beta`
is the tilt angle of the array.
Implementation is based on Loutzenhiser et al.
(2007) [3]_, Equation 8. The beam and ground reflectance portion of the
equation have been removed, therefore the model described here generates
ONLY the diffuse radiation from the sky and circumsolar, so the form of the
equation varies slightly from Equation 8 in [3]_.
References
----------
.. [1] Reindl, D. T., Beckmann, W. A., Duffie, J. A., 1990a. Diffuse
fraction correlations. Solar Energy 45(1), 1-7.
:doi:`10.1016/0038-092X(90)90060-P`
.. [2] Reindl, D. T., Beckmann, W. A., Duffie, J. A., 1990b. Evaluation of
hourly tilted surface radiation models. Solar Energy 45(1), 9-17.
:doi:`10.1016/0038-092X(90)90061-G`
.. [3] Loutzenhiser P. G. et. al., 2007. Empirical validation of models to
compute solar irradiance on inclined surfaces for building energy
simulation. Solar Energy 81(2), 254-267
:doi:`10.1016/j.solener.2006.03.009`
'''
cos_tt = aoi_projection(surface_tilt, surface_azimuth,
solar_zenith, solar_azimuth)
cos_tt = np.maximum(cos_tt, 0) # GH 526
# do not apply cos(zen) limit here (needed for HB below)
cos_solar_zenith = tools.cosd(solar_zenith)
# ratio of titled and horizontal beam irradiance
Rb = cos_tt / np.maximum(cos_solar_zenith, 0.01745) # GH 432
# Anisotropy Index
AI = dni / dni_extra
# DNI projected onto horizontal
HB = dni * cos_solar_zenith
HB = np.maximum(HB, 0)
# these are the () and [] sub-terms of the second term of eqn 8
term1 = 1 - AI
term2 = 0.5 * (1 + tools.cosd(surface_tilt))
with np.errstate(invalid='ignore', divide='ignore'):
hb_to_ghi = np.where(ghi == 0, 0, np.divide(HB, ghi))
term3 = 1 + np.sqrt(hb_to_ghi) * (tools.sind(0.5 * surface_tilt)**3)
sky_diffuse = dhi * (AI * Rb + term1 * term2 * term3)
sky_diffuse = np.maximum(sky_diffuse, 0)
return sky_diffuse
def king(surface_tilt, dhi, ghi, solar_zenith):
'''
Determine diffuse irradiance from the sky on a tilted surface using
the King model.
King's model determines the diffuse irradiance from the sky (ground
reflected irradiance is not included in this algorithm) on a tilted
surface using the surface tilt angle, diffuse horizontal irradiance,
global horizontal irradiance, and sun zenith angle. Note that this
model is not well documented and has not been published in any
fashion (as of January 2012).
Parameters
----------