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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 pvlib import atmosphere, solarposition, tools
# see References section of grounddiffuse function
SURFACE_ALBEDOS = {'urban': 0.18,
'grass': 0.20,
'fresh grass': 0.26,
'soil': 0.17,
'sand': 0.40,
'snow': 0.65,
'fresh snow': 0.75,
'asphalt': 0.12,
'concrete': 0.30,
'aluminum': 0.85,
'copper': 0.74,
'fresh steel': 0.35,
'dirty steel': 0.08,
'sea': 0.06}
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://solardat.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))
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 poa_horizontal_ratio(surface_tilt, surface_azimuth,
solar_zenith, solar_azimuth):
"""
Calculates the ratio of the beam components of the plane of array
irradiance and the horizontal irradiance.
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
-------
ratio : numeric
Ratio of the plane of array irradiance to the horizontal plane
irradiance
"""
cos_poa_zen = aoi_projection(surface_tilt, surface_azimuth,
solar_zenith, solar_azimuth)
cos_solar_zenith = tools.cosd(solar_zenith)
# ratio of tilted and horizontal beam irradiance
ratio = cos_poa_zen / cos_solar_zenith
try:
ratio.name = 'poa_ratio'
except AttributeError:
pass
return ratio
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
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=.25, surface_type=None,
model='isotropic',
model_perez='allsitescomposite1990', **kwargs):
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
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
ghi : numeric
Global horizontal irradiance
dhi : numeric
Diffuse horizontal irradiance
dni_extra : None or numeric, default None
Extraterrestrial direct normal irradiance
airmass : None or numeric, default None
Airmass
albedo : numeric, default 0.25
Surface albedo
surface_type : None or String, default None
Surface type. See grounddiffuse.
model : String, default 'isotropic'
Irradiance model.
model_perez : String, default 'allsitescomposite1990'
Used only if model='perez'. See :py:func:`perez`.
Returns
-------
total_irrad : OrderedDict or DataFrame
Contains keys/columns ``'poa_global', 'poa_direct', 'poa_diffuse',
'poa_sky_diffuse', 'poa_ground_diffuse'``.
"""
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
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
ghi : numeric
Global horizontal irradiance
dhi : numeric
Diffuse horizontal irradiance
dni_extra : None or numeric, default None
Extraterrestrial direct normal irradiance
airmass : None or numeric, default None
Airmass
model : String, default 'isotropic'
Irradiance model.
model_perez : String, default 'allsitescomposite1990'
See perez.
Returns
-------
poa_sky_diffuse : numeric
"""
model = model.lower()
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':
sky = perez(surface_tilt, surface_azimuth, dhi, dni, dni_extra,
solar_zenith, solar_azimuth, airmass,
model=model_perez)
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 (W/m^2), as measured from a TMY file or
calculated with a clearsky model.
poa_sky_diffuse : numeric
Diffuse irradiance (W/m^2) in the plane of the modules, as
calculated by a diffuse irradiance translation function
poa_ground_diffuse : numeric
Ground reflected irradiance (W/m^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 (W/m^2)
* ``poa_direct`` : Total in-plane beam irradiance (W/m^2)
* ``poa_diffuse`` : Total in-plane diffuse irradiance (W/m^2)
* ``poa_sky_diffuse`` : In-plane diffuse irradiance from sky (W/m^2)
* ``poa_ground_diffuse`` : In-plane diffuse irradiance from ground
(W/m^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):
'''
Estimate diffuse irradiance from ground reflections given
irradiance, albedo, and surface tilt
Function to determine the portion of irradiance on a tilted surface
due to ground reflections. Any of the inputs may be DataFrames or
scalars.
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 in W/m^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: None or string, default None
If not None, overrides albedo. String 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 irradiances in W/m^2.
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.
The calculation is the last term of equations 3, 4, 7, 8, 10, 11, and 12.
.. [2] albedos from:
http://files.pvsyst.com/help/albedo.htm
and
http://en.wikipedia.org/wiki/Albedo
and
https://doi.org/10.1175/1520-0469(1972)029<0959:AOTSS>2.0.CO;2
'''
if surface_type is not None:
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.
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 in W/m^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
.. [2] Hottel, H.C., Woertz, B.B., 1942. Evaluation of flat-plate solar
heat collector. Trans. ASME 64, 91.
'''
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 Klucher's 1979 model
.. math::
I_{d} = DHI \frac{1 + \cos\beta}{2} (1 + F' \sin^3(\beta/2))
(1 + F' \cos^2\theta\sin^3\theta_z)
where
.. math::
F' = 1 - (I_{d0} / GHI)^2
Klucher's 1979 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, global
horizontal irradiance, extraterrestrial irradiance, sun zenith
angle, and sun azimuth angle.
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 in W/m^2. DHI must be >=0.
ghi : numeric
Global irradiance in W/m^2. DNI must be >=0.
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.
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] Klucher, T.M., 1979. Evaluation of models to predict insolation on
tilted surfaces. Solar Energy 23 (2), 111-114.
'''
# 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):
r'''
Determine diffuse irradiance from the sky on a tilted surface using
Hay & Davies' 1980 model
.. math::
I_{d} = DHI ( A R_b + (1 - A) (\frac{1 + \cos\beta}{2}) )
Hay and Davies' 1980 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
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 in W/m^2.
dni : numeric
Direct normal irradiance in W/m^2.
dni_extra : numeric
Extraterrestrial normal irradiance in W/m^2.
solar_zenith : None or numeric, default None
Solar apparent (refraction-corrected) zenith angles in decimal
degrees. Must supply ``solar_zenith`` and ``solar_azimuth`` or
supply ``projection_ratio``.
solar_azimuth : None or numeric, default None
Solar azimuth angles in decimal degrees. Must supply
``solar_zenith`` and ``solar_azimuth`` or supply
``projection_ratio``.
projection_ratio : None or numeric, default None
Ratio of angle of incidence projection to solar zenith angle
projection. Must supply ``solar_zenith`` and ``solar_azimuth``
or supply ``projection_ratio``.
Returns
--------
sky_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
.. [2] 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.
'''
# 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))
sky_diffuse = dhi * (AI * Rb + term1 * term2)
sky_diffuse = np.maximum(sky_diffuse, 0)
return sky_diffuse
def reindl(surface_tilt, surface_azimuth, dhi, dni, ghi, dni_extra,
solar_zenith, solar_azimuth):
r'''
Determine diffuse irradiance from the sky on a tilted surface using
Reindl's 1990 model
.. math::
I_{d} = DHI (A R_b + (1 - A) (\frac{1 + \cos\beta}{2})
(1 + \sqrt{\frac{I_{hb}}{I_h}} \sin^3(\beta/2)) )
Reindl's 1990 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, global
horizontal irradiance, extraterrestrial 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 in W/m^2.
dni : numeric
direct normal irradiance in W/m^2.
ghi: numeric
Global irradiance in W/m^2.
dni_extra : numeric
Extraterrestrial normal irradiance in W/m^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 poa_sky_diffuse calculation is generated from the Loutzenhiser et al.
(2007) paper, equation 8. Note that I have removed the beam and ground
reflectance portion of the equation and this generates ONLY the diffuse
radiation from the sky and circumsolar, so the form of the equation
varies slightly from equation 8.
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] Reindl, D.T., Beckmann, W.A., Duffie, J.A., 1990a. Diffuse
fraction correlations. Solar Energy 45(1), 1-7.
.. [3] Reindl, D.T., Beckmann, W.A., Duffie, J.A., 1990b. Evaluation of
hourly tilted surface radiation models. Solar Energy 45(1), 9-17.
'''
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
----------
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)
dhi : numeric
Diffuse horizontal irradiance in W/m^2.
ghi : numeric
Global horizontal irradiance in W/m^2.
solar_zenith : numeric
Apparent (refraction-corrected) zenith angles in decimal degrees.
Returns
--------
poa_sky_diffuse : numeric
The diffuse component of the solar radiation.
'''
sky_diffuse = (dhi * (1 + tools.cosd(surface_tilt)) / 2 + ghi *
(0.012 * solar_zenith - 0.04) *
(1 - tools.cosd(surface_tilt)) / 2)
sky_diffuse = np.maximum(sky_diffuse, 0)
return sky_diffuse
def perez(surface_tilt, surface_azimuth, dhi, dni, dni_extra,
solar_zenith, solar_azimuth, airmass,
model='allsitescomposite1990', return_components=False):
'''
Determine diffuse irradiance from the sky on a tilted surface using
one of the Perez models.
Perez models determine 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, sun azimuth angle, and relative (not
pressure-corrected) airmass. Optionally a selector may be used to
use any of Perez's model coefficient sets.
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 in W/m^2. DHI must be >=0.
dni : numeric
Direct normal irradiance in W/m^2. DNI must be >=0.
dni_extra : numeric
Extraterrestrial normal irradiance in W/m^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).
airmass : numeric
Relative (not pressure-corrected) airmass values. If AM is a
DataFrame it must be of the same size as all other DataFrame
inputs. AM must be >=0 (careful using the 1/sec(z) model of AM
generation)
model : string (optional, default='allsitescomposite1990')
A string which selects the desired set of Perez coefficients. If
model is not provided as an input, the default, '1990' will be
used. All possible model selections are:
* '1990'
* 'allsitescomposite1990' (same as '1990')
* 'allsitescomposite1988'
* 'sandiacomposite1988'
* 'usacomposite1988'