def calc_surface_orientation(tracker_theta, axis_tilt=0, axis_azimuth=0):
"""
Calculate the surface tilt and azimuth angles for a given tracker rotation.
Parameters
----------
tracker_theta : numeric
Tracker rotation angle as a right-handed rotation around
the axis defined by ``axis_tilt`` and ``axis_azimuth``. For example,
with ``axis_tilt=0`` and ``axis_azimuth=180``, ``tracker_theta > 0``
results in ``surface_azimuth`` to the West while ``tracker_theta < 0``
results in ``surface_azimuth`` to the East. [degree]
axis_tilt : float, default 0
The tilt of the axis of rotation with respect to horizontal. [degree]
axis_azimuth : float, default 0
A value denoting the compass direction along which the axis of
rotation lies. Measured east of north. [degree]
Returns
-------
dict or DataFrame
Contains keys ``'surface_tilt'`` and ``'surface_azimuth'`` representing
the module orientation accounting for tracker rotation and axis
orientation. [degree]
References
----------
.. [1] William F. Marion and Aron P. Dobos, "Rotation Angle for the Optimum
Tracking of One-Axis Trackers", Technical Report NREL/TP-6A20-58891,
July 2013. :doi:`10.2172/1089596`
"""
with np.errstate(invalid='ignore', divide='ignore'):
surface_tilt = acosd(cosd(tracker_theta) * cosd(axis_tilt))
# clip(..., -1, +1) to prevent arcsin(1 + epsilon) issues:
azimuth_delta = asind(np.clip(sind(tracker_theta) / sind(surface_tilt),
a_min=-1, a_max=1))
# Combine Eqs 2, 3, and 4:
azimuth_delta = np.where(abs(tracker_theta) < 90,
azimuth_delta,
-azimuth_delta + np.sign(tracker_theta) * 180)
# handle surface_tilt=0 case:
azimuth_delta = np.where(sind(surface_tilt) != 0, azimuth_delta, 90)
surface_azimuth = (axis_azimuth + azimuth_delta) % 360
out = {
'surface_tilt': surface_tilt,
'surface_azimuth': surface_azimuth,
}
if hasattr(tracker_theta, 'index'):
out = pd.DataFrame(out)
return out
def physicaliam(K, L, n, aoi):
'''
Determine the incidence angle modifier using refractive
index, glazing thickness, and extinction coefficient
physicaliam calculates the incidence angle modifier as described in
De Soto et al. "Improvement and validation of a model for photovoltaic
array performance", section 3. The calculation is based upon a physical
model of absorbtion and transmission through a cover. Required
information includes, incident angle, cover extinction coefficient,
cover thickness
Note: The authors of this function believe that eqn. 14 in [1] is
incorrect. This function uses the following equation in its place:
theta_r = arcsin(1/n * sin(theta))
Parameters
----------
K : float
The glazing extinction coefficient in units of 1/meters. Reference
[1] indicates that a value of 4 is reasonable for "water white"
glass. K must be a numeric scalar or vector with all values >=0. If K
is a vector, it must be the same size as all other input vectors.
L : float
The glazing thickness in units of meters. Reference [1] indicates
that 0.002 meters (2 mm) is reasonable for most glass-covered
PV panels. L must be a numeric scalar or vector with all values >=0.
If L is a vector, it must be the same size as all other input vectors.
n : float
The effective index of refraction (unitless). Reference [1]
indicates that a value of 1.526 is acceptable for glass. n must be a
numeric scalar or vector with all values >=0. If n is a vector, it
must be the same size as all other input vectors.
aoi : Series
The angle of incidence between the module normal vector and the
sun-beam vector in degrees.
Returns
-------
IAM : float or Series
The incident angle modifier as specified in eqns. 14-16 of [1].
IAM is a column vector with the same number of elements as the
largest input vector.
Theta must be a numeric scalar or vector.
For any values of theta where abs(aoi)>90, IAM is set to 0. For any
values of aoi where -90 < aoi < 0, theta is set to abs(aoi) and
evaluated.
References
----------
[1] W. De Soto et al., "Improvement and validation of a model for
photovoltaic array performance", Solar Energy, vol 80, pp. 78-88,
2006.
[2] Duffie, John A. & Beckman, William A.. (2006). Solar Engineering
of Thermal Processes, third edition. [Books24x7 version] Available
from http://common.books24x7.com/toc.aspx?bookid=17160.
See Also
--------
getaoi
ephemeris
spa
ashraeiam
'''
thetar_deg = tools.asind(1.0 / n*(tools.sind(aoi)))
tau = ( np.exp(- 1.0 * (K*L / tools.cosd(thetar_deg))) *
((1 - 0.5*((((tools.sind(thetar_deg - aoi)) ** 2) /
((tools.sind(thetar_deg + aoi)) ** 2) +
((tools.tand(thetar_deg - aoi)) ** 2) /
((tools.tand(thetar_deg + aoi)) ** 2))))) )
zeroang = 1e-06
thetar_deg0 = tools.asind(1.0 / n*(tools.sind(zeroang)))
tau0 = ( np.exp(- 1.0 * (K*L / tools.cosd(thetar_deg0))) *
((1 - 0.5*((((tools.sind(thetar_deg0 - zeroang)) ** 2) /
((tools.sind(thetar_deg0 + zeroang)) ** 2) +
((tools.tand(thetar_deg0 - zeroang)) ** 2) /
((tools.tand(thetar_deg0 + zeroang)) ** 2))))) )
IAM = tau / tau0
IAM[abs(aoi) >= 90] = np.nan
IAM[IAM < 0] = np.nan
return IAM
def physical(aoi, n=1.526, K=4., L=0.002):
r"""
Determine the incidence angle modifier using refractive index ``n``,
extinction coefficient ``K``, and glazing thickness ``L``.
``iam.physical`` calculates the incidence angle modifier as described in
[1]_, Section 3. The calculation is based on a physical model of absorbtion
and transmission through a transparent cover.
Parameters
----------
aoi : numeric
The angle of incidence between the module normal vector and the
sun-beam vector in degrees. Angles of 0 are replaced with 1e-06
to ensure non-nan results. Angles of nan will result in nan.
n : numeric, default 1.526
The effective index of refraction (unitless). Reference [1]_
indicates that a value of 1.526 is acceptable for glass.
K : numeric, default 4.0
The glazing extinction coefficient in units of 1/meters.
Reference [1] indicates that a value of 4 is reasonable for
"water white" glass.
L : numeric, default 0.002
The glazing thickness in units of meters. Reference [1]_
indicates that 0.002 meters (2 mm) is reasonable for most
glass-covered PV panels.
Returns
-------
iam : numeric
The incident angle modifier
Notes
-----
The pvlib python authors believe that Eqn. 14 in [1]_ is
incorrect, which presents :math:`\theta_{r} = \arcsin(n \sin(AOI))`.
Here, :math:`\theta_{r} = \arcsin(1/n \times \sin(AOI))`
References
----------
.. [1] W. De Soto et al., "Improvement and validation of a model for
photovoltaic array performance", Solar Energy, vol 80, pp. 78-88,
2006.
.. [2] Duffie, John A. & Beckman, William A.. (2006). Solar Engineering
of Thermal Processes, third edition. [Books24x7 version] Available
from http://common.books24x7.com/toc.aspx?bookid=17160.
See Also
--------
pvlib.iam.martin_ruiz
pvlib.iam.ashrae
pvlib.iam.interp
pvlib.iam.sapm
"""
zeroang = 1e-06
# hold a new reference to the input aoi object since we're going to
# overwrite the aoi reference below, but we'll need it for the
# series check at the end of the function
aoi_input = aoi
aoi = np.where(aoi == 0, zeroang, aoi)
# angle of reflection
thetar_deg = asind(1.0 / n * (sind(aoi)))
# reflectance and transmittance for normal incidence light
rho_zero = ((1 - n) / (1 + n))**2
tau_zero = np.exp(-K * L)
# reflectance for parallel and perpendicular polarized light
rho_para = (tand(thetar_deg - aoi) / tand(thetar_deg + aoi))**2
rho_perp = (sind(thetar_deg - aoi) / sind(thetar_deg + aoi))**2
# transmittance for non-normal light
tau = np.exp(-K * L / cosd(thetar_deg))
# iam is ratio of non-normal to normal incidence transmitted light
# after deducting the reflected portion of each
iam = ((1 - (rho_para + rho_perp) / 2) / (1 - rho_zero) * tau / tau_zero)
with np.errstate(invalid='ignore'):
# angles near zero produce nan, but iam is defined as one
small_angle = 1e-06
iam = np.where(np.abs(aoi) < small_angle, 1.0, iam)
# angles at 90 degrees can produce tiny negative values,
# which should be zero. this is a result of calculation precision
# rather than the physical model
iam = np.where(iam < 0, 0, iam)
# for light coming from behind the plane, none can enter the module
iam = np.where(aoi > 90, 0, iam)
if isinstance(aoi_input, pd.Series):
iam = pd.Series(iam, index=aoi_input.index)
return iam
def physicaliam(K, L, n, aoi):
'''
Determine the incidence angle modifier using refractive
index, glazing thickness, and extinction coefficient
physicaliam calculates the incidence angle modifier as described in
De Soto et al. "Improvement and validation of a model for photovoltaic
array performance", section 3. The calculation is based upon a physical
model of absorbtion and transmission through a cover. Required
information includes, incident angle, cover extinction coefficient,
cover thickness
Note: The authors of this function believe that eqn. 14 in [1] is
incorrect. This function uses the following equation in its place:
theta_r = arcsin(1/n * sin(theta))
Parameters
----------
K : float
The glazing extinction coefficient in units of 1/meters. Reference
[1] indicates that a value of 4 is reasonable for "water white"
glass. K must be a numeric scalar or vector with all values >=0. If K
is a vector, it must be the same size as all other input vectors.
L : float
The glazing thickness in units of meters. Reference [1] indicates
that 0.002 meters (2 mm) is reasonable for most glass-covered
PV panels. L must be a numeric scalar or vector with all values >=0.
If L is a vector, it must be the same size as all other input vectors.
n : float
The effective index of refraction (unitless). Reference [1]
indicates that a value of 1.526 is acceptable for glass. n must be a
numeric scalar or vector with all values >=0. If n is a vector, it
must be the same size as all other input vectors.
aoi : Series
The angle of incidence between the module normal vector and the
sun-beam vector in degrees.
Returns
-------
IAM : float or Series
The incident angle modifier as specified in eqns. 14-16 of [1].
IAM is a column vector with the same number of elements as the
largest input vector.
Theta must be a numeric scalar or vector.
For any values of theta where abs(aoi)>90, IAM is set to 0. For any
values of aoi where -90 < aoi < 0, theta is set to abs(aoi) and
evaluated.
References
----------
[1] W. De Soto et al., "Improvement and validation of a model for
photovoltaic array performance", Solar Energy, vol 80, pp. 78-88,
2006.
[2] Duffie, John A. & Beckman, William A.. (2006). Solar Engineering
of Thermal Processes, third edition. [Books24x7 version] Available
from http://common.books24x7.com/toc.aspx?bookid=17160.
See Also
--------
getaoi
ephemeris
spa
ashraeiam
'''
thetar_deg = tools.asind(1.0 / n*(tools.sind(aoi)))
tau = ( np.exp(- 1.0 * (K*L / tools.cosd(thetar_deg))) *
((1 - 0.5*((((tools.sind(thetar_deg - aoi)) ** 2) /
((tools.sind(thetar_deg + aoi)) ** 2) +
((tools.tand(thetar_deg - aoi)) ** 2) /
((tools.tand(thetar_deg + aoi)) ** 2))))) )
zeroang = 1e-06
thetar_deg0 = tools.asind(1.0 / n*(tools.sind(zeroang)))
tau0 = ( np.exp(- 1.0 * (K*L / tools.cosd(thetar_deg0))) *
((1 - 0.5*((((tools.sind(thetar_deg0 - zeroang)) ** 2) /
((tools.sind(thetar_deg0 + zeroang)) ** 2) +
((tools.tand(thetar_deg0 - zeroang)) ** 2) /
((tools.tand(thetar_deg0 + zeroang)) ** 2))))) )
IAM = tau / tau0
IAM[abs(aoi) >= 90] = np.nan
IAM[IAM < 0] = np.nan
return IAM