"""
This module contains implementations of several PV module efficiency models.
These models have a common purpose, which is to predict the efficiency at
maximum power point as a function of the main operating conditions:
effective irradiance and module temperature.
A function to fit any of these models to measurements is also provided.
Copyright (c) 2019-2020 Anton Driesse, PV Performance Labs.
"""
import inspect
import numpy as np
import pandas as pd
from scipy.optimize import curve_fit
from pvpltools.iec61853 import BilinearInterpolator
[docs]
def fit_efficiency_model(irradiance, temperature, eta, model, p0=None,
**kwargs):
"""
Determine the parameters of a module efficiency model by non-linear
least-squares fit.
This is a convenience function that calls the scipy curve_fit function
with suitable parameters and defaults.
Parameters
----------
irradiance : non-negative numeric, W/m²
The effective irradiance incident on the PV module.
temperature : numeric, °C
The module operating temperature.
eta : numeric
The efficiency of the module at the specified irradiance and
temperature.
model : function
A PV module efficiency function such as `adr`. It must take
irradiance and temperature as the first two arguments and the
model-specific parameters as the remaining arguments.
p0 : array_like, optional
Initial guess for the parameters, which may speed up the fit process.
kwargs :
Optional keyword arguments passed to `curve_fit`.
Returns
-------
popt : array
Optimal values for the parameters so that the sum of the squared
residuals of ``model(irradiance, temperature, *popt) - eta`` is
minimized.
pcov : 2-D array
The estimated covariance of popt. See `curve_fit` for details.
Raises
------
(These errors and warnings are from `curve_fit`.)
ValueError
if either `ydata` or `xdata` contain NaNs, or if incompatible options
are used.
RuntimeError
if the least-squares minimization fails.
OptimizeWarning
if covariance of the parameters can not be estimated.
See also
--------
pvpltools.module_efficiency.adr
scipy.optimize.curve_fit
Author: Anton Driesse, PV Performance Labs
"""
if p0 is None:
# determine number of parameters by inspecting the function
# and set initial parameters all to 1
sig = inspect.signature(model)
p0 = np.zeros(len(sig.parameters) - 2)
if 'method' not in kwargs:
kwargs['method'] = 'trf'
def model_wrapper(xdata, *params):
return model(*xdata, *params)
popt, pcov = curve_fit(model_wrapper,
xdata=[irradiance, temperature],
ydata=eta,
p0=p0,
**kwargs
)
return popt, pcov
[docs]
def adr(irradiance, temperature, k_a, k_d, tc_d, k_rs, k_rsh):
'''
Calculate PV module efficiency using the ADR model.
The efficiency varies with irradiance and operating temperature
and is determined by 5 model parameters as described in [1]_.
Parameters
----------
irradiance : non-negative numeric, W/m²
The effective irradiance incident on the PV module.
temperature : numeric, °C
The module operating temperature.
k_a : float
Absolute scaling factor, which is equal to the efficiency at
reference conditions. This factor allows the model to be used
with relative or absolute efficiencies, and to accommodate data sets
which are not perfectly normalized but have a slight bias at
the reference conditions.
k_d : negative float
“Dark irradiance” or diode coefficient which influences the voltage
increase with irradiance.
tc_d : float
Temperature coefficient of the diode coefficient, which indirectly
influences voltage. Because it is the only temperature coefficient
in the model, its value will also reflect secondary temperature
dependencies that are present in the PV module.
k_rs and k_rsh : float
Series and shunt resistance loss factors. Because of the normalization
they can be read as power loss fractions at reference conditions.
For example, if k_rs is 0.05, the internal loss assigned to the
series resistance has a magnitude equal to 5% of the module output.
Returns
-------
eta : numeric
The efficiency of the module at the specified irradiance and
temperature.
Notes
-----
The efficiency values may be absolute or relative, and may be expressed
as percent or per unit. This is determined by the efficiency data
used to derive values for the 5 model parameters. The first model
parameter k_a is equal to the efficiency at STC and therefore
indicates the efficiency scale being used. k_a can also be changed
freely to adjust the scale, or to change the module class to a slightly
higher or lower efficiency.
References
----------
.. [1] A. Driesse and J. S. Stein, "From IEC 61853 power measurements
to PV system simulations", Sandia Report No. SAND2020-3877, 2020.
.. [2] A. Driesse, M. Theristis and J. S. Stein, "A New Photovoltaic
Module Efficiency Model for Energy Prediction and Rating",
forthcoming.
Author: Anton Driesse, PV Performance Labs
'''
g = np.asanyarray(irradiance)
t = np.asanyarray(temperature)
# normalize the irradiance
G_REF = 1000
s = g / G_REF
# obtain the difference from reference temperature
T_REF = 25
dt = t - T_REF
# equation 29 in JPV
s_o = 10**(k_d + (tc_d * dt))
s_o_ref = 10**(k_d)
# equation 28 and 30 in JPV
# the constant k_v does not appear here because it cancels out
v = np.log(s / s_o + 1)
v /= np.log(1 / s_o_ref + 1)
# equation 25 in JPV
eta = k_a * ((1 + k_rs + k_rsh) * v - k_rs * s - k_rsh * v**2)
return eta
[docs]
def heydenreich(irradiance, temperature, a, b, c, gamma_pmp):
"""
Calculate PV module efficiency using the Heydenreich model.
The efficiency varies with irradiance and operating temperature
and is determined by three parameters for irradiance dependency and
one for temperature dependency as described in [1]_.
Parameters
----------
irradiance : non-negative numeric, W/m²
The effective irradiance incident on the PV module.
temperature : numeric, °C
The module operating temperature.
a, b, c : float
Three model parameters usually determined by regression.
gamma_pmp : float
The temperature coefficient of power, which may be taken
from the module datasheet or also determined by regression.
Returns
-------
eta : numeric
The efficiency of the module at the specified irradiance and
temperature.
See also
--------
fit_efficiency_model
adr
Notes
-----
A comprehensive comparison of efficiency models is found in [2]_ and [3]_.
References
----------
.. [1] W. Heydenreich, et al., "Describing the world with three parameters:
a new approach to PV module power modelling," in 23rd European PV
Solar Energy Conference and Exhibition (EU PVSEC), 2008, pp. 2786-2789.
.. [2] A. Driesse and J. S. Stein, "From IEC 61853 power measurements
to PV system simulations", Sandia Report No. SAND2020-3877, 2020.
.. [3] A. Driesse, M. Theristis and J. S. Stein, "A New Photovoltaic
Module Efficiency Model for Energy Prediction and Rating",
forthcoming.
Author: Anton Driesse, PV Performance Labs
"""
from numpy import log, exp, square
g = np.asanyarray(irradiance)
t = np.asanyarray(temperature)
dt = t - 25
eta = (
# power loss in R series
a * g +
# power gain from voltage * current
b * log(g + 1) +
# power loss in R shunt (constant Rsh)
c * (square(log(g + exp(1))) / (g + 1) - 1)
)
eta *= 1 + gamma_pmp * dt
return eta
[docs]
def motherpv(irradiance, temperature, a, b, c, d, gamma_ref, aa, bb):
"""
Calculate PV module efficiency using the MotherPV model.
The efficiency varies with irradiance and operating temperature
and is determined by 7 parameters as described in [1]_.
Parameters
----------
irradiance : non-negative numeric, W/m²
The effective irradiance incident on the PV module.
temperature : numeric, °C
The module operating temperature.
a, b, c, d, aa, bb : float
Six model parameters usually determined by regression.
gamma_pmp : float
The temperature coefficient of power, which may be taken
from the module datasheet or also determined by regression.
Returns
-------
eta : numeric
The efficiency of the module at the specified irradiance and
temperature.
See also
--------
fit_efficiency_model
adr
Notes
-----
A comprehensive comparison of efficiency models is found in [2]_ and [3]_.
References
----------
.. [1] A. G. de Montgareuil, et al., "A new tool for the MotherPV method:
modelling of the irradiance coefficient of photovoltaic modules,"
in 24th European Photovoltaic Solar Energy Conference (EU PVSEC),
2009, pp. 21-25.
.. [2] A. Driesse and J. S. Stein, "From IEC 61853 power measurements
to PV system simulations", Sandia Report No. SAND2020-3877, 2020.
.. [3] A. Driesse, M. Theristis and J. S. Stein, "A New Photovoltaic
Module Efficiency Model for Energy Prediction and Rating",
forthcoming.
Author: Anton Driesse, PV Performance Labs
"""
from numpy import log
g = np.asanyarray(irradiance)
t = np.asanyarray(temperature)
s = g / 1000
dt = t - 25
eta = 1 + a * (s - 1) + b * log(s) + c * (s - 1) ** 2 + d * log(s) ** 2
gamma = gamma_ref * (1 + aa * (s - 1) + bb * log(s))
eta *= 1 + gamma * dt
return eta
[docs]
def pvgis(irradiance, temperature, k1, k2, k3, k4, k5, k6):
"""
Calculate PV module efficiency using the PVGIS model.
The efficiency varies with irradiance and operating temperature
and is determined by 6 parameters as described in [1]_.
Parameters
----------
irradiance : non-negative numeric, W/m²
The effective irradiance incident on the PV module.
temperature : numeric, °C
The module operating temperature.
k1, k2, k3, k4, k5, k6 : float
Six model parameters usually determined by regression.
Returns
-------
eta : numeric
The efficiency of the module at the specified irradiance and
temperature.
See also
--------
fit_efficiency_model
adr
Notes
-----
A comprehensive comparison of efficiency models is found in [2]_ and [3]_.
References
----------
.. [1] T. Huld, et al., "A power-rating model for crystalline silicon
PV modules," Solar Energy Materials and Solar Cells, vol. 95,
pp. 3359-3369, 2011.
.. [2] A. Driesse and J. S. Stein, "From IEC 61853 power measurements
to PV system simulations", Sandia Report No. SAND2020-3877, 2020.
.. [3] A. Driesse, M. Theristis and J. S. Stein, "A New Photovoltaic
Module Efficiency Model for Energy Prediction and Rating",
forthcoming.
Author: Anton Driesse, PV Performance Labs
"""
from numpy import log
g = np.asanyarray(irradiance)
t = np.asanyarray(temperature)
g = g / 1000
dt = t - 25
eta = (
1
+ k1 * log(g)
+ k2 * log(g) ** 2
+ dt * (k3 + k4 * log(g) + k5 * log(g) ** 2)
+ k6 * dt**2
)
return eta
[docs]
def mpm6(irradiance, temperature, c1, c2, c3, c4, c6=0.0):
"""
Calculate PV module efficiency using the MPM6 model (without windspeed).
The efficiency varies with irradiance and operating temperature
and is determined by 5 parameters as described in [1]_. A sixth
parameter captures the effect of windspeed but is not used in this
implementation.
Parameters
----------
irradiance : non-negative numeric, W/m²
The effective irradiance incident on the PV module.
temperature : numeric, °C
The module operating temperature.
c1, c2, c3, c4, c6 : float
Five model parameters usually determined by regression.
Returns
-------
eta : numeric
The efficiency of the module at the specified irradiance and
temperature.
See also
--------
mpm5
fit_efficiency_model
adr
Notes
-----
The author of MPM6 recommends the fitting constraint c6 <= 0.
A comprehensive comparison of efficiency models is found in [2]_ and [3]_.
References
----------
.. [1] S. Ransome and J. Sutterlueti, "How to Choose the Best Empirical
Model for Optimum Energy Yield Predictions," in 44th IEEE Photovoltaic
Specialist Conference (PVSC), 2017, pp. 652-657.
.. [2] A. Driesse and J. S. Stein, "From IEC 61853 power measurements
to PV system simulations", Sandia Report No. SAND2020-3877, 2020.
.. [3] A. Driesse, M. Theristis and J. S. Stein, "A New Photovoltaic
Module Efficiency Model for Energy Prediction and Rating",
forthcoming.
Author: Anton Driesse, PV Performance Labs
"""
g = np.asanyarray(irradiance)
t = np.asanyarray(temperature)
g = g / 1000
dt = t - 25
eta = (
# "actual/nominal"
c1
# loss due to temperature
+ c2 * dt
# loss at low light / due to Voc
+ c3 * np.log10(g)
# loss at high light / due to Rs
+ c4 * g
# loss due to Rsh
+ c6 / g
)
return eta
[docs]
def mpm5(irradiance, temperature, c1, c2, c3, c4):
"""
Call `mpm6` with one less parameter. See `mpm6` for more information.
"""
return mpm6(irradiance, temperature, c1, c2, c3, c4, c6=0.0)
[docs]
def fit_bilinear(irradiance, temperature, eta):
"""
Prepare a bilinear interpolant for module efficiency.
This function allows the class `pvpltools.iec61853.BilinearInterpolator`
to be used in a way that is compatible with other efficiency models
in this module.
Parameters
----------
irradiance : non-negative numeric, W/m²
The effective irradiance incident on the PV module.
temperature : numeric, °C
The module operating temperature.
eta : numeric
The efficiency of the module at the specified irradiance and
temperature.
Returns
-------
interpolator : object
A callable `BilinearInterpolator` object
See also
--------
pvpltools.module_efficiency.bilinear
pvpltools.iec61853.BilinearInterpolator
Notes
-----
Unlike the other efficiency models, bilinear interpolation only works
with a regular grid of measurements. Missing values at low irradiance
high temperature and vice versa are filled using the method described
in [1]_ and [2]_.
References
----------
.. [1] A. Driesse and J. S. Stein, "From IEC 61853 power measurements
to PV system simulations", Sandia Report No. SAND2020-3877, 2020.
.. [2] A. Driesse, M. Theristis and J. S. Stein, "A New Photovoltaic
Module Efficiency Model for Energy Prediction and Rating",
forthcoming.
Author: Anton Driesse, PV Performance Labs
"""
# (re)construct the matrix as a grid for the BilinearInterpolator
data = pd.DataFrame([irradiance, temperature, eta]).T
grid = data.pivot(
columns=data.columns[0], index=data.columns[1], values=data.columns[2]
)
# now create the interpolator object
interpolator = BilinearInterpolator(grid)
return interpolator
[docs]
def bilinear(irradiance, temperature, interpolator):
"""
Calculate PV module efficiency using bilinear interpolation/extrapolation.
This function allows the class `pvpltools.iec61853.BilinearInterpolator`
to be used in a way that is compatible with other efficiency models
in this module.
Parameters
----------
irradiance : non-negative numeric, W/m²
The effective irradiance incident on the PV module.
temperature : numeric, °C
The module operating temperature.
interpolator : object
A callable `BilinearInterpolator` object
Returns
-------
eta : numeric
The efficiency of the module at the specified irradiance and
temperature.
See also
--------
module_efficiency.fit_bilinear
pvpltools.iec61853.BilinearInterpolator
References
----------
.. [1] A. Driesse and J. S. Stein, "From IEC 61853 power measurements
to PV system simulations", Sandia Report No. SAND2020-3877, 2020.
.. [2] A. Driesse, M. Theristis and J. S. Stein, "A New Photovoltaic
Module Efficiency Model for Energy Prediction and Rating",
forthcoming.
Author: Anton Driesse, PV Performance Labs
"""
return interpolator(irradiance, temperature)
