"""
Low-level functions for solving the single diode equation.
"""
import numpy as np
import pandas as pd
from pvlib.tools import _golden_sect_DataFrame
from scipy.optimize import brentq, newton
from scipy.special import lambertw
# newton method default parameters for this module
NEWTON_DEFAULT_PARAMS = {
'tol': 1e-6,
'maxiter': 100
}
# intrinsic voltage per cell junction for a:Si, CdTe, Mertens et al.
VOLTAGE_BUILTIN = 0.9 # [V]
[docs]
def estimate_voc(photocurrent, saturation_current, nNsVth):
"""
Rough estimate of open circuit voltage useful for bounding searches for
``i`` of ``v`` when using :func:`~pvlib.pvsystem.singlediode`.
Parameters
----------
photocurrent : numeric
photo-generated current [A]
saturation_current : numeric
diode reverse saturation current [A]
nNsVth : numeric
product of thermal voltage ``Vth`` [V], diode ideality factor ``n``,
and number of series cells ``Ns``
Returns
-------
numeric
rough estimate of open circuit voltage [V]
Notes
-----
Calculating the open circuit voltage, :math:`V_{oc}`, of an ideal device
with infinite shunt resistance, :math:`R_{sh} \\to \\infty`, and zero
series resistance, :math:`R_s = 0`, yields the following equation [1]. As
an estimate of :math:`V_{oc}` it is useful as an upper bound for the
bisection method.
.. math::
V_{oc, est}=n Ns V_{th} \\log \\left( \\frac{I_L}{I_0} + 1 \\right)
.. [1] http://www.pveducation.org/pvcdrom/open-circuit-voltage
"""
return nNsVth * np.log(np.asarray(photocurrent) / saturation_current + 1.0)
[docs]
def bishop88(diode_voltage, photocurrent, saturation_current,
resistance_series, resistance_shunt, nNsVth, d2mutau=0,
NsVbi=np.inf, breakdown_factor=0., breakdown_voltage=-5.5,
breakdown_exp=3.28, gradients=False):
r"""
Explicit calculation of points on the IV curve described by the single
diode equation. Values are calculated as described in [1]_.
The single diode equation with recombination current and reverse bias
breakdown is
.. math::
I = I_{L} - I_{0} \left (\exp \frac{V_{d}}{nN_{s}V_{th}} - 1 \right )
- \frac{V_{d}}{R_{sh}}
- \frac{I_{L} \frac{d^{2}}{\mu \tau}}{N_{s} V_{bi} - V_{d}}
- a \frac{V_{d}}{R_{sh}} \left (1 - \frac{V_{d}}{V_{br}} \right )^{-m}
The input `diode_voltage` must be :math:`V + I R_{s}`.
.. warning::
* Usage of ``d2mutau`` is required with PVSyst
coefficients for cadmium-telluride (CdTe) and amorphous-silicon
(a:Si) PV modules only.
* Do not use ``d2mutau`` with CEC coefficients.
Parameters
----------
diode_voltage : numeric
diode voltage :math:`V_d` [V]
photocurrent : numeric
photo-generated current :math:`I_{L}` [A]
saturation_current : numeric
diode reverse saturation current :math:`I_{0}` [A]
resistance_series : numeric
series resistance :math:`R_{s}` [ohms]
resistance_shunt: numeric
shunt resistance :math:`R_{sh}` [ohms]
nNsVth : numeric
product of thermal voltage :math:`V_{th}` [V], diode ideality factor
:math:`n`, and number of series cells :math:`N_{s}` [V]
d2mutau : numeric, default 0
PVsyst parameter for cadmium-telluride (CdTe) and amorphous-silicon
(a-Si) modules that accounts for recombination current in the
intrinsic layer. The value is the ratio of intrinsic layer thickness
squared :math:`d^2` to the diffusion length of charge carriers
:math:`\mu \tau`. [V]
NsVbi : numeric, default np.inf
PVsyst parameter for cadmium-telluride (CdTe) and amorphous-silicon
(a-Si) modules that is the product of the PV module number of series
cells :math:`N_{s}` and the builtin voltage :math:`V_{bi}` of the
intrinsic layer. [V].
breakdown_factor : numeric, default 0
fraction of ohmic current involved in avalanche breakdown :math:`a`.
Default of 0 excludes the reverse bias term from the model. [unitless]
breakdown_voltage : numeric, default -5.5
reverse breakdown voltage of the photovoltaic junction :math:`V_{br}`
[V]
breakdown_exp : numeric, default 3.28
avalanche breakdown exponent :math:`m` [unitless]
gradients : bool
False returns only I, V, and P. True also returns gradients
Returns
-------
tuple
currents [A], voltages [V], power [W], and optionally
:math:`\frac{dI}{dV_d}`, :math:`\frac{dV}{dV_d}`,
:math:`\frac{dI}{dV}`, :math:`\frac{dP}{dV}`, and
:math:`\frac{d^2 P}{dV dV_d}`
Notes
-----
The PVSyst thin-film recombination losses parameters ``d2mutau`` and
``NsVbi`` should only be applied to cadmium-telluride (CdTe) and amorphous-
silicon (a-Si) PV modules, [2]_, [3]_. The builtin voltage :math:`V_{bi}`
should account for all junctions. For example: tandem and triple junction
cells would have builtin voltages of 1.8[V] and 2.7[V] respectively, based
on the default of 0.9[V] for a single junction. The parameter ``NsVbi``
should only account for the number of series cells in a single parallel
sub-string if the module has cells in parallel greater than 1.
References
----------
.. [1] J.W. Bishop, "Computer simulation of the effects of electrical
mismatches in photovoltaic cell interconnection circuits" Solar Cells,
vol. 25 no. 1, pp. 73-89, Oct. 1988.
:doi:`doi.org/10.1016/0379-6787(88)90059-2`
.. [2] J. Merten, J. M. Asensi, C. Voz, A. V. Shah, R. Platz and J. Andreu,
"Improved equivalent circuit and Analytical Model for Amorphous
Silicon Solar Cells and Modules." , IEEE Transactions
on Electron Devices, vol. 45, no. 2, pp. 423-429, Feb 1998.
:doi:`10.1109/16.658676`
.. [3] A. Mermoud and T. Lejeune, "Performance assessment of a simulation
model for PV modules of any available technology", In Proc. of the 25th
European PVSEC, Valencia, ES, 2010.
:doi:`10.4229/25thEUPVSEC2010-4BV.1.114`
"""
# calculate recombination loss current where d2mutau > 0
is_recomb = d2mutau > 0 # True where there is thin-film recombination loss
v_recomb = np.where(is_recomb, NsVbi - diode_voltage, np.inf)
i_recomb = np.where(is_recomb, photocurrent * d2mutau / v_recomb, 0)
# calculate temporary values to simplify calculations
v_star = diode_voltage / nNsVth # non-dimensional diode voltage
g_sh = 1.0 / resistance_shunt # conductance
brk_term = 1 - diode_voltage / breakdown_voltage
brk_pwr = np.power(brk_term, -breakdown_exp)
i_breakdown = breakdown_factor * diode_voltage * g_sh * brk_pwr
i = (photocurrent - saturation_current * np.expm1(v_star) # noqa: W503
- diode_voltage * g_sh - i_recomb - i_breakdown) # noqa: W503
v = diode_voltage - i * resistance_series
retval = (i, v, i*v)
if gradients:
# calculate recombination loss current gradients where d2mutau > 0
grad_i_recomb = np.where(is_recomb, i_recomb / v_recomb, 0)
grad_2i_recomb = np.where(is_recomb, 2 * grad_i_recomb / v_recomb, 0)
g_diode = saturation_current * np.exp(v_star) / nNsVth # conductance
brk_pwr_1 = np.power(brk_term, -breakdown_exp - 1)
brk_pwr_2 = np.power(brk_term, -breakdown_exp - 2)
brk_fctr = breakdown_factor * g_sh
grad_i_brk = brk_fctr * (brk_pwr + diode_voltage *
-breakdown_exp * brk_pwr_1)
grad2i_brk = (brk_fctr * -breakdown_exp # noqa: W503
* (2 * brk_pwr_1 + diode_voltage # noqa: W503
* (-breakdown_exp - 1) * brk_pwr_2)) # noqa: W503
grad_i = -g_diode - g_sh - grad_i_recomb - grad_i_brk # di/dvd
grad_v = 1.0 - grad_i * resistance_series # dv/dvd
# dp/dv = d(iv)/dv = v * di/dv + i
grad = grad_i / grad_v # di/dv
grad_p = v * grad + i # dp/dv
grad2i = -g_diode / nNsVth - grad_2i_recomb - grad2i_brk # d2i/dvd
grad2v = -grad2i * resistance_series # d2v/dvd
grad2p = (
grad_v * grad + v * (grad2i/grad_v - grad_i*grad2v/grad_v**2)
+ grad_i
) # d2p/dv/dvd
retval += (grad_i, grad_v, grad, grad_p, grad2p)
return retval
[docs]
def bishop88_i_from_v(voltage, photocurrent, saturation_current,
resistance_series, resistance_shunt, nNsVth,
d2mutau=0, NsVbi=np.inf, breakdown_factor=0.,
breakdown_voltage=-5.5, breakdown_exp=3.28,
method='newton', method_kwargs=None):
"""
Find current given any voltage.
Parameters
----------
voltage : numeric
voltage (V) in volts [V]
photocurrent : numeric
photogenerated current (Iph or IL) [A]
saturation_current : numeric
diode dark or saturation current (Io or Isat) [A]
resistance_series : numeric
series resistance (Rs) in [Ohm]
resistance_shunt : numeric
shunt resistance (Rsh) [Ohm]
nNsVth : numeric
product of diode ideality factor (n), number of series cells (Ns), and
thermal voltage (Vth = k_b * T / q_e) in volts [V]
d2mutau : numeric, default 0
PVsyst parameter for cadmium-telluride (CdTe) and amorphous-silicon
(a-Si) modules that accounts for recombination current in the
intrinsic layer. The value is the ratio of intrinsic layer thickness
squared :math:`d^2` to the diffusion length of charge carriers
:math:`\\mu \\tau`. [V]
NsVbi : numeric, default np.inf
PVsyst parameter for cadmium-telluride (CdTe) and amorphous-silicon
(a-Si) modules that is the product of the PV module number of series
cells ``Ns`` and the builtin voltage ``Vbi`` of the intrinsic layer.
[V].
breakdown_factor : float, default 0
fraction of ohmic current involved in avalanche breakdown :math:`a`.
Default of 0 excludes the reverse bias term from the model. [unitless]
breakdown_voltage : float, default -5.5
reverse breakdown voltage of the photovoltaic junction :math:`V_{br}`
[V]
breakdown_exp : float, default 3.28
avalanche breakdown exponent :math:`m` [unitless]
method : str, default 'newton'
Either ``'newton'``, ``'brentq'``, or ``'chandrupatla'``.
''method'' must be ``'newton'`` if ``breakdown_factor`` is not 0.
.. note::
``'chandrupatla'`` requires scipy 1.15 or greater.
method_kwargs : dict, optional
Keyword arguments passed to the root finder. For options, see:
* ``method='brentq'``: :py:func:`scipy:scipy.optimize.brentq`
* ``method='newton'``: :py:func:`scipy:scipy.optimize.newton`
* ``method='chandrupatla'``: :py:func:`scipy:scipy.optimize.elementwise.find_root`
``'full_output': True`` is allowed, and ``optimizer_output`` would be
returned. See examples section.
Returns
-------
current : numeric
current (I) at the specified voltage (V). [A]
optimizer_output : tuple, optional, if specified in ``method_kwargs``
see root finder documentation for selected method.
Found root is diode voltage in [1]_.
Examples
--------
Using the following arguments that may come from any
`calcparams_.*` function in :py:mod:`pvlib.pvsystem`:
>>> args = {'photocurrent': 1., 'saturation_current': 9e-10, 'nNsVth': 4.,
... 'resistance_series': 4., 'resistance_shunt': 5000.0}
Use default values:
>>> i = bishop88_i_from_v(0.0, **args)
Specify tolerances and maximum number of iterations:
>>> i = bishop88_i_from_v(0.0, **args, method='newton',
... method_kwargs={'tol': 1e-3, 'rtol': 1e-3, 'maxiter': 20})
Retrieve full output from the root finder:
>>> i, method_output = bishop88_i_from_v(0.0, **args, method='newton',
... method_kwargs={'full_output': True})
References
----------
.. [1] "Computer simulation of the effects of electrical mismatches in
photovoltaic cell interconnection circuits" JW Bishop, Solar Cell (1988)
:doi:`10.1016/0379-6787(88)90059-2`
""" # noqa: E501
# collect args
args = (photocurrent, saturation_current,
resistance_series, resistance_shunt, nNsVth, d2mutau, NsVbi,
breakdown_factor, breakdown_voltage, breakdown_exp)
method = method.lower()
# method_kwargs create dict if not provided
# this pattern avoids bugs with Mutable Default Parameters
if not method_kwargs:
method_kwargs = {}
def fv(x, v, *a):
# calculate voltage residual given diode voltage "x"
return bishop88(x, *a)[1] - v
if method == 'brentq':
# first bound the search using voc
voc_est = estimate_voc(photocurrent, saturation_current, nNsVth)
# start iteration slightly less than NsVbi when voc_est > NsVbi, to
# avoid the asymptote at NsVbi
xp = np.where(voc_est < NsVbi, voc_est, 0.9999*NsVbi)
# brentq only works with scalar inputs, so we need a set up function
# and np.vectorize to repeatedly call the optimizer with the right
# arguments for possible array input
def vd_from_brent(voc, v, iph, isat, rs, rsh, gamma, d2mutau, NsVbi,
breakdown_factor, breakdown_voltage, breakdown_exp):
return brentq(fv, 0.0, voc,
args=(v, iph, isat, rs, rsh, gamma, d2mutau, NsVbi,
breakdown_factor, breakdown_voltage,
breakdown_exp),
**method_kwargs)
vd_from_brent_vectorized = np.vectorize(vd_from_brent)
vd = vd_from_brent_vectorized(xp, voltage, *args)
elif method == 'newton':
x0, (voltage, *args), method_kwargs = \
_prepare_newton_inputs(voltage, (voltage, *args), method_kwargs)
vd = newton(func=lambda x, *a: fv(x, voltage, *a), x0=x0,
fprime=lambda x, *a: bishop88(x, *a, gradients=True)[4],
args=args, **method_kwargs)
elif method == 'chandrupatla':
try:
from scipy.optimize.elementwise import find_root
except ModuleNotFoundError as e:
# TODO remove this when our minimum scipy version is >=1.15
msg = (
"method='chandrupatla' requires scipy v1.15 or greater "
"(available for Python 3.10+). "
"Select another method, or update your version of scipy."
)
raise ImportError(msg) from e
voc_est = estimate_voc(photocurrent, saturation_current, nNsVth)
shape = _shape_of_max_size(voltage, voc_est)
vlo = np.zeros(shape)
vhi = np.full(shape, voc_est)
bounds = (vlo, vhi)
kwargs_trimmed = method_kwargs.copy()
kwargs_trimmed.pop("full_output", None) # not valid for find_root
result = find_root(fv, bounds, args=(voltage, *args), **kwargs_trimmed)
vd = result.x
if method_kwargs.get('full_output'):
vd = (vd, result) # mimic the other methods
else:
raise NotImplementedError("Method '%s' isn't implemented" % method)
# When 'full_output' parameter is specified, returned 'vd' is a tuple with
# many elements, where the root is the first one. So we use it to output
# the bishop88 result and return tuple(scalar, tuple with method results)
if method_kwargs.get('full_output') is True:
return (bishop88(vd[0], *args)[0], vd)
else:
return bishop88(vd, *args)[0]
[docs]
def bishop88_v_from_i(current, photocurrent, saturation_current,
resistance_series, resistance_shunt, nNsVth,
d2mutau=0, NsVbi=np.inf, breakdown_factor=0.,
breakdown_voltage=-5.5, breakdown_exp=3.28,
method='newton', method_kwargs=None):
"""
Find voltage given any current.
Parameters
----------
current : numeric
current (I) in amperes [A]
photocurrent : numeric
photogenerated current (Iph or IL) [A]
saturation_current : numeric
diode dark or saturation current (Io or Isat) [A]
resistance_series : numeric
series resistance (Rs) in [Ohm]
resistance_shunt : numeric
shunt resistance (Rsh) [Ohm]
nNsVth : numeric
product of diode ideality factor (n), number of series cells (Ns), and
thermal voltage (Vth = k_b * T / q_e) in volts [V]
d2mutau : numeric, default 0
PVsyst parameter for cadmium-telluride (CdTe) and amorphous-silicon
(a-Si) modules that accounts for recombination current in the
intrinsic layer. The value is the ratio of intrinsic layer thickness
squared :math:`d^2` to the diffusion length of charge carriers
:math:`\\mu \\tau`. [V]
NsVbi : numeric, default np.inf
PVsyst parameter for cadmium-telluride (CdTe) and amorphous-silicon
(a-Si) modules that is the product of the PV module number of series
cells ``Ns`` and the builtin voltage ``Vbi`` of the intrinsic layer.
[V].
breakdown_factor : float, default 0
fraction of ohmic current involved in avalanche breakdown :math:`a`.
Default of 0 excludes the reverse bias term from the model. [unitless]
breakdown_voltage : float, default -5.5
reverse breakdown voltage of the photovoltaic junction :math:`V_{br}`
[V]
breakdown_exp : float, default 3.28
avalanche breakdown exponent :math:`m` [unitless]
method : str, default 'newton'
Either ``'newton'``, ``'brentq'``, or ``'chandrupatla'``.
''method'' must be ``'newton'`` if ``breakdown_factor`` is not 0.
.. note::
``'chandrupatla'`` requires scipy 1.15 or greater.
method_kwargs : dict, optional
Keyword arguments passed to the root finder. For options, see:
* ``method='brentq'``: :py:func:`scipy:scipy.optimize.brentq`
* ``method='newton'``: :py:func:`scipy:scipy.optimize.newton`
* ``method='chandrupatla'``: :py:func:`scipy:scipy.optimize.elementwise.find_root`
``'full_output': True`` is allowed, and ``optimizer_output`` would be
returned. See examples section.
Returns
-------
voltage : numeric
voltage (V) at the specified current (I) in volts [V]
optimizer_output : tuple, optional, if specified in ``method_kwargs``
see root finder documentation for selected method.
Found root is diode voltage in [1]_.
Examples
--------
Using the following arguments that may come from any
`calcparams_.*` function in :py:mod:`pvlib.pvsystem`:
>>> args = {'photocurrent': 1., 'saturation_current': 9e-10, 'nNsVth': 4.,
... 'resistance_series': 4., 'resistance_shunt': 5000.0}
Use default values:
>>> v = bishop88_v_from_i(0.0, **args)
Specify tolerances and maximum number of iterations:
>>> v = bishop88_v_from_i(0.0, **args, method='newton',
... method_kwargs={'tol': 1e-3, 'rtol': 1e-3, 'maxiter': 20})
Retrieve full output from the root finder:
>>> v, method_output = bishop88_v_from_i(0.0, **args, method='newton',
... method_kwargs={'full_output': True})
References
----------
.. [1] "Computer simulation of the effects of electrical mismatches in
photovoltaic cell interconnection circuits" JW Bishop, Solar Cell (1988)
:doi:`10.1016/0379-6787(88)90059-2`
""" # noqa: E501
# collect args
args = (photocurrent, saturation_current,
resistance_series, resistance_shunt, nNsVth, d2mutau, NsVbi,
breakdown_factor, breakdown_voltage, breakdown_exp)
method = method.lower()
# method_kwargs create dict if not provided
# this pattern avoids bugs with Mutable Default Parameters
if not method_kwargs:
method_kwargs = {}
# first bound the search using voc
voc_est = estimate_voc(photocurrent, saturation_current, nNsVth)
# start iteration slightly less than NsVbi when voc_est > NsVbi, to avoid
# the asymptote at NsVbi
xp = np.where(voc_est < NsVbi, voc_est, 0.9999*NsVbi)
def fi(x, i, *a):
# calculate current residual given diode voltage "x"
return bishop88(x, *a)[0] - i
if method == 'brentq':
# brentq only works with scalar inputs, so we need a set up function
# and np.vectorize to repeatedly call the optimizer with the right
# arguments for possible array input
def vd_from_brent(voc, i, iph, isat, rs, rsh, gamma, d2mutau, NsVbi,
breakdown_factor, breakdown_voltage, breakdown_exp):
return brentq(fi, 0.0, voc,
args=(i, iph, isat, rs, rsh, gamma, d2mutau, NsVbi,
breakdown_factor, breakdown_voltage,
breakdown_exp),
**method_kwargs)
vd_from_brent_vectorized = np.vectorize(vd_from_brent)
vd = vd_from_brent_vectorized(xp, current, *args)
elif method == 'newton':
x0, (current, *args), method_kwargs = \
_prepare_newton_inputs(xp, (current, *args), method_kwargs)
vd = newton(func=lambda x, *a: fi(x, current, *a), x0=x0,
fprime=lambda x, *a: bishop88(x, *a, gradients=True)[3],
args=args, **method_kwargs)
elif method == 'chandrupatla':
try:
from scipy.optimize.elementwise import find_root
except ModuleNotFoundError as e:
# TODO remove this when our minimum scipy version is >=1.15
msg = (
"method='chandrupatla' requires scipy v1.15 or greater "
"(available for Python 3.10+). "
"Select another method, or update your version of scipy."
)
raise ImportError(msg) from e
shape = _shape_of_max_size(current, voc_est)
vlo = np.zeros(shape)
vhi = np.full(shape, voc_est)
bounds = (vlo, vhi)
kwargs_trimmed = method_kwargs.copy()
kwargs_trimmed.pop("full_output", None) # not valid for find_root
result = find_root(fi, bounds, args=(current, *args), **kwargs_trimmed)
vd = result.x
if method_kwargs.get('full_output'):
vd = (vd, result) # mimic the other methods
else:
raise NotImplementedError("Method '%s' isn't implemented" % method)
# When 'full_output' parameter is specified, returned 'vd' is a tuple with
# many elements, where the root is the first one. So we use it to output
# the bishop88 result and return tuple(scalar, tuple with method results)
if method_kwargs.get('full_output') is True:
return (bishop88(vd[0], *args)[1], vd)
else:
return bishop88(vd, *args)[1]
[docs]
def bishop88_mpp(photocurrent, saturation_current, resistance_series,
resistance_shunt, nNsVth, d2mutau=0, NsVbi=np.inf,
breakdown_factor=0., breakdown_voltage=-5.5,
breakdown_exp=3.28, method='newton', method_kwargs=None):
"""
Find max power point.
Parameters
----------
photocurrent : numeric
photogenerated current (Iph or IL) [A]
saturation_current : numeric
diode dark or saturation current (Io or Isat) [A]
resistance_series : numeric
series resistance (Rs) in [Ohm]
resistance_shunt : numeric
shunt resistance (Rsh) [Ohm]
nNsVth : numeric
product of diode ideality factor (n), number of series cells (Ns), and
thermal voltage (Vth = k_b * T / q_e) in volts [V]
d2mutau : numeric, default 0
PVsyst parameter for cadmium-telluride (CdTe) and amorphous-silicon
(a-Si) modules that accounts for recombination current in the
intrinsic layer. The value is the ratio of intrinsic layer thickness
squared :math:`d^2` to the diffusion length of charge carriers
:math:`\\mu \\tau`. [V]
NsVbi : numeric, default np.inf
PVsyst parameter for cadmium-telluride (CdTe) and amorphous-silicon
(a-Si) modules that is the product of the PV module number of series
cells ``Ns`` and the builtin voltage ``Vbi`` of the intrinsic layer.
[V].
breakdown_factor : numeric, default 0
fraction of ohmic current involved in avalanche breakdown :math:`a`.
Default of 0 excludes the reverse bias term from the model. [unitless]
breakdown_voltage : numeric, default -5.5
reverse breakdown voltage of the photovoltaic junction :math:`V_{br}`
[V]
breakdown_exp : numeric, default 3.28
avalanche breakdown exponent :math:`m` [unitless]
method : str, default 'newton'
Either ``'newton'``, ``'brentq'``, or ``'chandrupatla'``.
''method'' must be ``'newton'`` if ``breakdown_factor`` is not 0.
.. note::
``'chandrupatla'`` requires scipy 1.15 or greater.
method_kwargs : dict, optional
Keyword arguments passed to the root finder. For options, see:
* ``method='brentq'``: :py:func:`scipy:scipy.optimize.brentq`
* ``method='newton'``: :py:func:`scipy:scipy.optimize.newton`
* ``method='chandrupatla'``: :py:func:`scipy:scipy.optimize.elementwise.find_root`
``'full_output': True`` is allowed, and ``optimizer_output`` would be
returned. See examples section.
Returns
-------
tuple
max power current ``i_mp`` [A], max power voltage ``v_mp`` [V], and
max power ``p_mp`` [W]
optimizer_output : tuple, optional, if specified in ``method_kwargs``
see root finder documentation for selected method.
Found root is diode voltage in [1]_.
Examples
--------
Using the following arguments that may come from any
`calcparams_.*` function in :py:mod:`pvlib.pvsystem`:
>>> args = {'photocurrent': 1., 'saturation_current': 9e-10, 'nNsVth': 4.,
... 'resistance_series': 4., 'resistance_shunt': 5000.0}
Use default values:
>>> i_mp, v_mp, p_mp = bishop88_mpp(**args)
Specify tolerances and maximum number of iterations:
>>> i_mp, v_mp, p_mp = bishop88_mpp(**args, method='newton',
... method_kwargs={'tol': 1e-3, 'rtol': 1e-3, 'maxiter': 20})
Retrieve full output from the root finder:
>>> (i_mp, v_mp, p_mp), method_output = bishop88_mpp(**args,
... method='newton', method_kwargs={'full_output': True})
References
----------
.. [1] "Computer simulation of the effects of electrical mismatches in
photovoltaic cell interconnection circuits" JW Bishop, Solar Cell (1988)
:doi:`10.1016/0379-6787(88)90059-2`
""" # noqa: E501
# collect args
args = (photocurrent, saturation_current,
resistance_series, resistance_shunt, nNsVth, d2mutau, NsVbi,
breakdown_factor, breakdown_voltage, breakdown_exp)
method = method.lower()
# method_kwargs create dict if not provided
# this pattern avoids bugs with Mutable Default Parameters
if not method_kwargs:
method_kwargs = {}
# first bound the search using voc
voc_est = estimate_voc(photocurrent, saturation_current, nNsVth)
# start iteration slightly less than NsVbi when voc_est > NsVbi, to avoid
# the asymptote at NsVbi
xp = np.where(voc_est < NsVbi, voc_est, 0.9999*NsVbi)
def fmpp(x, *a):
return bishop88(x, *a, gradients=True)[6]
if method == 'brentq':
# break out arguments for numpy.vectorize to handle broadcasting
vec_fun = np.vectorize(
lambda voc, iph, isat, rs, rsh, gamma, d2mutau, NsVbi, vbr_a, vbr,
vbr_exp: brentq(fmpp, 0.0, voc,
args=(iph, isat, rs, rsh, gamma, d2mutau, NsVbi,
vbr_a, vbr, vbr_exp),
**method_kwargs)
)
vd = vec_fun(xp, *args)
elif method == 'newton':
# make sure all args are numpy arrays if max size > 1
# if voc_est is an array, then make a copy to use for initial guess, v0
x0, args, method_kwargs = \
_prepare_newton_inputs(xp, args, method_kwargs)
vd = newton(func=fmpp, x0=x0,
fprime=lambda x, *a: bishop88(x, *a, gradients=True)[7],
args=args, **method_kwargs)
elif method == 'chandrupatla':
try:
from scipy.optimize.elementwise import find_root
except ModuleNotFoundError as e:
# TODO remove this when our minimum scipy version is >=1.15
msg = (
"method='chandrupatla' requires scipy v1.15 or greater "
"(available for Python 3.10+). "
"Select another method, or update your version of scipy."
)
raise ImportError(msg) from e
vlo = np.zeros_like(photocurrent)
vhi = np.full_like(photocurrent, voc_est)
kwargs_trimmed = method_kwargs.copy()
kwargs_trimmed.pop("full_output", None) # not valid for find_root
result = find_root(fmpp,
(vlo, vhi),
args=args,
**kwargs_trimmed)
vd = result.x
if method_kwargs.get('full_output'):
vd = (vd, result) # mimic the other methods
else:
raise NotImplementedError("Method '%s' isn't implemented" % method)
# When 'full_output' parameter is specified, returned 'vd' is a tuple with
# many elements, where the root is the first one. So we use it to output
# the bishop88 result and return
# tuple(tuple with bishop88 solution, tuple with method results)
if method_kwargs.get('full_output') is True:
return (bishop88(vd[0], *args), vd)
else:
return bishop88(vd, *args)
def _shape_of_max_size(*args):
return max(((np.size(a), np.shape(a)) for a in args),
key=lambda t: t[0])[1]
def _prepare_newton_inputs(x0, args, method_kwargs):
"""
Make inputs compatible with Scipy's newton by:
- converting all arguments (`x0` and `args`) into numpy.ndarrays if any
argument is not a scalar.
- broadcasting the initial guess `x0` to the shape of the argument with
the greatest size.
Parameters
----------
x0: numeric
Initial guess for newton.
args: Iterable(numeric)
Iterable of additional arguments to use in SciPy's newton.
method_kwargs: dict
Options to pass to newton.
Returns
-------
tuple
The updated initial guess, arguments, and options for newton.
"""
if not (np.isscalar(x0) and all(map(np.isscalar, args))):
args = tuple(map(np.asarray, args))
x0 = np.broadcast_to(x0, _shape_of_max_size(x0, *args))
# set abs tolerance and maxiter from method_kwargs if not provided
# apply defaults, but giving priority to user-specified values
method_kwargs = {**NEWTON_DEFAULT_PARAMS, **method_kwargs}
return x0, args, method_kwargs
def _lambertw_v_from_i(current, photocurrent, saturation_current,
resistance_series, resistance_shunt, nNsVth):
# Record if inputs were all scalar
output_is_scalar = all(map(np.isscalar,
(current, photocurrent, saturation_current,
resistance_series, resistance_shunt, nNsVth)))
# This transforms Gsh=1/Rsh, including ideal Rsh=np.inf into Gsh=0., which
# is generally more numerically stable
conductance_shunt = 1. / resistance_shunt
# Ensure that we are working with read-only views of numpy arrays
# Turns Series into arrays so that we don't have to worry about
# multidimensional broadcasting failing
I, IL, I0, Rs, Gsh, a = \
np.broadcast_arrays(current, photocurrent, saturation_current,
resistance_series, conductance_shunt, nNsVth)
# Intitalize output V (I might not be float64)
V = np.full_like(I, np.nan, dtype=np.float64)
# Determine indices where 0 < Gsh requires implicit model solution
idx_p = 0. < Gsh
# Determine indices where 0 = Gsh allows explicit model solution
idx_z = 0. == Gsh
# Explicit solutions where Gsh=0
if np.any(idx_z):
V[idx_z] = a[idx_z] * np.log1p((IL[idx_z] - I[idx_z]) / I0[idx_z]) - \
I[idx_z] * Rs[idx_z]
# Only compute using LambertW if there are cases with Gsh>0
if np.any(idx_p):
# use only the relevant subset for what follows
I = I[idx_p]
IL = IL[idx_p]
I0 = I0[idx_p]
Rs = Rs[idx_p]
Gsh = Gsh[idx_p]
a = a[idx_p]
# LambertW argument, cannot be float128, may overflow to np.inf
# overflow is explicitly handled below, so ignore warnings here
with np.errstate(over='ignore'):
argW = I0 / (Gsh * a) * np.exp((-I + IL + I0) / (Gsh * a))
# lambertw typically returns complex value with zero imaginary part
# may overflow to np.inf
lambertwterm = lambertw(argW).real
# Record indices where lambertw input overflowed output
idx_inf = np.isinf(lambertwterm)
# Only re-compute LambertW if it overflowed
if np.any(idx_inf):
# Calculate using log(argW) in case argW is really big
logargW = (np.log(I0[idx_inf]) - np.log(Gsh[idx_inf]) -
np.log(a[idx_inf]) +
(-I[idx_inf] + IL[idx_inf] + I0[idx_inf]) /
(Gsh[idx_inf] * a[idx_inf]))
# Three iterations of Newton-Raphson method to solve
# w+log(w)=logargW. The initial guess is w=logargW. Where direct
# evaluation (above) results in NaN from overflow, 3 iterations
# of Newton's method gives approximately 8 digits of precision.
w = logargW
for _ in range(0, 3):
w = w * (1. - np.log(w) + logargW) / (1. + w)
lambertwterm[idx_inf] = w
# Eqn. 3 in Jain and Kapoor, 2004
# V = -I*(Rs + Rsh) + IL*Rsh - a*lambertwterm + I0*Rsh
# Recast in terms of Gsh=1/Rsh for better numerical stability.
V[idx_p] = (IL + I0 - I) / Gsh - I * Rs - a * lambertwterm
if output_is_scalar:
return V.item()
else:
return V
def _lambertw_i_from_v(voltage, photocurrent, saturation_current,
resistance_series, resistance_shunt, nNsVth):
# Record if inputs were all scalar
output_is_scalar = all(map(np.isscalar,
(voltage, photocurrent, saturation_current,
resistance_series, resistance_shunt, nNsVth)))
# This transforms Gsh=1/Rsh, including ideal Rsh=np.inf into Gsh=0., which
# is generally more numerically stable
conductance_shunt = 1. / resistance_shunt
# Ensure that we are working with read-only views of numpy arrays
# Turns Series into arrays so that we don't have to worry about
# multidimensional broadcasting failing
V, IL, I0, Rs, Gsh, a = \
np.broadcast_arrays(voltage, photocurrent, saturation_current,
resistance_series, conductance_shunt, nNsVth)
# Intitalize output I (V might not be float64)
I = np.full_like(V, np.nan, dtype=np.float64) # noqa: E741, N806
# Determine indices where 0 < Rs requires implicit model solution
idx_p = 0. < Rs
# Determine indices where 0 = Rs allows explicit model solution
idx_z = 0. == Rs
# Explicit solutions where Rs=0
if np.any(idx_z):
I[idx_z] = IL[idx_z] - I0[idx_z] * np.expm1(V[idx_z] / a[idx_z]) - \
Gsh[idx_z] * V[idx_z]
# Only compute using LambertW if there are cases with Rs>0
# Does NOT handle possibility of overflow, github issue 298
if np.any(idx_p):
# LambertW argument, cannot be float128, may overflow to np.inf
argW = Rs[idx_p] * I0[idx_p] / (
a[idx_p] * (Rs[idx_p] * Gsh[idx_p] + 1.)) * \
np.exp((Rs[idx_p] * (IL[idx_p] + I0[idx_p]) + V[idx_p]) /
(a[idx_p] * (Rs[idx_p] * Gsh[idx_p] + 1.)))
# lambertw typically returns complex value with zero imaginary part
# may overflow to np.inf
lambertwterm = lambertw(argW).real
# Eqn. 2 in Jain and Kapoor, 2004
# I = -V/(Rs + Rsh) - (a/Rs)*lambertwterm + Rsh*(IL + I0)/(Rs + Rsh)
# Recast in terms of Gsh=1/Rsh for better numerical stability.
I[idx_p] = (IL[idx_p] + I0[idx_p] - V[idx_p] * Gsh[idx_p]) / \
(Rs[idx_p] * Gsh[idx_p] + 1.) - (
a[idx_p] / Rs[idx_p]) * lambertwterm
if output_is_scalar:
return I.item()
else:
return I
def _lambertw(photocurrent, saturation_current, resistance_series,
resistance_shunt, nNsVth, ivcurve_pnts=None):
# collect args
params = {'photocurrent': photocurrent,
'saturation_current': saturation_current,
'resistance_series': resistance_series,
'resistance_shunt': resistance_shunt, 'nNsVth': nNsVth}
# Compute short circuit current
i_sc = _lambertw_i_from_v(0., **params)
# Compute open circuit voltage
v_oc = _lambertw_v_from_i(0., **params)
# Set small elements <0 in v_oc to 0
if isinstance(v_oc, np.ndarray):
v_oc[(v_oc < 0) & (v_oc > -1e-12)] = 0.
elif isinstance(v_oc, (float, int)):
if v_oc < 0 and v_oc > -1e-12:
v_oc = 0.
# Find the voltage, v_mp, where the power is maximized.
# use scipy.elementwise if available
# remove try/except when scipy>=1.15, and golden mean is retired
try:
from scipy.optimize.elementwise import find_minimum
# left negative to insure strict inequality
init = (-1., 0.8*v_oc, v_oc)
res = find_minimum(_vmp_opt, init,
args=(params['photocurrent'],
params['saturation_current'],
params['resistance_series'],
params['resistance_shunt'],
params['nNsVth'],))
v_mp = res.x
p_mp = -1.*res.f_x
except ModuleNotFoundError:
# switch to old golden section method
p_mp, v_mp = _golden_sect_DataFrame(params, 0., v_oc * 1.14,
_pwr_optfcn)
i_mp = _lambertw_i_from_v(v_mp, **params)
# Find Ix and Ixx using Lambert W
i_x = _lambertw_i_from_v(0.5 * v_oc, **params)
i_xx = _lambertw_i_from_v(0.5 * (v_oc + v_mp), **params)
out = (i_sc, v_oc, i_mp, v_mp, p_mp, i_x, i_xx)
# create ivcurve
if ivcurve_pnts:
ivcurve_v = (np.asarray(v_oc)[..., np.newaxis] *
np.linspace(0, 1, ivcurve_pnts))
ivcurve_i = _lambertw_i_from_v(ivcurve_v.T, **params).T
out += (ivcurve_i, ivcurve_v)
return out
def _vmp_opt(v, iph, io, rs, rsh, nNsVth):
'''
Function to find negative of power from ``i_from_v``.
'''
current = _lambertw_i_from_v(v, iph, io, rs, rsh, nNsVth)
return -v * current
def _pwr_optfcn(df, loc):
'''
Function to find power from ``i_from_v``.
'''
current = _lambertw_i_from_v(df[loc], df['photocurrent'],
df['saturation_current'],
df['resistance_series'],
df['resistance_shunt'], df['nNsVth'])
return current * df[loc]
[docs]
def batzelis(photocurrent, saturation_current, resistance_series,
resistance_shunt, nNsVth):
"""
Estimate maximum power, open-circuit, and short-circuit points from
single-diode equation parameters using Batzelis's method.
This method is described in Section II.B of [1]_.
Parameters
----------
photocurrent : numeric
Light-generated current. [A]
saturation_current : numeric
Diode saturation current. [A]
resistance_series : numeric
Series resistance. [Ohm]
resistance_shunt : numeric
Shunt resistance. [Ohm]
nNsVth : numeric
The product of the usual diode ideality factor (n, unitless),
number of cells in series (Ns), and cell thermal voltage at
specified effective irradiance and cell temperature. [V]
Returns
-------
dict
The returned dict-like object contains the keys/columns:
* ``p_mp`` - power at maximum power point. [W]
* ``i_mp`` - current at maximum power point. [A]
* ``v_mp`` - voltage at maximum power point. [V]
* ``i_sc`` - short circuit current. [A]
* ``v_oc`` - open circuit voltage. [V]
References
----------
.. [1] E. I. Batzelis, "Simple PV Performance Equations Theoretically Well
Founded on the Single-Diode Model," Journal of Photovoltaics vol. 7,
no. 5, pp. 1400-1409, Sep 2017, :doi:`10.1109/JPHOTOV.2017.2711431`
"""
# convenience variables
Iph = photocurrent
Is = saturation_current
Rsh = resistance_shunt
Rs = resistance_series
a = nNsVth
# Eqs 3-4
isc = Iph / (Rs / Rsh + 1) # manipulated to handle Rsh=np.inf correctly
with np.errstate(divide='ignore'): # zero Iph
voc = a * np.log(Iph / Is)
# Eqs 5-8
w = np.real(lambertw(np.e * Iph / Is))
# vmp = (1 + Rs/Rsh) * a * (w - 1) - Rs * Iph * (1 - 1/w) # not needed
with np.errstate(divide='ignore', invalid='ignore'): # zero Iph -> zero w
imp = Iph * (1 - 1/w) - a * (w - 1) / Rsh
vmp = a * (w - 1) - Rs * imp
vmp = np.where(Iph > 0, vmp, 0)
voc = np.where(Iph > 0, voc, 0)
imp = np.where(Iph > 0, imp, 0)
isc = np.where(Iph > 0, isc, 0)
out = {
'p_mp': imp * vmp,
'i_mp': imp,
'v_mp': vmp,
'i_sc': isc,
'v_oc': voc,
}
# if pandas in, ensure pandas out
pandas_inputs = [
x for x in [Iph, Is, Rsh, Rs, a] if isinstance(x, pd.Series)]
if pandas_inputs:
out = pd.DataFrame(out, index=pandas_inputs[0].index)
return out
