import numpy as np
from scipy import constants
from pvlib.pvsystem import calcparams_pvsyst
from pvlib.singlediode import bishop88_mpp
from pvlib.ivtools.utils import rectify_iv_curve
from pvlib.ivtools.sde import _fit_sandia_cocontent
from pvlib.tools import _first_order_centered_difference
from pvlib.ivtools.sdm._fit_desoto_pvsyst_sandia import (
_extract_sdm_params, _initial_iv_params, _update_iv_params
)
from pvlib.pvsystem import (
_pvsyst_Rsh, _pvsyst_IL, _pvsyst_Io, _pvsyst_nNsVth, _pvsyst_gamma
)
CONSTANTS = {'E0': 1000.0, 'T0': 25.0, 'k': constants.k, 'q': constants.e}
[docs]
def fit_pvsyst_sandia(ivcurves, specs, const=None, maxiter=5, eps1=1.e-3):
"""
Estimate parameters for the PVsyst module performance model.
Parameters
----------
ivcurves : dict
i : array
One array element for each IV curve. The jth element is itself an
array of current for jth IV curve (same length as v[j]) [A]
v : array
One array element for each IV curve. The jth element is itself an
array of voltage for jth IV curve (same length as i[j]) [V]
ee : array
effective irradiance for each IV curve, i.e., POA broadband
irradiance adjusted by solar spectrum modifier [W / m^2]
tc : array
cell temperature for each IV curve [C]
i_sc : array
short circuit current for each IV curve [A]
v_oc : array
open circuit voltage for each IV curve [V]
i_mp : array
current at max power point for each IV curve [A]
v_mp : array
voltage at max power point for each IV curve [V]
specs : dict
cells_in_series : int
number of cells in series
alpha_sc : float
temperature coefficient of isc [A/C]
const : dict
E0 : float
effective irradiance at STC, default 1000 [W/m^2]
T0 : float
cell temperature at STC, default 25 [C]
k : float
Boltzmann's constant [J/K]
q : float
elementary charge [Coulomb]
maxiter : int, default 5
input that sets the maximum number of iterations for the parameter
updating part of the algorithm.
eps1: float, default 1e-3
Tolerance for the IV curve fitting. The parameter updating stops when
absolute values of the percent change in mean, max and standard
deviation of Imp, Vmp and Pmp between iterations are all less than
eps1, or when the number of iterations exceeds maxiter.
Returns
-------
dict
I_L_ref : float
light current at STC [A]
I_o_ref : float
dark current at STC [A]
EgRef : float
effective band gap at STC [eV]
R_s : float
series resistance at STC [ohm]
R_sh_ref : float
shunt resistance at STC [ohm]
R_sh_0 : float
shunt resistance at zero irradiance [ohm]
R_sh_exp : float
exponential factor defining decrease in shunt resistance with
increasing effective irradiance
gamma_ref : float
diode (ideality) factor at STC [unitless]
mu_gamma : float
temperature coefficient for diode (ideality) factor [1/K]
cells_in_series : int
number of cells in series
iph : array
light current for each IV curve [A]
io : array
dark current for each IV curve [A]
rs : array
series resistance for each IV curve [ohm]
rsh : array
shunt resistance for each IV curve [ohm]
u : array
boolean for each IV curve indicating that the parameter values
are deemed reasonable by the private function ``_filter_params``
Notes
-----
The PVsyst module performance model is described in [1]_, [2]_, and [3]_.
The fitting method is documented in [4]_, [5]_, and [6]_.
Ported from PVLib Matlab [7]_.
References
----------
.. [1] K. Sauer, T. Roessler, C. W. Hansen, Modeling the Irradiance and
Temperature Dependence of Photovoltaic Modules in PVsyst, IEEE Journal
of Photovoltaics v5(1), January 2015.
:doi:`10.1109/JPHOTOV.2014.2364133`
.. [2] A. Mermoud, PV Modules modeling, Presentation at the 2nd PV
Performance Modeling Workshop, Santa Clara, CA, May 2013
.. [3] A. Mermoud, T. Lejeuene, Performance Assessment of a Simulation
Model for PV modules of any available technology, 25th European
Photovoltaic Solar Energy Conference, Valencia, Spain, Sept. 2010
.. [4] C. Hansen, Estimating Parameters for the PVsyst Version 6
Photovoltaic Module Performance Model, Sandia National Laboratories
Report SAND2015-8598. :doi:`10.2172/1223058`
.. [5] C. Hansen, Parameter Estimation for Single Diode Models of
Photovoltaic Modules, Sandia National Laboratories Report SAND2015-2065.
:doi:`10.2172/1177157`
.. [6] C. Hansen, Estimation of Parameters for Single Diode Models using
Measured IV Curves, Proc. of the 39th IEEE PVSC, June 2013.
:doi:`10.1109/PVSC.2013.6744135`
.. [7] PVLib MATLAB https://github.com/sandialabs/MATLAB_PV_LIB
"""
if const is None:
const = CONSTANTS
ee = ivcurves['ee']
tc = ivcurves['tc']
tck = tc + 273.15
isc = ivcurves['i_sc']
voc = ivcurves['v_oc']
imp = ivcurves['i_mp']
vmp = ivcurves['v_mp']
# Cell Thermal Voltage
vth = const['k'] / const['q'] * tck
n = len(ivcurves['v_oc'])
# Initial estimate of Rsh used to obtain the diode factor gamma0 and diode
# temperature coefficient mu_gamma. Rsh is estimated using the co-content
# integral method.
rsh = np.ones(n)
for j in range(n):
voltage, current = rectify_iv_curve(ivcurves['v'][j], ivcurves['i'][j])
# initial estimate of Rsh, from integral over voltage regression
# [5] Step 3a; [6] Step 3a
_, _, _, rsh[j], _ = _fit_sandia_cocontent(
voltage, current, vth[j] * specs['cells_in_series'])
gamma_ref, mu_gamma = _fit_pvsyst_sandia_gamma(voc, isc, rsh, vth, tck,
specs, const)
badgamma = np.isnan(gamma_ref) or np.isnan(mu_gamma) \
or not np.isreal(gamma_ref) or not np.isreal(mu_gamma)
if badgamma:
raise RuntimeError(
"Failed to estimate the diode (ideality) factor parameter;"
" aborting parameter estimation.")
gamma = gamma_ref + mu_gamma * (tc - const['T0'])
nnsvth = gamma * (vth * specs['cells_in_series'])
# For each IV curve, sequentially determine initial values for Io, Rs,
# and Iph [5] Step 3a; [6] Step 3
iph, io, rs, u = _initial_iv_params(ivcurves, ee, voc, isc, rsh,
nnsvth)
# Update values for each IV curve to converge at vmp, imp, voc and isc
iph, io, rs, rsh, u = _update_iv_params(voc, isc, vmp, imp, ee,
iph, io, rs, rsh, nnsvth, u,
maxiter, eps1)
# get single diode models from converged values for each IV curve
pvsyst = _extract_sdm_params(ee, tc, iph, io, rs, rsh, gamma, u,
specs, const, model='pvsyst')
# Add parameters estimated in this function
pvsyst['gamma_ref'] = gamma_ref
pvsyst['mu_gamma'] = mu_gamma
pvsyst['cells_in_series'] = specs['cells_in_series']
return pvsyst
def _fit_pvsyst_sandia_gamma(voc, isc, rsh, vth, tck, specs, const):
# Estimate the diode factor gamma from Isc-Voc data. Method incorporates
# temperature dependence by means of the equation for Io
y = np.log(isc - voc / rsh) - 3. * np.log(tck / (const['T0'] + 273.15))
x1 = const['q'] / const['k'] * (1. / (const['T0'] + 273.15) - 1. / tck)
x2 = voc / (vth * specs['cells_in_series'])
uu = np.logical_or(np.isnan(y), np.isnan(x1), np.isnan(x2))
x = np.vstack((np.ones(len(x1[~uu])), x1[~uu], -x1[~uu] *
(tck[~uu] - (const['T0'] + 273.15)), x2[~uu],
-x2[~uu] * (tck[~uu] - (const['T0'] + 273.15)))).T
alpha = np.linalg.lstsq(x, y[~uu], rcond=None)[0]
gamma_ref = 1. / alpha[3]
mu_gamma = alpha[4] / alpha[3] ** 2
return gamma_ref, mu_gamma
[docs]
def pvsyst_temperature_coeff(alpha_sc, gamma_ref, mu_gamma, I_L_ref, I_o_ref,
R_sh_ref, R_sh_0, R_s, cells_in_series,
R_sh_exp=5.5, EgRef=1.121, irrad_ref=1000,
temp_ref=25):
r"""
Calculates the temperature coefficient of power for a pvsyst single
diode model.
The temperature coefficient is determined as the numerical derivative
:math:`\frac{dP}{dT}` at the maximum power point at reference conditions
[1]_.
Parameters
----------
alpha_sc : float
The short-circuit current temperature coefficient of the module. [A/C]
gamma_ref : float
The diode ideality factor. [unitless]
mu_gamma : float
The temperature coefficient for the diode ideality factor. [1/K]
I_L_ref : float
The light-generated current (or photocurrent) at reference conditions.
[A]
I_o_ref : float
The dark or diode reverse saturation current at reference conditions.
[A]
R_sh_ref : float
The shunt resistance at reference conditions. [ohm]
R_sh_0 : float
The shunt resistance at zero irradiance conditions. [ohm]
R_s : float
The series resistance at reference conditions. [ohm]
cells_in_series : int
The number of cells connected in series.
R_sh_exp : float, default 5.5
The exponent in the equation for shunt resistance. [unitless]
EgRef : float, default 1.121
The energy bandgap of the module's cells at reference temperature.
Default of 1.121 eV is for crystalline silicon. Must be positive. [eV]
irrad_ref : float, default 1000
Reference irradiance. [W/m^2].
temp_ref : float, default 25
Reference cell temperature. [C]
Returns
-------
gamma_pdc : float
Temperature coefficient of power at maximum power point at reference
conditions. [1/C]
References
----------
.. [1] K. Sauer, T. Roessler, C. W. Hansen, Modeling the Irradiance and
Temperature Dependence of Photovoltaic Modules in PVsyst, IEEE Journal
of Photovoltaics v5(1), January 2015.
"""
def maxp(temp_cell, irrad_ref, alpha_sc, gamma_ref, mu_gamma, I_L_ref,
I_o_ref, R_sh_ref, R_sh_0, R_s, cells_in_series, R_sh_exp, EgRef,
temp_ref):
params = calcparams_pvsyst(
irrad_ref, temp_cell, alpha_sc, gamma_ref, mu_gamma, I_L_ref,
I_o_ref, R_sh_ref, R_sh_0, R_s, cells_in_series, R_sh_exp, EgRef,
irrad_ref, temp_ref)
res = bishop88_mpp(*params)
return res[2]
args = (irrad_ref, alpha_sc, gamma_ref, mu_gamma, I_L_ref,
I_o_ref, R_sh_ref, R_sh_0, R_s, cells_in_series, R_sh_exp, EgRef,
temp_ref)
pmp = maxp(temp_ref, *args)
gamma_pdc = _first_order_centered_difference(maxp, x0=temp_ref, args=args)
return gamma_pdc / pmp
[docs]
def fit_pvsyst_iec61853_sandia_2025(effective_irradiance, temp_cell,
i_sc, v_oc, i_mp, v_mp,
cells_in_series, EgRef=1.121,
alpha_sc=None, beta_mp=None,
R_s=None, r_sh_coeff=0.12,
min_Rsh_irradiance=None,
irradiance_tolerance=20,
temperature_tolerance=1):
"""
Estimate parameters for the PVsyst module performance model using
IEC 61853-1 matrix measurements.
Parameters
----------
effective_irradiance : array
Effective irradiance for each test condition [W/m²]
temp_cell : array
Cell temperature for each test condition [C]
i_sc : array
Short circuit current for each test condition [A]
v_oc : array
Open circuit voltage for each test condition [V]
i_mp : array
Current at maximum power point for each test condition [A]
v_mp : array
Voltage at maximum power point for each test condition [V]
cells_in_series : int
The number of cells connected in series.
EgRef : float, optional
The energy bandgap at reference temperature in units of eV.
1.121 eV for crystalline silicon. EgRef must be >0.
alpha_sc : float, optional
Temperature coefficient of short circuit current. If not specified,
it will be estimated using the ``i_sc`` values at irradiance of
1000 W/m2. [A/K]
beta_mp : float, optional
Temperature coefficient of maximum power voltage. If not specified,
it will be estimated using the ``v_mp`` values at irradiance of
1000 W/m2. [1/K]
R_s : float, optional
Series resistance value. If not provided, a value will be estimated
from the input measurements. [ohm]
r_sh_coeff : float, default 0.12
Shunt resistance fitting coefficient. The default value is taken
from [1]_.
min_Rsh_irradiance : float, optional
Irradiance threshold below which values are excluded when estimating
shunt resistance parameter values. May be useful for modules
with problematic low-light measurements. [W/m²]
irradiance_tolerance : float, default 20
Tolerance for irradiance variation around the STC value.
The default value corresponds to a +/- 2% interval around the STC
value of 1000 W/m². [W/m²]
temperature_tolerance : float, default 1
Tolerance for temperature variation around the STC value.
The default value corresponds to a +/- 1 degree interval around the STC
value of 25 degrees. [C]
Returns
-------
dict
alpha_sc : float
short circuit current temperature coefficient [A/K]
gamma_ref : float
diode (ideality) factor at STC [unitless]
mu_gamma : float
temperature coefficient for diode (ideality) factor [1/K]
I_L_ref : float
light current at STC [A]
I_o_ref : float
dark current at STC [A]
R_sh_ref : float
shunt resistance at STC [ohm]
R_sh_0 : float
shunt resistance at zero irradiance [ohm]
R_sh_exp : float
exponential factor defining decrease in shunt resistance with
increasing effective irradiance
R_s : float
series resistance at STC [ohm]
cells_in_series : int
number of cells in series
EgRef : float
effective band gap at STC [eV]
See also
--------
pvlib.pvsystem.calcparams_pvsyst
pvlib.ivtools.sdm.fit_pvsyst_sandia
Notes
-----
Input arrays of operating conditions and electrical measurements must be
1-D with equal lengths.
Values supplied for ``alpha_sc``, ``beta_mp``, and ``R_s`` must be
consistent with the matrix data, as these values are used when estimating
other model parameters.
This method is non-iterative. In some cases, it may be desirable to
refine the estimated parameter values using a numerical optimizer such as
the default method in ``scipy.optimize.minimize``.
References
----------
.. [1] K. S. Anderson, C. W. Hansen, and M. Theristis, "A Noniterative
Method of Estimating Parameter Values for the PVsyst Version 6
Single-Diode Model From IEC 61853-1 Matrix Measurements," IEEE Journal
of Photovoltaics, vol. 15, 3, 2025. :doi:`10.1109/JPHOTOV.2025.3554338`
"""
is_g_stc = np.isclose(effective_irradiance, 1000, rtol=0,
atol=irradiance_tolerance)
is_t_stc = np.isclose(temp_cell, 25, rtol=0,
atol=temperature_tolerance)
if alpha_sc is None:
mu_i_sc = _fit_tempco_pvsyst_iec61853_sandia_2025(i_sc[is_g_stc],
temp_cell[is_g_stc])
i_sc_ref = float(i_sc[is_g_stc & is_t_stc].item())
alpha_sc = mu_i_sc * i_sc_ref
if beta_mp is None:
beta_mp = _fit_tempco_pvsyst_iec61853_sandia_2025(v_mp[is_g_stc],
temp_cell[is_g_stc])
R_sh_ref, R_sh_0, R_sh_exp = \
_fit_shunt_resistances_pvsyst_iec61853_sandia_2025(
i_sc, i_mp, v_mp, effective_irradiance, temp_cell, beta_mp,
coeff=r_sh_coeff, min_irradiance=min_Rsh_irradiance)
if R_s is None:
R_s = _fit_series_resistance_pvsyst_iec61853_sandia_2025(v_oc, i_mp,
v_mp)
gamma_ref, mu_gamma = \
_fit_diode_ideality_factor_pvsyst_iec61853_sandia_2025(
i_sc[is_t_stc], v_oc[is_t_stc], i_mp[is_t_stc], v_mp[is_t_stc],
effective_irradiance[is_t_stc], temp_cell[is_t_stc],
R_sh_ref, R_sh_0, R_sh_exp, R_s, cells_in_series)
I_o_ref = _fit_saturation_current_pvsyst_iec61853_sandia_2025(
i_sc, v_oc, effective_irradiance, temp_cell, gamma_ref, mu_gamma,
R_sh_ref, R_sh_0, R_sh_exp, R_s, cells_in_series, EgRef
)
I_L_ref = _fit_photocurrent_pvsyst_iec61853_sandia_2025(
i_sc, effective_irradiance, temp_cell, alpha_sc,
gamma_ref, mu_gamma,
I_o_ref, R_sh_ref, R_sh_0, R_sh_exp, R_s, cells_in_series, EgRef
)
gamma_ref, mu_gamma = \
_fit_diode_ideality_factor_post_pvsyst_iec61853_sandia_2025(
i_mp, v_mp, effective_irradiance, temp_cell, alpha_sc, I_L_ref,
I_o_ref, R_sh_ref, R_sh_0, R_sh_exp, R_s, cells_in_series, EgRef)
fitted_params = dict(
alpha_sc=alpha_sc,
gamma_ref=gamma_ref,
mu_gamma=mu_gamma,
I_L_ref=I_L_ref,
I_o_ref=I_o_ref,
R_sh_ref=R_sh_ref,
R_sh_0=R_sh_0,
R_sh_exp=R_sh_exp,
R_s=R_s,
cells_in_series=cells_in_series,
EgRef=EgRef,
)
return fitted_params
def _fit_tempco_pvsyst_iec61853_sandia_2025(values, temp_cell,
temp_cell_ref=25):
fit = np.polynomial.polynomial.Polynomial.fit(temp_cell, values, deg=1)
intercept, slope = fit.convert().coef
value_ref = intercept + slope*temp_cell_ref
return slope / value_ref
def _fit_shunt_resistances_pvsyst_iec61853_sandia_2025(
i_sc, i_mp, v_mp, effective_irradiance, temp_cell,
beta_v_mp, coeff=0.2, min_irradiance=None):
if min_irradiance is None:
min_irradiance = 0
mask = effective_irradiance >= min_irradiance
i_sc = i_sc[mask]
i_mp = i_mp[mask]
v_mp = v_mp[mask]
effective_irradiance = effective_irradiance[mask]
temp_cell = temp_cell[mask]
# Equation 10
Rsh_est = (
(v_mp / (1 + beta_v_mp * (temp_cell - 25)))
/ (coeff * (i_sc - i_mp))
)
Rshexp = 5.5
# Eq 11
y = Rsh_est
x = np.exp(-Rshexp * effective_irradiance / 1000)
fit = np.polynomial.polynomial.Polynomial.fit(x, y, deg=1)
intercept, slope = fit.convert().coef
Rshbase = intercept
Rsh0 = slope + Rshbase
# Eq 12
expRshexp = np.exp(-Rshexp)
Rshref = Rshbase * (1 - expRshexp) + Rsh0 * expRshexp
return Rshref, Rsh0, Rshexp
def _fit_series_resistance_pvsyst_iec61853_sandia_2025(v_oc, i_mp, v_mp):
# Stein et al 2014, https://doi.org/10.1109/PVSC.2014.6925326
# Eq 13
x = np.array([np.ones(len(i_mp)), i_mp, np.log(i_mp), v_mp]).T
y = v_oc
coeff, _, _, _ = np.linalg.lstsq(x, y, rcond=None)
R_s = coeff[1]
return R_s
def _fit_diode_ideality_factor_pvsyst_iec61853_sandia_2025(
i_sc, v_oc, i_mp, v_mp, effective_irradiance, temp_cell,
R_sh_ref, R_sh_0, R_sh_exp, R_s, cells_in_series):
NsVth = _pvsyst_nNsVth(temp_cell, gamma=1, cells_in_series=cells_in_series)
Rsh = _pvsyst_Rsh(effective_irradiance, R_sh_ref, R_sh_0, R_sh_exp)
term1 = (i_sc * (1 + R_s/Rsh) - v_oc / Rsh) # Eq 15
term2 = (i_sc - i_mp) * (1 + R_s/Rsh) - v_mp / Rsh # Eq 16
# Eq 14
x1 = NsVth * np.log(term2 / term1)
x = np.array([x1]).T
y = v_mp + i_mp*R_s - v_oc
coeff, _, _, _ = np.linalg.lstsq(x, y, rcond=None)
gamma_ref = coeff[0]
return gamma_ref, 0
def _fit_saturation_current_pvsyst_iec61853_sandia_2025(
i_sc, v_oc, effective_irradiance, temp_cell, gamma_ref, mu_gamma,
R_sh_ref, R_sh_0, R_sh_exp, R_s, cells_in_series, EgRef):
R_sh = _pvsyst_Rsh(effective_irradiance, R_sh_ref, R_sh_0, R_sh_exp)
gamma = _pvsyst_gamma(temp_cell, gamma_ref, mu_gamma)
nNsVth = _pvsyst_nNsVth(temp_cell, gamma, cells_in_series)
# Eq 17
I_o_est = (i_sc * (1 + R_s/R_sh) - v_oc/R_sh) / (np.expm1(v_oc / nNsVth))
x = _pvsyst_Io(temp_cell, gamma, I_o_ref=1, EgRef=EgRef)
# Eq 18
log_I_o_ref = np.mean(np.log(I_o_est) - np.log(x))
I_o_ref = np.exp(log_I_o_ref)
return I_o_ref
def _fit_photocurrent_pvsyst_iec61853_sandia_2025(
i_sc, effective_irradiance, temp_cell, alpha_sc, gamma_ref,
mu_gamma, I_o_ref, R_sh_ref, R_sh_0, R_sh_exp, R_s, cells_in_series,
EgRef):
R_sh = _pvsyst_Rsh(effective_irradiance, R_sh_ref, R_sh_0, R_sh_exp)
gamma = _pvsyst_gamma(temp_cell, gamma_ref, mu_gamma)
I_o = _pvsyst_Io(temp_cell, gamma, I_o_ref, EgRef)
nNsVth = _pvsyst_nNsVth(temp_cell, gamma, cells_in_series)
# Eq 19
I_L_est = i_sc + I_o * (np.expm1(i_sc * R_s / nNsVth)) + i_sc * R_s / R_sh
# Eq 20
x = np.array([effective_irradiance / 1000]).T
y = I_L_est - effective_irradiance / 1000 * alpha_sc * (temp_cell - 25)
coeff, _, _, _ = np.linalg.lstsq(x, y, rcond=None)
I_L_ref = coeff[0]
return I_L_ref
def _fit_diode_ideality_factor_post_pvsyst_iec61853_sandia_2025(
i_mp, v_mp, effective_irradiance, temp_cell, alpha_sc, I_L_ref,
I_o_ref, R_sh_ref, R_sh_0, R_sh_exp, R_s, cells_in_series, EgRef):
Rsh = _pvsyst_Rsh(effective_irradiance, R_sh_ref, R_sh_0, R_sh_exp)
I_L = _pvsyst_IL(effective_irradiance, temp_cell, I_L_ref, alpha_sc)
NsVth = _pvsyst_nNsVth(temp_cell, gamma=1, cells_in_series=cells_in_series)
Tref_K = 25 + 273.15
Tcell_K = temp_cell + 273.15
# Eq 21
k = constants.k # Boltzmann constant in J/K
q = constants.e # elementary charge in coulomb
numerator = (
(q * EgRef / k) * (1/Tref_K - 1/Tcell_K)
+ (v_mp + i_mp*R_s) / NsVth
)
denominator = (
np.log((I_L - i_mp - (v_mp+i_mp*R_s) / Rsh) / I_o_ref)
- 3 * np.log(Tcell_K / Tref_K)
)
gamma_est = numerator / denominator
# Eq 22
x = np.array([np.ones(len(i_mp)), temp_cell - 25]).T
y = gamma_est
coeff, _, _, _ = np.linalg.lstsq(x, y, rcond=None)
gamma_ref, mu_gamma = coeff
return gamma_ref, mu_gamma
