def graphite_LGM50_ocp_Chen2020(sto):
"""
LG M50 graphite open circuit potential as a function of stochiometry, fit taken
from [1].
References
----------
.. [1] Chang-Hui Chen, Ferran Brosa Planella, Kieran O’Regan, Dominika Gastol, W.
Dhammika Widanage, and Emma Kendrick. "Development of Experimental Techniques for
Parameterization of Multi-scale Lithium-ion Battery Models." Journal of the
Electrochemical Society 167 (2020): 080534.
Parameters
----------
sto: :class:`pybamm.Symbol`
Electrode stochiometry
Returns
-------
:class:`pybamm.Symbol`
Open circuit potential
"""
u_eq = (1.9793 * exp(-39.3631 * sto) + 0.2482 -
0.0909 * tanh(29.8538 * (sto - 0.1234)) -
0.04478 * tanh(14.9159 * (sto - 0.2769)) -
0.0205 * tanh(30.4444 * (sto - 0.6103)))
return u_eq
def nmc_LGM50_ocp_Chen2020(sto):
"""
LG M50 NMC open circuit potential as a function of stochiometry, fit taken
from [1].
References
----------
.. [1] Chang-Hui Chen, Ferran Brosa Planella, Kieran O’Regan, Dominika Gastol, W.
Dhammika Widanage, and Emma Kendrick. "Development of Experimental Techniques for
Parameterization of Multi-scale Lithium-ion Battery Models." Journal of the
Electrochemical Society 167 (2020): 080534.
Parameters
----------
sto: :class:`pybamm.Symbol`
Electrode stochiometry
Returns
-------
:class:`pybamm.Symbol`
Open circuit potential
"""
u_eq = (
-0.8090 * sto
+ 4.4875
- 0.0428 * tanh(18.5138 * (sto - 0.5542))
- 17.7326 * tanh(15.7890 * (sto - 0.3117))
+ 17.5842 * tanh(15.9308 * (sto - 0.3120))
)
return u_eq
def graphite_ocp_Ecker2015_function(sto):
"""
Graphite OCP as a function of stochiometry [1, 2, 3].
References
----------
.. [1] Ecker, Madeleine, et al. "Parameterization of a physico-chemical model of
a lithium-ion battery i. determination of parameters." Journal of the
Electrochemical Society 162.9 (2015): A1836-A1848.
.. [2] Ecker, Madeleine, et al. "Parameterization of a physico-chemical model of
a lithium-ion battery ii. model validation." Journal of The Electrochemical
Society 162.9 (2015): A1849-A1857.
.. [3] Richardson, Giles, et. al. "Generalised single particle models for
high-rate operation of graded lithium-ion electrodes: Systematic derivation
and validation." Electrochemica Acta 339 (2020): 135862
Parameters
----------
sto: :class:`pybamm.Symbol`
Electrode stochiometry
Returns
-------
:class:`pybamm.Symbol`
Open circuit potential
"""
# Graphite negative electrode from Ecker, Kabitz, Laresgoiti et al.
# Analytical fit (WebPlotDigitizer + gnuplot)
a = 0.716502
b = 369.028
c = 0.12193
d = 35.6478
e = 0.0530947
g = 0.0169644
h = 27.1365
i = 0.312832
j = 0.0199313
k = 28.5697
m = 0.614221
n = 0.931153
o = 36.328
p = 1.10743
q = 0.140031
r = 0.0189193
s = 21.1967
t = 0.196176
u_eq = (
a * exp(-b * sto)
+ c * exp(-d * (sto - e))
- r * tanh(s * (sto - t))
- g * tanh(h * (sto - i))
- j * tanh(k * (sto - m))
- n * exp(o * (sto - p))
+ q
)
return u_eq
def test_tanh(self):
a = pybamm.InputParameter("a")
fun = pybamm.tanh(a)
self.assertEqual(fun.evaluate(inputs={"a": 3}), np.tanh(3))
h = 0.0000001
self.assertAlmostEqual(
fun.diff(a).evaluate(inputs={"a": 3}),
(pybamm.tanh(pybamm.Scalar(3 + h)).evaluate() -
fun.evaluate(inputs={"a": 3})) / h,
places=5,
)
def nco_ocp_Ecker2015_function(sto):
"""
NCO OCP as a function of stochiometry [1, 2, 3].
References
----------
.. [1] Ecker, Madeleine, et al. "Parameterization of a physico-chemical model of
a lithium-ion battery i. determination of parameters." Journal of the
Electrochemical Society 162.9 (2015): A1836-A1848.
.. [2] Ecker, Madeleine, et al. "Parameterization of a physico-chemical model of
a lithium-ion battery ii. model validation." Journal of The Electrochemical
Society 162.9 (2015): A1849-A1857.
.. [3] Richardson, Giles, et. al. "Generalised single particle models for
high-rate operation of graded lithium-ion electrodes: Systematic derivation
and validation." Electrochemica Acta 339 (2020): 135862
Parameters
----------
sto: double
Stochiometry of material (li-fraction)
"""
# LiNiCo from Ecker, Kabitz, Laresgoiti et al.
# Analytical fit (WebPlotDigitizer + gnuplot)
a = -2.35211
c = 0.0747061
d = 31.886
e = 0.0219921
g = 0.640243
h = 5.48623
i = 0.439245
j = 3.82383
k = 4.12167
m = 0.176187
n = 0.0542123
o = 18.2919
p = 0.762272
q = 4.23285
r = -6.34984
s = 2.66395
t = 0.174352
u_eq = (
a * sto
- c * tanh(d * (sto - e))
- r * tanh(s * (sto - t))
- g * tanh(h * (sto - i))
- j * tanh(k * (sto - m))
- n * tanh(o * (sto - p))
+ q
)
return u_eq
def graphite_mcmb2528_ocp_Dualfoil1998(sto):
"""
Graphite MCMB 2528 Open Circuit Potential (OCP) as a function of the
stochiometry. The fit is taken from Dualfoil [1]. Dualfoil states that the data
was measured by Chris Bogatu at Telcordia and PolyStor materials, 2000. However,
we could not find any other records of this measurment.
References
----------
.. [1] http://www.cchem.berkeley.edu/jsngrp/fortran.html
"""
u_eq = (
0.194
+ 1.5 * exp(-120.0 * sto)
+ 0.0351 * tanh((sto - 0.286) / 0.083)
- 0.0045 * tanh((sto - 0.849) / 0.119)
- 0.035 * tanh((sto - 0.9233) / 0.05)
- 0.0147 * tanh((sto - 0.5) / 0.034)
- 0.102 * tanh((sto - 0.194) / 0.142)
- 0.022 * tanh((sto - 0.9) / 0.0164)
- 0.011 * tanh((sto - 0.124) / 0.0226)
+ 0.0155 * tanh((sto - 0.105) / 0.029)
)
return u_eq
def graphite_ocp_Kim2011(sto):
"""
Graphite Open Circuit Potential (OCP) as a function of the stochiometry [1].
References
----------
.. [1] Kim, G. H., Smith, K., Lee, K. J., Santhanagopalan, S., & Pesaran, A.
(2011). Multi-domain modeling of lithium-ion batteries encompassing
multi-physics in varied length scales. Journal of The Electrochemical
Society, 158(8), A955-A969.
"""
u_eq = (
0.124
+ 1.5 * exp(-70 * sto)
- 0.0351 * tanh((sto - 0.286) / 0.083)
- 0.0045 * tanh((sto - 0.9) / 0.119)
- 0.035 * tanh((sto - 0.99) / 0.05)
- 0.0147 * tanh((sto - 0.5) / 0.034)
- 0.102 * tanh((sto - 0.194) / 0.142)
- 0.022 * tanh((sto - 0.98) / 0.0164)
- 0.011 * tanh((sto - 0.124) / 0.0226)
+ 0.0155 * tanh((sto - 0.105) / 0.029)
)
return u_eq
def sigmoid(left, right, k):
"""
Sigmoidal approximation to the heaviside function. k is the smoothing parameter,
set by `pybamm.settings.heaviside_smoothing`. The recommended value is k=10.
Note that the concept of deciding which side to pick when left=right does not apply
for this smooth approximation. When left=right, the value is (left+right)/2.
"""
return (1 + pybamm.tanh(k * (right - left))) / 2
def lico2_ocp_Dualfoil1998(sto):
"""
Lithium Cobalt Oxide (LiCO2) Open Circuit Potential (OCP) as a a function of the
stochiometry. The fit is taken from Dualfoil [1]. Dualfoil states that the data
was measured by Oscar Garcia 2001 using Quallion electrodes for 0.5 < sto < 0.99
and by Marc Doyle for sto<0.4 (for unstated electrodes). We could not find any
other records of the Garcia measurements. Doyles fits can be found in his
thesis [2] but we could not find any other record of his measurments.
References
----------
.. [1] http://www.cchem.berkeley.edu/jsngrp/fortran.html
.. [2] CM Doyle. Design and simulation of lithium rechargeable batteries,
1995.
Parameters
----------
sto : :class:`pybamm.Symbol`
Stochiometry of material (li-fraction)
"""
stretch = 1.062
sto = stretch * sto
u_eq = (2.16216 + 0.07645 * tanh(30.834 - 54.4806 * sto) +
2.1581 * tanh(52.294 - 50.294 * sto) -
0.14169 * tanh(11.0923 - 19.8543 * sto) +
0.2051 * tanh(1.4684 - 5.4888 * sto) + 0.2531 * tanh(
(-sto + 0.56478) / 0.1316) - 0.02167 * tanh(
(sto - 0.525) / 0.006))
return u_eq
def graphite_ocp_PeymanMPM(sto):
"""
Graphite Open Circuit Potential (OCP) as a function of the
stochiometry. The fit is taken from Peyman MPM [1].
References
----------
.. [1] Peyman Mohtat et al, MPM (to be submitted)
"""
u_eq = (0.063 + 0.8 * pybamm.exp(-75 * (sto + 0.001)) -
0.0120 * pybamm.tanh((sto - 0.127) / 0.016) - 0.0118 * pybamm.tanh(
(sto - 0.155) / 0.016) - 0.0035 * pybamm.tanh(
(sto - 0.220) / 0.020) - 0.0095 * pybamm.tanh(
(sto - 0.190) / 0.013) - 0.0145 * pybamm.tanh(
(sto - 0.490) / 0.020) - 0.0800 * pybamm.tanh(
(sto - 1.030) / 0.055))
return u_eq
