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Parameterisation#

In this notebook, we show how to find which parameters are needed in a model and define them.

For other notebooks about parameterization, see:

The API documentation of Parameters
Setting parameter values can be found at pybamm/docs/source/examples/notebooks/getting_started/tutorial-4-setting-parameter-values.ipynb. This explains the basics of how to set the parameters of a model (in less detail than here).
parameter-values.ipynb can be found at pybamm/examples/notebooks/parameterization/parameter-values.ipynb. This explains the basics of the ParameterValues class.

Adding your own parameter sets (using a dictionary)#

We will be using the model defined and explained in more detail in 3-negative-particle-problem.ipynb example notebook. We begin by importing the required libraries

[47]:

%pip install "pybamm[plot,cite]" -q    # install PyBaMM if it is not installed
import pybamm
import numpy as np
import matplotlib.pyplot as plt

/bin/bash: warning: setlocale: LC_ALL: cannot change locale (en_US.UTF-8)
Note: you may need to restart the kernel to use updated packages.

Setting up the model#

We define all the parameters and variables using pybamm.Parameter and pybamm.Variable respectively.

[48]:

c = pybamm.Variable("Concentration [mol.m-3]", domain="negative particle")

R = pybamm.Parameter("Particle radius [m]")
D = pybamm.FunctionParameter(
    "Diffusion coefficient [m2.s-1]", {"Concentration [mol.m-3]": c}
)
j = pybamm.InputParameter("Interfacial current density [A.m-2]")
c0 = pybamm.Parameter("Initial concentration [mol.m-3]")
c_e = pybamm.Parameter("Electrolyte concentration [mol.m-3]")

Now we define our model equations, boundary and initial conditions. We also add the variables required using the dictionary model.variables

[49]:

model = pybamm.BaseModel()

# governing equations
N = -D * pybamm.grad(c)  # flux
dcdt = -pybamm.div(N)
model.rhs = {c: dcdt}

# boundary conditions
lbc = pybamm.Scalar(0)
rbc = -j
model.boundary_conditions = {c: {"left": (lbc, "Neumann"), "right": (rbc, "Neumann")}}

# initial conditions
model.initial_conditions = {c: c0}

model.variables = {
    "Concentration [mol.m-3]": c,
    "Surface concentration [mol.m-3]": pybamm.surf(c),
    "Flux [mol.m-2.s-1]": N,
}

We also define the geometry, since there are parameters in the geometry too

[50]:

r = pybamm.SpatialVariable(
    "r", domain=["negative particle"], coord_sys="spherical polar"
)
geometry = pybamm.Geometry(
    {"negative particle": {r: {"min": pybamm.Scalar(0), "max": R}}}
)

Finding the parameters required#

To know what parameters are required by the model and geometry, we can do

[51]:

model.print_parameter_info()
geometry.print_parameter_info()

┌─────────────────────────────────────┬────────────────────────────────────────────────────────────┐
│ Parameter                           │ Type of parameter                                          │
├─────────────────────────────────────┼────────────────────────────────────────────────────────────┤
│ Initial concentration [mol.m-3]     │ Parameter                                                  │
│ Interfacial current density [A.m-2] │ InputParameter                                             │
│ Diffusion coefficient [m2.s-1]      │ FunctionParameter with inputs(s) 'Concentration [mol.m-3]' │
└─────────────────────────────────────┴────────────────────────────────────────────────────────────┘

Particle radius [m] (Parameter)

This tells us that we need to provide parameter values for the initial concentration and Faraday constant, an InputParameter at solve time for the interfacial current density, and diffusivity as a function of concentration. Since the electrolyte concentration does not appear anywhere in the model, there is no need to provide a value for it.

Adding the parameters#

Now we can proceed to the step where we add the parameter values using a dictionary. We set up a dictionary with parameter names as the dictionary keys and their respective values as the dictionary values.

[52]:

def D_fun(c):
    return 3.9  # * pybamm.exp(-c)


values = {
    "Particle radius [m]": 2,
    "Diffusion coefficient [m2.s-1]": D_fun,
    "Initial concentration [mol.m-3]": 2.5,
}

Now we can pass this dictionary in pybamm.ParameterValues class which accepts a dictionary of parameter names and values. We can then print param to check if it was initialised.

[53]:

param = pybamm.ParameterValues(values)

param

[53]:

{'Boltzmann constant [J.K-1]': 1.380649e-23,
 'Diffusion coefficient [m2.s-1]': <function D_fun at 0x7f3267f12480>,
 'Electron charge [C]': 1.602176634e-19,
 'Faraday constant [C.mol-1]': 96485.33212,
 'Ideal gas constant [J.K-1.mol-1]': 8.314462618,
 'Initial concentration [mol.m-3]': 2.5,
 'Particle radius [m]': 2}

Updating the parameter values#

The parameter values or param can be further updated by using the update function of ParameterValues class. The update function takes a dictionary with keys being the parameters to be updated and their respective values being the updated values. Here we update the "Particle radius [m]" parameter’s value. Additionally, a function can also be passed as a parameter’s value which we will see ahead, and a new parameter can also be added by passing check_already_exists=False in the update function.

[54]:

param.update({"Initial concentration [mol.m-3]": 1.5})
param

[54]:

{'Boltzmann constant [J.K-1]': 1.380649e-23,
 'Diffusion coefficient [m2.s-1]': <function D_fun at 0x7f3267f12480>,
 'Electron charge [C]': 1.602176634e-19,
 'Faraday constant [C.mol-1]': 96485.33212,
 'Ideal gas constant [J.K-1.mol-1]': 8.314462618,
 'Initial concentration [mol.m-3]': 1.5,
 'Particle radius [m]': 2}

Solving the model#

Finding the parameters in a model#

The parameter function of the BaseModel class can be used to obtain the parameters of a model.

[55]:

parameters = model.parameters
parameters

[55]:

[FunctionParameter(0x5151bd3d66db3d0a, Diffusion coefficient [m2.s-1], children=['Concentration [mol.m-3]'], domains={'primary': ['negative particle']}),
 Parameter(0x1643caef07df6bfd, Initial concentration [mol.m-3], children=[], domains={}),
 InputParameter(-0x8e0791036f53f59, Interfacial current density [A.m-2], children=[], domains={})]

As explained in the 3-negative-particle-problem.ipynb example, we first process both the model and the geometry.

[56]:

param.process_model(model)
param.process_geometry(geometry)

We can now set up our mesh, choose a spatial method, and discretise our model

[57]:

submesh_types = {"negative particle": pybamm.Uniform1DSubMesh}
var_pts = {r: 20}
mesh = pybamm.Mesh(geometry, submesh_types, var_pts)

spatial_methods = {"negative particle": pybamm.FiniteVolume()}
disc = pybamm.Discretisation(mesh, spatial_methods)
disc.process_model(model);

We choose a solver and times at which we want the solution returned, and solve the model. Here we give a value for the current density j.

[58]:

# solve
solver = pybamm.ScipySolver()
t = np.linspace(0, 3600, 600)
solution = solver.solve(model, t, inputs={"Interfacial current density [A.m-2]": 1.4})

# post-process, so that the solution can be called at any time t or space r
# (using interpolation)
c = solution["Concentration [mol.m-3]"]
c_surf = solution["Surface concentration [mol.m-3]"]

# plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(13, 4))

ax1.plot(solution.t, c_surf(solution.t))
ax1.set_xlabel("Time [s]")
ax1.set_ylabel("Surface concentration [mol.m-3]")

rsol = mesh["negative particle"].nodes  # radial position
time = 1000  # time in seconds
ax2.plot(rsol * 1e6, c(t=time, r=rsol), label=f"t={time}[s]")
ax2.set_xlabel("Particle radius [microns]")
ax2.set_ylabel("Concentration [mol.m-3]")
ax2.legend()

plt.tight_layout()
plt.show()

../../../../_images/source_examples_notebooks_parameterization_parameterization_27_0.png

Using pre-defined models in `PyBaMM`#

In the next few steps, we will be showing the same workflow with the Single Particle Model (SPM). We will also see how you can pass a function as a parameter’s value and how to plot such parameter functions.

We start by initializing our model

[59]:

spm = pybamm.lithium_ion.SPM()

Finding the parameters in a model#

We can print the parameters of a model by using the print_parameter_info function.

[60]:

spm.print_parameter_info()

┌───────────────────────────────────────────────────────────┬─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐
│ Parameter                                                 │ Type of parameter                                                                                                                                                                                           │
├───────────────────────────────────────────────────────────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│ Electrode width [m]                                       │ Parameter                                                                                                                                                                                                   │
│ Number of cells connected in series to make a battery     │ Parameter                                                                                                                                                                                                   │
│ Maximum concentration in positive electrode [mol.m-3]     │ Parameter                                                                                                                                                                                                   │
│ Positive electrode Bruggeman coefficient (electrode)      │ Parameter                                                                                                                                                                                                   │
│ Number of electrodes connected in parallel to make a cell │ Parameter                                                                                                                                                                                                   │
│ Initial concentration in electrolyte [mol.m-3]            │ Parameter                                                                                                                                                                                                   │
│ Positive electrode Bruggeman coefficient (electrolyte)    │ Parameter                                                                                                                                                                                                   │
│ Nominal cell capacity [A.h]                               │ Parameter                                                                                                                                                                                                   │
│ Electrode height [m]                                      │ Parameter                                                                                                                                                                                                   │
│ Upper voltage cut-off [V]                                 │ Parameter                                                                                                                                                                                                   │
│ Negative electrode Bruggeman coefficient (electrode)      │ Parameter                                                                                                                                                                                                   │
│ Faraday constant [C.mol-1]                                │ Parameter                                                                                                                                                                                                   │
│ Positive electrode thickness [m]                          │ Parameter                                                                                                                                                                                                   │
│ Separator Bruggeman coefficient (electrolyte)             │ Parameter                                                                                                                                                                                                   │
│ Negative electrode Bruggeman coefficient (electrolyte)    │ Parameter                                                                                                                                                                                                   │
│ Reference temperature [K]                                 │ Parameter                                                                                                                                                                                                   │
│ Maximum concentration in negative electrode [mol.m-3]     │ Parameter                                                                                                                                                                                                   │
│ Lower voltage cut-off [V]                                 │ Parameter                                                                                                                                                                                                   │
│ Negative electrode thickness [m]                          │ Parameter                                                                                                                                                                                                   │
│ Separator thickness [m]                                   │ Parameter                                                                                                                                                                                                   │
│ Ideal gas constant [J.K-1.mol-1]                          │ Parameter                                                                                                                                                                                                   │
│ Current function [A]                                      │ FunctionParameter with inputs(s) 'Time[s]'                                                                                                                                                                  │
│ Negative particle radius [m]                              │ FunctionParameter with inputs(s) 'Through-cell distance (x) [m]'                                                                                                                                            │
│ Negative electrode active material volume fraction        │ FunctionParameter with inputs(s) 'Through-cell distance (x) [m]'                                                                                                                                            │
│ Positive particle radius [m]                              │ FunctionParameter with inputs(s) 'Through-cell distance (x) [m]'                                                                                                                                            │
│ Positive electrode exchange-current density [A.m-2]       │ FunctionParameter with inputs(s) 'Electrolyte concentration [mol.m-3]', 'Positive particle surface concentration [mol.m-3]', 'Maximum positive particle surface concentration [mol.m-3]', 'Temperature [K]' │
│ Positive electrode porosity                               │ FunctionParameter with inputs(s) 'Through-cell distance (x) [m]'                                                                                                                                            │
│ Negative electrode OCP entropic change [V.K-1]            │ FunctionParameter with inputs(s) 'Negative particle stoichiometry', 'Maximum negative particle surface concentration [mol.m-3]'                                                                             │
│ Positive electrode OCP [V]                                │ FunctionParameter with inputs(s) 'Positive particle stoichiometry'                                                                                                                                          │
│ Initial concentration in positive electrode [mol.m-3]     │ FunctionParameter with inputs(s) 'Radial distance (r) [m]', 'Through-cell distance (x) [m]'                                                                                                                 │
│ Negative electrode porosity                               │ FunctionParameter with inputs(s) 'Through-cell distance (x) [m]'                                                                                                                                            │
│ Positive electrode OCP entropic change [V.K-1]            │ FunctionParameter with inputs(s) 'Positive particle stoichiometry', 'Maximum positive particle surface concentration [mol.m-3]'                                                                             │
│ Negative electrode OCP [V]                                │ FunctionParameter with inputs(s) 'Negative particle stoichiometry'                                                                                                                                          │
│ Positive electrode active material volume fraction        │ FunctionParameter with inputs(s) 'Through-cell distance (x) [m]'                                                                                                                                            │
│ Negative electrode exchange-current density [A.m-2]       │ FunctionParameter with inputs(s) 'Electrolyte concentration [mol.m-3]', 'Negative particle surface concentration [mol.m-3]', 'Maximum negative particle surface concentration [mol.m-3]', 'Temperature [K]' │
│ Initial concentration in negative electrode [mol.m-3]     │ FunctionParameter with inputs(s) 'Radial distance (r) [m]', 'Through-cell distance (x) [m]'                                                                                                                 │
│ Ambient temperature [K]                                   │ FunctionParameter with inputs(s) 'Distance across electrode width [m]', 'Distance across electrode height [m]', 'Time [s]'                                                                                  │
│ Positive particle diffusivity [m2.s-1]                    │ FunctionParameter with inputs(s) 'Positive particle stoichiometry', 'Temperature [K]'                                                                                                                       │
│ Separator porosity                                        │ FunctionParameter with inputs(s) 'Through-cell distance (x) [m]'                                                                                                                                            │
│ Negative particle diffusivity [m2.s-1]                    │ FunctionParameter with inputs(s) 'Negative particle stoichiometry', 'Temperature [K]'                                                                                                                       │
└───────────────────────────────────────────────────────────┴─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘

Note that there are no InputParameter objects in the default SPM. Also, note that if a FunctionParameter is expected, it is ok to provide a scalar (parameter) instead. However, if a Parameter is expected, you cannot provide a function instead.

Another way to view what parameters are needed is to print the default parameter values. This can also be used to get some good defaults (but care must be taken when combining parameters across datasets and chemistries)

[61]:

{k: v for k, v in spm.default_parameter_values.items() if k in spm.get_parameter_info()}

[61]:

{'Ideal gas constant [J.K-1.mol-1]': 8.314462618,
 'Faraday constant [C.mol-1]': 96485.33212,
 'Negative electrode thickness [m]': 0.0001,
 'Separator thickness [m]': 2.5e-05,
 'Positive electrode thickness [m]': 0.0001,
 'Electrode height [m]': 0.137,
 'Electrode width [m]': 0.207,
 'Nominal cell capacity [A.h]': 0.680616,
 'Current function [A]': 0.680616,
 'Maximum concentration in negative electrode [mol.m-3]': 24983.2619938437,
 'Negative particle diffusivity [m2.s-1]': <function pybamm.input.parameters.lithium_ion.Marquis2019.graphite_mcmb2528_diffusivity_Dualfoil1998(sto, T)>,
 'Negative electrode OCP [V]': <function pybamm.input.parameters.lithium_ion.Marquis2019.graphite_mcmb2528_ocp_Dualfoil1998(sto)>,
 'Negative electrode porosity': 0.3,
 'Negative electrode active material volume fraction': 0.6,
 'Negative particle radius [m]': 1e-05,
 'Negative electrode Bruggeman coefficient (electrolyte)': 1.5,
 'Negative electrode Bruggeman coefficient (electrode)': 1.5,
 'Negative electrode exchange-current density [A.m-2]': <function pybamm.input.parameters.lithium_ion.Marquis2019.graphite_electrolyte_exchange_current_density_Dualfoil1998(c_e, c_s_surf, c_s_max, T)>,
 'Negative electrode OCP entropic change [V.K-1]': <function pybamm.input.parameters.lithium_ion.Marquis2019.graphite_entropic_change_Moura2016(sto, c_s_max)>,
 'Maximum concentration in positive electrode [mol.m-3]': 51217.9257309275,
 'Positive particle diffusivity [m2.s-1]': <function pybamm.input.parameters.lithium_ion.Marquis2019.lico2_diffusivity_Dualfoil1998(sto, T)>,
 'Positive electrode OCP [V]': <function pybamm.input.parameters.lithium_ion.Marquis2019.lico2_ocp_Dualfoil1998(sto)>,
 'Positive electrode porosity': 0.3,
 'Positive electrode active material volume fraction': 0.5,
 'Positive particle radius [m]': 1e-05,
 'Positive electrode Bruggeman coefficient (electrolyte)': 1.5,
 'Positive electrode Bruggeman coefficient (electrode)': 1.5,
 'Positive electrode exchange-current density [A.m-2]': <function pybamm.input.parameters.lithium_ion.Marquis2019.lico2_electrolyte_exchange_current_density_Dualfoil1998(c_e, c_s_surf, c_s_max, T)>,
 'Positive electrode OCP entropic change [V.K-1]': <function pybamm.input.parameters.lithium_ion.Marquis2019.lico2_entropic_change_Moura2016(sto, c_s_max)>,
 'Separator porosity': 1.0,
 'Separator Bruggeman coefficient (electrolyte)': 1.5,
 'Initial concentration in electrolyte [mol.m-3]': 1000.0,
 'Reference temperature [K]': 298.15,
 'Ambient temperature [K]': 298.15,
 'Number of electrodes connected in parallel to make a cell': 1.0,
 'Number of cells connected in series to make a battery': 1.0,
 'Lower voltage cut-off [V]': 3.105,
 'Upper voltage cut-off [V]': 4.1,
 'Initial concentration in negative electrode [mol.m-3]': 19986.609595075,
 'Initial concentration in positive electrode [mol.m-3]': 30730.7554385565}

We now define a dictionary of values for ParameterValues as before (here, a subset of the Chen2020 parameters)

[62]:

def graphite_mcmb2528_diffusivity_Dualfoil1998(sto, T):
    D_ref = 3.9 * 10 ** (-14)
    E_D_s = 42770
    arrhenius = exp(E_D_s / constants.R * (1 / 298.15 - 1 / T))
    return D_ref * arrhenius


neg_ocp = np.array(
    [
        [0.0, 1.81772748],
        [0.03129623, 1.0828807],
        [0.03499902, 0.99593794],
        [0.0387018, 0.90023398],
        [0.04240458, 0.79649431],
        [0.04610736, 0.73354429],
        [0.04981015, 0.66664314],
        [0.05351292, 0.64137149],
        [0.05721568, 0.59813869],
        [0.06091845, 0.5670836],
        [0.06462122, 0.54746181],
        [0.06832399, 0.53068399],
        [0.07202675, 0.51304734],
        [0.07572951, 0.49394092],
        [0.07943227, 0.47926274],
        [0.08313503, 0.46065259],
        [0.08683779, 0.45992726],
        [0.09054054, 0.43801501],
        [0.09424331, 0.42438665],
        [0.09794607, 0.41150269],
        [0.10164883, 0.40033659],
        [0.10535158, 0.38957134],
        [0.10905434, 0.37756538],
        [0.1127571, 0.36292541],
        [0.11645985, 0.34357086],
        [0.12016261, 0.3406314],
        [0.12386536, 0.32299468],
        [0.12756811, 0.31379458],
        [0.13127086, 0.30795386],
        [0.13497362, 0.29207319],
        [0.13867638, 0.28697687],
        [0.14237913, 0.27405477],
        [0.14608189, 0.2670497],
        [0.14978465, 0.25857493],
        [0.15348741, 0.25265783],
        [0.15719018, 0.24826777],
        [0.16089294, 0.2414345],
        [0.1645957, 0.23362778],
        [0.16829847, 0.22956218],
        [0.17200122, 0.22370236],
        [0.17570399, 0.22181271],
        [0.17940674, 0.22089651],
        [0.1831095, 0.2194268],
        [0.18681229, 0.21830064],
        [0.19051504, 0.21845333],
        [0.1942178, 0.21753715],
        [0.19792056, 0.21719357],
        [0.20162334, 0.21635373],
        [0.2053261, 0.21667822],
        [0.20902886, 0.21738444],
        [0.21273164, 0.21469313],
        [0.2164344, 0.21541846],
        [0.22013716, 0.21465495],
        [0.22383993, 0.2135479],
        [0.2275427, 0.21392964],
        [0.23124547, 0.21074206],
        [0.23494825, 0.20873788],
        [0.23865101, 0.20465319],
        [0.24235377, 0.20205732],
        [0.24605653, 0.19774358],
        [0.2497593, 0.19444147],
        [0.25346208, 0.19190285],
        [0.25716486, 0.18850531],
        [0.26086762, 0.18581399],
        [0.26457039, 0.18327537],
        [0.26827314, 0.18157659],
        [0.2719759, 0.17814088],
        [0.27567867, 0.17529686],
        [0.27938144, 0.1719375],
        [0.28308421, 0.16934161],
        [0.28678698, 0.16756649],
        [0.29048974, 0.16609676],
        [0.29419251, 0.16414985],
        [0.29789529, 0.16260378],
        [0.30159806, 0.16224113],
        [0.30530083, 0.160027],
        [0.30900361, 0.15827096],
        [0.31270637, 0.1588054],
        [0.31640913, 0.15552238],
        [0.32011189, 0.15580869],
        [0.32381466, 0.15220118],
        [0.32751744, 0.1511132],
        [0.33122021, 0.14987253],
        [0.33492297, 0.14874637],
        [0.33862575, 0.14678037],
        [0.34232853, 0.14620776],
        [0.34603131, 0.14555879],
        [0.34973408, 0.14389819],
        [0.35343685, 0.14359279],
        [0.35713963, 0.14242846],
        [0.36084241, 0.14038612],
        [0.36454517, 0.13882096],
        [0.36824795, 0.13954628],
        [0.37195071, 0.13946992],
        [0.37565348, 0.13780934],
        [0.37935626, 0.13973714],
        [0.38305904, 0.13698858],
        [0.38676182, 0.13523254],
        [0.3904646, 0.13441178],
        [0.39416737, 0.1352898],
        [0.39787015, 0.13507985],
        [0.40157291, 0.13647321],
        [0.40527567, 0.13601512],
        [0.40897844, 0.13435452],
        [0.41268121, 0.1334765],
        [0.41638398, 0.1348317],
        [0.42008676, 0.13275118],
        [0.42378953, 0.13286571],
        [0.4274923, 0.13263667],
        [0.43119506, 0.13456447],
        [0.43489784, 0.13471718],
        [0.43860061, 0.13395369],
        [0.44230338, 0.13448814],
        [0.44600615, 0.1334765],
        [0.44970893, 0.13298023],
        [0.45341168, 0.13259849],
        [0.45711444, 0.13338107],
        [0.46081719, 0.13309476],
        [0.46451994, 0.13275118],
        [0.46822269, 0.13443087],
        [0.47192545, 0.13315202],
        [0.47562821, 0.132713],
        [0.47933098, 0.1330184],
        [0.48303375, 0.13278936],
        [0.48673651, 0.13225491],
        [0.49043926, 0.13317111],
        [0.49414203, 0.13263667],
        [0.49784482, 0.13187316],
        [0.50154759, 0.13265574],
        [0.50525036, 0.13250305],
        [0.50895311, 0.13324745],
        [0.51265586, 0.13204496],
        [0.51635861, 0.13242669],
        [0.52006139, 0.13233127],
        [0.52376415, 0.13198769],
        [0.52746692, 0.13254122],
        [0.53116969, 0.13145325],
        [0.53487245, 0.13298023],
        [0.53857521, 0.13168229],
        [0.54227797, 0.1313578],
        [0.54598074, 0.13235036],
        [0.5496835, 0.13120511],
        [0.55338627, 0.13089971],
        [0.55708902, 0.13109058],
        [0.56079178, 0.13082336],
        [0.56449454, 0.13011713],
        [0.5681973, 0.129869],
        [0.57190006, 0.12992626],
        [0.57560282, 0.12942998],
        [0.57930558, 0.12796026],
        [0.58300835, 0.12862831],
        [0.58671112, 0.12656689],
        [0.59041389, 0.12734947],
        [0.59411664, 0.12509716],
        [0.59781941, 0.12110791],
        [0.60152218, 0.11839751],
        [0.60522496, 0.11244226],
        [0.60892772, 0.11307214],
        [0.61263048, 0.1092165],
        [0.61633325, 0.10683058],
        [0.62003603, 0.10433014],
        [0.6237388, 0.10530359],
        [0.62744156, 0.10056993],
        [0.63114433, 0.09950104],
        [0.63484711, 0.09854668],
        [0.63854988, 0.09921473],
        [0.64225265, 0.09541635],
        [0.64595543, 0.09980643],
        [0.64965823, 0.0986612],
        [0.653361, 0.09560722],
        [0.65706377, 0.09755413],
        [0.66076656, 0.09612258],
        [0.66446934, 0.09430929],
        [0.66817212, 0.09661885],
        [0.67187489, 0.09366032],
        [0.67557767, 0.09522548],
        [0.67928044, 0.09535909],
        [0.68298322, 0.09316404],
        [0.686686, 0.09450016],
        [0.69038878, 0.0930877],
        [0.69409156, 0.09343126],
        [0.69779433, 0.0932404],
        [0.70149709, 0.09350762],
        [0.70519988, 0.09339309],
        [0.70890264, 0.09291591],
        [0.7126054, 0.09303043],
        [0.71630818, 0.0926296],
        [0.72001095, 0.0932404],
        [0.72371371, 0.09261052],
        [0.72741648, 0.09249599],
        [0.73111925, 0.09240055],
        [0.73482204, 0.09253416],
        [0.7385248, 0.09209515],
        [0.74222757, 0.09234329],
        [0.74593034, 0.09366032],
        [0.74963312, 0.09333583],
        [0.75333589, 0.09322131],
        [0.75703868, 0.09264868],
        [0.76074146, 0.09253416],
        [0.76444422, 0.09243873],
        [0.76814698, 0.09230512],
        [0.77184976, 0.09310678],
        [0.77555253, 0.09165615],
        [0.77925531, 0.09159888],
        [0.78295807, 0.09207606],
        [0.78666085, 0.09175158],
        [0.79036364, 0.09177067],
        [0.79406641, 0.09236237],
        [0.79776918, 0.09241964],
        [0.80147197, 0.09320222],
        [0.80517474, 0.09199972],
        [0.80887751, 0.09167523],
        [0.81258028, 0.09322131],
        [0.81628304, 0.09190428],
        [0.81998581, 0.09167523],
        [0.82368858, 0.09285865],
        [0.82739136, 0.09180884],
        [0.83109411, 0.09150345],
        [0.83479688, 0.09186611],
        [0.83849965, 0.0920188],
        [0.84220242, 0.09320222],
        [0.84590519, 0.09131257],
        [0.84960797, 0.09117896],
        [0.85331075, 0.09133166],
        [0.85701353, 0.09089265],
        [0.86071631, 0.09058725],
        [0.86441907, 0.09051091],
        [0.86812186, 0.09033912],
        [0.87182464, 0.09041547],
        [0.87552742, 0.0911217],
        [0.87923019, 0.0894611],
        [0.88293296, 0.08999555],
        [0.88663573, 0.08921297],
        [0.89033849, 0.08881213],
        [0.89404126, 0.08797229],
        [0.89774404, 0.08709427],
        [0.9014468, 0.08503284],
        [1.0, 0.07601531],
    ]
)

pos_ocp = np.array(
    [
        [0.24879728, 4.4],
        [0.26614516, 4.2935653],
        [0.26886763, 4.2768621],
        [0.27159011, 4.2647018],
        [0.27431258, 4.2540312],
        [0.27703505, 4.2449446],
        [0.27975753, 4.2364879],
        [0.28248, 4.2302647],
        [0.28520247, 4.2225528],
        [0.28792495, 4.2182574],
        [0.29064743, 4.213294],
        [0.29336992, 4.2090373],
        [0.29609239, 4.2051239],
        [0.29881487, 4.2012677],
        [0.30153735, 4.1981564],
        [0.30425983, 4.1955218],
        [0.30698231, 4.1931167],
        [0.30970478, 4.1889744],
        [0.31242725, 4.1881533],
        [0.31514973, 4.1865883],
        [0.3178722, 4.1850228],
        [0.32059466, 4.1832285],
        [0.32331714, 4.1808805],
        [0.32603962, 4.1805749],
        [0.32876209, 4.1789522],
        [0.33148456, 4.1768146],
        [0.33420703, 4.1768146],
        [0.3369295, 4.1752872],
        [0.33965197, 4.173111],
        [0.34237446, 4.1726718],
        [0.34509694, 4.1710877],
        [0.34781941, 4.1702285],
        [0.3505419, 4.168797],
        [0.35326438, 4.1669831],
        [0.35598685, 4.1655135],
        [0.35870932, 4.1634517],
        [0.3614318, 4.1598248],
        [0.36415428, 4.1571712],
        [0.36687674, 4.154079],
        [0.36959921, 4.1504135],
        [0.37232169, 4.1466532],
        [0.37504418, 4.1423388],
        [0.37776665, 4.1382346],
        [0.38048913, 4.1338248],
        [0.38321161, 4.1305799],
        [0.38593408, 4.1272392],
        [0.38865655, 4.1228104],
        [0.39137903, 4.1186109],
        [0.39410151, 4.114182],
        [0.39682398, 4.1096005],
        [0.39954645, 4.1046948],
        [0.40226892, 4.1004758],
        [0.4049914, 4.0956464],
        [0.40771387, 4.0909696],
        [0.41043634, 4.0864644],
        [0.41315882, 4.0818448],
        [0.41588129, 4.077683],
        [0.41860377, 4.0733309],
        [0.42132624, 4.0690737],
        [0.42404872, 4.0647216],
        [0.4267712, 4.0608654],
        [0.42949368, 4.0564747],
        [0.43221616, 4.0527525],
        [0.43493864, 4.0492401],
        [0.43766111, 4.0450211],
        [0.44038359, 4.041986],
        [0.44310607, 4.0384736],
        [0.44582856, 4.035171],
        [0.44855103, 4.0320406],
        [0.45127351, 4.0289288],
        [0.453996, 4.02597],
        [0.45671848, 4.0227437],
        [0.45944095, 4.0199757],
        [0.46216343, 4.0175133],
        [0.46488592, 4.0149746],
        [0.46760838, 4.0122066],
        [0.47033085, 4.009954],
        [0.47305333, 4.0075679],
        [0.47577581, 4.0050669],
        [0.47849828, 4.0023184],
        [0.48122074, 3.9995501],
        [0.48394321, 3.9969349],
        [0.48666569, 3.9926589],
        [0.48938816, 3.9889555],
        [0.49211064, 3.9834003],
        [0.4948331, 3.9783037],
        [0.49755557, 3.9755929],
        [0.50027804, 3.9707632],
        [0.50300052, 3.9681098],
        [0.50572298, 3.9635665],
        [0.50844545, 3.9594433],
        [0.51116792, 3.9556634],
        [0.51389038, 3.9521511],
        [0.51661284, 3.9479132],
        [0.51933531, 3.9438281],
        [0.52205777, 3.9400866],
        [0.52478024, 3.9362304],
        [0.52750271, 3.9314201],
        [0.53022518, 3.9283848],
        [0.53294765, 3.9242232],
        [0.53567012, 3.9192028],
        [0.53839258, 3.9166257],
        [0.54111506, 3.9117961],
        [0.54383753, 3.90815],
        [0.54656, 3.9038739],
        [0.54928247, 3.8995597],
        [0.55200494, 3.8959136],
        [0.5547274, 3.8909314],
        [0.55744986, 3.8872662],
        [0.56017233, 3.8831048],
        [0.5628948, 3.8793442],
        [0.56561729, 3.8747628],
        [0.56833976, 3.8702576],
        [0.57106222, 3.8666878],
        [0.57378469, 3.8623927],
        [0.57650716, 3.8581741],
        [0.57922963, 3.854146],
        [0.5819521, 3.8499846],
        [0.58467456, 3.8450022],
        [0.58739702, 3.8422534],
        [0.59011948, 3.8380919],
        [0.59284194, 3.8341596],
        [0.5955644, 3.8309333],
        [0.59828687, 3.8272109],
        [0.60100935, 3.823164],
        [0.60373182, 3.8192315],
        [0.60645429, 3.8159864],
        [0.60917677, 3.8123021],
        [0.61189925, 3.8090379],
        [0.61462172, 3.8071671],
        [0.61734419, 3.8040555],
        [0.62006666, 3.8013639],
        [0.62278914, 3.7970879],
        [0.62551162, 3.7953317],
        [0.62823408, 3.7920673],
        [0.63095656, 3.788383],
        [0.63367903, 3.7855389],
        [0.6364015, 3.7838206],
        [0.63912397, 3.78111],
        [0.64184645, 3.7794874],
        [0.64456893, 3.7769294],
        [0.6472914, 3.773608],
        [0.65001389, 3.7695992],
        [0.65273637, 3.7690265],
        [0.65545884, 3.7662776],
        [0.65818131, 3.7642922],
        [0.66090379, 3.7626889],
        [0.66362625, 3.7603791],
        [0.66634874, 3.7575538],
        [0.66907121, 3.7552056],
        [0.67179369, 3.7533159],
        [0.67451616, 3.7507198],
        [0.67723865, 3.7487535],
        [0.67996113, 3.7471499],
        [0.68268361, 3.7442865],
        [0.68540608, 3.7423012],
        [0.68812855, 3.7400677],
        [0.69085103, 3.7385788],
        [0.6935735, 3.7345319],
        [0.69629597, 3.7339211],
        [0.69901843, 3.7301605],
        [0.7017409, 3.7301033],
        [0.70446338, 3.7278316],
        [0.70718585, 3.7251589],
        [0.70990833, 3.723861],
        [0.71263081, 3.7215703],
        [0.71535328, 3.7191267],
        [0.71807574, 3.7172751],
        [0.72079822, 3.7157097],
        [0.72352069, 3.7130945],
        [0.72624317, 3.7099447],
        [0.72896564, 3.7071004],
        [0.7316881, 3.7045615],
        [0.73441057, 3.703588],
        [0.73713303, 3.70208],
        [0.73985551, 3.7002664],
        [0.74257799, 3.6972122],
        [0.74530047, 3.6952841],
        [0.74802293, 3.6929362],
        [0.7507454, 3.6898055],
        [0.75346787, 3.6890991],
        [0.75619034, 3.686522],
        [0.75891281, 3.6849759],
        [0.76163529, 3.6821697],
        [0.76435776, 3.6808143],
        [0.76708024, 3.6786573],
        [0.7698027, 3.6761947],
        [0.77252517, 3.674763],
        [0.77524765, 3.6712887],
        [0.77797012, 3.6697233],
        [0.78069258, 3.6678908],
        [0.78341506, 3.6652565],
        [0.78613753, 3.6630611],
        [0.78885999, 3.660274],
        [0.79158246, 3.6583652],
        [0.79430494, 3.6554828],
        [0.79702741, 3.6522949],
        [0.79974987, 3.6499848],
        [0.80247234, 3.6470451],
        [0.8051948, 3.6405547],
        [0.80791727, 3.6383405],
        [0.81063974, 3.635076],
        [0.81336221, 3.633549],
        [0.81608468, 3.6322317],
        [0.81880714, 3.6306856],
        [0.82152961, 3.6283948],
        [0.82425208, 3.6268487],
        [0.82697453, 3.6243098],
        [0.829697, 3.6223626],
        [0.83241946, 3.6193655],
        [0.83514192, 3.6177621],
        [0.83786439, 3.6158531],
        [0.84058684, 3.6128371],
        [0.84330931, 3.6118062],
        [0.84603177, 3.6094582],
        [0.84875424, 3.6072438],
        [0.8514767, 3.6049912],
        [0.85419916, 3.6030822],
        [0.85692162, 3.6012688],
        [0.85964409, 3.5995889],
        [0.86236656, 3.5976417],
        [0.86508902, 3.5951984],
        [0.86781149, 3.593843],
        [0.87053395, 3.5916286],
        [0.87325642, 3.5894907],
        [0.87597888, 3.587429],
        [0.87870135, 3.5852909],
        [0.88142383, 3.5834775],
        [0.8841463, 3.5817785],
        [0.88686877, 3.5801177],
        [0.88959124, 3.5778842],
        [0.89231371, 3.5763381],
        [0.8950362, 3.5737801],
        [0.89775868, 3.5721002],
        [0.90048116, 3.5702102],
        [0.90320364, 3.5684922],
        [0.90592613, 3.5672133],
        [1.0, 3.52302167],
    ]
)

from pybamm import exp, constants


def graphite_LGM50_electrolyte_exchange_current_density_Chen2020(
    c_e, c_s_surf, c_n_max, T
):
    m_ref = 6.48e-7  # (A/m2)(m3/mol)**1.5 - includes ref concentrations
    E_r = 35000
    arrhenius = exp(E_r / constants.R * (1 / 298.15 - 1 / T))

    return m_ref * arrhenius * c_e**0.5 * c_s_surf**0.5 * (c_n_max - c_s_surf) ** 0.5


def nmc_LGM50_electrolyte_exchange_current_density_Chen2020(c_e, c_s_surf, c_p_max, T):
    m_ref = 3.42e-6  # (A/m2)(m3/mol)**1.5 - includes ref concentrations
    E_r = 17800
    arrhenius = exp(E_r / constants.R * (1 / 298.15 - 1 / T))

    return m_ref * arrhenius * c_e**0.5 * c_s_surf**0.5 * (c_p_max - c_s_surf) ** 0.5


values = {
    "Negative electrode thickness [m]": 8.52e-05,
    "Separator thickness [m]": 1.2e-05,
    "Positive electrode thickness [m]": 7.56e-05,
    "Electrode height [m]": 0.065,
    "Electrode width [m]": 1.58,
    "Nominal cell capacity [A.h]": 5.0,
    "Typical current [A]": 5.0,
    "Current function [A]": 5.0,
    "Maximum concentration in negative electrode [mol.m-3]": 33133.0,
    "Negative particle diffusivity [m2.s-1]": 3.3e-14,
    "Negative electrode OCP [V]": ("graphite_LGM50_ocp_Chen2020", neg_ocp),
    "Negative electrode porosity": 0.25,
    "Negative electrode active material volume fraction": 0.75,
    "Negative particle radius [m]": 5.86e-06,
    "Negative electrode Bruggeman coefficient (electrolyte)": 1.5,
    "Negative electrode Bruggeman coefficient (electrode)": 1.5,
    "Negative electrode electrons in reaction": 1.0,
    "Negative electrode exchange-current density [A.m-2]": graphite_LGM50_electrolyte_exchange_current_density_Chen2020,
    "Negative electrode OCP entropic change [V.K-1]": 0.0,
    "Maximum concentration in positive electrode [mol.m-3]": 63104.0,
    "Positive particle diffusivity [m2.s-1]": 4e-15,
    "Positive electrode OCP [V]": ("nmc_LGM50_ocp_Chen2020", pos_ocp),
    "Positive electrode porosity": 0.335,
    "Positive electrode active material volume fraction": 0.665,
    "Positive particle radius [m]": 5.22e-06,
    "Positive electrode Bruggeman coefficient (electrolyte)": 1.5,
    "Positive electrode Bruggeman coefficient (electrode)": 1.5,
    "Positive electrode electrons in reaction": 1.0,
    "Positive electrode exchange-current density [A.m-2]": nmc_LGM50_electrolyte_exchange_current_density_Chen2020,
    "Positive electrode OCP entropic change [V.K-1]": 0.0,
    "Separator porosity": 0.47,
    "Separator Bruggeman coefficient (electrolyte)": 1.5,
    "Typical electrolyte concentration [mol.m-3]": 1000.0,
    "Reference temperature [K]": 298.15,
    "Ambient temperature [K]": 298.15,
    "Number of electrodes connected in parallel to make a cell": 1.0,
    "Number of cells connected in series to make a battery": 1.0,
    "Lower voltage cut-off [V]": 2.5,
    "Upper voltage cut-off [V]": 4.4,
    "Initial concentration in electrolyte [mol.m-3]": 1000,
    "Initial concentration in negative electrode [mol.m-3]": 29866.0,
    "Initial concentration in positive electrode [mol.m-3]": 17038.0,
    "Initial temperature [K]": 298.15,
}
param = pybamm.ParameterValues(values)
param

[62]:

{'Ambient temperature [K]': 298.15,
 'Boltzmann constant [J.K-1]': 1.380649e-23,
 'Current function [A]': 5.0,
 'Electrode height [m]': 0.065,
 'Electrode width [m]': 1.58,
 'Electron charge [C]': 1.602176634e-19,
 'Faraday constant [C.mol-1]': 96485.33212,
 'Ideal gas constant [J.K-1.mol-1]': 8.314462618,
 'Initial concentration in electrolyte [mol.m-3]': 1000,
 'Initial concentration in negative electrode [mol.m-3]': 29866.0,
 'Initial concentration in positive electrode [mol.m-3]': 17038.0,
 'Initial temperature [K]': 298.15,
 'Lower voltage cut-off [V]': 2.5,
 'Maximum concentration in negative electrode [mol.m-3]': 33133.0,
 'Maximum concentration in positive electrode [mol.m-3]': 63104.0,
 'Negative electrode Bruggeman coefficient (electrode)': 1.5,
 'Negative electrode Bruggeman coefficient (electrolyte)': 1.5,
 'Negative electrode OCP [V]': ('graphite_LGM50_ocp_Chen2020',
                                ([array([0.        , 0.03129623, 0.03499902, 0.0387018 , 0.04240458,
       0.04610736, 0.04981015, 0.05351292, 0.05721568, 0.06091845,
       0.06462122, 0.06832399, 0.07202675, 0.07572951, 0.07943227,
       0.08313503, 0.08683779, 0.09054054, 0.09424331, 0.09794607,
       0.10164883, 0.10535158, 0.10905434, 0.1127571 , 0.11645985,
       0.12016261, 0.12386536, 0.12756811, 0.13127086, 0.13497362,
       0.13867638, 0.14237913, 0.14608189, 0.14978465, 0.15348741,
       0.15719018, 0.16089294, 0.1645957 , 0.16829847, 0.17200122,
       0.17570399, 0.17940674, 0.1831095 , 0.18681229, 0.19051504,
       0.1942178 , 0.19792056, 0.20162334, 0.2053261 , 0.20902886,
       0.21273164, 0.2164344 , 0.22013716, 0.22383993, 0.2275427 ,
       0.23124547, 0.23494825, 0.23865101, 0.24235377, 0.24605653,
       0.2497593 , 0.25346208, 0.25716486, 0.26086762, 0.26457039,
       0.26827314, 0.2719759 , 0.27567867, 0.27938144, 0.28308421,
       0.28678698, 0.29048974, 0.29419251, 0.29789529, 0.30159806,
       0.30530083, 0.30900361, 0.31270637, 0.31640913, 0.32011189,
       0.32381466, 0.32751744, 0.33122021, 0.33492297, 0.33862575,
       0.34232853, 0.34603131, 0.34973408, 0.35343685, 0.35713963,
       0.36084241, 0.36454517, 0.36824795, 0.37195071, 0.37565348,
       0.37935626, 0.38305904, 0.38676182, 0.3904646 , 0.39416737,
       0.39787015, 0.40157291, 0.40527567, 0.40897844, 0.41268121,
       0.41638398, 0.42008676, 0.42378953, 0.4274923 , 0.43119506,
       0.43489784, 0.43860061, 0.44230338, 0.44600615, 0.44970893,
       0.45341168, 0.45711444, 0.46081719, 0.46451994, 0.46822269,
       0.47192545, 0.47562821, 0.47933098, 0.48303375, 0.48673651,
       0.49043926, 0.49414203, 0.49784482, 0.50154759, 0.50525036,
       0.50895311, 0.51265586, 0.51635861, 0.52006139, 0.52376415,
       0.52746692, 0.53116969, 0.53487245, 0.53857521, 0.54227797,
       0.54598074, 0.5496835 , 0.55338627, 0.55708902, 0.56079178,
       0.56449454, 0.5681973 , 0.57190006, 0.57560282, 0.57930558,
       0.58300835, 0.58671112, 0.59041389, 0.59411664, 0.59781941,
       0.60152218, 0.60522496, 0.60892772, 0.61263048, 0.61633325,
       0.62003603, 0.6237388 , 0.62744156, 0.63114433, 0.63484711,
       0.63854988, 0.64225265, 0.64595543, 0.64965823, 0.653361  ,
       0.65706377, 0.66076656, 0.66446934, 0.66817212, 0.67187489,
       0.67557767, 0.67928044, 0.68298322, 0.686686  , 0.69038878,
       0.69409156, 0.69779433, 0.70149709, 0.70519988, 0.70890264,
       0.7126054 , 0.71630818, 0.72001095, 0.72371371, 0.72741648,
       0.73111925, 0.73482204, 0.7385248 , 0.74222757, 0.74593034,
       0.74963312, 0.75333589, 0.75703868, 0.76074146, 0.76444422,
       0.76814698, 0.77184976, 0.77555253, 0.77925531, 0.78295807,
       0.78666085, 0.79036364, 0.79406641, 0.79776918, 0.80147197,
       0.80517474, 0.80887751, 0.81258028, 0.81628304, 0.81998581,
       0.82368858, 0.82739136, 0.83109411, 0.83479688, 0.83849965,
       0.84220242, 0.84590519, 0.84960797, 0.85331075, 0.85701353,
       0.86071631, 0.86441907, 0.86812186, 0.87182464, 0.87552742,
       0.87923019, 0.88293296, 0.88663573, 0.89033849, 0.89404126,
       0.89774404, 0.9014468 , 1.        ])],
                                 array([1.81772748, 1.0828807 , 0.99593794, 0.90023398, 0.79649431,
       0.73354429, 0.66664314, 0.64137149, 0.59813869, 0.5670836 ,
       0.54746181, 0.53068399, 0.51304734, 0.49394092, 0.47926274,
       0.46065259, 0.45992726, 0.43801501, 0.42438665, 0.41150269,
       0.40033659, 0.38957134, 0.37756538, 0.36292541, 0.34357086,
       0.3406314 , 0.32299468, 0.31379458, 0.30795386, 0.29207319,
       0.28697687, 0.27405477, 0.2670497 , 0.25857493, 0.25265783,
       0.24826777, 0.2414345 , 0.23362778, 0.22956218, 0.22370236,
       0.22181271, 0.22089651, 0.2194268 , 0.21830064, 0.21845333,
       0.21753715, 0.21719357, 0.21635373, 0.21667822, 0.21738444,
       0.21469313, 0.21541846, 0.21465495, 0.2135479 , 0.21392964,
       0.21074206, 0.20873788, 0.20465319, 0.20205732, 0.19774358,
       0.19444147, 0.19190285, 0.18850531, 0.18581399, 0.18327537,
       0.18157659, 0.17814088, 0.17529686, 0.1719375 , 0.16934161,
       0.16756649, 0.16609676, 0.16414985, 0.16260378, 0.16224113,
       0.160027  , 0.15827096, 0.1588054 , 0.15552238, 0.15580869,
       0.15220118, 0.1511132 , 0.14987253, 0.14874637, 0.14678037,
       0.14620776, 0.14555879, 0.14389819, 0.14359279, 0.14242846,
       0.14038612, 0.13882096, 0.13954628, 0.13946992, 0.13780934,
       0.13973714, 0.13698858, 0.13523254, 0.13441178, 0.1352898 ,
       0.13507985, 0.13647321, 0.13601512, 0.13435452, 0.1334765 ,
       0.1348317 , 0.13275118, 0.13286571, 0.13263667, 0.13456447,
       0.13471718, 0.13395369, 0.13448814, 0.1334765 , 0.13298023,
       0.13259849, 0.13338107, 0.13309476, 0.13275118, 0.13443087,
       0.13315202, 0.132713  , 0.1330184 , 0.13278936, 0.13225491,
       0.13317111, 0.13263667, 0.13187316, 0.13265574, 0.13250305,
       0.13324745, 0.13204496, 0.13242669, 0.13233127, 0.13198769,
       0.13254122, 0.13145325, 0.13298023, 0.13168229, 0.1313578 ,
       0.13235036, 0.13120511, 0.13089971, 0.13109058, 0.13082336,
       0.13011713, 0.129869  , 0.12992626, 0.12942998, 0.12796026,
       0.12862831, 0.12656689, 0.12734947, 0.12509716, 0.12110791,
       0.11839751, 0.11244226, 0.11307214, 0.1092165 , 0.10683058,
       0.10433014, 0.10530359, 0.10056993, 0.09950104, 0.09854668,
       0.09921473, 0.09541635, 0.09980643, 0.0986612 , 0.09560722,
       0.09755413, 0.09612258, 0.09430929, 0.09661885, 0.09366032,
       0.09522548, 0.09535909, 0.09316404, 0.09450016, 0.0930877 ,
       0.09343126, 0.0932404 , 0.09350762, 0.09339309, 0.09291591,
       0.09303043, 0.0926296 , 0.0932404 , 0.09261052, 0.09249599,
       0.09240055, 0.09253416, 0.09209515, 0.09234329, 0.09366032,
       0.09333583, 0.09322131, 0.09264868, 0.09253416, 0.09243873,
       0.09230512, 0.09310678, 0.09165615, 0.09159888, 0.09207606,
       0.09175158, 0.09177067, 0.09236237, 0.09241964, 0.09320222,
       0.09199972, 0.09167523, 0.09322131, 0.09190428, 0.09167523,
       0.09285865, 0.09180884, 0.09150345, 0.09186611, 0.0920188 ,
       0.09320222, 0.09131257, 0.09117896, 0.09133166, 0.09089265,
       0.09058725, 0.09051091, 0.09033912, 0.09041547, 0.0911217 ,
       0.0894611 , 0.08999555, 0.08921297, 0.08881213, 0.08797229,
       0.08709427, 0.08503284, 0.07601531]))),
 'Negative electrode OCP entropic change [V.K-1]': 0.0,
 'Negative electrode active material volume fraction': 0.75,
 'Negative electrode electrons in reaction': 1.0,
 'Negative electrode exchange-current density [A.m-2]': <function graphite_LGM50_electrolyte_exchange_current_density_Chen2020 at 0x7f3267dc7f60>,
 'Negative electrode porosity': 0.25,
 'Negative electrode thickness [m]': 8.52e-05,
 'Negative particle diffusivity [m2.s-1]': 3.3e-14,
 'Negative particle radius [m]': 5.86e-06,
 'Nominal cell capacity [A.h]': 5.0,
 'Number of cells connected in series to make a battery': 1.0,
 'Number of electrodes connected in parallel to make a cell': 1.0,
 'Positive electrode Bruggeman coefficient (electrode)': 1.5,
 'Positive electrode Bruggeman coefficient (electrolyte)': 1.5,
 'Positive electrode OCP [V]': ('nmc_LGM50_ocp_Chen2020',
                                ([array([0.24879728, 0.26614516, 0.26886763, 0.27159011, 0.27431258,
       0.27703505, 0.27975753, 0.28248   , 0.28520247, 0.28792495,
       0.29064743, 0.29336992, 0.29609239, 0.29881487, 0.30153735,
       0.30425983, 0.30698231, 0.30970478, 0.31242725, 0.31514973,
       0.3178722 , 0.32059466, 0.32331714, 0.32603962, 0.32876209,
       0.33148456, 0.33420703, 0.3369295 , 0.33965197, 0.34237446,
       0.34509694, 0.34781941, 0.3505419 , 0.35326438, 0.35598685,
       0.35870932, 0.3614318 , 0.36415428, 0.36687674, 0.36959921,
       0.37232169, 0.37504418, 0.37776665, 0.38048913, 0.38321161,
       0.38593408, 0.38865655, 0.39137903, 0.39410151, 0.39682398,
       0.39954645, 0.40226892, 0.4049914 , 0.40771387, 0.41043634,
       0.41315882, 0.41588129, 0.41860377, 0.42132624, 0.42404872,
       0.4267712 , 0.42949368, 0.43221616, 0.43493864, 0.43766111,
       0.44038359, 0.44310607, 0.44582856, 0.44855103, 0.45127351,
       0.453996  , 0.45671848, 0.45944095, 0.46216343, 0.46488592,
       0.46760838, 0.47033085, 0.47305333, 0.47577581, 0.47849828,
       0.48122074, 0.48394321, 0.48666569, 0.48938816, 0.49211064,
       0.4948331 , 0.49755557, 0.50027804, 0.50300052, 0.50572298,
       0.50844545, 0.51116792, 0.51389038, 0.51661284, 0.51933531,
       0.52205777, 0.52478024, 0.52750271, 0.53022518, 0.53294765,
       0.53567012, 0.53839258, 0.54111506, 0.54383753, 0.54656   ,
       0.54928247, 0.55200494, 0.5547274 , 0.55744986, 0.56017233,
       0.5628948 , 0.56561729, 0.56833976, 0.57106222, 0.57378469,
       0.57650716, 0.57922963, 0.5819521 , 0.58467456, 0.58739702,
       0.59011948, 0.59284194, 0.5955644 , 0.59828687, 0.60100935,
       0.60373182, 0.60645429, 0.60917677, 0.61189925, 0.61462172,
       0.61734419, 0.62006666, 0.62278914, 0.62551162, 0.62823408,
       0.63095656, 0.63367903, 0.6364015 , 0.63912397, 0.64184645,
       0.64456893, 0.6472914 , 0.65001389, 0.65273637, 0.65545884,
       0.65818131, 0.66090379, 0.66362625, 0.66634874, 0.66907121,
       0.67179369, 0.67451616, 0.67723865, 0.67996113, 0.68268361,
       0.68540608, 0.68812855, 0.69085103, 0.6935735 , 0.69629597,
       0.69901843, 0.7017409 , 0.70446338, 0.70718585, 0.70990833,
       0.71263081, 0.71535328, 0.71807574, 0.72079822, 0.72352069,
       0.72624317, 0.72896564, 0.7316881 , 0.73441057, 0.73713303,
       0.73985551, 0.74257799, 0.74530047, 0.74802293, 0.7507454 ,
       0.75346787, 0.75619034, 0.75891281, 0.76163529, 0.76435776,
       0.76708024, 0.7698027 , 0.77252517, 0.77524765, 0.77797012,
       0.78069258, 0.78341506, 0.78613753, 0.78885999, 0.79158246,
       0.79430494, 0.79702741, 0.79974987, 0.80247234, 0.8051948 ,
       0.80791727, 0.81063974, 0.81336221, 0.81608468, 0.81880714,
       0.82152961, 0.82425208, 0.82697453, 0.829697  , 0.83241946,
       0.83514192, 0.83786439, 0.84058684, 0.84330931, 0.84603177,
       0.84875424, 0.8514767 , 0.85419916, 0.85692162, 0.85964409,
       0.86236656, 0.86508902, 0.86781149, 0.87053395, 0.87325642,
       0.87597888, 0.87870135, 0.88142383, 0.8841463 , 0.88686877,
       0.88959124, 0.89231371, 0.8950362 , 0.89775868, 0.90048116,
       0.90320364, 0.90592613, 1.        ])],
                                 array([4.4       , 4.2935653 , 4.2768621 , 4.2647018 , 4.2540312 ,
       4.2449446 , 4.2364879 , 4.2302647 , 4.2225528 , 4.2182574 ,
       4.213294  , 4.2090373 , 4.2051239 , 4.2012677 , 4.1981564 ,
       4.1955218 , 4.1931167 , 4.1889744 , 4.1881533 , 4.1865883 ,
       4.1850228 , 4.1832285 , 4.1808805 , 4.1805749 , 4.1789522 ,
       4.1768146 , 4.1768146 , 4.1752872 , 4.173111  , 4.1726718 ,
       4.1710877 , 4.1702285 , 4.168797  , 4.1669831 , 4.1655135 ,
       4.1634517 , 4.1598248 , 4.1571712 , 4.154079  , 4.1504135 ,
       4.1466532 , 4.1423388 , 4.1382346 , 4.1338248 , 4.1305799 ,
       4.1272392 , 4.1228104 , 4.1186109 , 4.114182  , 4.1096005 ,
       4.1046948 , 4.1004758 , 4.0956464 , 4.0909696 , 4.0864644 ,
       4.0818448 , 4.077683  , 4.0733309 , 4.0690737 , 4.0647216 ,
       4.0608654 , 4.0564747 , 4.0527525 , 4.0492401 , 4.0450211 ,
       4.041986  , 4.0384736 , 4.035171  , 4.0320406 , 4.0289288 ,
       4.02597   , 4.0227437 , 4.0199757 , 4.0175133 , 4.0149746 ,
       4.0122066 , 4.009954  , 4.0075679 , 4.0050669 , 4.0023184 ,
       3.9995501 , 3.9969349 , 3.9926589 , 3.9889555 , 3.9834003 ,
       3.9783037 , 3.9755929 , 3.9707632 , 3.9681098 , 3.9635665 ,
       3.9594433 , 3.9556634 , 3.9521511 , 3.9479132 , 3.9438281 ,
       3.9400866 , 3.9362304 , 3.9314201 , 3.9283848 , 3.9242232 ,
       3.9192028 , 3.9166257 , 3.9117961 , 3.90815   , 3.9038739 ,
       3.8995597 , 3.8959136 , 3.8909314 , 3.8872662 , 3.8831048 ,
       3.8793442 , 3.8747628 , 3.8702576 , 3.8666878 , 3.8623927 ,
       3.8581741 , 3.854146  , 3.8499846 , 3.8450022 , 3.8422534 ,
       3.8380919 , 3.8341596 , 3.8309333 , 3.8272109 , 3.823164  ,
       3.8192315 , 3.8159864 , 3.8123021 , 3.8090379 , 3.8071671 ,
       3.8040555 , 3.8013639 , 3.7970879 , 3.7953317 , 3.7920673 ,
       3.788383  , 3.7855389 , 3.7838206 , 3.78111   , 3.7794874 ,
       3.7769294 , 3.773608  , 3.7695992 , 3.7690265 , 3.7662776 ,
       3.7642922 , 3.7626889 , 3.7603791 , 3.7575538 , 3.7552056 ,
       3.7533159 , 3.7507198 , 3.7487535 , 3.7471499 , 3.7442865 ,
       3.7423012 , 3.7400677 , 3.7385788 , 3.7345319 , 3.7339211 ,
       3.7301605 , 3.7301033 , 3.7278316 , 3.7251589 , 3.723861  ,
       3.7215703 , 3.7191267 , 3.7172751 , 3.7157097 , 3.7130945 ,
       3.7099447 , 3.7071004 , 3.7045615 , 3.703588  , 3.70208   ,
       3.7002664 , 3.6972122 , 3.6952841 , 3.6929362 , 3.6898055 ,
       3.6890991 , 3.686522  , 3.6849759 , 3.6821697 , 3.6808143 ,
       3.6786573 , 3.6761947 , 3.674763  , 3.6712887 , 3.6697233 ,
       3.6678908 , 3.6652565 , 3.6630611 , 3.660274  , 3.6583652 ,
       3.6554828 , 3.6522949 , 3.6499848 , 3.6470451 , 3.6405547 ,
       3.6383405 , 3.635076  , 3.633549  , 3.6322317 , 3.6306856 ,
       3.6283948 , 3.6268487 , 3.6243098 , 3.6223626 , 3.6193655 ,
       3.6177621 , 3.6158531 , 3.6128371 , 3.6118062 , 3.6094582 ,
       3.6072438 , 3.6049912 , 3.6030822 , 3.6012688 , 3.5995889 ,
       3.5976417 , 3.5951984 , 3.593843  , 3.5916286 , 3.5894907 ,
       3.587429  , 3.5852909 , 3.5834775 , 3.5817785 , 3.5801177 ,
       3.5778842 , 3.5763381 , 3.5737801 , 3.5721002 , 3.5702102 ,
       3.5684922 , 3.5672133 , 3.52302167]))),
 'Positive electrode OCP entropic change [V.K-1]': 0.0,
 'Positive electrode active material volume fraction': 0.665,
 'Positive electrode electrons in reaction': 1.0,
 'Positive electrode exchange-current density [A.m-2]': <function nmc_LGM50_electrolyte_exchange_current_density_Chen2020 at 0x7f3267dc6e80>,
 'Positive electrode porosity': 0.335,
 'Positive electrode thickness [m]': 7.56e-05,
 'Positive particle diffusivity [m2.s-1]': 4e-15,
 'Positive particle radius [m]': 5.22e-06,
 'Reference temperature [K]': 298.15,
 'Separator Bruggeman coefficient (electrolyte)': 1.5,
 'Separator porosity': 0.47,
 'Separator thickness [m]': 1.2e-05,
 'Typical current [A]': 5.0,
 'Typical electrolyte concentration [mol.m-3]': 1000.0,
 'Upper voltage cut-off [V]': 4.4}

Here we would have got the same result by doing

[63]:

param_same = pybamm.ParameterValues("Chen2020")
{k: v for k, v in param_same.items() if k in spm.get_parameter_info()}

[63]:

{'Ideal gas constant [J.K-1.mol-1]': 8.314462618,
 'Faraday constant [C.mol-1]': 96485.33212,
 'Negative electrode thickness [m]': 8.52e-05,
 'Separator thickness [m]': 1.2e-05,
 'Positive electrode thickness [m]': 7.56e-05,
 'Electrode height [m]': 0.065,
 'Electrode width [m]': 1.58,
 'Nominal cell capacity [A.h]': 5.0,
 'Current function [A]': 5.0,
 'Maximum concentration in negative electrode [mol.m-3]': 33133.0,
 'Negative particle diffusivity [m2.s-1]': 3.3e-14,
 'Negative electrode OCP [V]': <function pybamm.input.parameters.lithium_ion.Chen2020.graphite_LGM50_ocp_Chen2020(sto)>,
 'Negative electrode porosity': 0.25,
 'Negative electrode active material volume fraction': 0.75,
 'Negative particle radius [m]': 5.86e-06,
 'Negative electrode Bruggeman coefficient (electrolyte)': 1.5,
 'Negative electrode Bruggeman coefficient (electrode)': 0,
 'Negative electrode exchange-current density [A.m-2]': <function pybamm.input.parameters.lithium_ion.Chen2020.graphite_LGM50_electrolyte_exchange_current_density_Chen2020(c_e, c_s_surf, c_s_max, T)>,
 'Negative electrode OCP entropic change [V.K-1]': 0.0,
 'Maximum concentration in positive electrode [mol.m-3]': 63104.0,
 'Positive particle diffusivity [m2.s-1]': 4e-15,
 'Positive electrode OCP [V]': <function pybamm.input.parameters.lithium_ion.Chen2020.nmc_LGM50_ocp_Chen2020(sto)>,
 'Positive electrode porosity': 0.335,
 'Positive electrode active material volume fraction': 0.665,
 'Positive particle radius [m]': 5.22e-06,
 'Positive electrode Bruggeman coefficient (electrolyte)': 1.5,
 'Positive electrode Bruggeman coefficient (electrode)': 0,
 'Positive electrode exchange-current density [A.m-2]': <function pybamm.input.parameters.lithium_ion.Chen2020.nmc_LGM50_electrolyte_exchange_current_density_Chen2020(c_e, c_s_surf, c_s_max, T)>,
 'Positive electrode OCP entropic change [V.K-1]': 0.0,
 'Separator porosity': 0.47,
 'Separator Bruggeman coefficient (electrolyte)': 1.5,
 'Initial concentration in electrolyte [mol.m-3]': 1000.0,
 'Reference temperature [K]': 298.15,
 'Ambient temperature [K]': 298.15,
 'Number of electrodes connected in parallel to make a cell': 1.0,
 'Number of cells connected in series to make a battery': 1.0,
 'Lower voltage cut-off [V]': 2.5,
 'Upper voltage cut-off [V]': 4.2,
 'Initial concentration in negative electrode [mol.m-3]': 29866.0,
 'Initial concentration in positive electrode [mol.m-3]': 17038.0}

Updating a specific parameter#

Once a parameter set has been defined (either via a dictionary or a pre-built set), single parameters can be updated

Using a constant value:#

[64]:

param.search("Current function [A]")

param.update({"Current function [A]": 4.0})

param["Current function [A]"]

Current function [A]    5.0

[64]:

4.0

Using a function:#

[65]:

def curren_func(time):
    return 1 + pybamm.sin(2 * np.pi * time / 60)

param.update({"Current function [A]": curren_func})

param["Current function [A]"]

[65]:

<function __main__.curren_func(time)>

Plotting parameter functions#

As seen above, functions can be passed as parameter values. These parameter values can then be plotted by using pybamm.plot

Plotting “Current function [A]”#

[66]:

currentfunc = param["Current function [A]"]
time = pybamm.linspace(0, 120, 60)
evaluated = param.evaluate(currentfunc(time))
evaluated = pybamm.Array(evaluated)
pybamm.plot(time, evaluated)

../../../../_images/source_examples_notebooks_parameterization_parameterization_47_0.png

[66]:

<Axes: >

Taking another such example:

Plotting “Negative electrode exchange-current density [A.m-2]”#

[67]:

negative_electrode_exchange_current_density = param[
    "Negative electrode exchange-current density [A.m-2]"
]
x = pybamm.linspace(3000, 6000, 100)
c_n_max = param["Maximum concentration in negative electrode [mol.m-3]"]
evaluated = param.evaluate(
    negative_electrode_exchange_current_density(1000, x, c_n_max, 300)
)
evaluated = pybamm.Array(evaluated)
pybamm.plot(x, evaluated)

../../../../_images/source_examples_notebooks_parameterization_parameterization_49_0.png

[67]:

<Axes: >

Simulating and solving the model#

Finally we can simulate the model and solve it using pybamm.Simulation and solve respectively.

[68]:

sim = pybamm.Simulation(spm, parameter_values=param)
t_eval = np.arange(0, 3600, 1)
sim.solve(t_eval=t_eval)
sim.plot()

[68]:

<pybamm.plotting.quick_plot.QuickPlot at 0x7f3267badc50>

References#

The relevant papers for this notebook are:

[69]:

pybamm.print_citations()

[1] Joel A. E. Andersson, Joris Gillis, Greg Horn, James B. Rawlings, and Moritz Diehl. CasADi – A software framework for nonlinear optimization and optimal control. Mathematical Programming Computation, 11(1):1–36, 2019. doi:10.1007/s12532-018-0139-4.
[2] 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(8):080534, 2020. doi:10.1149/1945-7111/ab9050.
[3] Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, and others. Array programming with NumPy. Nature, 585(7825):357–362, 2020. doi:10.1038/s41586-020-2649-2.
[4] Scott G. Marquis, Valentin Sulzer, Robert Timms, Colin P. Please, and S. Jon Chapman. An asymptotic derivation of a single particle model with electrolyte. Journal of The Electrochemical Society, 166(15):A3693–A3706, 2019. doi:10.1149/2.0341915jes.
[5] Valentin Sulzer, Scott G. Marquis, Robert Timms, Martin Robinson, and S. Jon Chapman. Python Battery Mathematical Modelling (PyBaMM). Journal of Open Research Software, 9(1):14, 2021. doi:10.5334/jors.309.
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Parameterisation#

Adding your own parameter sets (using a dictionary)#

Setting up the model#

Finding the parameters required#

Adding the parameters#

Updating the parameter values#

Solving the model#

Finding the parameters in a model#

Using pre-defined models in PyBaMM#

Finding the parameters in a model#

Updating a specific parameter#

Using a constant value:#

Using a function:#

Plotting parameter functions#

Plotting “Current function [A]”#

Plotting “Negative electrode exchange-current density [A.m-2]”#

Simulating and solving the model#

References#

Using pre-defined models in `PyBaMM`#