EIS behaviour for different charge rates

[Akila1993]

Hello @brosaplanella @tinosulzer and Everyone,

I am trying to generate the EIS using the following code, the EIS profile is shifting to the right side when we increase the charge C rate from 0.3 to 1C or 3C (refer to the figure below). How we can explain this behavior in DFN model? Please feel free to comment on it.

Initial conditions and parameter update

import pybamm
import matplotlib.pyplot as plt
import pbeis
import numpy as np
import matplotlib.pyplot as plt
import time as timer
from scipy.fft import fft

parameter_values = pybamm.ParameterValues("Chen2020")

pp = pybamm.ParameterValues("OKane2022")

model = pybamm.lithium_ion.DFN(
{
"thermal": "lumped", #lumped thermal submodel
"surface form": "differential", #Adding capacitance to the model
#"SEI": "solvent-diffusion limited",
#"SEI porosity change": "true",
#"lithium plating": "partially reversible",
#"lithium plating porosity change": "true", # alias for "SEI porosity change"
#"particle mechanics":"swelling only",
#"SEI on cracks": "true",
#"loss of active material": "stress-driven",

}

)
var_pts = {
"x_n": 5, # negative electrode
"x_s": 5, # separator
"x_p": 5, # positive electrode
"r_n": 30, # negative particle
"r_p": 30, # positive particle
}

def get_chen2020_parameters():
parameter_values = pybamm.ParameterValues("Chen2020")
parameters = parameter_values
return parameters

def get_okane2022_degradation_parameters():

degradation_parameters = {
'Dead lithium decay constant [s-1]': 1e-06,
'Dead lithium decay rate [s-1]':pp["Dead lithium decay rate [s-1]"],
'Initial plated lithium concentration [mol.m-3]': 0.0,
'Lithium metal partial molar volume [m3.mol-1]': 1.3e-05,
'Lithium plating kinetic rate constant [m.s-1]': 1e-09,
'Lithium plating transfer coefficient': 0.65,
'Negative electrode LAM constant exponential term': 2.0,
'Negative electrode LAM constant proportional term [s-1]': 2.7778e-07,
"Negative electrode Paris' law constant b": 1.12,
"Negative electrode Paris' law constant m": 2.2,
"Negative electrode Poisson's ratio": 0.3,
"Negative electrode Young's modulus [Pa]": 15000000000.0,
'Negative electrode cracking rate':pp["Negative electrode cracking rate"],
'Negative electrode critical stress [Pa]': 60000000.0,
'Negative electrode initial crack length [m]': 2e-08,
'Negative electrode initial crack width [m]': 1.5e-08,
'Negative electrode number of cracks per unit area [m-2]': 3180000000000000.0,
'Negative electrode partial molar volume [m3.mol-1]': 3.1e-06,
'Negative electrode reference concentration for free of deformation [mol.m-3]': 0.0,
'Negative electrode volume change':pp["Negative electrode volume change"],
'Positive electrode LAM constant exponential term': 2.0,
'Positive electrode LAM constant proportional term [s-1]': 2.7778e-07,
'Positive electrode OCP entropic change [V.K-1]': 0.0,
"Positive electrode Paris' law constant b": 1.12,
"Positive electrode Paris' law constant m": 2.2,
"Positive electrode Poisson's ratio": 0.2,
"Positive electrode Young's modulus [Pa]": 375000000000.0,
'Positive electrode active material volume fraction': 0.665,
'Positive electrode cracking rate':pp["Positive electrode cracking rate"],
'Positive electrode critical stress [Pa]': 375000000.0,
'Positive electrode initial crack length [m]': 2e-08,
'Positive electrode initial crack width [m]': 1.5e-08,
'Positive electrode number of cracks per unit area [m-2]': 3180000000000000.0,
'Positive electrode partial molar volume [m3.mol-1]': 1.25e-05,
'Positive electrode reference concentration for free of deformation [mol.m-3]': 0.0,
'Positive electrode volume change':pp["Positive electrode volume change"],
'Typical plated lithium concentration [mol.m-3]': 1000.0,
'Exchange-current density for stripping [A.m-2]':pp["Exchange-current density for stripping [A.m-2]"],
'Exchange-current density for plating [A.m-2]':pp["Exchange-current density for plating [A.m-2]"],

}

return degradation_parameters

def update_parameters():
# Load Chen2020 parameters
chen2020_params = get_chen2020_parameters()

# Load OKane2022 degradation parameters
okane2022_degradation_params = get_okane2022_degradation_parameters()

# Update Chen2020 parameters with OKane2022 degradation parameters
chen2020_params.update(okane2022_degradation_params, check_already_exists=False)

return chen2020_params

updated_parameters = update_parameters()

Set up experiment

cycle_number = 10

exp = pybamm.Experiment(
["Hold at 4.2 V until C/100",
"Rest for 4 hours",
"Discharge at 0.1C until 2.5 V", # initial capacity check
"Charge at 0.3C until 4.2 V",
"Hold at 4.2 V until C/100",]

• [("Discharge at 1C until 2.5 V", # ageing cycles
  "Charge at 0.3C until 4.2 V",
  "Hold at 4.2 V until C/100",)] * cycle_number

)

start_time = timer.time()

sim = pybamm.Simulation(model, parameter_values=updated_parameters,experiment=exp,solver=pybamm.CasadiSolver(mode='safe without grid'),var_pts=var_pts)

solution = sim.solve()

Set initial conditions of aged_model from the solution and EIS Running

aged_model = model.set_initial_conditions_from(solution)

#frequency range
frequencies = np.logspace(-4, 2, 30)

#amplitude of the current for EIS
I = 1 * 1e-3

number_of_periods = 20
samples_per_period = 20

#Current function
def current_function(t):
#return I * pybamm.sin(2 * np.pi * F * t)
return I * pybamm.sin(2 * np.pi * pybamm.InputParameter("Frequency [Hz]") * t)

updated_parameters["Current function [A]"] = current_function

sim_aged = pybamm.Simulation(aged_model, parameter_values=updated_parameters,solver=pybamm.CasadiSolver(mode='safe without grid'),var_pts=var_pts)

impedances_time = []

for frequency in frequencies:
# Solve
period = 1 / frequency
dt = period / samples_per_period
t_eval = np.array(range(0, 1 + samples_per_period * number_of_periods)) * dt
sol = sim_aged.solve(t_eval, inputs={"Frequency [Hz]": frequency},initial_soc=0.9)
# Extract final two periods of the solution
time = sol["Time [s]"].entries[-3 * samples_per_period - 1 :]
current = sol["Current [A]"].entries[-3 * samples_per_period - 1 :]
voltage = sol["Terminal voltage [V]"].entries[-3 * samples_per_period - 1 :]
# FFT
current_fft = fft(current)
voltage_fft = fft(voltage)
# Get index of first harmonic
idx = np.argmax(np.abs(current_fft))
impedance = -voltage_fft[idx] / current_fft[idx]
impedances_time.append(impedance)`