Battery Modeling with SIMBA

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This python script proposes an implementation of the Tremblay's model described in ¹.

Battery Model

The battery model considers following equations:

where:

• $R_{int}$ is the constant internal resistance,
• $OCV$ is the Open-Circuit Voltage,
• $R_{pol}$ a polarisation resistance.

Open-circuit Voltage ($OCV$) and polarisation resistance ($R_{pol}$) both depend on the State of Charge (or State Of Discharge). The OCV is given by:

Whereas $R_{pol}$ depends on the charge / discharge mode.

In discharge, it is given by:

In charge, the polarisation resistance increases until the battery is almost fully charged. It is given by:

where:

• $soc$ is the state of charge [0, 1],
• $sod$ is the state of discharge ($sod = 1 - soc$),
• $E_0$, $A$, $B$, $K$ are empirical parameters to be defined to fit experimental ,
• $soc_{limit}$ is a constant introduced here to allow the mathematical definition when $soc = 0$ and has been set to 0.005 here.

Extraction of Model Parameters

This step aims to determine the parameters of the battery voltage expression (written above) from a typical $(v, sod)$ discharge curve or for a typical set of several discharge curves for different current rates. It uses the curve_fit function (based on non-linear least squares) from Scipy.optimize module.

The principle is simple and an extract of the code is shown below:

• the function get_battery_voltage is defined to compute the battery voltage from model parameters $E_0$, $A$, $B$, $K$, $R_{int}$ and the battery current $i_{batt}$,
• the curve_fit function is then used to determine parameters to fit the experimental data, data_sod, data_current_rate and data_voltage.

def get_battery_voltage(XY, Eo, A, B, K, Rint):
    sod, current_rate = XY
    soc = 1 - sod
    soc_limit = 0.005
    ocv = Eo + A * np.exp(-B * (1 - soc)) - K * (1 / (soc + soc_limit) - 1)
    return ocv - Rint * current_rate - K / (soc + soc_limit) * current_rate

min_param = [2, 1e-9, 1e-9, 1e-9, 1e-9] # lower bounds for E0, A, B, K, Rint
max_param = [4.1, 1, 40, 1, 10] # upper bounds for E0, A, B, K, Rint

[Eo, A, B, K, Rint], _ = curve_fit(get_battery_voltage, (data_sod, data_current_rate), data_voltage, bounds=(min_param, max_param))

The figure below shows experimental data and model outputs where the parameters have been extracted to fit to a single discharge curve at a current rate of 0.5C:

As mentionned above, it is also possible to extract the model parameters to fit a surface which gathers all discharge curves, as shown below:

Implementation of the battery model

The figure below shows the proposed implementation in SIMBA using a C-code block.

The figure below shows the model results vs the battery data for the different discharge current rates.