Parameter maps

The electrical parameters of a battery are not constant throughout its operating range. As the State of Charge and temperature change, the battery exhibits different voltage characteristics, internal resistance, and polarization behaviour.

To accurately represent these variations, the identified parameters are organized into parameter maps. A parameter map expresses each model parameter as a function of one or more operating variables, allowing the battery model to adapt its behaviour to changing operating conditions.

State of Charge-dependent parameter maps

The electrical characteristics of a battery vary continuously with State of Charge. Consequently, each parameter of the second-order Thevenin model is represented as a function of State of Charge.

Temperature-dependent parameter maps

Battery characteristics are also influenced by temperature. Variations in temperature affect the battery's internal resistance, polarization behaviour, available capacity, and overall voltage response.

When characterization data are available at multiple temperatures, the parameters can be represented as functions of both State of Charge and temperature.

$$R_0=R_0(\mathrm{SoC},T)$$$$R_1=R_1(\mathrm{SoC},T)$$$$C_1=C_1(\mathrm{SoC},T)$$$$R_2=R_2(\mathrm{SoC},T)$$$$C_2=C_2(\mathrm{SoC},T)$$

These multidimensional parameter maps allow the battery model to account for changes in operating conditions and improve the accuracy of voltage prediction and State of Charge estimation.

Lookup tables

In practical battery models, parameter maps are commonly stored as lookup tables. During simulation or State of Charge estimation, the appropriate parameter values are obtained by interpolating between the measured operating points.

By using lookup tables, the battery model continuously updates its parameters according to the battery's operating condition while maintaining computational efficiency.

The completion of the parameter extraction process results in a fully identified second-order Thevenin Battery Model. The next chapter explains how this model is integrated with the Extended Kalman Filter to estimate the State of Charge of the battery in real time.