3 Battery Modeling

Many battery gauge algorithms use an equivalent circuit model for the battery to provide the desired information about the battery pack. The battery model used by the gauge algorithm is selected to manage a tradeoff between the accuracy to represent the battery voltage response to expected load currents and the computational complexity involved in estimation of the parameters. The gauge algorithm used by Impedance Track based gauges is optimized for applications with stable load currents. In these cases, there are long periods of time when the battery load does not vary significantly. A model that captures the low-frequency behavior of the battery is sufficient for battery gauging is shown in Figure 3-1. The nodes V_term and Gnd represent the positive voltage battery terminal and the battery terminal connected to ground. For a battery pack containing multiple cells or modules in series, only the terminal of the bottom battery in the stack is connected to the system ground. The resistor, R_S, represents the low-frequency battery resistance. The capacitor, C_S, represents toe charge storage of the battery. In a practical battery model, the R_S and C_S parameters depend on the battery conditions such as SoC and environmental conditions such as temperature.

The typical low-frequency battery model includes a resistor to model the effect of the load current on the battery voltage, and a variable capacitor to model the variation of the OCV of the battery as discharged. The OCV is the battery voltage after a long period of time without load current flowing into or out of the battery. Both the OCV and the low-frequency resistance of the battery are functions of the SoC, and the battery cell temperature.

The battery OCV is described as a function of the battery SoC, or equivalently, the DoD. The SoC is a ratio between the remaining charge in the battery and the total chemical charge storage capacity, or Qmax. The value of Qmax is the available charge that can be provided by the battery in the limit of very low discharge current. The OCV versus SOC curves for typical Li-Ion, NMC, and LFP battery chemistries do not shift significantly as the battery ages. Thus, the Qmax parameter captures the effect of battery aging for very low load currents. As the battery ages, the estimated Qmax value decreases, describing the loss of charge storage capacity.

The battery resistance captures the effect of large currents on the battery terminal voltage. For stable or slowly-varying load currents, the difference between the battery OCV and the measured battery voltage is proportional to the load current. The battery gauge maintains a resistance estimate of the battery and the temperature sensitivity as functions of SoC and DoD. For a stable load current, known temperature, and SoC, the battery model can predict the battery terminal voltage.

The initial parameters of the battery model are determined by characterizing a new battery. Since the OCV drifts a small amount versus SoC, this is treated as a fixed parameter of the battery model. The gauge tracks the Qmax and fixed temperature resistance parameters since these change significantly as the battery ages. For typical Li-Ion and related battery chemistries, the battery is considered at the end of useful life when the Qmax has declined to 80% of the original value.

The low-frequency resistance of the battery increases significantly with age, depending on the battery chemistry and usage as shown in Figure 3-2. Over 100 cycles, a typical value of the low-frequency resistance of the battery can increase by 60%, depending on the battery chemistry and usage patterns.

For stable load currents, modeling the battery impedance by a single resistor leads to a reasonable tradeoff between the computational complexity of estimating the model parameters and the accuracy of the gauge predictions of SoC and the remaining capacity of the battery.

The battery SoC in Impedance Track based gauges is estimated from either a voltage measurement or a current measurement, depending on the recent behavior of the load current. When the load current has been near zero for a sufficiently long period of time, the battery voltage matches the OCV. Since the OCV versus SoC curve is monotonic, the voltage measurement can be mapped to an estimate of SoC. When the battery is charging or discharging, the battery voltage does not map to the correct SOC value due to the effect of the battery impedance. To reduce the computational complexity of the gauging algorithm, the SoC is estimated using the ratio between the integrated current and the battery Qmax when current is flowing