Fractional modelling of rechargeable batteries

Abstract

To develop a compact battery model, many authors begin by measuring the impedance of a battery over a frequency range of interest. Most of the models in the literature are either Thevenin-style or Randles’ model consisting of one or two RC networks and sometimes a Warburg element. These models are usually based on frequency range that stretches to only 1 mHz. This explains why they require several parameters to accurately reproduce the measured impedance data. In most applications, a battery goes through a charge/discharge cycle daily or even longer. Therefore, it seems logical to measure the impedance of the cell at frequencies reciprocal of period of charge. This corresponds to approximately 11.6 µHZ or lower. The impedance data at lower frequencies shows that any rechargeable battery can be simply modelled with a constant phase element in series with a resistor. Based on this observation, an equivalent circuit model and a mathematical model were proposed in this study. Similar to Randles’ model presented in 1947, the proposed models do not contain a source or any purely reactive element. The models were then fitted to the measured impedance data of both lithium-ion and nickel-metal hydride cells, the linear region of the charge-voltage curve, and the transient recovery tail that results from a step-change in load current.