Passive, Fractional, Battery Equivalent-Circuit Model in Time and Frequency Domains Part 1: Linear Model

Existing time-domain fractional model simulations of batteries are either limited to short time sequences, frequently less than 100 s, if truly fractional or use low order RC-ladder fractional approximations to reduce computational burden. Here we present an entirely-passive, truly fractional, equivalent-circuit model of a battery. We rely on a Reimann-Louiville fractional order differintegral to account for long time-scales out to 12 days. An analytical solution is provided for the differintegral, subject to the constraint of piecewise constant current. We validate our model fitting against a multi-day sequence of measured time domain data and EIS measured to
$10~\mu $
Hz. The spectral content of the current waveform is identified as a crucial factor. The full evaluation of fractional elements leads to residual error of voltage waveforms that is amongst the best in the literature despite the model having only five parameters. In the time-domain, a root-mean-square error (RMSE) as low as 2.8 mV is achieved while maintaining a frequency-domain RMSE of 14 % from measured impedance values over a span of 6 decades. The use of time-weighted regression is shown to be important to the time-domain fit.