Zero-Free-Parameter, Mathematically Simple and Accurate Modeling of Battery Response to Discharge Profiles
Authors: Ulgut Burak , Ozdemir Elif 
Published: February 2020

Battery models are necessary since the response of batteries need to be predicted for a large number of applications. In automotive, consumer electronics and other fields, the AC response needs to be modeled in addition to the DC response of the battery for a full understanding of the voltages that will be encountered throughout the current profile that discharges(and potentially charges) the battery.
Current battery models used for this prediction are either based on very tedious algebraic models that are hard to follow and implement [1], or they are based on oversimplified models that fail to predict the accurate detail of the response [2]. These models can predict the response of the battery either through complicated math that requires the knowledge of a large number of parameters a priori, or through a number of assumptions regarding those parameters.
Both ways of modeling require kinetic and thermodynamic parameters that describe the various interfaces within the battery. One way of obtaining these parameters is through impedance spectroscopy. Typically, impedance spectra are fit to an equivalent circuit model in order to obtain these parameters. In some cases, these parameters are then further used to calculate the thermodynamic and kinetic parameters, in others, simple models are used as the circuit components that make up the battery in order to employ a circuit model to solve for the response of the battery.
The models that are complicated enough to accurately represent the impedance data are detailed and involve transmission line models that are somewhere between hard to impossible to properly represent in time domain. This is why most of the models use much more simplified models which can simply be integrated to get to time domain responses, however, these models fail to accurately represent the detailed structure of the impedance spectrum.
We are going to represent a brand new modeling paradigm where impedance data is not fit to any models, simple or complicated. Instead, the response of the battery to the desired time domain charge/discharge profile is calculated in the frequency domain. The discharge profile is transformed into the frequency domain, then, the frequency domain current signal is multiplied by the impedance data measured at every frequency of relevance in order to get the voltage as a function of frequency. This frequency domain voltage response is then brought to the time domain with an inverse Fourier transform.
The DC response of the battery is simply added in after the inverse Fourier transform through the use of a state-of-charge vs. DC potential curve that is measured for the battery of interest.
There are a number of advantages in this approach compared to the literature models. First, there are no free parameters in this model. The response calculated does not make any assumptions, just uses the impedance data as measured. The kinetic and the thermodynamic properties of the battery are not obtained individually, instead, their implications are used through the measured impedance at different frequencies. Furthermore, the math used in this approach is simple, straightforward and computationally cheap.
We will demonstrate that through the use of this approach, the response of batteries to a variety of current profiles can be calculated. The only requirement is the measured impedance values at every frequency that is relevant to the measured profile. In the figure attached, we are showing the reponse of an LFP cell to two profiles that are relevant in automotive applications.
[1]Fan G; Pan K; Canova M; Marcicki J; Yang XG; JECS
163(5)A666-A676 (2016)
[2]Castano S; Gauchia L; Voncila E; Sanz J; Ener. Conv. And Man., 92,396-405(2015)

Figure 1