Measurement of Non-Linear Impedances and Zero Free Parameter Modeling Approach for Predicting Battery Voltages
Authors: Uzundal Can Berk , Zabara Mohammed , Ulgut Burak 
Published: February 2020

Choosing the appropriate energy storage system for a given application is one of the most important and deciding issues that need to be tackled in designing electric powered equipment. In particular, for electric vehicles, a plethora of power consumption regimes are described by regulatory bodies. These power consumption profiles can be easily transformed to current requirements of the energy storage system which results in countless such current profiles. The problem is further complicated with the vast variety of available energy storage systems, that all need to be tested for a given application specified by the required current profile. Therefore, rather than testing every single combination, modelling approaches are utilized to predict various combinations. Models in the literature usually employ impedance measurements as their basis, either to obtain relevant parameters for a first principles model or to fit an equivalent circuit which is subsequently taken to the time domain.
In our modelling approach[1], the current profile to be modeled is converted to the frequency domain via a Fourier Transform. This frequency domain profile is multiplied by the impedance at each frequency, yielding a voltage response again in the frequency domain. This frequency domain voltage is then inverse Fourier transformed to obtain the time domain voltage response. After correcting for the state of charge with a simple DC discharge map, the voltage response of any electrochemical storage system can be obtained with errors less than 1%[1,2]. The developed modelling approach does not fit any models to the impedance data, thus has no free parameters.
Thus, modeling battery systems require the accurate measurement of impedance in both our method [1,2]and others. In order to acquire an accurate impedance spectrum, the system needs to be stable throughout the measurement and the sine wave applied needs to lead to a linear response.
The impedance spectrum measured can either have non-linearities due to the high amplitude used, or due to the chemistry of the system being irreversible. In either case, the measured impedance and/or the response of the battery would have contributions due to non-linearities.
EIS assumes linearity which is usually valid for small amplitudes (of both voltage and current) for energy storage systems. In real applications, however, larger amplitudes are required from these systems which is not easily modeled using equivalent circuit models. In our approach, however, the non-linearities can be accounted for by simply adding the relevant harmonics to the otherwise linear impedance data. We will show the applicability of our model for nonlinear systems using a supercapacitor under high current loads.
For the latter case, a primary chemistry, SOCl2 is investigated. In the literature the impedance of SOCl2 batteries is ill-characterized where the reports either have obvious non-linearities[3] or omit data [4]. If the following half reactions and the resulting total reaction is considered for the chemistry, a cell potential that is independent of any concentration is seen, which causes the cell voltage to stay constant at around 3.6V (minus a small decrease due to i.R) as shown in Figure a.

Li(s) --> Li+(dis.) + e- (Anode)

4Li+(dis.)+ 4e- + 2SOCl2 (l)--> 4LiCl(s) + SO2 (g)+ S(s) (Cathode)

Even a small perturbation such as the small amplitude voltage signal of a potentiostatic impedance measurement is not viable, where in the discharge region the cell potential drops below 3.6V only at full discharge and in the charge region, the reaction is not defined. Utilizing a galvanostatic measurement with a DC offset and dynamic adjustment of the measurement parameters, we will show Kramers-Kronig transformable impedance data for SOCl2 batteries as shown in Figure b.

References:

[1] E. Ozdemir, C. B. Uzundal, B. Ulgut, Zero-Free-Parameter Modeling Approach to Predict the Voltage of Batteries of Different Chemistries and Supercapacitors under Arbitrary Load, J. Electrochem. Soc. 164 (2017) 1274–1280. doi: 10.1149/2.1521706jes.
[2] B. Ulgut, C.B. Uzundal, E. Özdemir, Analysis of Errors in Zero-Free-Parameter Modeling Approach to Predict the Voltage of Electrochemical Energy Storage systems under Arbitrary Load, ECS Transactions 77 (2017) 99–104.
[3] M. Hughes, S.A.G.R. Karunathilaka, N.A. Hampson, T.J. Sinclair, The impedance of the lithium-thionyl chloride primary cell, J. Appl. Electrochem. 13 (1983) 669–678. doi:10.1007/BF00617825.
[4] B.N. Popov, Impedance Spectroscopy as a Nondestructive Health Interrogation Tool for Lithium-BCX Cells, J. Electrochem. Soc. 140 (1993) 3097. doi:10.1149/1.2220992.

Figure 1