Efficient Extraction of Electrochemical Impedance Spectra from Physical Models
Authors: Zhu Huayang , Weddle Peter , Vincent Tyrone , Kee Robert J. 
Published: February 2020

Electrochemical impedance spectroscopy (EIS) is a highly valuable and widely practiced approach to assist the interpretation of electrochemical device performance. In the laboratory, the device (e.g., fuel cell or battery) is actuated with a small-amplitude harmonic signal (e.g., current) and the response (e.g., voltage or temperature) is measured. The complex impedance is represented in terms of amplitude and phase-shift differences between actuation and response as functions of actuation frequency. Most often, the measured impedance is interpreted with equivalent circuit models [1].
The present approach focuses on the use of physical models to derive and interpret the impedance spectra. Assuming that a transient physical model has been developed (e.g., protonic-ceramic fuel cell), the model could be exercised just as done in the laboratory with small-perturbation actuation [2]. This approach certainly works, but is computationally expensive, especially in the low-frequency ranges.
Compared to the direct harmonic perturbation of the physical models, deriving state-space models from the physical models provides a much more efficient alternative to evaluate impedance spectra. State-space models take the form of two vector equations: dx/dt = Ax + Bu; y = Cx + Du. In these equations, x is the state vector, u is the actuation vector, and y is the vector of observables. A particular system is identified in terms of the four matrices A, B, C, and D. Once the state-space matrices are established, the complex impedance can be evaluated directly from the state-space matrices as Z(s;p) = C(sI – A)-1 B + D, where s=j, p is a vector of physical parameters, and I is the identity matrix. The complex impedance can be represented in various forms, such as in Nyquist plots.
The present paper develops two alternative approaches to establish the space-state models. The first approach is based upon actuation with pseudo-random binary sequences (PRBS), seeking to identify locally linear state-space models [3-4]. The PRBS actuation is composed of small-amplitude step-change actuations of random durations, with the model-predicted responses being recorded. The relationships between the PRBS actuation and the predicted responses can be used to establish matrix parameters in a state-space model. As an alternative to PRBS identification of the state models, the physical model can be interpreted directly as a large-scale state-space model. In this case, the A, B, C, and D matrices can be evaluated by numerical differentiation of the physical model.
Compared to the physical models that may have thousands of states (local composition, temperature, electrostatic potential, etc.), state-space models are typically reduced to only tens of states. In addition to evaluating the complex impedance, the reduced-order state-space models can play valuable roles in real-time model-predictive control algorithms.
References

[1]. E. Barsoukov, and J.R. Macdonald. Impedance Spectroscopy: Theory, Experiment, and Applications. Wiley, 2005

[2]. Zhu, and R.J. Kee, Modeling electrochemical impedance spectra in SOFC button cells with internal methane reforming. J. Electrochem. Soc., 153:A1765-A1772, 2006.

[3]. M. Sanandaji, T.L. Vincent, A.M. Colclasure, and R.J. Kee, Modeling and control of tubular solid-oxide fuel cell systems: II. Nonlinear model reduction and model predictive control. J. Power Sources, 196:208-217, 2011.

[4]. J. Weddle, R.J. Kee, and T.L. Vincent, A stitching algorithm to identify wide-bandwidth electrochemical impedance spectra for Li-ion batteries using binary perturbations. J. Electrochem. Soc., 165:A1679-A1684, 2018.

Figure 1