Electrochemical Impedance Spectroscopy (EIS) Parameter Estimation

Simscape™ Battery™ includes objects and functions for performing parameter estimation of a fractional-order equivalent circuit model (FOECM) from battery electrochemical impedance spectroscopy (EIS) data. This functionality allows you to process and tabulate battery cell test data, fit or regress parameters for FOECMs, verify the accuracy of the fit, visualize parameters trends, and simulate the frequency response of your system.

This figure shows the general workflow to derive or estimate model parameters from EIS data.

Electrochemical Impedance Spectroscopy

Electrochemical impedance spectroscopy is a powerful and versatile technique for battery design and modeling. EIS provides detailed information about the internal processes and performance characteristics of batteries. These characteristics are crucial for optimizing battery designs and improving efficiency and longevity. You can also use EIS techniques to study fuel cells, redox flow batteries, and generic electrochemical devices.

EIS allows you to:

• Understand battery internal mechanisms — You measure the battery impedance over a wide range of frequencies to differentiate between various processes such as charge transfer reactions, mass transport, and ion diffusion.

• Identify performance issues — You can identify performance issues and degradation mechanisms within batteries. An increase in charge transfer resistance can indicate issues with the electrode-electrolyte interface. An increase in the Warburg impedance might suggest issues with the ion diffusion.

• Select and optimize materials — By analyzing how different materials affect the impedance spectrum, you can identify the materials that minimize the resistive losses and enhance ion transport.

• Estimate state of charge (SOC) and state of health (SOH) — EIS provides valuable information into the changes in battery impedance with varying SOC and during aging. You can then correlate these changes to the battery SOH.

• Simulate and validate electrochemical models — EIS data is instrumental in developing and validating electrochemical models of batteries. You can use these models to simulate the battery behavior under various conditions, which is essential for predicting the battery performance and lifetime.

The EIS technique characterizes the battery impedance by using an AC current signal input. The corresponding voltage response is logged and analyzed to obtain the impedance of the sample. The input signal usually comprises a low C-rate AC current excitation of zero mean. This AC current pulse is repeated at different frequencies, SOC values, temperatures, SOH values, and more.

The low C-rate allows the overpotential response to stay inside the linear region and does not cause any harmonics or distortions that complicate the analysis.

This Nyquist plot shows the output from an EIS test.

Select Circuit Topology

To ensure that your model is computationally efficient, captures the necessary dynamics, and matches the goals of your analysis or application, you must appropriately select the model circuit topology according to the test data you want to use. The topology determines the complexity of the model, balancing between computational efficiency and accuracy. More complex topologies capture more detailed behaviors but require you to estimate more parameters.

To estimate the parameters for a battery FOECM, use the EISModel object. The EISModel object creates an FOECM object for analyzing battery impedance data. Use this object when you perform EIS tests to collect impedance data at different frequencies.

Electrochemical models provide a detailed representation of the physical and chemical process inside the battery. These models can represent the charge and mass transfer, reaction kinetics, and thermodynamics. They are computationally intensive and are challenging to parameterize.

This table shows the circuit elements that the EISModel object supports.

Circuit Element	Icon	Identifier String	Impedance Value	Mechanism
Resistor		R	$$R$$	Ohmic resistance of the electrolyte, electrodes, and other conductors
Capacitor		C	$$\frac{1}{j\cdot w\cdot C}$$	Double layer at the electrode or electrolyte interface, for example battery or supercapacitor
Inductor		L	$$j\cdot w\cdot L$$	Inductive effects due to leads, conductors, and wirings in the measuring device
Constant Phase Element		CPE	$$\frac{1}{{\left(j\cdot w\right)}^n\text{Q}}$$	Accounts for nonideal capacitive behavior, often due to surface roughness, inhomogeneity, or porous electrodes.
Finite-Space Warburg		FSW	$$\frac{Z}{\sqrt{j\cdot w\cdot t}}\cdot \coth \left(\sqrt{j\cdot w\cdot t}\right)$$	Diffusion processes that occur inside a finite region, for example within a solid-state battery electrolyte
Semi-Infinite Warburg		SIW	$$\frac{\text{W}}{\sqrt{w}}\left(1-j\right)$$	Diffusion processes in an unbounded medium, such as diffusion of redox species in the bulk electrolyte of a redox flow battery
Finite-Length Warburg		FLW	$$\frac{Z}{\sqrt{j\cdot w\cdot t}}\cdot \tanh \left(\sqrt{j\cdot w\cdot t}\right)$$	Diffusion processes in a medium with a defined length, for example in a thin-layer cell

For more information about the EISModel object and its properties and functions, see eisModel.

Process Test Data

To import, view, process, and store data from EIS experimental techniques in Simscape Battery, use the EISTest object. Alternatively, you can decide to fit your EIS data directly.

The EISTest object allows you to automatically extract and analyze individual impedance profiles from EIS frequency-based data. The EISTest object automatically detects every individual EIS profile and tabulates the data only if the test frequencies are the same for every test conducted at different conditions, including SOC, temperature, remaining capacity, and more.

For more information about the object and its properties and functions, see eisTest.

Estimate Model Parameters

After you choose the equivalent circuit model topology that reflects your battery behavior, you use the collected EIS test data to fit the model. This process involves adjusting the model parameters to minimize the difference between the model predictions and the experimental data. To find the best-fit parameters, you use optimization algorithms. You then validate the model by comparing the predictions against additional experimental data or different operating conditions to ensure accuracy and reliability.

For EIS data, the process of parameter estimation involves minimizing the error between the predicted impedance output of the FOECM and the measured battery impedance data from the EIS test. To minimize the error, you iteratively adjust the model parameters, such as the resistances, capacitances, and order of fractional elements, until you achieve a satisfactory error between the measured and simulated impedance.

The fitEISModel function performs impedance parameter estimation for a battery FOECM from frequency-based EIS data. The function then stores these parameters inside an EISModel object that you can use to analyze or interpret battery or fuel cell impedance data.

The fitEISModel function provides multiple fitting methods:

• "fminsearch" — Default fitting method in Simscape Battery.

• "fmincon" — Requires Optimization Toolbox™.

• "lsqnonlin" — Requires Optimization Toolbox.

• "patternsearch" — Requires Global Optimization Toolbox.

Optimization methods heavily rely on initial conditions and the values of the model parameters can vary by many orders of magnitude. To improve the accuracy of the fit, you can get a recommendation of the initial parameters of the FOECM by using the estimateBatteryEISParameters function. This function calculates the parameters based on a priori fit of several lithium-ion batteries and provides an approximation of the order of magnitude of the parameters only. The calculation of the parameter initial estimate depends on the battery capacity, temperature, and remaining capacity. These initial parameter estimates serve as an approximated initial guess for optimization-based fitting strategies of EIS parameters. The values that this function returns are only an approximation and they are not the final parameter values or representative of any system. As a result, expect great deviations between the simulated and the actual physical behavior of a battery system. To ensure the requisite accuracy, validate the simulated behavior against experimental data and refine the parameter values and models.

The eisParameterEstimator function in Simscape Battery also provides you with an interface to manually tune the parameter values of an FOECM. You can also graphically verify the accuracy of the fit against the measured data.

This figure shows the measured and the simulated impedance response of a battery ECM whose parameters are estimated using the fitECM function.

For more information about the fitEISModel function and its arguments, see fitEISModel.

Simulate Frequency Response

For EIS data, after you estimate the parameters of an FOECM, you can use the parameters to simulate the frequency response of your system in MATLAB^® or to analyze the internal processes that occur inside your cell. You can also obtain and tabulate the FOECM parameters for different operating conditions, including temperature, SOC, and more. To simulate the frequency response of your system, use the simulateFrequencyResponse. This function simulates the FOECM in the frequency domain and returns the real and imaginary impedance.