In the realm of modern energy infrastructure, the battery energy storage system plays a pivotal role in enabling renewable energy integration, grid stabilization, and peak shaving. As these systems proliferate, accurate and timely health assessment of storage batteries becomes critical to ensure reliability, safety, and longevity. Electrochemical impedance spectroscopy (EIS) has emerged as a powerful non-invasive diagnostic tool that can reveal intricate electrochemical parameters within batteries, such as charge transfer resistance, diffusion processes, and double-layer capacitance. However, conventional EIS measurement techniques, typically based on frequency-sweeping methods, are notoriously time-consuming. They involve sequentially injecting sinusoidal signals at discrete frequencies across a broad spectrum (e.g., 0.01 Hz to 1 kHz), which can take minutes to hours. This prolonged duration not only limits real-time monitoring capabilities but also risks state changes in the battery during measurement, thereby compromising accuracy for dynamic applications like online diagnostics in battery energy storage systems. To address this, we explore broadband excitation signal methods coupled with fast Fourier transform (FFT) decomposition to achieve rapid EIS acquisition. This approach aims to significantly reduce measurement time while maintaining high fidelity, paving the way for enhanced health management in battery energy storage systems.
The core principle of our proposed method lies in injecting a composite broadband signal into the battery energy storage system, instead of multiple single-frequency tones. By exciting the battery with a signal containing multiple frequency components simultaneously, we can capture the impedance response across a wide band in a single measurement cycle. This is analogous to compressing the entire frequency sweep into a brief time window. The response signal, along with the excitation, is then processed using FFT to decompose them into their constituent frequency components. From these, the impedance at each frequency point is computed as the ratio of the voltage and current phasors. Mathematically, for a linear time-invariant system, the impedance \(Z(f)\) at frequency \(f\) is given by:
$$ Z(f) = \frac{V(f)}{I(f)} $$
where \(V(f)\) and \(I(f)\) are the Fourier transforms of the time-domain voltage and current signals, respectively. The FFT algorithm enables efficient computation of these transforms, with a complexity of \(O(N \log N)\) for \(N\) sample points, compared to the \(O(N^2)\) of discrete Fourier transform (DFT). This efficiency is crucial for handling the large datasets inherent in broadband measurements. To validate the method, we consider two types of broadband excitation signals: square wave signals and equal-amplitude synthesized signals. Each offers distinct advantages in terms of frequency coverage and signal-to-noise ratio, which we analyze in the context of battery energy storage system diagnostics.
First, the square wave excitation method leverages the harmonic richness of a periodic square wave. A square wave with fundamental frequency \(f_0\) can be expressed via its Fourier series expansion:
$$ f(t) = \frac{4A}{\pi} \sum_{n=1}^{\infty} \frac{1}{2n-1} \sin[2\pi (2n-1) f_0 t] $$
where \(A\) is the amplitude. This decomposition yields odd harmonics at frequencies \(f_0, 3f_0, 5f_0, \ldots\), providing multiple frequency points per excitation. For EIS measurement in a battery energy storage system, we select square waves with fundamental frequencies corresponding to different decades (e.g., 0.01 Hz, 0.1 Hz, 1 Hz, 10 Hz, 100 Hz). Each square wave injection captures five frequency points within that decade, thereby reducing the number of required excitations. For instance, a 0.01 Hz square wave (period 100 s) gives access to frequencies 0.01 Hz, 0.03 Hz, 0.05 Hz, 0.07 Hz, and 0.09 Hz after FFT processing. However, a challenge arises from the amplitude disparity among harmonics: higher harmonics have smaller amplitudes, which can affect measurement precision for low-impedance batteries. To mitigate this, we adjust the excitation amplitude or employ signal conditioning, ensuring reliable extraction of impedance data across the spectrum.
Second, the equal-amplitude broadband synthesized excitation method takes this concept further by constructing a single signal that contains all desired frequency components with uniform amplitude. This signal is designed to span the entire measurement range (e.g., 0.01 Hz to 1 kHz) with equal energy per frequency bin, optimizing the signal-to-noise ratio and minimizing distortion. The synthesized signal \(s(t)\) can be represented as:
$$ s(t) = \sum_{k=1}^{M} A_k \sin(2\pi f_k t + \phi_k) $$
where \(f_k\) are the target frequencies, \(A_k\) are equal amplitudes, and \(\phi_k\) are phase offsets that can be randomized to reduce peak-to-average power ratio. Injecting this composite signal into the battery energy storage system elicits a response that, after FFT decomposition, yields impedance values at all \(M\) frequencies simultaneously. This method requires only one excitation event, drastically cutting measurement time. The trade-off lies in the complexity of signal generation and processing, but with modern digital signal processors, it is feasible for embedded applications in battery management systems.
To model the electrochemical behavior of batteries, we employ equivalent circuit models (ECMs) that approximate the impedance characteristics. A typical ECM for a battery energy storage system includes elements such as ohmic resistance \(R_s\), charge transfer resistance \(R_{ct}\), double-layer capacitance \(C_{dl}\), Warburg diffusion impedance \(Z_w\), and inductance \(L_s\). For example, a first-order ECM might consist of \(R_s\) in series with a parallel \(R_{ct}-C_{dl}\) combination, while higher-order models add more RC branches to capture multiple time constants. The impedance of a first-order ECM is given by:
$$ Z(f) = R_s + \frac{R_{ct}}{1 + j2\pi f R_{ct} C_{dl}} $$
where \(j = \sqrt{-1}\). These models serve as benchmarks in simulations to evaluate the accuracy of our broadband measurement methods against traditional sweep-based EIS.
We conducted extensive simulations using MATLAB-Simulink to compare the performance of broadband excitation methods with conventional frequency sweeping. The simulation setup included battery ECMs of first, second, and third orders, with parameters tuned to mimic real lithium-ion batteries. For the square wave method, we injected square waves at fundamental frequencies 0.01 Hz, 0.1 Hz, 1 Hz, 10 Hz, and 100 Hz, each with an amplitude of 10 V. The response currents were sampled at 10 kHz to avoid aliasing, and FFT was applied to extract harmonic components. For the equal-amplitude synthesized method, we generated a signal containing 46 frequency points logarithmically spaced from 0.01 Hz to 1 kHz, with each component amplitude set to 0.1 V. The impedance results were compared to those obtained from sweep measurements, where sinusoidal signals at each frequency were injected sequentially.
The simulation results demonstrated significant time savings. For the square wave method, measurement time was reduced by over 60% compared to sweeping, while the equal-amplitude synthesized method achieved savings exceeding 78%. The accuracy was quantified through relative errors in impedance magnitude, phase, real part, and imaginary part. Tables 1 and 2 summarize the error statistics and time comparisons for different ECMs.
| Frequency Range | Avg. Impedance Magnitude Error | Avg. Phase Error | Max. Real Part Error | Max. Imaginary Part Error |
|---|---|---|---|---|
| 0.01-0.09 Hz | 1.2% | 3.5% | 2.8% | 4.1% |
| 0.1-0.9 Hz | 1.5% | 4.2% | 3.1% | 5.0% |
| 1-9 Hz | 1.8% | 5.0% | 3.5% | 6.2% |
| 10-90 Hz | 2.0% | 5.8% | 3.8% | 7.0% |
| 100-900 Hz | 2.3% | 6.5% | 4.2% | 8.1% |
| ECM Type | Sweep Method | Square Wave Method | Equal-Amplitude Synthesized Method |
|---|---|---|---|
| First-Order | 634.33 | 149.61 | 114.00 |
| Second-Order | 510.33 | 150.81 | 111.00 |
| Third-Order | 505.33 | 155.11 | 110.00 |
The errors were generally within acceptable limits for most frequencies, though some outliers occurred at higher harmonics due to diminishing signal amplitudes. For the equal-amplitude method, errors were even lower, often below \(10^{-3}\) in relative terms, thanks to uniform excitation energy. This highlights the potential of broadband techniques for high-precision EIS in battery energy storage systems.
Beyond simulations, we validated the methods experimentally using a dedicated test platform. The setup included commercial lithium iron phosphate (LFP) batteries, a battery cycler for excitation, a high-performance data acquisition system, and an electrochemical workstation for reference measurements. The batteries were conditioned at 25°C in a thermal chamber to minimize temperature effects. We focused on the square wave method for experimental simplicity, injecting current signals (to avoid voltage compliance issues) and measuring voltage responses. The excitation currents were square waves with fundamental frequencies of 0.01 Hz and 0.1 Hz, covering the low-frequency range (0.01-0.9 Hz) critical for diffusion process analysis in battery energy storage systems. The signals were sampled at 10 kHz, and FFT was performed offline using custom software. For comparison, we used the electrochemical workstation to perform sweep measurements at the same frequencies, with excitation amplitudes matched to the FFT-derived harmonic amplitudes.
The experimental results aligned well with simulations. Impedance spectra from broadband measurements closely matched those from sweep methods, with deviations primarily attributed to noise, cable impedance, and instrument limitations. For four LFP batteries at full state of charge (SOC), the impedance real part and imaginary part errors were typically within ±0.4 mΩ and ±15%, respectively. The higher relative errors for imaginary parts at some frequencies stemmed from near-zero values, which amplify percentage errors. Nevertheless, the trends across frequencies were consistent, affirming the method’s reliability. Additionally, we applied the square wave method to track impedance changes during discharge, measuring EIS at SOC levels of 100%, 80%, 60%, 40%, and 20%. As expected, impedance magnitudes increased with decreasing SOC, reflecting the degradation of electrochemical kinetics. This demonstrates the utility of broadband EIS for real-time SOC estimation and health monitoring in battery energy storage systems.
The advantages of broadband EIS measurement are multifaceted. Firstly, the drastic reduction in measurement time enables near-real-time diagnostics, which is essential for dynamic applications like frequency regulation in grid-tied battery energy storage systems. Secondly, by capturing the entire impedance spectrum in one go, the method minimizes the risk of state drift during measurement, ensuring more consistent data. Thirdly, the use of FFT for signal processing is computationally efficient and widely supported by embedded hardware, facilitating integration into battery management systems. However, challenges remain, such as the need for high-fidelity signal generation and susceptibility to nonlinearities in batteries. For instance, the battery energy storage system may exhibit nonlinear behavior at high excitation amplitudes, violating the linearity assumption of EIS. To address this, we keep excitation signals small (typically less than 5% of the battery’s voltage or current rating) and employ distortion analysis techniques.
From a signal processing perspective, the choice of excitation waveform impacts performance. The square wave method offers simplicity and good frequency coverage per decade, but its harmonic amplitude decay can limit accuracy at high frequencies. The equal-amplitude synthesized method overcomes this but requires more sophisticated hardware. A hybrid approach could optimize both aspects. Moreover, advanced algorithms like maximum entropy spectral estimation or wavelet transforms could further enhance frequency resolution, though they increase computational load. For battery energy storage systems, a balance must be struck between speed, accuracy, and resource constraints.
The implications for battery energy storage system management are profound. Rapid EIS can facilitate online fault detection, such as identifying lithium plating, electrolyte dry-out, or electrode degradation. By integrating broadband measurements into routine operation, operators can perform continuous health assessment without interrupting service. This is particularly valuable for large-scale battery energy storage systems in renewable farms, where downtime is costly. Furthermore, the method can accelerate battery testing during manufacturing and sorting during second-life applications. For example, retired batteries from electric vehicles can be quickly characterized for reuse in stationary battery energy storage systems, leveraging EIS to gauge remaining capacity and internal resistance.
To quantify the time savings, consider a typical EIS sweep from 0.01 Hz to 1 kHz with 10 points per decade, totaling 46 points. If each point requires 10 seconds for stabilization and measurement, the sweep takes 460 seconds. In contrast, a broadband measurement with a 100-second excitation (covering low frequencies) and FFT processing might complete in 120 seconds, saving over 70% time. This efficiency scales with the number of points, making broadband methods increasingly advantageous for detailed spectra. The mathematical foundation for this relies on the convolution theorem: the response to a sum of sinusoids is the sum of individual responses, provided the system is linear. For a battery energy storage system under small-signal conditions, this holds reasonably well.
In terms of implementation, we propose a system architecture for embedded broadband EIS. It comprises a digital signal generator (e.g., a DAC) to produce excitation signals, a current or voltage sensor for response acquisition, an ADC for sampling, and a microcontroller running FFT algorithms. The impedance results can be transmitted to a central monitor for analysis. Key design considerations include anti-aliasing filters, synchronization between excitation and response, and calibration to remove system artifacts. For battery energy storage systems with multiple cells, multiplexing can enable sequential measurement of individual cells, though parallel excitation might be explored for further speed gains.
Looking ahead, future work will focus on refining the excitation signals to improve noise immunity and extend the frequency range. Techniques like pseudo-random binary sequences (PRBS) or chirp signals offer alternative broadband excitations with flat power spectral densities. Additionally, machine learning algorithms could be trained on broadband EIS data to predict state of health (SOH) or state of charge (SOC) directly, bypassing explicit impedance modeling. This would marry the speed of broadband measurement with the predictive power of data-driven approaches, creating a robust health management framework for battery energy storage systems.
In conclusion, broadband excitation methods for electrochemical impedance spectroscopy represent a significant advancement in battery diagnostics. By leveraging composite signals and FFT processing, we have demonstrated reductions in measurement time of 60-78% compared to traditional sweeping, with maintained accuracy. These methods are particularly suited for battery energy storage systems, where rapid, non-invasive health assessment can enhance reliability and lifespan. As the demand for energy storage grows, such innovations will be crucial for optimizing performance and safety. We envision widespread adoption in next-generation battery management systems, enabling real-time monitoring and predictive maintenance across diverse applications.