In modern energy systems, the cell energy storage system plays a pivotal role in applications ranging from renewable energy integration to grid stabilization. Monitoring the health and performance of these systems is critical, as degradation can lead to efficiency losses or catastrophic failures. Electrochemical impedance spectroscopy (EIS) has emerged as a powerful non-invasive technique for probing the internal electrochemical parameters of a cell energy storage system. By applying small-amplitude sinusoidal signals across a frequency spectrum, EIS decomposes complex electrochemical processes into relaxation-time-based components, offering insights into state-of-charge (SOC), state-of-health (SOH), and aging mechanisms. However, traditional EIS methods, such as frequency sweeping, are time-consuming, often taking minutes to hours to cover a broad frequency range like 0.01 Hz to 1 kHz. This latency limits real-time diagnostics and online monitoring for cell energy storage systems. To address this, broadband excitation signals combined with fast Fourier transform (FFT) analysis have been proposed to accelerate EIS measurements. This article explores these methods in depth, focusing on their principles, implementation, and validation through simulations and experiments, with an emphasis on enhancing the efficiency of cell energy storage system assessments.
Traditional EIS measurements rely on the sweep-frequency method, where single-frequency sinusoidal signals are sequentially injected into the cell energy storage system. For each frequency point, the impedance is calculated from the ratio of voltage and current responses. While accurate, this approach requires multiple measurements, leading to extended durations. For instance, covering 0.01 Hz to 1 kHz with 46 points might take over 10 minutes, during which the cell’s state may drift, compromising data reliability. In contrast, broadband excitation methods inject a composite signal containing multiple frequencies simultaneously, reducing the number of injections and overall measurement time. This is particularly beneficial for cell energy storage systems in dynamic environments, where rapid state changes occur. The core idea involves synthesizing a wideband signal, applying it to the cell, and using FFT to decompose the response into individual frequency components, thereby extracting impedance spectra in a single shot. This not only saves time but also maintains precision, enabling near-real-time monitoring of cell energy storage system parameters.
The electrochemical behavior of a cell energy storage system can be modeled using equivalent circuit models (ECMs), which simplify complex processes into electrical components. A typical ECM includes elements like solution resistance ($R_s$), charge-transfer resistance ($R_{ct}$), double-layer capacitance ($C_{dl}$), and Warburg impedance for diffusion effects. These components correspond to different frequency ranges in the EIS spectrum. For example, high frequencies reflect fast processes like ionic conduction, while low frequencies relate to slow diffusion phenomena. The impedance $Z(f)$ of such a model at frequency $f$ is given by:
$$Z(f) = R_s + \frac{1}{\frac{1}{R_{ct}} + j\omega C_{dl}} + Z_W(\omega)$$
where $\omega = 2\pi f$, $j$ is the imaginary unit, and $Z_W$ represents the Warburg impedance. This formulation allows for simulation-based validation of broadband methods. In this study, we consider first-, second-, and third-order ECMs to represent various cell energy storage system complexities. The broadband excitation signal, whether a square wave or a synthesized multi-frequency wave, is applied to these models, and the response is analyzed to compute impedance. The use of FFT is crucial here, as it efficiently transforms time-domain signals into frequency-domain components. Compared to discrete Fourier transform (DFT), FFT reduces computational load significantly, with operation counts summarized in Table 1.
| Operation Type | DFT | FFT |
|---|---|---|
| Additions | $N \times (N – 1)$ | $N \times \log_2 N$ |
| Multiplications | $N^2$ | $(N \times \log_2 N) / 2$ |
Here, $N$ denotes the number of sampling points. For typical EIS measurements in a cell energy storage system, $N$ might be 10,000 or more, making FFT the preferred choice to handle large datasets without excessive resource consumption.
Two primary broadband excitation methods are investigated for cell energy storage system EIS: square wave excitation and equal-amplitude wideband synthesized excitation. The square wave method leverages the harmonic content of a periodic square signal. A square wave with period $T$ and amplitude $A$ can be expressed via Fourier series as:
$$f(t) = \frac{4A}{\pi} \sum_{n=1}^{\infty} \frac{1}{2n-1} \sin((2n-1)\omega t)$$
where $\omega = 2\pi/T$. This decomposition yields odd harmonics (1st, 3rd, 5th, etc.), each corresponding to a frequency point. For instance, a 0.01 Hz square wave provides components at 0.01, 0.03, 0.05, 0.07, and 0.09 Hz, covering five points per decade. By using square waves at different base frequencies (e.g., 0.01, 0.1, 1, 10, and 100 Hz), we can uniformly sample the 0.01 Hz to 1 kHz range. However, a challenge arises: higher harmonic amplitudes diminish as $1/(2n-1)$, leading to small signals at high frequencies that may be susceptible to noise in a cell energy storage system. To mitigate this, we adjust the excitation amplitude or employ signal averaging. In practice, for a cell energy storage system, we inject square wave currents or voltages, record the responses, and apply FFT to extract magnitude and phase at each harmonic. The impedance $Z$ at frequency $f_k$ is then calculated as:
$$Z(f_k) = \frac{V(f_k)}{I(f_k)}$$
where $V(f_k)$ and $I(f_k)$ are the FFT-derived voltage and current components, respectively. This method reduces measurement time by about 60% compared to sweep-frequency, as shown in simulations.
The second method, equal-amplitude wideband synthesized excitation, further compresses the measurement process. Here, we construct a signal that combines sine waves at all desired frequencies (e.g., 46 points from 0.01 Hz to 1 kHz) with equal amplitudes. The synthesized signal $s(t)$ is:
$$s(t) = \sum_{k=1}^{M} A \sin(2\pi f_k t + \phi_k)$$
where $M$ is the number of frequencies, $A$ is a constant amplitude, and $\phi_k$ are phase offsets (often set to zero for simplicity). This signal is injected into the cell energy storage system in a single burst, and the response is captured. FFT decomposition then separates each frequency component, allowing impedance calculation across the entire spectrum. Since all components have equal amplitude, signal-to-noise ratio is consistent, enhancing accuracy for high-frequency points. This approach can cut measurement time by over 78%, making it highly efficient for rapid diagnostics of cell energy storage systems. However, it requires precise signal generation and processing capabilities, which may pose implementation challenges in embedded systems.
To validate these methods, we conducted simulations using MATLAB-Simulink. Three ECMs—first-, second-, and third-order—were used to represent diverse cell energy storage system behaviors. For the square wave method, we applied 0.01, 0.1, 1, 10, and 100 Hz square waves to each model, with an amplitude of 10 V. The FFT was performed on both excitation and response signals to compute impedance at 26 frequency points. Sweep-frequency measurements served as a benchmark. Results showed strong agreement: for the first-order model, impedance magnitude errors were below 3%, and phase errors were under 10% at most points. Similar trends held for higher-order models, though some frequency points exhibited larger relative errors due to near-zero imaginary parts. Table 2 summarizes the time savings for each model.
| Equivalent Circuit Model | Sweep-Frequency Time (s) | Square Wave Excitation Time (s) | Equal-Amplitude Wideband Time (s) |
|---|---|---|---|
| First-Order Model | 387.59 | 149.61 | 114 |
| Second-Order Model | 386.59 | 150.81 | 111 |
| Third-Order Model | 382.59 | 155.11 | 110 |
The equal-amplitude wideband method demonstrated even better accuracy, with errors on the order of $10^{-3}$ to $10^{-11}$ for impedance parameters, confirming its reliability for cell energy storage system applications. The key advantage lies in its single-injection nature, which minimizes state drift during measurement.
Experimental validation was performed on lithium iron phosphate (LiFePO4) batteries, common in cell energy storage systems. A test platform included a battery cycler, a high-precision measurement system, and an electrochemical workstation for reference. The environment was controlled at 25°C to eliminate thermal effects. Square wave currents at 0.01 and 0.1 Hz were injected into four batteries at full SOC, and responses were recorded. FFT processing yielded impedance at five frequency points per decade (0.01–0.9 Hz). These results were compared to sweep-frequency measurements from the electrochemical workstation. The impedance real part ($Z’$) and imaginary part ($Z”$) showed consistent trends, with absolute errors ranging from -0.4 to 0.3 mΩ and relative errors within ±15% for most points. Some outliers occurred where $Z”$ approached zero, inflating relative error. This highlights the importance of signal conditioning in practical cell energy storage system setups. Additionally, we tested the square wave method across different SOC levels (e.g., 100%, 80%, 50%). As SOC decreased, impedance increased, aligning with expected electrochemical behavior—lower SOC slows reaction rates, raising internal resistance. This demonstrates the method’s capability to track dynamic states in a cell energy storage system.
Error analysis for both broadband methods reveals several factors influencing accuracy. For square wave excitation, the diminishing amplitude of higher harmonics can reduce signal-to-noise ratio, especially in noisy environments typical of industrial cell energy storage systems. This can be mitigated by increasing the base amplitude or using filtering techniques. The equal-amplitude method avoids this issue but requires careful design to prevent intermodulation distortion and aliasing. In both cases, FFT resolution is critical; sampling frequency must exceed twice the highest frequency (e.g., >2 kHz for 1 kHz content) to satisfy the Nyquist criterion. Moreover, windowing functions (e.g., Hanning window) can reduce spectral leakage, improving impedance calculation. For a cell energy storage system with non-linearities, small-signal approximation holds if excitation amplitudes are kept low (typically <5% of nominal voltage), ensuring linear response. Our simulations assumed ideal conditions, but real-world cell energy storage systems may exhibit hysteresis or temperature dependencies, necessitating calibration.
The application of broadband EIS methods extends beyond laboratory settings. In battery management systems (BMS) for cell energy storage systems, real-time impedance monitoring can enable early fault detection, such as lithium plating or electrolyte degradation. By integrating broadband excitation circuits with microcontrollers, onboard EIS can be implemented without significant overhead. This facilitates predictive maintenance and optimized charging strategies, prolonging the lifespan of cell energy storage systems. For example, rapid EIS measurements during idle periods could assess SOH, informing capacity forecasts. In grid-scale cell energy storage systems, where thousands of cells are connected, broadband methods could streamline periodic testing, reducing downtime and operational costs. However, challenges remain in scaling these techniques: signal injection must be safe (avoiding overcurrents), and processing algorithms need to be efficient for embedded deployment. Future work could explore adaptive broadband signals tailored to specific cell energy storage system chemistries or machine learning for impedance interpretation.
In conclusion, broadband excitation methods combined with FFT analysis offer a viable path to rapid electrochemical impedance spectroscopy for cell energy storage systems. The square wave and equal-amplitude wideband approaches significantly reduce measurement time—by 60% and over 78%, respectively—while maintaining acceptable accuracy. Simulations on equivalent circuit models and experiments on lithium batteries confirm their effectiveness. These methods enable near-real-time diagnostics, enhancing the monitoring capabilities of cell energy storage systems in diverse applications, from electric vehicles to renewable energy farms. As cell energy storage system technology evolves, incorporating such fast EIS techniques into management systems will be crucial for reliability and efficiency. Further refinement in signal processing and hardware integration will unlock broader adoption, making cell energy storage systems smarter and more resilient.