A novel approach to data analysis for lithium-ion batteries using higher-order spectral analysis

The usage of lithium-ion batteries has become widespread across various applications, such as electronic devices, electric vehicles, and integrated renewable energy systems. Hence, there is a pressing need to enhance analytical techniques to optimize their performance and prolong their operational lifespan. This study introduces an innovative methodology that utilizes higher-order spectral analysis (HOSA) for evaluating data collected from lithium-ion batteries. The objective is to capture the inherent nonlinear and non-stationary behaviors displayed by battery data. The proposed approach mainly uses Bispectrum and Bicoherence to establish hypothesis tests for nonlinearity and non-Gaussianity of data. Through a customized algorithm, the higher-order spectrum of these refined data is calculated, uncovering intricate characteristics that may go unnoticed by conventional analytical methods. These findings have a wide range of potential applications, including battery modeling, fault detection, state-of-charge estimation, state-of-health estimation and the enhancement of battery performance. Finally, an experiment is conducted to validate the effectiveness of the method.


Introduction
Lithium-ion batteries have gained prominence as the primary energy storage technology due to their high energy density, long cycle life and environmental friendliness.They have been extensively employed in diverse fields, including consumer electronics, electric vehicles and integrated renewable energy systems.The battery management system (BMS) plays a crucial role in effectively monitoring and controlling the battery's operational status, relying heavily on the accuracy of the battery model to ensure safe operation.State of charge (SOC), state of health (SOH), remaining useful life (RUL), and other parameters are managed by the BMS.However, modeling lithium-ion batteries is challenging due to their highly nonlinear capacity behavior and non-Gaussian uncertainty.Specifically, the relationship between battery voltage and SOC is nonlinear, particularly at high and low charge levels.It is imperative that the model accurately captures this nonlinearity [1].Furthermore, uncertainties stemming from manufacturing inconsistencies and battery aging follow non-Gaussian distributions with longer tails.Given that traditional linear Gaussian models based on a normal distribution may 2. A novel approach for detecting and quantifying nonlinearity and non-Gaussian characteristics in collected battery data Within the multifaceted and dynamic operational milieu of lithium-ion batteries, the harvested data embodies not only the fundamental electrochemical behaviors but also the fluctuations inherent in operational processes and external environmental conditions.The advent of Higher-Order Spectral Analysis (HOSA) heralds a paradigm shift in the domain of data processing.Traditional data processing techniques adeptly dissect linear and Gaussian data using rudimentary statistical measures, namely mean, median, and mode, as well as secondary statistical parameters such as variance, covariance, correlation coefficients, autocorrelation functions, and power spectral density.Nonetheless, these conventional methodologies exhibit limitations in discerning the subtle nonlinear characteristics that emerge within the data.The integration of advanced higher-order spectra, specifically through the utilization of Bispectral and Bicoherence analysis, provides a strategic edge [3].This novel method significantly enhances the capacity to detect and quantify the intricate nonlinear and non-Gaussian attributes that are fundamental to the data collected from the operation of lithium-ion batteries.

Bispectrum and Bicoherence
The Bispectrum of a discrete data ( ) (t x ) is the expected value of the product of its Fourier transform at two distinct frequencies 1 f and 2 f , and the conjugate at frequency 2 1 f f * .The Bispectrum is mathematically represented as [4]: Here, ζ| √ E denotes the expectation operation, and ) , ( f .This metric reveals the phase relationships among the three frequency components, particularly when the data exhibits nonlinear or non-Gaussian processes.Non-zero Bispectrum values signify phase coupling between the frequencies, indicative of higher-order statistical relationships within the data.The Bicoherence squared is a normalization of the Bispectrum, measuring the phase constancy between frequency components 1 f and 2 f , as well as their sum frequency.The equation for Bicoherence squared is: In this equation, represents the expected value of the power spectral density at frequency f .The Bicoherence values, ranging from 0 to 1, facilitate the quantification of phase consistency across varying frequency components, with values nearing 1 indicating robust phase coupling.

Detecting and quantifying nonlinearity and non-Gaussian characteristics Assuming
) (t x is an ergodic stationary time series, which can be modeled as a convolution of the system's impulse response ) (σ h with a sequence of independent, identically distributed white noise ) (t a , the relationship is given by: denote the mean, variance, and third central moment of the white noise, respectively.The power spectrum, ) ( f S , and Bispectrum, ) , ( 21 f f B , are obtained as follows: where is the frequency response of the system, which is calculated as: Thus, Equation ( 2) can be rented: By employing Welch's periodogram method to estimate Bicoherence [5][6], we replace the expectation operator by averaging across multiple data segments, assuming ergodicity, as shown in Equation (8).
(8) For a linear system, substituting Equations ( 4) and (5) into Equation (7) demonstrates that: From Equation (9), it is apparent that the squared Bicoherence remains invariant across bifrequencies for any linear data.In the case of Gaussian data ) (t x , the squared Bicoherence is zero given that 3 λ is zero.To establish whether the data is Gaussian and has been generated by a linear process, we need to ascertain if the squared Bicoherence is zero.Conversely, if the squared Bicoherence is a nonzero constant, it suggests a non-Gaussian but linear generation process.Bicoherence estimations at bifrequencies are independent.Therefore, for linear data, the squared Bicoherence at each frequency follows a central chi-squared ( 2 β ) distribution with two degrees of freedom [3].Utilizing this, hypothesis tests for Bicoherence at each frequency can be formulated: where ϑ represents the number of data segments utilized in the estimation of Bicoherence, and β distribution with two degrees of freedom and a noncentrality parameter of γ .By estimating the sample interquartile range D of the squared Bicoherence and comparing it with the theoretical interquartile range of a 2 β distribution with the same degrees of freedom and noncentrality parameter γ , we can determine the validity of the linearity hypothesis.If the empirically determined interquartile range, denoted as D ˆ, significantly deviates from the expected theoretical range, it provides strong evidence to challenge and potentially reject the underlying hypothesis of linearity.

Experiment implementation
The dataset used in this study comes from [7], which included data on commercially available 18650type NCA cells with a nominal capacity of 3500 mAh.The cells were tested at an ambient temperature of 45°C.For the cycling tests, each cell was charged at a constant current up to 4.2 V with a rate of 0.5 C, then switched to a constant voltage phase at 4.2 V until the charge current dropped to 0.05 C (1 C being equal to 3500 mAh).Discharge cycles were performed at a constant current of 3500 mA until the cell voltage reached the lower threshold of 2.65 V.
Voltage and current were the primary parameters monitored and recorded during the experiment.Figure 1 shows representative voltage and current profiles from the 1 st to the 3 rd cycle of a CY45-0.5/1NCA battery, illustrating the charging and discharging phases.The current-voltage phase diagram in Figure 2 can be obtained based on the provided values.This phenomenon, resembling a limit cycle, may stem from nonlinearity in internal battery impedance, battery chemical dynamics, temperature effects, or measurement instrument delays.The dataset exhibits left-skewed and leptokurtic characteristics, with a mean of 3.69, variance of 0.22, normalized skewness of -0.46, and normalized kurtosis of -1.08.This indicates an asymmetrical distribution that is flatter compared to a normal distribution.After analyzing the battery voltage data, we obtained the Bispectrum and evaluated the Bicoherence.We used a Fast Fourier Transform (FFT) length of 128 and a segment length of 8.Both the Bispectrum and the Bicoherence were calculated by using a Hanning window.Measuring the voltage in battery systems is commonly done to study the system's output and understand its process characteristics.As shown in Figures 3 to 6, the Bispectrum exhibits sharp peaks around (0.07,0.07) and symmetric locations, suggesting possible quadratic frequency coupling.The squared Bicoherence plot of the voltage data also showed sharp peaks, indicating phase correlation among the different voltage components.The sharp peaks may stem from charging-discharging transitions, rapid discharge voltage drops, and swift charge voltage rises.
In this case, the STAT is 144883.91,degrees of freedom is 170, PFA is 0, D ˆ is 533.39,γ is 1602.14, the theoretical D is 108.01,N = 61.We can reject the Gaussianity assumption since the PFA is small.We cannot accept the linearity hypothesis since the estimated interquartile range is much larger than the theoretical value.Therefore, the voltage data exhibit nonlinear and non-Gaussian characteristics.

Conclusions
This study explores the higher-order spectral analysis of lithium-ion battery data to quantify nonlinearity and non-Gaussianity under specific operating conditions.The goal is to uncover the nonlinear and non-stationary characteristics of battery operation.The effectiveness of this method is validated through experimental data.The paper primarily utilizes techniques like Bispectrum and Bicoherence for hypothesis testing of data nonlinearity and non-Gaussianity.Building upon this bic 2 (-0.33594,-0.33594)= 0.98461

Figure 1 .
Figure 1.Voltage and current profiles from the 1 st to the 3 rd cycle of a CY45-0.5/1NCA battery.

Figure 3 .
Figure 3. Contour plot of magnitude bispectum of the voltage data.

Figure 4 .
Figure 4.The 3-D plot of the magnitudebispectum of the voltage data.

Figure 5 .
Figure 5. Contour plot of magnitude squared Bicoherence of the voltage data.

Figure 6 .
Figure 6.The 3-D plot of the magnitude-Bicoherence of the voltage data.
N is the number of STATs.This allows for a statistical test to evaluate the congruence of the observed STAT with a central 2 β distribution, which is quantified as the probability of false alarm (PFA).This represents the risk of incorrectly attributing a nonzero Bispectrum to the data.A PFA below 0.05, for example, would accept a null Bispectrum, supporting the Gaussianity premise.Empirically, the estimated Bicoherence may vary.We can approximate its constant value by calculating the mean of the Bicoherence across the nonredundant domain, denoted as γ .The squared