Lithium-ion battery state of health and failure analysis with mixture weibull and equivalent circuit model

Summary

Existing methods for interpreting Electrochemical Impedance Spectroscopy data involve various models, which face significant challenges in parameterization and physical interpretation and fail to comprehensively reflect the electrochemical behavior within batteries. To address these issues, this study proposes a Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model to capture the non-ideal capacitive characteristics of electrode surfaces. Additionally, the study employs a Copula based Joint Mixture Weibull Model and multi-output Gaussian Process Regression to enhance the precision in capturing the distribution of battery electrochemical parameters and predict SoH curves. Experimental validation shows that the model used in this article has an average RMSE error of 8.5%, and the prediction of the SoH curve after the 100th cycle can achieve an average RMSE error of 9.2%. These findings provide insightful implications for understanding the electrochemical complexities and parameter interdependencies in the battery aging process, offering a robust framework for future research in battery diagnostics.

Graphical abstract

Highlights

• •Constructing the Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model
• •Using Copula to construct a joint mixture model for parameter variation capture
• •SM-LMC Multi-Output Gaussian Process Regression for SoH curve prediction
• •In-depth degradation mechanism analysis based on joint mixture Weibull model

Electrochemistry; Energy systems

Introduction

Lithium-ion batteries, as indispensable energy storage units in modern technology and industry, have been widely used in a variety of fields, such as electric vehicles and renewable energy storage due to their high energy density, long cycle life, and environmental friendliness. They have made significant contributions to promoting green energy transformation and achieving carbon emission reduction. With the rapid advancement of globalization, the increasing energy demand poses a heavy challenge to traditional petroleum energy, which is rapidly depleting. The burning of fossil fuels will release greenhouse gases into the atmosphere, exacerbating global climate change and its related adverse effects.¹ To this end, the world is turning to reducing greenhouse gas emissions, mitigating global warming, adapting to climate change and energy shortages, and vigorously developing sustainable and renewable energy utilization.² In this context, the application of lithium-ion batteries with excellent ecological and environmental attributes has achieved rapid development and become the next-stage energy solution.

With the increasing reliance on lithium batteries, people are paying more and more attention to their safety and reliability issues. Overuse of batteries may have an adverse effect on the health of the battery during the charge and discharge cycle, which in turn leads to a sudden drop in battery performance. In fact, it is inevitable that battery performance will naturally decay over time. Factors such as charge and discharge cycles, temperature fluctuations, and working voltage range will all lead to performance degradation, affecting the service life and application efficiency of the battery system. The degradation of lithium batteries is mainly caused by factors such as the growth of the solid electrolyte interface (SEI) and the precipitation of lithium dendrites,^3,4 which will affect the charge and discharge efficiency and life of the battery. However, even lithium-ion batteries made of the same material may exhibit different levels of degradation at the same state of health (SoH), such as lithium dendrite deposition or electrode cracking. The impact of future performance and degradation path largely depends on the type of degradation that occurs.^5,6 This heterogeneity means that there may be significant differences in the degradation state of each battery, including the extent of lithium dendrite deposition or electrode cracking. Due to the unpredictability of battery degradation, accurately assessing the degradation risk of lithium batteries is a major challenge, which puts forward many research needs for battery management systems (BMS), including the estimation of battery state values (such as SoH, SoC, RUL), and fault detection. Therefore, it is crucial to comprehensively assess the risks of lithium batteries, deeply understand and analyze their degradation mechanisms, and improve battery performance, extend service life, and prevent the drawbacks of failures.

Battery management systems (BMS) need to be able to accurately reflect the actual health status and remaining useful life of the battery. High accuracy can minimize the occurrence of misjudgments and provide a reliable basis for subsequent battery maintenance and replacement decisions. At the same time, the model needs to have good interpretability, such as understanding the impact of failure mechanisms on features. Figure 1 shows the classification of risk prediction research on lithium batteries in the literature. Currently, these technologies can be divided into two categories: data-driven model methods and physics-based model methods.⁷ Physics-based models provide intuitive understanding by simulating the internal processes of the battery, relying on battery models that capture the electrochemical dynamics of the battery internally. In contrast, data-driven models do not rely on any specific assumptions about the battery model. Instead, they extract patterns and trends from historical data by training to establish the relationship between input and output data.

Figure 1.

The classification of risk prediction research on lithium batteries in the literature

DEM: Distributed Element Model, TLM:Transmission Line Model, ECM: Equivalent Circuit Model, EM: Electrochemical Model, ANN: Artificial Neural Network, LSTM: Long Short-term Memory, DNN: Deep Neural Network, CNN: Convolutional Neural Network, SVM: Support Vector Machine, EL: Ensemble learning.

Data-driven models avoid the complexity of parameter fitting by utilizing statistical or machine learning models based on experimental data. These models can predict the output state under given specific input conditions. Typical data used in these methods include voltage, current, temperature, time, number of charge/discharge cycles, and measurement values related to the output, such as the SOC value. Moreover, in big data-driven models, online health state estimation of the battery can be realized, such as the convolutional long short-term memory-Bayesian neural network structure proposed in,⁸ which is used for online health state and degradation rate prediction to detect accelerated aging.

Physics-based models mainly include the distributed element model (DEM), transmission line model (TLM), equivalent circuit model (ECM), and electrochemical model (EM).^9,10,11 Each model has its own unique advantages and disadvantages, which determine their applicability in different battery analysis scenarios. DEM is able to handle the complex electrochemical processes of batteries in detail, providing a comprehensive perspective for understanding the internal behavior of batteries. However, this type of model faces significant challenges in parameterization and physical interpretation, especially when analyzing battery aging and degradation. Its complexity and high demand for specialized knowledge may limit its universality in battery state estimation and prediction applications.¹² TLM is mainly used to simulate the charge transfer and diffusion processes inside the battery, and are particularly suitable for studying the physicochemical dynamics inside the battery.¹³ The PNP equation can be re-constructed into TLM, and further simplified based on the principle of electroneutrality to directly obtain quantities that can be observed experimentally.¹⁴ However, the complexity of this type of model and its high sensitivity to parameter estimation may also lead to limitations in practical applications, especially in terms of deeply understanding the battery degradation mechanism. EM uses high-order differential equations to describe the electrochemical changes inside lithium-ion batteries, and realizes prediction and health management by identifying key electrochemical parameters.¹⁵

Among these models, ECM stands out due to its relatively simple structure and intuitive physical interpretation. ECM uses components such as resistors and capacitors to simulate the dynamic behavior of the battery, and the health state of the battery is evaluated by identifying the parameters in the equivalent circuit. This method has been proposed and applied in multiple studies.¹⁶ ECM focuses on the external characteristics of the battery, such as resistance and capacitance, making it relatively easy to parameterize and validate. This has led to its widespread application in battery management systems, especially in battery state monitoring and performance prediction. It simulates the electrochemical process of the battery by connecting circuit elements (such as resistors, capacitors, and so forth) in series or parallel. This model structure is simple, the parameters are easy to estimate, and it can intuitively reflect the internal behavior of the battery, such as ohmic polarization, charge transfer, and double layer effect. However, ECM also has some inherent limitations, such as the difficulty in fully capturing the complex electrochemical process inside the battery, and the possible ambiguity of physical meaning when considering battery aging and degradation. For example, the Rint model, which only includes a resistance component, has limited ability to simulate the battery voltage response.¹⁷ To overcome these limitations, researchers have proposed various improved ECM. H.He¹⁸ conducted a comparative study of the Rint model, the first-order RC model, and the second-order RC battery model under electric vehicle dynamic conditions through a combination of experiments and simulations. They found that the accuracy of the second-order RC model is higher than that of the first-order model and the Rint model. More complex models, such as the PNGV model, although perform well in terms of accuracy and simulating battery dynamics, are limited in practical engineering applications due to the complex parameter identification process.¹⁹

Table 1 shows the current advantages and disadvantages of these two methods. Physics-based provide theoretical guidance for the application of lithium-ion batteries and are of great significance. However, traditional physical model rely on simulating microscopic degradation mechanisms, such as solid electrolyte interface growth, lithium dendrite deposition, and active material loss. Although these methods provide physical insights, it can be overly complex to characterize and simulate each degradation mechanism. These methods usually require stable operating conditions, high-precision data acquisition, and the model building and calculation can be challenging,²⁰ which makes it difficult to apply them in engineering applications. In recent years, the popularity of research on data-driven models, especially the fact that data-driven algorithms provide more usability than physics-based model, has made them more popular among researchers. However, collecting a large amount of data is both time-consuming and expensive,²¹ especially for aging tests that may require a long time. Moreover, data-driven models usually do not consider the various mechanisms that lead to battery aging, thus limiting their effectiveness in comprehensively evaluating and predicting battery health status under actual use conditions.

Table 1.

Physics-based model vs. Data-driven model

Method	Advantages	Disadvantages
Physics-based model	High accuracy	High computational complexity
	Strong interpretability	Parameter sensitivity
	Extensibility
Data-driven model	Low computational complexity	Large data requirement
	Easy to implement	Poor interpretability
	Does not require extensive domain knowledge

Addressing the limitations of existing lithium-ion battery risk prediction methods, this study aims to develop a more accurate and flexible model for more in-depth analysis and prediction of battery performance. In this context, this article proposes an innovative approach that combines physical models with data-driven models. In terms of physical model, since this study focuses on battery degradation mechanism analysis, it requires a deep understanding of the changes in the internal state of the battery. ECM parameters are directly related to the internal characteristics of the battery and can be conveniently used for battery state monitoring and control in BMS, such as charge transfer impedance and capacitance decay. Compared with electrochemical models that usually involve complex mathematical equations and parameters, ECM has an intuitive structure and is easier to understand and apply, making it a more suitable choice. Therefore, this study proposes a temperature-controlled second-order R-CPE ECM based on the second-order RC-ECM. In terms of data-driven models, this article introduces Copula theory to establish a joint mixture Weibull distribution of battery performance parameters. By applying the probability density model to the parameters obtained from the physical model, these measurement results are correlated with the battery health state, thus establishing a robust statistical model. This integrated approach more accurately reflects the behavior of batteries in real-world use and provides insights into the key drivers of battery performance degradation and aging. The contributions of this article are as follows.

• (1)Constructing the Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model. To better simulate the non-ideal electrochemical behavior at the electrode interface and the temperature dependence of the resistance, this study transformed the existing second-order RC equivalent circuit model into a temperature-controlled second-order R-CPE equivalent circuit model. The differential evolution algorithm was employed to accurately fit the electrochemical impedance spectroscopy (EIS) data.
• (2)Constructing a Joint Mixture Weibull Model with Gumbel Copula for capturing the dynamic variation of model parameters under different SoH. To address the challenges of poor interpretability and insufficient accuracy of traditional statistical methods, this study proposes a Joint Mixture Weibull model based on Gumbel Copula to capture the variation characteristics of the parameters of the temperature-controlled second-order R-CPE equivalent circuit model under different battery SoH. Based on the analysis of the parameters of the joint mixture Weibull distribution model, an early risk prediction model is constructed.
• (3)Sparse Spectral Mixture-Linear Model Coregionalization (SM-LMC) Multi-Output Gaussian Process Regression for SoH curve prediction. To tackle the challenge of low SoH prediction accuracy when using external input features such as battery voltage and current, this study proposes an SoH curve prediction approach based on the parameters of a joint distribution model and the SM-LMC algorithm. Compared with time series prediction models such as LSTM and GRU, the proposed method demonstrates higher prediction accuracy and reliability.
• (4)In-depth degradation mechanism analysis based on joint mixture Weibull distribution model. This study successfully applied the joint mixture Weibull distribution model to analyze the electrochemical impedance spectroscopy (EIS) data of commercial lithium-ion batteries under different frequency conditions. The detailed degradation mechanism was analyzed, and the specific effects of the solid electrolyte interface (SEI) film and electrochemical reaction process on battery degradation were clarified.

The structure of the article is organized as follows: Section Results describes the design, implementation process, and results of the experiments. In Section Discussion - analysis of battery degradation mechanisms, the study conducts an in-depth analysis of battery degradation mechanisms, with a focus on the SoH as a key metric for assessing aging. Section Conclusions summarizes the main findings of the study, and Section Limitations of the study discusses the study’s limitations and potential directions for future research. Section STAR Methods details the methods used in our work, including the Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model and the Joint Mixture Weibull Distribution based on Gumbel Copula.

Results

The experiment in this article is designed to construct an ECM-Copula-based mixture distribution model, combining the Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model with the mixture distribution model based on Gumbel Copula. This setup aims to evaluate the model’s accuracy in risk prediction for batteries based on SoH and provides new insights into the understanding of battery degradation mechanisms. The specific design process and steps of the experiment are detailed in Figure 2. The construction and parameterization process of the Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model are thoroughly discussed in Section Model fitting of electrochemical impedance spectroscopy data. In Section Copula function analysis and model fitting of the joint mixture Weibull distribution, we describe in detail the process of building the joint mixture Weibull distribution model using the Gumbel Copula function.

Figure 2.

Experimental flow chart for risk prediction and failure mode analysis of lithium-ion batteries based on the joint mixture Weibull model

Experiment dataset

The dataset selected for this study comprises 12 commercially available 45 mAh Eunicell LR2032 lithium-ion coin cells,²² which were cycled under multi-stage constant current charge and discharge conditions, with currents randomly varying between cycles to simulate real-world usage patterns. The chemical composition of these batteries is LiCoO2/graphite. The batteries were cycled in climate boxes set to 25°C, 35°C, and 45°C. Each cycle included a constant current-constant voltage (CC-CV) charge at a 1C rate (45mA) to 4.2V, followed by a constant current discharge at a 2C rate (90 mA) to 3V. The dataset includes EIS measurement data over a wide frequency range from 0.02 Hz to 20 kHz, using an excitation current of 5 mA.

Figure 3 shows the SoH changes for eight batteries from 25C01 to 25C08 under the same charge and discharge cycles. Although the SoH of all batteries decreased with the number of cycles, some batteries were able to maintain a higher SoH for a longer period before starting to degrade rapidly. It is evident that battery 08 maintained a higher SoH for most cycles before its rapid decline, indicating a late but fast degradation. Batteries 04 and 07 possibly indicate early failure or manufacturing defects, causing these batteries to either degrade faster initially or after a certain number of cycles. Battery 01 showed the best performance among all, as it maintained the highest SoH in most cycles. The traces of battery numbers 02, 03, 05, and 06 displayed typical medium performance, with a relatively stable SoH before gradually declining. Therefore, this article classifies the batteries labeled as 04, 07, and 08 as outlier data and selects the data from batteries 01, 02, 03, 05, and 06 under the 25°C cycle to build and validate the proposed model.

Figure 3.

The State of Health (SoH) changes of the eight batteries labeled 25C01 to 25C08 under the same charge and discharge cycle conditions

Frequency region analysis in electrochemical impedance spectroscopy nyquist diagram

Based on the EIS data measured from the frequency range of 0.02 Hz–20,000 Hz, we plotted the Nyquist diagram of lithium-ion batteries, as shown in Figure 4. In the diagram, several distinct semicircle features can be observed, each representing the characteristic response of different electrochemical processes within the battery. The semicircular shapes in different frequency regions reveal various dynamic processes inside the battery.

• (1)High-frequency region ($>$1 kHz): This region is indicated by a vertical line segment on the far right side of the impedance plot, representing the ohmic resistance (Rs) and inductive effects (L). It involves the resistance of the electrolyte, current collectors, electrodes, connectors, and inductance. The high-frequency region is often used to determine the internal resistance of the battery.
• (2)Mid-frequency region (10 Hz to 1 kHz): This region is characterized by a distinct semicircular structure in the impedance plot. The semicircle in the mid-frequency region represents the effects of the SEI layer, a crucial factor in lithium-ion batteries. The R-CPE unit in the model effectively represents the resistive and capacitive characteristics of the SEI layer, impacting the long-term stability and performance of the battery.
• (3)Low-frequency region ($<$10 Hz): This region appears as an extended part in the lower-left corner of the impedance plot. The semicircle at low frequencies indicates the charge transfer processes, which describe the rate of electrochemical reactions and diffusion processes at the electrode surface. The R-CPE unit in the model is used to simulate these kinetics, capturing the dynamics at the electrode-electrolyte interface with greater accuracy.

Figure 4.

The impedance curves from the experimental measurement EIS data within the frequency range of 0.02Hz–20,000Hz

Model fitting of electrochemical impedance spectroscopy data

In our study, we have performed precise fitting of the EIS data by synergistically utilizing the Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model and the Differential Evolution algorithm. This approach has provided innovative analysis of the impedance characteristics of lithium-ion batteries and their relationship with SoH. As shown in Figure 5, we compared the results of the model fitting with the experimentally measured EIS data over a frequency range from 0.02 Hz to 20,000 Hz. The observation results indicate that the ECM optimized by the Differential Evolution algorithm is highly consistent with the experimental data, confirming the accuracy and reliability of our proposed “Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model” in describing the behavior of lithium-ion batteries. Moreover, to comprehensively evaluate the quality of the model fitting, we used the Root-Mean-Square Error (RMSE) and the coefficient of determination ($R^2$) as the evaluation metrics, with their respective formulas shown as Equations 1 and 2. Table 2 presents the RMSE and R² evaluations of the ECM fitting results, demonstrating the model’s good capability in predicting the electrochemical behavior of batteries when dealing with complex electrochemical impedance.

$$\begin{gathered} \text{RMSE}=\frac{{\sum }_{i=1}^{cucle}\left(\sqrt{\frac{1}{n}{\sum }_{i=1}^n{\left(Z_{\exp ,i}-Z_{ \\ \text{model},i}\right)}^2}\right)}{cycle} \end{gathered}$$ (Equation 1)

$$\begin{gathered} R^2=\frac{{\sum }_{i=1}^{cucle}\left(1-\frac{{\sum }_{i=1}^n{\left(Z_{\exp ,i}-Z_{ \\ \text{model},i}\right)}^2}{{\sum }_{i=1}^n{\left(Z_{\exp ,i}-{\overline{Z}}_{\exp }\right)}^2}\right)}{cycle} \end{gathered}$$ (Equation 2)

Figure 5.

The impedance of battery serial number 25C01 measured and fitted under different aging conditions

Table 2.

ECM fitting results RMSE and $R^2$ evaluation

Battery Serial	RMSE	$R^2$
25C01	0.077	0.976
25C02	0.071	0.986
25C03	0.093	0.968
25C04	0.069	0.988
25C05	0.073	0.986
25C06	0.084	0.919
25C07	0.094	0.978
25C08	0.120	0.973

The results of the parameter fitting using the Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model are presented in Figure 6, displaying the variation curves of parameters $R_{anode}$, $E{a}_{anode}$, $R_{cathode}$, $E{a}_{cathode}$, $Q_{anode}$, $Q_{cathode}$, ${\alpha }_{anode}$, and ${\alpha }_{cathode}$ across the charge-discharge cycles. The initial gradual increase in $Q_{anode}$ suggests a growing non-ideal capacitive behavior of the battery, which after a certain number of cycles, exhibits a steeper rise indicating an acceleration of the degradation processes. $Q_{cathode}$ shows a similar trend to $Q_{anode}$. ${\alpha }_{anode}$ increases with the cycle count, reflecting an increase in the complexity and heterogeneity of the electrochemical processes at the electrode-electrolyte interface, potentially linked to changes in the SEI layer’s properties and the emergence of more complex charge transfer mechanisms. Conversely, ${\alpha }_{cathode}$ gradually decreases throughout the cycles, possibly indicating a shift in electrochemical dynamics toward less ideal behavior. This decrease could be attributed to the degradation of electrode materials or the cumulative effects of cycle-induced stresses. $R_{anode}$ parameter initially shows minor fluctuations followed by stabilization, reflecting slight changes in the internal resistance of the battery with the charge-discharge cycles. The stability of $R_{anode}$ indicates the reliability of the battery in the initial to mid-use phases. $R_{cathode}$ gradually increases with the cycles, indicating an increase in internal resistance, potentially due to the thickening of the SEI layer, aging of electrode materials, or other degradation phenomena. $E{a}_{anode}$ and $E{a}_{cathode}$ parameters exhibit complex and fluctuating upward trends, particularly under different charge-discharge conditions.

Figure 6.

Variation of the Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model parameters with cycle index

(A–H) represent the variation curves of R_anode, R_cathode, Ea_anode, Ea_cathode, Q_anode, Q_cathode, α_anode, α_cathode respectively.

Copula function analysis and model fitting of the joint mixture Weibull distribution

To gain a deeper understanding of the dependencies between battery parameters, we performed goodness-of-fit tests for the joint distributions of different Copula functions at 90% SoH, as shown in Table 3, including Kolmogorov–Smirnov (KS) test and Cramér-von Mises (CvM) test. Under the same conditions, the Gumbel–Hougaard (G-H) Copula demonstrated the best fitting performance. Figure 7 presents the probability-probability plot (PP-plot) for the best-performing G-H Copula, where the displayed good consistency further confirms the superiority of the G-H Copula in modeling ECM parameters with tail dependence. Such tail dependency is crucial for understanding the behavior of batteries nearing full charge state, especially in assessing the long-term reliability of batteries and predicting imminent performance declines. In summary, we have chosen the Gumbel Copula function to model the dependencies of ECM parameters. The Gumbel Copula offers multiple advantages when modeling battery ECM parameters, particularly in handling extreme dependencies and outliers. It is well-suited for capturing upper tail dependencies between variables, which is crucial for battery performance analysis, as the decay of key performance often manifests as extreme changes in parameters. Additionally, it can describe asymmetric dependencies, simplifying the representation of dependency structures with a single parameter, which facilitates analysis and interpretation, while also capturing complex dependencies, especially in the tails. In terms of robustness, it is sensitive to extreme values, providing a robust estimate even when data includes noise. Most importantly, the Gumbel Copula can join multiple variables, even if these variables have different marginal distributions, which is particularly important for the multi-parameter analysis of battery performance.

Table 3.

Goodness-of-fit tests for different copula functions at 90% SoH

Parameter Pair	Copula Type	First marginal KS Test p-value	Second marginal KS Test p-value	CvM Test p-value
$R_{\text{anode}}$	Gumbel-Hougaard	0.7	0.792	0.67
$R_{\text{anode}}$	Gaussian	0.341	0.058	0
$R_{\text{anode}}$	Frank	0.194	0.582	0.24
$R_{\text{cathode}}$	Gumbel-Hougaard	0.896	0.706	0.53
$R_{\text{cathode}}$	Gaussian	0.436	0.425	0
$R_{\text{cathode}}$	Frank	0.515	0.353	0.01
$Q_{\text{anode}}$	Gumbel-Hougaard	0.829	0.434	0.989
$Q_{\text{anode}}$	Gaussian	0.038	0.182	0
$Q_{\text{anode}}$	Frank	0.013	0.352	1
$Q_{\text{cathode}}$	Gumbel-Hougaard	0.415	0.523	0.77
$Q_{\text{cathode}}$	Gaussian	0.275	0.013	0
$Q_{\text{cathode}}$	Frank	0.364	0.073	0.10

Figure 7.

The probability-probability plot (PP-plot) comparing the theoretical and empirical joint distributions

(A–D) represent the pp-plots of R and E_a, and Q and ∝, respectively.

In Section SoH curve prediction, we will apply the methodologies outlined in Section STAR Methods to derive the joint distribution of eight ECM parameters from the battery’s EIS data, using the model. By setting a failure threshold for the battery when ECM parameters decrease below the 50% confidence interval of the joint probability distribution, we will verify the model’s capability for early risk prediction of battery performance degradation. Further discussions on the performance variations and degradation trends of batteries at different SoH stages will be presented in Section Discussion - analysis of battery degradation mechanisms, utilizing the joint distribution established.

SoH curve prediction

The accurate prediction of SoH for batteries is crucial for electric vehicles and their Battery Management Systems. Precise SoH prediction can alert potential failures, preventing unexpected downtimes and the consequent expensive maintenance costs. In this context, our study is dedicated to predicting the SoH curve of batteries using ECM parameters.

In our research, we employed a multi-output Gaussian process regression model, specifically the Sparse Spectral Mixture - Linear Model Coregionalization (SM-LMC), using ECM parameters obtained under different SOC for the same battery model (e.g., 25C01), with the corresponding cycle’s SoH as the output. To build an accurate battery SoH prediction model, we used the data from the first 50, 100, and 150 cycles as the training set, and data beyond these cycles as the prediction set. This strategy ensures that the model learns across a wide range of charging and discharging cycles while also being able to reliably evaluate its prediction capability on unknown data. In terms of feature selection, our study combined the use of Pearson correlation coefficients and f-regression methods. The Pearson correlation coefficient²³ is a statistical indicator used to measure the strength of the linear relationship between two variables. The Pearson correlation coefficient r is defined by Equation 3, where $x_i$ and $y_i$ are the sample values of the two variables, $\overline{x}$ and $\overline{y}$ are their sample means, and n is the size of the sample. The f-regression method,²⁴ based on Analysis of Variance (ANOVA), assesses their linear relationship by calculating the F-statistics between each input feature and the target variable. The goal is to identify features that have a significant linear relationship with the target variable. The F-statistic for regression is calculated by Equation 4. The combination of these two methods ensured the effectiveness and reliability of the selected features for predicting battery SoH. Figures 8 and 9 display the F-values, P-values, and the Pearson correlation coefficient matrix of the ECM parameters for the 25C01 battery data, respectively. From these figures, it is evident that parameters such as $Rs$, L, and ${\alpha }_{cathode}$ show a strong negative correlation with SoH; on the other hand, parameters such as $Q_{anode}$ and ${\alpha }_{anode}$ show a strong positive correlation. Based on this, we selected $E{a}_{anode}$, $R_{anode}$, $Rs$, L, $Q_{anode}$, $Q_{cathode}$, ${\alpha }_{anode}$, ${\alpha }_{cathode}$ as input features for the prediction model.

$$r=\frac{{\sum }_{i=1}^n\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}{\sqrt{{\sum }_{i=1}^n{\left(x_i-\overline{x}\right)}^2}\sqrt{{\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}}$$ (Equation 3)

$$F=\frac{\left(\text{SSR}/k\right)}{\left(\text{SSE}/\left(n-k-1\right)\right)}$$ (Equation 4)

Figure 8.

Comparative Analysis of Feature Importance of the ECM parameters for the 25C01 battery data Using F-values and P-values

Figure 9.

The Pearson correlation coefficient matrix of the ECM parameters for the 25C01 battery data

This study utilized a multi-output Gaussian process regression model, SM-LMC, capable of processing multiple related outputs and capturing the correlations between different outputs. Given a set of n observations $\mathbf{X}$ and corresponding multi-output responses $\mathbf{Y}$, SM-LMC model can be defined by Equation 5, where $\mathbf{f}\left(\mathbf{X}\right)$ represents a vector of latent functions, $\mathbf{B}$ is a coregionalization matrix that mixes the contributions of the latent functions to the observed outputs, and ϵ is a Gaussian noise vector. Each latent function $f_i\left(\mathbf{X}\right)$ is assumed to be drawn from a Gaussian Process Equation 6, where $m_i\left(\mathbf{x}\right)$ is a mean function and $k_i\left(\mathbf{x},{\mathbf{x}}^{\prime }\right)$ is a covariance function or kernel. The covariance of the observed outputs is then modeled by Equation 7, where $\mathbf{K}$ is a block-diagonal matrix of kernel matrices for each latent function and $\mathbf{\Sigma }$ is a diagonal matrix representing independent Gaussian noise for each observation.

$$\mathbf{Y}\left(\mathbf{X}\right)=\mathbf{\text{Bf}}\left(\mathbf{X}\right)+ϵ\text{,}$$ (Equation 5)

$$f_i\left(\mathbf{x}\right)\sim \mathcal{G}\mathcal{P}\left(m_i\left(\mathbf{x}\right),k_i\left(\mathbf{x},{\mathbf{x}}^{\prime }\right)\right)\text{,}$$ (Equation 6)

$$\text{Cov}\left(\mathbf{Y}\right)=\mathbf{\text{BK}}{\mathbf{B}}^T+\mathbf{\Sigma }\text{,}$$ (Equation 7)

To further validate the effectiveness of our method, we also conducted comparative experiments with LSTM and GRU time series prediction models. Different from our method, the input features of LSTM and GRU models are not the ECM parameters of the corresponding cycle, but the measured SoH of the corresponding cycle. We used the same training set and prediction set to train and test the LSTM and GRU models, and evaluated the performance of the models using multiple metrics, including Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Root-Mean-Square Error (RMSE), as shown in Table 4. Figure 10 compares the prediction capabilities of the SM-LMC model, LSTM, and GRU models using data from the first 50, 100, and 150 cycles as the training set and the corresponding actual measured values. The results show that our method significantly outperforms LSTM and GRU models in terms of prediction accuracy. At the cycle thresholds of 150 and 100, the proposed method exhibits high prediction accuracy and reliability, with an MAE of 0.0285, a MAPE of 6.118%, and an RMSE of 0.0414. As can be seen from Figure 10, the SM-LMC model has a higher prediction performance and a better fit with the actual data under different cycle thresholds.

Table 4.

Error metrics for the SM-LMC model, LSTM and GRU at different cycle thresholds

Cycle Threshold	Model	MAE	MAPE	RMSE	Model	MAE	MAPE	RMSE	Model	MAE	MAPE	RMSE
150	SM-LMC	0.028	6.118	0.041	LSTM	0.065	31.1	0.076	GRU	0.059	8.82	0.071
100	SM-LMC	0.080	16.34	0.092	LSTM	0.250	38.3	0.276	GRU	0.158	28.5	0.170
50	SM-LMC	0.326	89.08	0.391	LSTM	0.314	54.3	0.360	GRU	0.243	35.2	0.270

Figure 10.

The comparison between the predictive capabilities of the SM-LMC model, LSTM and GRU using data from the first 50, 100, and 150 cycles as the training set and the actual measured values

The gray area indicates the range of prediction uncertainty.

Battery risk prediction

Risk prediction is crucial in battery management systems, where identifying risks and performing preventative maintenance can reduce the costs of emergency repairs and battery replacements due to sudden failures. In this context, we have constructed a model for risk prediction based on ECM-Copula based mixture distribution.

To demonstrate the advantages of our model in the early detection of battery health, in the experimental design, we used five sets of normal datasets under 25°C conditions for training the model. These datasets include a wide range of charging and discharging cycles to ensure that the model comprehensively understands the behavior of batteries under standard operating conditions. In addition to the 25°C datasets, to verify the generalization ability and applicability of the model, this article further tested the model on four datasets under higher temperature conditions (35°C and 45°C). These test datasets were used to simulate more severe usage environments and test the model’s predictive ability for battery behavior at different temperatures.

By setting the threshold for battery failure as the ECM parameters falling below the 50% confidence level of the joint probability distribution, our model demonstrated the ability to early identify the decline in battery performance. In the experiments, we observed that some batteries performed well in most cycles but then experienced sudden performance declines, a phenomenon often overlooked by traditional methods. Through our approach, these abnormal changes were identified in a timely manner, thereby revealing the characteristics of batteries in the potential failure stage.

The experimental results displayed in Table 5 provide robust support for our model. Under standard training conditions at 25°C, the model successfully identified abnormal changes in batteries numbered 04, 07, and 08. This finding confirms the precision and sensitivity of the model in monitoring battery health status. When applied to test datasets at higher temperatures (35°C and 45°C), the model also exhibited excellent early detection capabilities. This not only affirms the model’s applicability under various temperature conditions but also highlights its ability to maintain accuracy in extreme environments. Furthermore, we conducted a comprehensive comparison of our approach with the univariate mixture Weibull distribution, traditional One-class SVM, and Isolation Forest methods. The comparison shows that our method significantly outperforms these traditional approaches in terms of accuracy and reliability in identifying battery performance degradation.

Table 5.

Performance comparison: Accuracy of SoH estimation by proposed method, one-class SVM, and isolation forest models

Battery ID	25C04	25C07	25C08	35C	45C
Proposed Method	91%	94%	94%	91%	91%
Weibull Distribution	85%	85%	82%	85%	82%
One-class SVM	91%	91%	91%	91%	91%
Isolation Forest	85%	85%	91%	75%	85%

Discussion - analysis of battery degradation mechanisms

In battery performance analysis, SoH is a crucial metric for assessing its degree of aging and predicting its remaining lifespan. By observing the battery’s internal resistance, shape parameters, and dependence parameters at different SoH stages, we gain insights into the battery’s performance variations and degradation trends during charging and discharging processes. These parameters intricately map the complex electrochemical reactions within the battery, such as the gradual consumption of active materials, the weakening of lithium-ion intercalation capability, the formation of the SEI layer, and structural changes in electrode materials. In this study, the exploration of the degradation mechanisms of lithium-ion batteries particularly focuses on the influence of the SEI layer and its behavior during the battery aging process. As one of the key components of the battery, the formation and performance of the SEI layer significantly affect the long-term stability and overall performance of the battery.

For the analysis of ECM parameters, including the anode impedance $R_{anode}$, anode activation energy $E{a}_{anode}$, anode CPE capacitance parameter $Q_{anode}$, and fractional phase element coefficient ${\alpha }_{anode}$, as well as the corresponding cathode parameters, we use the joint distribution of copula functions for fitting. Given the constraints of space, this article only presents the three-dimensional charts of the joint distribution of Copula function for parameters $R_{anode}$ and $E{a}_{anode}$ in Figure 11. Next, we will delve into two key aspects of battery aging: the electrochemical perspective and the interdependence of parameters. We will first analyze the changes in electrochemical characteristics of the battery at different SoH stages, and then discuss how these changes affect the overall performance and lifespan of the battery. After that, we will focus on the analysis of the interdependence of parameters in relation to the theta parameter.

Figure 11.

The joint distribution of Copula function for parameters $R_{\text{anode}}$ and $E{a}_{\text{anode}}$

(A–F) 3D images of the joint distribution under SoH from 90% to 76%.

Analysis of electrochemical aspects

In this section, we will delve into the electrochemical properties of lithium-ion batteries, particularly focusing on how changes at different SoH stages affect the battery’s overall performance and lifespan. By analyzing EIS data, we concentrate on the electrochemical behavior of the battery at various stages of aging, including the formation and evolution of the SEI layer, the increase in internal resistance, and the complexification of the charge transfer process.

• (1)In the initial phase (90% to 85%), the formation of the SEI layer and early degradation occurs. During the early stages of battery charging, the electrolyte decomposes on the anode surface to form the SEI layer, which selectively allows lithium ions to pass through while preventing direct contact between the electrolyte and the electrode. This study, through the analysis of EIS data, observes that the initial SoH decline (90%–85%) coincides with the formation of the SEI layer, which is reflected in a slight increase in anode resistance. Furthermore, the quality and thickness of the SEI layer are crucial in the initial stages of the battery. An ideal SEI layer should be thin and uniform to effectively protect the electrode and minimize the ineffective consumption of lithium ions. Under non-ideal charging and discharging conditions, such as overcharging or high-temperature environments, the SEI layer may become excessively thick or uneven, affecting battery efficiency. In the data, lower anode resistance $R_{anode}$ and lower activation energy $E{a}_{anode}$ indicate good initial formation of the SEI layer, implying less hindrance to lithium ion transport.
• (2)During the mid-stage SoH phase (85% to 75%), the SEI layer thickens, and micro-cracks form, leading to a decline in the stability of the battery’s charge-discharge cycles. In this mid-SoH stage, with repeated charge-discharge cycles of the battery, the SEI layer gradually thickens. This thickening is primarily due to the continuous decomposition of the electrolyte, particularly under high voltage or unstable electrochemical conditions. As the SEI layer thickens, its mechanical strength decreases, leading to the formation of microcracks during the charging and discharging processes. These cracks can expose the internal electrodes to the electrolyte, triggering more electrolyte decomposition reactions and further exacerbating the thickening of the SEI layer.

In EIS measurements, a significant increase in anode resistance indicates an intensified hindrance to lithium-ion transport by the SEI layer. Additionally, a substantial change in activation energy suggests that temperature’s impact on anodic electrochemical reactions becomes more pronounced. With the thickening of the SEI layer, a higher activation energy might be required to facilitate electrochemical reactions on the electrode surface.

The evolution of the SEI layer may lead to changes in the kinetics of electrochemical reactions on the electrode surface, as reflected in the changes in $Q_{\text{anode}}$ and ${\alpha }_{\text{anode}}$. These indicate that the charge transfer process becomes more complex and non-ideal. $Q_{\text{anode}}$ increases with SoH, suggesting that as the SEI layer thickens, the non-ideal capacitive behavior of the electrode surface intensifies. This phenomenon is due to the slow charge transfer process caused by lithium-ion diffusion limitations within the SEI layer, increasing the non-ideal capacitive behavior at the electrode interface. ${\alpha }_{\text{anode}}$ reflects the dispersion of the electrode response. As the SEI layer thickens and becomes more structurally complex, the electrode’s charge storage and release processes may become more dispersed, leading to an increase in e${\alpha }_{\text{anod}}$.

• (3)During the change from 75% to 70% SoH, there is a significant reduction in shape parameters across different SoH stages, indicating substantial changes in the internal electrochemical properties of the battery. This SoH range might represent a turning point in the battery’s aging process, where multiple degradation mechanisms could occur simultaneously, leading to a significant decline in battery performance. For example, as the battery undergoes more charge-discharge cycles, stress accumulation may increase micro-cracks and fractures in the electrode materials. Damage to these micro-structures can impact the macroscopic performance of the battery, especially when the degradation reaches a critical point, causing a sudden exacerbation in the loss of active material.

Furthermore, the growth of the SEI layer, formation of micro-cracks, or poor contact between the electrolyte and electrode interface can restrict the transfer of lithium ions between electrodes. A reduction in the lithium-ion intercalation capacity leads to a decrease in the actual usable capacity. When these factors combine, they can cause changes in battery performance, particularly in the mid-SoH stage, as the battery has undergone sufficient cycles for these degradation processes to start significantly affecting its performance.

Analysis of interdependency among parameters

In the analysis of the theta parameter, we observe dramatic changes across different SoH stages, particularly in the transition from 70% to 65% SoH. These variations in the dependency parameter may indicate changes in the rate of battery aging. Especially in the later stages of battery aging, the emergence of different degradation pathways or new electrochemical reactions exhibit a more complex dependency on the overall performance of the battery. Specifically, this could be due to.

• (1)Inhomogeneous growth of the SEI layer. The SEI layer may grow unevenly on the electrode surface, leading to changes in the electrochemical characteristics of local areas. This is reflected in the variations of ECM parameters. These local changes might become desynchronized, weakening the interdependency of parameters.
• (2)Complexification of electrochemical processes. As the battery ages, new electrochemical processes (such as the formation of lithium dendrites, decomposition of active materials, and so forth) may emerge. These processes could be independent of the original degradation processes, thereby increasing the complexity of internal processes within the battery.

These factors together contribute to a more complex and less predictable aging process, reflected in the changes of the theta parameter and its relationship with other parameters within the battery’s electrochemical model.

Conclusions

This study proposes a method for analyzing the health status of lithium-ion batteries by combining the ECM and a joint mixture Weibull model. The method establishes an accurate battery performance degradation monitoring system. The Gumbel Copula function is used to fit the ECM parameters under normal battery operation, demonstrating its powerful statistical analysis capability. This has rarely been applied in previous research and provides new insights into the analysis of the dependency relationship between parameters, such as the change in the relationship between Q and α during battery aging, revealing the complexity and decreasing the uniformity of the internal electrochemical process.

Our results show that the proposed method can accurately predict the battery SoH and can sensitively and accurately detect battery anomalies. This shows a significant advantage compared to other models commonly used in other research, such as time series prediction models that rely on a single electrochemical statistical indicator²⁵ or single statistical models.²⁶ Additionally, our method can deeply analyze the degradation mechanism of battery performance at different SoH stages by comprehensively considering the statistical models of multiple related parameters.

Limitations of the study

In future research, we will focus on further exploring and improving the methods of this study, and verifying and enhancing its applicability to different types and brands of lithium-ion batteries. Although the temperature-controlled second-order R-CPE ECM and Gumbel Copula joint mixture Weibull model used in our findings performed well on the current dataset, we are aware that these models may be specific. Therefore, we plan to expand the sample size, introduce more diverse battery types and brands, and richer environmental conditions to verify the generality and accuracy of the model. In particular, we will focus on the performance of batteries under extreme environmental conditions, such as high temperature, low temperature, and high load cycling conditions. These extreme conditions have a significant impact on the health state and life of the battery, so understanding the performance of the model under these conditions will be of great significance for practical applications. We plan to conduct systematic experiments to collect battery performance data under different environmental conditions and incorporate it into the model verification and optimization process.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Deposited data
45mAh Eunicell LR2032 lithium-ion coin cell	Yunwei. Zhang	https://doi.org/10.5281/zenodo.3633835
Software and algorithms
Python	Python Software Foundation	https://www.python.org/
PyTorch	The Linux Foundation	https://pytorch.org/
scikit-learn	Pedregosa et al.	https://scikit-learn.org/
Temperature-Controlled R-CPE ECM	This paper	https://zenodo.org/records/10963647
Joint Mixture Weibull Model	This paper	https://zenodo.org/records/10963647

Resource availability

Lead contact

Further information and requests for resources and reagents should be directed to and will be fulfilled by the Lead Contact, Quan Qian (qqian@shu.edu.cn).

Materials availability

This study did not generate new unique reagents.

Data and code availability

The battery dataset used in this study comes from Zhang et al.²² and is available at https://doi.org/10.1038/s41467-020-15235-7 or https://doi.org/10.5281/zenodo.3633835.

All original code has also been deposited at Zenodo https://zenodo.org/records/10963647 and is publicly available.

Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Experimental model and study participant details

The dataset adopted in this investigation comprises 12 commercially available 45 mAh Eunicell LR2032 lithium-ion coin cells, which were cycled under multi-stage constant current charge and discharge conditions, with currents randomly varying between cycles to simulate real-world usage patterns. The chemical composition of battery is LiCoO2/graphite.

Method details

Temperature-controlled second-order R-CPE ECM

This study introduces a novel equivalent circuit model, termed the “Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model”, whose structure is illustrated in Figure S1 of the supplemental information. This model innovates upon the traditional RC unit by incorporating two serially connected R-CPE units, thereby more accurately simulating the ion migration and charge transfer processes within lithium-ion battery. Additionally, the model integrates temperature-dependent resistance, enabling it to adjust its resistance values based on environmental temperature changes, thus more accurately simulating battery performance under various electrochemical conditions. The improvements in this model not only provide an efficient tool for analyzing battery charging and discharging behavior but also significantly enhance its feasibility and applicability in practical applications.

In the Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model, the Constant Phase Element (CPE) is used to replace the ideal capacitor to more accurately describe the non-ideal capacitive behavior at the electrode surface. These behaviors include capacitive performance variations caused by factors such as the roughness and porosity of the electrode surface. This model is particularly suitable for analyzing battery behavior in the mid to low-frequency range, effectively capturing charge transfer processes (in the mid-frequency area) and material transport effects (in the low-frequency area), such as the migration and diffusion of ions in the electrolyte. The introduction of CPE allows the model to characterize these processes in the form of non-ideal capacitors, providing more precise simulation results.

In the model, the anode and cathode resistances ($R_{anode}$ and $R_{cathode}$) are set as temperature-dependent resistances, dynamically adjusting with changes in temperature, providing us with a deeper understanding of battery behavior. The temperature dependence of these resistances is described by the Arrhenius equation, which is formulated as $R\left(T\right)=R_0{e}^{\frac{E_a}{k\left(T+273.15\right)}}$, where $R\left(T\right)$ represents the resistance at temperature T, $R_0$ is the resistance at a reference temperature, $Ea$ is the activation energy, k is Boltzmann’s constant, and T is the temperature in degrees Celsius. The introduction of temperature dependency not only enhances the applicability of the model but also allows it to more accurately reflect the actual electrochemical behavior.

The impedance characteristics of the ECM are defined by Equation 8. This formula illustrates the comprehensive electrochemical impedance behavior of the battery under alternating current voltage stimulation. Equation 9 presents the impedance of the CPE, where $Z_{CPE}$ is the impedance of the CPE, Q is the non-ideal capacitance measure of the CPE, ω is the angular frequency, j is the imaginary unit, and α is a constant between 0 and 1, representing the non-ideality of the CPE. Equation 10 converts the CPE into an equivalent capacitance value C, which helps to understand how the CPE simulates the electrochemical behavior of the battery at different frequencies.

$$Z\left(\omega \right)=R_s+j\omega L+\frac{R_{\text{anode}}}{1+{\left(j\omega \right)}^{{\alpha }_{\text{anode}}}{Q}_{\text{anode}}{R}_{\text{anode}}}+\frac{R_{\text{cathode}}}{1+{\left(j\omega \right)}^{{\alpha }_{\text{cathode}}}{Q}_{\text{cathode}}{R}_{\text{cathode}}}$$ (Equation 8)

$$Z_{\text{CPE}}=\frac{1}{Q{\left(j\omega \right)}^{\alpha }}$$ (Equation 9)

$$C=Q·{\omega }^{\left\{\alpha -1\right\}}$$ (Equation 10)

Based on the research work in,²⁷ during the constant current charge and discharge process, the charge and discharge current is the input to the model and remains constant, while the terminal voltage is the output and can be calculated using Equation 11. The relationships between $V_{anode}$, $V_{cathode}$ and the two R-CPE pairs are shown in Equations 12 and 13. By solving the above equations, the values of $V_{anode}$ and $V_{cathode}$ can be obtained, as indicated by Equations 14 and 15. During the constant voltage charging process, $V_t$ is the input to the model and remains unchanged. The current I is the output and is calculated using Equation 16.

By precisely measuring and adjusting these parameters, our Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model can be effectively used to simulate the performance of lithium-ion battery under EIS data. Specific parameters and descriptions of each element in the model are detailed in Table S1 of the supplemental information.

$$V_t=V_{OCV}-V_{anode}-V_{cathode}-V_L-IRs$$ (Equation 11)

$${\dot{V}}_{anode}=-\frac{V_{anode}}{R_{anode}{C}_{anode}}+\frac{I}{C_{anode}}$$ (Equation 12)

$${\dot{V}}_{cathode}=-\frac{V_{cathode}}{R_{cathode}{C}_{cathode}}+\frac{I}{C_{cathode}}$$ (Equation 13)

$$V_{anode}=I{R}_{anode}-D_{anode}\times e^{-\frac{t}{R_{anode}{C}_{anode}}}$$ (Equation 14)

$$V_{cathode}=I{R}_{cathode}-D_{cathode}\times e^{-\frac{t}{R_{cathode}{C}_{cathode}}}$$ (Equation 15)

$$I=\frac{1}{R_s}\left(V_{OCV}-V_{anode}-V_{cathode}-V_T\right)=\frac{1}{R_s}\left(V_{OCV}-i_{anode}{R}_{anode}-i_{cathode}{R}_{cathode}-V_T\right)$$ (Equation 16)

To effectively solve the “Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model” and ensure that the model parameters accurately match the EIS data of lithium-ion battery, this study employs the differential evolution (DE) algorithm.²⁸ This algorithm is an efficient population-based optimization technique, particularly suited for dealing with multi-parameter global optimization problems. The DE algorithm does not rely on initial parameter estimates, a feature that makes it an ideal choice for solving complex nonlinear problems, especially when there is a lack of physical insight into the parameters. The algorithm optimizes the candidate solution set of parameters through a series of mutation, crossover, and selection operations, effectively exploring the global optimum.

During the ECM parameter identification process, we have set logical bounds for each parameter to ensure the validity of the solutions. The bounds for the resistance values are set between 0.01 and 1 ohm, considering the general resistive characteristics of electrode materials. The range for activation energy is set between 0.001 and 0.1 electron volts, reflecting the typical magnitude of activation energy in electrochemical reactions. In the model fitting process, we focus on minimizing a complex loss function, which calculates the discrepancy between the impedance predicted by the model and the experimental data. With the global search capability of the differential evolution algorithm, we can precisely determine the ECM parameters, thereby reproducing the battery’s impedance behavior under different operating conditions and further revealing the underlying mechanisms of battery aging and performance changes. The DE algorithm is provided in Algorithm 1 of the supplemental information.

Copula based joint mixture weibull model

The statistical foundation of the joint mixture Weibull distribution model stems from its ability to provide a complete description of the uncertainty and variability of parameters, which is crucial for battery performance data. For different batches of the same type of batteries at the same SoH, each degradation process may not exhibit the same level, and the impact on future battery performance and degradation pathways significantly depends on the type of degradation that has occurred. This heterogeneity means that the ECM parameters for each batch of batteries might differ. Traditional univariate distributions struggle to capture these characteristics. The mixture Weibull distribution introduces mixing parameters, allowing for more flexible adaptation to the distribution of data, while the joint distribution links different Weibull components through the Copula, providing a multivariate distribution framework. Such a model not only fits the data better but also reveals dependencies between different parameters.

To establish the dependencies among ECM parameters and further understand their joint statistical characteristics, this paper chooses to analyze the joint distribution of certain parameters. Specifically, at every 3% SoH increment, the study establishes joint distributions of $R_{anode}$ with $E{a}_{anode}$, $R_{cathode}$ with $E{a}_{cathode}$, $Q_{anode}$ with ${\alpha }_{anode}$, and $Q_{cathode}$ with ${\alpha }_{cathode}$ to explore potential interactions and dependencies among them. The selection of these parameters for joint distribution analysis is based on the following reasons:

• (1)The anode resistance and anode activation energy reveal the electrochemical reaction kinetics of the anode at different temperatures. Anode resistance reflects the degree of hindrance to the electrochemical reactions on the electrode surface at a specific temperature, while anode activation energy indicates the sensitivity of this resistance to temperature changes. The joint distribution of these two parameters helps to understand the impact of temperature changes on the anode’s electrochemical behavior, especially during the battery’s charging and discharging processes. By analyzing the relationship between these two parameters, we can better assess and predict battery performance under different temperature conditions, which is particularly important for thermal management and safety performance optimization of the battery. Similarly, the cathode resistance and cathode activation energy together describe the electrochemical reaction characteristics of the cathode.
• (2)The joint distribution analysis of anode and cathode capacitance with their phase constants allows for an in-depth exploration of the non-ideal capacitive behavior of electrode materials. The capacitance parameter reflects the amount of charge that the electrode can store, while the phase constant reveals the dispersion level of the electrode response. The analysis of these two sets of parameters helps identify the charge storage and release capabilities of the electrode materials, as well as their behavioral characteristics at different frequencies. This is important for optimizing the electrochemical performance of batteries and enhancing their responsiveness under various operating conditions. Through such analysis, we can more precisely understand and improve the design of electrode materials, thereby enhancing the overall performance and efficiency of the battery.

Marginal distribution — Mixture weibull distribution

To construct the joint distribution, it is first necessary to determine the marginal distribution functions of the variables. in this work, we chose the mixture Weibull distribution to fit the marginal distributions of the ECM parameters. The probability density function of the mixture Weibull distribution is described in Equation 17, where $f\left(x\right)$ is the probability density function of the mixture Weibull distribution, ${\alpha }_i$ is the shape parameter of the ith component, ${\beta }_i$ is the scale parameter of the ith component, ${\omega }_i$ is the weight of the ith component, and these weights satisfy ${\sum }_{i=1}^n{\omega }_i=1$.

$$f\left(x\right)=\sum _{i=1}^n{\omega }_i\left(\frac{{\alpha }_i}{{\beta }_i}\right){\left(\frac{x}{{\beta }_i}\right)}^{{\alpha }_i-1}{e}^{-{\left(\frac{x}{{\beta }_i}\right)}^{{\alpha }_i}}$$ (Equation 17)

The reason this paper selects the mixture Weibull distribution as the marginal distribution is due to its strong adaptability and ability to describe extreme behaviors. The Weibull distribution can simulate a variety of distribution forms, from exponential to Rayleigh distributions, and this versatility allows it to accurately fit diverse data patterns. It also occupies a central position in extreme value theory, which is particularly important for the analysis of extreme values of ECM parameters, especially those closely related to battery performance degradation and failure modes. Moreover, the Weibull distribution can intricately analyze the tail behavior of data, which is crucial for the failure analysis of battery performance, as battery failures are often closely related to tail behaviors. By adjusting its shape parameter, the Weibull distribution can capture heavy-tailed or light-tailed characteristics of the tails. The Weibull distribution, widely used in material fatigue analysis, has the physical significance of statistically reflecting the physical degradation processes of batteries, such as the formation and growth of the SEI layer, and the fatigue and fracture of electrode materials.

The multimodal nature of the actual ECM parameters, due to different physical states or battery degradation processes, points out the limitations of single distribution models. Compared to single distribution models, mixture distribution models can improve the fitting accuracy to data by combining multiple distributions, thereby increasing the prediction accuracy for future battery performance.

Gumbel copula function

In this research, for the parameter analysis of the Temperature-Controlled Second-Order R-CPE Equivalent Circuit Model, we have replaced the traditional univariate analysis methods with multivariate evaluation techniques. This is particularly crucial when studying the degradation mechanisms of lithium-ion batteries, as considering the changes in internal resistance and capacitance jointly is essential. To achieve this goal, we have chosen the copula function as a modeling tool that can effectively transform univariate marginal distributions into multivariate distributions. The theory of Copula functions was originally proposed by Sklar in 1959²⁹ and has been widely used due to its versatility in constructing joint distribution functions, especially when simulating variables with different marginal distributions.

According to Sklar’s theorem, if there are two correlated variables, Q and W, representing two related parameters of the ECM, then their association can be characterized by a copula dependency function, which can be expressed as Equation 18. In this equation, u and v are the marginal cumulative distribution functions of the univariate random variables Q and W, respectively. Furthermore, if $F_Q\left(q\right)$ and $F_W\left(w\right)$ are continuous, then the joint function C is unique and captures the dependencies between the random variables.

$$F\left(q,w\right)=C_{\theta }\left(F_Q\left(q\right),F_W\left(w\right)\right)=C_{\theta }\left(u,v\right)$$ (Equation 18)

In our experiment, we conducted comparative tests on three types of Copula functions: Gumbel–Hougaard,³⁰ Gaussian,³¹ and Frank.³² The Gumbel–Hougaard Copula is the preferred tool for modeling variables with tail dependence, the Gaussian Copula is able to capture the linear correlation between variables, and the Frank Copula is suitable for analyzing the nonlinear dependency structures between variables. The expressions for the three copula functions are shown in Equations 19, 20, and 21. The reason for choosing these three Copula functions is that they can cover different types of dependency structures and provide a comprehensive analytical framework to identify and quantify the relationships between cathode and anode parameters.

$$C_{\theta }\left(u,v\right)=\exp \left(-{\left[{\left(-\ln \left(u\right)\right)}^{\theta }+{\left(-\ln \left(v\right)\right)}^{\theta }\right]}^{1/\theta }\right)$$ (Equation 19)

$$C_{\mathrm{\Sigma }}\left(u_1,\dots ,u_n\right)={\mathrm{\Phi }}_{\mathrm{\Sigma }}\left({\mathrm{\Phi }}^{-1}\left(u_1\right),\dots ,{\mathrm{\Phi }}^{-1}\left(u_n\right)\right)$$ (Equation 20)

$$C_{\theta }\left(u,v\right)=-\frac{1}{\theta }\ln \left(1+\frac{\left(\exp \left(-\theta u\right)-1\right)\left(\exp \left(-\theta v\right)-1\right)}{\exp \left(-\theta \right)-1}\right)$$ (Equation 21)

To determine which Copula function is best suited to describe the joint distribution of a specific pair of parameters, we use the Kolmogorov-Smirnov (KS) test to confirm the consistency of each marginal distribution with the observed data, and the multivariate Cramér–von Mises (CvM) test to verify the fit of the joint distribution model to the experimental data. The KS test is a non-parametric test used to compare the consistency between a univariate sample distribution and a reference distribution,³³ and the formula is as follows: $D_n={\sup }_x\left|F_n\left(x\right)-F\left(x\right)\right|$. The statistic for the multivariate CvM test is typically defined as the overall difference between the joint cumulative distribution function of the observed data and the theoretical joint distribution function.³⁴ For an n-dimensional joint distribution, the statistic $T_n$ can be represented as: $T_n={\int }_{-\infty }^{\infty }\cdots {\int }_{-\infty }^{\infty }{\left(F_n\left(x_1,\dots ,x_n\right)-F\left(x_1,\dots ,x_n\right)\right)}^{2}dF\left(x_1,\dots ,x_n\right)$.

Quantification and statistical analysis

Statistical analyses were conducted with Python 3.8. We performed goodness-of-fit tests for the joint distributions of different Copula functions at 90% SoH, as shown in Table 3, using Kolmogorov–Smirnov (KS) test and Cramér-von Mises (CvM) test. The sample size n of each group is 12. p>0.05 was used to indicate statistical significance. We use Pearson correlation coefficients and f-regression methods in terms of feature selection. In Figure 10, the gray area indicates the range of prediction uncertainty. The confidence level is set to 95%.

Published: May 20, 2024