Early-Stage Fault Diagnosis for Batteries Based on Expansion Force Prediction

Abstract

With the continuous expansion of the electric vehicle market, lithium-ion batteries have also been rapidly developed, but this has brought about concerns over the safety of lithium-ion batteries. Research on the correlation mechanism between the expansion and safety of lithium-ion batteries is a key step in the construction of a battery life cycle safety evaluation system. In this paper, the physicochemical mechanism of early safety faults in batteries was analyzed from three dimensions of electricity, heat, and force. The interactions of electrochemical side reactions, thermal runaway chain reactions, and mechanical fault mechanisms were analyzed, and the core induction of early safety risk was explored. A battery coupling model based on electrical, thermal, and mechanical dimensions was built, and the accuracy of the coupling model was verified by a variety of test conditions. Based on the coupling model, the stress distribution of the battery under different safety boundary conditions was simulated, and then the average expansion force of the battery surface was calculated through the stress distribution results. Through this process, a multi-parameter database based on the test and simulation data was obtained. According to the data of battery parameters at different times, an early safety classification method based on the battery expansion force was proposed, and a classification model between battery dimension data and safety level was proposed based on the nonlinear dynamic sparse regression method, and the classification accuracy was validated. From the perspective of fault warning, by establishing a multi-physical coupling model of electrical, thermal, and mechanical fields, the space-time evolution law of battery expansion under different working conditions can be dynamically monitored, and the fault criterion based on the expansion force can be established accordingly to provide quantitative indicators for safety risk classification warnings, and improve the battery’s reliability and durability.

1. Introduction

With the large-scale application of batteries in the fields of electric vehicles and energy storage, the demand for high specific energy batteries in the market continues to rise. While the energy density continues to increase, the risk of thermal fault in battery systems is growing exponentially, which has become a key bottleneck restricting the sustainable development of the industry [1]. Lithium-ion batteries commonly face fault modes such as accelerated capacity decay, gas expansion, internal short circuits, and mechanical deformation during service [2]. Among them, thermal runaway and the resulting combustion and explosion accidents pose the most serious threat to system safety [3]. When a battery is subjected to mechanical abuse, electrical abuse, or thermal abuse, the exothermic reaction between the positive and negative active substances can cause irreversible heat accumulation [4].

The safety of lithium-ion batteries is affected by various coupling factors, including electrical, mechanical, and thermal factors. The electrochemical model, as a mechanistic model, captures the complex processes inside lithium-ion batteries. The widely used pseudo two-dimensional (P2D) model was pioneered by the Newman team [5], which applied concentrated solution theory and porous electrode theory. Later, simplification was carried out, including polynomial fitting of porous electrode models and single particle electrochemical models, which provided accurate insights into the electrochemical mechanism and enhanced the understanding of the thermoelectric coupling properties during thermal runaway processes [6,7]. However, their complexity limits their application in large-scale battery systems. Therefore, electrochemical models mainly focus on the mechanism analysis of thermoelectric coupling performance under various electrical abuse conditions (such as overcharge, discharge, and short circuit), rather than result prediction [7,8].

The main method for describing the heat generation of abusive reactions is the reaction kinetics equation based on Arrhenius’ law, which was first developed by Hatchard et al. [9,10]. Subsequently, the researchers extended the model to include other exothermic reactions during thermal runaway [11,12]. The reaction kinetics model of a battery cell refers to the temperature rise curve of the entire cell measured under adiabatic conditions, which is used to estimate thermodynamic parameters [13,14]. This method divides the entire process into several stages based on the range of heating rates, and performs linear fitting for each stage to obtain dynamic parameters for different ranges of heating rates [15]. At present, the determination of thermodynamic parameters usually relies on a large number of calorimetry tests to be consistent with experimental results. Therefore, there is an urgent need to develop more robust and cost-effective parameter identification methods to improve model development efficiency [16,17].

The safety of batteries often involves thermal runaway [18,19]. Sarkar S et al. proposed a probabilistic method for safer operating area (SOA) based on abuse, fault mechanism, electrochemical parameters, and various health indicators [1]. Li W et al. used a large amount of data and machine learning to define the safety boundary of the mechanical load conditions of lithium-ion batteries [20]. In addition, battery aging will affect the safety limit of the battery, which is reflected in the battery aging and the operation mode of the battery near the end of its life [21]. Ji C et al. proposed a thermal runaway warning method based on heat transfer theory and a simplified lumped-parameter heat transfer model, which can replace the heat transfer finite element method with higher calculation requirements [22]. This simplified method is only based on the three temperature sensors in the battery module, which is designed to be included in the BMS to predict and calculate the temperature distribution in the battery module, so as to identify the faulty battery before thermal runaway and provide an early warning for the battery pack. Jia Y et al. developed an online safety classification method for the lithium-ion battery charging and discharging process by using machine learning, and defined four representative safety levels. The model only needs short-term current and voltage signals as inputs. However, the method does not consider capacity loss, which is the limitation of the model [23]. A simulation study based on voltage, temperature, and deformation eigenvalues to analyze the safety state shows that deformation will reach the early warning threshold before other features [24]. Therefore, battery deformation can be used for early detection and early warning of thermal runaway. However, when the overcharge contact heats out of control, the earliest detected signal is the voltage signal [25,26].

At present, research on the safety of a power battery mainly focuses on the internal reaction mechanism of the battery, while there is less research on the safety problems caused by the change in battery expansion force [27,28]. Lithium-ion battery expansion is closely related to battery safety [29,30,31]. If expansion occurs during use, it may lead to poor contact between the positive and negative materials inside the battery [32], which will affect the performance of the battery, thereby reducing the battery capacity, reducing the voltage, and shortening the cycle life of the battery [33,34]. In addition, expansion may also lead to excessive heat inside the battery, which may lead to overheating of the battery, which may lead to fire or explosion, causing personal injury and property damage. Therefore, it is of great significance to study the relationship between lithium-ion battery expansion and safety.

Battery expansion may lead to poor contact between the positive and negative materials inside the battery, including capacity reduction, voltage drop, and battery cycle life reduction, which will lead to the safety problem of overheating and out-of-control heating of the battery.

Challenges and problems:

(1)

The performance and safety of lithium-ion batteries are affected by many factors, which need to be considered in the research and controlled in the experiment in order to accurately evaluate the relationship between expansion and safety. A large number of experimental designs and complex experimental conditions are required.

(2)

The expansion mechanism of lithium-ion batteries is very complex, and is related to many factors, such as the chemical reaction inside the battery, ion diffusion, the deformation of electrode materials, and so on. At present, the understanding of the lithium-ion battery expansion mechanism is not completely clear.

(3)

The safety assessment of lithium-ion batteries needs to consider many aspects, including battery behavior under overcharge, over-discharge, high temperature, short circuit, and other conditions. When studying the relationship between lithium-ion battery expansion and safety, it is necessary to comprehensively consider many factors and evaluate the overall safety of the battery.

Contributions of this work:

(1)

This paper analyzes the research progress in the fields of electrical–thermal modeling, mechanical properties, and early warning at home and abroad, points out the problems of insufficient coupling of current models and limited early warning methods, and puts forward the research framework covering experimental research, multi-physical field modeling and early warning method development. The research results can provide theoretical support for the safety design of battery cells and systems.

(2)

The influence of electrical performance, thermal performance, and mechanical performance on battery safety was analyzed from the perspective of the mechanism. A multi-physical field coupling model integrating a thermal model and elastic mechanics model was established, and the distribution laws of the expansion force and thermal coupling effect of the battery under different working conditions were discussed.

(3)

The multi-dimensional data such as expansion force, temperature, voltage, current, and SOC are established to build the database, and a safety classification method based on the expansion force is proposed. The method is used to classify the safety level of the expansion force and realize the early warning of battery safety.

This paper mainly studies the expansion behavior of the battery under different working conditions, determines the possible safety problems of the battery in advance, and takes corresponding measures to reduce the risk, so as to realize the safety assessment of the battery and play the role of early warning. In addition, the expansion of the battery may lead to stress concentration and material rupture in the battery, which will affect the performance and life of the battery. Studying the relationship between expansion and safety of lithium-ion batteries can help optimize the assembly process and material selection of batteries, reduce the occurrence of expansion, and improve the reliability and durability of batteries.

Organization of the paper:

Section 2 presents the battery safety mechanistic analysis for electrochemical, thermal, and expansion forces; Section 3 presents the model establishment and validation; Section 4 shows the methods for early-stage fault diagnosis; and the key conclusions are summarized in Section 5. As shown in Figure 1.

2. Methodology

When a battery system encounters safety hazards, both its external characteristic parameters and internal state parameters will exhibit abnormal responses. Therefore, it is crucial to first analyze the battery safety characteristic parameters, with a focus on studying the evolution patterns of these parameters under different operating conditions. From a mechanistic perspective, the relationship between the three dimensions of electricity, heat, and force, and battery safety, should be analyzed.

2.1. Eelectrochemical and Safety

There exists a complex coupling relationship between the voltage, current, and safety of lithium-ion batteries. The underlying mechanism primarily involves three dimensions: electrochemical side reaction kinetics, thermal runaway critical conditions, and material structure stability.

In terms of voltage, the intrinsic stability of electrode materials determines the safe voltage window: taking the typical layered oxide cathode LiCoO₂ as an example, when the charging voltage exceeds 4.3 V, the oxidation potential of lattice oxygen is triggered, and the change in Gibbs free energy of the reaction can be characterized by the Nernst equation.

$$E_{O_2}=E_{O_2}^0+{\displaystyle \frac{RT}{4F}}\ln \left({\displaystyle \frac{a_{O_2}}{a_{O^{2-}}}}\right)$$

(1)

Among them, $E_{O_2}$ is the battery potential under non-standard conditions, $E_{O_2}^0$ is the standard electrode potential of the half cell, $R$ is the gas constant, $F$ is the Faraday constant, $T$ is the temperature, $a_{O_2}$ is the concentration of $O_2$, and $a_{O^{2-}}$ is the concentration of $O^{2-}$.

At the level of electrochemical kinetics, the reaction rate at the electrode interface is dominated by the Bulter-Volmer equation.

$$j=j_0\left[\exp \left({\displaystyle \frac{{\alpha }_{a}F\eta }{RT}}\right)-\exp \left(-{\displaystyle \frac{{\alpha }_{c}F\eta }{RT}}\right)\right]$$

(2)

where $j$ is the electrode reaction current density, $j_0$ is the exchange current density, $\eta$ is the overpotential, and ${\alpha }_a$ and ${\alpha }_c$ are the positive and negative charge transfer coefficients, with values ranging from 0 to 1.

When the charging voltage exceeds the stable window of the positive electrode material, the overpotential $\eta$ increases, which accelerates the oxidation and decomposition of the electrolyte. The reaction rate follows the Tafel equation.

$$\log j_{\mathrm{oxi}}={\displaystyle \frac{\alpha F}{2.303RT}}\eta +\mathrm{const}.$$

(3)

where $j_{\mathrm{oxi}}$ is the current density of electrolyte oxidation reaction, $\alpha$ is the transfer coefficient, usually taken as 0.5, and $\mathrm{const}.$ is a constant. For every 0.1 V increase in voltage, the oxidation and decomposition rate of the electrolyte on the positive electrode surface increases by about 3 times, leading to the accumulation of gases such as CO. At this time, the relationship between the internal pressure $P$ of the battery and the gas production $n$ can be expressed as follows.

$$P={\displaystyle \frac{nRT}{V}}-{\displaystyle \frac{2\gamma }{r}}$$

(4)

where $P$ is the internal air pressure of the battery, $n$ is the number of moles of gas, $V$ is the free volume inside the battery, $\gamma$ is the surface tension of the electrolyte, and $r$ is the pore radius of the electrode. When the internal pressure exceeds the opening threshold of the safety valve, it will cause a risk of electrolyte leakage, leading to physical short circuit and thermal runaway.

In terms of current, high rate charging and discharging will exacerbate electrode polarization effects. The critical conditions for lithium deposition on the negative electrode side can be determined by the Sand time model.

$$t_{\mathrm{plating}}={\displaystyle \frac{\pi D_{{\mathrm{Li}}^{+}}}{4}}{\left({\displaystyle \frac{zF{c}_0}{j}}\right)}^2$$

(5)

where $t_{\mathrm{plating}}$ is the critical time for lithium deposition, $D_{{\mathrm{Li}}^{+}}$ is the diffusion coefficient of lithium ions, $z$ is the number of lithium ion charges, with a value of 1, $c_0$ is the initial lithium ion concentration of the electrolyte, and $j$ is the applied current density. When the actual polarization time is $t<t_{\mathrm{plating}}$, the lithium ion concentration on the negative electrode surface tends to zero, leading to the deposition of metallic lithium. The sedimentary morphology is controlled by current density, and mossy lithium is often formed at a rate of 1C. However, the probability of dendrite growth significantly increases when it is greater than 3C. The relationship between the tip curvature radius and current density is as follows.

$$r\propto j^{-0.5}$$

(6)

2.2. Thermal and Safety

The correlation between the thermal behavior and safety of lithium-ion batteries is essentially a multi-field coupling problem of electrochemistry thermodynamics kinetics. The increase in temperature not only exacerbates heat generation through the internal resistance effect, but also triggers a chain reaction of exothermic reactions, ultimately leading to thermal runaway. The rate constant $k_i$ of each side reaction is characterized by the Arrhenius equation.

$$k_i=A_i\exp \left(-{\displaystyle \frac{E_{a,i}}{RT}}\right)$$

(7)

where $A_i$ is the pre exponential factor, $E_{a,i}$ is the activation energy, and $R$ is the ideal gas constant.

Under high temperature conditions, the chain reaction caused by the enhanced activity of the electrolyte system becomes the main cause of fault. According to the differences in thermal stability of material components, this process exhibits a staged evolution characteristic: after the SEI film structure collapses in the initial stage, it sequentially triggers a negative-electrode electrolyte exothermic reaction, membrane melting fault, positive-electrode active material decomposition, electrolyte cracking and combustion, and other progressive destruction processes.

2.3. Expansion Force and Safety

From a mechanistic perspective, the generation of expansion force is mainly due to the volume changes of electrode materials, the decomposition of electrolyte, and the thermal expansion effect caused by temperature changes during the charging and discharging process of the battery. These factors work together to increase the internal stress of the battery, which in turn affects its cycle life and safety.

During the charging and discharging process of lithium-ion batteries, the insertion and extraction of lithium ions in the positive and negative electrode materials can cause volume changes in the electrode materials. Taking a graphite negative electrode as an example, when lithium ions are embedded to form LiC6, the negative electrode volume will expand by about 10%. Similarly, positive electrode materials (such as NCM) also undergo volume changes during lithium-ion deintercalation. This volume change $\Delta V$ can be described as follows, where $V_{\mathrm{LI}}$ is the volume after lithium ion insertion, and $V_{\mathrm{ILI}}$ is the volume before lithium ion insertion.

$$\Delta V=V_{\mathrm{LI}}-V_{\mathrm{ILI}}$$

(8)

Under overcharge or high-temperature conditions, the electrolyte may undergo decomposition reactions, generating gases (such as CO₂, C₂H₄, etc.), leading to an increase in internal pressure of the battery and further exacerbating the expansion phenomenon. From a thermodynamic perspective, an increase in expansion force may cause internal short circuits in the battery, leading to thermal runaway. During the thermal runaway process, the battery temperature rises sharply, intensifying the decomposition reactions of electrode materials and the electrolyte.

Temperature changes can also cause stress in different materials inside the battery due to differences in thermal expansion coefficients. The thermal expansion effect can be described as follows.

$$\Delta L=\alpha L_0\Delta T$$

(9)

where $\alpha$ is the thermal expansion coefficient of the material, $L_0$ is the initial length or thickness, and $\Delta T$ is the temperature change.

At the mechanical level, the volume change and gas generation of electrode materials can lead to an increase in internal stress in the battery. This stress can be calculated as follows.

$$\sigma =E\epsilon$$

(10)

where $\sigma$ is stress, $E$ is the elastic modulus of the material, and $\epsilon$ is strain. When the internal stress exceeds the yield strength of the battery shell material, the shell may deform or break, causing electrolyte leakage and increasing the risk of short circuit and thermal runaway. The criterion for shell fault can be expressed as follows.

$$\sigma \ge \sigma \mathrm{yield}\mathrm{strength}$$

(11)

The expansion force also has a significant impact on the long-term cycle life of batteries. During the repeated expansion and contraction of electrode materials, particle breakage may occur, resulting in loss of active material and capacity degradation of the battery. Capacity decay can be described as follows.

$$C_n=C_0{\left(1-\alpha \right)}^n$$

(12)

where $C_n$ is the capacity after the nth cycle, $C_0$ is the initial capacity, and $\alpha$ is the attenuation coefficient.

The expansion force of lithium-ion batteries has a profound impact on the safety and cycle life of batteries through the combined action of electrochemical reactions, thermodynamic processes, and mechanical mechanisms. The volume change of electrode materials, electrolyte decomposition, thermal runaway, and internal stress increase are the main ways in which expansion force affects battery performance. A deep understanding of these mechanisms not only helps optimize battery material and structural design, but also provides theoretical support for improving battery safety and cycling stability.

3. ECM-Thermal-Expansion Force Coupling Model

The coupling model of batteries is divided into three parts, including the equivalent circuit model (ECM) model, thermal model, and force model.

3.1. ECM Modeling

Taking into account the accuracy and computational complexity of the model, a second-order ECM is adopted to simulate the electrical characteristics of the battery. The second-order RC model is shown in Figure 2, where $U_{OCV}$ is the ideal voltage source, $R_0$ is the ohmic resistance, $R_c,R_d$ are the second-order polarization resistance, and $C_c,C_d$ are capacitors connected in parallel with the polarization resistance.

Assuming that $U_c,U_d$ are the voltages across polarization resistance $R_c,R_d$,

$$\stackrel{•}{U_c}={\displaystyle \frac{1}{C_c}}-{\displaystyle \frac{U_c}{R_c{C}_c}}$$

(13)

$$\stackrel{•}{U_d}={\displaystyle \frac{1}{C_d}}-{\displaystyle \frac{U_d}{R_d{C}_d}}$$

(14)

The formula for calculating the voltage at both ends of a single battery is as follows.

$$U=U_{OCV}-U_c-U_d-I{R}_0$$

(15)

3.2. Thermal Modeling

The battery thermal model mainly consists of the heat generation equation, heat dissipation equation, and corresponding boundary conditions. The heat generation model is the Bernardi model, and the formula for calculating the heat generation rate is as follows.

$$q=I\left(U_{OCV}-U\right)-I{T}_K{\displaystyle \frac{\Delta U_{OCV}}{\Delta T}}$$

(16)

where $q$ is the heat generation rate, W; $U_{OCV}$ is the open circuit voltage of the cell, V; U is the terminal voltage of the cell, V; $T_K$ is the thermodynamic temperature of the cell, K; T is the cell temperature, °C. $I\left(U_{OCV}-U\right)$ describes the Joule heat of the cell; $I{T}_K\times \Delta U_{OCV}/\Delta T$ describes the reaction heat of the cell.

In practical applications, lithium-ion batteries mainly rely on convective heat transfer. Assuming the convective heat transfer coefficient between the battery and the environment is h (W/[m²·°C]), the calculation formula for convective heat transfer is as follows.

$$q=hS\left(T-T_0\right)$$

(17)

where $S$ is the surface area of the battery; $T$ is the surface temperature of the battery; and $T_0$ is the ambient temperature. Since this study considers the battery as a homogeneous model, the model can be simplified as a rectangular prism with a surface area:

$$S=2\left(HD+HL+DL\right)$$

(18)

where $H$ is the height of the cell; $H$ is the thickness of the cell; and $L$ is the length of the cell.

The temperature rise rate of the battery is:

$${\displaystyle \frac{dT}{dt}}={\displaystyle \frac{1}{mc}}\left[I\left(U_{OCV}-U\right)-I{T}_K{\displaystyle \frac{\Delta U_{OCV}}{\Delta T}}-hS\left(T-T_0\right)\right]$$

(19)

where $m$ is the battery mass and $c$ is the specific heat capacity.

3.3. Expansion Force Modeling

To simplify the model complexity, the battery is treated as a homogeneous model; that is, each part is treated as an isotropic linear elastic material. By using the generalized Hooke’s law, the constitutive equation is obtained, which represents the relationship between tensile and shear stress and strain in all directions, as follows.

$$\begin{array}{l}{\epsilon }_x={\displaystyle \frac{1}{E}}\left[{\sigma }_x-\mu \left({\sigma }_y+{\sigma }_z\right)\right]\hfill \\ {\epsilon }_y={\displaystyle \frac{1}{E}}\left[{\sigma }_y-\mu \left({\sigma }_x+{\sigma }_z\right)\right]\hfill \\ {\epsilon }_z={\displaystyle \frac{1}{E}}\left[{\sigma }_z-\mu \left({\sigma }_y+{\sigma }_x\right)\right]\hfill \\ {\gamma }_{yz}={\displaystyle \frac{1}{G}}{\tau }_{yz}\hfill \\ {\gamma }_{zx}={\displaystyle \frac{1}{G}}{\tau }_{zx}\hfill \\ {\gamma }_{xy}={\displaystyle \frac{1}{G}}{\tau }_{xy}\hfill \end{array}$$

(20)

where E is the tensile modulus of elasticity, G is the shear modulus of elasticity, and μ is Poisson’s ratio. The relationship is as follows.

$$G={\displaystyle \frac{E}{2\left(1+\mu \right)}}$$

(21)

Figure 3 is a schematic diagram of the battery model. Due to the fact that the expansion force of the battery almost occurs within the plane yz, the expansion forces in other planes can be ignored. Therefore, it can be assumed that the battery is subjected to a uniformly distributed force perpendicular to the z-axis direction at this time, assuming no deformation in the z-axis direction, as follows.

$$\begin{array}{l}{\epsilon }_z={\displaystyle \frac{\partial w}{\partial z}}=0\hfill \\ {\gamma }_{xz}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial u}{\partial z}}+{\displaystyle \frac{\partial w}{\partial x}}\right)=0\hfill \\ {\gamma }_{yz}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\partial v}{\partial z}}+{\displaystyle \frac{\partial w}{\partial y}}\right)=0\hfill \end{array}$$

(22)

At this point, deformation only occurs in the x-axis and y-axis directions. By substituting the conditions into the generalized Hooke’s law:

$${\sigma }_z=\mu \left({\sigma }_y+{\sigma }_x\right)$$

(23)

Thus, we obtain the constitutive equation under plane strain state:

$$\begin{array}{l}{\epsilon }_x={\displaystyle \frac{1-{\mu }^2}{E}}\left({\sigma }_x-{\displaystyle \frac{\mu }{1-\mu }}{\sigma }_y\right)\hfill \\ {\epsilon }_y={\displaystyle \frac{1-{\mu }^2}{E}}\left({\sigma }_y-{\displaystyle \frac{\mu }{1-\mu }}{\sigma }_x\right)\hfill \\ {\gamma }_{xy}={\displaystyle \frac{1}{G}}{\tau }_{xy}={\displaystyle \frac{2\left(1+\mu \right)}{E}}{\tau }_{xy}\hfill \end{array}$$

(24)

Converted into the stress–strain equation, we get:

$$\left\{\begin{array}{l}{\sigma }_x\\ {\sigma }_y\\ {\tau }_{xy}\end{array}\right\}={\displaystyle \frac{E}{\left(1+\mu \right)\left(1-2\mu \right)}}\left[\begin{array}{ccc}1-\mu & \mu & 0\\ \mu & 1-\mu & 0\\ 0& 0& {\displaystyle \frac{1-2\mu }{2}}\end{array}\right]\left\{\begin{array}{l}{\epsilon }_x\\ {\epsilon }_y\\ {\gamma }_{xy}\end{array}\right\}$$

(25)

Due to the need to establish a coupling relationship between mechanics and thermodynamics in this study, it is necessary to consider the effect of thermal expansion in the equation and revise the stress–strain equation shown as follows.

$$\left\{\begin{array}{l}{\sigma }_x\\ {\sigma }_y\\ {\tau }_{xy}\end{array}\right\}={\displaystyle \frac{E}{\left(1+\mu \right)\left(1-2\mu \right)}}\left[\begin{array}{ccc}1-\mu & \mu & 0\\ \mu & 1-\mu & 0\\ 0& 0& {\displaystyle \frac{1-2\mu }{2}}\end{array}\right]\left\{\begin{array}{l}{\epsilon }_x\\ {\epsilon }_y\\ {\gamma }_{xy}\end{array}\right\}+{\displaystyle \frac{E\alpha T}{\left(1-2\mu \right)}}\left\{\begin{array}{l}1\\ 1\\ 0\end{array}\right\}$$

(26)

where α is the coefficient of thermal expansion, E represents the elastic modulus, and μ represents Poisson’s ratio.

3.4. ECM-Thermal-Expansion Force Coupling

To clarify the interaction among the electrochemical, thermal, and mechanical domains, Figure 4 provides a structured overview of the parameter-transfer mechanisms within the proposed electro-thermo-mechanical coupling framework.

Table 1. ECM parameters.

SOC	OCV (V)	R0/Ω	R1/Ω	τ1/Ω	R2/Ω	τ2/Ω
1	4.1829	0.001081	0.000158	5.021341	0.000297	82.23048
0.95	4.1163	0.001094	0.000156	4.866847	0.000298	78.82343
0.9	4.0555	0.001106	0.000155	4.712353	0.000299	75.41638
0.85	3.9986	0.001116	0.00016	4.990518	0.000297	75.86599
0.8	3.9449	0.001128	0.00016	5.04003	0.000299	75.39546
0.75	3.8938	0.001146	0.000159	4.995256	0.000308	74.97147
0.7	3.8461	0.001163	0.000161	4.927039	0.000304	75.18536
0.65	3.801	0.001176	0.000164	5.204565	0.000301	78.26622
0.6	3.7456	0.001196	0.000165	5.366052	0.00028	77.31309
0.55	3.7028	0.001214	0.000147	5.165138	0.000252	67.93488
0.5	3.6745	0.001233	0.00014	5.072755	0.000238	60.10544
0.45	3.6528	0.001256	0.000135	5.068357	0.000235	57.85831
0.4	3.6353	0.001281	0.000133	5.045888	0.00024	57.58777
0.35	3.6199	0.001305	0.000135	5.367482	0.000241	60.87374
0.3	3.6024	0.00133	0.000139	5.433208	0.000239	60.44693
0.25	3.5702	0.001359	0.000141	5.735522	0.000253	69.79126
0.2	3.5424	0.001391	0.000143	5.582792	0.000272	72.51261
0.15	3.5091	0.001429	0.000156	6.092303	0.000279	81.0445
0.1	3.4692	0.001478	0.00017	5.729504	0.000281	78.29128
0.05	3.439	0.001539	0.000205	5.461715	0.000394	98.92885
0	3.1413	0.0016	0.00024	5.193926	0.000507	119.5664

provides ECM parameters.

Table 2. Thermal model and force model key parameters.

m (kg)	c (/J/(kg.))	h (W/(m2·K))	T0 (K)	S (m2)	a (1/K)	E (Pa)	μ
0.972	385	2	298.15	0.0414	7E-5	3.2E9	0.35

provides thermal model and force model key parameters. The coupling between the ECM and the thermal sub-model is bidirectional: the open-circuit voltage and terminal voltage computed from the second-order ECM serve as direct inputs for evaluating the heat generation rate, while the temperature field obtained from the thermal model is fed back to the ECM through Arrhenius-type corrections to all temperature-dependent circuit parameters ($R_0,R_c,C_c,C_d$). In contrast, the information flow from the thermal sub-model to the mechanical expansion-force sub-model is unidirectional. The temperature rise induces thermal strain, which is subsequently used to compute the stress–strain response of the cell. Although mechanical deformation can, in principle, increase contact resistance and thereby influence ECM behavior and heat generation, this reverse mechanical feedback is negligible during the early safety stage that this study focuses on.

According to the geometric size and structure of the battery NCM622 used in the test, the geometric model of the battery is built and the corresponding materials are set. Then, based on the electrical model, thermal model, and force model, relevant variables are defined, the heat source of the battery and the heat dissipation relationship between the battery and the environment are set, and finally, the geometric structure as shown in Figure 5 is formed, in which the length, height, and thickness of the battery are 150.0 mm, 90.0 mm, and 30.00 mm, respectively.

In order to obtain the temperature and stress changes in various parts of the battery, and taking into account the symmetry of the battery, three probes are uniformly set on the side of the model. The specific location distribution of the probes is shown in Figure 6.

3.5. Model Validation

The test conditions of the battery coupling model are carried out, and the results of the battery expansion force are used as the verification basis. The test equipment includes a thermostat, expansion force acquisition equipment, and charge and discharge equipment. The test object is a NCM622 power lithium battery, with a nominal voltage of 3.6 V and a nominal capacity of 40 Ah. The structure of the test platform is shown in Figure 7.

After 1C constant current charging of the battery model, the stress distribution is mainly low near the electrode at the upper end and high at the lower end. This is because the mechanical model in this study is elastic mechanics, and the thermal expansion of the battery is also considered, so the stress distribution is strongly related to the temperature and material properties. When the battery works, although the current density at the upper part near the pole ear is large, and the high temperature area is concentrated at the upper pole ear, the thermal expansion of the cell material is stronger because the expansion coefficient of the material at the pole ear is small, and finally, the expansion force distribution with a certain gradient is formed.

Because the final surface stress distribution is inconsistent, it is necessary to deal with the stress distribution results when studying the expansion force. Firstly, three probes were set on the surface of the battery during modeling, and the average of the three values was taken as the average stress. Finally, the average expansion force was obtained by multiplying it with the surface area, and the value was compared with the experimental value to verify the accuracy of the model. The comparison between the simulation results and the test results of the battery expansion force is shown in Figure 8.

It can be seen from Figure 8 that when 1C is charged at 25 °C and 10 °C, the average absolute percentage error (MAPE) in this case can be calculated to be 7.14% and 8.08%. After the above model verification, it can be considered that the accuracy of the current model can support the subsequent research work.

It is worth noting that, in this study, the battery is modeled as a homogenized isotropic linear elastic material. This assumption is made to capture the macroscopic expansion behavior rather than the detailed layer-level deformation inside the electrode stack. Since the dominant deformation associated with the expansion force occurs mainly within the yz-plane direction, and the out-of-plane deformation is constrained under the plane-strain condition, the isotropic assumption provides a reasonable approximation of the global mechanical response. Moreover, the model is experimentally validated, and its accuracy is adequate for safety-trend prediction. Nevertheless, anisotropic or multilayer structural models will be considered in future work when more detailed deformation analysis is required.

4. Fault Diagnosis Method

4.1. Database Construction

Through testing and simulation, the data of current, voltage, temperature, expansion force, and SOC are obtained, and the multidimensional measurement database is established. The battery was tested at 0 °C, 10 °C, 25 °C, and 45 °C.

It can be seen from Figure 9 that when the temperature decreases or increases, the maximum value and change rate of the battery expansion force will increase to a certain extent. This is because when the temperature rises, it may accelerate the decomposition of the electrolyte, produce gas, and increase the internal pressure of the battery, resulting in the increase in the expansion force of the battery. In addition, the increase in temperature will also cause the release of lattice oxygen in the ternary material to produce oxygen, resulting in volume expansion. For active particles, a high temperature is more prone to volume change, resulting in the overall deformation of the electrode. On the contrary, when the temperature decreases, the electrolyte viscosity increases, which causes lithium ions to be embedded and detached only in local areas, resulting in uneven expansion of the electrode. At the same time, low temperature will also reduce the ionic conductivity of the SEI film, which makes lithium ions accumulate at the SEI interface and promotes lithium metal deposition. Therefore, in the follow-up study of the early safety classification method based on the expansion force, it is necessary to include the temperature into the reference variable to make the model more accurate.

When the charge–discharge ratio of the battery is increased, the temperature and expansion force of the battery will change significantly. In order to avoid battery safety problems during charging, data acquisition under 2C, 3C, and 5C charging and discharging rates is carried out through model simulation, which provides a data basis for the subsequent verification of the safety classification method based on expansion force.

Figure 10 shows the simulation of 2C, 3C, and 5C charging rates at 25 °C, and the change in battery surface temperature during charging. Different from the test data, when the magnification is greater than 1C, the temperature has a significant change, and the temperature rise rate and the maximum temperature increase with the magnification. However, the maximum temperature at 5C is smaller than that at 3C, because the charging time at 5C is too short. Although the temperature rise rate is increased, the time is shorter, which makes the temperature rise smaller, so the maximum temperature is reduced.

Figure 11 shows the change in battery expansion force during charging. Similarly to the temperature change, the change rate and maximum value of the expansion force increase with the increase in charge–discharge ratio, and the maximum value at 5C is less than that at 3C due to the short charging time at 5C. Similar to the test data is the change rate of the expansion force, in which the expansion force changes slowly at the initial stage of charging, and the change rate of the expansion force gradually increases with the advancement of the charging process.

4.2. Data Preprocessing and Safety Classification

According to the change trend of each data parameter, the battery safety status is classified.

Due to the different sampling steps of the expansion force and the other four characteristic parameters, the structure length of each characteristic data parameter obtained is inconsistent. Therefore, it is necessary to adjust the length of each characteristic data parameter through a linear interpolation method to make the data structure standardized. The Min–Max method is used for standardization to reduce the impact of data magnitude on the research results.

First, determine the minimum value $X_{\min }$ and the maximum value $X_{\max }$ of the parameter, and then scale to the 0–1 interval according to Formula (27).

$$X_{\mathrm{norm}}={\displaystyle \frac{X-X_{\min }}{X_{\max }-X_{\min }}}$$

(27)

The peak expansion force of the ternary material (NCM) battery designed in the module is about 15 kN. At this time, the volume change rate of the battery has reached about 10%, and the battery can be considered to have undergone irreversible deformation. However, this study only focuses on the early safety of the battery, that is, the stage of preventing the safety problems of the battery. According to the current relevant research [35], this study sets the threshold value of the battery expansion force to 8 kN, divides the threshold value, and divides the early safety state into five levels. The specific level limit range is shown in

Table 3. Early safety classification method of battery based on expansion force.

Expansion Force Range (N)	Level
0~1000	1
1000~2500	2
2500~4000	3
4000~6000	4
6000~8000	5

.

4.3. Expansion Force Prediction Method

The nonlinear dynamics sparse regression SINDy [36] method is used to establish the control model between characteristic parameters such as the voltage, current, temperature, SOC, and expansion force. The battery charging safety level is classified based on the safety level classification method.

The core idea of sparse regression is to select the features that have a significant impact on the target variables while maintaining the prediction ability of the model [37], so as to improve the explanatory and generalization ability of the model. The control relationship between voltage, current, temperature, SOC, and expansion force can be expressed by Formula (28).

$$P\left(t\right)=f\left(x\left(t\right)\right),x\left(t\right)=\left[I\left(t\right),U\left(t\right),T\left(t\right),SOC\left(t\right)\right]$$

(28)

where $P\left(t\right),I\left(t\right),U\left(t\right),T\left(t\right),SOC\left(t\right)$ is the state of the expansion force, current, voltage, and temperature at time t, respectively. $f\left(x\left(t\right)\right)$ is the dynamic constraint that defines the equation.

In order to determine the function $f$, it is necessary to collect the historical time of $x\left(t\right)$ and establish the data matrix at time $t_1,t_2,\dots t_m$, as shown in Formulas (29) and (30).

$$X=\left[\begin{array}{c}{x}^T\left(t_1\right)\\ x^T\left(t_2\right)\\ ⋮\\ x^T\left(t_m\right)\end{array}\right]=\left[\begin{array}{cccc}I\left(t_1\right)& U\left(t_1\right)& T\left(t_1\right)& SOC\left(t_1\right)\\ I\left(t_2\right)& U\left(t_2\right)& T\left(t_2\right)& SOC\left(t_2\right)\\ ⋮& ⋮& ⋮& ⋮\\ I\left(t_m\right)& U\left(t_m\right)& T\left(t_m\right)& SOC\left(t_m\right)\end{array}\right]$$

(29)

$$P=\left[\begin{array}{c}P\left(t_1\right)\\ P\left(t_2\right)\\ ⋮\\ P\left(t_m\right)\end{array}\right]$$

(30)

Build the $\Theta \left(X\right)$ library using Formula (31):

$$\Theta \left(X\right)=\left[\begin{array}{ccccccc}1& X& X^{P_2}& X^{P_3}& \cdots & \sin (X)& \cos (X)\end{array}\right]$$

(31)

where the higher-order polynomial is recorded as $X^{P_2}$, $X^{P_3}$, etc., and where $X^{P_2}$ represents the quadratic nonlinear term under state x, i.e., Formula (32).

$$X^{P_2}=\left[\begin{array}{cccccc}{I}^2\left(t_1\right)& U^2\left(t_1\right)& T^2\left(t_1\right)& I\left(t_1\right)U\left(t_1\right)& I\left(t_1\right)T\left(t_1\right)& U\left(t_1\right)T\left(t_1\right)\\ I^2\left(t_2\right)& U^2\left(t_2\right)& T^2\left(t_2\right)& I\left(t_2\right)U\left(t_2\right)& I\left(t_2\right)T\left(t_2\right)& U\left(t_2\right)T\left(t_2\right)\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ I^2\left(t_m\right)& U^2\left(t_m\right)& T^2\left(t_m\right)& I\left(t_m\right)U\left(t_m\right)& I\left(t_m\right)T\left(t_m\right)& U\left(t_m\right)T\left(t_m\right)\end{array}\right]$$

(32)

Since there are only a limited number of nonlinear terms in each row of f, it is necessary to establish a sparse regression problem to determine a regression coefficient $\xi$ where $\xi =0.05$.

$$P=\Theta \left(X\right)\xi$$

(33)

where $\xi$ is a single column sparse coefficient vector, and the governing equation is Formula (34):

$$P=f\left(x\right)=\Theta \left(x^T\right)\xi ={\xi }^T{\left(\Theta \left(x^T\right)\right)}^T$$

(34)

The structure of the expansion force prediction method is shown in Figure 12.

The average value of each coefficient is processed, and a mathematical model with expansion force as the output is constructed with voltage, current, temperature, and SOC as inputs. The comparison between the estimated results and the actual results is shown in Figure 13.

The SINDy model is trained offline using the comprehensive multi-condition dataset described in Section 4.1. The resulting sparse coefficient vector and the governing equation are fixed and not updated online during battery operation. This fixed-model strategy guarantees real-time predictability, computational efficiency, and compliance with automotive-grade BMS determinacy requirements, while still achieving high accuracy across the tested temperature and rate ranges.

4.4. Early-Stage Fault Diagnosis

For the simulation results at 2C, 3C, and 5C magnification, the expansion force corresponds to the five safety levels to obtain the sample size distribution as shown in

Table 4. Distribution under 2C, 3C, and 5C charging rates.

Rate	Level 1	Level 2	Level 3	Level 4	Level 5
2C	195	519	155	1	0
3C	167	664	201	171	70
5C	69	247	70	64	1

. The reason for the small total number of samples at 2C magnification is the problem of sampling time.

Under the state of charge, the model is used to estimate the simulation results, and the estimated values are classified to obtain the confusion matrix as shown in Figure 14, Figure 15 and Figure 16. Each line represents the sample size of each safety level in the simulation results. The diagonal blue background indicates accurate evaluation results, while other areas indicate misjudgement results. For example, in the first line, there are 195 real level 1 sample sizes, while 182 safety levels corresponding to the expansion force calculated by using the regression equation obtained by SINDy are estimated accurately; 12 samples are estimated to be level 2, and 1 sample is estimated to be level 3.

For the 2C charging rate, level 5 is not shown in the confusion matrix because there are no simulation results and estimation results of level 5. The false alarm rates of the other four levels are 6.67%, 7.70%, 8.39% and 0%, respectively. For level 4, the sample size is small, so the reference value is small. For the other three levels, the accuracy rate has reached more than 90%. Although the accuracy of level 1 and level 2 is high, there are cross-level misjudgments, while the accuracy of level 3 is slightly low, but there are no cross-level misjudgments.

For the 3C charging rate, the misjudgment rates of the five levels are 16.17%, 7.98%, 7.46%, 8.77% and 7.14%, respectively. On the whole, the effect of level 3, level 4, and level 5 is better, the accuracy rate is high, and there is no cross-level misjudgement. Although the accuracy rate of level 2 is also high, it has cross-level misjudgement. For level 1, the accuracy rate is low and there is cross-level misjudgement.

For the 5C charging rate, the misjudgment rates of the five levels are 10.14%, 6.88%, 5.71%, 9.38%, and 0%, respectively. For level 5, too few samples have no reference value. Based on the estimation results of the above three charging rates, the accuracy of the intermediate level (level 3 and level 4) is high. The root mean square error (RMSE) is used to evaluate the classification estimation under three conditions. The root mean square errors under three charging rates (2C, 3C, and 5C) are 0.3051, 0.3478, and 0.2978, respectively.

5. Conclusions

This paper focuses on an early warning method for lithium-ion battery safety, and constructs a complete battery safety assessment through multi-dimensional parameter analysis and an intelligent algorithm.

(1)

The evolution law between charging and discharging performance and the expansion force of lithium batteries was revealed through systematic testing.

(2)

The study established an electrical–thermal–mechanical coupling model for lithium-ion battery cells. Based on the model, simulation tests were conducted near the safety boundary conditions to achieve high rate charging scenarios, and the evolution mechanism between surface temperature, expansion force, and charge–discharge rate was revealed from the simulation level.

(3)

A multi-parameter estimation algorithm and security level assessment method have been proposed by selecting five core monitoring indicators, namely current, voltage, temperature, expansion force, and SOC, and innovatively using expansion force as the main judgment basis.

By constructing a multi-physical field coupling model and proposing an early safety level classification method, this paper provides theoretical support and a technical path for the early safety assessment of batteries, and can achieve early safety level classification warning in batteries.