Research on Anisotropic Electro-Thermal Coupling Model for Large-Capacity Prismatic Lithium-Ion Power Batteries

1  Introduction

The rapid growth of renewable energy integration and the increasing demand for large-scale energy storage have driven significant interest in advanced battery technologies [1,2]. Energy storage systems play a pivotal role in improving grid stability, enabling peak shaving, and facilitating the utilization of intermittent renewable resources such as solar and wind [3]. Among the available technologies, lithium-ion batteries have emerged as the leading choice due to their high energy density, long cycle life, and scalability [4,5]. For stationary applications like grid storage, microgrids, and backup power, large-capacity prismatic batteries are being adopted more widely. A key feature of these batteries is their high energy density, with single-cell capacities frequently above 280 Ah [6].

However, the thermal behavior of large-capacity prismatic batteries is markedly different from that of conventional cylindrical or pouch cells, which have been the primary focus of most previous studies. Prismatic cells have a multilayer structure. This structure includes electrodes, separators, and current collectors. It makes their thermal conductivity strongly anisotropic. There is a significant difference between the through-thickness direction and the in-plane direction [7,8]. Such anisotropy leads to uneven heat diffusion during charge–discharge processes, generating non-uniform internal temperature distributions and pronounced internal–external gradients. Traditional surface-mounted temperature sensors are incapable of capturing these internal variations, thereby limiting the effectiveness of temperature monitoring and thermal management systems [9].

Another critical factor influencing the thermal behavior of prismatic cells is the variation of internal resistance with both temperature and SOC. The temperature-dependent and SOC-dependent characteristics of ohmic and polarization resistances lead to corresponding changes in heat generation, which further complicate accurate thermal prediction [10,11]. At low temperatures, increased internal resistance can cause significant heat generation and capacity fade. At higher temperatures, changes in capacitance accelerate dynamic responses. Additionally, the entropy heat effect is positive at mid-SOC and negative at low and high SOC. This leads to reversible heat generation and makes the thermal process more nonlinear [12]. Failure to properly capture these phenomena can result in inaccurate thermal models, leading to unsafe operation, reduced efficiency, and shortened battery lifetime [13].

Electrothermal modeling provides an effective approach for predicting battery temperature evolution and guiding the design of thermal management systems. Previous efforts have introduced lumped-parameter thermal models, equivalent circuit models (ECMs), and electrochemical-thermal coupled models to describe the heat generation and transfer behavior of lithium-ion batteries [14–16]. Although these methods have been successfully applied to small-capacity cylindrical and pouch cells, they are difficult to extend to large-capacity prismatic cells. This is because they neglect anisotropic heat transfer and lack sufficient consideration of temperature- and SOC-dependent parameters. Moreover, systematic parameter identification for large cells under realistic operating conditions remains underexplored [17].

To bridge these gaps, this study proposes an anisotropic electrothermal coupled model specifically designed for large-capacity prismatic batteries. First, an equivalent circuit model is established to characterize the electrical dynamics, followed by analytical derivation of heat generation and anisotropic heat transfer processes. Then, comprehensive experimental investigations are conducted to ensure the accuracy and reliability of the proposed model. The contributions of this work can be summarized as follows:

(1)   Considering the time-varying properties of the parameters in the electrochemical model and thermal model, an anisotropic electrothermal coupling model was constructed by combining the battery equivalent circuit model and the battery thermal model.

(2)   An experimental testing platform was established to acquire the fundamental parameters of the battery. Comprehensive mechanistic analyses were then carried out on its capacity, heat capacity, entropy coefficient, SOC–OCV relationship, and electro-thermal characteristics.

(3)   The proposed anisotropic electrothermal coupling model was experimentally verified through constant current charging conditions at different ambient temperatures to evaluate the performance of the proposed anisotropic electrothermal coupling model.

This work provides both theoretical and experimental insights into the electrothermal behavior of large-capacity prismatic batteries, addressing key gaps in anisotropic heat transfer and parameter dependency. The proposed model not only enhances the accuracy of temperature prediction but also contributes to the design of safer and more efficient thermal management strategies in energy storage systems.

2  Method

2.1 Battery Equivalent Circuit Model

Lithium-ion batteries can be described using two main categories of models: electrochemical models and equivalent circuit models [18]. Electrochemical models are based on the fundamental mechanisms of lithium-ion diffusion, reaction kinetics, and charge conservation in electrodes and electrolytes. These models provide high accuracy and allow for multiphysics coupling. However, they require solving partial differential equations. This makes the computation highly complex and time-consuming. Consequently, they are unsuitable for real-time applications such as BMS. In contrast, ECMs represent battery behavior using circuit elements such as resistors, capacitors, and voltage sources. Their parameters can be readily identified through experiments such as pulse tests or HPPC, offering high computational efficiency and fast response. While ECMs focus mainly on the input current–output voltage relationship, they are particularly suitable for charge–discharge control. Considering these advantages, this study adopts the equivalent circuit model for analysis.

Commonly used equivalent circuit models include the Rint model, the Thevenin model, and higher-order RC models with multiple time constants [19]. Higher-order RC models provide more accurate descriptions of the dynamic response of batteries but also increase model complexity and computational cost. This study employs a second-order RC model as shown in Fig. 1 to characterize the electrical behavior of large-capacity batteries. This approach is suitable for practical scenarios where current profiles involve fixed charging and discharging periods.

Figure 1: Second-order equivalent circuit model of the battery.

According to Kirchhoff’s current and voltage laws, the clockwise direction in Fig. 1 is defined as the positive current direction, i.e., IL > 0 during discharge and IL < 0 during charge. The second-order RC model can thus be expressed as Eq. (1).

{−Uoc+ILR0+U1+U2+Ut=0U1˙=ILC1−U1R1C1U2˙=ILC2−U2R2C2(1)

where Uoc is the open-circuit voltage, representing the equilibrium potential of the battery under rest conditions; R0 is the ohmic resistance of the battery, representing its instantaneous impedance, including the intrinsic resistance of the electrode materials, the ionic conduction resistance of the electrolyte, and the contact resistance between the current collectors and tabs. The first RC network represents the electrochemical polarization of the battery, R1 represents the electrode reaction kinetics resistance, which is associated with the charge-transfer resistance, while, C1 denotes the double-layer capacitance, reflecting the charge storage capability at the electrode–electrolyte interface; The second RC network characterizes the concentration polarization of the battery, where R2 corresponds to the diffusion resistance of lithium ions in the electrode or electrolyte, and C2 represents the equivalent capacitance associated with the formation of lithium-ion concentration gradients. The time constants are defined as τ1=R1C1 and τ2=R2C2, representing the fast dynamic process and the slow dynamic process of the battery, respectively. UL denotes the terminal voltage of the battery.

By applying the Laplace transform to Eq. (1), and defining U(s)=Ut(s)−Uoc(s), the equation can be rearranged into the transfer function form as follows:

G(s)=U(s)IL(s)=−R0⋅s2+R0τ1+R0τ2+R1τ2+R2τ1τ1τ2⋅s+R0+R1+R2τ1τ2s2+τ1+τ2τ1τ2⋅s+1τ1τ2(2)

For a discrete second-order system, the corresponding form can be expressed as:

G(z)=β0+β1⋅z−1+β2⋅z−21−α1⋅z−1−α2⋅z−2(3)

By applying the bilinear z-inverse transformation, the mapping in the s-plane can be obtained. Substituting z−1=2ΔT−s2ΔT+s into Eq. (3), where ΔT is the sampling interval, yields:

Gz(s)=β0−β1+β2α1−α2+1⋅s2+4⋅(β0−β2)ΔT(α1−α2+1)⋅s+4⋅(β0+β1+β2)ΔT2⋅(α1−α2+1)s2+4⋅(α2+1)ΔT⋅(α1−α2+1)⋅s+4⋅(1−α1−α2)ΔT2⋅(α1−α2+1)(4)

By matching the coefficients of Eqs. (2) and (4) term by term, can obtain:

{R0=−β0−β1+β2α1−α2+1R0τ1+R0τ2+R1τ2+R2τ1τ1τ2=−4⋅(β0−β2)ΔT(α1−α2+1)R0+R1+R2τ1τ2=−4⋅(β0+β1+β2)ΔT2⋅(α1−α2+1)τ1+τ2τ1τ2=4⋅(α2+1)ΔT⋅(α1−α2+1)1τ1τ2=4⋅(1−α1−α2)ΔT2⋅(α1−α2+1)(5)

Based on the above, can obtained:

{θ=[α1α2β0β1β2]φ(k)=[U(k−1)U(k−2)I(k)I(k−1)I(k−2)]U(k)=φ(k)θT+e(k)(6)

Here, θ denotes the system parameter vector, φ(k) represents the system state vector, and e(k) is the system noise.

2.2 Battery Thermal Model

The battery thermal model comprises heat generation and heat transfer components. Its development relies on specific assumptions, such as structural symmetry and uniform temperatures. Other premises include negligible tab heat, a compact surface film, and time-dependent external boundary conditions [20]. Based on these assumptions, the following sections detail the thermodynamic modeling of the battery.

2.2.1 Battery Heat Generation Model

During the charge–discharge process, heat generation in lithium-ion batteries can be classified into four components: irreversible heat, reversible heat, mixed heat, and side reaction heat [21]. In this study, since brand-new single cells are used, the effects of side reactions and mixed heat caused by battery aging are neglected. Only irreversible heat and reversible entropy heat are considered. Irreversible heat, including ohmic and polarization heat, originates from the various equivalent resistances in the second-order RC model and is always positive, occurring during both charging and discharging. Reversible entropy heat is related to the intrinsic thermodynamic properties of the electrode materials and reflects the entropy change during electrochemical reactions, i.e., the temperature dependence of the open-circuit voltage. It can be either positive or negative, indicating possible heat absorption during charge or discharge. The heat generation rate proposed by Bernardi et al. [22] is:

q=qir+qre(7)

qir=I⋅(U−OCV)(8)

qre=ITdOCVdT(9)

where, q denotes the total heat generation rate of the battery, qir represents the irreversible heat generated by various internal resistances, and qre corresponds to the reversible entropy heat. I is the operating current of the battery, U is the terminal voltage, OCV is the open-circuit voltage at the current SOC, and dOCVdT is the entropy coefficient of the battery.

2.2.2 Battery Heat Transfer Model

During charge and discharge processes, large-capacity energy storage batteries generate heat due to factors such as operating current and internal impedance [23]. To establish the relationship between battery operation and temperature evolution, it is necessary to analyze the heat transfer between the battery and its surrounding environment. According to the principle of energy conservation, the heat generated within the battery is partially absorbed by its equivalent heat capacity, leading to changes in battery temperature, while the remaining portion is dissipated to the external environment through heat transfer across the battery surface. The overall energy conservation of the battery can be expressed as:

Qp=Q−Qd(10)

Qp=CpmdTdt(11)

where, Qp denotes the heat absorbed by the battery leading to its temperature variation, Q represents the heat generation during charge and discharge processes, and Qd refers to the heat dissipated to the surrounding environment. Cp is the specific heat capacity, m is the battery mass, and dTdt represents the rate of change of temperature with respect to time. According to the principles of heat transfer, thermal energy is mainly transmitted through three mechanisms: (a) Heat conduction q1: the transfer of energy within a material or between bodies in direct contact, caused by the microscopic motion of particles, which can be described by Fourier’s law; (b) Heat convection q2: the transfer of thermal energy between a fluid and a solid surface due to the macroscopic movement of the fluid. Convection can be further categorized into natural convection, driven by temperature differences, and forced convection, induced by external forces such as fans or pumps. (c) Thermal radiation q3: the process by which a body emits energy in the form of electromagnetic waves due to its own temperature. These three heat transfer mechanisms can be expressed as follows:

q1=−λdtdx(12)

q2=h⋅(T−Ta)(13)

q3=εσ(T4−Ta4)(14)

where, λ is the thermal conductivity of the material, and the negative sign indicates that heat flows from the region of higher temperature to that of lower temperature. h is the convective heat transfer coefficient. T denotes the object temperature, Ta is the ambient temperature, ε represents the surface emissivity of the material, which depends on the material type and surface characteristics. σ is the Stefan-Boltzmann constant, a universal physical constant.

When heat is generated inside the battery, it is first conducted through the internal materials to the surface, a process referred to as internal heat conduction. Once the heat reaches the surface, it is exchanged with the surrounding air via convective heat transfer. In practical applications, large-capacity energy storage batteries are placed in enclosed spaces such as containers to form battery systems. In such systems, where forced convection dominates the cooling process, the contribution of thermal radiation is usually negligible. Therefore, in this study, the heat transfer analysis of large-capacity energy storage batteries considers only internal heat conduction and surface convection. The three-dimensional unsteady heat transfer process with a time-dependent internal heat source can be described as follows:

Cp∂T∂t=∂∂x(λx∂T∂x)+∂∂y(λy∂T∂y)+∂∂z(λz∂T∂z)+q˙(15)

where, λx, λy, λz represent the thermal conductivities along the three orthogonal directions. The effects of both thermal conductivity in heat conduction and convective heat transfer coefficient in convection can be uniformly expressed in terms of the equivalent thermal resistance per unit area. This approach is analogous to Ohm’s law in electrical conduction, where thermal resistance plays a role similar to that of electrical resistance. The expression can be written as follows:

RT,cd=1λ(16)

RT,cv=1h(17)

Based on the structure of large-capacity energy storage batteries, the front wide surface, the side narrow surfaces, and the bottom surface are defined as the X, Y, and Z directions, respectively. The heat transfer path of the internally generated heat is as follows: the heat produced within the battery is first conducted through the internal thermal resistances along the three directions to reach the surfaces, and then it is transferred to the external environment through the external thermal resistances of the three surfaces. The heat transfer path is illustrated in Fig. 2.

Figure 2: Heat transfer paths in the large-capacity energy storage battery.

The heat generated inside the battery is conducted through thermal resistances along the three directions to the battery’s aluminum casing surface, where it is then transferred to the external environment via convective heat transfer. The simplified equivalent thermal network model is shown in Fig. 3. Based on the equivalent thermal network model and the principle of energy conservation, the following expression can be derived:

{CidTidt=Q−Ti−TxRix−Ti−TyRiy−Ti−TzRizCxdTxdt=Ti−TxRix−Tx−TaRexCydTydt=Ti−TyRiy−Ty−TaReyCzdTzdt=Ti−TzRiz−Tz−TaRez(18)

here,

{Tia=Ti−TaTsax=Tx−TaTsay=Ty−TaTsaz=Tz−Ta(19)

Figure 3: Equivalent thermal network model considering anisotropic heat conduction.

The above equation can be discretized as follows:

{Tia(k)=(1−ΔtRixCi−ΔtRiyCi−ΔtRizCi)⋅Tia(k−1)+ΔtRixCi⋅Tsax(k−1)+ΔtRiyCi⋅Tsay(k−1)+ΔtRizCi⋅Tsaz(k−1)+ΔtCi⋅Q(k−1)Tsax(k)=ΔtRixCx⋅Tia(k−1)+(1−ΔtRixCx−ΔtRexCx)⋅Tsax(k−1)Tsay(k)=ΔtRiyCy⋅Tia(k−1)+(1−ΔtRiyCy−ΔtReyCy)⋅Tsay(k−1)Tsaz(k)=ΔtRizCz⋅Tia(k−1)+(1−ΔtRizCz−ΔtRezCz)⋅Tsaz(k−1)(20)

In this model, the state vector is defined as x=[TiaTsaxTsayTsaz]T, the input is the battery heat generation u=[Q], and the output yk corresponds to the state variables, representing the internal temperature of the battery and the temperatures of its three surfaces. Therefore, the anisotropic equivalent thermal network model can be reformulated in the discrete-time state-space representation. Based on the different surface temperature responses under various SOC and temperature conditions, the model parameters are identified using a genetic algorithm. In this study, a sampling interval of Δt is 1 s.

2.3 Anisotropic Electro-Thermal Coupling Model

Based on the single-cell equivalent circuit model and the anisotropic lumped-parameter thermal model, an anisotropic electrothermal coupled model for large-capacity energy storage batteries can be established. In this model, the electrical and thermal parameters depend on both the battery SOC and temperature and are mutually coupled [24]. Moreover, the external thermal boundary conditions of the battery are time-varying. As SOC and temperature change, the parameters in the second-order equivalent circuit model vary accordingly, which in turn affects the heat generation in the thermal model [25–27]. Simultaneously, variations in SOC and temperature also influence the internal thermal conductivities along different directions in the heat transfer model. Ultimately, the combined variations in heat generation and thermal parameters determine the temperature response in the electrothermal coupled model. Therefore, considering the time-varying nature of parameters in both the electrical and thermal models, the anisotropic electrothermal coupled model is established as illustrated in Fig. 4.

Figure 4: The electro-thermal coupled model incorporating anisotropic thermal properties.

The inputs to the model are the battery current I and the ambient temperature Ta. The parameters of the battery equivalent circuit model are determined from look-up tables based on the internal battery temperature Ti and the SOC. The OCV of the battery is calculated from the input current I and combined with the entropy coefficient dUdT obtained from preliminary experiments and the measured terminal voltage UL, it is used in the battery heat generation model to compute the heat generation Q, under the current operating conditions. The external thermal resistance parameters of the battery surfaces, Rex, Rey and Rez, are identified online, while other internal thermal parameters are obtained from offline look-up tables based on the current internal temperature and SOC. Finally, the calculated heat generation is input into the battery heat transfer model to obtain the internal and surface temperatures of the battery at the next time step.

3  Experiments and Results Analysis

The offline parameter identification of the proposed anisotropic electrothermal coupled model is conducted based on various preliminary battery tests. This includes the parameters of the second-order equivalent circuit model and the internal thermal parameters in the lumped-parameter thermal model, as these parameters are solely dependent on the battery SOC and temperature.

3.1 Experimental Platform Setup

In-line equations/expressions are embedded into the paragraphs of the text. For example, E=mc2. In-line equations or expressions should not be numbered and should use the same/similar font and size as the main text. In this study, a 310 Ah prismatic lithium iron phosphate energy storage battery was selected as the research object. The experimental design considered key operational parameters: typical battery C-rates and ambient temperatures during charge–discharge cycles in energy storage applications. It also accounted for the capabilities of the laboratory equipment, specifically the maximum current and sampling accuracy required. Accordingly, the setup integrated the following equipment: a LAND CT5002A battery test system, an AT2016B battery testing system, an SPX-50 high–low temperature test chamber, and a host computer. The specific specifications of the equipment are listed in Table 1, and the connection scheme among the devices is illustrated in Fig. 5.

Figure 5: Experimental platform for testing the large-capacity energy storage battery.

The T-type thermocouples of the AT2016B battery testing system temperature auxiliary channel were fixed onto the battery surface using insulating tape. The locations of the measurement points are shown in Fig. 6. Detailed testing procedures are outlined in Appendix A. Appendices A1 and A2 specifically describe the methods for battery basic parameter acquisition and the composite HPPC test, respectively.

Figure 6: Positions of thermocouples on the battery.

3.2 Results and Analysis

In this section, the test results of the battery’s electrical performance parameters will be analyzed in detail.

(1) Battery Capacity

As shown in Fig. 7, the capacity of the large-capacity battery decreases significantly at low temperatures. This is primarily due to hindered lithium-ion migration, increased electrolyte viscosity, reduced ionic conductivity, and slower diffusion rates within the electrode materials. Additionally, polarization effects in the electrochemical reactions are intensified, with both charge-transfer and ohmic resistances increasing significantly, resulting in a reduction of the available capacity. Conversely, at high temperatures, the electrolyte conductivity improves, ion diffusion is accelerated, and polarization is reduced, allowing more capacity to be delivered. However, prolonged operation at elevated temperatures can accelerate side reactions, such as consumption of active lithium, continuous growth of the SEI layer, and electrolyte decomposition with gas evolution, leading to irreversible capacity fade and potential thermal runaway risk [28–30].

Figure 7: Variation of battery capacity with temperature.

(2) Battery Heat Capacity

Based on the experimentally determined overall heat capacity of the battery and the manufacturer-provided casing thickness, the thicknesses of the aluminum shell in the X, Y, and Z directions are 0.6, 0.8, and 1.2 mm, respectively. The calculated internal and external heat capacities of the battery are summarized in Table 2.

(3) Entropy Heat Coefficient

As shown in Fig. 8, the measured relationship between the entropy coefficient dOCVdT and SOC indicates that it is positive in the mid-SOC range (0.3–0.6) and negative at low (0–0.2) and high (0.7–1) SOC ranges. This behavior determines whether the reversible heat generated by the battery is exothermic or endothermic.

Figure 8: Relationship between the battery entropy coefficient and SOC.

When dOCVdT<0, the reversible heat is exothermic during discharge, adding to the irreversible heat and increasing the total heat generation, while it is endothermic during charging, partially offsetting the irreversible heat and reducing the total heat generation. Conversely, when dOCVdT>0, the reversible heat is endothermic during discharge and exothermic during charging. These effects significantly influence battery heat generation at low charge/discharge rates, whereas at high rates where irreversible heat dominates, the impact of reversible heat is minimal and can be neglected.

(4) SOC-OCV Curves

Fig. 9 presents the SOC-OCV curves of the large-capacity energy storage battery at different temperatures. A flat voltage plateau is observed, with the OCV varying only slightly within the 20%–90% SOC range. This behavior is attributed to the two-phase reaction mechanism of the LiFePO4 cathode material. During lithium intercalation and deintercalation, the two phases coexist, resulting in an almost constant voltage throughout this SOC range.

Figure 9: SOC–OCV curves of the battery.

(5) Electrical Characteristic Parameters

Fig. 10 illustrates the relationships between the internal parameters of the second-order equivalent circuit model and the SOC and temperature. A significant increase in R0, R1, and R2 is observed with decreasing temperature. This means that low temperatures substantially elevate the overall conductive path resistance, the charge-transfer resistance, and the lithium-ion diffusion resistance in both the solid phase and electrolyte. In particular, R2 also increases markedly at low SOC, reflecting that at low SOC, a larger amount of lithium is extracted from the LiFePO4 cathode material, resulting in a significant decrease in the solid-phase lithium-ion concentration. Consequently, the diffusion path of lithium ions within the cathode material becomes longer, the diffusion driving force weakens, and the diffusion resistance rises correspondingly. Conversely, the double-layer capacitance C1 and diffusion capacitance C2 increase with rising temperature, indicating accelerated lithium-ion diffusion and a faster dynamic response.

Figure 10: Parameter identification results of the second-order equivalent circuit model of the battery: (a) ohmic resistance R0; (b) electrochemical polarization resistance R1; (c) concentration polarization resistance R2; (d) double-layer capacitance C1; (e) diffusion capacitance C2.

(6) Thermal Characteristic Parameters

Fig. 11 illustrates the internal thermal resistances of the large-capacity battery along three directions, highlighting the anisotropic thermal conductivity characteristics of the energy storage cell. Among the three directions, the internal thermal resistance in the X-direction (front surface) is the lowest, while that in the Z-direction (bottom surface) is the highest. Analysis of the overall battery structure and internal material properties indicates that the cell employs a wound design, where the distance from the central heat-generating region to the X and Y directions is shorter than to the Z direction. Furthermore, the copper/aluminum current collectors within the electrode layers are continuously arranged along the X-direction, forming an efficient heat conduction path. The minimal thermal resistance in the X-direction also suggests that a greater portion of the internally generated heat is transferred toward the front surface of the battery.

Figure 11: Internal thermal resistances of the large-capacity battery along different directions: (a) X-direction; (b) Y-direction; (c) Z-direction.

4  Validation and Analysis of the Electro-Thermal Coupling Model

To verify the reliability of the proposed anisotropic electro-thermal coupled model, a comprehensive evaluation was conducted by considering the application scenarios, validation objectives, and experimental conditions. In energy storage applications, the operating profile is typically characterized by distinct charging and discharging periods. During charging, the battery exhibits pronounced thermal effects due to the kinetic limitations of lithium-ion intercalation into the anode, making it more sensitive to temperature variations, which are critical for safety assessment. The continuous accumulation of heat during charging directly affects the thermal management design; therefore, it is essential to validate the model’s capability in predicting battery temperature evolution under constant-current charging conditions, which is crucial for evaluating the safety of energy storage systems. Accordingly, this study selected large-capacity energy storage batteries and conducted validation experiments under constant-current charging at 15°C, 25°C, and 35°C. Based on the anisotropic electro-thermal coupled model and the offline tabulated parameters identified through preparatory experiments, numerical verification was performed using MATLAB.

The specific validation procedure was as follows: the fully discharged battery (SOC = 0%) was placed in a controlled thermal environment at 15°C, 25°C, or 35°C and allowed to rest for 5 h to ensure thermal equilibrium. Subsequently, a constant-current charging test with a current of 150 A was applied to the large-capacity energy storage cell using a battery testing system, and the experiment was terminated once the cell voltage reached the upper cutoff voltage of 3.65 V. During the process, current, voltage, and temperature data were continuously recorded. The corresponding experimental and simulation results are presented in Figs. 12–14.

Figure 12: Experimental measurements and model predictions of the battery at 15°C.

Figure 13: Experimental measurements and model predictions of the battery at 25°C.

Figure 14: Experimental measurements and model predictions of the battery at 35°C.

Overall analysis indicates that during the charging process of the large-capacity cell, the temperature rise rate initially increases, then slows down in the middle and late stages, and finally accelerates again near the end of charging. This behavior can be attributed to the following factors: in the early stage, heat generation is dominated by both ohmic resistance and polarization resistance, leading to an increasing temperature rise rate at low-to-medium SOC. In the middle stage, the reversible heat associated with the entropy coefficient exhibits an endothermic effect, which offsets part of the irreversible heat generation and results in a plateau-like temperature response. In the final stage of charging, as the graphite anode approaches lithiation saturation, the resistance to lithium-ion intercalation increases significantly, causing a nonlinear growth in internal resistance and consequently a sharp rise in heat generation.

Comparative analysis demonstrates that the proposed anisotropic electro-thermal coupled model can accurately calculate the surface temperature of the battery, thereby verifying the reliability of the model. Nevertheless, certain deviations remain when compared with the experimental results. To quantify the model accuracy, the RMSE is employed, which is calculated according to the following expression:

RMSE=∑i=1n(xi−x~i)2n(21)

here, xi and x~i represent the measured and calculated values, respectively, while RMSE reflects the deviation between the two and serves as an indicator of their dispersion. The RMSE values of the surface temperature at different operating temperatures are summarized in Table 3, all of which are below 0.3°C, demonstrating the model’s capability to accurately predict the battery surface temperature. As illustrated in Figs. 12–14, the primary source of error arises in the mid-to-late charging stage, where a discrepancy is observed between the plateau region of the experimental measurements and the simulation results. This may be attributed to the complex variation of the entropy coefficient within the 60%–90% SOC range, leading to a mismatch between the calculated and actual heat generation. Future work will therefore require a finer SOC segmentation in experimental tests to obtain more precise offline tabulated parameters.

5  Conclusion

This work proposed and validated an anisotropic electro-thermal coupled model for large-capacity energy storage batteries. The model incorporates both equivalent circuit dynamics and anisotropic heat transfer characteristics and demonstrates high accuracy in reproducing surface temperature evolution under various operating conditions. The main conclusions are summarized as follows:

(1)   The ohmic resistance R0 and polarization resistances R1 and R2 increase significantly with decreasing temperature, resulting in noticeable capacity degradation at low temperatures. In contrast, the double-layer capacitance C1 and diffusion capacitance C2 increase with rising temperature, indicating an accelerated dynamic response of the battery.

(2)   The entropy coefficient is positive in the mid-SOC range (0.3–0.6) and negative at low/high SOC ranges, thereby influencing the reversible heat during charge and discharge processes.

(3)   The internal thermal resistance of the battery exhibits pronounced anisotropy. The front surface of X-direction shows the lowest thermal resistance (optimal heat conduction), whereas the bottom surface of Z-direction has the highest thermal resistance, confirming the impact of the wound structure on the heat conduction path.

(4)   During constant-current charging tests, the RMSE between predicted and measured surface temperatures was below 0.3°C. The model demonstrates high accuracy in the low-to-mid SOC range, while deviations at high SOC are mainly attributed to the nonlinear variation of the entropy coefficient. Finer SOC segmentation is required to improve the matching accuracy.

Acknowledgement: Not applicable.

Funding Statement: This research was funded by the National Natural Science Foundation of China (Grant No. 52477218) and the Doctoral Studio Construction Fund (Grant No. WP202517). The authors gratefully acknowledge the great help of the funds.

Author Contributions: The authors confirm contribution to the paper as follows: Xiang Chen: Conceptualization, methodology, validation, writing—original draft preparation, writing—review and editing. Shugang Sun: writing—review and editing. Xingxing Wang: formal analysis, writing—review and editing. Yelin Deng: Funding acquisition, supervision, writing—review and editing. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: Data will be made available on request.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest.

Appendix A

Appendix A.1 Acquisition of Battery Basic Parameters

(1) Battery Capacity Test

The battery capacity was calibrated using the constant current–constant voltage (CC-CV) method. This involved performing repeated full-charge and full-discharge cycles. The tests were conducted at different ambient temperatures (5°C, 15°C, 25°C, 35°C, 45°C) to obtain the average discharge capacity. This capacity provides a basis for SOC estimation, cell consistency evaluation, and battery health assessment. The entire calibration process is illustrated in Fig. A1.

Figure A1: Flowchart of the battery capacity calibration experiment.

(2) Battery Specific Heat Capacity Test

In the battery heat transfer model, the specific heat capacity represents the amount of heat required to raise the battery temperature by 1°C. Preliminary experiments are necessary to determine the range of the battery’s specific heat capacity to establish an accurate electrothermal coupled model. In this study, the overall specific heat capacity of the battery is taken as the weighted average of the internal components and the casing. The global specific heat capacity Cp,glb can be measured through an adiabatic temperature rise test, and the specific heat capacity of the casing can be calculated based on the known material properties, allowing the internal specific heat capacity Ci to be derived:

Cp,glb=Cimi+Csmsmglb(A1)

To create a quasi-adiabatic environment, the battery surface was fully wrapped with alumina ceramic fiber material which thermal conductivity of 0.085 W/m·K at 400°C. A continuous charge–discharge cycle of 10 s charge followed by 10 s discharge was applied using the battery test system to establish a stable internal heat source, ensuring that the heat generated during charging and discharging was entirely used to raise the battery’s temperature. Based on the heat generation power during operation, the temperature difference at thermal equilibrium, and the heat dissipation rate during cooling back to ambient temperature after the charge–discharge cycle, the overall specific heat capacity of the battery was calculated, as expressed in Eq. (A2).

Q−Qd=Cp,glb⋅mglb⋅ΔT(A2)

(3) Battery Entropy Heat Coefficient test

The entropic coefficient dOCVdTis an important parameter that describes the variation of the battery’s open−circuit voltage with temperature. Under low C−rate charge and discharge conditions, it can affect the thermal behavior of the battery during operation and serves as a key input parameter for constructing battery thermal models. According to the heat generation equation in the battery heat generation model, the entropic coefficient of the battery needs to be measured in advance. In this study, the entropic coefficient dOCVdT of large-capacity energy storage batteries will be determined using the direct measurement method:

(a)   Adjustment of battery SOC. The battery was placed in a thermal chamber, and its SOC was adjusted to 1, 0.9, …, 0 under standard ambient temperature 25°C using a 0.1 C current, followed by a 30 min rest period.

(b)   Modification of chamber temperature. The chamber temperature was then set to 10°C, and the battery was rested for 5 h to reach thermal equilibrium, after which the corresponding OCV was recorded. Subsequently, the chamber temperature was adjusted to 45°C, and the battery was again rested for 5 h before recording the OCV at this temperature.

(c)   Calculation of the entropic coefficient at the current SOC. The entropic coefficient was obtained by linear regression of OCV vs. temperature, where the slope was calculated as dOCVdT|SOC=ΔUΔT.

(d)   Steps 1–3 were repeated to calibrate the entropic coefficient over the full SOC range.

Appendix A.2 Composite HPPC Test

The HPPC test was conducted to assess the battery’s dynamic performance. It enables the accurate characterization of charge–discharge behavior and the identification of the second-order model’s equivalent resistances and capacitances. The test also extracts key parameters for use in the battery management system. Due to the risks of lithium plating at subzero temperatures and thermal runaway at high temperatures, the HPPC tests were restricted to a safe above-zero range. Consequently, the experiments were performed at 5°C, 15°C, 25°C, 35°C, and 45°C.

Prior to the start of the composite test sequence for each SOC point set, the battery underwent an extended resting period until thermal equilibrium was achieved. The equilibrium criteria were: the temperature difference between all surface measurement points and the ambient temperature remained consistently stable within ±0.3°C, while the rate of change of the open-circuit voltage was less than 0.1 mV/min. This ensured the uniformity of initial conditions for thermal parameter extraction. The HPPC test in Fig. A2 quantifies the equivalent circuit resistances and establishes the SOC-OCV relationship via pulse experiments and voltage recovery analysis, enhancing SOC estimation. Additionally, a composite HPPC procedure characterizes both electrical and thermal behavior. It applies a pulse sequence after an initial excitation to generate a stable internal heat source for thermal parameter identification across SOC conditions.

Figure A2: Voltage and current profiles of the composite HPPC test.

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