A Three-Stage Reaction-Process-Corrected Equivalent Circuit Model for Predicting External Short-Circuit Current in Lithium-Ion Batteries

Abstract

Accurate prediction of external short-circuit (ESC) current is important for battery safety analysis and protection design, but conventional equivalent circuit models have difficulty reproducing the strongly nonlinear current evolution under ESC conditions. This study proposes a reaction-process-corrected second-order RC model for ESC current prediction, based on ESC experiments on a 37 Ah commercial NCM pouch cell at different initial SOCs. The ESC process is described by three successive stages: bottleneck control, concentration-difference control, and separator pore closure. To represent the transport-related resistance deviation during this process, an additional correction resistance R_x and a queued-charge descriptor Q are introduced into the equivalent circuit framework. A segmented closed-loop simulation strategy is then developed to update R_x and predict the ESC current. Using the 50% SOC case as an unseen validation case, the proposed model captures the main nonlinear characteristics of ESC current, including rapid initial decay, secondary rebound, and subsequent attenuation. The proposed framework improves the physical interpretability of equivalent-circuit-based ESC simulation while retaining engineering simplicity, providing a practical approach for safety-boundary assessment and protection-oriented battery system design.

1. Introduction

Lithium-ion batteries are widely used in electric vehicles, stationary energy storage, and rail transit because of their high energy density and favorable cycle performance [1,2,3]. As battery systems continue to increase in size and energy level, safety issues have become a major constraint on large-scale deployment [4,5,6]. Among abuse conditions, external short circuit (ESC) is especially dangerous because it involves extremely high current, rapid energy release, and coupled electrical-thermal stress, all of which can trigger thermal runaway and system-level fire events [7,8]. Recent safety incidents in energy-storage systems further highlight the practical importance of accurately reproducing the ESC response of lithium-ion batteries. A reliable ESC simulation model is therefore of clear engineering value for safety-boundary assessment and protection-oriented system design.

Existing lithium-ion battery modeling approaches can be broadly classified as equivalent circuit models, electrochemical models, and data-driven models. Equivalent circuit models represent the battery by an open-circuit-voltage source, ohmic resistance, and one or more RC networks. Because of their simple structure, moderate computational cost, and comparatively straightforward parameter identification, they are widely used in state estimation and control applications [9,10]. Several studies have adapted equivalent circuit models for ESC fault diagnosis [11,12,13,14]. However, conventional equivalent circuit models are fundamentally electrical in nature and have limited capability to describe the internal reaction-process evolution that governs the full nonlinear ESC response. As a result, they are effective for rapid diagnosis in some cases, but they are less suitable for reproducing the complete short-circuit current trajectory.

Electrochemical models provide a more explicit physical description of ion transport, reaction kinetics, and heat generation. For example, Zavalis et al. [15] showed that electrolyte mass transport plays a decisive role in short-circuit current and temperature rise. Tran et al. [16] used a single-particle model with an electrolyte to describe diffusion-limited ESC dynamics, and An et al. [17] established an electrochemical-thermal coupled model for high-rate discharge and thermal-runaway analysis. Although these models offer strong mechanistic interpretability, they involve many coupled partial differential equations, require a large number of parameters, and are computationally expensive. These features limit their direct use in fast ESC-oriented engineering simulation.

Data-driven models provide another possible route for ESC prediction. Jia et al. [18] estimated short-circuit resistance using a convolutional neural network, and Yang et al. [19] developed an extreme-learning-machine thermal model for ESC conditions. Such methods can achieve good predictive performance, but they depend heavily on large quantities of high-risk and high-cost ESC data for training. Because comprehensive ESC datasets are difficult to obtain, the practical application of purely data-driven approaches remains constrained.

In summary, existing equivalent circuit models, electrochemical models, and data-driven models each have clear advantages, but none fully satisfies the combined requirements of mechanistic interpretability, engineering practicality, and accurate reproduction of the complete nonlinear ESC current evolution. Developing an ESC model that can capture the entire current trajectory while retaining the simplicity of an equivalent circuit model remains an important open problem.

To address this issue, this paper develops a reaction-process-corrected equivalent circuit model for ESC current prediction. The main contributions are as follows:

A three-stage interpretation of ESC current evolution is established. The ESC process is described in terms of a bottleneck-control stage, a concentration-difference-control stage, and a separator pore-closure stage, thereby linking the observed current waveform to the dominant internal transport and thermal constraints.

A queued-charge descriptor Q is introduced. By borrowing the bottleneck concept from transportation theory, the cumulative degree of lithium-ion transport congestion is quantified and related to the additional transport resistance that emerges under ESC conditions.

A stage-wise closed-loop simulation strategy is proposed. An additional correction resistance R_x is introduced into the equivalent circuit model, and a segmented closed-loop R_x prediction strategy is constructed, in which R_x is identified from the dominant variables of different control stages, thereby improving external short-circuit current prediction without sacrificing engineering practicality.

The remainder of this paper is organized as follows. Section 2 introduces the ESC experiments and analyzes the electro-thermal response under different initial SOCs. Section 3 develops the reaction-process-corrected equivalent circuit model, including the conventional second-order RC model, the three-stage mechanism interpretation, the queued-charge descriptor, and the stage-wise closed-loop simulation strategy. Section 4 summarizes the main conclusions.

2. Experimental Investigation of Battery External Short Circuits

2.1. Experimental Platform and Test Cell

A commercial NCM pouch cell was selected as the test object. The positive electrode material is a ternary material, the negative electrode material is graphite, and the rated capacity is 37 Ah. The main specifications of the cell are listed in

Table 1. Nominal specifications of the commercial NCM pouch cell.

Item	Specification
Positive Electrode Material	Ternary material
Negative Electrode Material	Graphite
Rated Capacity	37 Ah
Charge/Discharge Cut-off Voltage	4.2 V/2.75 V
Permissible Continuous Discharge C-rate	2 C
Maximum Discharge C-rate	10–15 C
Operating Temperature Range	0–55 °C

.

All ESC tests were conducted on the platform shown in Figure 1. The platform mainly consists of an ESC device unit, a control and monitoring unit, a data-acquisition unit, a video monitoring unit, and an explosion-proof chamber. The ESC device unit connects the short-circuit loop through a vacuum relay, and the data-acquisition unit uses a HIOKI LR8410 recorder (HIOKI E.E. CORPORATION, Ueda, Japan) to measure battery voltage, current, and temperature. During testing, the cell was placed in the explosion-proof chamber and the ESC event was triggered remotely through the relay while the electrical, thermal, and visual responses were continuously recorded.

2.2. External Short-Circuit Test Protocol

Long-duration ESC testing may lead to venting, rupture, or even thermal runaway. For safety, all tests were performed in the explosion-proof chamber shown in Figure 1. To evaluate the influence of initial SOC, the cells were tested at 100%, 75%, 50%, and 25% SOC. The external loop resistance ranged from 1.8 to 2.2 mΩ, and the ESC duration was set to 10 min so that the full response process, from current initiation to near-zero decay, could be recorded.

The external-loop resistance in the present ESC experiments lies within a low-resistance range of approximately 1.8–2.2 mΩ. Therefore, the current model construction and validation should be interpreted within this experimental range. Since the external-loop resistance can influence the short-circuit current magnitude, heating rate, and reaction-process evolution, validation over a wider range of external-loop resistances will be further considered in future work.

Considering the destructive nature and safety risks of ESC experiments on large-capacity pouch cells, each SOC condition was tested once in the present study. The obtained data are used to analyze the main ESC current, voltage, and temperature evolution under different initial SOCs and to support the development of the proposed correction framework. Repeated tests under identical conditions would be valuable for quantifying experimental variability and will be included in future work.

In practical applications, pouch cells are usually integrated into modules by laser welding and mechanically constrained by metal plates. To better reproduce this boundary condition, clamps were applied to the upper and lower surfaces of the test cell. A layer of thermal insulation cotton was inserted between the cell and the clamps to avoid direct contact effects on temperature measurement. As shown in Figure 2, the cell at 100% SOC first exhibited obvious smoke leakage near the positive tab after ESC initiation, indicating rapid local heating, electrolyte decomposition, and package rupture. With continued short circuit, smoke accumulated rapidly and visibility inside the chamber decreased substantially. At 75% SOC, gas generation and smoke were also observed, but the severity was lower than at 100% SOC. At 50% and 25% SOC, the smoke release weakened further. Because no ignition source was introduced during testing, no fire or explosion occurred in the experiments.

Figure 3 shows the current and voltage responses during ESC. Immediately after short-circuit initiation, the current rises sharply to a peak, then decreases rapidly, rebounds to a secondary peak, and finally decays continuously toward zero. The voltage follows a similar overall trend. This distinctive current waveform captures the complete ESC evolution and provides the key target for subsequent model development and validation.

Figure 4 presents the temperature evolution at the cell center and tabs during ESC. Right after short-circuit initiation, the tab temperatures increase rapidly and exceed the center temperature because the high current density near the tabs produces strong ohmic heating. The positive tab shows a larger temperature rise, which is consistent with the higher resistivity of the aluminum current collector. As the short-circuit current decreases after the secondary peak, the tab temperatures also decline and eventually become lower than the center temperature. In contrast, the center temperature changes more slowly because of thermal inertia and continues to rise even after the current has weakened. Under all tested SOC conditions, the center temperature exceeded 120 °C, far beyond the normal operating range of the cell.

These experiments provide complete current, voltage, and temperature data for ESC conditions at different SOCs. The current response clearly exhibits strong nonlinearity, especially the rapid initial decay followed by a secondary rebound. This behavior suggests that the internal electrochemical and thermal processes are not adequately represented by a conventional equivalent circuit model alone. Accordingly, the next section first establishes the baseline second-order RC model and then introduces a reaction-process-based correction method.

3. Reaction-Process-Corrected Equivalent Circuit Modeling for ESC

3.1. Conventional Equivalent Circuit Model and Parameter Identification

A second-order RC equivalent circuit model was selected as the baseline structure, as shown in Figure 5. In this model, OCV denotes the open-circuit voltage, R₀ is the ohmic resistance, R₁-C₁ and R₂-C₂ are the two polarization branches, R_ex is the external short-circuit resistance, and L is the total loop inductance.

The second-order RC equivalent circuit model is adopted because it is a widely used and relatively general model for lithium-ion battery electrical behavior. Compared with a first-order RC model, the second-order RC model can describe battery polarization dynamics more adequately by using two RC branches, while still maintaining a simple structure, clear physical meaning, and moderate computational cost. Therefore, it provides a suitable balance between model accuracy and practical applicability for the ESC simulation considered in this work.

Equivalent-circuit-based models have also been used in previous ESC studies after appropriate modification or parameter re-identification. Hong et al. [20] proposed an improved second-order RC fault model for external short-circuit diagnosis of lithium-ion batteries, while Chen et al. [12] used a modified first-order RC model to simulate the electrical behavior of lithium-ion batteries during ESC faults. These studies provide useful examples showing that RC equivalent circuit models can serve as practical reduced-order frameworks for ESC-related analysis when they are adapted to short-circuit characteristics.

Because the cell temperature changes significantly during ESC, Hybrid Pulse Power Characterization (HPPC) tests were carried out at 15, 25, 35, and 45 °C to support parameter identification over a range of thermal conditions. The parameters R₀, R₁, R₂, C₁, and C₂ at different SOCs were identified from the discharge-pulse data shown in Figure 6. Here, R₀ represents the ohmic resistance. Since ohmic polarization corresponds to the instantaneous voltage response to current, a short calculation interval was used. With a sampling interval of 10 ms in the HPPC test, R₀ was calculated according to Equation (1).

$$R_0=R_{10\mathrm{m}\mathrm{s}}={\displaystyle \frac{\Delta V_{10\mathrm{m}\mathrm{s}}}{\Delta I}}={\displaystyle \frac{OCV-V_{10\mathrm{m}\mathrm{s}}}{\Delta I}}$$

(1)

$$U=OCV-I{R}_0-I{R}_1\left(1-e^{-{\displaystyle \frac{t}{{\tau }_1}}}\right)-I{R}_2\left(1-e^{-{\displaystyle \frac{t}{{\tau }_2}}}\right)$$

(2)

$$\left\{\begin{array}{l}{C}_1={\displaystyle \frac{{\tau }_1}{R_1}}\\ C_2={\displaystyle \frac{{\tau }_2}{R_2}}\end{array}\right.$$

(3)

After R₀ had been obtained, the polarization resistances R₁ and R₂ and the polarization capacitances C₁ and C₂ were determined by fitting Equations (2) and (3). In these equations, U is the terminal voltage, OCV is the open-circuit voltage, I is the current, and t_au is the time constant. The fitting results are shown in Figure 6.

Because only a limited number of test temperatures were available, the Arrhenius equation was used to interpolate the identified parameters across a wider temperature range at each SOC. At a given SOC, the dependence of the model parameters on temperature was assumed to follow the Arrhenius form:

$$X=A\cdot \mathit{exp}\left(-{\displaystyle \frac{E_a}{RT}}\right)$$

(4)

where X represents the five identified battery parameters, A is the pre-exponential factor, E_a is the activation energy, R is the universal gas constant, and T is the absolute temperature. Taking the natural logarithm of Equation (4) yields Equation (5), where the constant term is denoted by C.

$$\mathit{ln}X={\displaystyle \frac{-E_a}{RT}}+C$$

(5)

The fitted parameter surfaces versus SOC and temperature are shown in Figure 7.

The OCV-SOC relationship was obtained by averaging the 0.05 C charge and discharge curves. Because of the entropy effect, OCV also varies with temperature. This variation can be described by the entropic coefficient dU_ocv/dT, namely the derivative of OCV with respect to temperature. To determine this coefficient, the cell was adjusted to a target SOC, placed in a temperature chamber, and allowed to rest under a programmed temperature sequence of 25, 23, 21, 19, 17, 15, 17, 19, 21, 23, and 25 °C. The chamber was held at each temperature for 2 h to ensure thermal equilibrium, while the OCV was continuously monitored. As shown in Figure 8 and Figure 9, the OCV–temperature relation is approximately linear within the tested range, enabling the entropic coefficient at each SOC to be obtained by linear fitting.

After identifying all parameters required by the equivalent circuit model, simulations were performed at 0.05 C and 1 C to verify the baseline model. The simulated voltages agree well with the experimental results in Figure 10, indicating that the model can reproduce the battery response under conventional discharge conditions.

To further examine whether the baseline second-order RC model can be directly extended to ESC conditions, an ESC simulation was carried out without introducing the reaction-process correction, with the external short-circuit resistance set to 2.2 mΩ. The comparison between the simulated and experimental ESC currents is shown in Figure 11. It can be seen that the baseline ECM cannot satisfactorily reproduce the measured short-circuit current response, especially the rapid initial decay, secondary rebound, and later attenuation. This result indicates that the baseline ECM is reasonable as a normal-operation electrical structure, but it is insufficient as a complete ESC current-prediction model by itself.

A clear mismatch can be observed in Figure 11. After the current peak, the measured ESC current drops to below 600 A within about 10 s and then rebounds noticeably, whereas the simulated current shows only small fluctuations and even a slight increase over the same interval. This result indicates that the conventional second-order RC model is not sufficient to describe the ESC response, especially the strong nonlinear current evolution after short-circuit initiation. A correction linked to the underlying reaction process is therefore required.

3.2. Reaction-Process-Based Model Correction

3.2.1. Mechanistic Interpretation of ESC Current Evolution

The discrepancy in Figure 11 suggests that ESC current evolution is governed not only by electrical polarization but also by internal transport and thermal constraints. Under ESC, a large number of lithium ions de-intercalate from the negative electrode, migrate through the electrolyte and separator, and intercalate into the positive electrode within a very short time. This transport process is limited by several factors, including solid-phase diffusion in active particles, liquid-phase diffusion in the electrolyte, ion transference behavior, concentration gradients, and separator porosity and permeability. Accordingly, lithium-ion transport is subject to a finite flux limit rather than being unbounded. During normal operation, this transport bottleneck is not prominent because the current remains far below the limiting level. During ESC, however, the current can approach or even exceed that limit, causing ion-transport congestion, a sharp increase in transport resistance, and a rapid initial current drop.

To capture this effect, an additional resistance R_x is introduced into the model to represent the reaction-process-induced transport constraint. The equivalent circuit is therefore modified from Figure 5 to Figure 12.

The correction resistance R_x represents the difference between the actual internal resistance during ESC and the resistance predicted by the baseline equivalent circuit model. To determine R_x, the real-time internal resistance R_in is first calculated from the measured ESC current I, the terminal voltage U, and the OCV data using Equation (6).

It should be noted that R_x is not intended to represent a pure ohmic resistance or a uniquely identifiable electrochemical parameter. Instead, it is introduced as a simplified lumped correction parameter outside the normal equivalent-circuit parameters. It reflects the additional obstruction effect that cannot be adequately described by the baseline second-order RC model under ESC conditions. This obstruction is associated with coupled effects such as lithium-ion transport limitation, charge accumulation, polarization enhancement, temperature rise, and thermally intensified separator pore closure. Therefore, R_x should be interpreted as a phenomenological composite parameter representing the overall degree of reaction-process-induced hindrance in the current path.

$$R_{in}={\displaystyle \frac{OCV-U}{I}}$$

(6)

Similarly, the modeled internal resistance R_sim can be obtained from the simulation results, and the time-dependent correction resistance R_x is then calculated according to Equation (7).

$$R_x=R_{in}-R_{sim}$$

(7)

Figure 13 shows the evolution of R_x during ESC at different initial SOCs. For all cells, R_x is close to zero at the onset of ESC, rises quickly to a local maximum within about 10 s, decreases gradually over roughly 10–50 s, and then increases sharply again after about 50 s. The first maximum remains below 5 mΩ, whereas the later increase reaches several tens of milliohms.

This behavior suggests that different mechanisms dominate R_x at different stages of ESC. In the first several seconds, rapid ion transport drives the system toward a flux bottleneck, causing ion accumulation and a marked increase in resistance. As ion accumulation builds up, the concentration gradient also increases, which enhances diffusion and partially relieves the transport constraint, so R_x decreases after the local maximum. At later times, the cell temperature rises to about 80–90 °C and side reactions together with separator shrinkage begin to influence transport, resulting in a renewed and accelerated increase in R_x. Because the amount of transportable lithium and the thermal path depend on SOC, the late-stage variation in R_x also shows strong nonlinearity.

The three ESC stages are separated according to the joint evolution of the measured short-circuit current and the identified correction resistance R_x. Stage I corresponds to the initial period in which the short-circuit current decreases rapidly and R_x increases sharply, indicating a rapid increase in lithium-ion transport obstruction under extremely high current demand. Stage II corresponds to the period in which the current shows a secondary rebound or slower variation while R_x gradually decreases, indicating that the enhanced concentration gradient helps relieve the initial transport congestion. Stage III corresponds to the later period in which R_x increases again and the current finally attenuates, which is associated with thermally intensified transport obstruction and separator pore closure. The specific transition points are characteristic of the tested cell and ESC conditions rather than universal fixed boundaries.

Based on the above observations, the evolution of the correction resistance during ESC can be interpreted in terms of three successive control effects:

(1)

Bottleneck-control effect. At the initial stage of ESC, the extremely high current approaches the lithium-ion transport bottleneck, leading to marked ion accumulation and a rapid increase in R_x. This stage corresponds to Stage I in Figure 14.

(2)

Concentration-difference-control effect. After the initial bottleneck has formed, the enhanced concentration gradient promotes lithium-ion migration and gradually reduces R_x. This stage corresponds to Stage II in Figure 14.

(3)

Separator pore-closure effect. With continued heating, separator shrinkage and porosity loss progressively hinder ion transport, causing R_x to increase sharply again. This stage corresponds to Stage III in Figure 14.

Because R_x is influenced by multiple coupled factors, it is difficult to derive from a single closed-form expression. Moreover, instantaneous external variables such as current and voltage cannot directly represent the cumulative effect of ion accumulation. To introduce a tractable descriptor of this accumulation process, the classical bottleneck model from transportation theory is adopted as an analogy for lithium-ion transport congestion.

In the classical bottleneck model, as shown in Figure 15, a traffic system contains a section with limited throughput capacity [21]. When demand exceeds this capacity, a queue forms and the waiting time depends on both the cumulative queue and the bottleneck throughput. By analogy, when the short-circuit-driven lithium-ion migration demand exceeds the maximum transport capability inside the battery, a transport queue can be considered to form, which manifests as additional ion-transport resistance. This analogy provides a useful basis for defining a cumulative descriptor of transport congestion.

$$Q\left(t\right)=\mathit{max}\left\{R\left(t\right)-s\left(t-t_s\right),0\right\}=\mathit{max}\left\{{\int }_{t_s}^{t}r\left(t\right)dt-s\left(t-t_s\right),0\right\}$$

(8)

The queuing time for a traveler departing at time t is expressed as:

$$T\left(t\right)={\displaystyle \frac{Q\left(t\right)}{S}}$$

(9)

According to this framework, queuing behavior is mainly determined by the cumulative queued amount and the maximum bottleneck throughput. A similar idea is used here for lithium-ion transport. Although the formation and relaxation of transport congestion in a battery are more complex than in a traffic system, defining a limiting throughput and a cumulative queued quantity still provides a practical descriptor for analyzing R_x. The corresponding quantity is calculated by Equation (10).

$$Q\left(t\right)=\mathit{max}\left\{{\int }_{t_s}^{t}I\left(t\right)dt-I_m\left(t-t_s\right),0\right\}$$

(10)

In Equation (10), I(t) is the instantaneous ESC current, I_m is the current associated with the transport bottleneck throughput, t_s is the time when I reaches I_m, and Q(t) is the cumulative queued charge at time t. According to the manufacturer specification, the maximum permissible discharge rate of the tested 37 Ah cell is 10 C-15 C, corresponding to a current range of approximately 370–555 A. Therefore, I_m was constrained within this physically admissible interval rather than treated as an arbitrary fitting parameter.

The queued-charge descriptor Q should be interpreted as a phenomenological variable rather than a directly measured electrochemical state. It is introduced to describe the cumulative tendency of lithium-ion transport congestion during ESC. The bottleneck analogy is used only to define a practical cumulative obstruction effect when the short-circuit-driven ion-transport demand exceeds the limiting transport capability. Therefore, Q is not intended to fully represent the internal concentration field, but to provide a compact descriptor linking the early transport bottleneck and subsequent concentration-gradient relaxation to the evolution of R_x.

To evaluate the influence of I_m on the final ESC-current prediction, a sensitivity analysis was carried out by varying I_m from 370 A to 550 A at intervals of 10 A. For each I_m setting, the complete modeling procedure was repeated, including the calculation of the queued-charge descriptor Q, the construction of the closed-loop R_x correction process, and the final ESC-current simulation. The resulting RMSE variation with I_m is shown in Figure 16.

As shown in Figure 16, the current-prediction RMSE first decreases and then increases as I_m increases. When I_m is lower than about 430 A, the bottleneck capacity is underestimated, causing excessive queued-charge accumulation and a larger prediction error. When I_m exceeds about 500 A, the transport bottleneck effect is weakened excessively in the model, and the prediction error increases rapidly. The minimum RMSE is obtained at I_m = 480 A, with an RMSE of 31.454 A. Therefore, I_m = 480 A is selected as the bottleneck-throughput current for the tested cell and used in the subsequent closed-loop simulation. The selected I_m value is optimal for the present validation condition and should be further examined under additional SOCs, loop resistances, and cell types.

Before the cell body temperature reaches 90 °C, the cumulative queued lithium-ion charge calculated according to Equation (10) is shown in Figure 17a, and the variation in R_x with Q is shown in Figure 17b. It can be seen from Figure 17 that, during the period from the onset of the short circuit until the temperature reaches 90 °C, the queued lithium-ion charge exhibits a two-stage growth pattern, while the resistance R_x first increases rapidly with Q and then decreases slowly. This is consistent with the previous theoretical analysis, suggesting that the cell successively underwent a bottleneck-controlled regime and a concentration-difference-controlled regime during this period.

The temperature of about 90 °C is interpreted here as a characteristic transition temperature identified from the present ESC dataset, rather than as a universal material threshold. Below this temperature, the evolution of R_x is more closely associated with the queued-charge-driven transport process, whereas above this temperature the temperature dependence of R_x becomes increasingly pronounced. As shown in Figure 18, R_x increases gradually from about 90 to 124 °C for all tested SOCs, indicating a thermally intensified regime associated with progressive separator pore closure. The temperature of about 124 °C is further interpreted as an apparent turning temperature at which the growth rate of R_x becomes markedly steeper, rather than as an exact universal threshold. Because the separator is mainly based on a PP/PE polymer membrane, this turning temperature is physically consistent with the thermal-shutdown or melting range of the PE component. Moreover, the measured value corresponds to the surface-center temperature rather than the internal hot-spot temperature, so the actual internal separator temperature is expected to be higher. The sharp increase in R_x near 124 °C is therefore consistent with accelerated separator pore closure and severe blockage of ion-transport pathways. At still higher temperatures, the dispersion of R_x among different SOCs increases, indicating that SOC should also be retained as an explanatory variable in the late stage.

It should be emphasized that I_m, the transition temperature near 90 °C, and the apparent turning temperature near 124 °C are treated here as characteristic quantities identified from the present ESC dataset rather than as universal constants. Their role in the model is to mark stage transitions and dominant trend changes in the evolution of R_x; accordingly, the mechanistic interpretation relies primarily on the consistency of stage behavior rather than on the exact numerical value of any single threshold. Therefore, the subsequent validation should be understood as validation of the calibrated reaction-process correction framework, rather than as a fully parameter-free prediction.

3.2.2. Closed-Loop Segmented Ridge Regression for R_x Prediction and External Short-Circuit Simulation

Before introducing R_x into the closed-loop equivalent-circuit simulation, its physical consistency was further examined from the viewpoint of thermal behavior. In a conventional equivalent circuit model, the irreversible heat generation is mainly associated with Joule heating. If R_x represents an additional transport-related resistance under ESC conditions, it should also contribute to the heat generation term rather than acting only as a mathematical correction for current fitting.

To verify this point, a lumped thermal model was established as

$$mc{\displaystyle \frac{dT}{dt}}=I^2\left(R_0+R_x\right)-h\left(T-T_{amb}\right)$$

(11)

where T is the cell surface temperature, T_amb is the initial ambient temperature, R₀ is obtained from the Arrhenius-parameterized equivalent circuit model, and R_x is the identified correction resistance. The parameters of the lumped thermal model were identified as mc = 857.197 J/°C and h = 1.620 W/°C. The same thermal parameters were used for all SOC conditions.

Figure 19 compares the measured temperature curves with the temperatures calculated by the lumped thermal model for the 25%, 50%, 75%, and 100% SOC ESC experiments. The calculated temperature trajectories reproduce the main heating trend under all SOC conditions, with RMSE values of 7.193 °C, 3.908 °C, 3.122 °C, and 5.289 °C, respectively. This agreement indicates that the additional correction resistance R_x is consistent not only with the electrical current response but also with the overall heat-generation trend. Therefore, introducing R_x into the equivalent circuit model is physically reasonable as a compact correction strategy. However, this thermal consistency check should be regarded as supporting evidence rather than proof that R_x uniquely corresponds to a single irreversible loss mechanism. R_x is better interpreted as a phenomenological composite parameter representing the overall reaction-process-induced obstruction effect during ESC.

This section establishes a closed-loop simulation strategy based on a modified equivalent circuit model. In this strategy, the resistance R_x is estimated in real time from the model’s own electrical state variables and immediately fed back into the equivalent-circuit calculation for the next time step, forming a closed-loop process of “prediction–simulation–update”.

In this work, the closed-loop process refers specifically to the recursive update of the electrical simulation variables, including current, SOC, Q, and R_x. The temperature sequence is not recursively updated within the same loop. Instead, before the closed-loop electrical simulation is performed, an independently trained LSTM model provides the temperature trajectory as an external thermal input. This temperature input is used to retrieve temperature-dependent equivalent-circuit parameters from the Arrhenius surfaces and to support the segmented prediction of R_x. This treatment reduces the accumulation of errors that could arise from simultaneously updating current, SOC, Q, R_x, and temperature in a fully coupled recursive loop. Therefore, the LSTM model is used to provide a stable thermal boundary for the current-prediction framework under the present experimental conditions, rather than to establish a fully self-contained electro-thermal closed-loop model.

The LSTM temperature model is trained using the temperature sequences from the 25%, 75%, and 100% SOC external short-circuit experiments, with the time sequence and the initial thermal condition as inputs and the cell center-point temperature as the output. The 50% SOC temperature sequence is then predicted independently and used as the temperature input in the subsequent closed-loop simulation, as shown in Figure 20. For the 50% SOC validation case, the predicted temperature agrees well with the measured center-point temperature, with an RMSE of 1.6954 °C. Since the temperature evolution under ESC is relatively smooth and this error is small compared with the overall temperature rise, the predicted temperature sequence provides a stable thermal boundary for R_x prediction and parameter interpolation. It should be noted that the LSTM model is used here to support the current-prediction framework under the present experimental conditions, rather than to establish a generally applicable ESC thermal prediction model.

The prediction of R_x adopts a segmented ridge-regression strategy. The segment boundaries are selected according to the three control effects identified above and the corresponding change points in the R_x evolution. In Stage I, governed by the bottleneck-control effect, R_x changes most violently because the short-circuit current is highest and lithium-ion accumulation develops rapidly; therefore, the initial 0–10 s period is divided into five short intervals: 0–1 s, 1–5 s, 5–6.5 s, 6.5–8 s, and 8–10 s. This fine segmentation captures the steep increase in R_x and the local inflection behavior during the formation of the transport bottleneck. In Stage II, the concentration-difference-control effect becomes dominant and the enhanced concentration gradient promotes ion migration, so the variation in R_x is relatively smoother; consequently, 10–40 s, 40–50 s, and 50–70 s are used to describe the gradual decrease and the transition from concentration-gradient relaxation to the subsequent thermal-dominated response. In Stage III, separator pore closure progressively restricts ion transport and causes R_x to rise again, so the later period is divided into 70–85 s, 85–100 s, and 100–200 s to resolve the onset of pore-closure-induced acceleration while avoiding unnecessary over-segmentation in the slowly varying tail. Thus, the 11 sub-intervals are physically linked to the bottleneck-control, concentration-difference-control, and separator pore-closure effects, while also providing sufficient local flexibility for ridge regression in the strongly nonlinear transition regions.

It should be noted that the segmented ridge-regression model is a semi-empirical correction strategy guided by the observed three-stage ESC behavior. The segment boundaries, feature terms, hinge-temperature terms, ridge regularization, and local smoothing are introduced to improve the local description of the strongly nonlinear evolution of R_x. These treatments are physically motivated by the bottleneck-controlled, concentration-difference-controlled, and separator pore-closure stages, but they still require calibration under the tested conditions. Thus, the segmented strategy is used to balance physical interpretability, fitting stability, and engineering simplicity, rather than to provide a fully first-principles derivation.

For each time segment, the correction resistance R_x is described as a linear function of the selected state variables:

$$R_{x,s}={\beta }_{0,s}+{\beta }_s^T{z}_s$$

(12)

where s denotes the segment index, β_0,s is the intercept, β_s is the coefficient vector, and z_s is the standardized feature vector. The feature vector includes the initial SOC, simulation time, battery temperature, temperature-change rate, simulated SOC, SOC decrease, and queued charge Q. In addition, nonlinear descriptors such as normalized time, square-root time, squared time, normalized Q, square-root Q, and the temperature hinge terms max(T − 90, 0) and max(T − 124, 0) are included to improve the description of the nonlinear evolution of R_x.

The coefficients of each segment are obtained by ridge regression:

$$\underset{{\beta }_{0,s},{\beta }_s}{\mathit{min}}\sum _{i\in {\Omega }_s}{\left[R_{x,i}-{\beta }_{0,s}-{\beta }_s^T{z}_i\right]}^2+\lambda {‖{\beta }_s‖}_2^2$$

(13)

where Ω_s is the set of training samples in the s-th time segment and λ is the regularization coefficient. During closed-loop simulation, the corresponding sub-model is selected according to the current simulation time, and a local smoothing transition is applied near adjacent segment boundaries to avoid discontinuities in the predicted R_x.

During the closed-loop simulation, R_x is predicted from the current simulated states and the independently predicted temperature sequence. The predicted R_x is then introduced into the modified equivalent circuit model to calculate the short-circuit current. The simulated current is subsequently used to update SOC by coulomb counting and Q by the bottleneck queued-charge equation with I_m = 480 A. The updated states are then used for the next-step R_x prediction, forming a recursive closed-loop simulation process.

Thus, R_x is not assigned arbitrarily in the simulation. It is predicted using the segmented ridge-regression model based on the simulated electrical states, the queued-charge descriptor Q, and the independently predicted temperature information, so that the main effects of charge accumulation, thermal evolution, and transport obstruction can be represented within a concise equivalent-circuit framework.

A local smoothing transition is applied near the boundaries between adjacent R_x sub-models to avoid discontinuities caused by segmentation. Within a narrow time window around each boundary, the predictions of the neighboring sub-models are blended smoothly. This treatment preserves the advantage of segmented modeling while improving the continuity of the predicted R_x trajectory.

In the present validation strategy, the ESC data at 25%, 75%, and 100% SOC are used for model construction, parameter identification, and reaction-process interpretation, whereas the 50% SOC case is reserved as an unseen validation case. The closed-loop simulation result for this 50% SOC validation case is shown in Figure 21, which compares the measured and predicted external short-circuit current. The predicted current successfully reproduces the main waveform characteristics of the nonlinear external short-circuit current, including the rapid initial drop, the secondary rebound, and the subsequent attenuation.

Prediction accuracy was evaluated using the root mean square error (RMSE) and normalized root mean square error (NRMSE). Under the 50% SOC validation condition, the proposed reaction-process-corrected second-order RC model achieved a current RMSE and NRMSE of 31.454 A and 2.50%, respectively. In contrast, the conventional second-order RC model yielded corresponding values of 986.60 A and 78.49%, respectively, indicating that the conventional model cannot reproduce the strongly nonlinear external short-circuit current evolution. Compared with the conventional model, the proposed model reduced the current RMSE by 96.81%.

These results demonstrate that the proposed model can accurately reproduce the highly nonlinear external short-circuit current evolution process.

4. Conclusions

A reaction-process-corrected equivalent circuit model for ESC current prediction has been developed for lithium-ion batteries. On the basis of ESC experiments and mechanism-guided model correction, the main conclusions are as follows:

• The ESC current evolution can be interpreted in three stages: bottleneck control, concentration-difference control, and separator pore closure. This three-stage framework explains the characteristic waveform of rapid initial current decay, secondary rebound, and final continuous attenuation, and provides the physical basis for correcting the equivalent circuit model.
• A queued-charge descriptor Q is proposed to characterize lithium-ion transport congestion under ESC conditions. By introducing the bottleneck-model concept, the cumulative transport accumulation inside the battery is related to the additional resistance R_x, thereby improving the physical interpretability of the correction strategy.
• A closed-loop segmented ridge regression correction strategy is constructed: the resistance R_x is predicted in real time from the state variables simulated internally by the equivalent circuit model and then fed back into the circuit calculation for the next time step, forming a closed-loop simulation. The prediction of R_x adopts a segmented ridge regression model divided according to the three-stage mechanism, with characteristic temperature points introduced into the input features to enhance stage-awareness capability; the battery temperature profile during the simulation is predicted by an LSTM network trained on temperature sequences from other SOC experiments.
• Compared with the conventional second-order RC model, the proposed corrected model reproduces the nonlinear ESC current trajectory much more faithfully while preserving the structural simplicity and engineering practicality of equivalent circuit models. The model is therefore promising for ESC fault analysis, safety-boundary evaluation, and protection-oriented design.

Overall, the proposed framework provides a practical compromise between physical insight and engineering efficiency for ESC simulation. The I_m sensitivity analysis shows that the current-prediction RMSE reaches its minimum at I_m = 480 A, and this value is therefore selected as the bottleneck-throughput current for the tested cell. Accordingly, I_m and the two temperature markers should be understood as cell-specific characteristic quantities for the present ESC conditions rather than universal thresholds. The transferable part of the proposed method is the modeling procedure, including the introduction of a reaction-process-related correction resistance, the definition of a cumulative transport descriptor, and the construction of a segmented closed-loop correction strategy. The specific characteristic quantities and regression parameters should be re-evaluated when the framework is applied to different cells or ESC conditions. The proposed framework shows promising performance under the present experimental conditions and provides a useful basis for ESC current prediction. Considering the destructive nature and safety risks of ESC experiments, future work will further extend the validation under broader experimental conditions to more comprehensively evaluate the repeatability, applicability, and robustness of the proposed framework.