Reliable assessment of the state of a lithium-ion battery is based on suitable battery models. Therefore, it is necessary to construct precise battery models that reflect both the internal and external characteristics of the battery accurately. These models can be categorized into electrochemical (Xin et al., 2020; Zhou et al., 2020; Hong et al., 2023; Yu et al., 2024), and equivalent circuit models (ECM) for batteries (Xu et al., 2024; Navas et al., 2023; Karimi et al., 2023; Cheng, 2024; Li et al., 2021).

Electrochemical models are mathematical frameworks used to describe the internal electrochemical processes of batteries. By simulating the reactions occurring at the electrodes, electrolyte, and interfaces, these models predict the battery’s performance, including voltage, capacity, energy efficiency, charge/discharge behavior, and aging characteristics. Common electrochemical models include the Pseudo-Two-Dimensional Model (P2D) and the Single Particle Model (SP). To adapt the SP model to high current conditions, an extended single-particle (ESP) model was proposed to improve model accuracy (Xin et al., 2020), but this method introduces complexity through curve fitting or approximate solutions to the differential equations (Zhou et al., 2020). Hong et al. (2023) integrated a portion of the equivalent circuit model with the SP model to effectively calculate the resistance in the electrolyte and utilized a second-order system dynamics approach to address the heat transfer delay from the center of the battery pack to the surface.

Equivalent circuit models can be used to simulate the external characteristics of the battery. A typical equivalent circuit model is the Thevenin model, which consists of an ideal voltage source, ohmic resistance, and an n-th order RC circuit. The accuracy of the model is directly proportional to the order of the RC circuit, while the model complexity is inversely proportional to the RC order. Model accuracy is proportionate to RC order, and model complexity is inversely proportional to RC order. The features of the various batteries can be accurately reflected by adjusting the order of the RC circuit (Li et al., 2021). Karimi et al. (2023) employed this model to develop an equivalent circuit model suitable for large currents, a wide temperature range, and various states of charge. Navas et al. (2023) used the same equations to simultaneously account for both the charging and discharging processes, resulting in a nonlinear relationship between the state of charge and the open-circuit voltage. This approach enabled the prediction of battery voltage and state of charge.

Although the equivalent circuit model can accurately simulate the electrical characteristics of the battery, it does not take into account the temperature increase due to the heat absorbed in the working process of the battery. Therefore, some researchers proposed to add a thermal model on the basis of the equivalent circuit model, so that the two models are coupled to each other to form a thermo-electric coupling model (Xu et al., 2024). At the same time, to describe the electrodynamic characteristics and thermal dynamic characteristics of the battery, providing the premise for the state prediction and performance optimization of the battery.

SoC of batteries is a crucial parameter for electric vehicle battery management systems. Its accuracy directly impacts energy management control strategy and performance, subsequently affecting reliability and cost. Various algorithms can be used to estimate SoC, including state-space estimation methods, and machine learning approaches.

State-space estimation method usually uses a battery model combined with the measurement data to estimate SoC, including Kalman filter (Wang X. et al., 2024; Hosseininasab et al., 2023; Lin et al., 2023), particle filter (Wang J. et al., 2024; Jia et al., 2023), least square filter (Diep et al., 2024; Liu F. et al., 2023). Diep et al. (2024) proposed a method for real-time estimation of the state of charge of lithium-ion batteries using the Variable Forgetting Factor Recursive Least Squares (VFFRLS) combined with the Unscented Kalman Filter (UKF) algorithm, without the need for offline battery test data. The results showed an error of 1.5% compared to the reference values. The state space estimation method applied to battery SoC has not only good accuracy and convergence, but also small computational amount and strong adaptability.

Machine learning methods, often referred to as black-box models, capture the input-output relationship of a lithium-ion battery without relying on a physics or chemistry-based model. These methods are typically trained using measured data such as voltage, current, temperature, to accurately estimate SoC (Chen et al., 2024; Qian et al., 2024; Zhang et al., 2024). Qian et al. (2024) developed a hybrid neural network model (CNN-SAM-LSTM) that combines Convolutional Neural Networks (CNN), self-attention mechanisms, and Long Short-Term Memory (LSTM) networks for the joint estimation of state parameters in lithium-ion batteries. Although battery SoC estimation using machine learning methods can enhance accuracy, it often requires the establishment of an offline database and involves computationally intensive training processes, which may lead to overfitting and local optimum phenomena. Zhang et al. (2024) employed a stacked ensemble learning paradigm to estimate the state of lithium batteries. By utilizing base learners with different architectures, they enhanced adaptability to various features and reduced the risk of overfitting.

A battery pack is typically composed of hundreds of individual cells connected in series and parallel. Ideally, each cell leaving the production line would exhibit consistent performance over its lifespan. However, in practice, ensuring uniform initial performance parameters for each battery cell is challenging due to the manufacturing process and other factors. Inconsistencies during usage further exacerbate differences between batteries, complicating battery pack management over its service life. Additionally, the capacity and SoC of the battery are not intrinsic properties but must be determined based on individual cell behavior within the pack. Managing individual cell inconsistencies within a battery pack makes it difficult to accurately ascertain the capacity and SoC of the entire pack. The capacity of a series battery pack is determined by the sum of the minimum rechargeable capacity and the minimum discharge capacity of the individual cells in the group. Methods such as the large battery method, important battery method, and mean difference method are commonly used to estimate the SoC of the battery pack.

Large battery method treats a battery pack as a single large-capacity cell because individual cells within the battery pack exhibit similar characteristics, as proposed in reference (Ramachandran et al., 2019). Tian et al. (2023) considered large-scale battery systems as a unified entity and proposed an online capacity estimation method based on Extended Kalman Particle Filtering (EKPF). Additionally, they introduced an improved Gaussian mixture model to visualize the inconsistencies within the battery pack. Although the large battery method has a small amount of calculation, it ignores the inconsistency among individual cells in the group and cannot accurately estimate the battery SoC.

Important battery method is to use a single major battery in the group to represent battery pack performance (Liu et al., 2024; Trevizan et al., 2024). Typically, the lowest battery is selected, that is, the battery with the lowest voltage during discharge and the highest voltage during charging. The important battery method is often used in combination with different filtering algorithms. For example, Yu et al. (2023) employed cosine similarity to quantify the inconsistency among battery cells, identifying the representative cell within the battery pack based on the maximum cosine similarity value. They further utilized a multi-branch fusion approach to estimate SoC of the representative cell using Bayesian probability formulas. The important battery method can effectively protect the operation safety of the power battery pack, but when the SoC works in the range of 30%–80%, this method will reduce the energy efficiency of the battery pack.

Mean difference method treats the battery pack as both a mean model and a difference model, estimating the mean model SoC and the difference model Δ SoC, where Δ SoC represents the deviation of each cell from the mean model. This method typically utilizes an equivalent circuit model to determine model parameters through fitting measured cell data. Wu et al. (2023) designed a multi-layer balancing circuit capable of achieving both intra-group and inter-group balancing. This circuit topology ensures fast balancing while maintaining high balancing efficiency. Van and Vinh (2020) employs micro and macro time scales to estimate SoC for selected and unselected cells, respectively. Liu Z. et al. (2023) implemented a fusion of data-driven autoregressive (AR) models with ECM to enhance the dynamic characteristics of the battery pack model under complex operating conditions. They employed a differentiated update strategy to online update the parameters of the average differential model. However, this method does not consider the influence of battery operating temperature on SoC estimation.

Lai et al. (2022) proposed a joint estimation method for SoC and state of health (SOH) that takes temperature effects into account. In scenarios with temperature variations of up to 35°C, the joint estimator achieved a SoC error of 2%. Generally, the mean difference method usually estimates the SoC of the battery pack with good accuracy and low computational complexity.

The paper’s major contributions are as follows:

(1) First, a mechanistic model was developed to explore the characteristics of lithium-ion battery packs. Based on the analysis of battery pack characteristics, it was identified that temperature variations during the operation of series-connected battery packs could lead to internal inconsistencies in battery characteristics. To address this issue, BP-MDM-TEM was proposed. This model considers the thermal-electric coupling effect to describe the mean characteristics of the battery pack, enabling the characterization of the overall relationship between the battery pack and temperature changes. The difference model then describes the discrepancies between individual cell characteristics and the overall battery pack characteristics.

Based on the operating conditions of the Hybrid Pulse Power Characterization (HPPC) (Belt, 2010), we found that the temperature of each individual cell in the battery pack rises during operation due to heat absorption.

When the battery is at rest, it no longer generates heat, and the temperature will slowly decrease. And, the temperature variation differs slightly due to the different positions of the cells. The temperature of the cells in the middle is slightly higher than those on the outer edges. This is because the heat generated by the middle cells is not dissipated as quickly, resulting in slightly higher heat absorption. However, this difference is minimal, and the overall temperature trend of the cells within the pack is consistent.

The basic operating principle for the thermal-electric coupling model is shown in the mean model section of the Figure 1, and the upper left section is the circuit model, as shown in Equation 1: U t = U o c − U 1 − U 2 − I R 0 U ̇ 1 = − 1 R 1 C 1 U 1 − 1 C 1 I U ̇ 2 = − 1 R 2 C 2 U 2 − 1 C 2 I (1)

Where, U o c is usually a nonlinear function of SoC and the battery operating temperature; I R 0 is the voltage difference across the ohmic resistance R 0 ; R 1 and C 1 are the internal resistance and capacitance of electrochemical polarization, respectively, with the voltage across the R 1 C 1 circuit being the electrochemical polarization voltage is U 1 ; R 2 and C 2 are the internal resistance and capacitance of concentration polarization, respectively, with the voltage across the R 2 C 2 circuit being the concentration polarization voltage is U 2 . R 1 , C 1 , R 2 , C 2 change with the operating temperature (Gu and Wang, 2000). A thermal model is illustrated in Equation 2. C c T ̇ c = Q + T s − T c R c C s T ̇ s = T f − T s R u + T s − T c R c (2)

Where, T s and T c represent the battery surface temperature and the battery core temperature; T f is the ambient temperature; C c and C s represent the heat capacity of the battery core and the battery surface; R c is the equivalent heat conduction resistance, and R u is the equivalent convective resistance used to simulate convective cooling of the battery surface, which is affected by the geometry of the battery pack, coolant type, and coolant flow; Q represent the heat production during the battery operation.

Then from Bernardi equation, the heat production of the battery can be obtained as shown in Equation 3: Q = I * U o c − U t + I * T * d U o c d T (3) Using Equation 2, the battery core temperature and surface temperature are computed as the battery operating temperature T as shown in Equation 4: T = T c + T s / 2 (4)

Then, the battery operating temperature is used to modify the parameters in the Equation 1 to form a thermal-electrical coupling model with high simulation precision.

The thermal-electrical coupling model has the following three features:

First, the thermal-electrical coupling model, designed for real-time simulation with the control algorithm, can avoid the limitation of the finite element method, which cannot realize the real-time simulation of the battery temperature control algorithm. This effectively shortens the cycle of battery thermal management in the development of the control strategy, saving time.

Secondly, the thermal properties of the battery are very complex. The thermal-electrical coupling model can balance both the precision and computational complexity of the model. With appropriate computational intensity, the battery core temperature and battery surface temperature can be monitored more accurately simultaneously.

Thirdly, the thermal-electrical coupling model uses the equivalent circuit model and thermal model to describe the relationship between the thermal power of the battery and the current and voltage. Without considering the specific physical significance, the calculation is simple and easy to implement widely.

When the working temperature of the battery changes, the circuit model parameters related to the operating temperature of the battery will also change, making the identification of the thermal-electric coupling model very complex, and the identification accuracy cannot be guaranteed. In view of this problem, the identification of thermal-electric coupling model is regarded as an optimization process to find the optimal parameter values of a model under a specific open circuit voltage, SoC and battery operating temperature, so that the model output result and the measurement result are close.

The conventional MDM consists of CMM and CDM. CMM is used to describe the mean state of the battery pack, generally Thevenin model; CDM is used to describe the difference between the cell and mean state in the group, generally Rint model.

From the study of the temperature characteristics of the battery pack, it can be seen that the difference between the batteries in the group in the working process is relatively small, and the number of temperature sampling points set up in the actual packaging process of the automotive battery pack is limited, so only the temperature change of the battery pack as a whole needs to be taken into account in the estimation of the SOC of the battery pack.

Through the study of the open-circuit voltage characteristics of the battery pack, it can be seen that the open-circuit voltage of the batteries in the group under different ambient temperatures has small differences, and the overall trend of change is consistent, so the open-circuit voltage can be normalized;

Based on the model of the MDM, BP-MDM-TEM which proposed in the paper changes the Thevenin model used in the mean model to the thermal-electrical coupling model. Where T ̄ is the mean working temperature of the battery pack and U ̄ o c is the mean open circuit voltage. The model can take into account the change in the operating temperature of the battery pack to obtain an accurate battery pack model.

For simplicity of the calculation, the difference model ignores the differences in polarized voltage and polarized capacitance. The mean differential model for thermal-electric coupling is shown in Figure 1.

If the object is modeled as shown in Equation 10: x k = f k , x k − 1 + w k y k = h k , x k − 1 + v k F k = ∂ f ∂ x ̂ k − 1 H k = ∂ h ∂ x ̂ k − 1 (10)

The algorithm steps of ISH-AEKF are as follows:

Step 1: Compute a priori estimates of the system x ̂ k | k − 1 , and update the system covariance matrix P ̂ k | k − 1 , as shown in Equation 11:

Where Q k is the covariance of the process noise w k .

Step 2: Adaptive measurement of noise covariance estimation, as shown in Equation 12:

Where d k is the adaptive factor; b is forgetting factor, the value range is [0,1], usually set to [0.95,0.99];

Step 3: Calculate the Kalman gain matrix K k , as shown in Equation 13:

Due to the inconsistency within the battery pack, the remaining charge/discharge capacity of each cell varies. However, users are generally only interested in the SoC of the entire battery pack. Therefore, a method that can accurately estimate the SoC of the battery pack is needed. The paper proposes a method that combines the thermal-electrical coupling model with mean difference method (IMDM), which has the following characteristics:

First, The thermal-electrical coupling model is applied to the mean model to account for the effects of temperature variations in the battery pack.

Secondly, a dual-time scale is used to estimate the mean model and the difference model separately, which can reduce the estimation frequency of the difference model, thereby reducing the computational load of the system.

Thirdly, Based on this, the method integrates the significant battery method to monitor the charge level of each cell in the pack. When a cell’s charge is particularly high or low, a charge alert is set to prevent issues such as overcharging or over-discharging within the battery pack.

Taking into account the fact that certain batteries within the battery pack may experience overcharging or over-discharging when the pack is close to reaching its charging/discharging cut-off voltage, the paper proposed the important battery method and improve the mean difference model, real-time monitoring within the monomer battery, when a battery is high or low, set the power remind, avoid the group overcharged battery, etc. The process is illustrated in Figure 2.

Step 1: Measure the current of series battery pack I ( k ) , battery end voltage of each cell U t i ( k ) , ambient temperature of battery pack T f ( k ) , and surface temperature of battery pack T s j ( k ) .

Assuming the sampling period of the mean model CMM is Δ T .

Set up the following matrix as shown in Equations 17–23: x k = U 1 k U 2 k S o C ̄ k T ̄ c k T ̄ s k T (17) u k = I k Q ̄ k T f k T (18) y k = U t ̄ k T ̄ s k T (19) A k = e − Δ T R 1 k C 1 k 0 0 0 0 0 e − Δ T R 2 k C 2 k 0 0 0 0 0 1 0 0 0 0 0 e − Δ T R c k C c k e Δ T R c k C c k 0 0 0 e Δ T R c k C s k e − Δ T R c k C s k + Δ T R u k C s k (20) B k = R 1 k ∗ 1 − e − Δ T R 1 k C 1 k 0 0 R 2 k ∗ 1 − e − Δ T R 2 k C 2 k 0 0 − Δ T C N 0 0 0 R c k ∗ 1 − e − Δ T R c k C c k R c k C c k R u k C s k e Δ T R c k C c k − 1 0 R c k C s k C c k e Δ T R c k C s k − 1 R c k R u k + R c k 1 − e − Δ T R c k C s k − Δ T R u k C s k (21) C k = − 1 − 1 0 0 0 0 0 0 0 1 (22) D k = − R 0 ̄ I k + U o c ̄ k 0 (23)

Then the mean model equation of state is shown in Equation 24: x k = A k x k -   1 +   B k u k + ω k y k = C k x k -   1 + D k + ν k (24)

Step 2.1: The mean voltage U t ̄ ( k ) and mean surface temperature T ̄ s ( k ) are calculated based on Equations 5, 7.

The sampling period of the Difference Model (CDM) is Δ T d . Δ S o C i k = Δ S o C i k − 1 − I k Δ T d C N − S o C k ̄ − S o C k − 1 ̄ Δ U t i k = Δ U o c i S o C k ̄ , Δ S o C i k − I k Δ R i (31)

Let: x k i = Δ S o C i k T y k i = Δ U t i k T (32)

For the difference model: F k i = 1 , H k i = ∂ Δ U o c i ∂ Δ S o C i (33)

Similarly, it can be obtained based on the ISH-AEKF method x k i .

Step 4: Monitor and handle battery pack SoC exceptions.

Due to the inconsistency of the battery pack, the battery is in danger of overcharging and overdischarge in the group, so when the battery pack is in a low/high power state, more attention should be paid to the two single batteries with the lowest/high remaining power in the battery pack. The open circuit voltage of the battery is higher and changes faster when the remaining power is higher than 80% and lower than 20%. Therefore, the paper proposes to set the monitoring module to remind the state of the important battery. When the battery with the lowest remaining power (high) in the process of battery discharge (charge) is low/above 20% (80%), that is min ( S o C i ) ≤ 20 % , max ( S o C i ) ≥ 80 % the monitoring module will remind the user and reduce the step size estimated by the difference model to avoid the problem of battery over discharge or overcharge.

Since the battery pack S o C pack is used for calculation S o C ̄ , the user is more concerned about the battery pack S o C ̄ than the individual monomer Δ S o C i ,so the step size of the difference model can be set as shown in Equation 34: Δ T d = M * Δ T , M > 1 (34)

Which can effectively reduce the calculation amount. However, when the battery pack is calculated by two limit cells, the requirements for the difference model are increased, so the step size of the difference model needs to be changed M = 1 , so that the estimation accuracy can be effectively improved.

The test object of the paper is the rechargeable lithium-ion battery pack produced by Jiawei Longneng Solid State Energy Storage Technology Rugao Co. Ltd. with the model number JL60V25Ah, which consists of 19 single cells connected in series. The battery cells are model 90133200F25. The product specifications of the battery pack are shown in Table 1.

Battery pack uses RS485 communication, with a baud rate of 9,600, 8 data bits, and 1 stop bit, with no parity check. All data is requested by the charging and discharging device from the protection board, which sends responses including the total battery current (high and low), the voltage of individual cells (high and low), battery SoC, battery temperature, and battery series.

To charge the battery pack, a Chroma 62100H-1000 is used; for discharging, a Chroma 63210E-1200-400 programmable DC electronic load is chosen. The upper machine uses BMS test software provided by Hangzhou Xinwei Technology Co., Ltd., which can monitor the voltage, current, temperature, and other data of the battery pack in real time.

Based on the above devices, a hardware test platform for the battery pack is built, and the battery pack is tested under different working conditions. The charging test platform is shown in Figure 3A, and the discharging test platform is shown in Figure 3B.

In order to establish the battery pack model and verify the accuracy of BP-MDM-TEM estimation of battery pack SoC method, the paper designed HPPC test, constant current constant voltage charging test and constant current discharge test experiments at different ambient temperatures. The purpose of the HPPC condition test is to establish the battery pack model and identify the parameters of the mean difference model. The advantage of the HPCC condition test is not removing the battery pack to obtain the characteristics of the constant current constant voltage charging test and constant current discharge test is to verify the BP-MDM-TEM to estimate the accuracy and robust of the battery SoC, which can effectively reflect the battery charge and discharge state.

(1) The HPPC test operating condition place the battery pack at the set ambient temperature for the following steps:

In the HPPC test condition, in addition to recording the open circuit voltage value of the cell within [0%, 100%], the battery current, the battery voltage, the cell voltage and the mean temperature of the group measured in the experiment should also be recorded.

(2) Constant current and constant pressure charging test condition.

The test condition of constant current and constant voltage charging is the standard charging mode of the battery pack. First, the constant current charging is charged at 0.33°C rate, and when the charging stop voltage reaches 69.35 V, the charging is charged at constant voltage of 69.35 V until the current is less than 0.05°C.

(3) Constant current discharge test working condition.

The constant discharge test condition is the standard discharge mode of the battery pack. 0.33°C ratio is used to discharge the full battery pack until the discharge cut-off voltage reaches 47.5 V.

In order to verify that BP-MDM-TEM has good SoC estimation accuracy in charge-discharge experiments, the paper tests standard charge-discharge of the battery pack and substitutes the SoC estimation data, such as measured voltage, measured current, and measured temperature obtained by the upper computer, into BP-MDM-TEM. The results are shown in Figures 8A, B.

The above figures compare the traditional MDM and the Battery Pack MDM with BP-MDM-TEM under constant current-constant voltage charging and constant current discharging conditions. The objective is to verify whether applying the thermal-electric coupling model to the mean model can effectively enhance the accuracy of SoC estimation for battery packs. Figures 9A, B depict error plots for this comparison.

As can be seen from the charge-discharge error diagrams in Figures 9A, B, the battery operating temperature of the model fluctuates throughout the SoC estimation. However, the BP-MDM-TEM applies the thermal-electric coupling model to the mean model and updates the model parameters with the battery’s operating temperature. Therefore, the BP-MDM-TEM shows a clear convergence trend in the early stages of estimation. This is because during the battery’s operation, part of the heat is not released in time and is absorbed by the battery itself, causing an increase in battery temperature. This temperature affects the battery parameters and changes its external characteristics, leading to a difference in temperature between MDM and the reference value. In the later stages of estimation, the BP-MDM-TEM’s SoC estimate can converge near the reference value.

According to the experimental results, the MAE of SoC estimated by the BP-MDM-TEM is 0.468% and RMSE is 0.982%, while the MAE of the traditional MDM method is 0.992% and RMSE is 1.279%. It is evident that the BP-MDM-TEM has higher accuracy, meeting the requirements for SoC estimation.