2025 Volume 93 Issue 8 Pages 087003
Accurate temperature prediction plays a vital role in the thermal management and safety assurance of lithium-ion battery systems. This study proposes a hybrid temperature prediction model that integrates Fully Connected Networks (FCN) and Gradient Boosting Machines (GBM) to capture temperature evolution under varying discharge rates. A nonlinear electrothermal simulation framework based on the Nonlinear Thermal Generalized Kalman (NTGKF) model is first constructed to analyze the electrothermal coupling behavior of batteries under discharge rates ranging from 1 to 5 times the nominal capacity. Leveraging the nonlinear feature extraction capability of FCN and the ensemble learning robustness of GBM, an FCN–GBM hybrid model is developed and evaluated using different input configurations, including voltage alone, internal resistance alone, and the combination of both. Simulation and prediction results demonstrate that the evolution of voltage and internal resistance closely aligns with temperature variation, indicating their suitability as key features for temperature modeling. The proposed FCN–GBM model is applied to predict discharge temperature profiles of LiFePO4 (LFP) and LiNi0.8Co0.15Al0.05O2 (NCA) cells. The combination of voltage and resistance as input features significantly enhances prediction performance. Under the 20 % training data condition, the LFP model achieves a mean absolute error (MAE) of 0.4576 K and a root mean square error (RMSE) of 0.5411 K, compared to 1.3404 K and 1.5727 K for the baseline GBM model. For the NCA battery, the FCN–GBM model achieves an MAE of 0.3025 K and an RMSE of 0.9973 K, also outperforming GBM with respective errors of 0.8447 K and 1.5878 K. Moreover, the use of single input features leads to larger prediction errors. These results confirm the effectiveness of the FCN-enhanced GBM model and the advantage of feature fusion using voltage and resistance, providing practical insights for improving thermal management and risk mitigation in lithium-ion battery systems.
Lithium-ion batteries are widely employed in electric vehicles, energy storage systems, and portable electronic devices.1,2 Among the many operational factors, working temperature exerts a significant influence on battery safety, lifespan, and efficiency.3 During high C-rate charge and discharge processes, substantial heat is generated, leading to rapid temperature rise, which may cause performance degradation and, in severe cases, trigger safety incidents.4 Consequently, accurate temperature prediction and effective thermal regulation have become critical areas of research within the development of battery management systems (BMS).5
Under high-rate discharge conditions, the heat generated within the battery increases dramatically, far exceeding that under conventional operating scenarios. This can rapidly elevate the cell temperature and trigger a cascade of side reactions, such as decomposition of the SEI layer, thermal breakdown of the electrolyte, and reactions between active lithium and the electrolyte, all of which significantly elevate the risk of thermal runaway.6 Such risks are exacerbated under limited heat dissipation conditions or when thermal management systems respond slowly,7 leading to the rapid propagation of local hotspots and the potential development of uncontrollable thermal chain reactions.
However, due to the rapid dynamics and the difficulty of real-time measurement of internal parameters under high C-rate conditions, experimental methods are often insufficient to capture the detailed internal temperature distribution, electrochemical kinetics, and structural evolution of the battery cells.8 To address this limitation, this study adopts a nonlinear thermal–electrochemical model based on the nonlinear thermal generalized Kalman filter (NTGKF) for multiphysics-coupled simulation. As a nonlinear electrothermal model, NTGKF has been widely applied in battery thermal management and state estimation. It can not only replicate high-rate discharge processes under controlled conditions but also systematically analyze key factors affecting thermal behavior, such as internal resistance evolution, voltage response, current density fluctuations, and polarization effects. This enables accurate prediction of temperature trends and quantitative assessment of thermal risks, thereby offering theoretical support and decision-making guidance for the design of thermal management systems and safety strategies.
Recent approaches to battery temperature prediction generally fall into two categories: physics-based modeling and data-driven techniques.9,10 While physics-based models offer higher interpretability, their prediction accuracy is often limited by parameter uncertainties and the complexity of real-world operating conditions.11,12 On the other hand, traditional data-driven models—such as support vector machines (SVM),13 artificial neural networks (ANN),14 and random forests (RF)—may offer higher accuracy in specific contexts but often suffer from poor generalization and limited effectiveness in handling abrupt thermal changes.15 This is especially problematic under high C-rate operations, where rapid temperature fluctuations can lead to significant prediction lag and compromise the reliability of BMS decision-making. For example, ANN and RF models typically exhibit temperature prediction errors exceeding 1.5 K during 4 C or 5 C discharges, posing serious safety risks.16 Studies have also pointed out the inability of conventional data-driven methods to capture the dynamic internal states of batteries. Therefore, the development of a more accurate and generalizable predictive model has become a pressing research challenge in the field of battery thermal management.
In recent years, several large-scale accidents in energy storage systems—such as the 2024 battery plant fire in Hwaseong, Korea,17 and the 2025 Moss Landing storage facility fire in the U.S.18—have exposed the significant safety risks associated with high-rate discharge conditions, particularly during discharge phases that are more prone to thermal runaway. However, due to the limited sampling rate and delayed response of field monitoring systems, high-resolution data reflecting the coupled evolution of voltage, internal resistance, and temperature are often unavailable, which hinders in-depth analysis of precursors to thermal runaway.
To compensate for the scarcity of high-fidelity measurement data, this study develops a high-accuracy electrothermal coupled simulation model under various C-rate discharge conditions. Systematic simulation results reveal that the voltage curve exhibits distinct drop points that closely coincide with temperature spikes. Concurrently, internal resistance also demonstrates abrupt changes during these phases, with all three phenomena showing strong temporal synchronicity. This suggests that voltage and resistance jointly serve as sensitive and predictive features for early identification of thermal escalation trends.
Based on these insights, this study proposes a hybrid model that integrates fully connected networks (FCN) with gradient boosting machines (GBM), using voltage (V) and internal resistance (Ro) as core input features. The FCN component enhances the model’s capability to express nonlinear thermal responses, while the GBM structure improves learning robustness—particularly under conditions with limited training data—thereby improving prediction accuracy and generalization.
This work makes three key contributions. First, it establishes a system-level electrothermal simulation framework that uncovers the dynamic interplay among voltage, internal resistance, and temperature, revealing a clear temporal alignment between voltage drops, temperature spikes, and resistance mutations—thus identifying crucial precursor features for thermal modeling. Second, it introduces a novel hybrid model that combines FCN with GBM, effectively leveraging voltage and resistance information to enhance the learning capacity and predictive accuracy for battery thermal behavior. Finally, the model is extensively validated using two publicly available battery datasets—LiFePO4 (LFP) and LiNi0.8Co0.15Al0.05O2 (NCA)—across various discharge rates and training ratios, demonstrating strong adaptability to different battery chemistries and confirming its generalization capability and practical application value.
The outcomes of this study offer a theoretical foundation for the construction of temperature prediction models in lithium-ion battery management systems and propose new perspectives for ensuring the safe and efficient operation of battery systems. The predicted temperature profiles can be integrated into real-time thermal management strategies, such as initiating cooling actions, adjusting current loads, or activating safety protocols when abnormal thermal behavior is detected. These practical applications enhance the reliability and safety of battery systems in electric vehicles and energy storage stations.
In the battery model simulation, a single cell from a 1P3S water-cooled lithium-ion battery module with a rated capacity of 60 Ah, used in an energy storage power station, was selected as the research object. The model retained most of the structural and thermal characteristics of the actual module, including the cell body, tabs, bus bars, insulation materials, casing, and water-cooling plate. The simulation was conducted using COMSOL Multiphysics and Fluent software based on the NTGKF model. The key input parameters included rated capacity, voltage range, discharge rates (from 1 C to 5 C), specific heat capacity, thermal conductivity, and internal thermal-physical properties. The initial depth of discharge (DOD) was set to 1, corresponding to a fully charged state. Detailed simulation parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| Minimum Cut-off Voltage (V) | 3.0 |
| Maximum Cut-off Voltage (V) | 4.2 |
| System Power (W) | 60.0 |
| External Resistance (Ω) | 1.0 |
| Specific Heat Capacity (J/kg K) | 871.0 |
| Thermal Conductivity (W/m K) | 20.0 |
| Electrical Conductivity (S/m) | 1.0 × 10−7 |
The simulation focused on evaluating impedance, voltage curves, and temperature distributions of the battery module.
For model validation, experimental data were obtained from two sources: a LFP battery provided by Toyota Research Lab and a NCA battery from NASA. Both datasets include time, voltage, current, internal resistance, and battery temperature data. The LFP cell, labeled 2017-05-12-CH35 (designated as Battery #1), had a rated capacity of 1.1 Ah.19 It was first charged using a fast-charging protocol, followed by a 1 C constant-current (CC) and constant-voltage (CV) charging cycle, and then discharged at 4 C. The cut-off voltages were set at 3.6 V (upper) and 2.0 V (lower), with tests conducted at 30 °C. The charging and discharging strategy are shown in Fig. 1a.
Charging and discharging strategy for Battery #1 (a) and Battery #2 (b).
The NCA battery, labeled B5 (designated as Battery #2), had a rated capacity of 2 Ah.20 It underwent 1 C constant-current discharging to 2.7 V, followed by 3/4 C charging to 4.2 V with CV hold until the current dropped below 20 mA. Tests were conducted at 24 °C. Its charging and discharging protocol is illustrated in Fig. 1b.
Prior to model training, standard preprocessing procedures were applied to improve data quality and ensure model stability. Specifically, the raw temperature data were smoothed using a moving average filter with a window size of 5 to suppress high-frequency noise. Outliers were detected and removed based on a three-sigma rule, in which any data point deviating more than three standard deviations from the local mean was excluded. All input features were scaled to the [0, 1] range using min–max normalization to reduce scale-related bias and promote convergence stability during training. These preprocessing steps were essential to enhance model generalization and ensure the robustness of the training process.
The NTGKF model is an empirical electrothermal framework calibrated using experimental data. It simulates heat generation and transfer within battery cells through coupled electrochemical-thermal equations. In this study, the NTGKF model was used to simulate battery behavior under 1 C to 5 C discharge conditions, aiming to investigate electrothermal responses and extract key features.
The simulation methodology of the NTGKF model is as follows.
First, the voltage and time data of the battery under different discharge rates are obtained. The DOD is calculated using Eq. 1,21 from which the variable migration current j is derived.
| \begin{equation} \mathit{DOD} = \frac{\textit{Vol}}{3600 \times Q_{\text{nominal}}}\int_{0}^{\text{t}}jdt \end{equation} | (1) |
Here, Vol denotes the battery voltage (unit: V), Qnominal represents the nominal capacity of the battery (unit: Ah), and j refers to the current density (unit: A/m2).
Subsequently, a fifth-order polynomial is employed to fit the battery’s open-circuit voltage (OCV) and impedance coefficients. The relationship between the open-circuit voltage (U) and thermal effects is described by Eq. 2,22 where U denotes the battery’s OCV, which is a function of both the DOD and temperature.
| \begin{equation} U = \left(\sum_{\text{n}=0}^{5}a_{\text{n}}(\mathit{DoD})^{\text{n}}\right) - C_{1}(T - T_{\text{ref}}) \end{equation} | (2) |
In this context, U denotes the open-circuit voltage (unit: V); an represents the fitting coefficients that characterize the OCV–SOC relationship; DOD refers to the depth of discharge (unit: %); C1 is the temperature correction factor that accounts for the effect of temperature on OCV; T is the current battery temperature (unit: K); and Tref is the reference test temperature (unit: K).
Equation 3 defines the parameter equation of the battery impedance model, which serves as an empirical expression describing the variation of polarization resistance with respect to DOD and temperature.23
| \begin{equation} Y = \left(\sum_{\text{n} = 0}^{5}b_{\text{n}}(\mathit{DoD})^{\text{n}}\right)^{-1}\exp \left[C_{2}\left(\frac{1}{T} - \frac{1}{T_{\text{ref}}}\right)\right] \end{equation} | (3) |
Here, Y denotes the impedance-related coefficient (unit: Ω); DOD refers to the depth of discharge (unit: %); bn represents the fitting coefficients; C2 is the temperature correction factor; T is the current battery temperature (unit: K); and Tref is the reference test temperature (unit: K).
In the third step, a thermal generation model is constructed based on Joule heat and entropic heat, and a heat conduction model is applied to simulate the internal temperature field of the battery. Equations 4 and 5 are used to calculate the heat generated by Joule heating (i.e., ohmic loss) and entropy change.24,25 These two components constitute the primary thermal sources in the battery, which significantly affect the internal temperature distribution and require regulation via a thermal management system. The Fluent software is employed to extract relevant data from the discharge rate curves, and both Y and U are fitted as functions of the DOD.
| \begin{equation} j_{\text{ech}} = \frac{Q_{\text{nominal}}}{Q_{\text{ref}} \times \textit{Vol}}Y \times [U - V] \end{equation} | (4) |
| \begin{equation} q_{\text{ech}} = j_{\text{ech}}\left[U - V - T\frac{\text{d}U}{\text{d}T}\right] \end{equation} | (5) |
Here, jech denotes the heat generation power associated with current density; Qnominal is the nominal capacity of the battery (unit: Ah); Qref is the reference capacity used for normalization (unit: Ah); Y represents the polarization-related term; U − V refers to the polarization loss, also known as voltage drop; qech denotes the total heat generation, including both Joule heat and entropic heat; and $T\frac{\text{d}U}{\text{d}T}$ accounts for the temperature effect on the battery.
Equation 6 represents the heat conduction equation of the battery, which is used to simulate the internal temperature distribution by considering both heat conduction and heat dissipation.26 This equation governs the thermal diffusion and heat release inside the battery and determines the spatial distribution of temperature:
| \begin{equation} \frac{\partial \rho C_{\text{P}}T}{\partial tT} + \nabla (\rho\overset{\rightharpoonup}{V}C_{\text{P}}T) = \nabla (k\nabla T) + \dot{q} \end{equation} | (6) |
Where ρ is the density of the battery (unit: kg/m3), CP is the specific heat capacity (unit: J/kg K), T is the temperature (unit: K), k is the thermal conductivity (unit: W/m K), and $\dot{q}$ is the heat generation rate, also referred to as the heat source term, which includes both Joule and entropic heating.
Equations 7 and 8 describe the conservation of current, where variations in temperature induce thermal expansion,27,28 which in turn leads to mechanical stress that may cause cell swelling or material degradation.
| \begin{equation} \nabla(\sigma_{+} \nabla\phi\sigma_{+}) - j = 0 \end{equation} | (7) |
| \begin{equation} \nabla(\sigma_{-} \nabla\phi\sigma_{-}) + j = 0 \end{equation} | (8) |
Here, σ+, σ− represent the thermal stress distributions, and j is the migration current.
The NTGKF model effectively characterizes the electro-thermal coupled behavior of lithium-ion batteries under various discharge rates, providing reliable simulation data to support the subsequent extraction of critical features and the development of temperature prediction models. As illustrated in Fig. 2, this model integrates the thermal and electrical domains into a unified simulation framework tailored for battery modules. Section 4.1 will present detailed analyses based on the simulation results, aiming to identify key factors that influence battery temperature variations.
Simulation framework of the NTGKF model for the battery module.
The GBM algorithm, proposed in 1999 by Jerome H. Friedman, is an ensemble learning method that enhances prediction performance by combining multiple decision trees.29 The overall process is illustrated in Fig. 3. GBM builds a sequence of models, each designed to correct the prediction errors made by the previous one. The final output is obtained by aggregating the predictions of all models through weighted averaging.
Structure of the GBM algorithm.
At the initial stage, the model output is usually set to the mean of the target variable y, as defined in Eq. 9.30
| \begin{equation} F_{0}(x) = \frac{1}{N}\sum_{\text{i}=1}^{\text{N}}y_{\text{i}} \end{equation} | (9) |
Here, N represents the total number of samples, and yi refers to the true value of the ith sample.
At each iteration m = 1, 2, …, M, the model is updated by first computing the residuals, which measure the discrepancy between the current prediction and the actual values, as shown in Eq. 10.31
| \begin{equation} r_{\text{i}}^{\text{(m)}} = y_{\text{i}} - F_{\text{m}-1}(x_{\text{i}}) \end{equation} | (10) |
In this equation, yi represents the true value, and Fm−1(xi) is the predicted value for the ith sample from the previous iteration m − 1.
Next, the residuals $r_{\text{i}}^{\text{(m)}}$ are used as the new target to train a regression tree hm(x), which aims to minimize the sum of squared residuals across all samples, as shown in Eq. 11.32
| \begin{equation} h_{\text{m}}(x) = \arg\min \end{equation} | (11) |
Here, hm(x) denotes the newly trained regression tree, $r_{\text{i}}^{\text{(m)}}$ is the residual of the ith sample at iteration m, and h(xi) represents the model fitted to the residuals during the current step.
By minimizing the objective function, the new tree hm(x) provides a better fit to the residuals, thereby improving the overall prediction.
The updated model after adding the new tree is given in Eq. 12.33
| \begin{equation} F_{\text{m}}(x) = F_{\text{m}-1}(x) + vh_{\text{m}}(x) \end{equation} | (12) |
The parameter v represents the learning rate, which controls the contribution of each decision tree to the overall model. A smaller learning rate generally leads to a more robust model but may require a greater number of iterations to achieve optimal performance.
This iterative process continues until further improvements in model performance become negligible. After M iterations, the final model is defined as shown in Eq. 13.34
| \begin{equation} F_{\text{M}}(x) = F_{0}(x) + \sum_{\text{m}=1}^{\text{M}}vh_{\text{m}}(x) \end{equation} | (13) |
In this equation, FM(x) represents the final predictive model after M iterations, while F0(x) serves as the initial model.
In the GBM algorithm, each iteration involves training a new regression tree to fit the residuals from the current model. This approach gradually reduces prediction errors and continuously improves the model’s overall performance.
The tree-based iterative structure allows GBM to handle complex data patterns and nonlinear relationships effectively.
However, when applied to battery temperature prediction tasks, traditional GBM still faces certain limitations. It tends to rely too heavily on individual features, making it difficult to capture the complex nonlinear interactions between temperature and multiple input parameters.
To address this issue, it is necessary to integrate GBM with a model that possesses stronger nonlinear representation and feature extraction capabilities. Such integration is expected to enhance prediction accuracy in scenarios involving multi-feature fusion, nonlinear modeling, and limited training data.
3.2.2 Fully connected networkTemperature prediction is typically influenced by multiple factors, including internal resistance, charge–discharge states, and ambient temperature, all of which often exhibit complex nonlinear relationships.
FCN offer strong nonlinear representation capabilities. By applying multilayer nonlinear transformations, FCN can extract deep relationships among input features. Each neuron in a given layer is connected to all neurons in the previous layer, enabling the network to effectively capture intricate dependencies between features and generate high-level representations.
This architecture significantly enhances feature learning performance and is particularly well-suited for battery temperature prediction tasks involving multiple influencing variables.
The first step involves the fusion of different input features. As shown in Eq. 14, two features are concatenated to construct a new feature representation.34
| \begin{equation} H = [H_{1}\parallel H_{2}] \end{equation} | (14) |
In this context, H1 and H2 represent Feature 1 and Feature 2, respectively, and the symbol “||” denotes vector concatenation.
The resulting fused feature vector H is then passed into the FCN for further processing.
The input vector is first processed by a fully connected layer, followed by a ReLU activation function, and then passed through a second fully connected layer. The final output is a transformed feature vector that captures higher-level representations. The mathematical formulation of the fully connected layers is presented in Eq. 15.
| \begin{equation} Y = f(W \times H + b) \end{equation} | (15) |
Here, H denotes the input feature vector, W is the weight matrix, and b is the bias vector.
The weight matrices W and biases b in each layer are optimized to minimize prediction error. The output vector is represented by Y, and f denotes the activation function, ReLU, which enhances the model’s nonlinearity by outputting only positive values. Given the input x, the final output is computed as shown in Eq. 16.
| \begin{equation} f = \max(0,x) \end{equation} | (16) |
The ReLU activation function introduces nonlinearity by truncating negative values while retaining positive inputs. This operation enhances the model’s expressive power and effectively captures nonlinear relationships among features.
The processed feature vector is passed through a sequence of fully connected layers in a recursive manner, where each layer performs a linear transformation followed by an activation function. This multilayer mapping extracts deeper feature representations, further improving the model’s learning capability. At the final stage, the output layer of the FCN generates a new feature vector Y, which is used for subsequent model prediction.
To prevent overfitting and enhance generalization, the FCN incorporates L2 regularization and an early stopping strategy during training. Regularization is achieved by adding a penalty term to the loss function to constrain model complexity. The early stopping mechanism monitors the validation loss and halts training when performance no longer improves, avoiding excessive training.
This feature fusion and FCN processing pipeline is illustrated in a simplified flowchart. Figure 4 highlights the key steps from feature concatenation to deep feature extraction, demonstrating the FCN’s ability to efficiently learn complex feature interactions.
Feature fusion and deep representation learning via fully connected network.
By integrating the FCN algorithm with GBM, the model benefits from the strong feature extraction and nonlinear modeling capabilities of FCN while retaining the efficiency and robustness of GBM’s ensemble learning framework.
The algorithmic flow of the FCN-GBM model is illustrated in Fig. 5. Initially, input features are fed into the FCN to perform nonlinear feature extraction. To optimize this process, the FCN’s learning rate, number of layers, and number of neurons were carefully tuned. The learning rate was set to 0.001 and adjusted dynamically using a learning rate scheduler under the Adam optimizer to ensure convergence. The number of layers was increased gradually from 3 to 5 to observe performance changes, and reduced when signs of overfitting appeared.
Workflow of the FCN-GBM algorithm.
The number of neurons was set to 128 in the first layer and gradually reduced in subsequent layers to form a pyramidal structure. To mitigate overfitting, L2 regularization was applied, with the regularization coefficient initially set to 32 or 64, and increased to 128 if necessary based on generalization performance. An early stopping strategy was also employed, halting training when validation loss no longer showed improvement, which ensured the stability and reliability of the extracted features.
Meanwhile, key parameters were tuned in the GBM training stage, including maximum depth and subsampling rate. The maximum depth was varied from 3 to 10, and cross-validation was used to determine the optimal value that balances model complexity and overfitting. The final depth was set to 5. The subsampling rate was set to 0.8 to maintain data diversity while preventing overfitting. The number of decision trees was initially set to 100. The minimum samples for node splitting and leaf nodes were set within the ranges of 2–10 and 5–20, respectively, to control the tree’s complexity and reduce overfitting. L2 regularization ranging from 0.01 to 0.1 was found effective in enhancing generalization and improving test performance.
Throughout the entire training process, all input features were normalized to ensure consistent scales across variables and to stabilize gradient updates during optimization.
3.3 Model evaluation metricsThree metrics are employed to evaluate the accuracy of the prediction results, namely Absolute Error (AE), Mean Absolute Error (MAE), and Root Mean Square Error (RMSE), as defined in Eqs. 17, 18, and 19, respectively.
| \begin{equation} \text{AE} = T_{\text{predicted,i}} - T_{\text{real,i}} \end{equation} | (17) |
| \begin{equation} \text{MAE} = \frac{\displaystyle\sum_{\text{i}=1}^{\text{n}}|T_{\text{predicted,i}} - T_{\text{real,i}}|}{n} \end{equation} | (18) |
| \begin{equation} \text{RMSE} = \sqrt{\frac{\displaystyle\sum_{\text{i}=1}^{\text{n}}(T_{\text{predicted,i}} - T_{\text{real,i}})^{2}}{n}} \end{equation} | (19) |
Here, n denotes the total number of cycles, i represents the ith cycle, Tpredicted,i is the predicted temperature, and Treal,i is the actual temperature.
This section analyzes key thermal characteristics based on the NTGKF model under varying discharge rates. Figure 6a illustrates the voltage-time curves at discharge rates of 1 C, 2 C, 3 C, 4 C, and 5 C. It is observed that as the discharge rate increases, the voltage drops more rapidly, particularly at the initial and later stages of discharge, due to enhanced polarization and diffusion effects.
(a) Battery voltage–time curve; (b) Battery voltage–DOD curve.
Using Eq. 20, the voltage-DOD curves were derived, as shown in Fig. 6b, where DOD ranges from 0 % (fully charged) to nearly 100 % (fully discharged).
| \begin{equation} \mathit{DOD} = \frac{Q_{\text{discharge}}}{Q_{\text{total}}} \times 100\,\% \end{equation} | (20) |
Here, Qdischarge represents the discharged capacity (unit: Ah), and Qtotal refers to the nominal total capacity of the battery (unit: Ah).
Subsequently, the DOD range from 0 % to 90 % was evenly divided into nine consecutive intervals, each spanning 10 %. Voltage data corresponding to each interval were extracted for further analysis. Figure 7 presents the voltage–discharge rate curves at different DOD levels. For each fixed DOD, the voltage values at various discharge rates were linearly fitted. The y-intercept of each fitted line is regarded as the OCV at that DOD level, corresponding to parameter U in the NTGKF model. The absolute value of the slope reflects the polarization resistance at that DOD level, which corresponds to the model parameter Y. The extracted values of U and Y for all DOD intervals are summarized in Table 2.
Voltage–discharge rate curves under different DOD levels.
| DOD (%) | U (V) | |Y| (Ω) |
|---|---|---|
| 0 | 4.1183 | 0.0125 |
| 10 | 4.0478 | 0.0214 |
| 20 | 3.9917 | 0.0232 |
| 30 | 3.9428 | 0.0221 |
| 40 | 3.8940 | 0.0225 |
| 50 | 3.8384 | 0.0243 |
| 60 | 3.7688 | 0.0251 |
| 70 | 3.6782 | 0.0232 |
| 80 | 3.5595 | 0.0216 |
| 90 | 3.4052 | 0.0319 |
To enable continuous use in the NTGKF model, the discrete U(DOD) and Y(DOD) values were fitted using polynomial functions, shown in Eqs. 21 and 22. The fitted curves are illustrated in Fig. 8.
| \begin{align} U &= 4.12\mathit{DOD}^{0} - 7.89e^{-1}\mathit{DOD}^{1} + 1.17\mathit{DOD}^{2} - 1.68\mathit{DOD}^{3} \\ &\quad + 9.07e^{-1}\mathit{DOD}^{4} - 4.49e^{-1}\mathit{DOD}^{5} \end{align} | (21) |
| \begin{align} Y &= 4.87e^{3}\mathit{DOD}^{0} - 3.67e^{4}\mathit{DOD}^{1} + 2.13e^{5}\mathit{DOD}^{2} \\ &\quad - 5.39e^{5}\mathit{DOD}^{3} + 6.04e^{5}\mathit{DOD}^{4} - 2.44e^{5}\mathit{DOD}^{5} \end{align} | (22) |
Fitted curves of U and Y functions of DOD.
To represent the evolution of polarization resistance over time, the function Y(DOD) was transformed into Y(t) based on the linear relationship between depth of discharge and time under constant C-rate discharge, as defined in Eqs. 23 and 24.
| \begin{equation} \mathit{DOD}(t) = \frac{t}{t_{\text{total}}} \end{equation} | (23) |
Here, t denotes the current time, and ttotal is the total discharge time under the given C-rate. Both are measured in seconds. The calculation is shown in Eq. 24.
| \begin{equation} t_{\text{total}} = \frac{Q_{\text{nominal}}}{I} \times 3600 \end{equation} | (24) |
In this equation, Qnominal is the nominal capacity of the battery in Ah, and I is the constant discharge current in A.
A pronounced drop in resistance can be observed in the Fig. 9a, which shows a strong correlation with the subsequent rapid temperature rise. Accordingly, the inflection points corresponding to these sharp drops are marked in the Fig. 9a to support subsequent feature extraction and thermal behavior analysis.
Analysis of battery parameter variations over time and inflection point responses under different discharge rates: (a) Variation of battery resistance coefficient Y with time; (b) Voltage variation with time.
The discharge voltage curves at different C-rates were fitted using the Bacon-Watts model via nonlinear least squares regression implemented in Python, in order to extract the inflection point associated with the onset of rapid voltage decline. The mathematical form of the Bacon-Watts model is given in Eq. 25.
| \begin{equation} V(t) = a + bt + c(t - \tau) \times \frac{1 + \tan h[d(t - \tau)]}{2} \end{equation} | (25) |
Where V(t) represents the discharge voltage at time t; a and b denote the intercept and slope of the initial linear segment; c characterizes the change in slope after the inflection; τ is the inflection point indicating the beginning of the steep voltage drop; and d is a smoothing parameter that controls the sharpness of the transition. This model enables a continuous and differentiable connection between two linear regimes, and is well-suited to describing nonlinear voltage decay behavior.
Table 3 summarizes the inflection times τ extracted by fitting the Bacon-Watts model to the discharge data under various C-rates. Figure 9b presents the voltage profiles at different discharge rates, with the inflection points identified by the model clearly annotated. The results indicate that the inflection time τ shifts markedly earlier as the discharge rate increases, exhibiting a strong inverse correlation. This observation implies that high-rate discharges accelerate the transition from the quasi-stable voltage plateau to the rapid decay phase. Such a trend is attributed to enhanced polarization and intensified electrochemical reactions under higher current densities, which lead to earlier onset of terminal degradation and a shortened effective discharge duration. Based on the NTGKF model and the previously defined parameters, thermal simulations were carried out to evaluate the temperature evolution and spatial distribution of the battery module under different discharge rates. The results are summarized in Fig. 10. On the left side of the figure, temperature–time curves for each discharge rate are presented, with two critical points clearly marked: the onset of rapid temperature rise and the final temperature point at the end of discharge.
| C-rate | Inflection Time τ (s) |
|---|---|
| 1.00 | 3392.36 |
| 2.00 | 1663.11 |
| 3.00 | 1084.24 |
| 4.00 | 804.39 |
| 5.00 | 635.80 |
Simulated temperature–time curves under different discharge rates, along with temperature field distributions corresponding to the onset of rapid temperature rise and the final temperature point.
The timing of these inflection points shifts earlier as the discharge rate increases, revealing a consistent trend that reflects the enhanced electrothermal coupling at high-rate operation. This behavior indicates that stronger electrochemical activity leads to faster heat accumulation and earlier thermal response.
On the right side of Fig. 10, the corresponding simulated temperature field distributions are shown for the two marked points. At the onset of rapid temperature rise, localized heating begins to emerge, while by the end of discharge, the overall temperature is significantly elevated and more unevenly distributed. The progression from a relatively stable to a highly heterogeneous temperature field illustrates the dynamic spatial effects of electrothermal behavior during high-rate discharge.
Figure 11 compares the timing of resistance inflection points, voltage drops, and temperature spikes under different discharge rates. As the discharge rate increases, these key transitions occur earlier. Notably, their temporal order remains consistent: resistance change precedes voltage drop, which in turn precedes temperature rise. This stable sequence suggests that resistance and voltage signals serve as early indicators for temperature evolution and potential thermal risk, providing a physical foundation for predictive modeling.
Comparison of the response times of inflection points in resistance, voltage, and temperature under different discharge rates.
Following feature extraction, this section evaluates the predictive performance of the proposed FCN-GBM model compared to a baseline GBM, using V, Ro, and their combination (V + Ro) as input features. Model performance is tested under varying training data proportions (80 %, 60 %, 40 %, and 20 %). The influence of various models and input parameter combinations on prediction performance is detailed in Table 4.
| Battery Number |
Model | Input Parameters | Input Parameters Proportion of Training Set (%) |
AEMax (K) | MAE (K) | RMSE (K) |
|---|---|---|---|---|---|---|
| Battery #1 | FCN-GBM | V | 80 | 0.8920 | 0.3563 | 0.3929 |
| 60 | 1.2506 | 0.5770 | 0.6239 | |||
| 40 | 2.2705 | 0.4951 | 0.6396 | |||
| 20 | 3.4236 | 1.0666 | 1.3158 | |||
| Ro | 80 | 0.6907 | 0.2172 | 0.2744 | ||
| 60 | 0.8071 | 0.2516 | 0.3458 | |||
| 40 | 1.6759 | 0.4732 | 0.4853 | |||
| 20 | 3.1158 | 0.8941 | 1.1478 | |||
| V + Ro | 80 | 0.4054 | 0.1405 | 0.1653 | ||
| 60 | 0.7271 | 0.2251 | 0.2792 | |||
| 40 | 1.5075 | 0.2044 | 0.2794 | |||
| 20 | 1.3177 | 0.4567 | 0.5411 | |||
| GBM | V + Ro | 80 | 1.3108 | 0.5375 | 0.6736 | |
| 60 | 2.1993 | 0.6492 | 0.7258 | |||
| 40 | 2.3082 | 0.6611 | 0.8407 | |||
| 20 | 3.8155 | 1.3404 | 1.5727 | |||
| Battery #2 | FCN-GBM | V | 80 | 0.1749 | 0.2148 | 0.4836 |
| 60 | 0.2282 | 0.2694 | 0.5857 | |||
| 40 | 0.5456 | 0.5955 | 1.1798 | |||
| 20 | 0.6952 | 0.7538 | 1.4654 | |||
| Ro | 80 | 0.1125 | 0.1465 | 0.3359 | ||
| 60 | 0.1412 | 0.1814 | 0.5574 | |||
| 40 | 0.3038 | 0.3909 | 1.1502 | |||
| 20 | 0.5667 | 0.6076 | 1.2494 | |||
| V + Ro | 80 | 0.0941 | 0.1045 | 0.1941 | ||
| 60 | 0.1002 | 0.1138 | 0.2005 | |||
| 40 | 0.2235 | 0.2991 | 0.6815 | |||
| 20 | 0.2753 | 0.3025 | 0.9973 | |||
| GBM | V + Ro | 80 | 0.2112 | 0.2482 | 0.5344 | |
| 60 | 0.2384 | 0.2826 | 0.7636 | |||
| 40 | 0.5454 | 0.5992 | 1.2451 | |||
| 20 | 0.7901 | 0.8447 | 1.5878 |
Figure 12 compares the temperature prediction performance of FCN-GBM and GBM using V + Ro as inputs. FCN-GBM consistently outperforms GBM across all training ratios, with the performance gap widening as the training set size decreases. This demonstrates the FCN-GBM model’s robustness and superior feature representation capacity, especially in low-data scenarios.
Comparison of prediction performance between the FCN-GBM and GBM models using voltage and internal resistance as input features.
Further comparison in Fig. 13 reveals that using both V and Ro as input features significantly improves prediction accuracy over using either feature alone. For instance, in Battery #1 with 80 % training data, using only V yields a MAE of 0.3563 K, and using only Ro yields an MAE of 0.2172 K. However, the combined V + Ro input reduces the MAE to 0.1405 K. In Battery #2, with 60 % training data, the RMSE drops from 0.5857 K (V only) and 0.5574 K (Ro only) to just 0.2005 K when both features are used—a 66 % error reduction.
Prediction performance of the FCN-GBM model under different combinations of input parameters.
Figure 14 presents the AE distributions under different training proportions. FCN-GBM shows highly concentrated AE distributions, with most errors within ±1.0 K, even at 20 % training data. This indicates strong generalization and stability. In contrast, GBM exhibits large errors (>3 K) when data are scarce, suggesting high sensitivity to sample size. FCN-GBM models using single features (V or Ro) also show more fluctuation than the combined input version.
Temperature prediction errors for Battery #1 and Battery #2 under varying training set proportions and different input features.
In summary, the FCN-GBM model demonstrates clear advantages in temperature prediction by effectively integrating dynamic (V) and static (Ro) characteristics. This enhances generalization and accuracy, making it well-suited for real-world battery thermal monitoring and safety management.
In addition to performance comparison, the temporal sequence among resistance, voltage, and temperature responses under high C-rate discharge also reveals important implications for early fault detection. In particular, the Toyota dataset includes discharge conditions up to 6 C, which impose substantial thermal and electrochemical stress on the battery. Under these conditions, a consistent response pattern is observed: internal resistance begins to rise before noticeable voltage drops occur, and both precede the rapid increase in temperature. This temporal alignment reflects the early development of thermal abnormalities and provides a physical basis for predictive modeling. Therefore, although the present study focuses on normal operational data without explicit fault labeling, the proposed FCN–GBM model may hold potential for recognizing early-stage fault signatures, such as sudden resistance surges or voltage instability. These results suggest that the model could be extended to support early thermal warning and safety intervention in real-world battery management systems, especially under high-stress conditions.
This study proposed a hybrid model that integrates FCN with GBM for predicting the temperature evolution of lithium-ion batteries under high C-rate discharge conditions, using V and Ro as core input features. By analyzing both simulation and experimental data from LFP and NCA cells under various discharge rates, the effects of model structure and feature combinations on prediction accuracy and generalization were systematically explored. The main findings are summarized as follows.
Furthermore, the consistent temporal patterns observed under high C-rate discharge suggest that the model could be extended to support early thermal risk identification in practical applications.
Although the proposed model achieves notable improvements in prediction accuracy and generalization, this study is currently limited to LFP and NCA batteries under fixed ambient temperature conditions. The diversity of datasets and operational scenarios does not yet fully reflect the complexity of real-world battery environments. In addition, the real-time applicability and deployment feasibility of the model in engineering settings require further evaluation and optimization.
battery management system
NTGKFnonlinear thermal generalized Kalman filter
SVMsupport vector machine
ANNartificial neural network
RFrandom forest
FCNfully connected network
GBMgradient boosting machine
LFPLiFePO4
NCALiNi0.8Co0.15Al0.05O2
DODdepth of discharge
CCconstant-current
CVconstant-voltage
OCVopen-circuit voltage
RMSEroot mean square error
AEabsolute error
MAEmean absolute error
This work was sponsored by the Science and Technology Commission of Shanghai Municipality (19DZ2271100) and Shanghai Key Laboratory of Materials Protection and Advanced Materials in Electric Power, China.
Luyan Wang: Conceptualization (Lead)
Hongliang Hao: Methodology (Lead)
Zhongkang Zhou: Data curation (Lead)
Huimin Ma: Data curation (Lead)
Jin Zhao: Validation (Lead)
Zeyang Liu: Formal analysis (Lead)
Qiangqiang Liao: Writing – review & editing (Lead)
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Science and Technology Commission of Shanghai Municipality: 19DZ2271100
