As a researcher in energy storage technologies, the accurate and reliable operation of lithium-ion batteries is paramount to my work. Among the various factors influencing their performance and safety, temperature stands out as a critical state variable. The electrochemistry, power capability, degradation rate, and ultimately the safety of a li ion battery are intimately tied to its thermal state. Precise temperature estimation, therefore, is not merely a monitoring task but a foundational requirement for advanced Battery Management Systems (BMS) to enable optimal charging, power limits, lifespan prediction, and, most crucially, the prevention of catastrophic thermal runaway.
Traditional approaches to li ion battery temperature management can be broadly categorized into model-based and data-driven methods. Model-based strategies, such as those employing equivalent circuit models (ECMs) or detailed electrochemical-thermal models, offer valuable physical insight. For instance, a common technique involves coupling an electrical model (like a first- or second-order RC network) with a thermal model (representing heat capacity and resistance). The electrical model estimates losses (I²R, polarization), which serve as the heat generation input for the thermal model to compute temperature rise. While elegant and physically interpretable, these lumped-parameter models inevitably simplify the complex, multi-dimensional heat transfer within a li ion battery. They often struggle to capture dynamic thermal behaviors accurately, especially under high-rate or rapidly changing loads, leading to estimation gaps.
On the other hand, purely data-driven methods, such as those employing Artificial Neural Networks (ANNs), Long Short-Term Memory (LSTM) networks, or other machine learning architectures, learn the relationship between operational data (current, voltage, ambient temperature) and battery temperature directly. These models can achieve high accuracy if trained on comprehensive datasets. However, they typically function as “black boxes,” lacking physical interpretability. Their performance is heavily reliant on the quantity and quality of training data, and they may generate physically implausible predictions when operating outside their training domain—a significant risk for safety-critical li ion battery systems.
To bridge this gap, I propose a novel hybrid framework that synergistically combines the strengths of a physics-based model with the adaptive learning power of deep learning. This method aims to deliver accurate, robust, and physically consistent temperature estimates for li ion battery systems under dynamic operating conditions.
1. The Proposed Hybrid Estimation Framework
The core of my methodology is a two-stage process. First, a coupled thermoelectric model is established and its parameters are identified online using a robust recursive algorithm. Second, the outputs and physical structure of this model are embedded as constraints within a specially designed neural network, creating an Adaptive Weighted Physics-Informed Neural Network (AWPINN). This fusion ensures that the final estimator respects the fundamental laws of thermodynamics while leveraging data to correct for the simplifications inherent in the lumped model.
2. Stage 1: Thermoelectric Model & Online Parameter Identification
2.1 Coupled Model Structure
I construct a coupled model from two core components: an electrical sub-model and a thermal sub-model. For the electrical behavior of the li ion battery, I adopt the first-order RC equivalent circuit model. This model offers a good balance between complexity and accuracy, capturing the dominant dynamic voltage response.
The governing equations for the electrical sub-model are:
$$I = C_p \frac{dU_p}{dt} + \frac{U_p}{R_p}$$
$$U_{ocv} = U_{batt} + I R_0 + U_p$$
where \(I\) is the load current, \(U_{batt}\) is the terminal voltage, \(U_{ocv}\) is the open-circuit voltage (a function of State of Charge, SOC, and temperature), \(R_0\) is the Ohmic resistance, and \(R_p\) and \(C_p\) are the polarization resistance and capacitance, respectively. The heat generation rate \(Q\) within the li ion battery is primarily calculated from the losses:
$$Q = I^2 R_e + I T \frac{dU_{ocv}}{dT}$$
where \(R_e = R_0 + R_p\) is the total internal resistance, and the second term represents the reversible entropic heat. For many operational estimates, the irreversible \(I^2R\) term dominates.
For the thermal dynamics, I use a first-order lumped thermal model, treating the entire li ion battery as a single node with uniform temperature \(T\):
$$C_T \frac{dT}{dt} = Q – \frac{T – T_{amb}}{R_T}$$
Here, \(C_T\) is the lumped thermal capacity, \(R_T\) is the thermal resistance to the ambient environment at temperature \(T_{amb}\), and \(Q\) is the heat generation from the electrical model. This coupled structure is illustrated in the conceptual diagram below, where the electrical model feeds heat generation into the thermal model, and the estimated temperature can theoretically feedback to update \(U_{ocv}\) and resistance parameters.
2.2 Online Parameter Identification via VFFRLS
The parameters of the coupled model (\(R_0, R_p, C_p, C_T, R_T\)) are not constant; they vary with SOC, temperature, aging, and load history. To achieve accurate estimation in real-time, online identification is essential. I employ a Variant Forgetting Factor Recursive Least Squares (VFFRLS) algorithm for this purpose.
First, the electrical model equation is discretized using the bilinear transform. Defining \(U’ = U_{ocv} – U_{batt}\), we get a linear-in-parameters form:
$$U'(k) = a_1 U'(k-1) + a_2 I(k) + a_3 I(k-1)$$
The parameter vector to identify is \(\theta_e = [a_1, a_2, a_3]^T\). The original electrical parameters can be recovered from these. Similarly, discretizing the thermal model equation yields:
$$T(k+1) – T(k) = b_1 Q(k) + b_2 (T_{amb}(k) – T(k))$$
with the thermal parameter vector \(\theta_t = [b_1, b_2]^T = [1/C_T, 1/(R_T C_T)]^T\).
The standard RLS update equations with a forgetting factor \(\lambda\) are:
$$
\begin{aligned}
K(k) &= \frac{P(k-1) \varphi(k)}{\lambda + \varphi^T(k) P(k-1) \varphi(k)} \\
\hat{\theta}(k) &= \hat{\theta}(k-1) + K(k) [y(k) – \varphi^T(k) \hat{\theta}(k-1)] \\
P(k) &= \frac{1}{\lambda} [I – K(k) \varphi^T(k)] P(k-1)
\end{aligned}
$$
Here, \(y(k)\) is the measurement (e.g., \(U'(k)\) or \(\Delta T(k)\)), and \(\varphi(k)\) is the regressor vector. A fixed \(\lambda\) (e.g., 0.995) works for steady conditions but can fail during dynamic transients. In the VFFRLS algorithm, the forgetting factor is adapted online:
$$\lambda(k) = 1 – \frac{e^2(k)}{1 + K^T(k)P(k)K(k)}$$
where \(e(k) = y(k) – \varphi^T(k) \hat{\theta}(k-1)\) is the prediction error. This formulation automatically reduces \(\lambda\) when the error is large (indicating a change in system parameters), allowing the algorithm to “forget” old data and track the new parameters quickly. As the estimate converges, the error decreases, and \(\lambda\) returns towards 1, ensuring stability and noise suppression. This adaptive mechanism is crucial for accurately identifying the time-varying parameters of a li ion battery under dynamic loads.
3. Stage 2: The Adaptive Weighted Physics-Informed Neural Network (AWPINN)
While the VFFRLS-identified thermoelectric model provides a physically-grounded estimate, its accuracy is limited by the fidelity of the first-order lumped thermal model. To overcome this limitation without abandoning physics, I design an AWPINN.
3.1 Network Architecture and Physical Embedding
The AWPINN is built upon a standard Fully Connected Neural Network (FCNN). The key innovation lies in how physics is incorporated. The inputs to the network are chosen to be relevant operational parameters: a time sequence of current \(I\), voltage \(U\), ambient temperature \(T_{amb}\), and the heat generation rate \(Q\) calculated from the online-identified electrical parameters (\(Q \approx I^2 \hat{R}_e\)). The network’s primary task is to output the estimated core temperature \(T_{est}\).
Physics is embedded via the loss function. Beyond the standard data loss \(L_{data} = \frac{1}{N} \sum ||T_{est} – T_{meas}||^2\), we introduce a physics-informed loss \(L_{PDE}\) based on the discretized thermal model:
$$L_{PDE} = \frac{1}{N} \sum \left\| \frac{T_{est}(k+1) – T_{est}(k)}{\Delta t} – \omega_1 Q(k) + \omega_2 (T_{est}(k) – T_{amb}(k)) \right\|^2$$
Crucially, \(\omega_1\) and \(\omega_2\) are not fixed constants from the VFFRLS stage. Instead, they are trainable parameters within the AWPINN. However, to maintain physical consistency, they are constrained. The VFFRLS identification provides nominal values and bounds (e.g., \(\hat{b}_1 \pm \delta_1\), \(\hat{b}_2 \pm \delta_2\)). I set the bounds for \(\omega_1\) and \(\omega_2\) as a relaxed version (e.g., 1.5 times the nominal range) of these identified parameters. This ensures the network learns corrections to the lumped model but within a physically plausible subspace, preventing it from converging to non-physical solutions. An initial condition loss \(L_{init} = ||T_{est}(0) – T_{amb}(0)||^2\) is also added.
The total loss function is a weighted sum:
$$L_{total} = L_{data} + \alpha_{PDE} L_{PDE} + \alpha_{init} L_{init}$$
3.2 Adaptive Loss Balancing Strategy
A major challenge in training PINNs is manually tuning the weights \(\alpha_{PDE}\) and \(\alpha_{init}\), as the losses \(L_{data}\), \(L_{PDE}\), and \(L_{init}\) often differ by orders of magnitude. To solve this, I implement an adaptive balancing strategy based on gradient statistics.
Let \(\hat{T}\) be the network’s output. We compute the gradients of each loss with respect to \(\hat{T}\): \(\nabla_{\hat{T}} L_{data}\), \(\nabla_{\hat{T}} L_{PDE}\), and \(\nabla_{\hat{T}} L_{init}\). At each training step \(t\), we calculate instantaneous weights:
$$\tilde{\alpha}_{PDE}(t) = \frac{\max(|\nabla_{\hat{T}} L_{data}|)}{\text{mean}(|\nabla_{\hat{T}} L_{PDE}|) + \epsilon}, \quad \tilde{\alpha}_{init}(t) = \frac{\max(|\nabla_{\hat{T}} L_{data}|)}{\text{mean}(|\nabla_{\hat{T}} L_{init}|) + \epsilon}$$
where \(\epsilon\) is a small constant for numerical stability. These instantaneous weights are then smoothed using an Exponential Moving Average (EMA) to avoid oscillations:
$$\alpha_{PDE}(t) = (1-\gamma) \alpha_{PDE}(t-1) + \gamma \tilde{\alpha}_{PDE}(t)$$
$$\alpha_{init}(t) = (1-\gamma) \alpha_{init}(t-1) + \gamma \tilde{\alpha}_{init}(t)$$
with smoothing factor \(\gamma\). This strategy automatically scales the physics-based losses to have a comparable influence on the gradient updates as the data loss, dramatically improving training stability and convergence without manual hyperparameter search.
4. Experimental Validation and Analysis
To validate the proposed framework, tests were conducted on commercial 3.8V, 3850mAh lithium polymer li ion battery cells. Data was collected under Hybrid Pulse Power Characterization (HPPC) profiles at various ambient temperatures (0°C, 10°C, 20°C, 30°C, 40°C) in a thermal chamber. Surface temperature was measured and used as a proxy for the core temperature target.
4.1 Performance of the VFFRLS Identification
The VFFRLS algorithm successfully identified the electrical parameters online. The terminal voltage simulated using the identified parameters closely matched the measured data. The adaptive forgetting factor effectively handled the sudden changes during current pulses.
Table 1: Voltage Simulation Error for VFFRLS Identification (20°C HPPC)
| Metric | Value |
|---|---|
| Maximum Absolute Error | ~0.12 V |
| Root Mean Square Error (RMSE) | ~1.84% of voltage range |
The thermal model parameters (\(C_T\), \(R_T\)) were also identified. While the simulated temperature trend followed the measurements, clear discrepancies were observed, especially during fast temperature rise phases, highlighting the limitation of the lumped model.
4.2 AWPINN Prediction Results and Comparative Study
The AWPINN was trained on 80% of the HPPC data at 20°C and tested on the remaining 20%. Its performance was compared against three pure data-driven models: a standard FCNN, an LSTM network, and a basic PINN with fixed loss weights. The results are summarized below.
Table 2: Temperature Prediction Performance Comparison at 20°C
| Model | MAE (°C) | RMSE (°C) | R² Score |
|---|---|---|---|
| Proposed AWPINN | 0.242 | 0.4069 | 0.9693 |
| Basic PINN | 0.6053 | 1.0130 | 0.8090 |
| LSTM | 1.3910 | 1.1794 | 0.7419 |
| FCNN | 1.0986 | 1.3652 | 0.6543 |
The AWPINN demonstrated superior accuracy, with a 52.4% lower MAE and a 58.7% lower RMSE compared to the basic PINN. More importantly, its prediction curves showed excellent tracking of rapid temperature changes, a scenario where the other models, particularly the pure data-driven ones, exhibited significant lag or error. The AWPINN’s constraints and adaptive loss balancing effectively guided the learning towards physically plausible and accurate solutions.
4.3 Generalization Across Temperature Ranges
A critical test for any li ion battery model is performance across different thermal environments. The AWPINN framework was evaluated using data from all five ambient temperatures. The results confirm its robust generalization capability.
Table 3: AWPINN Performance Across Different Ambient Temperatures
| Ambient Temperature | MAE (°C) | RMSE (°C) | R² Score |
|---|---|---|---|
| 0°C | 0.392 | 0.573 | 0.943 |
| 10°C | 0.279 | 0.436 | 0.955 |
| 30°C | 0.278 | 0.371 | 0.972 |
| 40°C | 0.196 | 0.264 | 0.977 |
While minor error spikes occurred during the steepest parts of temperature transients—a challenge for any estimator—the overall performance remained excellent. The highest RMSE was 0.573°C at 0°C, and the lowest R² was 0.943, demonstrating that the hybrid model maintains high accuracy and reliability across the typical operating range of a li ion battery.
5. Conclusion
In this work, I have presented a novel hybrid framework for accurate temperature estimation in li ion battery systems. The method integrates an online-identified thermoelectric model with an Adaptive Weighted Physics-Informed Neural Network (AWPINN). The VFFRLS algorithm provides real-time, adaptive parameter identification for a coupled electrical-thermal model. The AWPINN then uses these parameters as physically-constrained, trainable variables within its loss function, while an adaptive weighting strategy balances the data-driven and physics-based learning objectives automatically.
Experimental validation under dynamic HPPC profiles across a 0°C to 40°C range demonstrates the framework’s superiority. It significantly outperforms pure data-driven models (FCNN, LSTM) and a basic PINN in terms of MAE, RMSE, and R² score. The model successfully tracks rapid temperature changes and generalizes well across different ambient conditions, addressing key limitations of both traditional lumped models and black-box neural networks.
This approach offers a powerful pathway for developing BMS algorithms that are both accurate and physically interpretable, enhancing the safety, performance, and longevity of li ion battery packs in electric vehicles and grid storage applications. Future work will focus on extending the model to account for multi-dimensional temperature gradients within a battery pack and integrating it with state-of-charge and state-of-health co-estimation algorithms.