Technical Summary: Single-fibre internal observability in lithium-ion batteries

Problem Statement

Lithium-ion battery management systems (BMS) currently rely on a narrow set of external observables—surface temperature, terminal voltage, and current—to infer internal safety states. This "observability gap" prevents the detection of critical internal phenomena such as localized lithium plating, separator fatigue, and thermal runaway precursors until they manifest as external warning signs. Consequently, BMS strategies must operate with conservative safety margins, sacrificing usable energy, charging speed, and cycle life.

While embedded Fiber Bragg Grating (FBG) sensors offer a direct route to internal states, they face a fundamental physical limitation: a single FBG wavelength shift ($\Delta\lambda_B$) is an underdetermined measurement that conflates temperature ($\Delta T$) and mechanical strain ($\varepsilon$). For battery-relevant operating ranges, these two contributions are of comparable magnitude, making it impossible to decouple them using algebraic methods alone without introducing complex hardware or violating physical laws. Existing data-driven models often fail to produce physically admissible reconstructions, while hardware-based decoupling introduces packaging complexity and stress concentrations.

Methodology: FBG-PhysNet

The authors propose FBG-PhysNet, a physics-informed, dual-branch recurrent framework designed to solve the single-fibre observability problem in real-time. The architecture integrates three core components:

1. Dual-Branch RWKV-7 Backbone:

   • The model utilizes two independently parameterized RWKV-7 (Recurrent Weighted K-Vector) branches: Branch A for thermal estimation and Branch B for strain estimation.
   • Each branch consists of 4 layers with a model dimension ($d_{model}$) of 128 and 4 attention heads.
   • Memory Efficiency: The RWKV-7 architecture maintains a fixed recurrent state size of 64 KB per block (totaling 512 KB for the dual-branch state), independent of sequence length. This $O(1)$ memory footprint enables deployment on automotive-grade microcontrollers (e.g., STM32H7 with 1 MB SRAM), unlike Transformer-based models which require memory scaling with context length.

2. Q-FiLM Conditioning:

   • To address the thirteen-fold variation in electrochemical heat generation ($Q_{gen}$) across different drive cycles (from urban driving to 2C fast charging), the model employs a Q-FiLM (Feature-wise Linear Modulation) module.
   • This module injects $Q_{gen}$ estimates (derived from terminal voltage and current) as scale and shift vectors into the latent representation at each layer. This allows the network to adapt its feature space dynamically to the current thermal operating regime without requiring separate models for each cycle type.

3. Cross-Gating and Nine-Term Physics Loss:

   • Cross-Gating: A mechanism exchanges boundary-condition information between the thermal and strain branches. This captures the physically necessary coupling ($\varepsilon_{th} = \alpha_{CTE}\Delta T$) without conflating the degrees of freedom, enabling the resolution of mechanically independent strain ($\varepsilon_{mech}$).
   • Physics-Informed Loss Hierarchy: The training objective combines supervised data loss with nine differentiable physics constraints:
     1. Supervised MSE on $\Delta T$ and $\varepsilon$.
     2. FBG Optical Consistency: Enforces that predictions lie on the Bragg constraint manifold ($\Delta\lambda_B = K_T \Delta T + K_\varepsilon \varepsilon$).
     3. Heat Conduction PDE Residual: Enforces the lumped-parameter heat equation.
     4. Global Energy Conservation: Ensures heat balance over the entire time window.
     5. Strain-Rate Bounds: Limits the rate of strain change.
     6. Yield Feasibility: Ensures strain remains within material yield limits.
     7. State-of-Charge (SOC) Feasibility: Constrains SOC estimates to $[0,1]$.
     8. Temporal Smoothing.
     9. L2 Regularization.
   • Training follows a "physics curriculum," starting with data fitting and gradually amplifying physics weights (by a factor of 40) to transition to physics-constrained refinement.

Key Results

The framework was validated on 75,120 hardware-in-the-loop (HIL) windows derived from a calibrated P2D Doyle-Fuller-Newman electrochemical model, covering five operating regimes: UDDS, WLTP, Highway, 2C fast charging, and emergency braking.

• Accuracy: FBG-PhysNet achieved a Mean Absolute Error (MAE) of 2.270 K for internal temperature and 21.94 $\mu\varepsilon$ for strain.
• Decoupling Performance (Cross-Talk): The model reduced the cross-talk ratio (CT) to 58.6%. This is a significant improvement over the best physics-aware baseline (PINN-MLP at 77.3%) and falls below the metric-specific thermal-routing reference bound of 71.3%. This indicates the model successfully recovers mechanically independent strain rather than merely subtracting thermal expansion.
• Physical Convergence: Over 47 training epochs, the model demonstrated mutual reinforcement between physics compliance and prediction accuracy. The energy-conservation residual decreased by 75%, and the PDE residual by 44%. The FBG optical self-consistency metric ($\chi^2_{FBG}$) improved from 0.773 to 0.906, approaching the ideal value of 1.0.
• Ablation Studies: Removing all physical constraints (Abl-A) increased the temperature MAE by 46.6% and the cross-talk ratio to 87.2%. Removing Cross-Gating (Abl-C) resulted in a cross-talk ratio of exactly 71.3%, confirming that without this mechanism, the model cannot resolve the mechanically independent strain component.
• Robustness: While the model showed higher temperature error (4.280 K) in extreme 2C fast-charging scenarios (where $Q_{gen}$ exceeded the training mean by 13x), the cross-talk ratio remained invariant at 58.6%, demonstrating that the physical decoupling quality is robust to drive-cycle intensity.

Significance and Claims

The paper posits that internal battery safety can be reframed as an observability problem. By reconstructing internal temperature and strain in real-time from a single embedded optical signal, the framework enables a shift from static, worst-case safety margins to adaptive control based on the actual thermo-mechanical state of the cell.

The authors highlight three specific implications:

1. Battery Longevity and Capacity Utilization: Real-time strain awareness allows BMS to tighten current limits only when strain approaches safety boundaries (e.g., the 20,000 $\mu\varepsilon$ yield strain of graphite), potentially recovering 3–8% of usable capacity in high-power applications compared to constant derating.
2. DC Fast Charging (DCFC): The model's ability to predict core temperature with low latency supports rapid thermal monitoring, potentially reducing charging interruptions and increasing throughput in existing infrastructure without new hardware.
3. Battery Passport and Second-Life Economics: The physics-anchored, self-consistent tracking of internal states provides the regulatory-grade lifecycle data required for "Battery Passport" certification, enabling more accurate valuation of used packs for second-life applications.

Crucially, the authors claim that the widely assumed "irreducible coupling" limit in single-sensor FBG systems is not a fundamental physical barrier but a limitation of existing modeling approaches. By learning a representation where mechanically independent strain is explicitly resolved, FBG-PhysNet operates beyond conventional limits. The work concludes that the fundamental limitation of current battery systems is not material performance, but observability; resolving this gap allows operational limits to be redefined without changing chemistry or hardware.