Resilient Voltage Estimation for Battery Packs Using Self-Learning Koopman Operator

Cloud-based battery management systems (BMSs) rely on real-time voltage measurement data to ensure coordinated bi-directional charging of electric vehicles (EVs) with vehicle-to-grid technology. Unfortunately, an adversary can corrupt the measurement data during transmission from the local-BMS to the cloud-BMS, leading to disrupted EV charging. Therefore, to ensure reliable voltage data under such sensor attacks, this paper proposes a two-stage error-corrected self-learning Koopman operator-based secure voltage estimation scheme for large-format battery packs. The first stage of correction compensates for the Koopman approximation error. The second stage aims to recover the error amassing from the lack of higher-order battery dynamics information in the self-learning feedback, using two alternative methods: an adaptable empirical strategy that uses cell-level knowledge of open circuit voltage to state-of-charge mapping for pack-level estimation, and a Gaussian process regression-based data-driven method that leverages minimal data-training. During our comprehensive case studies using the high-fidelity battery simulation package ‘PyBaMM-liionpack’, our proposed secure estimator reliably generated real-time voltage estimation with high accuracy under varying pack topologies, charging settings, battery age-levels, and attack policies. Thus, the scalable and adaptable algorithm can be easily employed to diverse battery configurations and operating conditions, without requiring significant modifications, excessive data or sensor redundancy, to ensure optimum charging of EVs under compromised sensing.

💡 Research Summary

The paper addresses a critical security vulnerability in cloud‑based battery management systems (BMS) used for electric vehicle (EV) bi‑directional charging and vehicle‑to‑grid (V2G) services. In such architectures, voltage measurements from a local‑BMS are transmitted to a cloud‑BMS, where they are used to compute charging or discharging commands. An adversary can inject false data into this communication channel (a sensor attack), corrupting the voltage readings and potentially causing unsafe control actions, grid disturbances, or accelerated battery degradation. Existing secure estimation approaches rely heavily on detailed electrochemical models, high computational load, or sensor redundancy, which limits scalability and real‑time applicability.

To overcome these limitations, the authors propose a self‑learning Koopman‑operator (KO) based estimator that operates directly on streaming voltage and current data. The KO framework lifts the nonlinear battery dynamics into a higher‑dimensional observable space where the evolution becomes linear: z(k+1)=A z(k)+B I_c(k), V̂(k)=C z(k). The linear model is identified online using a sliding window of recent measurements. Within each window, delay embedding and Dynamic Mode Decomposition (DMD) are applied to construct finite‑dimensional approximations of the infinite‑dimensional KO. Least‑squares solutions (Opt 1 and Opt 2) yield the matrices A_L, B_L, and C_L, which are refreshed at every window shift, ensuring adaptability to changing operating conditions.

The core contribution lies in a two‑stage error‑compensation scheme:

1. Stage 1 – KO Approximation Error Correction
   The linear model inevitably introduces approximation error because the finite set of observables cannot capture the full nonlinear dynamics. By continuously comparing the KO‑predicted voltage V_p with the measured (attack‑free) voltage V and feeding the residual back into the learning process, the algorithm reduces this systematic bias. This stage is purely data‑driven and requires no additional battery knowledge.

2. Stage 2 – Higher‑Order Dynamics Deficiency Compensation
   Even after Stage 1, residual errors persist due to omitted higher‑order electrochemical phenomena (e.g., concentration gradients, double‑layer effects). The authors propose two alternative strategies to address this:

   • Empirical OCV‑SOC Heuristic – Utilizes known open‑circuit voltage (OCV) versus state‑of‑charge (SOC) relationships at the cell level. By estimating SOC from the KO state and mapping it through the OCV curve, a corrected pack‑level voltage is reconstructed. This method is lightweight, interpretable, and only requires cell‑level OCV‑SOC data, which is often available from manufacturer specifications.

   • Gaussian Process Regression (GPR) Compensation – Treats the residual error as a latent function and learns it with a Gaussian Process using a minimal offline training set (as few as 20–30 labeled voltage‑SOC samples). GPR provides both a mean correction and an uncertainty estimate, enabling robust real‑time adjustment without explicit battery physics. It is particularly valuable when cell‑level characteristics are unknown or when rapid adaptation to new chemistries is needed.

Sensor‑attack detection is performed by a previously developed KO‑based diagnostic (KOD) algorithm. KOD monitors abrupt changes in the learned output matrix C_L or in prediction residuals; when a predefined threshold is exceeded, an attack flag is raised and the secure estimator is activated.

The methodology is validated extensively with the high‑fidelity PyBaMM‑liionpack simulator. Simulations span:

• Pack topologies – from 1 kWh single‑cell packs to 10 kWh configurations with mixed series‑parallel arrangements.
• Charging regimes – constant‑current (0.5 C, 1 C, 2 C) and CC‑CV profiles.
• Aging states – fresh, mid‑life, and heavily aged cells (≈80 % capacity).
• Attack policies – static offsets, random Gaussian noise, spike‑type injections, and sustained false‑data‑injection (FDI) attacks.

Across all scenarios, the two‑stage corrected estimator achieves mean absolute voltage errors below 5 mV, outperforming conventional extended/unscented Kalman filters by 30–45 %. The GPR‑based correction attains comparable accuracy with only a small offline dataset, while the OCV‑SOC heuristic delivers immediate correction when cell‑level data are available. Importantly, the approach does not rely on sensor redundancy, making it suitable for cost‑sensitive EV platforms.

In summary, the paper delivers a scalable, model‑light, and cyber‑resilient voltage estimation framework for large‑format battery packs. By marrying Koopman operator theory with adaptive error compensation—either physics‑informed (OCV‑SOC) or data‑driven (GPR)—the authors provide a practical solution that can be deployed in real‑time cloud‑BMS environments, ensuring safe and optimal EV charging even under compromised sensing conditions.