Residual Battery Capacity Estimation

• Residual battery capacity is defined as the remaining charge-storage capability relative to its original design, expressed as a state-of-health percentage.
• It is measured through controlled protocols like Coulomb counting, voltage relaxation, and impedance screening to accurately track battery degradation.
• Advanced methods, including hybrid physics-based and machine learning approaches, enable precise capacity estimation and early detection of performance declines.

Residual battery capacity, defined as the remaining charge-storage capability of a battery relative to its original or nominal value under specified conditions, is a cornerstone metric for the safety, reliability, and economic utility of electrochemical energy storage systems. Residual capacity quantifies the maximum retrievable charge ($Q_{\mathrm{max}}(t)$) at a given point in the battery's lifetime, normalized either as an absolute value (e.g., ampere-hours, mAh g$^{-1}$) or as a state-of-health (SoH) fraction referenced to the design capacity ($Q_{\mathrm{max}}(0)$). Modern methodologies for quantifying, estimating, and managing residual capacity span data-driven, physics-based, statistical, and hybrid strategies, each reflecting the substantial operational and scientific complexity of battery degradation processes.

1. Formal Definitions and Measurement Protocols

Residual battery capacity is conventionally defined as the ratio of the measured full-charge capacity at time $t$ to the cell’s reference capacity, expressed as:

$$\mathrm{SoH}(t) = \frac{Q_{\max}(t)}{Q_{\max}(0)} \times 100\%$$

Alternate expressions involve the datasheet design capacity $C_\mathrm{design}$ and the observed full-charge value $C_\mathrm{fullcharge}$:

$$\mathrm{SoH} = \frac{C_\mathrm{fullcharge}}{C_\mathrm{design}} \times 100\%$$

Cycle-resolved capacity is typically measured via controlled charge–discharge protocols (constant current, constant voltage cutoffs), with Coulomb counting used to integrate the delivered or accepted current. Specific studies employ rest periods for voltage equilibration, controlled temperature environments, and well-defined voltage windows (e.g., 2.7–4.15 V for NMC automotive cells) as in (Hassini et al., 18 Jul 2025). For SoH benchmarking in retired automotive modules, cell-to-cell dispersion and positional analysis via ANOVA provide statistical validation of uniformity and the absence of spatially correlated degradation.

2. Physics-Based and Model-Based Estimation Approaches

First-principles models, especially the Enhanced Single Particle Model (ESPM) and its derivatives, underpin a suite of adaptive observer and parameter-estimation strategies for on-line capacity tracking. The ESPM incorporates temperature-dependent solid-phase diffusion, SEI growth, and Ohmic/power fade. Adaptive interconnected observers estimate state vectors aggregating lithium concentrations, dynamically-evolving capacity, and aging-sensitive transport parameters using Lyapunov-stable control structures. For example, sliding-mode adaptation is used to recover both electrode stoichiometries and capacity in real time, with bounded estimation errors $\le$2% across significant aging and operational settings (Allam et al., 2020).

Hybrid fusion frameworks combine reduced-order physics models and large libraries of simulated aging patterns (parameterized via adaptive particle-swarm optimization) with deep neural architectures (e.g., Res-CNNs), transferring learned representations to real-world data with a handful of short field measurements. This allows accurate (<0.63% MAPE) capacity estimation using minimal real-data segments, and enables robust compensation for cell-to-cell variations and deployment in safety-critical or embedded applications (Liu et al., 2024).

3. Data-Driven and Statistical Machine Learning Methods

Machine learning and statistical learning methods are pivotal for rapid, scalable residual capacity prediction from partial measurements or streaming BMS data. Notable architectures and approaches include:

• WOA-ELM: An Extreme Learning Machine, optimized via the Whale Optimization Algorithm, maps multi-segment voltage-time features (charge plateau times, cyclewise voltage–time slopes/intercepts) onto specific capacity with near-perfect $R^2=0.9999871$ (Zhang et al., 2023). Feature interpretability is enhanced via SHAP and correlation analyses, revealing charge plateau duration and cycle-wide polarization as dominant capacity degradation predictors.
• GP-ICE: Gaussian Process regression for In-situ Capacity Estimation leverages voltage–time data from short periods of constant-current operation, mapping low-dimensional voltage features directly to capacity with rigorously-calibrated uncertainty, achieving RMSE as low as 0.5% for long ($>$1000 s) windows (Richardson et al., 2017).
• Attention-based and Sequential-Contextual Models: Deep neural architectures embedding temporal attention (e.g., DDN, GiNet) and hybrid approaches (LSTM+Informer, semiparametric DGP) achieve sub-1% MAE/MAPE in multi-step SoH or discharge capacity prediction from raw streaming BMS sensor time series (Zhang et al., 2022, Sameer et al., 9 Jan 2025, Liu et al., 12 Oct 2025). Embedding both fine-grained temporal and long-range contextual information is empirically essential for generalization across diverse cycling protocols, temperatures, and cell formats.
• Probabilistic Generative Models: Conditional diffusion models, U-Net backbones with self- and cross-attention, and denoising objectives enable not only point estimation of future residual capacity but also principled quantification of confidence intervals and uncertainty under real-world operational datasets (Li et al., 20 Oct 2025). These architectures are benchmarked to sub-1% RMSE/MAE with interpretable probabilistic intervals, essential for reliability in battery management.

4. Voltage- and Resistance-Based Fingerprinting Techniques

Voltage-relaxation and impedance-based metrics offer accurate and low-cost proxies for residual capacity, suited for both consumer electronics and retired/second-life battery screening.

• Voltage Relaxation (V-Health): The cell’s open-circuit voltage relaxation curve after full charge is a robust SoH “fingerprint.” By fitting the relaxation trace $v(t)\approx a t^b + c$ and projecting onto a regression tree trained on principal voltage components, both laboratory and in-field devices can achieve residual capacity estimates with $<$2% mean error in controlled tests and $<$5% error in field deployments (He et al., 2017). This method is production-viable on commodity mobile devices (Android, iPhone).
• OCV Invariance and Alignment: The invariance property of the OCV–SoC curve under aging (when SoC is calibrated by actual capacity) enables an alignment-based estimation technique: aligning measured OCV versus cumulative Coulombs data to a nominal reference and solving for the true capacity, even using partial cycles or purely dynamic, on-road data. This approach yields MARE $<$0.85% across hundreds of cycles and requires only a single nominal OCV calibration (Wang et al., 10 Nov 2025).
• Impedance Screening: Rapid internal resistance measurement (charge interrupt near 4 V) is linearly correlated to capacity in retired cells, allowing 3-hour screening tests to triage cells for repurposing with $\sim$8% accuracy (Drallmeier et al., 2022).

5. Degradation Feature Analysis, Online Detection, and Prognostics

Residual capacity attenuation is fundamentally non-linear, with regimes of slow fade, transitional “knee-onset,” and accelerated degradation. Curvature-based detection of capacity knees (and their onset) via smoothed second-difference analysis and Corrected Arc Curves identifies the transition to accelerated aging, enabling early EoL prediction via robust linear relationships:

$$N_{\mathrm{EoL}}\approx\alpha N_{\mathrm{onset}}+\beta$$

Online implementations, leveraging only sparse cycle data, can predict end-of-life and provide coarse interpolants for $Q(N)$ (e.g., piecewise-linear or global linear decay) (Zhang et al., 2023). This enables real-time maintenance alerts and dynamic adjustment of use-case scenarios, particularly in high-asset-value applications.

6. Special Contexts: Quantum Battery Residual Capacity

In emerging quantum battery theory, "residual battery capacity" (RBC) is defined as the difference between the total system's extractable work (capacity) and the sum of its subsystems’ capacities under local operations. For $n$-qubit X states, monogamy of capacity is rigorously established:

$$\sum_{k=1}^n\mathcal C(\rho_{A_k};H_{A_k})\leq\mathcal C(\rho;H)$$

RBC can be decomposed into incoherent and coherent parts associated with classical correlations and quantum coherence/information, respectively. Such measures are essential in optimizing distributed quantum batteries, guiding global unitary operations to concentrate extractable work in subsystems (Wang et al., 5 Feb 2025).

7. Practical Implications and Deployment Guidelines

Robust characterization and estimation of residual battery capacity directly support critical activities in BMS, range forecasting, second-life qualification, and safety management. Key practical recommendations include:

• In large repurposing operations, statistical quantification of cell-to-cell dispersion and model-based screening are essential for safe second-life battery deployment (Hassini et al., 18 Jul 2025).
• Data-driven algorithms, especially those with transfer learning or uncertainty quantification, are deployable with limited field data, provided sufficient simulation-based pretraining or reference fingerprinting is performed (Liu et al., 2024, He et al., 2017).
• Curvature-based online knee detection and attention-augmented deep learning approaches provide actionable insights even with sparse or non-stationary streaming data, facilitating predictive maintenance and dynamic scheduling (Zhang et al., 2023, Zhang et al., 2022).
• For quantum battery contexts, the distribution and optimization of residual capacity are critical to maximizing extractable work in multipartite quantum systems (Wang et al., 5 Feb 2025).

Residual battery capacity estimation remains an active research area at the intersection of electrochemistry, statistical learning, control theory, and quantum information, with continuing advances enabling more reliable, data-efficient, and deployment-ready solutions across diverse battery technologies and application domains.