Estimating the Health and State of Charge of Each Cell in a Second-Life Battery System from Field Data

INTRODUCTION

• Heterogeneous degradation and SOC imbalance let weaker cells reach operational limits sooner, constraining pack capability and potentially causing shortfalls or conservative underuse.Cell histories and current degradation states are typically unknown at repurposing, and cells continue to diverge during second life.Source: INTRODUCTION · pp. 1–2
• Field measurements are limited and noisy, while purely data-driven approaches can require large labeled datasets and offer limited physical interpretability outside controlled laboratory conditions.Kalman-filter variants are established model-based estimators, but their accuracy depends on proper tuning.Source: INTRODUCTION · p. 2
• Model-based estimation depends on an accurate ECM, but degradation changes its parameters and fixed parametric forms can poorly represent SOC- and temperature-dependent behavior.The OCV is nonlinear in SOC, and the beginning-of-life OCV curve may not remain valid as cells age.Source: INTRODUCTION · p. 2
• Prior OCV-reconstruction methods can require rest periods, low currents, chemistry-specific calibration, or laboratory measurements, and earlier arbitrary-cycling work was validated only on individual laboratory cells.The earlier approach reported below 10 mV terminal-voltage RMSE on NMC cells across SOH 72% to 100%.Source: INTRODUCTION · p. 2
• The framework jointly estimates SOC and ECM parameters from field data, reconstructs OCV without prior cell characterization, and aggregates individual cell models to module and system levels.It is demonstrated on a 27-module, 324-cell second-life system and benchmarked against a lumped-module model fitted without individual cell voltages.Source: INTRODUCTION · pp. 2–3

Battery system and dataset

• Two cells diverge sharply from their module peers, and P1M6 and P3M5 deliver only 95 Ah and 129 Ah, versus 217 Ah to 238 Ah for most modules.The data were collected over 12.5 hours of field operation; the two outlier cells constrain their modules’ usable capacity.Source: Battery system and dataset · pp. 3–4, Figure 1 · p. 4

Battery model and parametrization

• The recursive GP-ECM jointly estimates cell states and model parameters from field measurements, reconstructing OCV curves without prior cell characterization.It uses an extended Kalman filter and corrects resistance parameters for temperature using a jointly estimated Arrhenius factor.Source: Battery model and parametrization · pp. 4–5

SOH heterogeneity and SOC imbalance

• The 324 cells share an OCV shape but differ in curve coverage; aligned curves agree with the composite at 1.95 mV median RMSE.Most cells cluster near 75.4% SOH, while P1M6C1 and P3M5C12 are estimated at 47.8% and 38.0%.Source: Battery model and parametrization · p. 5, SOH heterogeneity and SOC imbalance · p. 7
• A systematic 6% to 11% SOC imbalance places cells C1–C3 below their module peers, leaving 3.5 Ah to 6.5 Ah inaccessible in most modules.Balancing could recover most of this capacity in principle, but 0.5 Ah to 1.6 Ah per module remains inaccessible from SOH heterogeneity; losses in P1M6 and P3M5 are entirely irreversible.Source: SOH heterogeneity and SOC imbalance · pp. 7–8, Figure 4 · p. 8
• 24% and 31%: lumped-module SOH estimates exceed cell-aggregated estimates in P1M6 and P3M5, despite comparable fit quality.For the other 25 modules, the overestimate is 4% to 16%; the two outlier-module fit residuals are only slightly above the median.Source: SOH heterogeneity and SOC imbalance · p. 8, Figure 5 · p. 9

Validation against reference measurements

• Reference comparisons across 96 cells show 4.0 mV median OCV-curve RMSE, 1.25 Ah capacity RMSE, and 1.1% SOC-imbalance RMSE.Capacity deviations from each module mean agree to 0.55 Ah RMSE; the SOC comparison spans experiments about seven weeks apart.Source: Validation against reference… · pp. 10–11, Figure 6 · p. 10

Voltage prediction accuracy and SOC estimation

• Open-loop voltage RMSE is comparable for cell-combined and lumped-module fits, but lumped-model SOC deviations reach 23.1% across modules.With measurement updates, EKF charge-estimation error remains below 1 Ah for all twelve P1M9 cells over 12.5 hours.Source: Voltage prediction accuracy and SOC… · pp. 11–12, Figure 7 · p. 12

DISCUSSION

• From 12.5 h of field data, cell models predicted measured voltage at 5.09 mV median RMSE and reconstructed OCV curves agreed with reference measurements at 4.0 mV median RMSE.Estimated cell capacities agreed to 1.25 Ah and SOC imbalance to 1.1%.Source: DISCUSSION · p. 13
• Across 324 cells, estimated capacities varied substantially, two cells fell far below the population, and every module showed systematic SOC imbalance.The two outliers had roughly half the population-median capacity and twice its resistance; inspection traced both faults to busbar contacts, which were fixed.Source: DISCUSSION · pp. 13–14
• The lumped-module model overestimates SOH by up to 31% and SOC by up to 23%, despite comparable voltage-prediction accuracy.It follows average cell behavior and cannot identify the cells that set module limits.Source: DISCUSSION · p. 14

Limitations of the study

• SOH relative to nominal capacity requires extrapolating beyond the measured voltage range, leaving the absolute capacity scale unverified without individual laboratory characterization.The reference measurement shares onboard sensors and extrapolated reference points.Source: Limitations of the study · p. 14
• The short measurement window produces a snapshot of system states and parameters rather than tracking degradation over months or years.Long-term monitoring is identified as a next step.Source: Limitations of the study · p. 14
• Cell-level models require every cell voltage to be transmitted, stored, and processed, so telemetry and computing costs can become substantial over years.Tracking only limiting cells is proposed as a cost-accuracy trade-off to explore.Source: Limitations of the study · p. 14

Hybrid battery model

• The first-order ECM couples charge-based OCV(q) and R1(q) recursive GPs to an EKF that jointly estimates hidden states and ECM parameters, with R0 and τ held SOC-independent.R0 and R1 share an Arrhenius temperature factor with jointly estimated κ.Source: Hybrid battery model · pp. 14–15

Filter parametrization and initialization

• The filter uses 21 equally spaced GP basis points, while the initial parametrization run fixes q0=0 Ah with zero uncertainty and zero process noise.Default GP hyperparameters are adjusted through a population-based procedure for cases where they do not produce a consistent OCV reconstruction.Source: Filter parametrization and… · pp. 16–17

OCV curve alignment and SOH estimation

• Pairwise comparisons of inverted q(v) curves over shared voltages form a least-squares alignment whose scales and shifts yield each cell’s capacity and initial SOC.The estimates are aligned against the population; the common SOC span relies on a low-power reference curve extended linearly beyond its measured range.Source: OCV curve alignment and SOH estimation · pp. 18–19

Simulation and SOC estimation

• After fitting, frozen parameters support open-loop voltage simulation; SOC estimation restores EKF corrections for q and vrc, and module aggregation follows the cells’ dischargeable capacity.The SOC run gives q nonzero initial uncertainty and process noise so voltage corrections can adjust integrated charge.Source: Simulation and SOC estimation · p. 20

Materials availability

• The study did not generate new materials.Source: Materials availability · p. 21

Data and code availability

• The Julia software is available on GitHub, and the paper’s version and dataset are archived on Zenodo.Source: Data and code availability · p. 21

• Document S1 contains Figures S1–S9 and Notes S1 and S2.Source: SUPPLEMENTAL INFORMATION · p. 21

• Across the system, cell voltages spread apart within modules, most visibly near discharge end; divergent cells are marked in P1M6 and P3M5.Source: Figure S1 · p. 26
• Modules follow different dynamic load profiles as the converter redistributes phase current, while module temperatures vary and drift over the measurement window.Source: Figure S2 · p. 27
• Sampling intervals for current, voltage, cell voltage, and temperature do not always meet their nominal 1 s, 1 s, 10 s, and 15 s intervals.Source: Figure S3 · p. 28
• Cell SOH has a 75.4% median, with outliers P1M6C1 at 47.8% and P3M5C12 at 38.0%.Source: Figure S4 · p. 29
• Most validated reconstructed-versus-reference pseudo-OCV RMSE values fall between 3 mV and 5 mV; the tail comprises low-SOC cells reaching the steep OCV knee.Source: Figure S5 · p. 29
• For module P1M9, cell-level EKF SOC error stays within about ±1.3% of each cell’s estimated capacity over 12.5 h, or less than 1 Ah.Source: Figure S6 · p. 30
• With a 3.0 Ah initial offset and 0.1 A current bias, Coulomb-count error reaches about 4 Ah, while the EKF corrects the offset and tracks the reference closely.Source: Figure S7 · p. 30

Note S1: Recursive Gaussian process regression

• GP regression reconstructs functions from observed data using a kernel to encode input correlations, then predicts outputs and observation-calibrated uncertainty by conditioning on the data.The recursive formulation carries the fitted function in its state, allowing online updates as measurements arrive.Source: Note S1: Recursive Gaussian process… · p. 31
• Recursive GP regression represents a function with values at fixed basis inputs and updates them sequentially with a Kalman filter, avoiding storage of the full observation history.With m basis inputs, each update costs O(m^2) and processing n observations costs O(nm^2), compared with O(n^3) for full GP regression.Source: Note S1: Recursive Gaussian process… · pp. 31–32

Note S2: GP hyperparameter selection

• Poor GP length scales can smooth away real OCV features or fit measurement noise, while limited-window voltage data can support multiple plausible-looking or implausible curve shapes.Joint estimation also allows a poorly chosen hyperparameter to let one model parameter absorb variation that belongs to another.Source: Note S2: GP hyperparameter selection · p. 33
• The selection procedure scores default fits against a population composite, rebuilds the reference without outliers, then retunes only discarded fits against that reference.The outlier thresholds are 4 mV for cells and 50 mV for modules; candidate length scales are searched on a grid.Source: Note S2: GP hyperparameter selection · p. 33
• Module and cell GP hyperparameters coincide when module values are expressed per cell equivalent, so the same parametrization serves both model levels.Figure S8 displays the hyperparameters in physical units.Source: Note S2: GP hyperparameter selection · p. 33, Figure S8 · p. 34
• After adaptation, 316 of 324 cells and 25 of 27 modules met the consistency threshold, versus 289 of 324 cells and 23 of 27 modules initially; most curves regained plausible shapes.P1M6C1 improved from 42 mV to 10 mV only at ℓOCV = 15 and still exceeded the threshold; the default-fit median deviation was 1.9 mV.Source: Note S2: GP hyperparameter selection · p. 33, Figure S9 · p. 35