Why the Middle curve is the preferred ATL20 OCV backbone
This page addresses the OCV/model branch of the repository’s broader SOC drift diagnostic. An OCV backbone can improve model consistency, but it cannot by itself correct sensor bias or poor initialization.
The Middle method is retained as the preferred OCV backbone because it preserves the expected macroscopic monotonicity and weak LFP two-phase plateau while avoiding the interpretation of finite-rate hysteresis and polarization as equilibrium OCV structure.
This page documents a modelling choice for the ATL20 LiFePO\(_4\) (LFP) cell. The available identification inputs are slow, but still finite-rate, charge/discharge trajectories. They are therefore evidence for a useful macroscopic model representation, not a direct measurement of a unique thermodynamic equilibrium curve.
Why OCV identification is difficult for LFP
For an equivalent-circuit model, the measured terminal voltage is better written as a decomposition such as
\[ V = U_{\mathrm{eq}}(SOC,T) + U_{\mathrm{hys}} - I R_0 - V_{\mathrm{polarization}} + \epsilon . \]
The exact signs depend on the current convention; this repository uses +I = discharge. The important point is that a voltage observed while a cell is being driven contains more than its equilibrium potential. Ohmic drop, charge-transfer and diffusion polarization, relaxation, hysteresis, thermal drift, SOC/capacity misalignment, current-integration error, and smoothing or interpolation choices can all be folded into an unconstrained curve if the identification method is asked to explain every feature.
The repository’s own OCV documentation makes the same limitation explicit: the method-comparison metrics use a reconstructed charge/discharge-envelope reference and are not absolute physical-ground-truth OCV errors. A useful name for the selected curve is therefore mean OCV backbone, pseudo-OCV, or regularized macroscopic OCV. Calling it the exact thermodynamic equilibrium OCV would overstate what these data establish.
What the Middle curve represents
The implementation in middleCurvePolylineDtw.m does more than take a pointwise voltage average. The common preprocessing first extracts the slow discharge and charge branches, smooths them in the voltage domain, and puts them on a consistent SOC orientation. The Middle engine then uses a bounded dynamic-time-warping alignment of the two polylines, takes the geometric midpoint of matched points, and resamples the result on a regular SOC grid.
That construction makes the curve a candidate mean backbone between branches that are known to contain path and rate effects. Hysteresis and dynamic states remain available to explain the charge/discharge separation in the ECM instead of being silently baked into the static SOC-to-voltage map. It is a modelling compromise, not a claim that every midpoint is an equilibrium state.
Thermodynamic constraint: macroscopic monotonicity
Under the repository’s normal full-cell SOC convention, the macroscopic full-cell equilibrium voltage is expected to progress monotonically with SOC (with flat or weakly sloped regions). That statement is about the voltage function used by a full-cell ECM or BMS. It is not a claim that every microscopic chemical potential in a phase-separating particle must be monotonic.
An unconstrained local negative slope should therefore be treated as a diagnostic question, not immediately as a new equilibrium feature. Possible causes include:
- hysteresis or branch-dependent history;
- ohmic, charge-transfer, and diffusion polarization;
- incomplete relaxation or temperature drift;
- SOC/capacity misalignment or current-integration error; and
- interpolation, smoothing, or endpoint artifacts.
This distinction matters because the macroscopic curve is a state relation that other model states are expected to complement. If the static OCV curve absorbs branch-specific terminal-voltage effects, the dynamic and hysteresis states are forced to explain less of the behaviour for the wrong reason.
Why LFP has a weakly sloped two-phase plateau
The positive-electrode reaction can be represented schematically as a two-phase transformation between FePO\(_4\) and LiFePO\(_4\). In the ideal coexistence region, SOC changes mainly by changing the relative phase fractions while the lithium chemical potential changes only weakly. The result is a nearly constant electrode potential and the familiar LFP plateau (Roberts et al. 2014).
A real graphite/LFP full cell need not have a mathematically constant plateau. The negative-electrode contribution, electrode balancing, finite solid-solution regions, particle-size distributions, coherency and strain, temperature, ageing, and manufacturing variation can all give the full-cell curve a small slope. The defensible target is therefore a monotonic, weakly sloped plateau, not a forced constant and not an arbitrary shape that follows every feature in a finite-rate trace.
The phase-coexistence picture also does not eliminate hysteresis. A multi-particle insertion electrode can exhibit path-dependent apparent equilibria, and LFP is known for pronounced charge/discharge separation (Dreyer et al. 2010; Barai et al. 2015). That is precisely why the static backbone and the hysteresis/dynamic terms should be kept conceptually distinct.
Why not force every C-rate feature into OCV?
There are two modelling failures at opposite extremes:
- An aggressively flexible fit can reproduce finite-rate terminal voltage by assigning polarization, relaxation, or hysteresis to the OCV map.
- An exactly flat OCV plateau can discard the small but repeatable full-cell slope and leaves no local voltage sensitivity at all.
The Middle curve is a sensible compromise when it is used with a separate hysteresis/dynamic description:
- it favours macroscopic monotonicity;
- it retains a practical weak slope over the LFP plateau;
- it does not privilege the charge or discharge branch as the static truth;
- it avoids overstating voltage-based SOC observability; and
- it leaves room for hysteresis and polarization states to explain the branch separation.
Benchmark: ATL20 at +25 °C
The central visual benchmark is the repository’s existing inspection figure. It overlays the raw discharge and charge branches, the Middle-based metrics reference, and the enabled candidate methods.
How to read the figure
The useful questions are structural rather than purely visual:
- Does the green Middle curve stay between the branch behaviour through the main plateau rather than tracking one branch’s endpoint artefact?
- Is the plateau weakly rising and monotonic at the macroscopic scale?
- Do alternative methods introduce sharper endpoint turns or visibly absorb charge/discharge separation into the static map?
- Is any apparent local negative slope large enough to be physically meaningful, or is it at the scale expected from finite-rate data and interpolation?
At +25 °C, the displayed Middle curve is smooth and non-decreasing across the plotted SOC interval, with a gently rising central plateau and sharper changes near the SOC ends. The charge and discharge traces remain visibly separated, which is evidence that a single static curve should not be asked to explain all of their voltage difference. The figure supports the choice; it does not prove thermodynamic equilibrium, and it should not be used to claim that every alternative has a negative slope at every temperature.
The repository’s promoted inspection record reports the selected engine as middleCurve and lists reference-relative metrics across eight temperatures. Those metrics are useful for method comparison, but they are not ground-truth OCV accuracy metrics.
What the −25 °C MAT file contains
The requested ATL_OCV_N25.mat was inspected with SciPy. Its top-level payload is OCVData, containing four scripts. Each script has the fields time, step, current, voltage, chgAh, and disAh.
| variable | samples | step == 2 samples |
role in repository preprocessing |
|---|---|---|---|
OCVData.script1 |
7,188 | 7,058 | slow discharge branch |
OCVData.script2 |
21,837 | 8,359 | intervening capacity/current sequence |
OCVData.script3 |
5,278 | 5,148 | slow charge branch |
OCVData.script4 |
27,816 | 14,358 | intervening capacity/current sequence |
Using the same bookkeeping as prepareOcvBranches.m gives an efficiency scaling of approximately 0.995874 and a normalization quantity named Q25 of approximately 19.338 Ah for this dataset. The extracted slow branches cover only about 25.6–100.0% SOC on discharge and 0.0–54.1% SOC on charge, leaving a partial overlap at this cold-temperature condition.
Two cautions follow directly from those facts:
- The N25 MAT file is a raw trajectory source, not a saved Middle-curve representation. It cannot by itself certify the monotonicity of the final temperature-regressed Middle model.
- The P25 figure is a different dataset (
ATL_OCV_P25.mat) and is used as the requested illustrative benchmark. It must not be confused with the N25 raw-file inventory above.
The P25 Middle curve is verified visually as non-decreasing in the rendered inspection figure. A numerical monotonicity statistic for the exact temperature-regressed model should be generated by rerunning the repository MATLAB inspection pipeline in an environment with the required MATLAB/DTW toolbox; the local MATLAB executable was present but could not check out its license during this documentation build.
Estimator implications of plateau slope
The quantity
\[ \frac{dU}{dSOC} \]
is the local voltage sensitivity to SOC. If a fitted plateau slope is made too large, an estimator is told that voltage contains more SOC information than the physical LFP cell actually provides. This tends to make voltage updates look more informative than they are and can hide an SOC observability problem in the tuning results.
At the other extreme, a perfectly flat plateau gives essentially zero local voltage observability. The practical target is therefore:
monotonic + weakly sloped + regularized toward flatness
rather than arbitrarily flat, aggressively sloped, or locally negative. The Middle curve gives the estimator a defensible mean voltage sensitivity while leaving branch history and transient polarization to their own model states.
What the alternative methods reveal
The P25 overlay is useful precisely because it displays the alternatives on the same axes. Vavg and SOCavg provide different averaging choices; Resistance blend explicitly mixes voltage with a resistance-related correction. Their deviations near the ends and their different placement relative to the raw branches are reminders that a low residual against finite-rate data is not automatically a better equilibrium OCV model.
The diagonal methods are evaluated in the repository summary but are hidden from the default inspection plots. This is appropriate for a first visual comparison: the page’s decision is about the physical role of the static backbone, not about selecting the curve with the most freedom to reproduce one branch.
Validation across the remaining models
The +25 °C example is illustrative, not the only proof. The repository has inspection figures for −25, −15, −5, 5, 15, 25, 35, 45 °C and the promoted summary records all eight input files and method-comparison cases:
- all ATL20 OCV inspection figures
- promoted OCV inspection summary
- OCV modelling workflow and rerun instructions
Across those cases, the validation question should be whether the selected backbone remains defensible: nearly monotonic, weakly sloped through the LFP plateau, resistant to artificial negative-slope features, and consistent across temperatures and model variants without absorbing branch-specific voltage effects. That broader consistency argument is stronger than claiming that one attractive P25 overlay proves the method universally correct.
Modelling recommendation
For the intended ATL20 ECM/BMS use, retain middleCurve as the default OCV identification backbone, and represent hysteresis and dynamic polarization separately where the model requires them. Describe the result as a thermodynamically consistent pseudo-OCV / mean macroscopic OCV backbone.
Revisit the choice only when a broader temperature and ageing study shows a repeatable feature that survives SOC alignment, relaxation, branch-aware analysis, and sensitivity checks. A single local negative slope in a finite-rate identification is not enough evidence to override the macroscopic monotonicity prior.
References
The scientific basis for this conservative wording includes the distinction between equilibrium voltage and measured/rest-voltage procedures, the two-phase origin of the LFP plateau, and the thermodynamic and practical origins of LFP hysteresis (Barai et al. 2015; Dreyer et al. 2010; Roberts et al. 2014). The repository’s model implementation and data provenance remain the primary references for the ATL20-specific claims.