Li-ion Voltage Under Load & Runtime: 2RC Cell Model

The reference cell is a Panasonic NCR18650PF. Positive current means discharge. A charge state and two polarization voltages obey

For one constant-current interval, each branch advances exactly as $v_j^{+}=v_j e^{-\Delta t/\tau_j}+I R_j(1-e^{-\Delta t/\tau_j})$. SOC parameters are held at the interval's beginning, then refreshed; terminal OCV and R₀ use the new SOC. The calculation uses intervals of at most 5 s, reduced to 2.5 s within 0.15 V of cutoff, splits them at every load change, and bisects an interval containing cutoff. Both sides of current discontinuities appear in the trace. Delivered charge integrates I; delivered energy integrates each exponential polarization voltage analytically. The OCV and ohmic terms use exact antiderivatives of their piecewise-linear SOC curves, including intervals crossing table knots. All internal state arrays use normalized SI values, restored to explicit units in the named steps.

The first crossing of terminal cutoff or zero SOC permanently opens the load. If neither occurs before the maximum load duration, the status reports an observation limit. Relaxation then holds SOC and OCV fixed and lets both RC voltages decay. Every integration endpoint is cached in compact state blocks. Playback reads the immediately preceding integration checkpoint and applies only the remaining fraction of that same interval, with the same frozen RC parameters as the simulation. It evaluates OCV and the ohmic resistance at the returned SOC. Chart thinning is separate from this full state cache; moving the time cursor does not rerun the discharge. Plot sampling retains both sides of every load change and a regular grid of at most about 480 intervals. Internal integration remains at most 5 s, with refinement near cutoff. Bounded cache blocks keep every intermediate within the engine's unit-check limit.

For each RC branch, $\int_0^h v_j\,dt=IR_jh+(v_j(0)-IR_j)\tau_j(1-e^{-h/\tau_j})$. Subtracting these voltage integrals from the integrated OCV-minus-ohmic term and multiplying by current gives delivered joules. This resolves fast transients even when their time constants are shorter than the state integration interval.

Parameter identification. Own positive constrained least-squares fits use Kollmeyer's public 2.9 A, 10 s HPPC pulses and following relaxation at 0 °C and 25 °C. OCV is the pre-pulse rested voltage. At 25 °C the first sustained 1C discharge constrains the sum of the three resistances, preventing the slow branch from becoming poorly identified by a short pulse alone. The second 1C discharge and the 5.8 A pulses are held out. Attached source s3, pages 1–2, records filenames, SHA256 hashes, extraction rows, fitting bounds and derived parameters. The nominal rating is 2.9 Ah; the C/20 charge difference is 2.99732 Ah and defines the SOC capacity. Chen and Rincón-Mora supply the model architecture, not these fitted parameters.

Temperature linearly interpolates the six circuit parameters between 0 °C and 25 °C. Capacity ageing multiplies the measured capacity; resistance ageing multiplies all R values while keeping both time constants fixed. There is no additional Peukert penalty: the rate-capacity effect comes from current-dependent voltage loss and early cutoff, avoiding double counting. Constant current, pulses and the six-stage drive-style profile are prescribed current loads, not constant-power loads.

Numerical and measurement checks. At defaults, halving the integration ceilings from 5/2.5 s to 2.5/1.25 s changes cutoff by about 0.34 s and delivered charge by about 0.00038 Ah. The held-out 25 °C 1C case is accepted within 120 s (about 3.5%) and 0.10 Ah, with 20 mV tolerance at 1000 s. Two held-out 5.8 A pulse endpoints use 30 mV tolerance. These tolerances describe these checks, not a guaranteed error bound for every profile.

Playback continuity checks compare both sides of saved checkpoints whenever current is unchanged; real current-step voltage jumps are retained. The voltage-balance property verifies that all displayed voltage drops refer to the same playback state.

The independent constant-parameter 5.8 A, 1 s pulse at 0 °C delivers 0.005678006654465168 Wh by the closed-form RC integral; the energy reference case checks this value to 1 nWh.

One initially relaxed cell, with v₁ = v₂ = 0; prescribed nonnegative discharge current; fixed cell temperature. SOC is the fraction of the cell's low-rate charge capacity. The 25 °C zero-SOC OCV uses the C/20 one-hour-rest endpoint, 2.86117 V. Below the lowest 25 °C HPPC point (about 8% SOC), resistances continue the last rising depletion slope and time constants hold. Below the lowest cold point (about 17.7% SOC), cold resistances and time constants hold; cold OCV follows the warm endpoint shape with the last measured cold offset. These extensions are assumptions, not measurements.

The drive-style cycle repeats six current fractions 0.1, 0.4, 1.0, 0.2, 0.7, 0.3 for fractions 1/12, 1/6, 1/4, 1/6, 1/6, 1/6 of its period. It is an authored scenario, not UDDS, US06 or measured vehicle demand. Factors for ageing are independent sensitivity controls, not a fitted degradation law.

Use as a fitted Panasonic NCR18650PF reference, not as a universal chemistry or another cell's datasheet. Temperature is restricted to 0–25 °C. The quantitative held-out validation is at 25 °C; intermediate temperatures interpolate, and cold behavior and ageing factors need application-specific verification. The fit uses 2.9 A pulses and sustained discharge; held-out 5.8 A pulses test larger current. Treat results above 5.8 A and near depleted SOC as extrapolations. A 2.5–4.15 V cutoff is available, subject to the initial OCV condition. High cutoffs can legitimately give zero runtime at the first load step.

The observation window is at most 180 min. A result ending at this window is censored; it does not establish when cutoff would occur. A low-rate delivered/rated percentage above 100% is possible because the measured C/20 capacity exceeds the nominal rating. Very short pulses below 1 s, impedance at switching frequencies, charging and regeneration, hysteresis, self-discharge, thermal evolution, protection delays and automatic restart are excluded. No pack topology, cell imbalance, balancing or pack protection is modeled. No safety certification or prediction of cycle ageing is implied.

For readable full-profile charts, the loaded interval is limited to 300 pulse cycles or 100 drive-style cycles. Shorten the observation window or lengthen the period if this limit is reached.