Battery Pack Cell Imbalance, 2RC Models and SoC Balancing

Each cell uses discharge-positive current $i$: $v=U(z)-R_0i-v_1-v_2$, $\dot v_k=-v_k/(R_kC_k)+i/C_k$, and $\dot z=-(\eta(i)i+I_{sd})/Q$. Charge efficiency applies only for negative cell current. Leakage removes stored charge internally; it does not create an additional terminal RC drop.

For PCM groups, $v_g=(\sum E_j/R_{0j}-I_{pack}-I_{bal,g})/\sum(1/R_{0j})$ and $i_j=(E_j-v_g)/R_{0j}$, with $E_j=U(z_j)-v_{1j}-v_{2j}$. For SCM strings, sum the cell source voltages and resistances, including each cell shunt drop, then solve the parallel-string network. This follows Plett and Klein (2009), attached pp. 4–5; the present cell dynamics retain two independent RC branches and omit ESC hysteresis.

The integrator advances both RC voltages with their exponential response. Their interval-average voltages and midpoint OCV determine the average branch currents through two linearized network iterations. SoC is coulomb-counted using those currents. The terminal network is solved again at the endpoint. Steps start small after phase changes and grow to the chosen leg interval; a bracketed refinement shortens threshold-crossing steps while keeping every accepted state inside the SoC and phase limits.

A PCM balancing channel acts across an entire parallel group, using capacity-weighted group SoC. An SCM channel acts across one cell. At each balancing step, higher channels drain toward the minimum channel SoC plus the deadband. The required current is the worst pre-window excess charge divided by the window, plus positive differential leakage relative to that target. This is a charge budget, rather than an optimized controller design. Plett's balancing notes, pp. 5–23 to 5–24, provide the independent 7.7 Ah / 77 mAh benchmark.

The SoC domain 0.1–1.0 is enforced in every phase. An outward boundary crossing stops the whole simulation immediately. Charging stops as impossible when even its idealized net charge input cannot cover leakage. A powered leg reaching its finite time horizon or step limit without a cell cutoff is explicitly incomplete. Cell limit identifiers are assigned only after an actual cutoff; completed-cycle counts exclude interrupted cycles.

All cells share one OCV curve and fixed, positive parameters at an implicitly fixed temperature. The NMC OCV values are Liu and Zhang (2022), Table 2, measured after two hours of rest; monotone cubic interpolation is used over SoC 0.1–1.0. The R₁, C₁, R₂, C₂ defaults use their Table 3 at SoC 0.5. The editable 30 mΩ R₀ is an illustrative scenario input, independent of that paper's fitted R₀.

Capacity, R₀, all four RC parameters, leakage and initial SoC receive independent reproducible bounded variations. Half-ranges describe this constructed sample, not a normal distribution or a confidence interval. One additional capacity outlier is optional through zero capacity loss. Cell numbering runs along each series string, then across parallel branches.

Connections are ideal apart from resistance included in R₀. Every cell begins with relaxed polarization. Shunts draw the specified average current while enabled, limited to the remaining removable charge. There is no shunting during charge or discharge.

Use a measured OCV curve and RC parameters for the chemistry, temperature, ageing state and current range being investigated. The custom affine OCV option is a local approximation; it does not represent a general LFP or NMC discharge curve. No temperature or progressive degradation model is included. Capacity mismatch creates within-cycle SoC spread; secular drift requires effects such as unequal leakage or charge losses.

The default is a 3S2P pack over two cycles with 100 integration intervals per nominal leg. This resolution and compact default pack keep browser evaluation responsive while retaining every cell and both RC branches. The default first-discharge capacity agrees with an independent SciPy DOP853 solution of the coupled ODEs to within 0.00002 Ah; doubling the requested interval count to 200 changes it by less than 0.00002 Ah. For larger packs or longer studies, the interval count is automatically capped by a workload budget, with a floor of 12 intervals per nominal leg; rest and balancing windows use correspondingly fewer intervals. Coarser intervals can shift cutoff timing and average fast current-sharing changes. Increase intervals and compare shorter studies when near a cutoff or when fast circulating-current transients matter; very short RC modes are exponentially integrated, but their current changes within a step are averaged.

Only the last attempted cycle is retained for playback, independently of cycle-summary storage. Discharge and charge each retain 31 base samples, rest retains 11, and balancing retains 21; every visited phase also retains its boundaries, cell-SoC extrema, envelope extrema and maximum-spread time. Interrupted phases keep their actual endpoint. Resampling and displayed interpolation are performed separately within each phase, and plot times include all retained timestamps. This can smooth brief voltage/current changes, so peak current comes from integration endpoints and phase starts, not from the picture.

Limits stop the whole pack at the first cell event. CC/CV taper, string contactor logic, fault propagation, hysteresis, thermal design, current-dependent capacity and cell-level ageing are outside scope. Any phase reaching an outward OCV-domain boundary stops the whole simulation and leaves the current cycle incomplete. A powered leg has a time horizon of 2.5 × largest cell capacity × parallel count / applied pack current; reaching that horizon without a cutoff also leaves the cycle incomplete. Increase charging current relative to leakage when charge cannot finish. Installation work belongs to a qualified person. Balancing-current budget and shunt power do not establish protection settings or component thermal ratings.