Battery AC Self-Heating at −30 °C: Plating-Limited Warmup

The frequency-domain approach follows Zhang et al. (2015), the constant-polarization optimization concept of Ruan et al. (2016), and the electrode-potential constraint of Ge et al. (2016). Liu et al. (2023), pp. 4–7, gives the open-access sinusoidal heat and anode-potential equations used here.

For each electrode, $Z_k=[R_{ct,k}+Z_{D,k}]/[1+Y_{dl,k}(R_{ct,k}+Z_{D,k})]$. Reflecting planar diffusion is $Z_D=R_D/[s\tanh(s)]$, $s=\sqrt{j\omega\tau_D}$, as implemented in impedance.py. The CPE admittance is $Y=C_*\omega_* (j\omega/\omega_*)^a$ with $\omega_*=1\,\mathrm{s}^{-1}$; the F-valued input is an equivalent capacitance at that angular rate. A separate $R_{SEI}\parallel$CPE film branch, $R_0$ and $j\omega L$ form $Z_{cell}$.

Arrhenius scaling is $g(E,T)=\exp[(E/R)(1/T-1/T_{ref})]$ with absolute temperatures and $T_{ref}=253.15$ K. Charge-transfer and film resistances use their own activation energies. Both diffusion resistance scale and diffusion time use the diffusion factor, keeping low-frequency chemical capacitance unchanged.

The peak current is the smallest of $I_{max}$, $(U_n-U_{reserve})/|Z_n|$, $V_{pol}/|Z_n+Z_p+Z_{SEI}|$, and $V_{AC}/|Z_{cell}|$. Thus the cycle-minimum model anode potential stays at or above the reserve. The anode branch includes diffusion, so its voltage contains both kinetic and concentration polarization. Terminal headroom must be supplied relative to OCV.

$P=I_{peak}^2\operatorname{Re}(Z_{cell})/2=I_{rms}^2\operatorname{Re}(Z_{cell})$, and $C_{th}dT/dt=P-H(T-T_{amb})$. The frequency sweep integrates $dt/dT=C_{th}/[P-H(T-T_{amb})]$ with the logarithmic integral of each locally linear net-heat segment, and rejects stalled paths. The logarithmic form resolves the rapid growth of time near zero net heating without a trapezoidal endpoint singularity. Bounded frequency minimization uses adaptive local refinement of this same integral, evaluated from the actual temperature-dependent operating point.

The selected manual, fixed or adaptive schedule is integrated with interval bisection until the local coarse/fine difference meets a relative tolerance of $10^{-5}$. A second run starts with twice as many intervals and a four-times tighter tolerance. Unresolved error is flagged. For adaptive control, the interpolated frequency candidate is compared with the best fixed frequency at each evaluated temperature, retaining the greater heat generation.

One operating-point function supplies the thermal ODE, plotted power, current and anode potential, including temperatures below the starting grid. The Dormand–Prince ODE integrates temperature, cumulative generated heat and cumulative ambient loss together. Reachable cases cover the complete warmup, independent of elapsed-time playback position.

Real and imaginary parts are evaluated analytically with scaled hyperbolic expressions to avoid overflow and repeated complex arithmetic. A direct complex-impedance calculation is retained as an arithmetic cross-check.

• One graphite-anode cell, symmetric sinusoidal current, no DC bias, nearly fixed SOC, uniform cell temperature and constant heat capacity. The source cell is graphite/NMC at 50% SOC.
• Liu et al., Table 1 (p. 7): mass 56 g and heat capacity 1.24 J/(g·K). Anode OCP 0.1354 V is from p. 5. Charge-transfer activation energies 76.8 and 81.9 kJ/mol are from p. 9.
• At −20 °C, approximate readings from Liu et al., Fig. 4 (p. 10): anode $R_{ct}=0.58$ Ω, CPE scale 0.72 F and exponent 0.73; cathode 0.14 Ω, 4.5 F and 0.85; full-cell ohmic resistance 0.012 Ω and film resistance 0.0075 Ω, film CPE scale 0.2 F and exponent 0.87. Values are rounded plot readings, not a published parameter table. CPE scales/exponents are held constant with temperature.
• Example assumptions requiring measurement: diffusion scales 0.10/0.05 Ω and times 100/100 s; diffusion activation energy 30 kJ/mol; ohmic/film activation energies 10/20 kJ/mol; inductance 1 µH; heat loss conductance 0.05 W/K; current and voltage limits. These do not constitute a characterized commercial cell.
• Zero-mean reversible entropic heat cancels over complete cycles. Converter losses and energy taken from a separate power supply are excluded; generated heat is the cell's real electrical loss.

This predicts a model anode-potential constraint, not the absence of lithium plating in an uncharacterized cell. The electrode potential, impedance, reserve and terminal headroom must be measured for the intended chemistry, SOC, aging state and temperature. Linear EIS can cease to describe large excitation; spatial electrolyte and particle gradients are not resolved. Start with a ramp, maintain zero DC bias, and validate the profile with cell-level temperature and voltage monitoring.

The published EIS example spans −20 to 25 °C and 0.01 Hz to 10 kHz. Operation at −30 °C is Arrhenius extrapolation. The reflecting planar diffusion extension and assumed diffusion parameters are not a reproduction of Liu's fitted model, and the paper's experimental heating times are not claimed as validation. The two diffusion benchmarks verify independent published numerical values; the thermal-capacitance check verifies source data and units.

The frequency search uses 501 logarithmic frequencies and 151 temperature knots, then refines the best fixed-frequency interval with adaptive time quadrature. The adaptive frequency candidate is interpolated between solved knots; grid spacing contributes up to about 1.2% frequency uncertainty near an optimum over the default five-decade band, and wider bands coarsen it. Heating power is recalculated from the impedance at every quadrature and ODE evaluation; it is never clamped to its starting value. Frequency is held at the schedule endpoint outside its temperature range, with the fixed-frequency fallback still compared during adaptive operation.

Time integration adaptively bisects intervals and compares the logarithmic integral for one segment with two half segments. It allows at most 24 refinement passes and 4097 stored temperature nodes; the calculation warns if the error estimate or tighter selected-strategy check remains unresolved. Near a thermal stall, the result also remains highly sensitive to cell and heat-loss parameters. Unreachable frequencies are intentional gaps. If no fixed frequency reaches the target, the optimum result is blank and playback uses the grid frequency giving the greatest initial heat.

Reachable cases show the full warmup in both charts and playback, including durations longer than 200000 s; generated and lost heat cover that complete interval. Stalled cases show a one-hour transient, and their energy results cover that hour. The input's elapsed time only selects a position within the cached solution and never truncates the energy integration. Playback interpolates cached thermal and electrical histories; it is a slow warmup visualization, not an AC-cycle waveform. Lumped temperature can miss internal hot spots, especially for thick cells or rapid heating. Pack gradients, connections, balancing, switching harmonics, electrode nonlinearities and plating kinetics require further modeling.