BESS Ageing Calculator: FCR, Peak Shaving and SEI Fade

Profiles and rainflow. A seeded Park–Miller generator and Box–Muller transform drive a stationary Gaussian AR(1) frequency process, with $\rho=\exp(-\Delta t/\tau_f)$. A warm-up of at least six hours and six recovery time constants precedes seven days sampled every 15 s. FCR follows Celi Cortés et al. (2025), Eq. (9): zero inside the deadband, $\Delta f/\Delta f_{full}$ outside it, saturated at ±1. High frequency charges the battery. SoC recovery uses $x_{k+1}=e^{-\Delta t/\tau_{bal}}x_k+(P_{FCR}/E_0)u_k\tau_{bal}(1-e^{-\Delta t/\tau_{bal}})$, clipped to the SoC window. Recovery is an additional exchange, not deducted from the initial FCR command. The peak-shaving day is an exact piecewise-linear charge–hold–discharge trace. A three-point stack rainflow algorithm retains full cycles and residual half cycles. A cycle's C-rate is mean absolute SoC movement per active time within its endpoint interval.

LFP, Naumann 2018/2020. Capacity calendar loss is $q_{cal}=k_{T,Q}[2.8575(z-0.5)^3+0.60225]\sqrt{t_s}$, with $k_{T,Q}=1.2571\times10^{-5}\exp[-17126/R(1/T-1/298.15)]$. Resistance calendar growth is $r_{cal}=k_{T,R}[-3.3903(z-0.5)^2+1.5604]t_s$, with $k_{T,R}=3.4194\times10^{-10}\exp[-71827/R(1/T-1/298.15)]$. Cycle coefficients, in fractional loss per √EFC and gain per EFC, are $b_Q=(0.0630C+0.0971)[4.0253(d-0.6)^3+1.0923]/100$ and $b_R=(-0.0020C+0.0021)[6.8477(d-0.5)^3+0.91882]/100$. Calendar resistance uses the quadratic SoC factor in Naumann's dissertation (2018), Eq. (5.14); the cycle-resistance coefficient 0.91882 follows Table 5.3. The 0.6 offset follows the TUM implementation and the explicit reproduction in doi:10.1016/j.est.2026.121158; it is not the 0.5 offset appearing in some secondary accounts. The SEI-associated calendar capacity loss is the √t term, not an extra additive loss.

NMC, Schmalstieg 2014. $q_{cal}=\alpha_Qt_d^{0.75}$ and $r_{cal}=\alpha_Rt_d^{0.75}$, with $\alpha_Q=(7.543U-23.75)10^6e^{-6976/T}$ and $\alpha_R=(5.270U-16.32)10^5e^{-5986/T}$. Cyclic terms are $q_{cyc}=\beta_Q\sqrt{Ah}$ and $r_{cyc}=\beta_R Ah$, with $\beta_Q=0.007348(U_{rms}-3.667)^2+0.000760+0.004081d$ and $\beta_R=0.0002153(U_{rms}-3.725)^2-0.00001521+0.0002798d$. The source cell is 2.15 Ah, hence $Ah=2(2.15)\,EFC$ independent of pack size. Open-circuit voltage uses the NREL source digitization with PCHIP interpolation; five-point Simpson integration estimates cycle RMS voltage.

Accumulation. For a power law $L=kX^p$, evolve $L^{1/p}$: $\Delta(L^{1/p})=k^{1/p}\Delta X$. Thus square-root cycle laws add $b_Q^2 n d$ to their capacity-loss clock, not a fresh loss per counted cycle. Time-integrated calendar clocks and daily cycle clocks are mixed by operating-day share, with storage included. The stationary-duty forecast evaluates the resulting power laws exactly at 241 ages; bisection locates capacity and resistance limits. Capacity losses and resistance gains are added separately.

Fresh cells start at normalized capacity and resistance of 1. Temperature is constant and uniform. SoC is a fraction of current available capacity. The same fractional SoC profiles repeat throughout life, so absolute service power and energy scale with remaining capacity; the initial FCR power is used only to establish that profile. No annual reset, augmentation, thermal feedback or stochastic failure is included.

FCR uses an illustrative Gaussian core with user-specified correlation; the 35 mHz spread and 120 s correlation are scenario defaults, not a fit to a particular grid. Coulombic efficiency is taken as unity and voltage is treated as approximately constant when converting pack MW/MWh into C-rate. Peak shaving performs one daily excursion on each allocated day. Storage days remain at target SoC. Mixed-service equivalent-age averaging assumes day types are well interleaved. Transfers between the starting SoCs of different day types are excluded; the share is an exposure mixture rather than a chronological dispatch schedule.

The film in the schematic is an exaggerated proxy for calendar ageing, not a calculated SEI thickness in nanometres. Capacity lost through SEI cannot identify physical film thickness without electrode area, composition and density. The NMC exponent 0.75 is an empirical calendar fit; it is not relabelled as pure diffusion growth.

These coefficients describe specific legacy cells, not all NMC or LFP products. Long-horizon results extrapolate experiments lasting roughly 29 months. Use the model principally up to the first end-of-life threshold; subsequent curves are scenario extrapolations. No knee, plating, gas evolution, cell imbalance, reversible shallow-cycle LFP capacity loss, SEI cracking feedback or pack thermal gradients are modelled. The original NMC cycle fit was obtained near 1C and 35 °C, and the LFP cycling data cover warm operation. FCR microcycles and lower rates are extrapolations. This tool does not establish a warranty or the reliability of an installation.

The Gaussian frequency model omits heavy tails, long biases and deterministic trading-hour events. Fifteen-second sampling resolves the default 120 s correlation with eight samples, but misses faster current pulses and very shallow cycles. A finite seven-day realization has sampling uncertainty: vary the seed and frequency assumptions. The example-day chart shows one-minute samples of the full 15-second solution; rainflow and damage retain every solved sample. The depth chart groups EFC into 5 percentage-point bins; all individual counted cycles, not bin centres, enter the ageing calculation. Five-point integration approximates cycle RMS OCV in NMC. Lifetime curves use the exact power laws, not a coarse annual interpolation.

Negative cyclic resistance coefficients produced by extrapolating empirical polynomials are constrained to zero; the model never treats cycling as resistance healing. This particularly affects shallow NMC cycles and changes their predicted resistance growth. SoC clipping reduces delivered service; the condition flags appreciable clipping, and predictions then describe the clipped trace. Recovery power assumes separate converter headroom and unrestricted recovery transactions.

The FCR characteristic follows the research example in Celi Cortés et al. (2025), describing 2022 Continental Europe requirements; it is not a check of current market rules or prequalification. The forecast is an ageing model, not a dispatch feasibility, fire-safety or electrical installation assessment.

Service-specific input checks apply only to allocated operating days. Inactive service traces rest at target SoC. The share-sensitivity curve omits counterfactual shares whose service profile violates the model's SoC, rate or droop limits.