Grid Frequency Drop Calculator & Sudden Trip Simulator

Fleet mix and physical inertia

For synchronous rating share $x$, $S_s=xS_b$ on the fixed post-trip reference fleet MVA base $S_b$. The replacement fleet is split as $S_v=(1-x)q_vS_b$ for GFM and $S_l=(1-x)(1-q_v)S_b$ for GFL. Inverter rated MW is its MVA rating times the entered active-MW/MVA ratio. These are capacity scenarios, not dispatch reconstruction.

A necessary capacity check requires $0\le P_{init}\le P_{cap}$, where $P_{init}=P_L-P_{loss}$ is initial generation from the surviving fleet and $P_{cap}=S_b[x+k_{inv}(1-x)]$ is its active-power ceiling, explicitly converting MVA to MW in the formula. For generation loss this is the surviving pre-trip output, $P_L-P_{trip}$; for load loss it is $P_L+P_{trip}$. The synchronous contribution uses an optimistic unity-power-factor ceiling and the inverter contribution uses the entered active-MW/MVA ratio. No external supply is represented. This is necessary, not sufficient: actual synchronous rated MW, reactive-power capability, unit dispatch and headroom may impose tighter limits. The same bound masks both comparison frequency and initial-RoCoF curves, and infeasible comparison scenarios are skipped before integration.

Physical energy is $E_s=H_sS_s$, using the conventional MVA basis, converted to MJ; the physical swing coefficient is $M_s=2E_s/f_0$. Use $H_s=\sum H_iS_i/\sum S_i$, excluding the tripped machine. ERCOT (2018), p. 5 and footnote 2, defines this normalization. An H value quoted on a rated MW base must first be rebased: $H_{MVA}=H_{MW}P_{rated,MW}/S_{rated,MVA}$. Rated active MW and apparent MVA are separate bases.

Three response mechanisms

The coherent single-area balance is

$P_{loss}$ is positive for generation loss and negative for load loss; $K_D=D(P_L-P_{shed})/f_0$.

Synchronous governor gain is $K_g=xS_g/(Rf_0)$ and reserve is $xP_{res}$. Continuous no-step deadband gives $u_g=\operatorname{clip}[-K_g\operatorname{sign}(\Delta f)\max(|\Delta f|-f_{db},0),-xP_{res},xP_{res}]$, with $T_g\dot P_g=u_g-P_g$. NERC Primary Frequency Control (September 2023), attached pp. 8 and 17–19, supports sustained droop and no-step deadband. This is one aggregate lag, not a turbine-specific model.

GFM droop gain is $K_v=P_{v,rated}/(R_vf_0)$. In VSM mode $M_v=2H_vS_v/f_0$; in ideal droop-only mode $M_v=0$. Programmed virtual energy $H_vS_v$ is a coefficient, never added to the displayed physical rotor energy. The combined GFM response is

The reduced coherent-frequency model is derived here from swing emulation and droop laws described by D’Arco and Suul (2014), Eqs. 1–6; Rathnayake et al. (2021), attached pp. 6 and 8, Eqs. 4 and 8–9; and the NREL GFM roadmap, attached pp. 24–28 (printed pp. 13–17). It is not a reproduction of those papers' converter or network simulations.

Let $B=P_g+P_{FFR}+P_{shed}-P_{loss}-K_D\Delta f$. Solve the unsaturated candidate $r_u=(B-K_v\Delta f)/(M_s+M_v)$, calculate and clamp $P_{v,u}=-K_v\Delta f-M_vr_u$, then use $\dot{\Delta f}=(B+P_v)/M_s$. This solves the implicit common-power-limit equation exactly for $M_s>0$; it does not retain virtual inertia when its requested MW cannot be delivered. At t=0, unsaturated VSM response gives $r_0=-P_{loss}/(M_s+M_v)$; droop alone gives $-P_{loss}/M_s$.

The GFL block is the separately delayed event-triggered boost $P_{FFR}=sP_F[1-\exp(-\max(0,t-t_d)/T_F)]$, capped at the GFL active rating and held until 30 s. $s$ is the event direction. NERC FFR (March 2020), attached pp. 13–14, describes fast response and contingency-based triggering. This selected implementation assumes knowledge of event direction; it does not simulate a PLL or frequency-trigger relay.

Solution, comparison and playback

The frequency and governor states form a two-state affine linear system while the governor deadband/saturation and shared GFM power-limit branches stay fixed. Within each segment, write $\dot y=Ay+c+q\exp(-\lambda u)$, where $u$ is time from the segment start. Its matrix-exponential solution and forcing integrals are evaluated in closed form. Modal constants are cached once per segment; zero eigenvalues, coincident eigenvalues and a forcing decay equal to a mode use their analytic limits. This removes the small stability-limited steps that an explicit ODE solver needs for fast ideal droop at low physical inertia.

The solver restarts at governor deadband/reserve crossings, GFM power-limit crossings and the delayed GFL activation time. A mode-scaled scan brackets the first control-boundary crossing. Stationary points are refined where a modal curvature bound cannot exclude an intervening grazing crossing. Boundary roots use safeguarded interpolation and bisection, with about 1e-10 s time resolution and 1e-12 normalized residual. These numerical tolerances do not remove model uncertainty.

One pre-shedding solution supplies the selected trace, UFLS event evaluation and the matching configured share scenario. Identical passive/configured trajectories are reused. Frequency-rate sign changes bracket extrema of the continuous pre-shedding solution; threshold roots split the monotone intervals into continuous under-frequency durations. The timer resets on recovery. Operation occurs at the first downward crossing plus the entered delay only when that interval lasts long enough. This detects the reference 13.28 ms dip even when both adjacent plotted frequency samples are above pickup. At operation, the state is evaluated directly from its analytic segment, the one load block is disconnected, and a new segment solution starts with reduced load and damping. There is no reclose.

The plotted trace has 1,000 samples over 30 s; the UFLS event replaces the nearest interior sample. Plotted extrema retain approximately 30 ms resolution, but UFLS threshold and timer times are refined on the continuous solution. RoCoF is evaluated from accelerating power rather than a finite-difference measurement window. The heavy segment solutions and share comparison do not read the playback input. Independent high-accuracy SciPy Radau integrations cross-check the default, fast-droop and power-limited generation/load-loss cases; these are numerical regression checks, separate from the published NREL method benchmarks.

The main trace evaluates both inverter blocks at the chosen share. Seven comparison scenarios from 10% to 100% synchronous rating use the same full-fleet base, event, load and response parameters, with UFLS disabled. The passive scenario sets both GFM and GFL support to zero. Replacement increases GFM/GFL ratings at the selected allocation while governor rating and reserve fall with synchronous share. Each worst-frequency comparison is withheld if its entire sampled trace leaves ±5%; the initial RoCoF remains a local t=0 result, available only for a capacity-feasible scenario.

The physical rotor display uses $E_s(f/f_0)^2$ and integrates the speed ratio for its slowed orientation. The solver linearizes its swing coefficient at nominal frequency. The heavy propagation solutions and share sweep do not read playback time.

The GFM energy integral executes only when the selected run meets both capacity and frequency validity. An if branch skips trapz for unavailable runs; the output step returns an intentional unavailable value. Invalid-run regression probes depend on successful evaluation of this energy result, and the complete reviewer trials are checked for unexpected engine errors in addition to the intended input-condition messages.

• One coherent area, one instantaneous trip at t=0, initially at nominal frequency and active-power balance.
• Fleet shares refer to surviving MVA ratings. H uses the same MVA base as the synchronous machines; motor inertia may be included only through a consistent equivalent stored energy. Inverters have no implicit physical inertia.
• Changing synchronous share replaces capacity on a fixed reference base; governor participating MW and reserve scale with that share. A necessary active-power capacity bound is enforced, but there is no dispatch, commitment or pre-event power-flow calculation. The synchronous MVA contribution is an optimistic unity-power-factor active-power ceiling; verify actual rated MW and headroom. No unmodelled external supply is included.
• GFM droop and ideal VSM dynamics act at the common area frequency. Their common symmetric power clamp is an aggregate headroom model. It is not a converter current limiter or a claim of stable voltage formation. Ideal droop-only deliberately neglects filter and network dynamics.
• GFL FFR is a prescribed contingency-triggered exponential boost with its own delay. Requested boost is capped at the GFL active rating; actual resource headroom must be confirmed independently.
• Both inverter groups must sustain the requested exchange through 30 seconds. GFM energy is the integral of absolute delivered power, counting both directions; GFL energy is the same for its unidirectional boost. No storage depletion, withdrawal, recharge or energy reserve enforcement is simulated.
• Governor reserve is symmetric upward/downward; aggregate headroom/footroom must be verified. Load relief is linear, instantaneous and decreases with the shed block.
• One illustrative UFLS stage has a resettable continuous pickup timer and a permanent trip latch. No reclose, relay filtering, extra breaker travel time, or multiple stages.
• The GFM aggregation and simultaneous saturation algebra are derived here from the cited general laws; source benchmark cases verify limiting behavior, not a specific vendor controller.
• Defaults describe an illustrative mixed 60 Hz system, not a PJM or Iberian blackout reconstruction.

Use for educational comparisons and early coherent single-area sensitivity studies over 0–30 s. At least 10% of the reference MVA fleet remains synchronous. Fully inverter-based operation, synchronization and voltage formation require a different model.

The selected transient is supported only when the initial surviving generation is between zero and the fleet active-power ceiling, and the entire sampled frequency trace stays within ±5% of nominal. Otherwise its result cards, main traces, GFM energy and rotor playback are withheld. Both comparison frequency and initial-RoCoF curves omit scenarios with insufficient fleet capacity; comparison frequency curves also show gaps outside the frequency range. Initial RoCoF is a local t=0 slope, not a prediction of subsequent stability. Each comparison disables UFLS regardless of the main setting.

Nadir and peak are sampled extrema within the horizon, including the initial state. A trace still moving away from nominal at 30 s has no demonstrated arrested nadir or peak. Primary droop need not restore nominal frequency. UFLS timing is refined independently of the approximately 30 ms plot sampling.

Exclude voltage/reactive instability, transmission constraints and losses, inter-area modes, island formation, changing topology, converter PLL and inner voltage/current loops, current limiting and loss of GFM synchronism, detailed turbine dynamics, generator protection/cascading trips, AGC and secondary recovery. The power clamp is not a physical converter fault model. A smaller nadir excursion does not prove overall grid stability.

Instantaneous RoCoF is not a filtered relay measurement or compliance test. UFLS inputs are illustrative, not regional rules or recommended field settings. NERC editions cited here explain the concepts; the calculator makes no regulatory compliance claim.