Which kicks does a grid node survive without crossing the angle limit?

An interactive companion to Jaiyong Lee, Daekyung Lee, Seung-Woo Son, Sang Hoon Lee, Mi Jin Lee and Heetae Kim, “Energy-barrier characterization of secured basins in coupled oscillators with inertia” (arXiv:2610.11147). Perturb one node of a three-oscillator second-order Kuramoto network, see which perturbations keep every line below a phase difference of π, and read the result as two energy thresholds.

Coupling K Inertia M Damping D Load at node 0, P₀

Node 0

Between E^S and E^U, energy alone does not decide the outcome: the direction of the kick matters. Disparity Δ measures how wide that band is on a log scale:

\overline{E} = \sqrt{E^{S} E^{U}}, Δ = \sqrt{\frac{E^{U}}{E^{S}}}, [E^{S}, E^{U}] = [\frac{\overline{E}}{Δ}, \overline{E} \cdot Δ]

Single kick

Click anywhere on the plane to fire one perturbation (Δθ, Δω) at the selected node and trace it.

angle limit π

How to read this. Each dot is a perturbation applied to the selected node at its synchronized state; the horizontal axis is the phase kick Δθ and the vertical axis the frequency kick Δω. Basin stability only asks whether the network eventually re-synchronizes. The secured basin also asks whether any connected pair ever reached a phase difference of π on the way, the point at which real protection equipment would act. Orange dots are the gap between the two: perturbations that basin stability counts as safe but that the grid would not have survived intact.

The perturbation energy puts every dot on one physical scale:

E = \frac{1}{2} M Δ ω^{2} + Δ V (Δ θ)

with the potential change of the kicked node i, its neighbors held at the synchronized angles θ*,

Δ V (Δ θ) = - \sum_{j} K_{i j} [\cos (θ_{i}^{*} + Δ θ - θ_{j}^{*}) - \cos (θ_{i}^{*} - θ_{j}^{*})] - P_{i} Δ θ

E^S is the lowest-energy unsecured outcome found and E^U the highest-energy secured outcome; the green and red curves are those two energy levels drawn back onto the plane. The hatched strip lies beyond the node's local unstable equilibrium and is left out of the energy analysis, as in the paper.

Uniform random sampling

Boundary-focused search

Relative error of each threshold estimate

Error against the best estimate found by either method (lower E^S, higher E^U), as in the paper's Fig. 7.

uniform, E^S boundary-focused, E^S uniform, E^U boundary-focused, E^U

Runs needed until both thresholds are within…

Method	5%	2%	1%

A dash means the level was not reached within the budget. One run is noisy; repeat it a few times to see the spread. The paper reports that the advantage is clearest down to the 2% level and can shrink or reverse at 1% in some grids.

E = \frac{1}{2} M {[\frac{Δ P}{D} (1 - e^{- D Δ t / M})]}^{2}

Do state-space thresholds carry over?

The violation rate

ε = \frac{N^{miss}}{N^{match} + N^{miss}}

counts pulses outside the mixed band that land on the wrong side: unsecured below the green curve, or secured above the red one. Only pulses shorter than a quarter of the local oscillation period are scored, because the mapping ignores the restoring force; longer ones are drawn faded.

Single pulse

Click on the plane to apply one pulse and trace it.