Sandpile cascades on oscillator networks: The BTW model meets Kuramoto

Abstract

Cascading failures abound in complex systems and the Bak–Tang–Weisenfeld (BTW) sandpile model provides a theoretical underpinning for their analysis. Yet, it does not account for the possibility of nodes having oscillatory dynamics, such as in power grids and brain networks. Here, we consider a network of Kuramoto oscillators upon which the BTW model is unfolding, enabling us to study how the feedback between the oscillatory and cascading dynamics can lead to new emergent behaviors. We assume that the more out-of-sync a node is with its neighbors, the more vulnerable it is and lower its load-carrying capacity accordingly. Also, when a node topples and sheds load, its oscillatory phase is reset at random. This leads to novel cyclic behavior at an emergent, long timescale. The system spends the bulk of its time in a synchronized state where load builds up with minimal cascades. Yet, eventually, the system reaches a tipping point where a large cascade triggers a “cascade of larger cascades,” which can be classified as a dragon king event. The system then undergoes a short transient back to the synchronous, buildup phase. The coupling between capacity and synchronization gives rise to endogenous cascade seeds in addition to the standard exogenous ones, and we show their respective roles. We establish the phenomena from numerical studies and develop the accompanying mean-field theory to locate the tipping point, calculate the load in the system, determine the frequency of the long-time oscillations, and find the distribution of cascade sizes during the buildup phase.

Synchronization dynamics on the EU and US power grids

“…Recently these kings, also called "Dragon Kings" (DK), were shown to exist in sand-pile models, following self-organized bistability [43], which is expected to be a common phenomenon in nature. Very recently, in a sand-pile model, coupled to massless Kuramoto oscillators, designed to model power-grids the existence of DK-s has also been shown [44]. We also show the appearance of DK-s in our power-grid simulations, based on the swing equations of the AC circuits.…”

“…The small threshold distributions resemble to the so called "Dragon King" events described in Refs. [42,44], while in the 0.5 ≤ T ≤ 0.4 region, the continuously changing exponents 1.4 ≤ τ ≤ 2.4 suggest Griffihts effects as the consequence of the heterogeneity. Note, that Griffihts effects may also occur near mixed order transitions, where the steady state values jump, but dynamical scaling persists [59].…”