Power-Law Distributions of Dynamic Cascade Failures in Power-Grid Models

Abstract

Power-law distributed cascade failures are well known in power-grid systems. Understanding this phenomena has been done by various DC threshold models, self-tuned at their critical point. Here, we attempt to describe it using an AC threshold model, with a second-order Kuramoto type equation of motion of the power-flow. We have focused on the exploration of network heterogeneity effects, starting from homogeneous two-dimensional (2D) square lattices to the US power-grid, possessing identical nodes and links, to a realistic electric power-grid obtained from the Hungarian electrical database. The last one exhibits node dependent parameters, topologically marginally on the verge of robust networks. We show that too weak quenched heterogeneity, coming solely from the probabilistic self-frequencies of nodes (2D square lattice), is not sufficient for finding power-law distributed cascades. On the other hand, too strong heterogeneity destroys the synchronization of the system. We found agreement with the empirically observed power-law failure size distributions on the US grid, as well as on the Hungarian networks near the synchronization transition point. We have also investigated the consequence of replacing the usual Gaussian self-frequencies to exponential distributed ones, describing renewable energy sources. We found a drop in the steady state synchronization averages, but the cascade size distribution, both for the US and Hungarian systems, remained insensitive and have kept the universal tails, being characterized by the exponent τ ≃ 1.8 . We have also investigated the effect of an instantaneous feedback mechanism in case of the Hungarian power-grid.

Synchronization dynamics on the EU and US power grids

“…where α is the damping parameter, describing the power dissipation, or an instantaneous feedback [36], K is a global coupling, related to the maximum transmitted power between nodes and A ij , which is the adjacency matrix of the network containing admittance elements.…”

“…In our simulations the following parameter settings were used: the α dissipation factor, which is chosen to be equal to 0.4 to meet expectations for power-grids, with the [1/s] inverse time physical dimension assumption. For modeling instantaneous feedback, or increased damping parameter we applied: α = 3.0[1/s] similarly as in [36].…”

“…These rare regions can also lead to frustrated synchronization and chimera states [30][31][32], which result in non-universal PL distributions of the desynchronization events below the transition point [3,[33][34][35]. The authors have previously provided evidence for this using the second order Kuramoto model on 2d lattices and large synthetic power system topologies [3] and on the realistic representation of the Hungarian power system while also allowing line outages (line-cuts) [36].…”

“…The second order Kuramoto model with power transmission thresholds (line capacities) has been also introduced in [36,37] for the dynamical modelling of cascade events. Identification of critical lines of transmission in different national power grids has been determined.…”

Sandpile cascades on oscillator networks: the BTW model meets Kuramoto

“…Avalanches of desynchronization on networks of oscillators have attracted increasing interest in recent years. There are two general classes of oscillator cascades: phase dynamics of oscillators with complex patterns (mainly desynchronization bursts and spontaneous synchrony) [38][39][40][41][42][43][44][45][46][47][48] and threshold oscillators with more-or-less uniform driving [49][50][51][52][53][54][55][56][57][58] . The cascades are in terms of either synchronization/desynchronization patterns, or simultaneous firing in the case of threshold dynamics, so, none of these models account for separate degrees of freedom for load and for phase.…”