Multistability in lossy power grids and oscillator networks

Abstract

Networks of phase oscillators are studied in various contexts, in particular in the modeling of the electric power grid. A functional grid corresponds to a stable steady state, such that any bifurcation can have catastrophic consequences up to a blackout. But also the existence of multiple steady states is undesirable, as it can lead to sudden transitions or circulatory flows. Despite the enormous practical importance there is still no general theory of the existence and uniqueness of steady states in such systems. Analytic results are mostly limited to grids without Ohmic losses. In this article, we introduce a method to systematically construct the solutions of the real power load-flow equations in the presence of Ohmic losses and explicitly compute them for tree and ring networks. We investigate different mechanisms leading to multistability and discuss the impact of Ohmic losses on the existence of solutions.The stable operation of the electric power grid relies on a precisely synchronized state of all generators and machines. All machines rotate at exactly the same frequency with fixed phase differences, leading to steady power flows throughout the grid. Whether such a steady state exists for a given network is of eminent practical importance. The loss of a steady state typically leads to power outages up to a complete blackout. But also the existence of multiple steady states is undesirable, as it can lead to sudden transitions, circulating flows and eventually also to power outages. Steady states are typically calculated numerically, but this approach gives only limited insight into the existence and (non-)uniqueness of steady states. Analytic results are available only for special network configuration, in particular for grids with negligible Ohmic losses or radial networks without any loops. In this article, we introduce a method to systematically construct the solutions of the real power loadflow equations in the presence of Ohmic losses. We calculate the steady states explicitly for elementary networks demonstrating different mechanisms leading to multistability. Our results also apply to models of coupled oscillators which are widely used in theoretical physics and mathematical biology.

Multistability and anomalies in oscillator models of lossy power grids

“…However, a large proportion of the work mentioned above relies on the lossless line assumption, namely, neglecting dissipation, voltage amplitude dynamics, and reactive power flows. Recently, there has been a common effort in trying to pursue a more realistic mathematical analysis of power grids, by incorporating reactive power flows [16,17], voltage amplitude dynamics [54,55], and dissipation [25,26]. Despite all this work, there is still no clear extension of the winding partition to the full active-reactive power flows, even though there are some notable related preliminary works [56,57].…”

“…The presence of cycles in the network can induce the existence of multiple solutions to the Dissipative Flow Network Problem (see Fig. 3 or [25]). We rigorously show here that winding vectors characterize these solutions for sufficiently moderate dissipation.…”

“…The importance of understanding the more realistic case of dissipative couplings motivated the early work by Sakaguchi and Kuramoto [23,24] and is still an active field of research. Recent numerical investigations [25,26] as well as analytical studies in regular systems [27][28][29] are beginning to shed light on a more A winding number q ∈ Z is associated to each equilibrium, counting the number of times its angles wind counterclockwise around the origin. The three dots (left) are equilibria of the spring network, labeled with their winding numbers q.…”

Multistability and anomalies in oscillator models of lossy power grids

“…The importance of understanding the more realistic case of dissipative couplings motivated the early work by Sakaguchi and Kuramoto 25 , 26 and is still an active field of research. Recent numerical investigations 27 , 28 as well as analytical studies in regular systems 29 – 31 are beginning to shed light on a more in-depth understanding of dissipative networks. More generally, the extension of standard approaches to more realistic systems is gaining momentum in the fields of synchronization and complex networks 32 – 34 .…”

“…The mathematical challenges encountered when relaxing the lossless line assumption confined most of the literature to numerical investigations 27 , 28 .…”