Propagation of Disturbances in AC Electricity Grids

Abstract

The energy transition towards high shares of renewable energy will affect the stability of electricity grids in many ways. Here, we aim to study its impact on propagation of disturbances by solving nonlinear swing equations describing coupled rotating masses of synchronous generators and motors on different grid topologies. We consider a tree, a square grid and as a real grid topology, the german transmission grid. We identify ranges of parameters with different transient dynamics: the disturbance decays exponentially in time, superimposed by oscillations with the fast decay rate of a single node, or with a smaller decay rate without oscillations. Most remarkably, as the grid inertia is lowered, nodes may become correlated, slowing down the propagation from ballistic to diffusive motion, decaying with a power law in time. Applying linear response theory we show that tree grids have a spectral gap leading to exponential relaxation as protected by topology and independent on grid size. Meshed grids are found to have a spectral gap which decreases with increasing grid size, leading to slow power law relaxation and collective diffusive propagation of disturbances. We conclude by discussing consequences if no measures are undertaken to preserve the grid inertia in the energy transition.

Power grid stability under perturbation of single nodes: Effects of heterogeneity and internal nodes

“…Two further variants "Hom±IN" (with and without internal nodes) are obtained by homogenising the transmission line properties as well as the generated and consumed powers. Corresponding homogeneous power grid structures have been investigated in various previous studies [7,12,13,14,30,31], typically for the case of loss-free transmission lines. For the homogenisation, we here also consider a loss-free situation, i.e.…”

“…All four models, Het±IN and Hom±IN are illustrated in figure 2(c) and the relevant parameters entering equation (7) are plotted in figure 3. For the heterogeneous models, the P (l) j = 0 and therefore act like passive nodes in the synchronous state of operation, but not in the dynamical case with an overall imbalance of mechanical and electrical power.…”

“…2 of coupled harmonic oscillators, which characterises the system in the neighbourhood of the fixed point, see equation (7). If the phase space trajectory {(θ j (t), ω j (t))} enters the nearest neighbourhood, this energy decreases monotonically with time t.…”

“…With respect to the grid representation, often simplifications are made for specific purposes, e.g. by using artificial grid structures, and/or by taking real network graph structures, but neglecting heterogeneities in properties of generators, loads and transmission lines [7,12,13,14]. A further simplification frequently made is the neglect of reactances between points of power production/consumption (internal nodes) and power injection/ejection (terminal nodes).…”

The key player problem in complex oscillator networks and electric power grids: Resistance centralities identify local vulnerabilities

“…change of the network-averaged frequency [27,28] or other dynamical quantities such as network susceptibilities [29] and the wave dynamics following disturbances [30]. Here, we quantify the total transient excursion through performance measures that are time-integrated quadratic forms in the system's degrees of freedom (supplementary materials, materials and methods).…”