Fluctuation-induced distributed resonances in oscillatory networks

A framework for synthetic power system dynamics

We present a modular framework for generating synthetic power grids that consider the heterogeneity of real power grid dynamics but remain simple and tractable. This enables the generation of large sets of synthetic grids for a wide range of applications. For the first time, our synthetic model also includes the major drivers of fluctuations on short-time scales and a set of validators that ensure the resulting system dynamics are plausible. The synthetic grids generated are robust and show good synchronization under all evaluated scenarios, as should be expected for realistic power grids. A software package that includes an efficient Julia implementation of the framework is released as a companion to the paper.

Double resonance induced by group coupling with quenched disorder

Results show that the astrocytes can not only listen to the talk of large assemble of neurons but also give advice to the conversations and are significant sources of heterogeneous couplings as well. In the present work, we focus on such regulation character of astrocytes and explore the role of heterogeneous couplings among interacted neuron-astrocyte components in a signal response. We consider reduced dynamics in which the listening and advising processes of astrocytes are mapped into the form of group coupling, where the couplings are normally distributed. In both globally coupled overdamped bistable oscillators and an excitable FitzHugh-Nagumo (FHN) neuron model, we numerically and analytically demonstrate that two types of bell-shaped collective response curves can be obtained as the ensemble coupling strength or the heterogeneity of group coupling rise, respectively, which can be seen as a new type of double resonance. Furthermore, through the bifurcation analysis, we verify that these resonant signal responses stem from the competition between dispersion and aggregation induced by heterogeneous group and positive pairwise couplings, respectively. Our results contribute to a better understanding of the signal propagation in coupled systems with quenched disorder.

Linear response theory for renewable fluctuations in power grids with transmission losses

We study the spreading of renewable power fluctuations through grids with Ohmic losses on the lines. By formulating a network-adapted linear response theory, we find that vulnerability patterns are linked to the left Laplacian eigenvectors of the overdamped eigenmodes. We show that for tree-like networks, fluctuations are amplified in the opposite direction of the power flow. This novel mechanism explains vulnerability patterns that were observed in previous numerical simulations of renewable microgrids. While exact mathematical derivations are only possible for tree-like networks with a homogeneous response, we show that the mechanisms discovered also explain vulnerability patterns in realistic heterogeneous meshed grids by studying the IEEE RTS-1996 test system.

Diversity-induced resonance in a globally coupled bistable system with diversely distributed heterogeneity

A moderate degree of diversity, in form of quenched noise or intrinsic heterogeneity, can significantly strengthen the collective response of coupled extended systems. As yet, related discoveries on diversity-induced resonance are mainly concentrated on symmetrically distributed heterogeneity, e.g., the Gaussian or uniform distributions with zero-mean. The necessary conditions that guarantee the arise of resonance phenomenon in heterogeneous oscillators remain largely unknown. In this work, we show that the standard deviation and the ratio of negative entities of a given distribution jointly modulate diversity-induced resonance and the concomitance of negative and positive entities is the prerequisite for this resonant behavior emerging in diverse symmetrical and asymmetrical distributions. Particularly, for a proper degree of diversity of a given distribution, the collective signal response behaves like a bell-shaped curve as the ratio of negative oscillator increases, which can be termed negative-oscillator-ratio induced resonance. Furthermore, we analytically reveal that the ratio of negative oscillators plays a gating role in the resonance phenomenon on the basis of a reduced equation. Finally, we examine the robustness of these results in globally coupled bistable elements with asymmetrical potential functions. Our results suggest that the phenomenon of diversity-induced resonance can arise in arbitrarily distributed heterogeneous bistable oscillators by regulating the ratio of negative entities appropriately.

Absence of pure voltage instabilities in the third-order model of power grid dynamics

Secure operation of electric power grids fundamentally relies on their dynamical stability properties. For the third-order model, a paradigmatic model that captures voltage dynamics, three routes to instability are established in the literature: a pure rotor angle instability, a pure voltage instability, and one instability induced by the interplay of both. Here, we demonstrate that one of these routes, the pure voltage instability, requires infinite voltage amplitudes and is, thus, nonphysical. We show that voltage collapse dynamics nevertheless exist in the absence of any voltage instabilities.

Predictable topological sensitivity of Turing patterns on graphs

Reaction-diffusion systems implemented as dynamical processes on networks have recently renewed the interest in their self-organized collective patterns known as Turing patterns. We investigate the influence of network topology on the emerging patterns and their diversity, defined as the variety of stationary states observed with random initial conditions and the same dynamics. We show that a seemingly minor change, the removal or rewiring of a single link, can prompt dramatic changes in pattern diversity. The determinants of such critical occurrences are explored through an extensive and systematic set of numerical experiments. We identify situations where the topological sensitivity of the attractor landscape can be predicted without a full simulation of the dynamical equations, from the spectrum of the graph Laplacian and the linearized dynamics. Unexpectedly, the main determinant appears to be the degeneracy of the eigenvalues or the growth rate and not the number of unstable modes.

Oscillatory spreading and inertia in power grids

The increase in variable renewable generators (VRGs) in power systems has altered the dynamics from a historical experience. VRGs introduce new sources of power oscillations, and the stabilizing response provided by synchronous generators (SGs, e.g., natural gas, coal, etc.), which help avoid some power fluctuations, will lessen as VRGs replace SGs. These changes have led to the need for new methods and metrics to quickly assess the likely oscillatory behavior for a particular network without performing computationally expensive simulations. This work studies the impact of a critical dynamical parameter-the inertia value-on the rest of a power system's oscillatory response to representative VRG perturbations. We use a known localization metric in a novel way to quantify the number of nodes responding to a perturbation and the magnitude of those responses. This metric allows us to relate the spread and severity of a system's power oscillations with inertia. We find that as inertia increases, the system response to node perturbations transitions from localized (only a few close nodes respond) to delocalized (many nodes across the network respond). We introduce a heuristic computed from the network Laplacian to relate this oscillatory transition to the network structure. We show that our heuristic accurately describes the spread of oscillations for a realistic power-system test case. Using a heuristic to determine the likely oscillatory behavior of a system given a set of parameters has wide applicability in power systems, and it could decrease the computational workload of planning and operation.

Recurrent dynamics in the cerebral cortex: Integration of sensory evidence with stored knowledge

Current concepts of sensory processing in the cerebral cortex emphasize serial extraction and recombination of features in hierarchically structured feed-forward networks in order to capture the relations among the components of perceptual objects. These concepts are implemented in convolutional deep learning networks and have been validated by the astounding similarities between the functional properties of artificial systems and their natural counterparts. However, cortical architectures also display an abundance of recurrent coupling within and between the layers of the processing hierarchy. This massive recurrence gives rise to highly complex dynamics whose putative function is poorly understood. Here a concept is proposed that assigns specific functions to the dynamics of cortical networks and combines, in a unifying approach, the respective advantages of recurrent and feed-forward processing. It is proposed that the priors about regularities of the world are stored in the weight distributions of feed-forward and recurrent connections and that the high-dimensional, dynamic space provided by recurrent interactions is exploited for computations. These comprise the ultrafast matching of sensory evidence with the priors covertly represented in the correlation structure of spontaneous activity and the context-dependent grouping of feature constellations characterizing natural objects. The concept posits that information is encoded not only in the discharge frequency of neurons but also in the precise timing relations among the discharges. Results of experiments designed to test the predictions derived from this concept support the hypothesis that cerebral cortex exploits the high-dimensional recurrent dynamics for computations serving predictive coding.

NetworkDynamics.jl-Composing and simulating complex networks in Julia

NetworkDynamics.jl is an easy-to-use and computationally efficient package for simulating heterogeneous dynamical systems on complex networks, written in Julia, a high-level, high-performance, dynamic programming language. By combining state-of-the-art solver algorithms from DifferentialEquations.jl with efficient data structures, NetworkDynamics.jl achieves top performance while supporting advanced features such as events, algebraic constraints, time delays, noise terms, and automatic differentiation.

Open database analysis of scaling and spatio-temporal properties of power grid frequencies

The electrical energy system has attracted much attention from an increasingly diverse research community. Many theoretical predictions have been made, from scaling laws of fluctuations to propagation velocities of disturbances. However, to validate any theory, empirical data from large-scale power systems are necessary but are rarely shared openly. Here, we analyse an open database of measurements of electric power grid frequencies across 17 locations in 12 synchronous areas on three continents. The power grid frequency is of particular interest, as it indicates the balance of supply and demand and carries information on deterministic, stochastic, and control influences. We perform a broad analysis of the recorded data, compare different synchronous areas and validate a previously conjectured scaling law. Furthermore, we show how fluctuations change from local independent oscillations to a homogeneous bulk behaviour. Overall, the presented open database and analyses constitute a step towards more shared, collaborative energy research.

Power grid frequencies mirror the state of the grid. Here, Rydin Gorjão et al. analyse measurements of power grid frequencies across areas and continents and uncover scaling laws of their fluctuations and spatio-temporal dynamics, which could aid the design, operation and control of power systems.

Topological Determinants of Perturbation Spreading in Networks

Spreading phenomena essentially underlie the dynamics of various natural and technological networked systems, yet how spatiotemporal propagation patterns emerge from such networks remains largely unknown. Here we propose a novel approach that reveals universal features determining the spreading dynamics in diffusively coupled networks and disentangles them from factors that are system specific. In particular, we first analytically identify a purely topological factor encoding the interaction structure and strength, and second, numerically estimate a master function characterizing the universal scaling of the perturbation arrival times across topologically different networks. The proposed approach thereby provides intuitive insights into complex propagation patterns as well as accurate predictions for the perturbation arrival times. The approach readily generalizes to a wide range of networked systems with diffusive couplings and may contribute to assess the risks of transient influences of ubiquitous perturbations in real-world systems.

Vulnerability in dynamically driven oscillatory networks and power grids

Vulnerability of networks has so far been quantified mainly for structural properties. In driven systems, however, vulnerability intrinsically relies on the collective response dynamics. As shown recently, dynamic response patterns emerging in driven oscillator networks and AC power grid models are highly heterogeneous and nontrivial, depending jointly on the driving frequency, the interaction topology of the network, and the node or nodes driven. Identifying which nodes are most susceptible to dynamic driving and may thus make the system as a whole vulnerable to external input signals, however, remains a challenge. Here, we propose an easy-to-compute Dynamic Vulnerability Index (DVI) for identifying those nodes that exhibit largest amplitude responses to dynamic driving signals with given power spectra and thus are most vulnerable. The DVI is based on linear response theory, as such generic, and enables robust predictions. It thus shows potential for a wide range of applications across dynamically driven networks, for instance, for identifying the vulnerable nodes in power grids driven by fluctuating inputs from renewable energy sources and fluctuating power output to consumers.

Time delay effects in the control of synchronous electricity grids

The expansion of inverter-connected generation facilities (i.e., wind and photovoltaics) and the removal of conventional power plants is necessary to mitigate the impacts of climate change, whereas conventional generation with large rotating generator masses provides stabilizing inertia, inverter-connected generation does not. Since the underlying power system and the control mechanisms that keep it close to a desired reference state were not designed for such a low inertia system, this might make the system vulnerable to disturbances. In this paper, we will investigate whether the currently used control mechanisms are able to keep a low inertia system stable and how this is affected by the time delay between a frequency deviation and the onset of the control action. We integrate the control mechanisms used in Continental Europe into a model of coupled oscillators which resembles the second order Kuramoto model. This model is then used to investigate how the interplay of changing inertia, network topology, and delayed control affects the stability of the interconnected power system. To identify regions in the parameter space that make stable grid operation possible, the linearized system is analyzed to create the system's stability chart. We show that lower and distributed inertia could have a beneficial effect on the stability of the desired synchronous state.

Network desynchronization by non-Gaussian fluctuations

Many networks must maintain synchrony despite the fact that they operate in noisy environments. Important examples are stochastic inertial oscillators, which are known to exhibit fluctuations with broad tails in many applications, including electric power networks with renewable energy sources. Such non-Gaussian fluctuations can result in rare network desynchronization. Here we build a general theory for inertial oscillator network desynchronization by non-Gaussian noise. We compute the rate of desynchronization and show that higher moments of noise enter at specific powers of coupling: either speeding up or slowing down the rate exponentially depending on how noise statistics match the statistics of a network's slowest mode. Finally, we use our theory to introduce a technique that drastically reduces the effective description of network desynchronization. Most interestingly, when instability is associated with a single edge, the reduction is to one stochastic oscillator.

Low-frequency oscillations in coupled phase oscillators with inertia

This work considers a second-order Kuramoto oscillator network periodically driven at one node to model low-frequency forced oscillations in power grids. The phase fluctuation magnitude at each node and the disturbance propagation in the network are numerically analyzed. The coupling strengths in this work are sufficiently large to ensure the stability of equilibria in the unforced system. It is found that the phase fluctuation is primarily determined by the network structural properties and forcing parameters, not the parameters specific to individual nodes such as power and damping. A new "resonance" phenomenon is observed in which the phase fluctuation magnitudes peak at certain critical coupling strength in the forced system. In the cases of long chain and ring-shaped networks, the Kuramoto model yields an important but somehow counter-intuitive result that the fluctuation magnitude distribution does not necessarily follow a simple attenuating trend along the propagation path and the fluctuation at nodes far from the disturbance source could be stronger than that at the source. These findings are relevant to low-frequency forced oscillations in power grids and will help advance the understanding of their dynamics and mechanisms and improve the detection and mitigation techniques.

Impact of network topology on the stability of DC microgrids

We probe the stability of Watts-Strogatz DC microgrids, in which droop-controlled producers and constant power load consumers are homogeneously distributed and obey Kirchhoff's circuit laws. The concept of survivability is employed to evaluate the system's response to Dirac-delta voltage perturbations at single nodes. A fixed point analysis of the power grid model yields that there is only one relevant attractor. Using a set of simulations with random networks, we investigate correlations between survivability and three topological network measures: the share of producers in the network and the degree and the average neighbor degree of the perturbed node. Depending on the imposed voltage and current limits, the stability is optimized for low node degrees or a specific share of producers. Based on our findings, we provide an insight into the local dynamics of the perturbed system and derive explicit guidelines for the design of resilient DC power grids.