arXiv subjects

Dirk Witthaut

Publications and source records attributed to Dirk Witthaut.

90 records · Page 5Linked to original sources

arXiv2017

A universal order parameter for synchrony in networks of limit cycle oscillators

We analyze the properties of order parameters measuring synchronization and phase locking in complex oscillator networks. First, we review network order parameters previously introduced and reveal several shortcomings: none of the introduced order parameters capture all transitions from incoherence over phase locking to full synchrony for arbitrary, finite networks. We then introduce an alternative, universal order parameter that accurately tracks the degree of partial phase locking and synchronization, adapting the traditional definition to account for the network topology and its influence on the phase coherence of the oscillators. We rigorously proof that this order parameter is strictly monotonously increasing with the coupling strength in the phase locked state, directly reflecting the dynamic stability of the network. Furthermore, it indicates the onset of full phase locking by a diverging slope at the critical coupling strength. The order parameter may find applications across systems where different types of synchrony are possible, including biological networks and power grids.

nlin.AO↗

arXiv2017

Escape Routes, Weak Links, and Desynchronization in Fluctuation-driven Networks

Shifting our electricity generation from fossil fuel to renewable energy sources introduces large fluctuations to the power system. Here, we demonstrate how increased fluctuations, reduced damping and reduced intertia may undermine the dynamical robustness of power grid networks. Focusing on fundamental noise models, we derive analytic insights into which factors limit the dynamic robustness and how fluctuations may induce a system escape from an operating state. Moreover, we identify weak links in the grid that make it particularly vulnerable to fluctuations. These results thereby not only contribute to a theoretical understanding of how fluctuations act on distributed network dynamics, they may also help designing future renewable energy systems to be more robust.

nlin.AO↗

arXiv2017

From State Estimation to Network Reconstruction

We develop methods to efficiently reconstruct the topology and line parameters of a power grid from the measurement of nodal variables. We propose two compressed sensing algorithms that minimize the amount of necessary measurement resources by exploiting network sparsity, symmetry of connections and potential prior knowledge about the connectivity. The algorithms are reciprocal to established state estimation methods, where nodal variables are estimated from few measurements given the network structure. Hence, they enable an advanced grid monitoring where both state and structure of a grid are subject to uncertainties or missing information.

eess.SY↗

arXiv2017

Dual theory of transmission line outages

A new graph dual formalism is presented for the analysis of line outages in electricity networks. The dual formalism is based on a consideration of the flows around closed cycles in the network. After some exposition of the theory is presented, a new formula for the computation of Line Outage Distribution Factors (LODFs) is derived, which is not only computationally faster than existing methods, but also generalizes easily for multiple line outages and arbitrary changes to line series reactance. In addition, the dual formalism provides new physical insight for how the effects of line outages propagate through the network. For example, in a planar network a single line outage can be shown to induce monotonically decreasing flow changes, which are mathematically equivalent to an electrostatic dipole field.

nlin.AO↗

arXiv2016

Cycle flows and multistabilty in oscillatory networks: an overview

The functions of many networked systems in physics, biology or engineering rely on a coordinated or synchronized dynamics of its constituents. In power grids for example, all generators must synchronize and run at the same frequency and their phases need to appoximately lock to guarantee a steady power flow. Here, we analyze the existence and multitude of such phase-locked states. Focusing on edge and cycle flows instead of the nodal phases we derive rigorous results on the existence and number of such states. Generally, multiple phase-locked states coexist in networks with strong edges, long elementary cycles and a homogeneous distribution of natural frequencies or power injections, respectively. We offer an algorithm to systematically compute multiple phase- locked states and demonstrate some surprising dynamical consequences of multistability.

nlin.AO↗

arXiv2016

Network susceptibilities: theory and applications

We introduce the concept of network susceptibilities quantifying the response of the collective dy- namics of a network to small parameter changes. We distinguish two types of susceptibilities: vertex susceptibilities and edge susceptibilities, measuring the responses due to changes in the properties of units and their interactions, respectively. We derive explicit forms of network susceptibilities for oscillator networks close to steady states and offer example applications for Kuramoto-type phase- oscillator models, power grid models and generic flow models. Focusing on the role of the network topology implies that these ideas can be easily generalized to other types of networks, in particular those characterizing flow, transport, or spreading phenomena. The concept of network susceptibil- ities is broadly applicable and may straightforwardly be transferred to all settings where networks responses of the collective dynamics to topological changes are essential.

nlin.AO↗

arXiv2016

A Dual Method for Computing Power Transfer Distribution Factors

Power Transfer Distribution Factors (PTDFs) play a crucial role in power grid security analysis, planning, and redispatch. Fast calculation of the PTDFs is therefore of great importance. In this paper, we present a non-approximative dual method of computing PTDFs. It uses power flows along topological cycles of the network but still relies on simple matrix algebra. At the core, our method changes the size of the matrix that needs to be inverted to calculate the PTDFs from $N\times N$, where $N$ is the number of buses, to $(L-N+1)\times (L-N+1)$, where $L$ is the number of lines and $L-N+1$ is the number of independent cycles (closed loops) in the network while remaining mathematically fully equivalent. For power grids containing a relatively small number of cycles, the method can offer a speedup of numerical calculations.

eess.SY↗

arXiv2016

Taming Instabilities in Power Grid Networks by Decentralized Control

Renewables will soon dominate energy production in our electric power system. And yet, how to integrate renewable energy into the grid and the market is still a subject of major debate. Decentral Smart Grid Control (DSGC) was recently proposed as a robust and decentralized approach to balance supply and demand and to guarantee a grid operation that is both economically and dynamically feasible. Here, we analyze the impact of network topology by assessing the stability of essential network motifs using both linear stability analysis and basin volume for delay systems. Our results indicate that if frequency measurements are averaged over sufficiently large time intervals, DSGC enhances the stability of extended power grid systems. We further investigate whether DSGC supports centralized and/or decentralized power production and find it to be applicable to both. However, our results on cycle-like systems suggest that DSGC favors systems with decentralized production. Here, lower line capacities and lower averaging times are required compared to those with centralized production.

nlin.AO↗

arXiv2015

Curing critical links in oscillator networks as power grid models

Modern societies crucially depend on the robust supply with electric energy. Blackouts of power grids can thus have far reaching consequences. During a blackout, often the failure of a single infrastructure, such as a critical transmission line, results in several subsequent failures that spread across large parts of the network. Preventing such large-scale outages is thus key for assuring a reliable power supply. Here we present a non-local curing strategy for oscillatory power grid networks based on the global collective redistribution of loads. We first identify critical links and compute residual capacities on alternative paths on the remaining network from the original flows. For each critical link, we upgrade lines that constitute bottlenecks on such paths. We demonstrate the viability of this strategy for random ensembles of network topologies as well as topologies derived from real transmission grids and compare the nonlocal strategy against local back-ups of critical links. These strategies are independent of the detailed grid dynamics and combined may serve as an effective guideline to reduce outages in power grid networks by intentionally strengthen optimally selected links.

nlin.AO↗

arXiv2015

Critical links and nonlocal rerouting in complex supply networks

Link failures repeatedly induce large-scale outages in power grids and other supply networks. Yet, it is still not well understood, which links are particularly prone to inducing such outages. Here we analyze how the nature and location of each link impact the network's capability to maintain stable supply. We propose two criteria to identify critical links on the basis of the topology and the load distribution of the network prior to link failure. They are determined via a link's redundant capacity and a renormalized linear response theory we derive. These criteria outperform critical link prediction based on local measures such as loads. The results not only further our understanding of the physics of supply networks in general. As both criteria are available before any outage from the state of normal operation, they may also help real-time monitoring of grid operation, employing counter-measures and support network planning and design.

physics.soc-ph↗

arXiv2015

Nonlocal effects and counter measures in cascading failures

We study the propagation of cascading failures in complex supply networks with a focus on nonlocal effects occurring far away from the initial failure. It is shown that a high clustering and a small average path length of a network generally suppress nonlocal overloads. These properties are typical for many real-world networks, often called small-world networks, such that cascades propagate mostly locally in these networks. Furthermore, we analyze the spatial aspects of countermeasures based on the intentional removal of additional edges. Nonlocal actions are generally required in networks which have a low redundancy and are thus especially vulnerable to cascades.

nlin.AO↗

arXiv2014

Tunneling in the self-trapped regime of a two-well BEC

Starting from a mean-field model of the Bose-Einstein condensate dimer, we reintroduce classically forbidden tunneling through a Bohr-Sommerfeld quantization approach. We find closed-form approximations to the tunneling frequency more accurate than those previously obtained using different techniques. We discuss the central role that tunneling in the self-trapped regime plays in a quantitatively accurate model of a dissipative dimer leaking atoms to the environment. Finally, we describe the prospects of experimental observation of tunneling in the self-trapped regime, both with and without dissipation.

quant-ph↗

arXiv2013

Decay and fragmentation in an open Bose-Hubbard chain

We analyze the decay of ultracold atoms from an optical lattice with loss form a single lattice site. If the initial state is dynamically stable a suitable amount of dissipation can stabilize a Bose-Einstein condensate, such that it remains coherent even in the presence of strong interactions. A transition between two different dynamical phases is observed if the initial state is dynamically unstable. This transition is analyzed here in detail. For strong interactions, the system relaxes to an entangled quantum state with remarkable statistical properties: The atoms bunch in a few "breathers" forming at random positions. Breathers at different positions are coherent, such that they can be used in precision quantum interferometry and other applications.

cond-mat.quant-gas↗

arXiv2013

Nonlocal failures in complex supply networks by single link additions

How do local topological changes affect the global operation and stability of complex supply networks? Studying supply networks on various levels of abstraction, we demonstrate that and how adding new links may not only promote but also degrade stable operation of a network. Intriguingly, the resulting overloads may emerge remotely from where such a link is added, thus resulting in nonlocal failure. We link this counter-intuitive phenomenon to Braess' paradox originally discovered in traffic networks. We use elementary network topologies to explain its underlying mechanism for different types of supply networks and find that it generically occurs across these systems. As an important consequence, upgrading supply networks such as communication networks, biological supply networks or power grids requires particular care because even adding only single connections may destabilize normal network operation and induce disturbances remotely from the location of structural change and even global cascades of failures.

nlin.AO↗

arXiv2013

Kuramoto dynamics in Hamiltonian systems

The Kuramoto model constitutes a paradigmatic model for the dissipative collective dynamics of coupled oscillators, characterizing in particular the emergence of synchrony. Here we present a classical Hamiltonian (and thus conservative) system with 2N state variables that in its action-angle representation exactly yields Kuramoto dynamics on N-dimensional invariant manifolds. We show that the synchronization transition on a Kuramoto manifold emerges where the transverse Hamiltonian action dynamics becomes unstable. The uncovered Kuramoto dynamics in Hamiltonian systems thus distinctly links dissipative to conservative dynamics.

nlin.CD↗

arXiv2013

Impact of network topology on synchrony of oscillatory power grids

Replacing conventional power sources by renewable sources in current power grids drastically alters their structure and functionality. The resulting grid will be far more decentralized with an distinctly different topology. Here we analyze the impact of grid topologies on spontaneous synchronization, considering regular, random and small-world topologies and focusing on the influence of decentralization. We employ the classic model of power grids modeling consumers and sources as second order oscillators. We first analyze the global dynamics of the simplest non-trivial (two-node) network that exhibit a synchronous (normal operation) state, a limit cycle (power outage) and coexistence of both. Second, we estimate stability thresholds for the collective dynamics of small network motifs, in particular star-like networks and regular grid motifs. For larger networks we numerically investigate decentralization scenarios finding that decentralization itself may support power grids in reaching stable synchrony for lower lines transmission capacities. Decentralization may thus be beneficial for power grids, regardless of their special resulting topology. Regular grids exhibit a specific transition behavior not found for random or small-world grids.

nlin.AO↗

arXiv2012

Global Phase Space of Coherence and Entanglement in a double-well BEC

Ultracold atoms provide an ideal system for the realization of quantum technologies, but also for the study of fundamental physical questions such as the emergence of decoherence and classicality in quantum many-body systems. Here, we study the global structure of the quantum dynamics of bosonic atoms in a double-well trap and analyze the conditions for the generation of many-particle entanglement and spin squeezing which have important applications in quantum metrology. We show how the quantum dynamics is determined by the phase space structure of the associated mean-field system and where true quantum features arise beyond this `classical' approximation.

quant-ph↗

arXiv2003

Relation between discrete and continuous teleportation using linear elements

We discuss the relation between discrete and continuous linear teleportation, i.e. teleportation schemes that use only linear optical elements and photodetectors. For this the existing qubit protocols are generalized to qudits with a discrete and finite spectrum but with an arbitrary number of states or alternatively to continuous variables. Correspondingly a generalization of linear optical operations and detection is made. It is shown that linear teleportation is only possible in a probabilistic sense. A general expression for the success probability is derived which is shown to depend only on the dimensions of the input and ancilla Hilbert spaces. From this the known results $p=1/2$ and $p=1$ for the discrete and continuous cases can be recovered. We also discuss the probabilistic teleportation scheme of Knill, Laflamme and Milburn and argue that it does not make optimum use of resources.

quant-ph↗