Dual Theory of Transmission Line Outages

Ronellenfitsch, Henrik; Manik, Debsankha; Hörsch, Jonas; Brown, Tom; Witthaut, Dirk

doi:10.1109/tpwrs.2017.2658022

Cited by 35 publications

(35 citation statements)

References 27 publications

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“…The power flow configuration f ( ) can be efficiently used to determine which lines subsequently fail, by checking for which k we have | f ( ) k | ≥ 1, see [33]. There is much evidence that failures propagate nonlocally in power grids [48][49][50][51][52]. To analyze this in our framework we first consider a ring network with µ = 0 and Σ p = I.…”

Section: Fig S2mentioning

confidence: 99%

Emergent Failures and Cascades in Power Grids: A Statistical Physics Perspective

2018

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We model power grids transporting electricity generated by intermittent renewable sources as complex networks, where line failures can emerge indirectly by noisy power input at the nodes. By combining concepts from statistical physics and the physics of power flows and taking weather correlations into account, we rank line failures according to their likelihood and establish the most likely way such failures occur and propagate. Our insights are mathematically rigorous in a small-noise limit and are validated with data from the German transmission grid.

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Section: Fig S2mentioning

confidence: 99%

Emergent Failures and Cascades in Power Grids: A Statistical Physics Perspective

2018

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“…Here, [ ] B k is the Laplacian of the network after removal of link k and ( ) F k 0 is the pre-outage flow on link k. This equation was studied in the past in different settings [35,22,36,33]. The failure of single links is thus comparably well understood [22,37,38], whereas the simultaneous failure of multiple links was not yet studied to the same extend on a theoretical level.…”

Section: Single and Double Link Failuresmentioning

confidence: 99%

Collective effects of link failures in linear flow networks

2020

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The reliable operation of supply networks is crucial for the proper functioning of many systems, ranging from biological organisms such as the human blood transport system or plant leaves to manmade systems such as power grids or gas pipelines. Whereas the failure of single transportation links has been analysed thoroughly, the understanding of multiple failures is becoming increasingly important to prevent large scale damages. In this publication, we examine the collective nature of the simultaneous failure of several transportation links. In particular, we focus on the difference between single link failures and the collective failure of several links. We demonstrate that collective effects can amplify or attenuate the impacts of multiple link failures-and even lead to a reversal of flows on certain links. A simple classifier is introduced to predict the overall strength of collective effects that we demonstrate to be generally stronger if the failing links are close to each other. Finally, we establish an analogy between link failures in supply networks and dipole fields in discrete electrostatics by showing that multiple failures may be treated as superpositions of multiple electrical dipoles for lattice-like networks. Our results show that the simultaneous failure of multiple links may lead to unexpected effects that cannot be easily described using the theoretical framework for single link failures.

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“…(17). The kernel of the matrix I corresponds exactly to cycle flows: A cycle flow being a constant flow along a cycle; with no in-or out-flow [45][46][47] . The kernel has dimension M − N + 1, which reflects the fact that the cycles in a graph forms a vector space of dimension M − N + 1 48 , a basis set of this space is called a fundamental cycle basis.…”

Section: A Constructing Solutionsmentioning

confidence: 99%

Multistability in lossy power grids and oscillator networks

Balestra

Kaiser

Manik

et al. 2019

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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.

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Dual Theory of Transmission Line Outages

Cited by 35 publications

References 27 publications

Emergent Failures and Cascades in Power Grids: A Statistical Physics Perspective

Emergent Failures and Cascades in Power Grids: A Statistical Physics Perspective

Collective effects of link failures in linear flow networks

Multistability in lossy power grids and oscillator networks

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