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72
Kron Reduction of Graphs with Applications to Electrical Networks
"... Consider a weighted undirected graph and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to selfloops. The Kron reduction of this graph is again a graph whose Laplacian matrix is obtained by the Schur complement of the original Laplacian mat ..."
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Cited by 39 (16 self)
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Consider a weighted undirected graph and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to selfloops. The Kron reduction of this graph is again a graph whose Laplacian matrix is obtained by the Schur complement of the original Laplacian matrix with respect to a specified subset of nodes. The Kron reduction process is ubiquitous in classic circuit theory and in related disciplines such as electrical impedance tomography, smart grid monitoring, transient stability assessment, and analysis of power electronics. Kron reduction is also relevant in other physical domains, in computational applications, and in the reduction of Markov chains. Related concepts have also been studied as purely theoretic problems in the literature on linear algebra. In this paper we analyze the Kron reduction process from the viewpoint of algebraic graph theory. Specifically, we provide a comprehensive and detailed graphtheoretic analysis of Kron reduction encompassing topological, algebraic, spectral, resistive, and sensitivity analyses. Throughout our theoretic elaborations we especially emphasize the practical applicability of our results to various problem setups arising in engineering, computation, and linear algebra. Our analysis of Kron reduction leads to novel insights both on the mathematical and the physical side.
On the Critical Coupling for Kuramoto Oscillators
 SIAM Journal on Applied Dynamical Systems
"... Abstract. The celebrated Kuramoto model captures various synchronization phenomena in biological and manmade dynamical systems of coupled oscillators. It is wellknown that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronizatio ..."
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Cited by 15 (9 self)
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Abstract. The celebrated Kuramoto model captures various synchronization phenomena in biological and manmade dynamical systems of coupled oscillators. It is wellknown that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronization occurs. This paper features four contributions. First, we characterize and distinguish the different notions of synchronization used throughout the literature and formally introduce the concept of phase cohesiveness as an analysis tool and performance index for synchronization. Second, we review the vast literature providing necessary, sufficient, implicit, and explicit estimates of the critical coupling strength in the finite and infinitedimensional case and for both firstorder and secondorder Kuramoto models. Third, we present the first explicit necessary and sufficient condition on the critical coupling strength to achieve synchronization in the finitedimensional Kuramoto model for an arbitrary distribution of the natural frequencies. The multiplicative gap in the synchronization condition yields a practical stability result determining the admissible initial and the guaranteed ultimate phase cohesiveness as well as the guaranteed asymptotic magnitude of the order parameter. As supplementary results, we provide a statistical comparison of our synchronization condition with other conditions proposed in the literature, and we show that our results also hold for switching and
Spontaneous synchrony in powergrid networks
 IEEE 57 〈z(0)z(t)〉, ε≈0.02 p.u. 〈xT(0)x(t)〉, ε≈0.02 p.u. 〈z(0)z(t)〉, ε≈0.17 p.u. 〈xT(0)x(t)〉, ε≈ 0.17
, 2013
"... An imperative condition for the functioning of a powergrid network is that its power generators remain synchronized. Disturbances can prompt desynchronization, which is a process that has been involved in large power outages. Here we derive a condition under which the desired synchronous state of ..."
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Cited by 15 (1 self)
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An imperative condition for the functioning of a powergrid network is that its power generators remain synchronized. Disturbances can prompt desynchronization, which is a process that has been involved in large power outages. Here we derive a condition under which the desired synchronous state of a power grid is stable, and use this condition to identify tunable parameters of the generators that are determinants of spontaneous synchronization. Our analysis gives rise to an approach to specify parameter assignments that can enhance synchronization of any given network, which we demonstrate for a selection of both test systems and real power grids. Because our results concern spontaneous synchronization, they are relevant both for reducing dependence on conventional control devices, thus offering an additional layer of protection given that most power outages involve equipment or operational errors, and for contributing to the development of “smart grids ” that can recover from failures in real time. 1 ar
Controllability Metrics, Limitations and Algorithms for Complex Networks
"... Abstract — This paper studies the problem of controlling stable and symmetric complex networks, that is, the joint problem of selecting a set of control nodes and of designing a control input to drive a network to a target state. We adopt the smallest eigenvalue of the controllability Gramian as met ..."
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Cited by 14 (2 self)
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Abstract — This paper studies the problem of controlling stable and symmetric complex networks, that is, the joint problem of selecting a set of control nodes and of designing a control input to drive a network to a target state. We adopt the smallest eigenvalue of the controllability Gramian as metric for the controllability degree of a network, as it identifies the energy needed to accomplish the control task. In the first part of the paper we characterize tradeoffs between the control energy and the number of control nodes, based on the network topology and weights. Our bounds show for instance that, if the number of control nodes is constant, then the control energy increases exponentially with the number of network nodes. Consequently, despite the classic controllability notion, few nodes cannot in practice arbitrarily symmetric control complex networks. In the second part of the paper we propose a distributed openloop strategy with performance guarantees for the control of complex networks. In our strategy we select control nodes based on network partitioning, and we design the control input based on optimal and distributed control techniques. For our control strategy we show that the control energy depends on the controllability properties of the clusters and on their coupling strength, and it is independent of the network dimension. I.
Synchronization in Complex Networks of Phase Oscillators: A Survey
, 2014
"... The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in realworl ..."
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Cited by 13 (1 self)
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The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in realworld synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology. We review the history and the countless applications of this model throughout science and engineering. We justify the importance of the widespread coupled oscillator model as a locally canonical model and describe some selected applications relevant to control scientists, including vehicle coordination, electric power networks, and clock synchronization. We introduce the reader to several synchronization notions and performance estimates. We propose analysis approaches to phase and frequency synchronization, phase balancing, pattern formation, and partial synchronization. We present the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finitedimensional and infinitedimensional settings. We conclude by summarizing the limitations of existing analysis methods and by highlighting some directions for future research.
Exploring Synchronization in Complex Oscillator Networks
"... Abstract — The emergence of synchronization in a network of coupled oscillators is a pervasive topic in various scientific disciplines ranging from biology, physics, and chemistry to social networks and engineering applications. A coupled oscillator network is characterized by a population of hetero ..."
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Cited by 11 (1 self)
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Abstract — The emergence of synchronization in a network of coupled oscillators is a pervasive topic in various scientific disciplines ranging from biology, physics, and chemistry to social networks and engineering applications. A coupled oscillator network is characterized by a population of heterogeneous oscillators and a graph describing the interaction among the oscillators. These two ingredients give rise to a rich dynamic behavior that keeps on fascinating the scientific community. In this article, we present a tutorial introduction to coupled oscillator networks, we review the vast literature on theory and applications, and we present a collection of different synchronization notions, conditions, and analysis approaches. We focus on the canonical phase oscillator models occurring in countless realworld synchronization phenomena, and present their rich phenomenology. We review a set of applications relevant to control scientists. We explore different approaches to phase and frequency synchronization, and we present a collection of synchronization conditions and performance estimates. For all results we present selfcontained proofs that illustrate a sample of different analysis methods in a tutorial style. I.
A FlockingBased Dynamical Systems Paradigm for Smart Power System Analysis
"... Abstract—We propose a paradigm to model the cyberphysical interactions related to transient stability in a smart grid. In our multiagent framework each node, representing both electrical and information system components, is modeled as having dynamics that synergistically describe physical and inf ..."
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Cited by 11 (11 self)
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Abstract—We propose a paradigm to model the cyberphysical interactions related to transient stability in a smart grid. In our multiagent framework each node, representing both electrical and information system components, is modeled as having dynamics that synergistically describe physical and information couplings with neighboring agents. Physical behaviors are described through the application of swing equations to a reduced model of the electrical grid. Cyberphysical integrated action is formulated as a flocking control problem to achieve transient stability. Analysis and simulation demonstrate the potential of the paradigm to model cyberphysical smart grid dynamics as well as highlight strategies for effective distributed control. I.
Design of Optimal Sparse Interconnection Graphs for Synchronization of Oscillator Networks
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2014
"... We study the optimal design of a conductance network as a means for synchronizing a given set of oscillators. Synchronization is achieved when all oscillator voltages reach consensus, and performance is quantified by the meansquare deviation from the consensus value. We formulate optimization probl ..."
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Cited by 10 (3 self)
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We study the optimal design of a conductance network as a means for synchronizing a given set of oscillators. Synchronization is achieved when all oscillator voltages reach consensus, and performance is quantified by the meansquare deviation from the consensus value. We formulate optimization problems that address the tradeoff between synchronization performance and the number and strength of oscillator couplings. We promote the sparsity of the coupling network by penalizing the number of interconnection links. For identical oscillators, we establish convexity of the optimization problem and demonstrate that the design problem can be formulated as a semidefinite program. Finally, for special classes of oscillator networks we derive explicit analytical expressions for the optimal conductance values.
Novel Insights into Lossless AC and DC Power Flow
, 2013
"... A central question in the analysis and operation of power networks is feasibility of the power flow equations subject to security constraints. For largescale networks the solution of the nonlinear AC power flow equations can be constructed only numerically or approximated through the linear DC pow ..."
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Cited by 7 (3 self)
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A central question in the analysis and operation of power networks is feasibility of the power flow equations subject to security constraints. For largescale networks the solution of the nonlinear AC power flow equations can be constructed only numerically or approximated through the linear DC power flow. The latter serves as wellused approximation to obtain the phase angle differences near an acceptable operating point, but the accuracy of the DC approximation drops in a stressed grid. Here, we propose a modified DC approximation that applies to lossless networks with parametric uncertainties and with voltage magnitudes bounded within security constraints. We show that the phase angle differences can xbe well approximated by the solution to a set of linear and intervalvalued equations reminiscent of the DC power flow equations. Our proposed approximation improves upon the standard DC approximation, is computationally attractive and provably exact for a broad range of network topologies and parameters. We validate the accuracy of our approximation through standard power network test cases with randomized power demand at the loads.
Topological Equivalence of a StructurePreserving Power Network Model and a NonUniform Kuramoto Model of Coupled Oscillators
"... Abstract — We study synchronization in the classic structurepreserving power network model proposed by Bergen and Hill. We find that, locally near the synchronization manifold, the phase and frequency dynamics of the power network model are topologically conjugate to the phase dynamics of a nonunif ..."
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Cited by 5 (0 self)
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Abstract — We study synchronization in the classic structurepreserving power network model proposed by Bergen and Hill. We find that, locally near the synchronization manifold, the phase and frequency dynamics of the power network model are topologically conjugate to the phase dynamics of a nonuniform Kuramoto model and decoupled exponentially stable dynamics for the frequencies. This topological conjugacy implies the equivalence of local exponential synchronization in power networks and in nonuniform Kuramoto oscillators. Hence, we can harness the results available for Kuramoto oscillators to analyze synchronization in power networks. We establish necessary and sufficient conditions for phase synchronization, sufficient conditions for frequency synchronization, and necessary and sufficient conditions for frequency synchronization with a uniform topology. These conditions also extend the results known for the classic firstorder Kuramoto model and the secondorder linear consensus protocols. Our synchronization conditions all share a common physical interpretation: the ratio of power inputs and dissipation has to be sufficiently uniform and the coupling in the network has to be sufficiently strong. I.