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Synchronization and Power Sharing for DroopControlled Inverters in Islanded Microgrids ⋆
"... Motivated by the recent and growing interest in smart grid technology, we study the operation of DC/AC inverters in an inductive microgrid. We show that a network of loads and DC/AC inverters equipped with powerfrequency droop controllers can be cast as a Kuramoto model of phasecoupled oscillators ..."
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Cited by 27 (8 self)
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Motivated by the recent and growing interest in smart grid technology, we study the operation of DC/AC inverters in an inductive microgrid. We show that a network of loads and DC/AC inverters equipped with powerfrequency droop controllers can be cast as a Kuramoto model of phasecoupled oscillators. This novel description, together with results from the theory of coupled oscillators, allows us to characterize the behavior of the network of inverters and loads. Specifically, we provide a necessary and sufficient condition for the existence of a synchronized solution that is unique and locally exponentially stable. We present a selection of controller gains leading to a desirable sharing of power among the inverters, and specify the set of loads which can be serviced without violating given actuation constraints. Moreover, we propose a distributed integral controller based on averaging algorithms, which dynamically regulates the system frequency in the presence of a timevarying load. Remarkably, this distributedaveraging integral controller has the additional property that it preserves the power sharing properties of the primary droop controller. Our results hold without assumptions on identical line characteristics or voltage magnitudes. Key words: inverters; powersystem control, smart power applications, synchronization, coupled oscillators, Kuramoto model, distributed control. 1
Secondorder consensus for multiagent systems with directed topologies and nonlinear dynamics
 IEEE Transactions on Automatic Control
"... Abstract—This paper considers a secondorder consensus problem for multiagent systems with nonlinear dynamics and directed topologies where each agent is governed by both position and velocity consensus terms with a timevarying asymptotic velocity. To describe the system’s ability for reaching co ..."
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Cited by 20 (4 self)
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Abstract—This paper considers a secondorder consensus problem for multiagent systems with nonlinear dynamics and directed topologies where each agent is governed by both position and velocity consensus terms with a timevarying asymptotic velocity. To describe the system’s ability for reaching consensus, a new concept about the generalized algebraic connectivity is defined for strongly connected networks and then extended to the strongly connected components of the directed network containing a spanning tree. Some sufficient conditions are derived for reaching secondorder consensus in multiagent systems with nonlinear dynamics based on algebraic graph theory, matrix theory, and Lyapunov control approach. Finally, simulation examples are given to verify the theoretical analysis. Index Terms—Algebraic connectivity, directed spanning tree, multiagent system, secondorder consensus, strongly connected network. I.
2013 Universality in network dynamics
 Nat. Phys. 9
"... Despite significant advances in characterizing the structural properties of complex networks, a mathematical framework that uncovers the universal properties of the interplay between the topology and the dynamics of complex systems continues to elude us. Here we develop a selfconsistent theory of d ..."
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Cited by 16 (1 self)
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Despite significant advances in characterizing the structural properties of complex networks, a mathematical framework that uncovers the universal properties of the interplay between the topology and the dynamics of complex systems continues to elude us. Here we develop a selfconsistent theory of dynamical perturbations in complex systems, allowing us to systematically separate the contribution of the network topology and dynamics. The formalism covers a broad range of steadystate dynamical processes and offers testable predictions regarding the system's response to perturbations and the development of correlations. It predicts several distinct universality classes whose characteristics can be derived directly from the continuum equation governing the system's dynamics and which are validated on several canonical networkbased dynamical systems, from biochemical dynamics to epidemic spreading. Finally, we collect experimental data pertaining to social and biological systems, demonstrating that we can accurately uncover their universality class even in the absence of an appropriate continuum theory that governs the system's dynamics. I.
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 16 (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
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 (0 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.
V.: Measuring the effect of social communications on individual working rhythms: A case study of open source software
 In the proceedings of 2012 ASE/IEEE International Conference on Social Informatics
, 2012
"... Abstract—This paper proposes novel quantitative methods to measure the effects of social communications on individual working rhythms by analyzing the communication and code committing records in tens of Open Source Software (OSS) projects. Our methods are based on complex network and timeseries ana ..."
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Cited by 9 (6 self)
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Abstract—This paper proposes novel quantitative methods to measure the effects of social communications on individual working rhythms by analyzing the communication and code committing records in tens of Open Source Software (OSS) projects. Our methods are based on complex network and timeseries analysis. We define the notion of a working rhythm as the average time spent on a commit task and we study the correlation between working rhythm and communication frequency. We build communication networks for code developers, and find that the developers with higher social status, represented by the nodes with larger number of outgoing or incoming links, always have faster working rhythms and thus contribute more per unit time to the projects. We also study the dependency between work (committing) and talk (communication) activities, in particular the effect of their interleaving. We introduce multiactivity timeseries and quantitative measures based on activity latencies to evaluate this dependency. Comparison of simulated timeseries with the real ones suggests that when work and talk activities are in proximity they may accelerate each other in OSS systems. These findings suggest that frequent communication before and after committing activities is essential for effective software development in distributed systems. I.
SYNCHRONIZATION VIA PINNING CONTROL ON GENERAL COMPLEX NETWORKS
, 2013
"... This paper studies synchronization via pinning control on general complex dynamical networks, such as strongly connected networks, networks with a directed spanning tree, weakly connected networks, and directed forests. A criterion for ensuring network synchronization on strongly connected network ..."
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Cited by 6 (0 self)
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This paper studies synchronization via pinning control on general complex dynamical networks, such as strongly connected networks, networks with a directed spanning tree, weakly connected networks, and directed forests. A criterion for ensuring network synchronization on strongly connected networks is given. It is found that the vertices with very small indegrees should be pinned first. In addition, it is shown that the original condition with controllers can be reformulated such that it does not depend on the form of the chosen controllers, which implies that the vertices with very large outdegrees may be pinned. Then, a criterion for achieving synchronization on networks with a directed spanning tree, which can be composed of many strongly connected components, is derived. It is found that the strongly connected components with very few connections from other components should be controlled and the components with many connections from other components can achieve synchronization even without controls. Moreover, a simple but effective pinning algorithm for reaching synchronization on a general complex dynamical network is proposed. Finally, some simulation examples are given to verify the proposed pinning scheme.
2010a). Spectral analysis of synchronization in a lossless structurepreserving power network model
 In IEEE International Conference on Smart Grid Communications. Gaithersburg, MD. Submitted
"... Abstract—This paper considers the synchronization and transient stability analysis in a simple model of a structurepreserving power system. We derive sufficient conditions relating synchronization in a power network directly to the underlying network state, parameters, and topology. In particular, ..."
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Cited by 5 (4 self)
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Abstract—This paper considers the synchronization and transient stability analysis in a simple model of a structurepreserving power system. We derive sufficient conditions relating synchronization in a power network directly to the underlying network state, parameters, and topology. In particular, we provide a spectral condition based on the algebraic connectivity of the network and a second condition based on the effective resistance among generators. These conditions build upon the authors’ earlier results on synchronization in networkreduced power system models. Central to our analysis is the reduced admittance matrix of the network, which is obtained by a Schur complement of the network’s topological admittance matrix with respect to its bus nodes. This networkreduction process, termed Kron reduction, relates the structurepreserving and the networkreduced power system model. We provide a detailed graphtheoretic, algebraic, and spectral analysis of the Kron reduction process leading directly to the novel synchronization conditions. I.
Distributed Higher Order Consensus Protocols in Multiagent Dynamical Systems
, 2011
"... This paper studies general higher order distributed consensus protocols in multiagent dynamical systems. First, network synchronization is investigated, with some necessary and sufficient conditions derived for higher order consensus. It is found that consensus can be reached if and only if all sub ..."
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Cited by 5 (1 self)
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This paper studies general higher order distributed consensus protocols in multiagent dynamical systems. First, network synchronization is investigated, with some necessary and sufficient conditions derived for higher order consensus. It is found that consensus can be reached if and only if all subsystems are asymptotically stable. Based on this result, consensus regions are characterized. It is proved that for the thorder consensus, there are at most ( +1) 2 disconnected stable and unstable consensus regions. It is shown that consensus can be achieved if and only if all the nonzero eigenvalues of the Laplacian matrix lie in the stable consensus regions. Moreover, the ratio of the largest to the smallest nonzero eigenvalues of the Laplacian matrix plays a key role in reaching consensus and a scheme for choosing the coupling strength is derived. Furthermore, a leaderfollower control problem in multiagent dynamical systems is considered, which reveals that to reach consensus the agents with very small degrees must be informed. Finally, simulation examples are given to illustrate the theoretical analysis.