1. F. Dörfler. Dynamics and Control in Power Grids and Complex Oscillator Networks. Ph.D. thesis, University of California at Santa Barbara, September 2013. Keyword(s): Power Networks, Microgrids, Complex Oscillator Networks, Algebraic Graph Theory, Microgrids. Abstract:

   The eefficient production, transmission and distribution of electrical power underpins our technological civilization. Public policy and environmental concerns are leading to an increasing adoption of renewable energy sources and the deregulation of energy markets. These trends, together with an ever-growing power demand, are causing power networks to operate increasingly closer to their stability margins. Recent scientific advances in complex networks and cyber-physical systems along with the technological re-instrumentation of the grid provide promising opportunities to handle the challenges facing our future energy supply. In this thesis, we discuss the synchronization problem in power networks, which is central to their operation and functionality. We identify and exploit a close connection between the mathematical models for power networks and complex oscillator networks. Our main contributions are concise, sharp, and purely-algebraic conditions that relate synchronization in a power grid to graph-theoretical properties of the underlying electric network. Our novel conditions hold for arbitrary interconnection topologies and network parameters, and they significantly improve upon previously-available tests. We illustrate how our results help in the analysis of large-scale transmission systems and lead to novel control strategies and their implementation in microgrids. Our approach combines traditional power engineering methods, synchronization theory for coupled oscillators, and control in multi-agent dynamical systems. Beside their applications in power networks, our mathematically-appealing results are also broadly applicable in synchronization phenomena ranging from natural and life sciences to engineering disciplines.

   
   

   @phdthesis{FD-PhD,
   abstract = {The eefficient production, transmission and distribution of electrical power underpins our technological civilization. Public policy and environmental concerns are leading to an increasing adoption of renewable energy sources and the deregulation of energy markets. These trends, together with an ever-growing power demand, are causing power networks to operate increasingly closer to their stability margins. Recent scientific advances in complex networks and cyber-physical systems along with the technological re-instrumentation of the grid provide promising opportunities to handle the challenges facing our future energy supply. In this thesis, we discuss the synchronization problem in power networks, which is central to their operation and functionality. We identify and exploit a close connection between the mathematical models for power networks and complex oscillator networks. Our main contributions are concise, sharp, and purely-algebraic conditions that relate synchronization in a power grid to graph-theoretical properties of the underlying electric network. Our novel conditions hold for arbitrary interconnection topologies and network parameters, and they significantly improve upon previously-available tests. We illustrate how our results help in the analysis of large-scale transmission systems and lead to novel control strategies and their implementation in microgrids. Our approach combines traditional power engineering methods, synchronization theory for coupled oscillators, and control in multi-agent dynamical systems. Beside their applications in power networks, our mathematically-appealing results are also broadly applicable in synchronization phenomena ranging from natural and life sciences to engineering disciplines.},
   advisor = {F. Bullo},
   author = {F. Dörfler},
   date-added = {2013-09-10 21:17:53 +0000},
   date-modified = {2018-05-01 12:18:42 +0000},
   keywords = {Power Networks, Microgrids, Complex Oscillator Networks, Algebraic Graph Theory, Microgrids},
   month = {September},
   pdf = {http://people.ee.ethz.ch/~floriand/docs/Theses/Dorfler_Thesis.pdf},
   school = {University of California at Santa Barbara},
   title = {{Dynamics and Control in Power Grids and Complex Oscillator Networks}},
   type = {{Ph.D.} thesis},
   year = {2013} 
   }
   

1. F. Dörfler and F. Bullo. Kron Reduction of Graphs with Applications to Electrical Networks. IEEE Transactions on Circuits and Systems I: Regular Papers, 60(1):150-163, January 2013. Keyword(s): Power Networks, Algebraic Graph Theory. Abstract:

   Consider a weighted undirected graph and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to self-loops. 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 graph-theoretic 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.

   
   

   @article{FD-FB:11d,
   abstract = {Consider a weighted undirected graph and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to self-loops. 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 graph-theoretic 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.},
   author = {F. D{ö}rfler and F. Bullo},
   date-added = {2013-02-06 17:54:41 +0000},
   date-modified = {2013-10-18 17:51:17 +0000},
   funding = {IIS-0904501, CPS-1135819},
   journal = {IEEE Transactions on Circuits and Systems~I: Regular Papers},
   keywords = {Power Networks, Algebraic Graph Theory},
   month = {January},
   number = 1,
   pages = {150-163},
   pdf = {http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6316101},
   title = {{K}ron Reduction of Graphs with Applications to Electrical Networks},
   url = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6316101&contentType=Journals+%26+Magazines&queryText%3Dkron+reduction+of+graphs},
   volume = 60,
   year = 2013,
   bdsk-url-1 = {http://dx.doi.org/10.1109/TCSI.2012.2215780} 
   }
   




2. F. Dörfler, M. Chertkov, and F. Bullo. Synchronization in Complex Oscillator Networks and Smart Grids. Proceedings of the National Academy of Sciences, 110(6):2005-2010, February 2013. Keyword(s): Power Networks, Complex Oscillator Networks.
   

   @article{FD-MC-FB:12c,
   author = {F. D{ö}rfler and M. Chertkov and F. Bullo},
   date-modified = {2013-10-01 04:32:11 +0000},
   funding = {CPS-1135819, IIS-0904501},
   journal = {Proceedings of the National Academy of Sciences},
   keywords = {Power Networks, Complex Oscillator Networks},
   month = {February},
   number = 6,
   pages = {2005-2010},
   pdf = {http://www.pnas.org/content/early/2013/01/11/1212134110.full.pdf+html?with-ds=yes},
   title = {Synchronization in Complex Oscillator Networks and Smart Grids},
   url = {http://www.pnas.org/content/early/2013/01/11/1212134110},
   volume = 110,
   year = 2013,
   bdsk-url-1 = {http://www.pnas.org/content/early/2013/01/11/1212134110} 
   }
   




3. F. Dörfler, F. Pasqualetti, and F. Bullo. Continuous-Time Distributed Observers with Discrete Communication. IEEE Journal of Selected Topics in Signal Processing, 7(2):296-304, April 2013. Keyword(s): Distributed Control Algorithms, Secure Control Systems, Linear Control Design. Abstract:

   This work presents a distributed algorithm for observer design for linear continuous-time systems. We assume the dynamical system to be partitioned into disjoint areas, and we let each area be equipped with a control center. Each control center knows local dynamics, collects local observations, performs local computation, and communicates with neighboring control centers at discrete times. For our continuous-discrete estimation algorithm we prove convergence, we characterize its convergence rate, and we show robustness against discretization and communication errors. Our technical approach is inspired by waveform relaxation methods and combines tools from estimation theory, decentralized control theory, and parallel computation. We illustrate the effectiveness of our algorithm with illustrative examples in sensor networks and electric power systems.

   
   

   @article{FD-FP-FB:12b,
   abstract = {This work presents a distributed algorithm for observer design for linear continuous-time systems. We assume the dynamical system to be partitioned into disjoint areas, and we let each area be equipped with a control center. Each control center knows local dynamics, collects local observations, performs local computation, and communicates with neighboring control centers at discrete times. For our continuous-discrete estimation algorithm we prove convergence, we characterize its convergence rate, and we show robustness against discretization and communication errors. Our technical approach is inspired by waveform relaxation methods and combines tools from estimation theory, decentralized control theory, and parallel computation. We illustrate the effectiveness of our algorithm with illustrative examples in sensor networks and electric power systems.},
   author = {F. D{ö}rfler and F. Pasqualetti and F. Bullo},
   date-added = {2012-07-02 15:36:38 -0600},
   date-modified = {2013-10-01 04:59:30 +0000},
   journal = {IEEE Journal of Selected Topics in Signal Processing},
   keywords = {Distributed Control Algorithms, Secure Control Systems, Linear Control Design},
   month = {April},
   number = {2},
   pages = {296-304},
   pdf = {http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6450033},
   title = {Continuous-Time Distributed Observers with Discrete Communication},
   url = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6450033},
   volume = {7},
   year = {2013},
   bdsk-url-1 = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6450033} 
   }
   




4. F. Pasqualetti, F. Dörfler, and F. Bullo. Attack Detection and Identification in Cyber-Physical Systems. IEEE Transactions on Automatic Control, 58(11):2715-2729, November 2013. Keyword(s): Secure Control Systems, Power Networks, Distributed Control Algorithms, Linear Control Design. Abstract:

   Cyber-physical systems are ubiquitous in power systems, transportation networks, industrial control processes, and critical infrastructures. These systems need to operate reliably in the face of unforeseen failures and external malicious attacks. In this paper (i) we propose a mathematical framework for cyber-physical systems, attacks, and monitors; (ii) we characterize fundamental monitoring limitations from system-theoretic and graph-theoretic perspectives; and (iii) we design centralized and distributed attack detection and identification monitors. Finally, we validate our findings through compelling examples.

   
   

   @article{FP-FD-FB:10y,
   abstract = {Cyber-physical systems are ubiquitous in power systems, transportation networks, industrial control processes, and critical infrastructures. These systems need to operate reliably in the face of unforeseen failures and external malicious attacks. In this paper (i) we propose a mathematical framework for cyber-physical systems, attacks, and monitors; (ii) we characterize fundamental monitoring limitations from system-theoretic and graph-theoretic perspectives; and (iii) we design centralized and distributed attack detection and identification monitors. Finally, we validate our findings through compelling examples.},
   author = {F. Pasqualetti and F. D{ö}rfler and F. Bullo},
   date-added = {2014-05-21 18:35:09 +0000},
   date-modified = {2014-05-21 18:35:09 +0000},
   funding = {CNS-1135819, W911NF-09-0001},
   journal = {IEEE Transactions on Automatic Control},
   keywords = {Secure Control Systems, Power Networks, Distributed Control Algorithms, Linear Control Design},
   month = {November},
   number = {11},
   pages = {2715-2729},
   pdf = {http://motion.me.ucsb.edu/~florian/Downloads/FP-FD-FB-12a.pdf},
   title = {Attack Detection and Identification in Cyber-Physical Systems},
   url = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6545301&queryText%3DAttack+Detection+and+Identification},
   volume = {58},
   year = 2013,
   bdsk-url-1 = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6545301&queryText%3DAttack+Detection+and+Identification} 
   }
   




5. J. W. Simpson-Porco, F. Dörfler, and F. Bullo. Synchronization and Power Sharing for Droop-Controlled Inverters in Islanded Microgrids. Automatica, 49(9):2603-2611, 2013. Keyword(s): Complex Oscillator Networks, Microgrids, Distributed Control Algorithms, Frequency Control, Power Electronics Control. Abstract:

   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 power-frequency droop controllers can be cast as a Kuramoto model of phase-coupled 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 time-varying load. Remarkably, this distributed-averaging integral controller has the additional property that it preserves the power sharing properties of the primary droop controller. Our results hold for any acyclic network topology, and hold without assumptions on identical line admittances or voltage magnitudes.

   
   

   @article{JWSP-FD-FB:12u,
   abstract = {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 power-frequency droop controllers can be cast as a Kuramoto model of phase-coupled 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 time-varying load. Remarkably, this distributed-averaging integral controller has the additional property that it preserves the power sharing properties of the primary droop controller. Our results hold for any acyclic network topology, and hold without assumptions on identical line admittances or voltage magnitudes. },
   author = {J. W. Simpson-Porco and F. D{ö}rfler and F. Bullo},
   date-added = {2012-11-10 20:00:57 +0000},
   date-modified = {2018-05-01 12:02:29 +0000},
   funding = {CPS-1135819},
   journal = {Automatica},
   keywords = {Complex Oscillator Networks, Microgrids, Distributed Control Algorithms, Frequency Control, Power Electronics Control},
   number = {9},
   pages = {2603-2611},
   pdf = {http://ac.els-cdn.com/S0005109813002884/1-s2.0-S0005109813002884-main.pdf?_tid=d0737ec2-2a52-11e3-99aa-00000aab0f01&acdnat=1380602272_20d1262a181b1a249c2c128a09eeae2c},
   title = {Synchronization and Power Sharing for Droop-Controlled Inverters in Islanded Microgrids},
   url = {http://www.sciencedirect.com/science/article/pii/S0005109813002884},
   volume = {49},
   year = 2013,
   bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/S0005109813002884} 
   }
   

1. H. Bouattour, J. W. Simpson-Porco, F. Dörfler, and F. Bullo. Further Results on Distributed Secondary Control in Microgrids. In IEEE Conf. on Decision and Control, pages 1514-1519, March 2013. Note: Extended manuscript available at http://motion.me.ucsb.edu/pdf/2013j-bsdb.pdf. Keyword(s): Microgrids, Distributed Control Algorithms, Complex Oscillator Networks, Frequency Control.
   

   @inproceedings{HB-JWSP-FD-FB:13a,
   author = {H. Bouattour and J. W. Simpson-Porco and F. D{ö}rfler and F. Bullo},
   booktitle = {{IEEE} Conf.\ on Decision and Control},
   date-added = {2013-03-11 03:58:50 +0000},
   date-modified = {2016-05-31 09:48:57 +0000},
   keywords = {Microgrids, Distributed Control Algorithms, Complex Oscillator Networks, Frequency Control},
   month = {March},
   note = {Extended manuscript available at {http://motion.me.ucsb.edu/pdf/2013j-bsdb.pdf}},
   pages = {1514--1519},
   pdf = {http://motion.me.ucsb.edu/pdf/2013j-bsdb.pdf},
   title = {Further Results on Distributed Secondary Control in Microgrids},
   year = {2013} 
   }
   




2. F. Dörfler and F. Bullo. Novel Insights into Lossless AC and DC Power Flow. In IEEE Power & Energy Society General Meeting, July 2013. Keyword(s): Power Networks.
   

   @inproceedings{FD-FB:PES12a,
   author = {F. D{ö}rfler and F. Bullo},
   booktitle = {IEEE Power \& Energy Society General Meeting},
   date-added = {2012-11-21 22:31:59 +0000},
   date-modified = {2015-01-16 18:31:42 +0000},
   keywords = {Power Networks},
   month = {July},
   title = {{Novel Insights into Lossless AC and DC Power Flow}},
   url = {http://ieeeexplore.info/xpl/articleDetails.jsp?reload=true&tp=&arnumber=6672260&queryText%3DNovel+Insights+into+Lossless+AC+and+DC+Power+Flow},
   year = {2013},
   bdsk-url-1 = {http://ieeeexplore.info/xpl/articleDetails.jsp?reload=true&tp=&arnumber=6672260&queryText%3DNovel+Insights+into+Lossless+AC+and+DC+Power+Flow} 
   }
   




3. F. Dörfler, M. R. Jovanovic, M. Chertkov, and F. Bullo. Sparse and optimal wide-area damping control in power networks. In American Control Conference, Washington, DC, USA, pages 4295-4300, June 2013. Keyword(s): Linear Control Design, Wide-Area Control, Power Networks.
   

   @inproceedings{FD-MJ-MC-FB:12s,
   address = {Washington, DC, USA},
   author = {F. D{ö}rfler and M. R. Jovanovic and M. Chertkov and F. Bullo},
   booktitle = {{A}merican {C}ontrol {C}onference},
   date-added = {2012-10-17 18:17:05 +0000},
   date-modified = {2013-12-08 21:55:41 +0000},
   funding = {IIS-0904501, CPS-1135819},
   keywords = {Linear Control Design, Wide-Area Control, Power Networks},
   month = jun,
   pages = {4295-4300},
   pdf = {http://motion.me.ucsb.edu/pdf/2012s-djcb.pdf},
   title = {Sparse and optimal wide-area damping control in power networks},
   url = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6580499&queryText%3DSparse+and+optimal+wide-area+damping+control+in+power+networks},
   year = 2013,
   bdsk-url-1 = {http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6580499&queryText%3DSparse+and+optimal+wide-area+damping+control+in+power+networks} 
   }
   




4. D. Romeres, F. Dörfler, and F. Bullo. Novel results on slow coherency in consensus and power networks. In European Control Conference, Zürich, Switzerland, pages 742-747, July 2013. Keyword(s): Wide-Area Control, Power Networks, Complex Oscillator Networks.
   

   @inproceedings{DR-FD-FB:12q,
   address = {Zürich, Switzerland},
   author = {D. Romeres and F. D{ö}rfler and F. Bullo},
   booktitle = {{E}uropean {C}ontrol {C}onference},
   date-added = {2012-11-10 20:03:37 +0000},
   date-modified = {2014-11-04 15:53:46 +0000},
   funding = {IIS-0904501, CPS-1135819},
   keywords = {Wide-Area Control, Power Networks, Complex Oscillator Networks},
   month = jul,
   nopdf = {http://motion.me.ucsb.edu/pdf/2012q-rdb.pdf},
   pages = {742--747},
   pdf = {http://motion.me.ucsb.edu/~florian/Downloads/Romeres_ECC2013.pdf},
   title = {Novel results on slow coherency in consensus and power networks},
   year = 2013 
   }
   




5. J. W. Simpson-Porco, F. Dörfler, and F. Bullo. Voltage Stabilization in Microgrids via Quadratic Droop Control. In IEEE Conf. on Decision and Control, pages 7582-7589, February 2013. Keyword(s): Microgrids, Distributed Control Algorithms, Complex Polynomial Networks, Voltage Control.
   

   @inproceedings{JWSP-FD-FB:13a,
   author = {J. W. Simpson-Porco and F. D{ö}rfler and F. Bullo},
   booktitle = {{IEEE} Conf.\ on Decision and Control},
   date-added = {2013-02-27 00:38:16 +0000},
   date-modified = {2016-05-31 09:49:10 +0000},
   keywords = {Microgrids, Distributed Control Algorithms, Complex Polynomial Networks, Voltage Control},
   month = {February},
   pages = {7582--7589},
   pdf = {http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6761093},
   title = {Voltage Stabilization in Microgrids via Quadratic Droop Control},
   url = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6761093},
   year = {2013},
   bdsk-url-1 = {http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6761093} 
   }
   




6. J. W. Simpson-Porco, F. Dörfler, Q. Shafiee, J. M. Guerrero, and F. Bullo. Stability, power sharing, & distributed secondary control in droop-controlled microgrids. In IEEE Int. Conf. on Smart Grid Communications, Vancouver, BC, Canada, pages 672-677, October 2013. Keyword(s): Microgrids, Distributed Control Algorithms, Complex Oscillator Networks, Power Networks, Power Electronics Control.
   

   @inproceedings{JWSP-FD-QS-JMG-FB:13e,
   address = {Vancouver, BC, Canada},
   author = {J. W. Simpson-Porco and F. D{ö}rfler and Q. Shafiee and J. M. Guerrero and F. Bullo},
   booktitle = {IEEE Int.\ Conf.\ on Smart Grid Communications},
   date-added = {2013-05-21 17:36:17 +0000},
   date-modified = {2018-05-01 12:16:51 +0000},
   keywords = {Microgrids, Distributed Control Algorithms, Complex Oscillator Networks, Power Networks, Power Electronics Control},
   month = oct,
   pages = {672--677},
   pdf = {http://motion.me.ucsb.edu/pdf/2013e-sdsgb.pdf},
   title = {Stability, power sharing, \& distributed secondary control in droop-controlled microgrids},
   year = 2013 
   }
   

1. F. Dörfler, M. R. Jovanovic, M. Chertkov, and F. Bullo. Sparsity-Promoting Optimal Wide-Area Control of Power Networks, July 2013. Note: Available at http://arxiv.org/abs/1307.4342. Keyword(s): Wide-Area Control, Power Networks, Linear Control Design.
   

   @misc{FD-MJ-MC-FB:13a-arxiv,
   author = {F. D{ö}rfler and M. R. Jovanovi{\'c} and M. Chertkov and F. Bullo},
   date-added = {2012-10-17 18:24:08 +0000},
   date-modified = {2013-11-13 02:22:28 +0000},
   keywords = {Wide-Area Control, Power Networks, Linear Control Design},
   month = jul,
   note = {Available at {http://arxiv.org/abs/1307.4342}},
   pdf = {http://arxiv.org/pdf/1307.4342v2.pdf},
   title = {Sparsity-Promoting Optimal Wide-Area Control of Power Networks},
   year = 2013 
   }
   




2. S. Dhople, B. Johnson, F. Dörfler, and A. Hamadeh. Synchronization of Nonlinear Circuits in Dynamic Electrical Networks with General Topologies, October 2013. Note: Available at http://arxiv.org/abs/1310.4550. Keyword(s): Complex Oscillator Networks, Microgrids, Algebraic Graph Theory, Virtual Oscillator Control, Power Electronics Control.
   

   @misc{SD-BJ-FD-AH:13b,
   author = {S. Dhople and B. Johnson and F. D{ö}rfler and A. Hamadeh},
   date-added = {2013-10-18 02:06:17 +0000},
   date-modified = {2018-05-01 12:23:46 +0000},
   funding = {UCLA Startup Funds},
   keywords = {Complex Oscillator Networks, Microgrids, Algebraic Graph Theory, Virtual Oscillator Control, Power Electronics Control},
   month = {October},
   note = {Available at {http://arxiv.org/abs/1310.4550}},
   pdf = {http://arxiv.org/pdf/1310.4550v1.pdf},
   title = {Synchronization of Nonlinear Circuits in Dynamic Electrical Networks with General Topologies},
   url = {http://arxiv.org/abs/1310.4550},
   year = 2013,
   bdsk-url-1 = {http://arxiv.org/abs/1310.4550} 
   }
   

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