Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> A state-space model of an interconnected power system having both generator and load nodes is proposed. The resulting system of equations is interpreted as the degenerate limit of a singularly perturbed system. The model is used to derive a condition for the (local) asymptotic stability of an equilibrium. This condition decomposes in an intuitive way for subsystems interconnected via a backbone network. The model is used to formulate the problem of steering the power system from a postdisturbance alert state to a secure state, and a solution to the steering problem is also proposed. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> A new model for the study of power system stability via Lyapunov functions is proposed. The key feature of the model is an assumption of frequency-dependent load power, rather than the usual impedance loads which are subsequently absorbed into a reduced network. The original network topology is explicitly represented. This approach has the important advantage of rigorously accounting for real power loads in the Lyapunov functions. This compares favorably with existing methods involving approximations to allow for the significant transfer conductances in reduced network models. The preservation of network topology can be exploited in stability analysis, with the concepts of critical and vulnerable cutsets playing central roles in dynamic and transient stability evaluation respectively. Of fundamental importance is the feature that the Lyapunov functions give a true representation of the spatial distribution of stored energy in the system <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> Abstract We analyze a large system of nonlinear phase oscillators with sinusoidal nonlinearity, uniformly distributed natural frequencies and global all-to-all coupling, which is an extension of Kuramoto's model to second-order systems. For small coupling, the system evolves to an incoherent state with the phases of all the oscillators distributed uniformly. As the coupling is increased, the system exhibits a discontinuous transition to the coherently synchronized state at a pinning threshold.of the coupling strength, or to a partially synchronized oscillation coherent state at a certain threshold below the pinning threshold. If the coupling is decreased from a strong coupling with all the oscillators synchronized coherently, this coherence can persist until the depinning threshold which is less than the pinning threshold, resulting in hysteretic synchrony depending on the initial configuration of the oscillators. We obtain analytically both the pinning and depinning threshold and also expalin the discontinuous transition at the thresholds for the underdamped case in the large system size limit. Numerical exploration shows the oscillatory partially coherent state bifurcates at the depinning threshold and also suggests that this state persists independent of the system size. The system studied here provides a simple model for collective behaviour in damped driven high-dimensional Hamiltonian systems which can explain the synchronous firing of certain fireflies or neural oscillators with frequency adaptation and may also be applicable to interconnected power systems. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> 1. Introduction. 2. Electromagnetic Transients. 3. Synchronous Machine Modeling. 4. Synchronous Machine Control Models. 5. Single-Machine Dynamic Models. 6. Multimachine Dynamic Models. 7. Multimachine Simulation. 8. Small-Signal Stability. 9. Energy Function Methods. Appendix A: Integral Manifolds for Model. Bibliography. Index. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> The theory of power systems dynamics has been developed largely from detailed studies of the dynamics of simple system structures with emphasis on the effects of modeling details of the dynamics. In contrast, the recent study in the science of complex networks, motivated by numerous real-world examples in various areas including power systems, employs rather simple dynamics and yet places much more emphasis on network structures. It appears that the two approaches can usefully inform each other for further progress into an integrated study of very large (or massive) systems as well as very complex dynamics. This paper reviews recent progress in these areas and suggests fruitful lines of future research. <s> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> We show that there is a link between the Kuramoto paradigm and another system of synchronized oscillators, namely an electrical power distribution grid of generators and consumers. The purpose of this work is to show both the formal analogy and some practical consequences. The mapping can be made quantitative, and under some necessary approximations a class of Kuramoto-like models, those with bimodal distribution of the frequencies, is most appropriate for the power-grid. In fact in the power-grid there are two kinds of oscillators: the “sources" delivering power to the “consumers". <s> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> We derive a Kuramoto-like equation from the Cardell-Ilic distributed electrical generation network and use the resulting model to simulate the phase stability and the synchronization of a small electrical grid. It is well-known that a major problem for distributed generation is the frequency stability. This is a non linear problem and proper models for analysis are sorely lacking. In our model nodes are arranged in a regular lattice; the strength of their couplings are randomly chosen and allowed to vary as square waves. Although the system undergoes several synchronization losses, nevertheless it is able to quickly resynchronize. Moreover, we show that the synchronization rising-time follows a power-law. <s> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> We analyze populations of Kuramoto oscillators with a particular distribution of natural frequencies. Inspired by networks where there are two groups of nodes with opposite behaviors, as for instance in power-grids where energy is either generated or consumed at different locations, we assume that the frequencies can take only two different values. Correlations between the value of the frequency of a given node and its topological localization are considered in both regular and random topologies. Synchronization is enhanced when nodes are surrounded by nodes of the opposite frequency. We find analytical estimations for the minimum value of the coupling strength between oscillators that guarantees the achievement of a globally synchronized state, getting a very good agreement with the numerical simulations. <s> BIB008 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> Motivated by recent interest for multi-agent systems and smart power grid architectures, we discuss the synchronization problem for the network-reduced model of a power system with non-trivial transfer conductances. Our key insight is to exploit the relationship between the power network model and a first-order model of coupled oscillators. Assuming overdamped generators (possibly due to local excitation controllers), a singular perturbation analysis shows the equivalence between the classic swing equations and a non-uniform Kuramoto model. Here, non-uniform Kuramoto oscillators are characterized by multiple time constants, non-homogeneous coupling, and non-uniform phase shifts. Extending methods from transient stability, synchronization theory, and consensus protocols, we establish sufficient conditions for synchronization of non-uniform Kuramoto oscillators. These conditions reduce to and improve upon previously-available tests for the standard Kuramoto model. Combining our singular perturbation and Kuramoto analyses, we derive concise and purely algebraic conditions that relate synchronization and transient stability of a power network to the underlying system parameters and initial conditions. <s> BIB009 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> 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 maintains the power sharing properties of the primary droop controller. Our results hold without assumptions on identical line characteristics or voltage magnitudes. <s> BIB010 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> Robust synchronization (phase locking) of power plants and consumers centrally underlies the stable operation of electric power grids. Despite current attempts to control large-scale networks, even their uncontrolled collective dynamics is not fully understood. Here we analyze conditions enabling self-organized synchronization in oscillator networks that serve as coarse-scale models for power grids, focusing on decentralizing power sources. Intriguingly, we find that whereas more decentralized grids become more sensitive to dynamical perturbations, they simultaneously become more robust to topological failures. Decentralizing power sources may thus facilitate the onset of synchronization in modern power grids. <s> BIB011 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> A novel and simple network model that is capable to reproduce quasi-stationary behavior and propagation phenomena in electric power grid dynamics is introduced. A new Kuramoto approximation to distributed generator dynamics is obtained from combining a continuous spatial interaction function with a discrete lattice model representing generator positions and network structure over a continuous spatial domain. At hand of model properties and a numerical study quasi-stationarity of electric power grid dynamics results, being related to non-vanishing fluctuation and oscillations in stationary state. <s> BIB012 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> Consider a weighted and undirected graph, possibly with self-loops, and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to the 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 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 in power networks, or analysis and simulation of induction motors and power electronics. More general applications of Kron reduction occur in sparse matrix algorithms, multi-grid solvers, finite--element analysis, and Markov chains. The Schur complement of a Laplacian matrix and related concepts have also been studied under different names and as purely theoretic problems in the literature on linear algebra. In this paper we propose a general graph-theoretic framework for Kron reduction that leads to novel and deep insights both on the mathematical and the physical side. We show the applicability of our framework to various practical problem setups arising in engineering applications and computation. Furthermore, we provide a comprehensive and detailed graph-theoretic analysis of the Kron reduction process encompassing topological, algebraic, spectral, resistive, and sensitivity analyses. Throughout our theoretic elaborations we especially emphasize the practical applicability of our results. <s> BIB013 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> The emergence of synchronization in a network of coupled oscillators is a fascinating topic in various scientific disciplines. A widely adopted model of a coupled oscillator network is characterized by a population of heterogeneous phase oscillators, a graph describing the interaction among them, and diffusive and sinusoidal coupling. It is known that a strongly coupled and sufficiently homogeneous network synchronizes, but the exact threshold from incoherence to synchrony is unknown. Here, we present a unique, concise, and closed-form condition for synchronization of the fully nonlinear, nonequilibrium, and dynamic network. Our synchronization condition can be stated elegantly in terms of the network topology and parameters or equivalently in terms of an intuitive, linear, and static auxiliary system. Our results significantly improve upon the existing conditions advocated thus far, they are provably exact for various interesting network topologies and parameters; they are statistically correct for almost all networks; and they can be applied equally to synchronization phenomena arising in physics and biology as well as in engineered oscillator networks, such as electrical power networks. We illustrate the validity, the accuracy, and the practical applicability of our results in complex network scenarios and in smart grid applications. <s> BIB014 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> An imperative condition for the functioning of a power-grid 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. <s> BIB015 </s> Synchronization in complex networks of phase oscillators: A survey <s> Electric Power Networks with Synchronous Generators and DC/AC Inverters <s> This paper presents a new method for analysis of frequency synchronization of lossless power networks whose sources are frequency-droop controlled inverters. Unlike most existing approaches, our method focuses on the interaction between inverters and the network to provide new physical insight into how those interactions create frequency synchronization. We introduce two structure-preserving models of such a network (one bus-oriented and one line-oriented), and show that frequency synchronization corresponds to convergence to an equilibrium of these models. We derive a necessary condition for existence of such equilibria, and determine a simple test for their local stability. Finally, we introduce a sufficient condition for frequency synchronization of such networks, and show that it consists of a set of local criteria which can each be determined from local measurements. <s> BIB016
Consider an AC power network modeled as an undirected, connected, and weighted graph with n nodes V = {1, . . . , n}, transmission lines E ⊂ V × V, and admittance matrix Y = Y T ∈ C n×n . For each node, consider the voltage phasor V i = |V i |e iθi corresponding to the phase θ i ∈ S 1 and magnitude |V i | ≥ 0 of the sinusoidal solution to the circuit equations. If the network is lossless, then the active power flow from node i to j is a ij sin(θ i − θ j ), where we adopt the shorthand Fig. 3.(a) . The node set is partitioned as V = V 1 ∪ V 2 ∪ V 3 , where V 1 are load buses, V 2 are synchronous generators, and V 3 are grid-connected direct current (DC) power sources.  If the generator reactances are absorbed into the admittance matrix, then the electromechanical swing dynamics of the synchronous generator i ∈ V 2 are where θ i ∈ S 1 andθ i ∈ R 1 are the generator rotor angle and frequency, P m,i > 0 is the mechanical power input, and M i > 0, and D i > 0 are the inertia and damping coefficients. The dynamics (8)-(9) constitute the structurepreserving power network model, proposed by BIB002 . A derivation from first principles can be found in BIB004 , Chapter 7). We assume that each DC source is connected to the AC grid via a DC/AC inverter, the inverter output impedances are absorbed into the admittance matrix, and each inverter is equipped with a conventional droop controller. For a droop-controlled inverter i ∈ V 3 with droop-slope 1/D i > 0, the deviation of the power output n j=1 a ij sin(θ i − θ j ) from its nominal value P d,i > 0 is proportional to the frequency deviation D iθi . This gives rise to the droop-controlled inverter dynamics BIB010 ) These power network devices are illustrated as circuit elements in panels (a)-(d) of Fig. 3 . Panels (e) and (f) show a high-voltage transmission network and a microgrid. We remark that different loads such as constant power/current/susceptance loads and synchronous motors can be modeled by the same set of equations (8)- (10), see BIB001  BIB004 BIB013 . Synchronization is pervasive in the operation of power networks. All generating units of an interconnected grid must remain in strict frequency synchronism while continuously following demand and rejecting disturbances. Notice that, with the exception of the inertial terms M iθi and the possibly non-unit coefficients D i , the power network dynamics (8)-(10) are a perfect electrical analog of the coupled oscillator model (1) with ω i ∈ {−P l,i , P m,i , P d,i }. Thus, it is not surprising that scientists from different disciplines recently advocated coupled oscillator approaches to analyze synchronization in power networks BIB003  BIB005 BIB006 BIB008 BIB007 BIB010 BIB009 BIB011 BIB014 BIB012 BIB015 BIB016 . The theoretical tools presented in this article establish how frequency synchronization in power networks depend on the nodal parameters (P l,i , P m,i , P d,i ) as well as the interconnecting electrical network with weights a ij .