Synchronization in complex networks of phase oscillators: A survey <s> Sufficient Synchronization Conditions <s> The flow function of a power system is the vector of node powers expressed in terms of the node angles. The Jacobian is the matrix of partial derivatives of the flow vector with respect to the angle vector. The ratio of the determinant of the Jacobian to the value which it has when the node angles are set to zero is a dimensionless stability margin which lies between one and zero (steady-state in- stability) and is easily computed from a load flow. The margin of torque is the least change in power flows that will cause instability. Maximizing the torque margin or maximizing the stability margin with a load constraint yields the optimum dispatching change to maximize security. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Sufficient Synchronization Conditions <s> Practical experience with load-flow solutions has indicated that stable solutions are probably unique, given sufficient data about a network to determine a solution from the standpoint of dimensionality, i.e., having as many equations as variables for the remaining unknowns. A counterexample is given here for which two stable solutions exist, and an approach towards the analysis of uniqueness is presented that adds some insight into the nature of load-flow solutions in general. <s> BIB002 </s> Synchronization in complex networks of phase oscillators: A survey <s> Sufficient Synchronization Conditions <s> Abstract The qualitative properties of the power-flow equation for a transmission network are studied in terms of global and local aspects, as well as stability. Global aspects include estimates of the number of solutions and topological properties of the stable region. Local aspects are examined through bifurcations of the flow equation. Examples are given to counter some intuitive conjectures, and other conjectures are made. <s> BIB003 </s> Synchronization in complex networks of phase oscillators: A survey <s> Sufficient Synchronization Conditions <s> Electric utility analysts today face an increasingly difficult task of formulating both long and short term operating plans which will provide at the same time efficient and economical operation while delivering reliable and uninterrupted service to electricity users. One of the key ingredients in this planning is a set of large scale simulations of the steady-state network performance under various anticipated operating conditions. Central to these analyses are the classical "load flow" equations which are the equilibrium equations for the "swing equations" which are a physically based model of the dynamic operation of an n -node power system. Despite the long standing and widespread use of these equations, there remain a number of very basic open questions: What are the number and nature of the equilibria of the swing equations? How many stable equilibrium operating points are there in an n -node electric power grid? In this paper some powerful analytical tools from topology and geometry are used to answer certain of these questions. It is well documented that the load flow equations comprise a formidable large scale system but what is interesting, and perhaps surprising, is that even for a small number of buses, these equations possess a rather rich and intricate qualitative behavior which has heretofor been only partially understood. Indeed, until now there was no complete statement in the literature concerning the number of load flows in a general three-bus network. In Theorem 2.7, we state that for the "generic" three-bus network there are, for sufficiently small power injections, either four or six real load flows and that, in either case, exactly one of those load flows is stable. This is a special case of the results we derive for a general n -bus powergrid. Our method consists in first transforming the load flow equations for a lossless electric power network by trigonometric substitutions into algebraic equations. This makes it possible to apply some deep and powerful results from algebraic geometry and intersection theory to study these equations. An obvious tool for determining the number of solutions is provided by the classical theorem of Bezout, but it is shown that for systems describing an n -machine network with n \geq 4 , this result cannot be directly applied because the solutions contain solution components of positive dimension "at infinity." A major result in this paper is a modified Bezout technique which allows us to compute the number of complex (and a fortiori an upper bound on the number of real) solutions to the load flow equations. Combining this with the classical Morse inequalities, we obtain very explicit results regarding the number of stable load flows for a given network topology and set of power injections. The cases of three and four machine networks are considered in detail. <s> BIB004 </s> Synchronization in complex networks of phase oscillators: A survey <s> Sufficient Synchronization Conditions <s> We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for all-to-all networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value, the synchronized state is locally asymptotically stable, resulting in convergence of all phase differences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We further explain the behavior of the system as the number of oscillators grows to infinity. <s> BIB005 </s> Synchronization in complex networks of phase oscillators: A survey <s> Sufficient Synchronization Conditions <s> In this technical note we study the problem of exponential synchronization for one of the most popular models of coupled phase oscillators, the Kuramoto model. We consider the special case of finite oscillators with distinct, bounded natural frequencies. Our first result derives a lower bound on the coupling gain which is necessary for the onset of synchronization. This bound improves the one derived by Jadbabaie . We then calculate a lower bound on the coupling gain that is sufficient to guarantee oscillator synchronization and derive further sufficient conditions to ensure exponential synchronization of the angular frequencies of all oscillators to the mean natural frequency of the group. We also characterize the coupling gain that is sufficient for the oscillator phase differences to approach any desired compact set in finite time. <s> BIB006 </s> Synchronization in complex networks of phase oscillators: A survey <s> Sufficient Synchronization Conditions <s> We study the stationary points of what is known as the lattice Landau gauge fixing functional in one-dimensional compact U(1) lattice gauge theory, or as the Hamiltonian of the one-dimensional random phase XY model in statistical physics. An analytic solution of all stationary points is derived for lattices with an odd number of lattice sites and periodic boundary conditions. In the context of lattice gauge theory, these stationary points and their indices are used to compute the gauge fixing partition function, making reference in particular to the Neuberger problem. Interpreted as stationary points of the one-dimensional XY Hamiltonian, the solutions and their Hessian determinants allow us to evaluate a criterion which makes predictions on the existence of phase transitions and the corresponding critical energies in the thermodynamic limit. <s> BIB007 </s> Synchronization in complex networks of phase oscillators: A survey <s> Sufficient Synchronization Conditions <s> We present the long-time dynamics of unidirectionally coupled identical Kuramoto oscillators in a ring, when each oscillator is influenced sinusoidally by a single preassigned oscillator. In this situation, for a large system with $N \geq 5$, it is well known that the synchronized states and the splay-state are the only stable equilibria by Gershgorin's theorem and linear stability analysis, whereas for low-dimensional systems with $N=2,3$ the synchronized state is the unique stable equilibrium. We present nontrivial proper subsets of synchronized and splay-state basins with positive Lebesgue measure in $N$-phase space. For the threshold case $N=4$, we show that the splay-state is nonlinearly unstable by explicit construction of perturbations converging toward the synchronized state asymptotically. <s> BIB008 </s> Synchronization in complex networks of phase oscillators: A survey <s> Sufficient Synchronization Conditions <s> 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 real-world 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 self-contained proofs that illustrate a sample of different analysis methods in a tutorial style. <s> BIB009
For arbitrary network topologies and weights the equilibrium and potential energy landscape of the oscillator network (1) has been studied by different communities, see BIB001 BIB002 BIB003 BIB004 BIB007 . We particularly recommend the article BIB003 , where various surprising and counter-intuitive examples are reported. To the best of the authors' knowledge, the conditions (27)- (28) in Lemma 4.2 are the best known explicit necessary conditions for the existence of equilibria for arbitrary topologies and weights. In what follows, we focus on sufficient conditions guaranteeing frequency synchronization, and we restrict ourselves to phase cohesive synchronous solutions within the set ∆ G (π/2). There are two reasons for this choice. First, as discussed in Subsection 4.1, the equilibria in ∆ G (π/2) are exponentially stable, and the forward invariance of the set ∆ G (π/2) leads to stable synchronization by incremental stability or frequency dynamics arguments. Second, from a pragmatic point of view, there are few analysis results and conditions for equilibria outside ∆ G (π/2), with the treatment of (directed) ring graphs in BIB008 being a notable exception. The approaches to phase synchronization (in Section 5) and to frequency synchronization in complete graphs (in Section 6) are generally not applicable to dissimilar natural frequencies and sparse coupling graphs, or are so only under very conservative conditions. For example, in the presence of dissimilar natural frequencies ω ∈ 1 ⊥ n , a Lyapunov analysis of the forced system (26) via the trigonometric potential function U (θ) is very involved since the level sets of U (θ) are hard to characterize. Likewise, the contraction Lyapunov analysis based on definition (32) inherently requires arc-invariance of all angles, and does not easily extend to arbitrary topologies. One quadratic Lyapunov function advocated by BIB005 ; BIB006  This Lyapunov function is useful to analyze the more general oscillator network (1), and yields the following result found in (Dörfler and Bullo, 2012a, Theorem 4.6) and (Dörfler and Bullo, 2012b, Theorem 4.4) . Theorem 7.1 (Practical phase synchronization in sparse graphs I) Consider the coupled oscillator model (1) with a connected graph G(V, E, A) and frequencies ω ∈ 1 ⊥ n . There exists a locally exponentially stable equilibrium manifold [θ] ∈ ∆ G (π/2) if the algebraic connectivity is larger than a critical value, that is, Moreover, if condition (49) holds, then the coupled oscillator model (1) achieves practical phase synchronization in the following sense. Given The analysis leading to Theorem 7.1 is similar to the proof of Theorem 6.6: the Lyapunov function (48) is used to guarantee the ultimate boundedness of the phases in θ ∈ Arc n (π) | θ E cplt ,2 ≤ γ min ⊂ ∆ G (γ min ), and the Jacobian arguments in Subsection 4.1 guarantee frequency synchronization. For classic Kuramoto oscillators (3), condition (49) reduces to K > ω E cplt ,2 ; this condition is more conservative than the tight bound (46) which reads K > ω E,∞ = ω max − ω min . One reason for this conservatism is that condition (49) guarantees that all phase differences |θ i −θ j | are bounded, not only those along the edges of the graph. However, by Lemma 4.1, we know that bounded phase differences |θ i − θ j | only for {i, j} ∈ E, are sufficient to establish the existence of a locally exponentially stable synchronized solution. In what follows we adopt a fixed-point approach to the study of the equilibrium equations for the coupled oscillator model (1). In matrix notation, these equilibrium equations read as where A = diag({a ij } {i,j}∈E ) is the diagonal matrix of weights. We next follow the ingenious analysis of (50) suggested in (Jadbabaie et al., 2004, Section IIV.B) . For the sake of a streamlined presentation, we treat the angles θ as vectors in 1 ⊥ n . Recall the state-dependent weights c ij (θ) = a ij sinc(θ i − θ j ) from the consensus formulation (31), and define the state-dependent Laplacian Hence, equations (50) can be written compactly as The following result has been obtained in BIB009 , Theorem 4.7) by applying to equation (51) a fixed point theorem in the incremental two norm · E,2 . Theorem 7.2 (Practical phase synchronization in sparse graphs II) Consider the coupled oscillator model (1) with a connected graph G(V, E, A) and frequencies ω ∈ 1 ⊥ n . There exists a locally exponentially Moreover, if condition (52) holds, then [θ * ] is phase cohesive in the following sense: Clearly, condition (52) is sharper than condition (49), but the stability result is only local. The synchronization condition (52) is the sharpest sufficient condition for general graphs known to the authors, but it is still a conservative estimate for most network topologies and weights. Indeed, the necessary condition (28) and sufficient condition (52) are separated by a tremendous gap for n > 2 oscillators. The reasons for this conservatism are manifold. First, the derivation of the conditions (27), (28), (49), and (52) involves conservative bounding of the trigonometric nonlinearities and network interactions. Second, by Theorem 7.2, the condition (52) guarantees that the 2-norm of all phase distances between neighboring oscillators is bounded as θ * E,2 ≤ arcsin(λ critical /λ 2 (L)). On the other hand, to conclude frequency synchronization by Lemma 4.1, phase synchronization by Theorem 5.2, or frequency synchronization (in the complete graph case) by Theorem 6.6, only the worst-case phase distance (between neighbors) θ E,∞ needs to be bounded. We conclude that the incremental 2-norm metric leads to overly strong phase cohesiveness requirements and accordingly to conservative conditions. All our previous results hint at the incremental ∞-norm as a natural metric.