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.nonlinear dynamics | power grids T he scientific interest in the synchronization of coupled oscillators can be traced back to Christiaan Huygens' seminal work on "an odd kind sympathy" between coupled pendulum clocks (1), and it continues to fascinate the scientific community to date (2, 3). A mechanical analog of a coupled oscillator network is shown in Fig. 1A and consists of a group of particles constrained to rotate around a circle and assumed to move without colliding. Each particle is characterized by a phase angle θ i and has a preferred natural rotation frequency ω i . Pairs of interacting particles i and j are coupled through an elastic spring with stiffness a ij . Intuitively, a weakly coupled oscillator network with strongly heterogeneous natural frequencies ω i does not display any coherent behavior, whereas a strongly coupled network with sufficiently homogeneous natural frequencies is amenable to synchronization. These two qualitatively distinct regimes are illustrated in Fig. 1 B and C. Formally, the interaction among n such phase oscillators is modeled by a connected graph G(V, E, A) with nodes V = {1, . . ., n}, edges E ⊂ V × V, and positive weights a ij > 0 for each undirected edge {i, k} ∈ E. For pairs of noninteracting oscillators i and j, the coupling weight a ij is 0. We assume that the node set is partitioned as V = V 1 ∪ V 2 , and we consider the following general coupled oscillator model:The coupled oscillator model [1] consists of the second-order oscillators V 1 with Newtonian dynamics, inertia coefficients M i , and viscous damping D i . The remaining oscillators V 2 feature first-order dynamics with time constants D i . A perfect electrical analog of t...

Abstract. When uncontrollable resources fluctuate, optimal power flow (OPF), routinely used by the electric power industry to redispatch hourly controllable generation (coal, gas, and hydro plants) over control areas of transmission networks, can result in grid instability and, potentially, cascading outages. This risk arises because OPF dispatch is computed without awareness of major uncertainty, in particular fluctuations in renewable output. As a result, grid operation under OPF with renewable variability can lead to frequent conditions where power line flow ratings are significantly exceeded. Such a condition, which is borne by our simulations of real grids, is considered undesirable in power engineering practice. Possibly, it can lead to a risky outcome that compromises grid stability-line tripping. Smart grid goals include a commitment to large penetration of highly fluctuating renewables, thus calling to reconsider current practices, in particular the use of standard OPF. Our chance-constrained (CC) OPF corrects the problem and mitigates dangerous renewable fluctuations with minimal changes in the current operational procedure. Assuming availability of a reliable wind forecast parameterizing the distribution function of the uncertain generation, our CC-OPF satisfies all the constraints with high probability while simultaneously minimizing the cost of economic redispatch. CC-OPF allows efficient implementation, e.g., solving a typical instance over the 2746-bus Polish network in 20 seconds on a standard laptop.Key words. optimization, power flows, uncertainty, wind farms, networks AMS subject classifications. 15A15, 15A09, 15A23 DOI. 10.1137/130910312The power grid, one of the greatest engineering achievements of the 20th century, delivers social development and resulting political stability of billions of people around the globe through control sophistication and careful long-term planning, with only very rare disruptions. *

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For a delta-correlated velocity field, simultaneous correlation functions of a passive scalar satisfy closed equations. We analyze the equation for the four-point function. To describe a solution completely, one has to solve the matching problems at the scale of the source and at the diffusion scale. We solve both the matching problems and thus find the dependence of the four-point correlation function on the diffusion and pumping scale for large space dimensionality d. It is shown that anomalous scaling appears in the first order of 1/d perturbation theory. Anomalous dimensions are found analytically both for the scalar field and for it's derivatives, in particular, for the dissipation field.

A new phenomenological model of turbulent fluctuations is constructed by considering the Lagrangian dynamics of 4 points (the tetrad). The closure of the equations of motion is achieved by postulating an anisotropic, i.e. tetrad shape dependent, relation of the local pressure and the velocity gradient defined on the tetrad. The non-local contribution to the pressure and the incoherent small scale fluctuations are modeled as Gaussian white "noise". The resulting stochastic model for the coarse-grained velocity gradient is analyzed approximately, yielding predictions for the probability distribution functions of different 2nd and 3d order invariants. The results are compared with the direct numerical simulation of the Navier-Stokes. The model provides a reasonable representation of the non-linear dynamics involved in energy transfer and vortex stretching and allows to study interesting aspects of the statistical geometry of turbulence, e.g. vorticity/strain alignment. In a state with a constant energy flux (and K41 power spectrum) it exhibits the anomalous scaling of high moments associated with formation of high gradient sheets -events associated with large energy transfer. An approach to the more complete analysis of the stochastic model, properly including the effect of fluctuations, is outlined and will enable further quantitative juxtaposition of the model with the results of the DNS.

We examine classical, transient fluctuation theorems within the unifying framework of Langevin dynamics. We explicitly distinguish between the effects of non-conservative forces that violate detailed balance, and non-autonomous dynamics arising from the variation of an external parameter.When both these sources of nonequilibrium behavior are present, there naturally arise two distinct fluctuation theorems.

In this paper we present the derivation details, logic, and motivation for the loop calculus introduced in [1]. Generating functions for three inter-related discrete statistical models are expressed in terms of a finite series. The first term in the series corresponds to the Bethe-Peierls/BeliefPropagation (BP) contribution, the other terms are labeled by loops on the factor graph. All loop contributions are simple rational functions of spin correlation functions calculated within the BP approach. We discuss two alternative derivations of the loop series. One approach implements a set of local auxiliary integrations over continuous fields with the BP contribution corresponding to an integrand saddle-point value. The integrals are replaced by sums in the complimentary approach, briefly explained in [1]. Local gauge symmetry transformations that clarify an important invariant feature of the BP solution, are revealed in both approaches. The individual terms change under the gauge transformation while the partition function remains invariant. The requirement for all individual terms to be non-zero only for closed loops in the factor graph (as opposed to paths with loose ends) is equivalent to fixing the first term in the series to be exactly equal to the BP contribution. Further applications of the loop calculus to problems in statistical physics, computer and information sciences are discussed.PACS numbers: 05.50.+q,89.70.+C One practically useful yet generally heuristic approach used for calculations of observables (correlation functions) in discrete statistical physics models, e.g. Ising model, is related to the so-called Bethe-Peierls (BP) approximation [2,3,4]. The BP approach is exact for graphs that do not contain loops, usually referred to as trees; otherwise the approach is approximate. The ad-hoc approach can also be re-stated in a variational form [5,6,7]. A similar tree-based method in information science has been developed by Gallager [8,9] in the context of error-correction theory. Gallager introduced the so-called Low-Density-Parity-Check (LDPC) codes, defined on locally tree-like Tanner graphs. The problem of ideal decoding, i.e. restoring the most probable pre-image out of the exponentially large pool of candidates, is identical to solving a statistical model on the graph [10]. An approximate yet efficient Belief-Propagation decoding algorithm introduced by Gallager constitutes an iterative solution of the Bethe-Peierls equations derived as if the statistical problem was defined on a tree that locally represents the Tanner graph. We utilize this abbreviation coincidence to call Bethe-Peierls and Belief-Propagation equations by the same acronym -BP. Recent resurgence of interest to LDPC codes [11,12], as well as proliferation of the BP approach to other areas of information and computer science, e.g. artificial intelligence [13] and combinatorial optimization [14,15,16], where interesting statistical models on graphs with long loops appear, made the BP approach to be one of the most interesting and h...

Abstract. When uncontrollable resources fluctuate, Optimum Power Flow (OPF), routinely used by the electric power industry to re-dispatch hourly controllable generation (coal, gas and hydro plants) over control areas of transmission networks, can result in grid instability, and, potentially, cascading outages. This risk arises because OPF dispatch is computed without awareness of major uncertainty, in particular fluctuations in renewable output. As a result, grid operation under OPF with renewable variability can lead to frequent conditions where power line flow ratings are significantly exceeded. Such a condition, which is borne by simulations of real grids, would likely resulting in automatic line tripping to protect lines from thermal stress, a risky and undesirable outcome which compromises stability. Smart grid goals include a commitment to large penetration of highly fluctuating renewables, thus calling to reconsider current practices, in particular the use of standard OPF. Our Chance Constrained (CC) OPF corrects the problem and mitigates dangerous renewable fluctuations with minimal changes in the current operational procedure. Assuming availability of a reliable wind forecast parameterizing the distribution function of the uncertain generation, our CC-OPF satisfies all the constraints with high probability while simultaneously minimizing the cost of economic re-dispatch. CC-OPF allows efficient implementation, e.g. solving a typical instance over the 2746-bus Polish network in 20s on a standard laptop.