We consider the synchronization problem of swing equations, a second-order nonuniform Kuramoto model, with general connectivity and dampings . This is motivated by its relevance to the dynamics of power grids. As an important topic in power grids, people have been paying special attention to the transient stability which concerns the system’s ability to reach an acceptable synchronism after a transient disturbance. For this concern, an important problem is to determine whether the post-fault state is located in the basin of attraction of synchronous states (sync basin). Recently this issue is becoming more and more challenging since the highly stochastic renewable power sources exert more transient disturbances on the power grids with increasing size and complexity. In Dörfler et al (2013 Proc. Natl Acad. Sci. USA 110 2005–10), it was pointed out that the sync basin is an important unsolved problem. In Li et al (2014 SIAM J. Control Optim . 52 2482–511), an explicit estimate on the region of attraction of coupled oscillators with homogenous damping for network with diameter less than two was obtained. However, it turns out that these assumptions are too restrictive in many real situations. The purpose of this work is to study the emergence of synchronization and give an estimate for sync basin for the nonuniform Kurmaoto model on connected graphs with general dampings , which is the most general setting for a connected power grid. Our strategy is based on the gradient-like formulation and energy estimate.

We study an emergent dynamics of the Kuramoto oscillators with adaptive couplings. In the Kuramoto model, pairwise coupling strengths are assumed to be constant and uniform over all interaction pairs. This assumption simplifies the analysis, but it is too restrictive to describe the real applications. In this paper, we relax this uniform strength ansatz by adopting a dynamic feedback law depending on the relative phase differences and discuss two types of adaptive rules for the couplings of oscillators. As a first adaptive model, we consider the adaptive law introduced by Picallo and Riecke and present several sufficient frameworks leading to the asymptotic synchronization. For the second model, we consider Model A introduced in [S.-Y. Ha, S. E. Noh, and J. Park, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 162--194]. We introduce a small parameter that diversifies time scales for the dynamics of the states of the model. By this slow-fast setting, coupling strengths become the fast variables, whereas the phas...

We study the stability of an oscillatory associative memory network consisting of $N$ coupled Kuramoto oscillators with applications in binary pattern retrieve. In this model, the coupling function...

In this paper we are concerned with a coupled system of Kuramoto oscillators for power grids with dynamic voltages. Among various models describing the dynamics of synchronous generators, analytic results are available mainly for the second-order model with constant voltages which describes only the dynamics of rotor angles. It is known that two of the most important forms for power system stability are rotor angle stability and voltage stability. Schmietendorf et al (2014 Eur. Phys. J. 223 2577–2592) derived a model of adaptive Kuramoto oscillators for the power grids with dynamic voltages and carried out some numerical studies. In this model, the transient dynamics of voltages is incorporated and the voltage dynamics could be considered together with rotor angle dynamics. In this article, we will consider this model and derive some analytic results for the synchronization of phase angles and stabilization of voltages. We will find a region of attraction for a class of steady states which is explicitly expressed in the parameters of system.

We continue the study on the Hebbian network of Kuramoto oscillators with a second-order Fourier term, aiming to apply the system to the binary pattern retrieve task with nonorthogonal standard bin...

Delay in feedback is inevitable in a multi-agent system due to time lags in information processing for self-organization. The well-known Cucker-Smale model incorporated with this information processing delay has been recently studied, and it was shown (at least for a two-agent system) that as long as the delay is below a threshold value, the system exhibits the flocking behavior where the agents ultimately reach the same velocity. Numerical studies however suggest that the threshold value established for the delay is not optimal, and the Cucke-Smale system remains to have the flocking property for large delay. Here, we rigorously prove that flocking behavior emerges for a large information processing delay, and we establish the relationship between the admissible size of time delay and the system’s parameters and initial data.