Second-order Kuramoto Oscillators Research Articles

On synchronization of third-order power systems governed by second-order networked Kuramoto oscillators incorporating first-order magnitude dynamics

This study explores the sufficient conditions for achieving synchronization in the third-order one-axis generator models in modern power systems, characterized by the second-order networked Kuramoto oscillators integrating the first-order amplitude dynamics. First, we provide detailed descriptions of dynamic models for this particular class of third-order power systems, which are governed by second-order networked Kuramoto oscillators incorporating first-order voltage magnitude dynamics. Then, we show that if the phase difference of oscillators is less than π2 and the initial amplitudes are positive, then the amplitudes are bounded and possess a positive lower bound. Furthermore, under specific conditions on the coupling strength, inertia, and oscillator parameters, we prove that the third-order model exhibits a frequency synchronization and the derivatives of amplitudes converge to zero. Numerical simulations are performed to support our main results.

Event-triggered synchronization control of complex networks with adaptive coupling strength

The event-triggered synchronization control problem is concerned for a class of complex networks with nonlinearly coupling function and adaptive coupling strength. Given a state-based event-trigger mechanism and the threshold, an event-triggered control method is introduced to make complex networks achieve exponential synchronization. By combining the Lyapunov stability theory and the knowledge of graph theory, a sufficient condition is established such that complex networks can achieve exponential synchronization. Then, the feasibility of the event-triggered control is analyzed. Moreover, the second-order Kuramoto oscillators is taken into account. And the event-triggered control strategy is used to make the oscillators achieve exponential synchronization. Meanwhile, two simulation results about the second-order Kuramoto oscillators are given to show the effectiveness of results.

Intermittent Control Strategy for Synchronization Analysis of Time-Varying Complex Dynamical Networks

The pth moment exponential synchronization (PMES) problem for time-varying complex dynamical networks (CDNs) with multiple time-varying delays (TCDNMTDs) by aperiodically intermittent control is studied in this paper. Compared with related work, the coupling structure of CDNs is time varying. Also, internal time-varying delays and coupling time-varying delays are taken into consideration and they are different from each other. It is worth emphasizing that the intermittent control scheme proposed here is nonperiodical. Some criteria are derived to make TCDNMTDs achieve PMES by using the Lyapunov method and graph theory. Additionally, we also consider the synchronization of second-order Kuramoto oscillators and coupled Chua's circuits and some numerical simulations are presented to verify the validity of our results.

Spanning-Tree-Based Synchronization Conditions for Second-Order Kuramoto Networks

This brief studies the synchronization of second-order phase-coupled Kuramoto oscillators. Under the effects of non-uniformity among the system parameters, we propose novel spanning-tree-based conditions for the oscillators to remain phase cohesive over time, which further guarantees the synchronization. First, by choosing a spanning tree in the underlying network graph, we consider the phase cohesiveness as a 2-norm bound on the tree-induced phase differences and an infinity norm bound on the relative phases that are excluded from the spanning tree. Energy functions are constructed to derive two sets of conditions on achieving these two types of boundedness, respectively. Then, we prove that presented phase cohesive conditions guarantee the asymptotic synchronization for a second-order Kuramoto network. Our method is less conservative than those in literature on estimating a region of permissible initial states for oscillators to achieve synchronization, which is verified by numerical examples.

Onset of synchronization in networks of second-order Kuramoto oscillators with delayed coupling: Exact results and application to phase-locked loops

We consider the inertial Kuramoto model of $N$ globally coupled oscillators characterized by both their phase and angular velocity, in which there is a time delay in the interaction between the oscillators. Besides the academic interest, we show that the model can be related to a network of phase-locked loops widely used in electronic circuits for generating a stable frequency at multiples of an input frequency. We study the model for a generic choice of the natural frequency distribution of the oscillators, to elucidate how a synchronized phase bifurcates from an incoherent phase as the coupling constant between the oscillators is tuned. We show that in contrast to the case with no delay, here the system in the stationary state may exhibit either a subcritical or a supercritical bifurcation between a synchronized and an incoherent phase, which is dictated by the value of the delay present in the interaction and the precise value of inertia of the oscillators. Our theoretical analysis, performed in the limit $N \to \infty$, is based on an unstable manifold expansion in the vicinity of the bifurcation, which we apply to the kinetic equation satisfied by the single-oscillator distribution function. We check our results by performing direct numerical integration of the dynamics for large $N$, and highlight the subtleties arising from having a finite number of oscillators.

Synchronization of stochastic complex networks with time delay via feedback control based on discrete-time state observations

Abstract In this paper, synchronization of stochastic complex networks (SCNs) with time delay is researched. A feedback control method is introduced to drive such SCNs to achieve inner synchronization in mean square. Different from previous researches, the feedback control we designed is based on discrete-time state observations. Moreover, by exploiting Lyapunov method and Kirchhoff’s Matrix Tree Theorem in graph theory, some sufficient criteria are obtained to guarantee synchronization in mean square and asymptotical synchronization in mean square of SCNs. What is more, we make use of the theoretical results to analyze asymptotical synchronization in mean square of second-order Kuramoto oscillators with time delay and stochastic disturbances and get a sufficient criterion. At the end of this paper, a numerical example is provided to validate the effectiveness and feasibility of our theoretical results.

Synchronization analysis of stochastic coupled systems with time delay on networks by periodically intermittent control and graph-theoretic method

In this paper, the issue of inner exponential synchronization (IES) for a class of stochastic coupled systems with time-varying delay on networks (SCSTVDNs) is investigated. A periodically intermittent control strategy is introduced to drive such SCSTVDNs to achieve IES. By exploiting Lyapunov method and Kirchhoffs matrix tree theorem in graph theory, two types of sufficient IES criteria are respectively established in the form of Lyapunov-type theorem and coefficients-type theorem. Furthermore, to show the practicability of the theoretical results, the analysis of IES for the second-order Kuramoto oscillators with stochastic perturbations and time-varying delay is performed. A numerical example is finally provided to demonstrate the validity and feasibility of our analytical results.

Influence of stochastic perturbations on the cluster explosive synchronization of second-order Kuramoto oscillators on networks.

Explosive synchronization in networked second-order Kuramoto oscillators has been well studied recently and it is revealed that the synchronization process is featured by cluster explosive synchronization. However, little attention has been paid to the influence of noise or perturbation. We here study this problem and discuss the influences of noise and perturbation. For the former, we interestingly find that noise has significant influence on the cluster explosive synchronization of those nodes with smaller degrees, i.e., their synchronization will change from the first-order to second-order transition and the critical points for both the forward and backward synchronization depend on the strength of noise. Especially, when the strength of noise is in an optimal range, a synchronization of the nodes with smaller degrees will be induced in the region of coupling strength where they do not display synchronization in the absence of noise. For the latter, we find that the effect of perturbation is similar to that of noise when its duration W is small. However, the perturbation will induce a change from cluster explosive synchronization to explosive synchronization when W is large. Furthermore, a brief theory is provided to explain the influence of perturbations on the critical points.

Multiplexing induced explosive synchronization in Kuramoto oscillators with inertia

Explosive synchronization (ES) of coupled oscillators on networks is shown to be originated from the existence of correlation between natural frequencies of oscillators and degrees of corresponding nodes. Here, we demonstrate that ES is a generic feature of multiplex network of second-order Kuramoto oscillators and can exist in the absence of a frequency-degree correlation. A monoplex network of second-order Kuramoto oscillators bearing homogeneous (heterogeneous) degree distribution is known to display the first-order (second-order) transition to synchronization. We report that multiplexing of two such networks having homogeneous degree distribution support the first-order transition in both the layers thereby facilitating ES. More interesting is the multiplexing of a layer bearing heterogeneous degree distribution with another layer bearing homogeneous degree distribution, which induces a first-order (ES) transition in the heterogeneous layer which was incapable of showing the same in isolation. Further, we report that such induced ES transition in the heterogeneous layer of multiplex networks can be controlled by varying inter- and intra-layer coupling strengths. Our findings emphasize the importance of multiplexing or the impact of one layer on the dynamical evolution of other layers of systems having inherent multiplex or multilevel architecture.

Synchronization of stochastic coupled systems via feedback control based on discrete-time state observations

In this paper, we formulate and investigate the synchronization of stochastic coupled systems via feedback control based on discrete-time state observations (SCSFD). The discrete-time state feedback control is used in the drift parts of response system. Combining Lyapunov method with graph theory, the upper bound of duration between two consecutive state observations is provided. And a global Lyapunov function of SCSFD is presented, which derives some sufficient criteria to guarantee the synchronization of drive–response systems in the sense of mean-square asymptotical synchronization. In addition, the theoretical results are applied to stochastic coupled oscillators and second-order Kuramoto oscillators. Finally, two numerical examples are given to verify the effectiveness of the theoretical results.

On synchronization in power-grids modelled as networks of second-order Kuramoto oscillators.

This work concerns analytical results on the role of coupling strength in the phenomenon of onset of complete frequency locking in power-grids modelled as a network of second-order Kuramoto oscillators. Those results allow estimation of the coupling strength for the onset of complete frequency locking and to assess the features of network and oscillators that favor synchronization. The analytical results are evaluated using an order parameter defined as the normalized sum of absolute values of phase deviations of the oscillators over time. The investigation of the frequency synchronization within the subsets of the parameter space involved in the synchronization problem is also carried out. It is shown that the analytical results are in good agreement with those observed in the numerical simulations. In order to illustrate the methodology, a case study is presented, involving the Brazilian high-voltage transmission system under a load peak condition to study the effect of load on the syncronizability of the grid. The results show that both the load and the centralized generation might have concurred to the 2014 blackout.

Analysis of cluster explosive synchronization in complex networks.

Correlations between intrinsic dynamics and local topology have become a new trend in the study of synchronization in complex networks. In this paper, we investigate the influence of topology on the dynamics of networks made up of second-order Kuramoto oscillators. In particular, based on mean-field calculations, we provide a detailed investigation of cluster explosive synchronization (CES) [Phys. Rev. Lett. 110, 218701 (2013)] in scale-free networks as a function of several topological properties. Moreover, we investigate the robustness of discontinuous transitions by including an additional quenched disorder, and we show that the phase coherence decreases with increasing strength of the quenched disorder. These results complement the previous findings regarding CES and also fundamentally deepen the understanding of the interplay between topology and dynamics under the constraint of correlating natural frequencies and local structure.