Synchronous Steady State Research Articles

Nonlinear Stability of Complex Droop Control in Converter-Based Power Systems

In this letter, we study the nonlinear stability problem of converter-based power systems, where the converter dynamics are governed by a complex droop control. This complex droop control augments the well-known power-frequency (p-f) droop control, and it proves to be equivalent to the state-of-the-art dispatchable virtual oscillator control (dVOC). In this regard, it is recognized as a promising grid-forming solution to address the high penetration of converters in future power systems. In previous work, the global stability of dVOC (i.e., complex droop control) has been proven by prespecifying a nominal synchronous steady state. For a general case of non-nominal (i.e., drooped) synchronous steady states, however, the stability problem requires further investigation. In this letter, we provide parametric conditions under which a non-nominal synchronous steady state exists and the system is almost globally asymptotically stable with respect to this non-nominal synchronous steady state.

Frequency Synchronization of a High-Order Multiconverter System

We investigate the stability properties of a multi-converter power system model, defined on a high-order manifold. For this, we identify its symmetry (i.e., rotational invariance) generated by a static angle shift and rotation of AC signals. We characterize the steady state set, primarily determined by the steady state angles and DC power input. Based on eigenvalue conditions of its Jacobian matrix, we show asymptotic stability of the multi-converter system in a neighborhood of the synchronous steady state set by applying the center manifold theory. We guarantee the eigenvalue conditions via an explicit approach. Finally, we demonstrate our results based on a numerical example involving a network of identical DC/AC converter systems.

Synchronization Stability Model of Complex Brain Networks: An EEG Study.

In this paper, from the perspective of complex network dynamics we investigated the formation of the synchronization state of the brain networks. Based on the Lyapunov stability theory of complex networks, a synchronous steady-state model suitable for application to complex dynamic brain networks was proposed. The synchronization stability problem of brain network state equation was transformed into a convex optimization problem with Block Coordinate Descent (BCD) method. By using Random Apollo Network (RAN) method as a node selection rule, the brain network constructs its subnet work dynamically. We also analyzes the change of the synchronous stable state of the subnet work constructed by this method with the increase of the size of the network. Simulation EEG data from alcohol addicts patients and Real experiment EEG data from schizophrenia patients were used to verify the robustness and validity of the proposed model. Differences in the synchronization characteristics of the brain networks between normal and alcoholic patients were analyzed, so as differences between normal and schizophrenia patients. The experimental results indicated that the establishment of a synchronous steady state model in this paper could be used to verify the synchronization of complex dynamic brain networks and potentially be of great value in the further study of the pathogenic mechanisms of mental illness.

Ghost Attractors in Spontaneous Brain Activity: Recurrent Excursions Into Functionally-Relevant BOLD Phase-Locking States.

Functionally relevant network patterns form transiently in brain activity during rest, where a given subset of brain areas exhibits temporally synchronized BOLD signals. To adequately assess the biophysical mechanisms governing intrinsic brain activity, a detailed characterization of the dynamical features of functional networks is needed from the experimental side to constrain theoretical models. In this work, we use an open-source fMRI dataset from 100 healthy participants from the Human Connectome Project and analyze whole-brain activity using Leading Eigenvector Dynamics Analysis (LEiDA), which serves to characterize brain activity at each time point by its whole-brain BOLD phase-locking pattern. Clustering these BOLD phase-locking patterns into a set of k states, we demonstrate that the cluster centroids closely overlap with reference functional subsystems. Borrowing tools from dynamical systems theory, we characterize spontaneous brain activity in the form of trajectories within the state space, calculating the Fractional Occupancy and the Dwell Times of each state, as well as the Transition Probabilities between states. Finally, we demonstrate that within-subject reliability is maximized when including the high frequency components of the BOLD signal (>0.1 Hz), indicating the existence of individual fingerprints in dynamical patterns evolving at least as fast as the temporal resolution of acquisition (here TR = 0.72 s). Our results reinforce the mechanistic scenario that resting-state networks are the expression of erratic excursions from a baseline synchronous steady state into weakly-stable partially-synchronized states – which we term ghost attractors. To better understand the rules governing the transitions between ghost attractors, we use methods from dynamical systems theory, giving insights into high-order mechanisms underlying brain function.

Parametric local stability condition of a multi-converter system

We study local (also referred to as small-signal) stability of a network of identical DC/AC converters having a rotating degree of freedom. We develop a stability theory for a class of partitioned linear systems with symmetries that has natural links to classical stability theories of interconnected systems. We find stability conditions descending from a particular Lyapunov function involving an oblique projection onto the complement of the synchronous steady state set and enjoying insightful structural properties. Our sufficient and explicit stability conditions can be evaluated in a fully decentralized fashion, reflect a parametric dependence on the converter's steady-state variables, and can be one-to-one generalized to other types of systems exhibiting the same behavior, such as synchronous machines. Our conditions demand for sufficient reactive power support and resistive damping. These requirements are well aligned with practitioners' insights.

Synchronization in Multiplex Networks of Chua’s Circuits: Theory and Experiments

In many natural and man-made networks, nodes communicate with each other through different types of interconnections giving rise to so-called multiplex networks. In this paper, we study the interplay between synchronous steady state solutions and the topology of multiplex networks of Chua’s circuits. In particular, we consider all circuits to be simultaneously interconnected through a network of resistors (static diffusive coupling) and a network of capacitors (dynamic diffusive coupling) each with its own structure. We find that the presence of dynamic links and the structure of the multiplex network affect the stability of the emerging synchronous solutions. Theoretical results are discussed and validated using a dedicated experimental set-up.

Amplitude-phase coupling drives chimera states in globally coupled laser networks.

For a globally coupled network of semiconductor lasers with delayed optical feedback, we demonstrate the existence of chimera states. The domains of coherence and incoherence that are typical for chimera states are found to exist for the amplitude, phase, and inversion of the coupled lasers. These chimera states defy several of the previously established existence criteria. While chimera states in phase oscillators generally demand nonlocal coupling, large system sizes, and specially prepared initial conditions, we find chimera states that are stable for global coupling in a network of only four coupled lasers for random initial conditions. The existence is linked to a regime of multistability between the synchronous steady state and asynchronous periodic solutions. We show that amplitude-phase coupling, a concept common in different fields, is necessary for the formation of the chimera states.

On Synchronous Steady States and Internal Models of Diffusively Coupled Systems

We investigate the problem of synchronizing nonidentical nonlinear dynamical systems by means of generalized diffusive couplings. The focus is not on the actual solution to the problem but on the derivation of necessary conditions for the existence of such a solution despite the systems possessing nonidentical models. We show that for the problem to be solvable a synchronous steady state needs to exist. This condition leads to the requirement that all individual system models need to embed an internal model of some common endosystem. The latter condition is expressed in terms of nonlinear partial differential equations. The conditions derived in this paper are related to those known from the theory of output regulation.

On Synchronous Steady States and Internal Models of Diffusively Coupled Systems

We investigate the problem of synchronizing non-identical nonlinear dynamical systems by means of generalized diffusive couplings. The focus is not on the actual solution to the problem but on the derivation of necessary conditions for the existence of such a solution despite the systems possessing non-identical models. We show that for the problem to be solvable a synchronous steady state needs to exist. This condition leads to the requirement that all individual system models need to embed an internal model of some virtual exosystem. The latter condition is expressed in terms of nonlinear partial differential equations. The conditions derived in this paper are related to those known from the theory of output regulation.

Non-linear resonance of an unbalanced rotating shaft with internal damping

The dynamic response of an unbalanced rotating shaft with internal damping is considered. Special attention is paid to the interaction between the primary resonance and the destabilizing influence of internal damping. Global behavior in terms of the existence and stability of synchronous steady state solutions is examined. This is followed by an investigation of local behavior near instabilities by an application of the center manifold theorem. It is shown that the non-linear resonance for a rotating shaft is much more complicated than a simple Duffing-type resonance when any amount of internal damping is present. In fact, the resonance is an example of a periodically perturbed Hopf bifurcation which results in saddle-node, Hopf and saddle-connection bifurcations which, in turn, give rise to a variety of interesting shaft responses. Specificaly, it is shown that the shaft may exhibit a whirling type motion in which the whirl amplitude undergoes a low frequency modulation, resulting in quasi-periodic motion. the interplay between these motions and the constant amplitude whirl are described in terms of a series of frequency response diagrams for a range of unbalance magnitudes.

Synchronous steady state response of an overhung rotor with squeeze film damping

An analytical method is developed for treatment of a typical system in which the outrigger bearing of an overhung rotor is provided with a squeeze film damper supported in a flexible casing. The rotor and stator of multiple degrees-of-freedom are handled with convenience by a “polar receptance matrix” method. A characteristic equation is derived which governs rotor-stator interaction either with dry contact or through a squeeze film damper. For the nonlinear squeeze film action solutions based on “mobility” information of a dynamically loaded journal bearing are obtained facilitating a general approach. A computer programme is written in Fortran for steady state response of the system in terms of whirl, position and force vectors, trial runs of which indicate complex behaviour of a squeeze film damped system.