Kick synchronization versus diffusive synchronization

The paper provides an introductory discussion about two fundamental models of oscillator synchronization: the (continuous-time) diffusive model, that dominates the mathematical literature on synchronization, and the (hybrid) kick model, that accounts for most popular examples of synchronization, but for which only few theoretical results exist. The paper stresses fundamental differences between the two models, such as the different contraction measures underlying the analysis, as well as important analogies that can be drawn in the limit of weak coupling.Peer reviewe

Uncovering Droop Control Laws Embedded Within the Nonlinear Dynamics of Van der Pol Oscillators

This paper examines the dynamics of power-electronic inverters in islanded microgrids that are controlled to emulate the dynamics of Van der Pol oscillators. The general strategy of controlling inverters to emulate the behavior of nonlinear oscillators presents a compelling time-domain alternative to ubiquitous droop control methods which presume the existence of a quasi-stationary sinusoidal steady state and operate on phasor quantities. We present two main results in this work. First, by leveraging the method of periodic averaging, we demonstrate that droop laws are intrinsically embedded within a slower time scale in the nonlinear dynamics of Van der Pol oscillators. Second, we establish the global convergence of amplitude and phase dynamics in a resistive network interconnecting inverters controlled as Van der Pol oscillators. Furthermore, under a set of non-restrictive decoupling approximations, we derive sufficient conditions for local exponential stability of desirable equilibria of the linearized amplitude and phase dynamics

Oscillation modes in symmetrical wireless-locked systems

Time synchronization of multiple elements of a wireless network can be achieved through the wireless coupling of their oscillator circuits. Most previous works on wireless locking of oscillators analyze the system in an idealized manner, representing the oscillator elements with phase models and describing the propagation effects with constant scalar coefficients and time delays. Here, a realistic analysis of the wireless system is presented, which relies on the extraction of the oscillator models from harmonic-balance (HB) simulations and takes into account the antenna gains and propagation effects. The most usual network configurations, corresponding to ring, fully connected, and star topologies, are investigated in detail. In symmetric conditions, the oscillation modes are detected through an eigenvalue/eigenvector calculation of an equivalent coupling matrix. For each particular mode, the system is analyzed in the following manners: by means of an analytical formulation, able to provide all the coexistent solutions, and through a circuit-level HB simulation of an equivalent system with a reduced number of oscillator elements. The stability properties are determined by means of a perturbation system of general application to any coupled structure. A specific formulation is also derived to predict the impact of discrepancies between the oscillator elements. All the results have been validated with independent circuit-level simulations and measurements.This work was supported in part by the Spanish Ministry of Economy and Competitiveness under the research project TEC2017-88242-C3-1-R, in part by the European Regional Development Fund (ERDF/FEDER), in part by Juan de la Cierva Research Program under IJCI-2014-19141, and in part by the Parliament of Cantabria under the project Cantabria Explora 12.JP02.64069

Wireless-coupled oscillator systems with an injection-locking signal

A detailed analysis of wireless-coupled oscillator systems under the effect of an injection-locking signal is presented. The injection source of high spectral purity is introduced at a single node and enables a reduction of the phase-noise spectral density. Under this injection source, the behavior of the coupled system is qualitatively different from the one obtained in free-running conditions. Two cases are considered: bilateral synchronization, in which an independent source is connected to a particular system oscillator, coupled to the other oscillator elements, and unilateral synchronization, in which one of these elements is replaced by an independent source that cannot be influenced by the rest. The two cases are illustrated through the analysis of a wireless-coupled system with a star topology, such that the injection signal is introduced at the central node. The investigation involves an insightful analytical calculation of the coexisting steady-state solutions, as well as a determination of their stability and bifurcation properties and phase noise. The injection signal stabilizes the system in a large and continuous distance interval, enabling a more robust operation than in autonomous (noninjected) conditions. A coupled system operating at 2.45 GHz has been manufactured and experimentally characterized, obtaining a very good agreement between simulations and measurements.This work was supported by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under research projects TEC2014-60283-C3-1-R and TEC2017-88242-C3-1-R

Mathematical frameworks for oscillatory network dynamics in neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience

Advanced Inverter control for mixed source microgrids

This thesis focuses on investigating virtual oscillator control (VOC) and applying it to mixed source microgrids to address several stability issues. A detailed comparison between VOC and droop control in a three-phase system is presented in terms of transient responses of a single inverter under small load disturbances and the synchronization speed in multiple paralleled inverters under various inverter terminal voltage amplitude and frequency regulation settings. In the single-inverter microgrid, it is demonstrated in both simulation and experiment that the two control models produce similar transient responses in the output voltage and current amplitudes. However, VOC has a faster instantaneous frequency transient response whilst still maintaining the terminal voltage amplitude transient response of the droop controller. In microgrids with multiple inverters, the synchronization speed of the VOC is faster than that of the droop control when the terminal voltage’s frequency regulation range is allowed to be wide. The conclusion is verified with different types of loads. A virtual inertia design method for the VOC inverter with a mixed source microgrid is presented to improve the frequency stability issues of the system. The per unit inertia constant of a VOC inverter is derived when coupled with a synchronous generator in an islanded microgrid. The control parameters of the virtual inertia are designed via small-signal analysis. A dual second order generalized integrator - frequency locked loop (DSOGI-FLL) is adopted for digital implementation of proposed virtual inertia based VOC. With the use of virtual inertia block, the frequency nadir is improved by 22% and rate of change of frequency is improved by 29% compared with the unmodified VOC inverter during the transient period induced by load disturbances. Simulation and experimental results verify the enhanced transient response of system frequency. A voltage and current dual-loop control structure is added to the VOC inverter to solve the voltage drop issues at the inverter terminals caused by the inverter dead-time effects, non-ideal semiconductor and LCL filter. A complete small-signal model for a multiple-inverters microgrid with the proposed control structure is presented in order to assess system stability using eigenvalue and participation factor analysis. Analytical results show that the parameter related to the frequency regulation and the integral gain of the voltage controller affect the location of the system’s dominant modes significantly. The stability margin is determined by modifying these control parameters. Experimental results on a laboratory test microgrid verify the predication from the small-signal analysis and time-domain simulations. Finally, a method to limit current in the VOC inverter under large disturbances in a mixed source microgrid is proposed. During a large load change in the islanded microgrid, the inverter based sources may get temporarily overloaded until other generations with sufficient power margin take the remaining load burden. The original VOC inverter lacks the ability to constrain the current within limits during the transient period. The dual-loop structure proposed in this thesis can limit the transient current with the use of virtual impedance. Such virtual impedance is presented by the desired maximum current magnitude and virtual voltage drop. Compared with a recently proposed fault ride through VOC inverter, the proposed virtual impedance based current limitation method can effectively constrain the inverter current within the pre-set value under large disturbances, which augments the range of application of VOC and enhances its robustness