Optimal entrainment with smooth, pulse, and square signals in weakly forced nonlinear oscillators

Tanaka, Hisaaki

doi:10.1016/j.physd.2014.07.003

Cited by 33 publications

(39 citation statements)

References 45 publications

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“…It appears as if all the TCLs are now oscillating with the same frequency ω 0 . This phenomenon is known in the study of rhythmic systems as frequency entrainment [40,46,47]. The phase model representation of nonlinear oscillators becomes highly valuable in this case because the phase sensitivity function Z(φ) can be used to provide the theoretical linearized limits of the entrainment region commonly referred to as Arnold tongue (see Fig.…”

Section: Analysis Of Temporary Synchronization In Tcl Ensemblesmentioning

confidence: 99%

“…where v is 2π-periodic with unit energy [47]. The region of existence of a synchronized state is called Arnold tongue [49,54].…”

Section: Appendix C: Entrainment Regionmentioning

confidence: 99%

“…Without loss of generality, let A = 1 and consider the 1 : 1 entrainment i.e., n = m = 1 and write Γ m/n (ψ) as simply Γ(ψ). It can be shown that when the condition min Γ(ψ) < −∆ < max Γ(ψ), is satisfied, (C2) has at least two fixed points at which dψ(t)/dt = 0 holds, and one of them is stable [29,47]. The interval ∆ of phase locking for a fixed input strength A decreases as A → 0.…”

Section: Appendix C: Entrainment Regionmentioning

confidence: 99%

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A phase model approach for thermostatically controlled load demand response

Bomela

Zlotnik

2018

Applied Energy

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A significant portion of electricity consumed worldwide is used to power thermostatically controlled loads (TCLs) such as air conditioners, refrigerators, and water heaters. Because the shortterm timing of operation of such systems is inconsequential as long as their long-run average power consumption is maintained, they are increasingly used in demand response (DR) programs to balance supply and demand on the power grid. Here, we present an ab initio phase model for general TCLs, and use the concept to develop a continuous oscillator model of a TCL and compute its phase response to changes in temperature and applied power. This yields a simple control system model that can be used to evaluate control policies for modulating the power consumption of aggregated loads with parameter heterogeneity and stochastic drift. We demonstrate this concept by comparing simulations of ensembles of heterogeneous loads using the continuous state model and an established hybrid state model. The developed phase model approach is a novel means of evaluating DR provision using TCLs, and is instrumental in estimating the capacity of ancillary services or DR on different time scales. We further propose a novel phase response based open-loop control policy that effectively modulates the aggregate power of a heterogeneous TCL population while maintaining load diversity and minimizing power overshoots. This is demonstrated by low-error tracking of a regulation signal by filtering it into frequency bands and using TCL sub-ensembles with duty cycles in corresponding ranges. Control policies that can maintain a uniform distribution of power consumption by aggregated heterogeneous loads will enable distribution system management (DSM) approaches that maintain stability as well as power quality, and further allow more integration of renewable energy sources.

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Section: Analysis Of Temporary Synchronization In Tcl Ensemblesmentioning

confidence: 99%

“…where v is 2π-periodic with unit energy [47]. The region of existence of a synchronized state is called Arnold tongue [49,54].…”

Section: Appendix C: Entrainment Regionmentioning

confidence: 99%

Section: Appendix C: Entrainment Regionmentioning

confidence: 99%

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A phase model approach for thermostatically controlled load demand response

Bomela

Zlotnik

2018

Applied Energy

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“…The phase reduction theory, originally developed for finite-dimensional smooth limit-cycle oscillators, has recently been generalized to non-conventional limitcycling systems such as collectively oscillating populations of coupled oscillators [26], systems with time delay [27][28][29], reaction-diffusion systems [30], oscillatory fluid convection [31], and hybrid dynamical systems [32]. Recently, methods for optimizing periodic external driving signals for efficient injection locking and controlling of a single nonlinear oscillator (or a population of uncoupled oscillators) have also been proposed on the basis of the phase reduction theory [33][34][35][36][37][38][39][40][41][42][43][44]. In this study, we consider a pair of coupled limit-cycle oscillators and try to optimize the linear stability of the synchronized state * nakao@mei.titech.ac.jp (corresponding author) using the phase reduction theory.…”

Section: Introductionmentioning

confidence: 99%

Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling

et al. 2017

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We consider optimization of linear stability of synchronized states between a pair of weakly coupled limit-cycle oscillators with cross coupling, where different components of state variables of the oscillators are allowed to interact. On the basis of the phase reduction theory, the coupling matrix between different components of the oscillator states that maximizes the linear stability of the synchronized state under given constraints on overall coupling intensity and on stationary phase difference is derived. The improvement in the linear stability is illustrated by using several types of limit-cycle oscillators as examples.

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“…Regarding optimization of synchronization, optimal input signals that efficiently entrains a limit-cycle oscillator described by ordinary differential equations have been obtained for various situations [33][34][35][36][37][38][39][40][41][42]. Also, in our preceding article, we derived optimal cross-coupling matrices that maximize linear stability of the synchronized states in a pair of diffusively coupled limit-cycle oscillators described by ordinary differential equations [43].…”

Section: Introductionmentioning

confidence: 99%

Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems

et al. 2017

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Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case where the two systems are nonlocally coupled. The optimal nonlinear interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in FitzHugh-Nagumo systems as examples.

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Optimal entrainment with smooth, pulse, and square signals in weakly forced nonlinear oscillators

Cited by 33 publications

References 45 publications

A phase model approach for thermostatically controlled load demand response

A phase model approach for thermostatically controlled load demand response

Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling

Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems

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