Topological equivalence of a structure-preserving power network model and a non-uniform Kuramoto model of coupled oscillators

Abstract

Abstract-We study synchronization in the classic structurepreserving power network model proposed by Bergen and Hill. We find that, locally near the synchronization manifold, the phase and frequency dynamics of the power network model are topologically conjugate to the phase dynamics of a non-uniform Kuramoto model and decoupled exponentially stable dynamics for the frequencies. This topological conjugacy implies the equivalence of local exponential synchronization in power networks and in non-uniform Kuramoto oscillators. Hence, we can harness the results available for Kuramoto oscillators to analyze synchronization in power networks. We establish necessary and sufficient conditions for phase synchronization, sufficient conditions for frequency synchronization, and necessary and sufficient conditions for frequency synchronization with a uniform topology. These conditions also extend the results known for the classic first-order Kuramoto model and the second-order linear consensus protocols. Our synchronization conditions all share a common physical interpretation: the ratio of power inputs and dissipation has to be sufficiently uniform and the coupling in the network has to be sufficiently strong.

Distance-based undirected formations of single-integrator and double-integrator modeled agents in n -dimensional space

“…It has been revealed that parameterized systems of the form have the identical equilibrium set and the identical local stability properties for all λ ∈ [0,1] as stated in the following theorem: Theorem For the one‐parameter family of dynamical systems in , the following statements hold independently of the parameter λ ∈ [0,1]. Equilibrium set: For all λ ∈ [0,1], the equilibrium set of is given by the set of critical points of the potential function ψ , that is, E p , v = {[ p T v T ] T : ∇ ψ = 0}. Local stability: For any equilibrium [ p T v T ] T ∈ E p , v and for all λ ∈ [0,1], the numbers of the stable, neutral, and unstable eigenvalues of the Jacobian of are not dependent on λ .…”

“…Formation dynamics of the agents under the control law can be described as a dissipative Hamiltonian system. On the basis of the topological equivalence of the dissipative Hamiltonian system and a gradient system, which has been studied in [9], we show that the local asymptotic stability of the undirected formations of double-integrator modeled agents is achieved. Although the author of [10] has addressed the stability of undirected formations of double-integrator modeled agents by means of the LaSalle invariance principle, as pointed out in [1], it is not certain whether the principle can be applied because the equilibrium set of the formation dynamics is not compact.…”

Two-tier hierarchical cyber-physical security analysis framework for smart grid

“…IDENTIFICATION To identify generator coherency, we make use of insights recently presented by Dörfler and Bullo in [4] whereby it is shown via singular perturbation analysis that Eq. (1) is equivalent to:…”

Quasi-stationarity of electric power grid dynamics based on a spatially embedded Kuramoto model

“…Consequently the degrading of coupling strength with distance is lowered. The peak magnitude appears to be bounded despite further increase in R. Remark 7: Non-uniform Kuramoto dynamics (10) are related to those of network models for power systems via a toplogical equivlance only of the equilibria, not the full dynamics [14]. In that light the phenomenon of peaking becomes more interesting, because it develops on grounds of an "almost" (i.e.…”

“…Condition (26b) suggests quasi-stationarity as varying χ Ω (t) where in intervals n 2π ω ≤ t < (n + 1) 2π ω , n ∈ N attraction and repulsion balance up to a constant to yield arbitrary small oscillations as follows: Frequency entrainment corresponds to exponential synchronization to some bounded frequencyΘ s ∈ [Θ min ,Θ max ], and phase locking to positive invariance of the set of bounded phase differences Θ ∈ T N : max ik∈E |Θ ik | ≤ γ , see [5]. Assume exponentially small differences of the form Θ i − Θ i−k = Θ i − Θ i+k , and substituting into the lumped parameter dynamics (22) gives together with (14) for the sinusoidal phase interaction while the cosine interaction term expanded similarly results in…”