Analysis of the dynamics and topology dependencies of small perturbations in electric transmission grids

Abstract

Through an eigenanalysis of small perturbations, as typically done in small-signal stability studies, we intend to discover the underlying reasons that make those perturbations propagate in some way or another in the grid. To this end, we establish connections between the perturbations time-scale and topological metrics. Namely, the algebraic connectivity and the Fiedler vector of a generalized/weighted Laplacian matrix that depends on the stationary phase solutions of the system and is thereby inherently conditioned by the topology and the power distribution. Then, we aim to find out the isolated influence of topology on the perturbations when the network interacting agents have, in principle, opposite behaviors (i.e. producers and consumers). To do so, we study three networks: Small-world, Random, German grid. Furthermore, we tackle the effect of machine clustering on small perturbations and the influence of the network's average clustering coefficient on the intensity localization of the generalized Fiedler vector. Finally, we propose ways in which future (dynamic topology control) and existing (power system stabilizer) grid control strategies can adapt their response to comprehensively consider the topology and remote signals in the system.

Propagation of Disturbances in AC Electricity Grids

“…The dimensionless disturbance function is defined by δΠ i (t) = JδP i (t)/(γ 2 ω). We expand the phase deviation α i (t) and the disturbance δΠ i (t) in a generalized Fourier series in terms of the Eigenvectors φ n of Λ, as defined by Λφ n = Λ n φ n , where Λ n is its Eigenvalue [8,16,15,14], see Suppl. III for a detailed derivation.…”

“…Thus, we can write the phase deviation α i (t) as a generalized Fourier series by writing its time dependence as a Fourier integral, and expanding its spatial dependence in terms of the Eigenvectors φ n of the generalized Laplace operator Λ, defined by Λφ n = Λ n φ n , where Λ n are its Eigenvalues [15,8,14]. Thereby we obtain [8,16]…”

“…Thus, the initially localized perturbation spreads diffusively with diffusion constant D = v 2 0 τ /2 = Π K (1 − σ 2 /σ 2 c ) 1/2 a 2 /(2τ ). (16) Diffusion causes very slow power law relaxation of the disturbance at the initial site, and an initial increase, followed by a slow power law decay at other sites. The resulting power law relaxation of the change in transmitted power between nodes k and l is then obtained to be [8] δF kl (t) = ±δP A kl π 2 a 2 ω 0 Dt 2 exp − (r i − r l ) 2 4Dt .…”

Transient Dynamics and Propagation of Voltage and Frequency Fluctuations in Transmission Grids

“…The spatial profile of the lowest Eigenmodes (Fiedler mode) of power grid graphs was analysed previously in Refs. [25] and [36] as reviewed in Ref. [5] and [37].…”