Performance Measures in Electric Power Networks Under Line Contingencies

Abstract

Classes of performance measures expressed in terms of H2-norms have been recently introduced to quantify the response of coupled dynamical systems to external perturbations. So far, investigations of these performance measures have been restricted to nodal perturbations. Here, we go beyond these earlier works and consider the equally important, but so far neglected case of line perturbations. We consider a networkreduced power system, where a Kron reduction has eliminated passive buses. Identifying the effect that a line fault in the physical network has on the Kron-reduced network, we find that performance measures depend on whether the faulted line connects two passive, two active buses or one active to one passive bus. In all cases, performance measures depend quadratically on the original load on the faulted line times a topology dependent factor. Our theoretical formalism being restricted to Dirac-δ perturbations, we investigate numerically the validity of our results for finite-time line faults. For uniform damping over inertia ratios, we find good agreement with theoretical predictions for longer fault durations in systems with more inertia, for which eigenmodes of the network are harder to excite.

The key player problem in complex oscillator networks and electric power grids: Resistance centralities identify local vulnerabilities

“…They are similar to performance measures based on L 2 -norms previously considered in the context of electric power networks [27,[36][37][38][39][40] but differ from them in two respects. First, here we subtract the averages ∆(t) = n −1 j δθ j (t) and∆(t) = n −1 j δθ j (t) because the synchronous state does not change under a constant angle shift.…”

Optimal Placement of Inertia and Primary Control: A Matrix Perturbation Theory Approach

“…When the damping ratio is constant, d i /m i = γ i = γ, ∀i, (2) can be integrated exactly [7], [12]. To see this we first transform angle coordinates as δθ = M −1/2 δθ M to obtaiṅ…”

“…In the context of electric power grids, the method was applied for instance in Ref. [12], where quadratic performance measures similar to those discussed below were calculated following a line fault, starting from the eigenvalues and eigenvectors of the network Laplacian before the fault.…”

Transient Performance of Electric Power Networks Under Colored Noise

“…Frequency nadir and RoCoF however only partially characterize the transient excursion away from the previously operating synchronous state. Inspired by consensus and synchronization studies [3], [4], [5], [6], recent works have introduced new transient performance measures following disturbances on busses [7], [8], [9], [10], [11] and on power lines [12]. The method is particularly appealling because (i) these performance measures have a clear physical meaning, quantifying the additional ohmic losses [7], [8], [9], the primary control effort [10], [12] originating from the transient or the phase coherence of the grid [3], [4], [12]; (ii) they are integrated quadratic forms in phase or frequency excursions that can be expressed as L 2 norms of the system's T. Coletta output.…”

“…Technically speaking, the advantage of the L 2 -based approach over infinity norm measures (frequency nadir and RoCoF) is not only that it is mathematically tractable, but also that it provides combined information on both the amplitude and the duration of the transient in both voltage phases and frequencies. Its drawback is that so far it has been applied to Dirac-δ impulse disturbances only [7], [8], [9], [10], [11], [12], which is equivalent to considering long time output variances for stochastic white noise power demand/generation fluctuations. Fluctuations of photovoltaic or wind turbine power generation have however finite correlation times [14], typically on the order of minutes or more, i.e.…”