Graph Laplacian Spectrum and Primary Frequency Regulation

Abstract

We present a framework based on spectral graph theory that captures the interplay among network topology, system inertia, and generator and load damping in determining the overall grid behavior and performance. Specifically, we show that the impact of network topology on a power system can be quantified through the network Laplacian eigenvalues, and such eigenvalues determine the grid robustness against low frequency disturbances. Moreover, we can explicitly decompose the frequency signal along scaled Laplacian eigenvectors when damping-inertia ratios are uniform across buses. The insight revealed by this framework partially explains why load-side participation in frequency regulation not only makes the system respond faster, but also helps lower the system nadir after a disturbance. Finally, by presenting a new controller specifically tailored to suppress high frequency disturbances, we demonstrate that our results can provide useful guidelines in the controller design for load-side primary frequency regulation. This improved controller is simulated on the IEEE 39-bus New England interconnection system to illustrate its robustness against high frequency oscillations compared to both the conventional droop control and a recent controller design.

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

“…where f α = 4λ α − γ 2 and P α = i u αi δP i /m 1/2 i . This result generalizes Theorem III.3 of [14].…”

“…The spectral decomposition approach used here has recently drawn the attention of a number of groups and has been used to calculate performance measures in power grids and consensus algorithms e.g. in [7], [13], [14], [15].…”

Global Analysis of Synchronization Performance for Power Systems: Bridging the Theory-Practice Gap

“…• We have only provided here the solution ofw(t) for the (practically more relevant) under-damped case. • Interestingly, (35) shows that the system may become overdamped by either increasing m, or decreasing m! However, the behavior is different for each case: in the very high inertia case the Nadir disappears; whereas when m goes to zero, there is an overshoot in the overdamped response.…”

“…Given a power system under Assumption 2 with generators containing first order turbine dynamics (g i (s) given by (4)). Then under the under-damped condition (35), the Nadir of the system frequency w(t) is given by…”

“…Given a power system under Assumption 2 with generators containing first order turbine dynamics (g i (s) given by (4)). Then under the under-damped condition (35), the maximum frequency deviation ||w|| ∞ is a decreasing function of m, i.e., ∂ ∂m ||w|| ∞ < 0. We now turn to the maximum rate of change of frequency.…”