Grouping of $N-1$ Contingencies for Controller Synthesis: A Study for Power Line Failures

TL;DR

The study tackles maintaining power-system stability after any single-line failure ($N-1$ contingencies) by partitioning contingencies into groups with similar control dynamics and designing one centralized controller per group. It introduces distance metrics (FR, SR, PSN) and clustering methods (k-centers, k-medoids, divisive) to form groups, then optimizes each group's controller to minimize the worst-case transfer norm $\|\mathcal{F}(P_i,K)\|_{\mathcal{H}_\infty}$ within that group. Through simulations on IEEE 39-bus and 68-bus systems under $\mathcal{H}_\infty$ and $\mathcal{H}_2$ control, the approach achieves near-optimal performance with far fewer controllers than contingency-specific designs and reveals severe contingencies for targeted analysis. The method provides a practical, offline-online framework to balance computation time and stability performance in dynamic power networks, with potential extensions to other contingency types and distributed energy resources.

Abstract

The problem of maintaining power system stability and performance after the failure of any single line in a power system (an "N-1 contingency") is investigated. Due to the large number of possible N-1 contingencies for a power network, it is impractical to optimize controller parameters for each possible contingency a priori. A method to partition a set of contingencies into groups of contingencies that are similar to each other from a control perspective is presented. Design of a single controller for each group, rather than for each contingency, provides a computationally tractable method for maintaining stability and performance after element failures. The choice of number of groups tunes a trade-off between computation time and controller performance for a given set of contingencies. Results are simulated on the IEEE 39-bus and 68-bus systems, illustrating that, with controllers designed for a relatively small number of groups, power system stability may be significantly improved after an N-1 contingency compared to continued use of the nominal controller. Furthermore, performance is comparable to that of controllers designed for each contingency individually.

Figures (11)

• Figure 1: Illustration of this grouping method on a 3 generator system considering 3 possible line failures, and grouping into 2 groups. Pairwise distances between perturbed systems are computed, followed by clustering of the perturbed systems, and finally controller design for each group of systems.
• Figure 2: The IEEE 39-bus system with lines that may fail labeled. Only single-line failures that do not disconnect the grid are considered.
• Figure 3: Comparison of metrics and clustering algorithms on IEEE 39-bus system by scaled $\mathcal{H}_\infty$ norm averaged over contingencies versus number of groups. FR is the frequency response metric, SR is the step response metric, and PSN is the perturbation spectral norm metric.
• Figure 4: Scaled $\mathcal{H}_\infty$ norm of each contingency when applying the step response metric and k-medoids clustering algorithm for various numbers of groups. The "nominal" column is the scaled $\mathcal{H}_\infty$ norm when applying the nominal controller to each contingency.
• Figure 5: Scaled $\mathcal{H}_\infty$ norm of each contingency for various numbers of groups using step response metric and k-medoids clustering. The scaled $\mathcal{H}_\infty$ norm of the nominal controller is cutoff above 1.1.