Generalized Spectral Clustering of Low-Inertia Power Networks

TL;DR

The paper addresses partitioning low-inertia power networks into dynamically coherent zones to enable scalable distributed control. It develops a generalized spectral clustering framework built on the generalized eigenproblem $(\tilde{L},D)$ derived from the linearized non-uniform Kuramoto dynamics, and then embeds nodes via the associated eigenvectors for clustering with $k$-means. The approach balances dynamic coupling and damping across clusters, demonstrated on the IEEE 30-bus system with robustness to operating-point variations and potential applications in distributed control and model reduction. This yields a practical, scalable method for structuring control and communication in grids with high penetrations of DERs.

Abstract

Large-scale integration of distributed energy resources has led to a rapid increase in the number of controllable devices and a significant change in system dynamics. This has necessitating the shift towards more distributed and scalable control strategies to manage the increasing system complexity. In this work, we address the problem of partitioning a low-inertia power network into dynamically coherent subsystems to facilitate the utilization of distributed control schemes. We show that an embedding of the power network using the spectrum of the linearized synchronization dynamics matrix results in a natural decomposition of the network. We establish the connection between our approach and the broader framework of spectral clustering using the Laplacian matrix of the admittance network. The proposed method is demonstrated on the IEEE 30-bus test system, and numerical simulations show that the resulting clusters using our approach are dynamically coherent. We consider the robustness of the clusters identified in the network by analyzing the sensitivity of the small eigenvalues and their corresponding eigenspaces, which determines the coherency structure of the oscillator dynamics, to variations in the steady-state operating points of the network.

Figures (8)

• Figure 1: $2$-D Embedding of the IEEE 30-bus test network using the (a) eigenvectors of $\tilde{L}$, (b) the generalized eigenvectors of $(\tilde{L}, D)$. The two clusters identified by running $k$-means on the respective embeddings of the network are highlighted. The total edge weights cut by the spectral and generalized spectral clustering solutions on $\tilde{\mathcal{G}}$, are 5.04 and 12.55, respectively, and the corresponding total damping in each cluster is 157.89 and 30.01 for spectral clustering, and 91.33 and 96.57 for generalized spectral clustering.
• Figure 2: Relative spectral gap of $\tilde{L}x = \lambda D x$.
• Figure 3: The dynamic graph $\tilde{G}$ of the IEEE case$30$ showing five clusters. Note that the position of the nodes on the graph do not reflect the physical proximity of the buses in the network.
• Figure 4: Total edges cut and total cluster damping for different values of $k$.
• Figure 5: The optimal value $\rho^*(k)$ of (\ref{['eqn:clustering_opt']}) and objective value $\hat{\rho}(k)$ of the generalized spectral clustering solution for different values of $k$.