Discovering Clusters in Power Networks From Orthogonal Structure of Spectral Embedding

Abstract

This paper presents an integrated approach to parti-5 tion similarity graphs, the task that arises in various contexts in 6 power system studies. The approach is based on orthogonal trans-7 formation of row-normalized eigenvectors obtained from spectral 8 clustering to closely fit the axes of the canonical coordinate system. 9 We select the number of clusters as the number of eigenvectors 10 that allows the best alignment with the canonical coordinate axes, 11 which is a more informative approach than the popular spectral 12 eigengap heuristic. We show a link between the two relevant meth-13 ods from the literature and on their basis construct a robust and 14 time-efficient algorithm for eigenvector alignment. Furthermore, 15 a graph partitioning algorithm based on the use of aligned eigen-16 vector columns is proposed, and its efficiency is evaluated by com-17 parison with three other methods. Finally, the proposed integrated 18 approach is applied to the adaptive reconfiguration of secondary 19 voltage control helping to achieve demonstrable improvements in 20 control performance. 21 Index Terms-Power network partitioning, spectral clustering, 22 number of clusters, adaptive network zone division.23

Slow Coherency Identification and Power System Dynamic Model Reduction by Using Orthogonal Structure of Electromechanical Eigenvectors

“…Orthogonal linear transformations can be used for the alignment of spectral embeddings with the canonical coordinate system of R k to facilitate the group assignment process (e.g., see Figure 1). In [22], we have proposed a robust algorithm to align normalized spectral embeddings with the standard basis. A normalized spectral embedding corresponds to the matrix X that is obtained from Z by normalizing the rows of Z to length one:…”

“…2) Align the normalized spectral embedding in X with the standard basis using the robust alignment algorithm from [22]. Store the final aligning orthogonal matrix as R * .…”

“…The first step of the new slow coherency grouping algorithm shares similarities with the cluster core estimation process from our previous work [22]. To avoid repetition, this section mostly focuses on the crucial differences of the group cores estimation process that is summarized as Algorithm 1 from the previously proposed cluster cores estimation process.…”

“…For Algorithm 1, the best results are obtained with γ being set to the lowest reliable value of √ 2/2. As γ is directly set to √ 2/2, the gains achievable by the cluster core refinement described in [22] diminish. Consequently, the refinement stage is not used to simplify the algorithm.…”

“…Our findings show that Ncut values close to the global minimum are crucial for the high fidelity of slow coherency groupings. Because of this, some elements of our previously published highly efficient Ncut minimization approach [22] are adapted from minimizing standard Ncuts in sparse graphs to minimizing generic Ncuts in complete graphs that are related to slow coherency. Besides the precise slow coherency grouping, we propose an improvement to the inertial aggregation algorithm (see Section VI-A) to reduce its so-called stiffening effect [18], which also helps to further check the validity of coherency groupings through nonlinear dynamic reduction case studies.…”