Interpolatory Approximations of PMU Data: Dimension Reduction and Pilot Selection

TL;DR

The paper addresses real-time PMU data management by introducing interpolatory matrix decompositions (IDs) to compress PMU data using a small set of pilot rows/columns. It integrates the discrete empirical interpolation method (DEIM) to adaptively select pilots and establish computable error bounds that guide online monitoring and fault detection. The ID-DEIM framework enables significant bandwidth reduction while preserving reconstruction quality and supports online detection and localization of disturbances, demonstrated on synthetic PMU data. This approach offers a mathematically grounded, scalable path to bandwidth-efficient, data-driven wide-area monitoring and rapid fault localization in power systems.

Abstract

This work investigates the reduction of phasor measurement unit (PMU) data through low-rank matrix approximations. To reconstruct a PMU data matrix from fewer measurements, we propose the framework of interpolatory matrix decompositions (IDs). In contrast to methods relying on principal component analysis or singular value decomposition, IDs recover the complete data matrix using only a few of its rows (PMU datastreams) and/or a few of its columns (snapshots in time). This compression enables the real-time monitoring of power transmission systems using a limited number of measurements, thereby minimizing communication bandwidth. The ID perspective gives a rigorous error bound on the quality of the data compression. We propose selecting rows and columns used in an ID via the discrete empirical interpolation method (DEIM), a greedy algorithm that aims to control the error bound. This bound leads to a computable estimate for the reconstruction error during online operations. A violation of this estimate suggests a change in the system's operating conditions, and thus serves as a tool for fault detection. Numerical tests using synthetic PMU data illustrate DEIM's excellent performance for data compression, and validate the proposed DEIM-based fault-detection method.

Figures (5)

• Figure 1: Sketch of four low-rank approximations ($\mathbf{Y}_{K}$, $\mathbf{Y}_{\mathcal{S}}$, $\mathbf{Y}^{\mathcal{T}}$, and $\mathbf{Y}^{\mathcal{T}}_{\mathcal{S}}$) to the PMU data matrix $\mathbf{Y}$. The bottom three are IDs.
• Figure 2: Relative errors for rank $k=2,3\ldots,20$ interpolatory matrix approximations $\mathbf{Y}_{\mathcal{S}}$ and $\mathbf{Y}^{\mathcal{T}}$ of the data matrix $\mathbf{Y} \in\mathop{\mathrm{\mathbb{R}}}\nolimits^{68\times 6000}$ generated using the $68$-bus, $16$-machine NETSNYPS test system. The size of the column-based data prevents use of MILP in the right plot.
• Figure 3: Evolution of the error factor $\eta_{\mathcal{S}}$ and the upper bound $\eta_{\mathcal{S}}\sigma_{K+1}$ throughout the adaptive training as $K$ pilots are chosen.
• Figure 4: Interpolatory reconstruction and true data for the non-pilot datastreams at bus $57$ (left) and bus $63$ (right) prior to and during a three-phase fault of the line between buses $28$ and $29$.
• Figure 5: Percentage of event simulations (83 experiments) in which the indicated method correctly identified both buses associated with the faulted line in the first $k$ indices for $k=2,\ldots,8$.