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Complex-Value Spatio-temporal Graph Convolutional Neural Networks and its Applications to Electric Power Systems AI

Tong Wu, Anna Scaglione, Daniel Arnold

TL;DR

It is proved that complex-valued GCNs can be stable with respect to perturbations of the underlying graph support, by bounding of the error propagation through multiple NN layers, and is generalized to handle spatio-temporal complex network processes.

Abstract

The effective representation, precessing, analysis, and visualization of large-scale structured data over graphs are gaining a lot of attention. So far most of the literature has focused on real-valued signals. However, signals are often sparse in the Fourier domain, and more informative and compact representations for them can be obtained using the complex envelope of their spectral components, as opposed to the original real-valued signals. Motivated by this fact, in this work we generalize graph convolutional neural networks (GCN) to the complex domain, deriving the theory that allows to incorporate a complex-valued graph shift operators (GSO) in the definition of graph filters (GF) and process complex-valued graph signals (GS). The theory developed can handle spatio-temporal complex network processes. We prove that complex-valued GCNs are stable with respect to perturbations of the underlying graph support, the bound of the transfer error and the bound of error propagation through multiply layers. Then we apply complex GCN to power grid state forecasting, power grid cyber-attack detection and localization.

Complex-Value Spatio-temporal Graph Convolutional Neural Networks and its Applications to Electric Power Systems AI

TL;DR

It is proved that complex-valued GCNs can be stable with respect to perturbations of the underlying graph support, by bounding of the error propagation through multiple NN layers, and is generalized to handle spatio-temporal complex network processes.

Abstract

The effective representation, precessing, analysis, and visualization of large-scale structured data over graphs are gaining a lot of attention. So far most of the literature has focused on real-valued signals. However, signals are often sparse in the Fourier domain, and more informative and compact representations for them can be obtained using the complex envelope of their spectral components, as opposed to the original real-valued signals. Motivated by this fact, in this work we generalize graph convolutional neural networks (GCN) to the complex domain, deriving the theory that allows to incorporate a complex-valued graph shift operators (GSO) in the definition of graph filters (GF) and process complex-valued graph signals (GS). The theory developed can handle spatio-temporal complex network processes. We prove that complex-valued GCNs are stable with respect to perturbations of the underlying graph support, the bound of the transfer error and the bound of error propagation through multiply layers. Then we apply complex GCN to power grid state forecasting, power grid cyber-attack detection and localization.
Paper Structure (29 sections, 6 theorems, 66 equations, 4 figures, 3 tables)

This paper contains 29 sections, 6 theorems, 66 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Contributions
  4. Preliminaries on Power Systems
  5. Complex-Valued Spatio-Temporal Graph Convolution Neural Network
  6. Grid-Graph Signal Processing
  7. Complex-valued Graph Convolution Neural Network
  8. Discussion about Cplx-STGCN vs Real-STGCN
  9. Analysis of the Cplx-GCN sensitivity
  10. Spatiotemporal Graph Convolutional Neural Network
  11. Applications of Cplx-STGCN
  12. Power System State Estimation and Forecasting
  13. Methodology
  14. False Data Detection and Localization
  15. Methodology
  16. ...and 14 more sections

Key Result

Theorem 1

Consider graph filter $\bm{h} = [h_0, \cdots, h_{K}]$ along with shift operator $\bf{S}$ having $\lvert\mathcal{V}\rvert$ nodes. Let $\mathbf{E} \in \mathbb{C}^{\lvert\mathcal{V}\rvert\times \lvert\mathcal{V}\rvert}$ denote the matrix perturbation with $\left\lVert\mathbf{E}\right\rVert \le {\epsil The following bound holds:

Figures (4)

  • Figure 1: The Architecture of Cplx-STGCN.
  • Figure 2: Power System State Estimation and Forecasting in the IEEE 118-bus system
  • Figure 3: False Data Injection Localization in the IEEE 118-bus system.
  • Figure 4: An example of PSSE and forecasting.

Theorems & Definitions (11)

  • Theorem 1
  • Proof 1
  • Theorem 2
  • Proof 2
  • Corollary 1: The Bound of Cplx-GCN Perturbation
  • Remark 1
  • Remark 2
  • Lemma 1
  • Proof 3
  • Corollary 2
  • ...and 1 more