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Tracking disturbances in transmission networks

Jean-Guy Caputo, Adel Hamdi

Abstract

We study the nonlinear inverse source problem of detecting, localizing and identifying unknown accidental disturbances on forced and damped transmission networks. A first result is that strategic observation sets are enough to guarantee detection of disturbances. To localize and identify them, we additionally need the observation set to be absorbent. If this set is dominantly absorbent, then detection, localization and identification can be done in "quasi real-time". We illustrate these results with numerical experiments.

Tracking disturbances in transmission networks

Abstract

We study the nonlinear inverse source problem of detecting, localizing and identifying unknown accidental disturbances on forced and damped transmission networks. A first result is that strategic observation sets are enough to guarantee detection of disturbances. To localize and identify them, we additionally need the observation set to be absorbent. If this set is dominantly absorbent, then detection, localization and identification can be done in "quasi real-time". We illustrate these results with numerical experiments.

Paper Structure

This paper contains 24 sections, 6 theorems, 89 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical modeling and problem statement
  3. Detection of accidental disturbances
  4. Strategic set of network vertices
  5. Procedure for detecting disturbances
  6. Inversion using dominantly absorbent observations
  7. Definition and practical guidelines
  8. Identifiability-Identification of detected disturbances
  9. Inversion using absorbent observations
  10. Orthogonal family of Legendre polynomials
  11. Localization-Identification of detected disturbances
  12. Localization of detected disturbances
  13. Identification of detected-localized disturbances
  14. Numerical experiments
  15. Determination of the healthy network state
  16. ...and 9 more sections

Key Result

Theorem 3.2

Let $X_0\in I\!\!R^N$ and $\bar{X}_0\in I\!\!R^N$ be unknowns and $T^0\in(0,T)$. Provided ${\cal{S}}$ is a strategic set of vertices, if the solution $X^{R}(t)=(x^{R}_1(t),\dots,x^{R}_N(t))^{\top}$ of the problem: fulfills $x^{R}_n(t)=0, \forall t\in (0,T^0), \forall n\in {\cal{S}}$, then the initial conditions $X_0=\bar{X}_0=0$ in $I\!\!R^N$.

Figures (5)

  • Figure 1: A graph with a joint.
  • Figure 2: A graph with five vertices.
  • Figure 3: Left panel: reconstructed disturbances in vertices 2 and 3, Right panel: reconstructed and exact disturbances in vertex 2.
  • Figure 4: Reconstructed Residuals $x_n^R(t)$ for disturbances located in vertex: 2(left), 3(middle) and 5(right).
  • Figure 5: Reconstruction of disturbances $F_m^{dis}(t)$ located in vertex: 2(left), 3(middle) and 5(right).

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Definition 4.1
  • Definition 4.3
  • Theorem 4.4
  • Theorem 4.5
  • ...and 4 more