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Fault Localisation in Infinite-Dimensional Linear Electrical Networks

Daniel Selvaratnam, Alessio Moreschini, Amritam Das, Thomas Parisini, Henrik Sandberg

Abstract

We present a novel fault localisation methodology for linear time-invariant electrical networks with infinite-dimensional edge dynamics and uncertain fault dynamics. The theory accommodates instability and also bounded propagation delays in the network. The goal is to estimate the location of a fault along a given network edge, using sensors positioned arbitrarily throughout the network. Passive faults of unknown impedance are considered, along with stable faults of known impedance. To illustrate the approach, we tackle a significant use-case: a multi-conductor transmission line, with dynamics modelled by the Telegrapher's equation, subject to a line-to-ground fault. Frequency-domain insights are used to reformulate the general fault localisation problem into a non-convex scalar optimisation problem, of which the true fault location is guaranteed to be a global minimiser. Numerical experiments are run to quantify localisation performance over a range of fault resistances.

Fault Localisation in Infinite-Dimensional Linear Electrical Networks

Abstract

We present a novel fault localisation methodology for linear time-invariant electrical networks with infinite-dimensional edge dynamics and uncertain fault dynamics. The theory accommodates instability and also bounded propagation delays in the network. The goal is to estimate the location of a fault along a given network edge, using sensors positioned arbitrarily throughout the network. Passive faults of unknown impedance are considered, along with stable faults of known impedance. To illustrate the approach, we tackle a significant use-case: a multi-conductor transmission line, with dynamics modelled by the Telegrapher's equation, subject to a line-to-ground fault. Frequency-domain insights are used to reformulate the general fault localisation problem into a non-convex scalar optimisation problem, of which the true fault location is guaranteed to be a global minimiser. Numerical experiments are run to quantify localisation performance over a range of fault resistances.

Paper Structure

This paper contains 23 sections, 13 theorems, 78 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Nomenclature
  3. Sets and numbers
  4. Linear algebra
  5. Functions
  6. Problem Formulation
  7. Network modelling
  8. Fault localisation
  9. Admittance modelling
  10. Transmission line element
  11. Growth bounds
  12. Admittance matrix composition
  13. Growth bounds
  14. Localisation algorithm
  15. Algebraic analysis
  16. ...and 8 more sections

Key Result

Lemma 1

The tuple $(i_1,v_2,v_4,i_4)$ solves Problem prob:port if and only if

Figures (4)

  • Figure 1: Four-port LTI Network $\mathcal{N}$ with fault.
  • Figure 2: Diagram of Case Study \ref{['ex:tline']}, an illustrative use-case.
  • Figure 3: Variation of cost functions with fault resistance: fixed fault location $\ell = 7.7$ km, and $R_f$ logarithmically spaced from $10^{-1} \ \Omega$ to $10^4 \ \Omega$.
  • Figure 4: Effect of unknown fault resistance on localisation performance.

Theorems & Definitions (22)

  • Definition 1: Exponential order signals
  • Definition 2: H-infinity
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Corollary 1: Uniqueness of estimates
  • Definition 3: Fault signal estimator
  • Theorem 1
  • Lemma 3
  • Remark 2: Sensing information
  • ...and 12 more