Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topology Reconstruction of a Class of Electrical Networks with Limited Boundary Measurements

Shivanagouda Biradar, Deepak U Patil

TL;DR

This work addresses reconstructing the topology and edge conductances of unknown circular planar passive electrical networks from limited boundary Thevenin impedances by linking $z^{th}_{i,j}$ to the Laplacian $\\mathcal{L}_{\\Gamma}$ and formulating a multivariate polynomial system solvable with Gröbner bases. The approach imposes triangle and Kalmanson inequalities to constrain the solution set and introduces a robust method to handle missing boundary data via the construction of a feasible set $\\widehat{\\mathcal{C}}_{\\mathcal{K}}$ using augmented problems. Theoretical contributions show that these inequalities hold for general RL and RC networks under convex-cone boundary conditions, and the algorithm yields all networks consistent with the measurements, as demonstrated on a 5-node numerical example. This topology-reconstruction framework enables set characterization and fault-detection-oriented analysis in scenarios with partial boundary access, with potential extensions to noisy data and broader RL/RC classes.

Abstract

We consider the problem of recovering the topology and the edge conductance value, as well as characterizing a set of electrical networks that satisfy the limitedly available Thevenin impedance measurements. The measurements are obtained from an unknown electrical network, which is assumed to belong to a class of circular planar passive electrical network. This class of electrical networks consists of R, RL, and RC networks whose edge impedance values are equal, and the absolute value of the real and the imaginary part of the edge impedances are also equal. To solve the topology reconstruction and the set characterization problem, we establish a simple relation between Thevenin impedance and the Laplacian matrix and leverage this relation to get a system of multivariate polynomial equations, whose solution is a set of all electrical networks satisfying the limited available Thevenin's impedance measurements. To confine the search space and generate valid electrical networks, we impose the triangle and Kalmanson's inequality as constraints. The solution to a constrained system of multivariate polynomial equations is a set of reconstructed valid electrical networks. For simple algorithmic solutions, we use Gröbner basis polynomials. This paper shows that the triangle and the Kalmanson's inequality holds for general circular planar passive R, RL, and RC electrical networks if certain boundary conditions lie within a convex cone. Numerical examples illustrate the developed topology reconstruction method.

Topology Reconstruction of a Class of Electrical Networks with Limited Boundary Measurements

TL;DR

This work addresses reconstructing the topology and edge conductances of unknown circular planar passive electrical networks from limited boundary Thevenin impedances by linking to the Laplacian and formulating a multivariate polynomial system solvable with Gröbner bases. The approach imposes triangle and Kalmanson inequalities to constrain the solution set and introduces a robust method to handle missing boundary data via the construction of a feasible set using augmented problems. Theoretical contributions show that these inequalities hold for general RL and RC networks under convex-cone boundary conditions, and the algorithm yields all networks consistent with the measurements, as demonstrated on a 5-node numerical example. This topology-reconstruction framework enables set characterization and fault-detection-oriented analysis in scenarios with partial boundary access, with potential extensions to noisy data and broader RL/RC classes.

Abstract

We consider the problem of recovering the topology and the edge conductance value, as well as characterizing a set of electrical networks that satisfy the limitedly available Thevenin impedance measurements. The measurements are obtained from an unknown electrical network, which is assumed to belong to a class of circular planar passive electrical network. This class of electrical networks consists of R, RL, and RC networks whose edge impedance values are equal, and the absolute value of the real and the imaginary part of the edge impedances are also equal. To solve the topology reconstruction and the set characterization problem, we establish a simple relation between Thevenin impedance and the Laplacian matrix and leverage this relation to get a system of multivariate polynomial equations, whose solution is a set of all electrical networks satisfying the limited available Thevenin's impedance measurements. To confine the search space and generate valid electrical networks, we impose the triangle and Kalmanson's inequality as constraints. The solution to a constrained system of multivariate polynomial equations is a set of reconstructed valid electrical networks. For simple algorithmic solutions, we use Gröbner basis polynomials. This paper shows that the triangle and the Kalmanson's inequality holds for general circular planar passive R, RL, and RC electrical networks if certain boundary conditions lie within a convex cone. Numerical examples illustrate the developed topology reconstruction method.

Paper Structure

This paper contains 24 sections, 10 theorems, 62 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Network Structure
  4. Thevenin's Impedance & Laplacian Matrix
  5. Triangle & Kalmanson's Inequality on Networks in $\Upsilon$
  6. Ordering
  7. Triangle Inequality
  8. Kalmanson's Inequality
  9. Problem Formulation
  10. Construction of a set of Kalmanson's Inequalities
  11. Construction of $\widehat{\mathcal{C}}_{\mathcal{K}}$: A Combinatorial Approach
  12. Gröbner Basis
  13. Topology Reconstruction Algorithm
  14. Numerical Example
  15. Discussion
  16. ...and 9 more sections

Key Result

Theorem 1

Consider a circular planar passive electrical network $\Gamma=(\mathcal{G},\gamma )$, where $\Gamma \in \Gamma_{\beta}$. Let $\mathcal{G}=(\mathcal{V_B},\mathcal{E})$ be a simple, connected graph on $n$ vertices, where $n\ge3$, corresponding to $\Gamma$. Then, the Thevenin's impedance $z^{th}_{j,k}$

Figures (5)

  • Figure 1: Computing Thevenin's impedance $z^{th}_{j,k}$.
  • Figure 2: A flow of Topology reconstruction algorithm for constructing set $\mathcal{N}^{i}$.
  • Figure 3: A valid electrical network $\widehat{\Gamma}$ in set $\mathcal{N}^1$
  • Figure 4: A valid electrical network in set $\mathcal{N}^1$
  • Figure 5: Kron reduced network $\mathcal{G}^{kron}$.

Theorems & Definitions (22)

  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Corollary 7
  • Remark 1
  • Corollary 8
  • ...and 12 more