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Simultaneous recovery of a sparse topology and the admittance of an electrical network

Álvaro Samperio

TL;DR

The paper tackles the problem of simultaneously recovering a network's topology and admittance from voltage and power data, revealing that the exact formulation is often ill-posed. It introduces a sparse-recovery reformulation and proves bounds on how sparsification affects data fitting for both DC and AC networks, enabling a robust iterative algorithm that alternates recovery and edge-sparsification. The proposed selfhealing algorithm leverages nonnegative least squares and spectral sparsification to produce sparse networks whose fit remains within a prescribed tolerance tol, even in the presence of data noise. Experimental results across DC and AC networks, including Kron reductions and notable test cases, show that the method can recover near-true topology with dramatically lower condition numbers, yielding computationally efficient sparse representations suitable for downstream tasks such as power-flow optimization and failure identification.

Abstract

We show that the problem of recovering the topology and admittance of an electrical network from power and voltage data at all vertices is often ill-posed, and sometimes it even has multiple solutions. We reformulate the problem to seek for a sparse network, i.e., with few edges, which fits the data up to a given tolerance. We propose an algorithm to solve this reformulated problem. It combines, in an iterative procedure, the resolution of non-negative linear regression problems, and techniques of spectral graph sparsification. The algorithm is based on original results bounding the fitting error of a sparse approximation of a network. We illustrate our techniques with several experimental results in which we are able to recover a sparse network.

Simultaneous recovery of a sparse topology and the admittance of an electrical network

TL;DR

The paper tackles the problem of simultaneously recovering a network's topology and admittance from voltage and power data, revealing that the exact formulation is often ill-posed. It introduces a sparse-recovery reformulation and proves bounds on how sparsification affects data fitting for both DC and AC networks, enabling a robust iterative algorithm that alternates recovery and edge-sparsification. The proposed selfhealing algorithm leverages nonnegative least squares and spectral sparsification to produce sparse networks whose fit remains within a prescribed tolerance tol, even in the presence of data noise. Experimental results across DC and AC networks, including Kron reductions and notable test cases, show that the method can recover near-true topology with dramatically lower condition numbers, yielding computationally efficient sparse representations suitable for downstream tasks such as power-flow optimization and failure identification.

Abstract

We show that the problem of recovering the topology and admittance of an electrical network from power and voltage data at all vertices is often ill-posed, and sometimes it even has multiple solutions. We reformulate the problem to seek for a sparse network, i.e., with few edges, which fits the data up to a given tolerance. We propose an algorithm to solve this reformulated problem. It combines, in an iterative procedure, the resolution of non-negative linear regression problems, and techniques of spectral graph sparsification. The algorithm is based on original results bounding the fitting error of a sparse approximation of a network. We illustrate our techniques with several experimental results in which we are able to recover a sparse network.
Paper Structure (7 sections, 3 theorems, 46 equations, 7 figures, 7 tables, 2 algorithms)

This paper contains 7 sections, 3 theorems, 46 equations, 7 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction and preliminaries
  2. Ill-posedness of the network recovery problem
  3. Reformulation of the problem: Recovery of a sparse electrical network
  4. Spectral network sparsification
  5. Sparsification of recovered electrical networks
  6. Algorithm for sparse network recovery
  7. Experimental results and discussion

Key Result

Theorem 4.3

Let $\Gamma$ be a DC network, let $\varepsilon \in {\mathbb{R}}$, $0 < \varepsilon \leq 1$ and let $\Gamma'= \text{\sffamily\normalshape Sparsify} (\Gamma,\varepsilon)$. Then $\Gamma'$ is an $\varepsilon$-approximation of $\Gamma$ with probability at least $1/2$.

Figures (7)

  • Figure 1: Electrical network in Example \ref{['triangulo']}.
  • Figure 2: Topology of the network in Example \ref{['ejemplo']}.
  • Figure 3: Topology of the CIGRE Network. (Example \ref{['ej']}).
  • Figure 4: Evolution of $\varepsilon$ with the iteration number. Example \ref{['ej']}.
  • Figure 5: Iteration number: $5$, $|E(\Gamma)|=21$. (Heawood graph network).
  • ...and 2 more figures

Theorems & Definitions (18)

  • Definition 1.1
  • Remark 1.2
  • Example 2.1
  • Remark 3.2
  • Definition 4.1: boletin
  • Definition 4.2
  • Theorem 4.3: Batson, Spielman, Srivastava, Teng
  • Example 4.4
  • Theorem 5.1: Main theorem
  • proof
  • ...and 8 more