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Electric Grid Topology and Admittance Estimation: Quantifying Phasor-based Measurement Requirements

Norak Rin, Iman Shames, Ian R. Petersen, Elizabeth L. Ratnam

TL;DR

The paper tackles the problem of uniquely identifying electric grid topology and admittance parameters from phasor measurements without prior topology. It develops a theoretical link between the voltage-coefficient matrix used in current–voltage relations and a rigidity matrix from graph rigidity theory, deriving a rank-based identifiability condition. Under generic nodal voltages, it shows that the minimum number of measurements required is $τ=n-1$ for both DC conductance and AC admittance networks, with the edge parameters recoverable via a pseudoinverse when the condition holds. A numerical example on an IEEE 4-node network validates the bound, recovering a non-complete topology and nonzero edge admittances using exactly $τ=n-1$ measurements. This work provides a rigorous measurement-efficiency bound for real-time grid topology and admittance estimation and links structural identifiability in power systems to fundamental rigidity theory.

Abstract

In this paper, we quantify voltage and current phasor-based measurement requirements for the unique estimation of the electric grid topology and admittance parameters. Our approach is underpinned by the concept of a rigidity matrix that has been extensively studied in graph rigidity theory. Specifically, we show that the rank of the rigidity matrix is the same as that of a voltage coefficient matrix in a corresponding electric power system. Accordingly, we show that there is a minimum number of measurements required to uniquely estimate the admittance matrix and corresponding grid topology. By means of a numerical example on the IEEE 4-node radial network, we demonstrate that our approach is suitable for applications in electric power grids.

Electric Grid Topology and Admittance Estimation: Quantifying Phasor-based Measurement Requirements

TL;DR

The paper tackles the problem of uniquely identifying electric grid topology and admittance parameters from phasor measurements without prior topology. It develops a theoretical link between the voltage-coefficient matrix used in current–voltage relations and a rigidity matrix from graph rigidity theory, deriving a rank-based identifiability condition. Under generic nodal voltages, it shows that the minimum number of measurements required is for both DC conductance and AC admittance networks, with the edge parameters recoverable via a pseudoinverse when the condition holds. A numerical example on an IEEE 4-node network validates the bound, recovering a non-complete topology and nonzero edge admittances using exactly measurements. This work provides a rigorous measurement-efficiency bound for real-time grid topology and admittance estimation and links structural identifiability in power systems to fundamental rigidity theory.

Abstract

In this paper, we quantify voltage and current phasor-based measurement requirements for the unique estimation of the electric grid topology and admittance parameters. Our approach is underpinned by the concept of a rigidity matrix that has been extensively studied in graph rigidity theory. Specifically, we show that the rank of the rigidity matrix is the same as that of a voltage coefficient matrix in a corresponding electric power system. Accordingly, we show that there is a minimum number of measurements required to uniquely estimate the admittance matrix and corresponding grid topology. By means of a numerical example on the IEEE 4-node radial network, we demonstrate that our approach is suitable for applications in electric power grids.

Paper Structure

This paper contains 7 sections, 5 theorems, 13 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. Conductance matrix estimation
  5. Admittance matrix estimation
  6. Numerical example
  7. Conclusion

Key Result

Lemma 1

The matrix ${\textbf{A}(\textbf{v})}^{\top} \in \mathbb{R}^{e \times n\tau}$ has the same null space as the rigidity matrix $\textbf{R}(\textbf{x})\in \mathbb{R}^{e \times n\tau}$ when $\textbf{v} = \textbf{x}$.

Figures (3)

  • Figure 1: An example graph $\mathcal{G}(\mathcal{N,E})$: (a) where $n=3$; and (b) where $n=3$ and represents a network of admittance. The shaded area represents subgraph $\overline{\mathcal{G}}(\overline{\mathcal{N}},\overline{\mathcal{E}}) \subset \mathcal{G}(\mathcal{N,E})$, where network admittance parameters are represented by $y_{1,3}, y_{1,2}, y_{2,3}$.
  • Figure 2: Graphs in two dimensions: 1. flexible; 2. rigid; and 3. globally rigid. That is, 1(a)-(d) are flexible frameworks, 2(a)-(c) are rigid frameworks, and 3(a)-(b) are globally rigid frameworks.
  • Figure 3: (a) Admittance network as defined by subgraph $\overline{\mathcal{G}}(\overline{\mathcal{N}},\overline{\mathcal{E}})$, where $n = 4$ and; (b) the corresponding estimated topology and admittance parameters of the network.

Theorems & Definitions (23)

  • Definition 1: Graph Realization Conditions
  • Definition 2: Framework egres-14-12
  • Definition 3: Generic Frameworks gortler2014generic
  • Remark 1: A nongeneric framework example
  • Definition 4: Finite Flexing Conditions
  • Remark 2: Infinitesimal Motion Conditions
  • Definition 5: Trivial Infinitesimal Motion Conditions
  • Remark 3: Number of trivial infinitesimal motions
  • Definition 6: Infinitesimal Rigidity Conditions
  • Remark 4: Infinitesimal rigidity is (generic) rigidity
  • ...and 13 more