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arxiv: 2605.00404 · v1 · submitted 2026-05-01 · 📡 eess.SY · cs.SY

Electric Grid Topology and Admittance Estimation using Phasor Measurements

Norak Rin , Iman Shames , Ian Petersen , Elizabeth Ratnam This is my paper

Pith reviewed 2026-05-09 19:30 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords grid topology estimationadmittance matrix identificationphasor measurement unitspower system parameter estimationstructured total least squaresidentifiability conditionsdistribution and transmission networks
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The pith

Voltage and current phasor measurements from a sufficient number of independent operating points uniquely determine the topology and admittance of any unknown electric grid.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives necessary and sufficient conditions on the number of independent sets of voltage and current phasor measurements needed to recover both the topology and the admittance matrix of a grid with no prior information. This matters for real-time monitoring because phasor units now provide precise data at known bus locations. When partial topology knowledge is available the required number of measurements drops. Noisy measurements are addressed by formulating the problem as structured total least squares. Numerical tests on standard IEEE feeders confirm the conditions hold for both radial and meshed networks.

Core claim

We show necessary and sufficient conditions for the number of independent operating points (measurements) required to determine the topology and admittance of a completely unknown electric grid. With prior topology information, we also show that there is a minimum number of measurements required to uniquely determine the admittance matrix and corresponding grid topology. In the presence of noisy phasor measurements, we show that the admittance matrix can be estimated using a structured total least squares approach.

What carries the argument

The identifiability conditions on the count of independent operating points together with the structured total least squares estimator that recovers the admittance matrix from noisy phasors.

If this is right

  • A known threshold on the number of independent operating points guarantees unique recovery of both topology and admittances with no prior knowledge.
  • Partial topology information reduces the minimum number of required measurements.
  • Structured total least squares yields consistent admittance estimates under measurement noise.
  • The same framework applies to both radial distribution feeders and meshed transmission networks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The derived count of operating points can inform the minimum deployment density of phasor units in new installations.
  • Sequential application of the estimator could detect and localize topology changes by tracking shifts in the recovered admittance matrix.
  • Fusion with other sensor types might lower the operating-point requirement further in practice.

Load-bearing premise

Admittance values remain constant across the independent operating points while the system stays linear and measurements are taken at buses with known locations.

What would settle it

Two grids with different topologies or different admittance values that produce identical sets of phasor measurements for the claimed minimum number of operating points would show the conditions are not sufficient for uniqueness.

Figures

Figures reproduced from arXiv: 2605.00404 by Elizabeth Ratnam, Ian Petersen, Iman Shames, Norak Rin.

Figure 1
Figure 1. Figure 1: An example graph G(N, E), where n = 3. Admittance parameters are represented by y1,3, y1,2, y2,3. presentation and the underlying graph is assumed to be undi￾rected. The graph G(N, E) is also assumed to be a weighted graph where each edge (i, j) ∈ E is assigned a weight αi, j . Let e denote the cardinality of the set E, such that |E| = e. Then, by definition of a complete graph for G(N, E), e = n(n − 1)/2.… view at source ↗
Figure 2
Figure 2. Figure 2: Topology of: (a) a radial distribution feeder represented by a modified view at source ↗
Figure 3
Figure 3. Figure 3: Total absolute error between the estimated admittance (represented by view at source ↗
Figure 4
Figure 4. Figure 4: Nodes 2–3 of the IEEE 13-node network: 3-phase topology identifi view at source ↗
Figure 5
Figure 5. Figure 5: Total absolute error between estimated admittance and actual admit view at source ↗
Figure 6
Figure 6. Figure 6: Total absolute error between estimated admittance (represented by view at source ↗
read the original abstract

Recent advances in precise phasor measurement units are enabling new approaches to estimate distribution and transmission grid parameters in real-time. In this paper, we investigate voltage and current phasor measurement requirements to estimate the electric grid topology and admittance parameters. We show necessary and sufficient conditions for the number of independent operating points (measurements) required to determine the topology and admittance of a completely unknown electric grid. With prior topology information, we also show that there is a minimum number of measurements required to uniquely determine the admittance matrix and corresponding grid topology. In the presence of noisy phasor measurements, we show that the admittance matrix can be estimated using a structured total least squares approach. By means of numerical simulations on the IEEE 13-node distribution feeder, the IEEE 14-node transmission network, and the IEEE 123-node distribution feeder, we demonstrate our approach is suitable for applications in radial and mesh grid topologies in the presence of measurement noise.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to derive necessary and sufficient conditions on the number of independent operating points (phasor measurements) required to uniquely recover both the topology (via the zero pattern of Y) and the complex admittance values of a completely unknown electric grid modeled as I = YV. It also gives a separate result on the minimum number of measurements needed when topology is known a priori, proposes a structured total least-squares estimator that enforces symmetry and other admittance properties for the noisy case, and reports numerical demonstrations on the IEEE 13-, 14-, and 123-bus test systems for both radial and meshed topologies.

Significance. If the rank-based identifiability conditions and the structured estimator hold, the work supplies a useful theoretical and algorithmic foundation for PMU-enabled real-time topology and parameter estimation in distribution and transmission grids. Credit is due for grounding the count conditions in the Kronecker structure of the vectorized measurement equation and for preserving physical constraints (symmetry, etc.) inside the total-least-squares formulation; the choice of standard IEEE benchmarks also aids reproducibility.

major comments (2)
  1. [§3] §3 (identifiability analysis): the necessary-and-sufficient count on independent operating points is asserted via a rank condition on the stacked voltage matrices, yet the manuscript supplies neither a proof sketch nor an explicit derivation showing how the Kronecker structure yields the precise bound in terms of the number of buses and unknown admittance entries; this step is load-bearing for the central claim.
  2. [Numerical results] Numerical results section: the demonstrations on the IEEE 13-, 14-, and 123-bus systems are described only qualitatively; no error norms, success rates under varying noise levels, or baseline comparisons (e.g., ordinary least squares) are reported, leaving the practical performance claim for noisy measurements unsupported by visible quantitative evidence.
minor comments (2)
  1. [Assumptions] The assumption that admittance parameters remain constant across the independent operating points is stated but not accompanied by a brief discussion of how violations (e.g., due to temperature or tap changes) would affect the rank condition.
  2. [Notation] Notation for the vectorized equation and the dimension of the stacked measurement matrix should be introduced with an explicit small-grid example to illustrate the counting argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review. We address the major comments point by point below, agreeing where revisions are needed to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (identifiability analysis): the necessary-and-sufficient count on independent operating points is asserted via a rank condition on the stacked voltage matrices, yet the manuscript supplies neither a proof sketch nor an explicit derivation showing how the Kronecker structure yields the precise bound in terms of the number of buses and unknown admittance entries; this step is load-bearing for the central claim.

    Authors: We agree that Section 3 would be strengthened by an explicit derivation. The necessary and sufficient conditions follow from the rank of the stacked matrix obtained by vectorizing the measurement equation I = YV, which produces a Kronecker-structured linear system vec(I) = (V^T ⊗ I) vec(Y). In the revision we will insert a concise proof sketch that starts from this vectorized form, shows how the rank condition on the stacked voltage matrices determines the minimum number of independent operating points, and arrives at the bound in terms of the number of buses n and the number of free parameters in the (symmetric) admittance matrix. This will make the central identifiability claim fully self-contained. revision: yes

  2. Referee: [Numerical results] Numerical results section: the demonstrations on the IEEE 13-, 14-, and 123-bus systems are described only qualitatively; no error norms, success rates under varying noise levels, or baseline comparisons (e.g., ordinary least squares) are reported, leaving the practical performance claim for noisy measurements unsupported by visible quantitative evidence.

    Authors: We acknowledge that the current numerical section is primarily qualitative. In the revised manuscript we will augment the results with quantitative evidence: Frobenius-norm errors on the estimated admittance matrix, topology-recovery success rates (fraction of Monte-Carlo trials in which the zero pattern of Y is correctly recovered) across a range of noise levels, and direct side-by-side comparisons against ordinary least squares on the same IEEE 13-, 14-, and 123-bus cases for both radial and meshed topologies. These additions will provide visible support for the practical performance of the structured total-least-squares estimator. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from LTI model and rank conditions

full rationale

The paper starts from the standard LTI relation I = Y V with constant complex Y across operating points, vectorizes to obtain a Kronecker-structured linear system in the unknown entries of Y, and derives the minimal number of independent operating points needed for full column rank (hence unique recovery of topology via zero pattern and admittance values). The noisy estimator is a standard structured total-least-squares formulation that enforces known algebraic properties of Y (symmetry, zero diagonal blocks). Neither step reduces to a fitted parameter renamed as a prediction, nor relies on a load-bearing self-citation whose content is itself unverified. IEEE test-system simulations serve as external numerical checks rather than circular confirmation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard linear power-system modeling without introducing new entities or many fitted parameters.

axioms (2)
  • domain assumption The electric grid is a linear time-invariant network whose behavior is fully captured by a constant admittance matrix relating nodal voltages and currents.
    Invoked throughout the identifiability analysis and estimator derivation.
  • domain assumption Phasor measurements are available at known bus locations and correspond to independent operating points with distinct voltage/current profiles.
    Required for the necessary-and-sufficient conditions on the number of measurements.

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