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Plug-in Estimation Method for Topology and Admittance Estimation in Electric Grids

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

arXiv:2610.30434v1eess.SYcs.SY

TL;DR

Estimating grid topology and admittance as operating conditions change is important, but scalable plug-in estimation without topology information has had limited application. The paper combines graph rigidity theory with a two-stage plug-in method using phasor measurements. Simulations report better scalability to larger grids and lower computational time than structured total least squares, including tests with partial topology information.

  • Problem

    Scalable plug-in approaches have had limited application to admittance estimation without prior grid topology information.

    Source: §I · p. 1

  • Method

    The method combines graph rigidity theory for measurement sufficiency with averaged phasor-derived voltage coefficient matrices and current vectors, followed by least-squares admittance estimation and thresholding.

    Source: §I · p. 1

  • Results

    Across simulations on the IEEE 13-node, IEEE 123-node, and European low-voltage feeders, the method outperformed structured total least squares in scalability to larger grids and computational time.

    Source: §V · p. 8

  • Takeaways & Limitations

    The simulations demonstrate applicability to grids with hundreds of nodes and to systems with partial topology information.

    Source: B. Scalability · p. 7, C. Admittance estimation with prior… · p. 7

    Obtaining the required n −1 independent operating conditions from natural load fluctuations is difficult because regulated voltages yield highly collinear voltage vectors and ill-conditioned coefficient matrices.

    Source: B. Plug-in Estimation Method · p. 4

Abstract

from arXiv · show

This paper proposes a scalable method for topology and admittance estimation in electric power grids. Our proposed plug-in estimation method incorporates precise phasor measurement unit data to estimate planned or unplanned grid topology and admittance parameter changes. Our plug-in estimation method decouples the problem into two stages: (i) constructing and averaging voltage coefficient matrices and current vectors from phasor data, and (ii) estimating admittance parameters via a least-squares regression approach with thresholding to recover the network topology. Rigidity matrix theory underpins the design of our plug-in estimation method --- to determine the data requirements (e.g., necessary and sufficient conditions) for accurate estimation of the grid topology and admittance parameters. By means of numerical simulations on the IEEE 13-node, 123-node, and European low voltage test feeders, we demonstrate the scalability of our plug-in estimation method and benchmark the computational time against a structured total least squares method. Our simulation results highlight the practicality of the proposed method for topology and admittance estimation in large-scale electric networks, including electric grids with partial topology information.

I. INTRODUCTION

Synchrophasor data creates opportunities to track grid topology and admittance as operating conditions evolve, while joint identification supports important grid operations. The paper addresses limited scalable use of the plug-in principle for admittance estimation without topology information by combining it with graph rigidity theory.

I. INTRODUCTION

  • Increasing PMU deployments provide high-resolution voltage and current phasors that create opportunities to estimate topology and admittance under evolving grid conditions.The motivation is the changing operation of power systems as distributed and renewable energy resources are integrated.Source: §I · p. 1
  • Joint topology and admittance identification supports state estimation, outage restoration, and distributed-energy coordination, and can help recalibrate controllers and protection settings.Real-time topology identification can provide an operational model instead of outdated switch-state diagrams.Source: §I · p. 1
  • Existing regression methods estimate admittance with known topology, while scalable plug-in applications have had limited use without topology information.Prior examples include structured total least squares, least squares, and linear estimation approaches for admittance estimation.Source: §I · p. 1
  • The proposed method integrates graph rigidity theory with a two-stage plug-in approach: average voltage coefficient matrices and current vectors, then regress admittances and threshold nonexistent edges.The method avoids large matrix inversion and is described as scalable to distribution grids with hundreds of nodes and robust to measurement noise.Source: §I · p. 1

II. PRELIMINARIES

The preliminaries model the grid as a weighted graph and express current measurements through voltage-derived coefficients and admittance parameters. Multiple operating conditions and averaged time-series measurements provide the data used by the estimation method.

II. PRELIMINARIES

  • The network is represented by a complete undirected weighted graph, with node pairs as candidate edges and an incidence matrix encoding edge direction.The complete graph has e = n(n − 1)/2 edges; edge weights correspond to admittance parameters, and the model excludes a ground node.Source: §II · p. 2
  • Kirchhoff’s laws relate voltage and current through a complex symmetric admittance matrix whose row sums are zero.The admittance vector contains off-diagonal parameters, omitting redundant diagonal entries.Source: §II · p. 2
  • The voltage coefficient matrix is constructed from the incidence matrix and node voltages, yielding a linear relation between current and admittance.The relation is I = Ã(V)y.Source: §II · p. 2
  • The setup uses multiple distinct voltage-current operating points and assumes their measured pairs are known exactly for the identifiability formulation.The paper later accounts for noisy time-series measurements, assuming grid operating conditions change more slowly than measurement noise.Source: §II · pp. 2–3
  • At each operating condition, repeated noisy voltage and current samples are batch-averaged into voltage coefficient matrices and current vectors for estimation.The stacked averages are termed the average voltage coefficient matrix and average current vector.Source: §II · p. 3

III. PROBLEM FORMULATION

The problem is to estimate an unknown grid admittance vector and its corresponding topology from time-series current and voltage phasors across operating conditions. Graph rigidity theory assesses measurement sufficiency, while plug-in estimation provides a computationally efficient, scalable solution.

III. PROBLEM FORMULATION

  • The setup applies graph rigidity theory and plug-in estimation to synchrophasor-based topology and admittance estimation.Source: §III · p. 3
  • The estimation problem seeks an unknown admittance vector and corresponding grid topology from current and voltage phasors collected across nodes and operating conditions.The inputs are time-series measurements at each node for each of T operating conditions.Source: §III · p. 3
  • Graph rigidity theory determines whether the measurements are sufficient for estimation, while the plug-in method supplies a computationally efficient and scalable solution..Source: §III · p. 3

A. Graph Rigidity Theory

Graph rigidity theory links the rank of the voltage coefficient matrix to measurement sufficiency for topology and admittance estimation. Under generic nodal voltage vectors, the theorem gives a minimum of n − 1 phasor measurements for a unique solution.

A. Graph Rigidity Theory

  • The voltage coefficient matrix A and rigidity matrix R have equal rank, with rank nT − T(T + 1)/2 when n ≥ T.This rank connection underpins theoretical bounds on the number of measurements needed for unique identification.Source: A. Graph Rigidity Theory · p. 3
  • The analysis assumes generic nodal voltage vectors, defined as vectors that satisfy no nonzero algebraic equation with rational coefficients.Linearly dependent nodal voltage vectors are not generic.Source: A. Graph Rigidity Theory · p. 3
  • For generic nodal voltage vectors on a complete graph, T = n − 1 measurements suffice for a unique solution to i = Ay.The result concerns the noiseless equation; the paper states that the measurement requirement supports accurate estimation in the presence of noise.Source: A. Graph Rigidity Theory · p. 3
  • The prior structured total least squares formulation becomes computationally intractable on standard hardware for networks exceeding 34 nodes, motivating a more scalable approach.The paper attributes this limit to the computation and memory demands of high-dimensional optimization.Source: A. Graph Rigidity Theory · p. 4

B. Plug-in Estimation Method

The plug-in method averages voltage coefficient matrices and current vectors across time-series phasor measurements, then estimates admittances by least squares and recovers topology by thresholding. Voltage centering improves numerical conditioning, while the separated stages avoid large matrix inversions and high-dimensional optimization.

B. Plug-in Estimation Method

  • The method averages voltage coefficient matrices and current vectors across time-series measurements at each operating condition, then concatenates them for estimation.At least n − 1 distinct operating conditions are required under the stated measurement-sufficiency result.Source: B. Plug-in Estimation Method · p. 4
  • Voltage centering removes the static nominal profile without changing branch voltage drops, improving conditioning when natural load variations are small.This addresses the near-collinearity of regulated voltage measurements and helps expose operating-condition independence.Source: B. Plug-in Estimation Method · p. 4
  • Least-squares regression estimates the admittance vector, and thresholding entries below ε to zero identifies inactive edges and the estimated topology.The threshold is intended to exceed the estimation-error level while remaining below the minimum physical line-admittance magnitude.Source: B. Plug-in Estimation Method · pp. 4–5
  • Separating averaging from regression avoids large-scale matrix inversion and high-dimensional optimization, providing a scalable alternative to structured total least squares.The method computes the averaging and least-squares stages separately.Source: B. Plug-in Estimation Method · p. 4

IV. NUMERICAL SIMULATIONS

The simulations assess accuracy, computational cost, noise robustness, scalability, and performance with partial topology knowledge across three test feeders. The feeders range from a small benchmark case to a 907-node low-voltage network.

IV. NUMERICAL SIMULATIONS

  • The modified IEEE 13-node feeder is modeled as a 12-node balanced single-phase system and used to compare the plug-in method with STLS.The substation node is omitted.Source: §IV · p. 5
  • The IEEE 123-node feeder is modeled as a 122-node balanced single-phase system to assess scalability without prior topology information.The substation node is omitted.Source: §IV · p. 5
  • The 907-node European low-voltage feeder is used to evaluate scalability with some prior topology information.The paper describes this feeder as representative of a real-world power system.Source: §IV · p. 5

A. Comparison with STLS

On the modified IEEE 13-node feeder, plug-in and STLS estimation have similar error once operating conditions meet the minimum requirement, while STLS takes about twice the computation time. The plug-in method’s error also rises with noise and improves with larger batches.

A. Comparison with STLS

  • The benchmark modifies the IEEE 13-node feeder and adds measurement noise to simulated voltage and current phasors.The comparison removes the substation node and uses Gaussian noise in the constructed measurements.Source: A. Comparison with STLS · pp. 5–6
  • Above the minimum number of operating conditions, plug-in and STLS have similar maximum relative error, while STLS takes approximately 2 times longer.With too few operating conditions, both methods show large error; the comparison is on the IEEE 13-node feeder.Source: Figure 2 · p. 6, A. Comparison with STLS · p. 6
  • Higher noise increases estimation error, while increasing batch size improves plug-in estimation performance.The noise comparison uses B = 10000; the batch-size comparison uses noise with standard deviation 1.Source: Figure 3 · p. 6, A. Comparison with STLS · p. 6

B. Scalability

The plug-in method is evaluated on the larger IEEE 123-node feeder without prior topology information. It maintains high admittance-estimation accuracy when enough operating conditions are available, while edge identification at a small batch size is sensitive to noise.

B. Scalability

  • The IEEE 123-node feeder is used as a larger, more complex benchmark for computational performance and estimation accuracy.The simulation assumes no prior topology information.Source: B. Scalability · p. 7
  • With sufficient operating conditions, the plug-in method maintains high admittance-estimation accuracy on the IEEE 123-node feeder without prior topology information.The evaluation uses batch size B = 1000.Source: B. Scalability · p. 7, Figure 4 · p. 7
  • At B = 100, the IEEE 123-node feeder has 18 misidentified edges at noise standard deviation 3 and 326 at standard deviation 5.The simulation uses 148 operating conditions for this feeder.Source: B. Scalability · p. 7

C. Admittance estimation with prior topology information

The method incorporates known topology edges and applies estimation and thresholding only to unknown elements. On the 907-node European low-voltage feeder, partial topology information reduces the number of edges to estimate, and two operating conditions suffice for accurate estimation.

C. Admittance estimation with prior topology information

  • Known edges are incorporated by fixing their entries in the admittance vector and applying estimation and thresholding only to unknown elements.This scenario is evaluated on the IEEE European low-voltage test feeder.Source: C. Admittance estimation with prior… · p. 7
  • Using partial topology information reduces the estimation set from 410871 potential edges to 1406 on the 907-node feeder.The setup starts with 906 true radial edges and adds 500 false candidate edges.Source: C. Admittance estimation with prior… · p. 7
  • With 1406 edges requiring estimation, two operating conditions are sufficient for accurate estimation on the European low-voltage feeder.The reported maximum relative error decreases as the number of operating conditions increases.Source: C. Admittance estimation with prior… · p. 7

V. CONCLUSION

The conclusion presents the plug-in method as a scalable approach that separates estimation into two stages. Simulations across three feeders show better scalability to larger grids and lower computational time than structured total least squares.

V. CONCLUSION

  • The method constructs and averages voltage coefficient matrices and current vectors, then estimates admittance parameters by least squares with thresholding to recover topology.This two-stage separation avoids large matrix inversion.Source: §V · p. 7
  • The European low-voltage feeder evaluation reports maximum relative error between estimated and actual admittance parameters at batch size B = 1000.Source: Figure 5 · p. 8
  • Across the IEEE 13-node, IEEE 123-node, and European low-voltage feeders, the plug-in method outperforms structured total least squares in scalability to larger grids and computational time.The method avoids large matrix inversion and iterative optimization required by structured total least squares.Source: §V · pp. 7–8
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