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Urban MV and LV Distribution Grid Topology Estimation via Group Lasso

Yizheng Liao, Yang Weng, Guangyi Liu, Ram Rajagopal

arXiv:1611.01845v2stat.MLcs.SYeess.SYmath.OC

TL;DR

Topology information is needed for distribution-grid operation amid DER-related reliability risks, but urban bus connectivity is difficult to monitor. The paper uses a probabilistic graphical model and group lasso for topology estimation, reporting over 95% accuracy on large-scale MV/LV grids across varied scenarios.

  • Problem

    Topology information is a prerequisite for distribution-grid operation as DER penetration brings reliability risks.

    Source: §I · p. 1

  • Method

    The method models the grid as a probabilistic graphical model and uses historical voltage measurements in a grouped-variable linear regression.

    Source: §I · p. 2

  • Results

    Over 95% accuracy was reported for large-scale MV/LV grid topology estimation with or without DER penetration, with robust results across varied scenarios.

    Source: §VI · p. 12

  • Takeaways & Limitations

    The approach estimates radial and mesh networks using smart-meter voltage magnitudes without requiring admittance matrices or switch-location information.

    Source: §VI · p. 11

Abstract

from arXiv · show

The increasing penetration of distributed energy resources poses numerous reliability issues to the urban distribution grid. The topology estimation is a critical step to ensure the robustness of distribution grid operation. However, the bus connectivity and grid topology estimation are usually hard in distribution grids. For example, it is technically challenging and costly to monitor the bus connectivity in urban grids, e.g., underground lines. It is also inappropriate to use the radial topology assumption exclusively because the grids of metropolitan cities and regions with dense loads could be with many mesh structures. To resolve these drawbacks, we propose a data-driven topology estimation method for MV and LV distribution grids by only utilizing the historical smart meter measurements. Particularly, a probabilistic graphical model is utilized to capture the statistical dependencies amongst bus voltages. We prove that the bus connectivity and grid topology estimation problems, in radial and mesh structures, can be formulated as a linear regression with a least absolute shrinkage regularization on grouped variables (\textit{group lasso}). Simulations show highly accurate results in eight MV and LV distribution networks at different sizes and 22 topology configurations using PG\&E residential smart meter data.

I. INTRODUCTION

DER penetration raises distribution-grid reliability risks, while underground lines, frequent reconfiguration, and unavailable topology information make monitoring difficult. The paper models voltage dependencies and uses smart-meter data to estimate connectivity and topology, including in mesh grids.

I. INTRODUCTION

  • The method uses household smart-meter data to estimate bus connectivity and grid topology without admittance matrices or switch-location information, including for mesh networks.It models voltage dependencies probabilistically and formulates estimation as grouped-variable regularized regression.Source: §I · pp. 1–2

II. MV/LV DISTRIBUTION GRID MODEL

  • The MV/LV grid model represents each bus as a voltage random variable and connects bus pairs with statistical dependence in a graphical model.The stated model assumes noiseless steady-state measurements and a slack bus with constant voltage magnitude and phase.Source: §II · p. 2

A. Problem Formulation

  • Under assumptions of independent Gaussian current-injection increments, bus voltage increments are conditionally independent of distant buses given neighbor increments.The authors use real-data observations to motivate approximating voltage dependencies with neighbors; under that approximation, finding connectivity is equivalent to finding adjacent buses.Source: A. Problem Formulation · pp. 3, 5

B. Bus Connectivity Reconstruction via Linear Regression

The section models each bus’s incremental voltage as a linear function of the other buses’ increments and an error term. Nonzero regression coefficients identify neighboring buses, making connectivity estimation a sparse regression problem.

B. Bus Connectivity Reconstruction via Linear Regression

C. Bus Neighbors Estimation via Group Lasso

The section converts the complex-valued connectivity regression into a real-valued formulation and applies group lasso so the real and imaginary coefficient parts for a bus are selected together. The resulting estimates are used to identify bus neighbors.

C. Bus Neighbors Estimation via Group Lasso

  • The proposed approach converts complex-valued lasso into a real-valued lasso formulation for power-system variables.The section identifies complex voltage and admittance values as the reason real-number lasso solvers cannot directly solve the prior formulation.Source: C. Bus Neighbors Estimation via Group… · p. 6
  • Group lasso constrains the real and imaginary parts of a coefficient as a group, so both are zero when the bus is not connected.This addresses the ordinary L1 formulation’s inability to ensure the two parts are jointly zero or nonzero.Source: C. Bus Neighbors Estimation via Group… · p. 6
  • Algorithm 1 estimates each bus’s parameter vector by solving the group-lasso problem, then uses that estimate to determine neighboring buses.Source: C. Bus Neighbors Estimation via Group… · p. 6

D. Bus Connectivity Estimation using Voltage Magnitude Only

The section adapts connectivity estimation to cases where voltage phase angles are unavailable by using voltage-magnitude increments. It states that these measurements can still provide accurate topology estimates.

D. Bus Connectivity Estimation using Voltage Magnitude Only

  • When only voltage magnitudes are available, the group-lasso objective reduces to ordinary lasso because each γ(i) coefficient becomes scalar.The increments are defined as changes in voltage magnitude between successive time steps.Source: D. Bus Connectivity Estimation using… · p. 6
  • Voltage magnitudes can still yield accurate topology estimates when phase data are unavailable, according to the paper’s simulations.The section attributes stronger statistical correlations to voltage magnitudes than to phase angles in distribution grids, citing.Source: D. Bus Connectivity Estimation using… · p. 6

E. Choice of the Regularization Parameter λ

The regularization parameter λ controls coefficient sparsity, so the paper selects it using BIC while favoring a sparse estimate. The LAR path provides BIC values across candidate λ values.

E. Choice of the Regularization Parameter λ

  • Larger λ values can zero coefficients or coefficient groups, whereas small λ values leave the solution close to ordinary least squares.The number of nonzero coefficients or groups depends on λ.Source: E. Choice of the Regularization… · p. 6
  • BIC is used to choose λ for each bus, and the selection requires solving the lasso or group-lasso problem at multiple λ values.LAR solvers can compute solutions for multiple λ values at once, avoiding additional computation for the selection process.Source: E. Choice of the Regularization… · pp. 6–7
  • The selected λ balances a significant BIC reduction with high sparsity rather than choosing the minimum BIC when that estimate is non-sparse.The paper describes this selection along the LAR path.Source: E. Choice of the Regularization… · p. 7, Figure 8 · p. 7

IV. GRID TOPOLOGY ESTIMATION VIA GROUP LASSO

The paper extends bus-level neighbor estimates to network topology reconstruction by combining AND and OR rules. Its AND-OR procedure adjusts the estimated edges using a voltage-based check intended to avoid disconnected graphs.

IV. GRID TOPOLOGY ESTIMATION VIA GROUP LASSO

  • The paper extends group-lasso bus connectivity estimates to reconstruct unknown branches across a subnetwork.Each bus's neighbor set is estimated, and pairs of buses are then used to identify branches.Source: §IV · p. 7
  • The AND rule estimates a branch only when both endpoint-level estimates support the connection, so an error at either endpoint can omit a true edge.The paper proposes the OR rule to reduce missing edges by accepting a connection when either endpoint estimate supports it.Source: §IV · p. 7
  • The AND-OR rule starts with the AND estimate, then uses voltage magnitudes to identify buses that may be isolated and add a neighbor through a modified OR rule.A load bus is expected to have a neighbor with higher average voltage because most load buses absorb power.Source: §IV · p. 7
  • The AND-OR rule addresses the fact that AND and OR estimates alone are not proven to produce a connected graph.The method uses physical voltage-magnitude information to adjust the AND estimate.Source: §IV · p. 7
  • The method applies the rules using voltage magnitude |V| rather than incremental voltage magnitude Δ|V|.The AND-OR steps are summarized in Algorithm 2.Source: §IV · p. 7

A. Grid Topology Estimation via Group Lasso

For bus connectivity, the paper formulates voltage-based estimation as a group-lasso problem. Grouping coefficients supports a pairwise connectivity decision, while L1 regularization yields the OR-rule formulation.

A. Grid Topology Estimation via Group Lasso

  • A single group-lasso optimization is proposed to use interactions between bus measurements and avoid solving multiple problems for each bus.The section introduces the approach using voltage-magnitude changes in a fully connected three-bus example.Source: A. Grid Topology Estimation via Group… · p. 7
  • With L1 regularization, the bus-level linear optimization is equivalent to the OR rule because individual coefficients may be zero.The paper notes that the predictor matrix is sparse and contains only M unique values, allowing efficient storage and processing.Source: A. Grid Topology Estimation via Group… · p. 8
  • The group-lasso formulation groups bus-pair coefficients so they become zero or nonzero together, making the resulting connectivity decision equivalent to the AND rule.The paper applies the same steps as Algorithm 2 to combine the AND and OR rules.Source: A. Grid Topology Estimation via Group… · p. 8

V. SIMULATION AND RESULTS

Across radial and mesh comparisons, the proposed lasso-based method reports nearly 0% topology error, while the correlation-based method’s detection ability drops. Connectivity estimates are mostly error-free in smaller systems and nearly perfect in 123-bus networks; topology error also remains nearly perfect in most tested MV/LV systems.

A. Estimation Error of Bus Connectivity

B. Network Topology Reconstruction Error Rate

  • Most tested MV and LV systems have zero error with PMUs, while the AND-OR rule achieves nearly perfect performance in most cases using voltage magnitude alone.In Urban, four branches are incorrectly estimated; known transformer locations reduce the error rate to 0.6%. The error does not scale by grid size in the artificial LV large system.Source: B. Network Topology Reconstruction… · p. 9, Table II · p. 9
  • The topology error rate does not change when the slack-bus voltage is nonconstant in the tested Urban LV grid.The feeder is the slack bus, and its voltage is determined by the upper MV grid.Source: B. Network Topology Reconstruction… · p. 9
  • Nearly 0% error rate is reported for the proposed method across radial and mesh comparisons, comparable with, while detection ability drops.[32] is excluded from mesh comparisons because it applies only to radial networks; the comparison averages 100 iterations.Source: B. Network Topology Reconstruction… · p. 9, Figure 10 · p. 9

C. Networks with DER Integration

With rooftop PV integration, the AND-OR rule remains the strongest reconstruction rule, and the method’s candidate-bus selection reduces computation while retaining accuracy. The section also evaluates the method across PV settings and reports runtime results for mesh networks.

C. Networks with DER Integration

  • Rooftop PV tests use hourly NREL PVWatts profiles for 5 kW LV and 20 kW MV systems, with power factor fixed at 0.90 lagging.The profiles are based on PG&E service-zone weather history.Source: C. Networks with DER Integration · p. 10
  • AND-OR has the lowest error rate with rooftop PV, and different PV penetration levels do not significantly affect performance.For Urban, with transformer locations known, it misses 22 of 3,237 branches (0.68%) with no false estimations; LV large has an error rate similar to the no-DER case.Source: C. Networks with DER Integration · p. 10, Table III · p. 10

D. Computational Complexity

  • For each bus, the method ranks candidate buses by pairwise mutual information, then applies group lasso to the top K to identify neighbors.This selection reduces the connectivity regression from the full M-bus system to K candidate buses.Source: D. Computational Complexity · p. 10
  • Selecting the K most relative buses makes the proposed algorithm 12 times faster while achieving the same accuracy; on the mesh 8-bus system, recovery takes 0.3 seconds versus over 100 seconds for the extended information-theory algorithm [50].For Urban system with K = 30, average computational time is less than 270 seconds.Source: D. Computational Complexity · p. 10

E. Sensitivity Analysis

Across the reported settings, the conclusion describes robust topology estimation across noise, data duration, load patterns, and sampling resolutions. Sensitivity tests identify when accurate reconstructions occur and where measurement processing or switching can constrain performance.

1) Sensitivity to Data Accuracy:

  • The AND-OR rule reconstructs the 123-bus loopy and MV urban networks without error at all tested noise levels, while LV suburban accuracy falls at high noise.The passage identifies ANSI C12.20 Class 0.5 meter accuracy as an error of less than ±0.5%.Source: 1) Sensitivity to Data Accuracy · p. 11, Table IV · p. 11
  • Rounding voltage measurements to integers can create repeated identical values, weaken statistical relationships, and make the algorithm perform poorly.The method can recommend the data resolution needed to use smart meters beyond billing.Source: 1) Sensitivity to Data Accuracy · p. 11
  • Measurements collected during temporary switch-status changes may reduce accuracy; the passage recommends removing them before topology estimation.It also suggests comparing estimates from consecutive time slots for consistency.Source: 1) Sensitivity to Data Accuracy · p. 11

2) Sensitivity to Data Length:

3) Sensitivity to Load Pattern:

4) Sensitivity to Data Resolution:

  • At 1-minute sampling, about 2 hours of voltage profiles yield perfect estimation; at 30-minute sampling, about 50 hours are needed.The passage states that estimation time can be reduced with higher sampling frequency and compares these durations with reconfiguration frequencies ranging from hours to weeks.Source: 4) Sensitivity to Data Resolution · p. 11, Figure 14 · p. 12

VI. CONCLUSION

  • The conclusion reports over 95% accuracy across eight MV/LV networks and 21 configurations, with or without DER penetration, and states that the method outperforms existing methods.The evaluations use real PG&E and NREL data and consider noise, data resolution, duration, and load patterns.Source: §VI · pp. 11–12

APPENDIX

The appendix supports the paper’s conditional-independence arguments: coefficient constraints govern independence in the two-variable examples, while the grid proof establishes dependence for neighbors and independence for buses beyond two hops.

A. Proof of Lemma 1

  • When both coefficients in the linear constraint are nonzero, X1 depends on X2 given Y; conditional independence therefore requires at least one coefficient to be zero.The proof uses a counterexample to establish this necessity.Source: A. Proof of Lemma 1 · p. 12

B. Proof of Lemma 2

  • In the second lemma’s construction, setting either c1 or c2 to zero makes the corresponding variable constant given Z, yielding conditional independence.When both coefficients are zero, both variables become constants, so their statistical dependency cannot be determined.Source: B. Proof of Lemma 2 · p. 12

C. Proof of Theorem 1

  • In the grid proof, bus i’s current injection is conditionally dependent on those of its neighbors, given voltages within two hops, but independent of injections beyond two hops.The proof applies the lemmas to establish the conditional-independence structure used for voltage-based connectivity reasoning.Source: C. Proof of Theorem 1 · pp. 12–13

D. Proof of (16)

The proof argues that voltage-magnitude increments preserve the linear relation used for lasso connectivity estimation. The European network descriptions cover representative LV and MV grids, including mesh variants, and report typical grid ratios and neighbor counts.

D. Proof of (16)

  • Voltage-magnitude increments preserve the linear relationship in equation (12), allowing lasso to estimate bus connectivity; nonconnected pairs have γik = 0.The lasso formulation approximates the derived relation by treating γik as real, which the proof says does not affect the zero solution for nonconnected pairs.Source: D. Proof of (16) · p. 13

E. European Representative Distribution Networks

  • The European examples span LV suburban mesh networks, MV suburban and urban grids, rural MV networks with switches closed, and an urban system with its switches closed.The described grids include underground branches; the paper reports X/R ratios and average and maximum neighbor counts as representative of typical MV/LV grids and as support for choosing candidate buses.Source: E. European Representative… · pp. 13–14
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