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PaToPa: A Data-Driven Parameter and Topology Joint Estimation Framework in Distribution Grids

Jiafan Yu, Yang Weng, Ram Rajagopal

arXiv:1705.08870v1eess.SY

TL;DR

Distribution-grid planning and operation need topology and line parameters that may be missing or inaccurate, while existing regression approaches do not adequately handle noise in both input and output measurements. PaToPa uses an EIV maximum-likelihood framework with generalized low-rank approximation and iterative topology–parameter estimation. It reports accurate joint estimation across IEEE systems and real feeder data, including zero topology error under varying noise levels and 0% topology error in the demonstrated real feeder case.

  • Problem

    Distribution-grid topology and line parameters may be unavailable or inaccurate, while regression-based estimation is inadequate when both input and output measurements are noisy.

  • Method

    PaToPa models input and output errors jointly with EIV maximum likelihood, transforms and decorrelates the problem into GLRA, and iteratively exchanges information between parameter and topology estimation.

  • Results

    PaToPa outperforms other methods for parameter and topology estimation; topology error remains zero across tested error levels, and the real feeder demonstration reconstructs topology with 0% error.

  • Takeaways & Limitations

    The framework supports joint topology and line-parameter estimation under measurement constraints and system complexity, including real distribution-feeder data.

Abstract

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The increasing integration of distributed energy resources (DERs) calls for new planning and operational tools. However, such tools depend on system topology and line parameters, which may be missing or inaccurate in distribution grids. With abundant data, one idea is to use linear regression to find line parameters, based on which topology can be identified. Unfortunately, the linear regression method is accurate only if there is no noise in both the input measurements (e.g., voltage magnitude and phase angle) and output measurements (e.g., active and reactive power). For topology estimation, even with a small error in measurements, the regression-based method is incapable of finding the topology using non-zero line parameters with a proper metric. To model input and output measurement errors simultaneously, we propose the error-in-variables (EIV) model in a maximum likelihood estimation (MLE) framework for joint line parameter and topology estimation. While directly solving the problem is NP-hard, we successfully adapt the problem into a generalized low-rank approximation problem via variable transformation and noise decorrelation. For accurate topology estimation, we let it interact with parameter estimation in a fashion that is similar to expectation-maximization fashion in machine learning. The proposed PaToPa approach does not require a radial network setting and works for mesh networks. We demonstrate the superior performance in accuracy for our method on IEEE test cases with actual feeder data from South California Edison.

I. INTRODUCTION

Distribution grids increasingly need accurate topology and line parameters for DER planning and operation, but these data are often unavailable or unreliable. PaToPa addresses noisy measurements by jointly estimating line parameters and topology through an EIV-based, iteratively interacting framework.

  • DER integration creates two-way power flows that increase distribution-grid stability concerns and motivate improved planning and operational tools.
  • Outdated or missing topology and line parameters make real-time monitoring and operational planning difficult in distribution grids.
  • Linear-regression approaches are limited because distribution-grid measurements can contain errors in both inputs and outputs, while some methods assume line measurements are broadly available.
  • PaToPa uses an EIV maximum-likelihood model, variable transformation, noise decorrelation, and generalized low-rank approximation to estimate line parameters under measurement noise.
  • Topology estimation interacts iteratively with parameter estimation, identifying connected and disconnected buses in an approach similar to expectation-maximization.
  • Across IEEE 8- and 123-bus systems and intermediate scales, PaToPa outperforms other state-of-the-art methods for both parameter and topology estimation and is tested under varying measurement errors.

III. LINE PARAMETER ESTIMATION THROUGH ERROR-IN-VARIABLE MODEL

The noiseless power-flow relation permits direct linear estimation of line parameters, but practical measurement errors require a statistical estimation model. The section establishes maximum-likelihood estimation and contrasts it with least squares when only outputs are noisy.

  • In the noiseless case, historical measurement matrices X and y satisfy the power-flow equation exactly, enabling linear estimation of line parameters g and b.
  • Measurement errors are unavoidable, so line-parameter estimation requires a statistical model such as maximum likelihood estimation.
  • When noise affects only the dependent variable y, the estimation problem reduces to least squares.
  • With i.i.d. Gaussian errors in y, the output-only estimation problem has a closed-form solution.

B. Measurement Errors on Both Input/Output: The EIV Model

Because both power and voltage measurements are noisy, PaToPa models induced errors in transformed inputs and outputs within an EIV likelihood framework. Variable transformations and rank constraints convert the nonlinear estimation structure into a generalized low-rank approximation problem.

  • Both power injections and voltage phasors contain measurement noise, so the transformed input X also has induced error rather than being error-free.
  • The transformed EIV likelihood remains difficult because nonlinear variable transformation makes the induced noises non-Gaussian, so noise decorrelation is used before the GLRA formulation.
  • A nonlinear transformation makes induced errors functions of direct voltage and phase-angle errors, motivating first-order Taylor approximation under small-noise assumptions.
  • After linearization, the transformed measurement errors become linear combinations of Gaussian errors, allowing a covariance-weighted likelihood norm.
  • The nonlinear equality constraint is reformulated as a matrix low-rank constraint, yielding a generalized low-rank approximation problem.

IV. SOLVING THE GENERALIZED LOW-RANK APPROXIMATION PROBLEM FOR PARAMETER ESTIMATION

The generalized low-rank approximation is solved through structured covariance relaxations. The identity relaxation enables a theoretical guarantee via total least squares, while preserving more covariance structure motivates a diagonal alternative.

  • An arbitrary covariance matrix makes the generalized optimization difficult, motivating structured relaxations based on power-flow measurement patterns.
  • Assuming equal uncorrelated variances reduces the covariance matrix to the identity and converts the problem into total least squares.
  • The total-least-squares formulation has a closed-form solution based on the smallest singular value of the expanded sample matrix.
  • Norm equivalence yields a bound showing that the identity relaxation provides a guaranteed sub-optimal solution relative to the original problem.

B. Diagonal Matrix Relaxation for Numerically Enhanced Solution

The diagonal relaxation preserves heterogeneous measurement accuracy while supporting iterative topology updates, whose likelihood criterion is theoretically justified for small measurement errors.

  • B. Diagonal Matrix Relaxation for Numerically Enhanced Solution: The diagonal matrix relaxation preserves heterogeneity in covariance accuracy by replacing the original matrix norm with a diagonal-matrix norm.At least 2nT covariance blocks are scalar, motivating the relaxation.
  • B. Diagonal Matrix Relaxation for Numerically Enhanced Solution: PaToPa alternates parameter estimation with topology estimation, using updated topology to improve subsequent parameter estimates.The iterations continue until the estimated branch number matches the expected number, after which disconnected-line conductances are set to zero.
  • B. Diagonal Matrix Relaxation for Numerically Enhanced Solution: The topology step sorts estimated conductances and uses binary search to remove candidate lines with small conductances, reducing model complexity iteratively.The reduced model is then used for more accurate line parameter estimation.
  • B. Diagonal Matrix Relaxation for Numerically Enhanced Solution: Theorem 2 states that disconnecting a connected edge yields a lower best-fit log-likelihood than retaining it when measurement error is small.The result underpins the topology update rule’s protection against mistakenly removing connected branches.
  • B. Diagonal Matrix Relaxation for Numerically Enhanced Solution: The likelihood analysis models measurements as noiseless values plus measurement noise and evaluates the optimal likelihood as a function of those perturbations.In the noiseless case, the true parameters achieve an optimal log-likelihood of zero.

A. Recover Admittance Matrix from Joint Topology and Line Parameter Estimation

After joint estimation, the recovered topology and line parameters can be converted into the admittance matrix using incidence and indexing matrices. The framework applies to both radial and mesh networks.

  • A. Recover Admittance Matrix from Joint Topology and Line Parameter Estimation: The estimated topology E and line parameters g and b are used with incidence matrix S and indexing matrix U to recover admittance matrix Y.This recovery follows the joint topology and parameter estimation stage.
  • A. Recover Admittance Matrix from Joint Topology and Line Parameter Estimation: The admittance-matrix recovery uses the estimated network structure together with associated conductance and susceptance parameters for each line.The recovered matrix is represented through the incidence and indexing matrices introduced earlier.
  • A. Recover Admittance Matrix from Joint Topology and Line Parameter Estimation: PaToPa supports mesh as well as radial networks because it treats edges equally without relying on radial-network properties.This broadens the network settings addressed by the recovery framework.

B. Recover Equivalent Admittance Matrix when There are Partial Measurements

The framework can handle distribution grids with measurements concentrated at root and leaf buses, including cases with missing PMU measurements at intermediate buses. Under these constraints, equivalent-network recovery and flexible input-error modeling preserve computability.

  • B. Recover Equivalent Admittance Matrix when There are Partial Measurements: When measurements exist only at root and leaf buses and hidden buses have no power injections, Kron reduction permits an equivalent network involving the measured nodes.An equivalent admittance matrix represents the connected graph among those remaining nodes.
  • B. Recover Equivalent Admittance Matrix when There are Partial Measurements: Missing PMU measurements at one or more buses can be accommodated by treating unknown phase angles as additional input noise.The subsequent derivation, including the first-order Taylor expansion, remains applicable after modifying the observed phase angle.
  • B. Recover Equivalent Admittance Matrix when There are Partial Measurements: When a phase-angle measurement is unavailable, its value can be treated as zero because distribution-grid angle differences are comparatively small, allowing X to remain computable.The associated covariance matrix is then derived from the first-order Taylor expansion at the modified operating point.
  • B. Recover Equivalent Admittance Matrix when There are Partial Measurements: The framework is reported as robust against system complexity and measurement constraints and is tested on IEEE feeders with Southern California Edison data.The experiments include IEEE 8-, 16-, 32-, 64-, 96-, and 123-bus feeders.

B. Accuracy of Joint Parameter and Topology Estimation

PaToPa jointly estimates line parameters and topology under measurement noise, retaining accurate results where regression and EIV-based alternatives degrade. It combines parameter estimation with topology identification and is demonstrated on synthetic and real distribution-grid data.

  • Joint estimation accuracy: In an 8-bus system, PaToPa perfectly identifies topology and accurately estimates line parameters even with 10% relative measurement-error standard deviation.The setup includes noise in both input measurements (θ, v) and output measurements (p, q).
  • Parameter estimation under noise: PaToPa has lower line-parameter estimation error than linear regression and EIV across tested measurement-error levels.Each error level uses 30 historical datasets with 500 historical data points.
  • Topology estimation under noise: PaToPa's topology estimation maintains zero error as measurement error increases, whereas regression and EIV degrade, with EIV especially vulnerable to large errors.Topology is evaluated using a conductance threshold for the comparison methods and the Jaccard Index for accuracy assessment.
  • Real-data demonstration: On an 8-bus Bakersfield feeder using real SCE data, topology reconstruction has 0% error and estimated line parameters have 1% relative error.The experiment adds errors to all measurement variables and uses a real-time online dashboard.
  • Matrix relaxation: Diagonal matrix relaxation exceeds identity matrix relaxation in log-likelihood after two iterations, while reaching a condition number on the scale of 10^15 after 100 iterations.The identity relaxation has an analytical solution, whereas the diagonal relaxation uses numerical iteration and requires constraint satisfaction over more iterations.

APPENDIX A LEAST SQUARES AND TOTAL LEAST SQUARES

Traditional regression-based MLE assumes measurement noise only in the dependent variable, enabling a least-squares solution when the mapping is linear. The appendix contrasts this with the broader error structure addressed later.

  • Traditional Least Squares: Traditional regression models place measurement error only in the dependent variable y.The state measurements X are treated as noiseless in this formulation.
  • Maximum Likelihood Estimation: Under i.i.d. noise, MLE selects parameters by maximizing the conditional likelihood log P(y|X, g, b).The likelihood is formulated from measurement series y and X.
  • Least Squares: With i.i.d. Gaussian errors and a mapping linear in [g; b], the likelihood becomes a sum of squared errors.This produces the standard least-squares objective.
  • Least Squares: The resulting MLE problem has a closed-form least-squares solution.This closed form depends on the traditional one-sided error assumption.

B. Error-In-Variables: Maximum Likelihood Estimation with Measurement Errors on All Variables

The error-in-variables formulation accounts for noise in both power and voltage measurements, but nonlinear power-flow relationships make the resulting MLE difficult to solve. Approximation and variable transformations provide a route toward tractable low-rank formulations.

  • Measurement Errors on All Variables: Both power injections and voltage phasors are measured, so measurement noise affects input and output variables simultaneously.This extends the traditional model, which places error only in the dependent variable.
  • Maximum Likelihood Estimation: The EIV MLE problem becomes difficult because the power-flow mapping is nonlinear.The nonlinear constraint is the direct source of the stated difficulty.
  • Low-Rank Reformulation: Under i.i.d. Gaussian noise and a linear mapping, the MLE objective reduces to minimizing the Frobenius norm of a measurement-matrix error.This reduction connects the likelihood problem to low-rank approximation.
  • Low-Rank Reformulation: A matrix constraint replaces the original parameter relation, making the augmented measurement matrix rank deficient with [g; b; −1] in its null space.The transformation expresses the parameter-estimation problem as a low-rank approximation problem.
  • Approximation: First-order Taylor expansion with truncated-normal modeling approximates induced measurement errors as a linear function of direct measurement errors.The approximation is justified when the truncation range is small and the relevant gradient is non-zero.

RANK APPROXIMATION PROBLEM

The transformed estimation problem is solved as a generalized low-rank approximation with either heterogeneous diagonal weighting or a simpler identity-style relaxation. An iterative algorithm updates the transformed variables and scale parameters.

  • Diagonal Matrix Relaxation: Diagonal covariance relaxation weights element-wise matrix errors through positive coefficients, producing a weighted low-rank objective.The low-rank constraint is represented by a nonzero vector in the null space of the approximated matrix.
  • Iterative Solution: The generalized low-rank approximation is converted into an optimization problem and solved iteratively after QR decomposition and variable substitution.The algorithm updates transformed variables through the resulting QR-based relations.
  • Iterative Solution: The iterative procedure updates l, c, and σ, with Algorithm 3 specifying the diagonal-matrix relaxation steps.The variables are updated from the transformed optimization relations and current iterates.
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