Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Power Grid Parameter Estimation Without Phase Measurements: Theory and Empirical Validation

Jean-Sébastien Brouillon, Keith Moffat, Florian Dörfler, Giancarlo Ferrari-trecate

TL;DR

The paper tackles parameter estimation for distribution grids without phase measurements, addressing a practical bottleneck in modeling large-scale grids. It develops a statistical sensor model and analyzes four admittance-estimation methods, contrasting them with impedance-based approaches, and derives biases and consistency properties under Total Least Squares (TLS). A key finding is that impedance estimates can be made unbiased with TLS even without phase data, whereas admittance estimates inherently incur biases unless phase information is incorporated or the problem is reframed (e.g., reduced regression or impedance-first inversion). Empirical validation on both synthetic data and real measurements from the Walenstadt grid confirms the theoretical insights and highlights practical implications for DSOs and researchers pursuing cost-effective grid-modeling with abundant smart-meter data.

Abstract

Reliable integration and operation of renewable distributed energy resources requires accurate distribution grid models. However, obtaining precise models is often prohibitively expensive, given their large scale and the ongoing nature of grid operations. To address this challenge, considerable efforts have been devoted to harnessing abundant consumption data for automatic model inference. The primary result of the paper is that, while the impedance of a line or a network can be estimated without synchronized phase angle measurements in a consistent way, the admittance cannot. Furthermore, a detailed statistical analysis is presented, quantifying the expected estimation errors of four prevalent admittance estimation methods. Such errors constitute fundamental model inference limitations that cannot be resolved with more data. These findings are empirically validated using synthetic data and real measurements from the town of Walenstadt, Switzerland, confirming the theory. The results contribute to our understanding of grid estimation limitations and uncertainties, offering guidance for both practitioners and researchers in the pursuit of more reliable and cost-effective solutions.

Power Grid Parameter Estimation Without Phase Measurements: Theory and Empirical Validation

TL;DR

The paper tackles parameter estimation for distribution grids without phase measurements, addressing a practical bottleneck in modeling large-scale grids. It develops a statistical sensor model and analyzes four admittance-estimation methods, contrasting them with impedance-based approaches, and derives biases and consistency properties under Total Least Squares (TLS). A key finding is that impedance estimates can be made unbiased with TLS even without phase data, whereas admittance estimates inherently incur biases unless phase information is incorporated or the problem is reframed (e.g., reduced regression or impedance-first inversion). Empirical validation on both synthetic data and real measurements from the Walenstadt grid confirms the theoretical insights and highlights practical implications for DSOs and researchers pursuing cost-effective grid-modeling with abundant smart-meter data.

Abstract

Reliable integration and operation of renewable distributed energy resources requires accurate distribution grid models. However, obtaining precise models is often prohibitively expensive, given their large scale and the ongoing nature of grid operations. To address this challenge, considerable efforts have been devoted to harnessing abundant consumption data for automatic model inference. The primary result of the paper is that, while the impedance of a line or a network can be estimated without synchronized phase angle measurements in a consistent way, the admittance cannot. Furthermore, a detailed statistical analysis is presented, quantifying the expected estimation errors of four prevalent admittance estimation methods. Such errors constitute fundamental model inference limitations that cannot be resolved with more data. These findings are empirically validated using synthetic data and real measurements from the town of Walenstadt, Switzerland, confirming the theory. The results contribute to our understanding of grid estimation limitations and uncertainties, offering guidance for both practitioners and researchers in the pursuit of more reliable and cost-effective solutions.
Paper Structure (29 sections, 1 theorem, 35 equations, 4 figures, 1 table)

This paper contains 29 sections, 1 theorem, 35 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. System model
  4. Distribution Network
  5. Measuring Devices
  6. Line parameter estimation
  7. Impedance Estimation
  8. Admittance Estimation
  9. Impedance Estimation Bias and Variance
  10. Admittance estimation bias
  11. Omitted-Phase Bias
  12. Real and Imaginary Simultaneity
  13. Endogeneity of $\delta_{hk}^\Re$ and $\delta_{hk}^{\Im}$
  14. Impedance Inversion Bias
  15. Network identification
  16. ...and 14 more sections

Key Result

Lemma 1

If a regression model $\tilde{z} = A (\tilde{x} - \epsilon_x) + \epsilon_z$, where $\epsilon_x, \epsilon_x \sim \mathcal{N}(0, \Sigma)$The RHS and LHS variables must be normalized to have the same noise variance. is fitted to $D$ datasets $\tilde{x}_d, \tilde{z}_d$, the bias of the TLS is given by

Figures (4)

  • Figure 1: Graph of the region of interest in the Walenstadt network. The lines that will be identified are in blue. The unobserved nodes are in red.
  • Figure 2: Estimates of the conductance and susceptance of a single line using (a) the admittance regression \ref{['eq_simple_ls_regression_noisy']}, (b) the joint estimation using \ref{['eq_simple_ls_regression_noisy_phase']}, (c) the reduced model \ref{['eq_line_flow_lin_real_reduced']}, (d) the impedance regression \ref{['eq_line_flow_z_lin_real_reduced_simple']} with real data. For each method, the points mark the estimates using each data subset and the line their average. The true conductance $b$ and susceptance $g$ of the line is not known, but $R \approx 0.1 \Omega$.
  • Figure 3: Estimates of the conductance and susceptance of a single line using the four methods (a)-(d) with synthetic data. For each method, the points mark the estimates affected by one of the 50 realizations of the noise generated for each noise level. The line in color shows their average and the black line shows the predicted bias. In case (d) the black line is a lower bound on the bias.
  • Figure 4: Heatmaps of the magitudes of the admittance matrix estimates given by the methods (c) and (d).

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof