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Power System State Estimation by Phase Synchronization and Eigenvectors

Iven Guzel, Richard Y. Zhang

TL;DR

The paper tackles the nonconvex PSSE problem by reformulating the angle estimation as a phase synchronization task, enabling a spectral initialization for the angles and a formal global optimality certificate. Under reasonably accurate voltage magnitudes, the spectral initialization yields an almost-perfect, single-shot angle estimate from $2n$ PQ measurements, often requiring only one Gauss--Newton iteration to reach a globally optimal solution with zero duality gap. The approach leverages chordal sparsity to compute eigenvectors efficiently via inverse iteration, achieving runtimes comparable to a few Gauss--Newton steps and enabling scalable certification on large networks (e.g., up to 13k buses). Experimental results on Polish, PEGASE, and RTE models show graceful degradation with magnitude errors and demonstrate the practical viability of provable angle recovery and verification in PSSE. Limitations include the lack of a magnitude initialization and the conditional nature of the certification, motivating future work on theoretical guarantees, extensions to weighted PQ measurements, and robust handling of bad data.

Abstract

To estimate accurate voltage phasors from inaccurate voltage magnitude and complex power measurements, the standard approach is to iteratively refine a good initial guess using the Gauss--Newton method. But the nonconvexity of the estimation makes the Gauss--Newton method sensitive to its initial guess, so human intervention is needed to detect convergence to plausible but ultimately spurious estimates. This paper makes a novel connection between the angle estimation subproblem and phase synchronization to yield two key benefits: (1) an exceptionally high quality initial guess over the angles, known as a \emph{spectral initialization}; (2) a correctness guarantee for the estimated angles, known as a \emph{global optimality certificate}. These are formulated as sparse eigenvalue-eigenvector problems, which we efficiently compute in time comparable to a few Gauss-Newton iterations. Our experiments on the complete set of Polish, PEGASE, and RTE models show, where voltage magnitudes are already reasonably accurate, that spectral initialization provides an almost-perfect single-shot estimation of $n$ angles from $2n$ moderately noisy bus power measurements (i.e. $n$ pairs of PQ measurements), whose correctness becomes guaranteed after a single Gauss--Newton iteration. For less accurate voltage magnitudes, the performance of the method degrades gracefully; even with moderate voltage magnitude errors, the estimated voltage angles remain surprisingly accurate.

Power System State Estimation by Phase Synchronization and Eigenvectors

TL;DR

The paper tackles the nonconvex PSSE problem by reformulating the angle estimation as a phase synchronization task, enabling a spectral initialization for the angles and a formal global optimality certificate. Under reasonably accurate voltage magnitudes, the spectral initialization yields an almost-perfect, single-shot angle estimate from PQ measurements, often requiring only one Gauss--Newton iteration to reach a globally optimal solution with zero duality gap. The approach leverages chordal sparsity to compute eigenvectors efficiently via inverse iteration, achieving runtimes comparable to a few Gauss--Newton steps and enabling scalable certification on large networks (e.g., up to 13k buses). Experimental results on Polish, PEGASE, and RTE models show graceful degradation with magnitude errors and demonstrate the practical viability of provable angle recovery and verification in PSSE. Limitations include the lack of a magnitude initialization and the conditional nature of the certification, motivating future work on theoretical guarantees, extensions to weighted PQ measurements, and robust handling of bad data.

Abstract

To estimate accurate voltage phasors from inaccurate voltage magnitude and complex power measurements, the standard approach is to iteratively refine a good initial guess using the Gauss--Newton method. But the nonconvexity of the estimation makes the Gauss--Newton method sensitive to its initial guess, so human intervention is needed to detect convergence to plausible but ultimately spurious estimates. This paper makes a novel connection between the angle estimation subproblem and phase synchronization to yield two key benefits: (1) an exceptionally high quality initial guess over the angles, known as a \emph{spectral initialization}; (2) a correctness guarantee for the estimated angles, known as a \emph{global optimality certificate}. These are formulated as sparse eigenvalue-eigenvector problems, which we efficiently compute in time comparable to a few Gauss-Newton iterations. Our experiments on the complete set of Polish, PEGASE, and RTE models show, where voltage magnitudes are already reasonably accurate, that spectral initialization provides an almost-perfect single-shot estimation of angles from moderately noisy bus power measurements (i.e. pairs of PQ measurements), whose correctness becomes guaranteed after a single Gauss--Newton iteration. For less accurate voltage magnitudes, the performance of the method degrades gracefully; even with moderate voltage magnitude errors, the estimated voltage angles remain surprisingly accurate.
Paper Structure (14 sections, 5 theorems, 45 equations, 4 figures, 3 tables, 1 algorithm)

This paper contains 14 sections, 5 theorems, 45 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. System Modeling and Quadratic Measurements
  4. PSSE via Gauss--Newton Method
  5. PSSE as Phase Synchronization Problem
  6. Solving Nonconvex Phase Synchronization
  7. Spectral initialization
  8. Global optimality certification
  9. Experimental Results
  10. Spectral initialization with accurate voltage magnitudes
  11. Optimality certification with accurate voltage magnitudes
  12. Performance with inaccurate voltage magnitudes
  13. Cost of Computing Eigenvectors
  14. Conclusions and future work

Key Result

Theorem 1

Define $\hat{v}$ as the global minimum to (eq:psse), denote $\hat{J}\equiv J(\hat{\theta},\hat{u})$ as the Jacobian evaluated at $\hat{v}$. If $\sigma<\lambda/2$, where then there exists a ball of radius $\rho>0$ about $\hat{v}$ such that where $v$ and $v_{+}$ are the initial and refined estimates in (eq:gn_iterate) respectively.

Figures (4)

  • Figure 1: Gauss--Newton can produce misleading spurious estimates over the voltage angles, even when the voltage magnitudes are perfectly measured. For a three-bus system with perfect voltage magnitude measurements (see Appendix A), the usual "cold start" initialization experiences convergence issues, resulting in slow convergence over many iterations to a spurious estimate (). In contrast, our spectral initialization technique gives a one-shot estimate () close to the maximum likelihood estimator () in a region where the convergence behavior is vastly improved. Dots ($\bullet$) show the first 5 Gauss--Newton iterations.
  • Figure 2: High quality of spectral initialization on 13k-bus system. (top) The initial guess chosen by spectral initialization is already within $3^{\circ}$ of ground truth, for noise of up to $\sigma_{\mathrm{noise}}\le0.1$. With just 1 iteration, GN converges to the MLE within $1^{\circ}$ of ground truth. (bottom) In contrast, GN with cold start takes 5 iterations to converge to the MLE.
  • Figure 3: Certifiably optimal estimates on 13k-bus system using certified lower bounds on the cost. (top) It takes only one GN iteration starting from the spectral initialization to certify global optimality on the average case, for noise of up to $\sigma_{\mathrm{noise}}\le0.1$. (bottom) The proposed method is applied to cold start initialization, requiring 5 GN iterations to certify global optimality.
  • Figure 4: Initialization and optimality certification with inaccurate voltage magnitudes on 13k-bus system. (top) Spectral initialization provides high quality initial guesses over the angles even with inaccurate voltage measurements. In fact, spectral initialization yields more accurate angle estimations compared to those achieved after 5 GN iterations when $\Delta u=0.04$ pu. (bottom) With moderate voltage magnitude error, the certification method verifies optimality of the angle estimates once voltage magnitude estimates are accurate enough, after 5 GN iterations.

Theorems & Definitions (5)

  • Theorem 1: Local linear convergence
  • Proposition 2: Bus complex power
  • Proposition 3: Branch complex power
  • Proposition 4: Phasor measurement
  • Theorem 5: Lagrange duality