Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cramér-Rao Bounds for the Simultaneous Estimation of Power System Electromechanical Modes and Forced Oscillations

Luke Dosiek

TL;DR

This work derives asymptotic Cramér-Rao Bounds for the simultaneous estimation of power-system electromechanical modes and forced oscillations using ARMA/ARMAX models, comparing two FO scenarios: steady-state only and with startup transients. The authors provide general CRB expressions and implement Monte Carlo methods to approximate the information matrices, showing that startup transients can dramatically improve the accuracy of mode estimation while steady-state FO does not affect mode CRBs. Numerical experiments on a minniWECC-based ARMAX model reveal that including transients yields substantial reductions in CRBs for both FO parameters and mode frequencies/damping, especially at higher SNRs and longer windows. The findings underscore the practical importance of incorporating transients in FO and mode estimation pipelines for PMU-based oscillation monitoring and highlight the CRB as a robust benchmark for estimator performance.

Abstract

In this paper, the Cramér-Rao Bounds (CRB) for the simultaneous estimation of power system electromechanical modes and forced oscillations (FO) are derived. Two cases are considered; in the first case only the steady-state response to the FO is present in the measured system output used by estimation algorithms. In the second, the startup transient of the FO is present in addition to the steady-state response. The CRBs are analyzed numerically to explore sensitivities to FO frequency, signal-to-noise ratio (SNR) and observation window length. It is demonstrated that 1) the CRB of FO parameters is not affected by the presence of the transient response, 2) the CRB of the system modes is not affected by the presence of an FO in steady-state and 3) the CRB of the system modes can be drastically reduced by the presence of a FO startup transient.

Cramér-Rao Bounds for the Simultaneous Estimation of Power System Electromechanical Modes and Forced Oscillations

TL;DR

This work derives asymptotic Cramér-Rao Bounds for the simultaneous estimation of power-system electromechanical modes and forced oscillations using ARMA/ARMAX models, comparing two FO scenarios: steady-state only and with startup transients. The authors provide general CRB expressions and implement Monte Carlo methods to approximate the information matrices, showing that startup transients can dramatically improve the accuracy of mode estimation while steady-state FO does not affect mode CRBs. Numerical experiments on a minniWECC-based ARMAX model reveal that including transients yields substantial reductions in CRBs for both FO parameters and mode frequencies/damping, especially at higher SNRs and longer windows. The findings underscore the practical importance of incorporating transients in FO and mode estimation pipelines for PMU-based oscillation monitoring and highlight the CRB as a robust benchmark for estimator performance.

Abstract

In this paper, the Cramér-Rao Bounds (CRB) for the simultaneous estimation of power system electromechanical modes and forced oscillations (FO) are derived. Two cases are considered; in the first case only the steady-state response to the FO is present in the measured system output used by estimation algorithms. In the second, the startup transient of the FO is present in addition to the steady-state response. The CRBs are analyzed numerically to explore sensitivities to FO frequency, signal-to-noise ratio (SNR) and observation window length. It is demonstrated that 1) the CRB of FO parameters is not affected by the presence of the transient response, 2) the CRB of the system modes is not affected by the presence of an FO in steady-state and 3) the CRB of the system modes can be drastically reduced by the presence of a FO startup transient.
Paper Structure (7 sections, 32 equations, 5 figures)

This paper contains 7 sections, 32 equations, 5 figures.

Table of Contents

  1. Introduction
  2. System Models
  3. Asymptotic Cramér-Rao Bounds
  4. Case 1
  5. Case 2
  6. Numerical Study
  7. Conclusions

Figures (5)

  • Figure 1: Power spectral density of the ARMA system used in the simulations.
  • Figure 2: Square root CRB for FO parameters as a function of FO frequency.
  • Figure 3: Square root CRB for FO parameters as a function of global SNR (top) and record length (bottom).
  • Figure 4: Square root CRB for the electromechanical mode frequency and damping as a function of FO frequency.
  • Figure 5: Square root CRB for electromechanical mode frequency and damping as a function of global SNR (top) and record length (bottom).