Instantaneous Frequency in Power Systems using the Teager-Kaiser Energy Operator

TL;DR

This work addresses the challenge of estimating instantaneous frequency in nonstationary power-system signals, especially under rapid magnitude changes and unbalances. It develops a Complex Teager–Kaiser Energy Operator (CTKEO) by extending the TKEO to complex signals, deriving a closed-form relation with complex frequency $\\bar{\\eta}=\\rho+\\mathrm{j}\\omega$ and a voltage phasor $\\bar{v}$. The key contributions are the bias-corrected IF expression $\\omega_{TV}=\\sqrt{\\frac{1}{2}[ \frac{\\Psi_{\\mathbb{C}}(\\bar{v})}{|\\bar{v}|^2} - (\\frac{|\\bar{v}|'}{|\\bar{v}|})^2 + \frac{|\\bar{v}|''}{|\\bar{v}|} ]}$ and the simpler special-case $\\omega_{TI}=\\sqrt{\\frac{\\Psi_{\\mathbb{C}}(\\bar{v})}{2|\\bar{v}|^2}}$, with $|\\bar{v}|$ constant. Through analytical examples and two case studies (IEEE 39-bus EMT simulation and PV plant measurements), the method demonstrates accurate, phase-unwrapping-free IF estimation that aligns with complex-frequency kinematics and provides a clear geometric interpretation of voltage trajectory curvature. The approach offers robust, real-time monitoring potential for power-system dynamics, complementing or surpassing traditional ESA/PLL-based methods in transients and unbalanced operation.

Abstract

This paper develops an instantaneous-frequency (IF) local estimator calculated with the complex Teager-Kaiser energy operator (CTKEO) and the dynamic-signal identity. The contribution is a novel IF expression that makes the envelope-curvature terms explicit, thus correcting the bias that affects conventional estimators used in power systems. The estimator aligns with complex-frequency (CF) kinematics and admits a geometric interpretation (curvature/torsion) without phase unwrapping. Simulations and data-driven examples demonstrate the accuracy of the proposed approach.

Figures (3)

• Figure 1: Frequency estimation results under stationary conditions with: (a, b) a magnitude unbalance of $|v_\alpha|=0.8|v_\beta|$; and (c, d) the presence of a balanced harmonic of order $h=5$ and magnitude $|\bar{v}_h|=0.01|\bar{v}|$.
• Figure 2: Frequency results bus $26$, of the IEEE 39-bus system for a secured three-phase fault under: (a) balanced conditions with Gaussian noise in the voltage; and (b) unbalanced operation.
• Figure 3: Frequency estimation results from the real voltage measurements at the point of connection to the grid of a PV power plant.