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

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

Problem Statement
The estimation of instantaneous frequency (IF) is critical for characterizing nonstationary phenomena in power systems. While recent interpretations link power system frequency to complex-frequency kinematics and the geometry of voltage trajectories, conventional estimation methods often rely on the Energy Separation Algorithm (ESA) based on the Teager-Kaiser Energy Operator (TKEO). These traditional estimators assume narrow-band signals and "slow-envelope" conditions (where amplitude and frequency modulation rates are small relative to the carrier frequency). In power systems, particularly during transients or when voltage magnitudes vary significantly, these assumptions are frequently violated. Consequently, conventional estimators can produce biased frequency estimates because they neglect the dynamic terms associated with voltage magnitude variations.

Methodology
The authors propose a novel local IF estimator derived by extending the TKEO to complex signals (CTKEO) and applying the dynamic-signal identity. The methodology proceeds as follows:

1. Complex TKEO Formulation: The standard real-valued TKEO is generalized to complex signals using a symmetric bilinear form. For a complex signal $\bar{x}$, the operator is defined as $\Psi_C(\bar{x}) = |\bar{x}'|^2 - \text{Re}(\bar{x}\bar{x}''^*)$. This formulation is additive on Cartesian components and invariant to constant phase shifts, avoiding the need for narrow-band assumptions.
2. Derivation of the Exact Identity: By expressing the complex signal in terms of its magnitude $|\bar{x}|$ and phase $\phi$, and utilizing the definition of complex frequency $\bar{\eta} = \rho + j\omega$ (where $\rho$ is the magnitude rate and $\omega$ is the angular frequency), the authors derive an exact continuous-time identity:
   $$\Psi_C(\bar{x}) = |\bar{x}|^2 (2\omega^2 - \rho')$$
   This equation explicitly separates the rotational energy component from the translation term associated with magnitude dynamics ($\rho'$).
3. Proposed Estimators:
   • $\omega_{TV}$ (Time-Varying): Solving the identity for $\omega$ yields an estimator that retains the magnitude-dynamics correction terms ($\rho'$ and $\rho^2$). This provides a bias-corrected estimate for conditions where voltage magnitude varies rapidly.
   • $\omega_{TI}$ (Time-Invariant): A simplified approximation is derived by assuming the magnitude is constant or varies slowly, effectively neglecting the correction terms. This results in $\omega_{TI} = \sqrt{\Psi_C(\bar{x}) / (2|\bar{x}|^2)}$, which is computationally simpler and less sensitive to noise but may introduce bias during significant magnitude transients.
4. Geometric Interpretation: The proposed estimator aligns with complex-frequency kinematics and admits a geometric interpretation related to the curvature of the voltage trajectory in the $(v_\alpha, v_\beta)$ plane, assuming zero torsion.

Key Contributions

• Novel Expression: The paper introduces a CTKEO-based IF expression that makes envelope-curvature terms explicit. This corrects the bias inherent in conventional estimators that neglect magnitude dynamics.
• Bias Correction: Unlike the classical ESA, the proposed formulation ($\omega_{TV}$) does not rely on the "slow-envelope" approximation, making it theoretically exact for smooth, nonzero, two-dimensional voltage trajectories in continuous time.
• Geometric Consistency: The estimator is consistent with complex-frequency kinematics and provides a geometric interpretation of frequency without requiring explicit phase unwrapping.
• Practical Approximation: The paper provides a computationally efficient approximation ($\omega_{TI}$) for scenarios where magnitude variations are slow, offering a trade-off between accuracy and noise sensitivity.

Results
The proposed methods were validated through two case studies:

1. EMT Simulation (IEEE 39-Bus System): Simulations of a three-phase fault showed that under balanced conditions with noise, the proposed estimators ($\omega_{TV}$ and $\omega_{TI}$) closely matched the geometric reference frequency ($\omega_v$). In unbalanced scenarios, the proposed estimators exhibited lower oscillations compared to the reference, suggesting they are less affected by curvature distortions introduced by imbalances.
2. Field Measurements (PV Power Plant): Tests on real voltage data from a PV plant in Spain demonstrated that both estimators followed the dominant trend of the geometric reference. While discrepancies appeared during intervals with high noise and derivative sensitivity (as $\omega_{TV}$ requires second-order derivatives), the results confirmed that the simplified $\omega_{TI}$ remains practically close to the full time-varying estimate under realistic operating conditions due to the inherently slow dynamics of magnitude variations.

Quantitative analysis on a synthetic amplitude-modulated signal (Table I) showed that the time-varying estimator ($\omega_{TV}$) significantly reduced Root Mean Square Error (RMSE) and Mean Bias Error (MBE) compared to the time-invariant estimator ($\omega_{TI}$), particularly at lower sampling frequencies.

Significance and Claims
The authors claim that the primary significance of this work lies in the derivation of an IF expression that explicitly accounts for voltage-magnitude dynamics, thereby correcting the bias found in simplified estimators used in power systems. The paper asserts that the proposed formulation unifies nonlinear energy and geometric interpretations. It does not claim to replace existing estimators in all contexts but highlights that $\omega_{TV}$ offers a precise local realization of the angular component of complex frequency, while $\omega_{TI}$ serves as a reliable, simpler approximation when magnitude dynamics are negligible. The work establishes a theoretical foundation for future extensions to three-dimensional trajectories (nonzero torsion) and broader validation against other frequency estimation techniques.