Assessment of Errors of Fundamental Frequency Estimation Methods in the Presence of Voltage Fluctuations and Distortions

Technical Summary: Assessment of Errors of Fundamental Frequency Estimation Methods in the Presence of Voltage Fluctuations and Distortions

Problem Statement
In modern power grids, power quality disturbances, particularly simultaneous voltage fluctuations and distortions, create complex and non-stationary voltage signals. While regulatory standards like EN 50160 define permissible fundamental frequency ranges (e.g., 49.5–50.5 Hz for 99.5% of the year), these standards typically assess frequency over 10-second intervals. However, power quality diagnostics often require estimating the fundamental frequency over much shorter windows (e.g., 200 ms) to identify disturbance sources and track rapid load changes. Under these dynamic conditions, existing estimation methods may exhibit significant errors. The paper addresses the lack of a complete, standardized measurement chain for fundamental frequency in complex signals and seeks to evaluate the accuracy of various estimation methods when subjected to simultaneous amplitude modulation, frequency deviation, and waveform distortion.

Methodology
The authors conducted numerical simulation studies using the Matlab environment. A test signal was generated to recreate realistic low-voltage network conditions, defined by a carrier signal with specific harmonic content and a modulating signal causing simultaneous amplitude and frequency variations.

• Signal Model: The test signal included a fundamental frequency ($f_0$) with deviations ($\Delta f_0$) and a modulating signal ($u_{mod}(t)$) with step changes to simulate abrupt load switching. Harmonic distortion was modeled using a "clipped cosine" signal to represent interactions with switched-mode power supplies. White noise was added at various Signal-to-Noise Ratio (SNR) levels (10, 0, and -10 dB) to simulate nondeterministic load changes.
• Estimation Methods Evaluated:
  1. IEC 61000-4-30 Method: A zero-crossing detection approach applied after band-pass filtering (46–54 Hz) to suppress harmonics.
  2. Autocorrelation Method: Estimates the fundamental period by averaging the distance between local maxima of the autocorrelation function.
  3. Hilbert Method: Calculates the average instantaneous frequency derived from the argument of the analytical signal.
  4. Modified ESPRIT Method: A subspace-based algorithm that models the signal as a superposition of complex exponentials, utilizing a modified scheme to reduce computational cost while maintaining accuracy.
• Experimental Setup: Simulations were run over a 10-second window with a sampling rate of 327,680 Sa/s. A 200 ms sliding window was used for calculations (shifted every 0.305 ms). The relative error ($\delta f_0$) was calculated by comparing the estimated frequency against a reference value derived from the known instantaneous frequency of the test signal.

Key Contributions

• Comprehensive Error Assessment: The study provides a comparative analysis of the IEC 61000-4-30 standard method against three alternative algorithms (Autocorrelation, Hilbert, and Modified ESPRIT) under conditions of simultaneous voltage fluctuations and distortions.
• Realistic Test Signal: The authors utilized a specific test signal model that combines amplitude modulation, frequency deviation, and harmonic distortion (clipped cosine), allowing for an unambiguous determination of the reference fundamental frequency.
• Parameter Sensitivity Analysis: The research systematically evaluates how estimation errors vary with modulating frequency ($f_m$), frequency deviation ($\Delta f_0$), carrier frequency ($f_0$), distortion levels ($m_c$), and noise levels (SNR).

Results
The simulation results, presented as error dispersion boxplots, indicate the following:

• Impact of Dynamics: For all methods, increasing the modulating frequency (rate of change) and the frequency deviation leads to a significant increase in estimation errors.
• Impact of Noise and Distortion: Different levels of noise and harmonic distortion had relatively little influence on the estimation results compared to the dynamics of the frequency changes.
• Method Performance:
  • The IEC 61000-4-30 method and the Modified ESPRIT method yielded the best and most stable results, with estimation errors generally reduced to a few percent.
  • The Modified ESPRIT method occasionally provided the lowest error values at specific modulating frequencies but involves higher computational complexity.
  • The IEC method is noted for its lower computational complexity.
• Outliers: All methods exhibited outliers with error values reaching tens of percent under certain conditions.
• Standard Compliance: Crucially, none of the evaluated methods consistently achieved error values below the limits defined in the IEC 61000-4-30 standard (±10 mHz or 0.02%) for the complex test signals used.

Significance and Claims
The paper concludes that current methods, including the standard IEC 61000-4-30 approach, are insufficient for accurately estimating fundamental frequency in signals characterized by simultaneous rapid fluctuations and distortions. The authors claim that the significant increase in errors observed with rapid frequency changes highlights a gap in current measurement capabilities. Consequently, the study asserts the necessity for:

1. Developing new, more effective methods for fundamental frequency estimation that can guarantee errors smaller than those defined in relevant standards under these complex conditions.
2. Conducting metrological evaluations of existing measuring and recording devices using the specific test signal proposed in this paper, which recreates the simultaneous occurrence of fluctuations and distortions found in modern power grids.