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arxiv: 1906.11154 · v1 · pith:OPNCYHOTnew · submitted 2019-06-26 · 📡 eess.SY · cs.SY

Beyond Phasors: Continuous-Spectrum Modeling of Power Systems using the Hilbert Transform

Asja Dervi\v{s}kadi\'c , Guglielmo Frigo , Mario Paolone This is my paper

Pith reviewed 2026-05-25 15:43 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords Hilbert Transformpower systemsphasor modelingFourier Transformelectromechanical transientssignal dynamicsspectrum analysiscontingency analysis
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The pith

Hilbert Transform identifies full signal spectrum to track power system dynamics during transients where Fourier phasors fall short.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Modern power systems with lower inertia experience broad-spectrum electromechanical transients after large contingencies. Traditional phasor models derived from the Fourier Transform can fail to extract accurate amplitude, frequency, and phase of the fundamental component in these conditions. The paper proposes the Hilbert Transform as an alternative that identifies the entire spectrum rather than a single frequency, thereby enabling continuous tracking of signal dynamics. Comparisons are shown for amplitude modulations, frequency ramps, and step changes in both synthetic and measured signals, with further validation on a contingency scenario for the IEEE 39-bus test system.

Core claim

During electromechanical transients triggered by large contingencies, transmission of electrical power may take place in a broad spectrum well beyond the single fundamental component. FT-based phasor analysis may fail to accurately identify the fundamental component parameters in terms of amplitude, frequency and phase, whereas the HT-based approach enables the tracking of signal dynamics by identifying the whole spectrum. The approaches are compared during representative operating conditions in synthetic and real-world datasets and further validated using a contingency analysis on the IEEE 39-bus system.

What carries the argument

The Hilbert Transform, which extracts the analytic signal to reveal the full instantaneous spectrum instead of a single-frequency phasor.

If this is right

  • HT modeling succeeds in tracking dynamics across amplitude modulations, frequency ramps, and step changes.
  • HT identifies the entire spectrum during conditions where FT phasors lose accuracy.
  • The method applies to both synthetic signals and measured data from real power systems.
  • Validation on the IEEE 39-bus system confirms utility for contingency analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Real-time HT implementations could support faster protective actions in low-inertia grids.
  • The continuous-spectrum view might integrate with existing monitoring hardware to flag spectrum broadening before instability develops.
  • Similar HT modeling could be tested on other engineering signals that exhibit rapid frequency content changes.

Load-bearing premise

The Fourier Transform-based phasor representation fails to accurately identify fundamental component parameters during large transients triggered by contingencies, while the Hilbert Transform succeeds by identifying the whole spectrum.

What would settle it

A recorded large transient event in which the amplitude, frequency, and phase values recovered by the Fourier method match the true fundamental component throughout the event while the Hilbert method does not, or vice versa.

Figures

Figures reproduced from arXiv: 1906.11154 by Asja Dervi\v{s}kadi\'c, Guglielmo Frigo, Mario Paolone.

Figure 1
Figure 1. Figure 1: Frequency estimated by PMUs during the Australian blackout on [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Block diagram of the adopted EMTP-RV simulation model. The [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Block diagram of the modified 39-bus power grid Opal-RT simulation [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Magnitude response of the filter approximating the ideal HT over [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Amplitude modulation: instantaneous single-phase active power (a) [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Frequency Ramp: instantaneous single-phase active power (a) and [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Amplitude Step: instantaneous single-phase active power (a) and three [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Australian blackout: instantaneous single-phase active power (a) and [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: European inter-area oscillation: instantaneous single-phase active [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: IEEE 39-bus: instantaneous single-phase active power (a) and three [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
read the original abstract

Modern power systems are at risk of largely reducing the inertia of generation assets and prone to experience extreme dynamics. The consequence is that, during electromechanical transients triggered by large contingencies, transmission of electrical power may take place in a broad spectrum well beyond the single fundamental component. Traditional modeling approaches rely on the phasor representation derived from the Fourier Transform (FT) of the signal under analysis. During large transients, though, FT-based analysis may fail to accurately identify the fundamental component parameters, in terms of amplitude, frequency and phase. In this paper, we propose an alternative approach relying on the Hilbert Transform (HT), that, in view of the possibility to identify the whole spectrum, enables the tracking of signal dynamics. We compare FT- and HT-based approaches during representative operating conditions, i.e., amplitude modulations, frequency ramps and step changes, in synthetic and real-world datasets. We further validate the approaches using a contingency analysis on the IEEE 39-bus.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes using the Hilbert Transform (HT) as an alternative to Fourier Transform (FT)-based phasor modeling for power system signals during large electromechanical transients. It argues that FT may fail to accurately identify fundamental component parameters (amplitude, frequency, phase) when power transmission occurs over a broad spectrum, while HT enables tracking of signal dynamics by identifying the whole spectrum. Validation is performed on synthetic signals (amplitude modulations, frequency ramps, step changes), real-world datasets, and a contingency analysis on the IEEE 39-bus system.

Significance. If the HT approach demonstrably outperforms FT for broad-spectrum transients without violating narrowband assumptions, it could improve dynamic modeling and monitoring in low-inertia systems. The inclusion of IEEE 39-bus contingency validation and real-world data strengthens applicability claims, but the absence of quantitative error metrics, spectrum width measurements, or direct comparison of instantaneous frequency validity limits assessment of practical impact.

major comments (2)
  1. [Abstract] Abstract: The central claim that 'FT-based analysis may fail to accurately identify the fundamental component parameters' during large transients while HT succeeds 'in view of the possibility to identify the whole spectrum' is load-bearing but unsupported by any quantitative results, error analysis, or spectrum characterization in the provided abstract. The tested synthetic cases (amplitude modulation, frequency ramps, step changes) are narrowband by construction; no evidence is given that the IEEE 39-bus post-contingency signals satisfy the conditions for valid HT instantaneous frequency extraction.
  2. [Abstract] Abstract (and implied methods): The HT approach for extracting instantaneous frequency from the analytic signal is valid only under the Bedrosian theorem (no spectral overlap between amplitude modulation and carrier) and for asymptotically narrowband signals. The manuscript invokes broad-spectrum transmission post-contingency but does not report spectrum occupancy, octave span, or incommensurate components for the IEEE 39-bus case, leaving open whether extracted 'whole spectrum' parameters are physical or artifacts.
minor comments (2)
  1. [Abstract] Abstract: The phrasing 'in view of the possibility to identify the whole spectrum' is vague; a brief statement of how the full spectrum is extracted and used for dynamics tracking would improve clarity.
  2. [Abstract] Abstract: No numerical results (e.g., tracking error, frequency deviation metrics) are reported despite claims of comparison and validation; adding at least one quantitative highlight would strengthen the summary.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below, focusing on the abstract and the conditions for HT applicability.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that 'FT-based analysis may fail to accurately identify the fundamental component parameters' during large transients while HT succeeds 'in view of the possibility to identify the whole spectrum' is load-bearing but unsupported by any quantitative results, error analysis, or spectrum characterization in the provided abstract. The tested synthetic cases (amplitude modulation, frequency ramps, step changes) are narrowband by construction; no evidence is given that the IEEE 39-bus post-contingency signals satisfy the conditions for valid HT instantaneous frequency extraction.

    Authors: The abstract summarizes the paper's scope, while quantitative error metrics (e.g., amplitude, frequency, and phase estimation errors) and direct FT-HT comparisons appear in Sections IV and V for the synthetic cases, real datasets, and IEEE 39-bus contingency. We agree that the abstract would benefit from explicit quantitative support for the central claim. We will revise the abstract to include key error reductions and performance highlights from the results. revision: yes

  2. Referee: [Abstract] Abstract (and implied methods): The HT approach for extracting instantaneous frequency from the analytic signal is valid only under the Bedrosian theorem (no spectral overlap between amplitude modulation and carrier) and for asymptotically narrowband signals. The manuscript invokes broad-spectrum transmission post-contingency but does not report spectrum occupancy, octave span, or incommensurate components for the IEEE 39-bus case, leaving open whether extracted 'whole spectrum' parameters are physical or artifacts.

    Authors: The synthetic test signals were constructed to comply with the Bedrosian theorem and narrowband assumptions to enable valid comparisons. For the IEEE 39-bus case, the manuscript shows practical tracking of dynamics via HT. We acknowledge the value of explicit spectrum characterization and will add quantitative measures of spectral occupancy and bandwidth for the post-contingency signals in the revised manuscript to confirm applicability. revision: yes

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper proposes an HT-based alternative to FT phasor modeling for tracking power system dynamics during broad-spectrum transients, validated on synthetic cases (amplitude modulation, frequency ramps, step changes) and IEEE 39-bus contingency. No equations, parameter fits, self-citations, or uniqueness theorems are quoted that reduce any claimed prediction or result to the inputs by construction. The central comparison rests on external benchmark datasets and operating conditions independent of the method itself, satisfying the criteria for a self-contained derivation with no load-bearing circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities; assessment limited to summary description.

pith-pipeline@v0.9.0 · 5711 in / 1021 out tokens · 26624 ms · 2026-05-25T15:43:57.990246+00:00 · methodology

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Reference graph

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