insight - Computational Complexity - # Instantaneous Frequency Estimation in Unbalanced and Single-Phase Power Systems

Core Concepts

A new instantaneous frequency estimation formula based on affine differential geometry that is particularly suited for unbalanced and single-phase power systems.

Abstract

The paper presents a novel approach to instantaneous frequency estimation in power systems using the concepts of affine differential geometry. The key contributions are:

- Derivation of expressions for the affine arc length and curvature in terms of the voltage of an AC system.
- Formulation of an instantaneous frequency estimation formula as a function of affine geometric invariants.
- Demonstration of the effectiveness of the proposed formula for unbalanced three-phase systems and single-phase systems, showing improved performance compared to conventional phase-locked loop (PLL) methods and the Frenet frame-based approach.

For stationary sinusoidal voltages, the paper shows that the proposed affine geometry-based formula can precisely estimate the exact frequency. For time-varying voltages, the formula provides a good approximation under certain conditions on the rate of change of voltage magnitude and frequency.

The paper also presents several examples and real-world measurements to illustrate the advantages of the proposed approach over PLL and Frenet frame-based methods, particularly in unbalanced and single-phase scenarios.

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arxiv.org

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The paper provides the following key equations and figures:
Equation (18): Instantaneous frequency formula for stationary sinusoidal voltages in terms of affine geometric invariants.
Equation (22): Approximated instantaneous frequency formula for time-varying unbalanced voltages.
Figures 1-5: Voltage waveforms and frequency estimation results for various balanced, unbalanced, and single-phase voltage scenarios.

Quotes

"The paper discusses the relationships between electrical and affine differential geometry quantities, establishing a link between frequency and time derivatives of voltage, through the utilization of affine geometric invariants."
"Based on this link, a new instantaneous frequency estimation formula is proposed, which is particularly suited for unbalanced and single-phase systems."

Key Insights Distilled From

by Ali Alshawab... at **arxiv.org** 09-26-2024

Deeper Inquiries

The proposed affine geometry-based frequency estimation approach can be extended to handle complex power system phenomena by incorporating additional geometric invariants and adapting the mathematical framework to account for non-linearities introduced by harmonics, inter-harmonics, and sub-synchronous oscillations.
Harmonics and Inter-Harmonics: To address harmonics, the affine curvature can be modified to include the effects of harmonic components in the voltage signals. This can be achieved by analyzing the voltage trajectory in the affine plane and decomposing it into its harmonic constituents using techniques such as Fourier analysis. By applying the affine differential geometry framework to each harmonic component, one can derive a more comprehensive expression for instantaneous frequency that accounts for the contributions of both fundamental and harmonic frequencies.
Sub-Synchronous Oscillations: For sub-synchronous oscillations, the approach can be enhanced by integrating time-varying parameters into the affine curvature calculations. This involves extending the model to include dynamic changes in voltage and current that occur during transient events, such as faults or system disturbances. By utilizing real-time measurements and applying adaptive filtering techniques, the affine geometry framework can be adjusted to track the instantaneous frequency variations caused by sub-synchronous oscillations.
Robustness and Filtering: Implementing robust filtering techniques to mitigate the effects of noise and disturbances will be crucial. The affine geometry-based method can be combined with advanced signal processing techniques, such as wavelet transforms or Kalman filtering, to enhance the accuracy of frequency estimation in the presence of harmonics and other complex phenomena.
By systematically addressing these complexities, the affine geometry-based frequency estimation approach can be made more versatile and applicable to a wider range of power system scenarios.

Affine differential geometry offers a rich mathematical framework that can be applied to various power system analysis and control tasks beyond frequency estimation. Some potential applications include:
Voltage Stability Analysis: The geometric properties of voltage trajectories can be utilized to assess voltage stability in power systems. By analyzing the curvature and other geometric invariants of voltage profiles, one can identify critical points and predict voltage collapse scenarios.
Dynamic System Modeling: Affine differential geometry can aid in the modeling of dynamic behaviors in power systems, such as transient stability and oscillatory dynamics. By representing system states as curves in an affine space, one can derive insights into system behavior during disturbances and design appropriate control strategies.
Control System Design: The principles of affine geometry can be applied to the design of advanced control systems for power converters and grid-connected devices. By leveraging geometric invariants, control algorithms can be developed to ensure robust performance under varying operating conditions, including unbalanced loads and harmonics.
State Estimation: Affine differential geometry can enhance state estimation techniques by providing a geometric interpretation of state variables. This can lead to improved algorithms for estimating system states in real-time, particularly in the presence of measurement noise and uncertainties.
Fault Detection and Diagnosis: The geometric framework can be employed to develop fault detection and diagnosis algorithms. By analyzing the geometric properties of voltage and current waveforms, one can identify anomalies and potential faults in the system, leading to timely interventions.
These applications highlight the versatility of affine differential geometry in addressing various challenges in power system analysis and control, ultimately contributing to more reliable and efficient power system operations.

Yes, the insights from the geometric interpretation of frequency can be leveraged to develop innovative power system monitoring and protection schemes. Here are several ways this can be achieved:
Real-Time Monitoring: The geometric approach provides a framework for real-time monitoring of voltage and frequency dynamics. By continuously analyzing the geometric properties of voltage trajectories, operators can gain insights into system behavior, detect anomalies, and respond to disturbances more effectively.
Adaptive Protection Schemes: The understanding of frequency as a geometric invariant allows for the design of adaptive protection schemes that can adjust their settings based on real-time system conditions. For instance, protection relays can utilize geometric metrics to differentiate between normal operating conditions and fault scenarios, leading to more accurate and timely tripping decisions.
Enhanced Fault Detection: By applying the geometric interpretation of frequency, new algorithms can be developed for fault detection that are sensitive to changes in the curvature of voltage trajectories. This can improve the detection of transient faults and reduce the risk of false positives in protection systems.
Integration with Smart Grid Technologies: The insights gained from affine differential geometry can be integrated into smart grid technologies, enabling more sophisticated monitoring and control capabilities. For example, the geometric framework can be used to enhance the functionality of phasor measurement units (PMUs) and other smart sensors, providing a deeper understanding of system dynamics.
Predictive Maintenance: The geometric analysis of frequency can also contribute to predictive maintenance strategies. By monitoring the geometric characteristics of voltage and current waveforms, operators can identify trends and patterns that indicate potential equipment failures, allowing for proactive maintenance actions.
In summary, the geometric interpretation of frequency not only enriches the theoretical understanding of power system dynamics but also paves the way for practical advancements in monitoring and protection schemes, ultimately enhancing the reliability and resilience of power systems.

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