Dynamic Lissajous patterns for real time identification and localization of power quality disturbance

Lissajous-based frequency-dependent analysis for power quality event detection

The Lissajous patterns of voltage and current signals exhibit frequency-dependent changes under various power quality (PQ) events. By introducing dynamic modulation and advanced frequency mapping techniques, the method enhances real-time identification and localization of PQ disturbances, including inter-harmonics, frequency drift, and transient frequency deviations. To capture frequency variations in real-time, a time-varying modulation approach is applied to the Lissajous figure. This dynamic modulation adapts the geometric attributes of the Lissajous figure to represent instantaneous changes in frequency. The area and shape of the Lissajous figure evolve dynamically as the frequency of the signal changes, providing insights into transient and steady-state frequency deviations. To capture frequency variations in real-time, a time-varying modulation approach is applied to the Lissajous figure. This dynamic modulation adapts the geometric attributes of the Lissajous figure to represent instantaneous changes in frequency. The area and shape of the Lissajous figure evolve dynamically as the frequency of the signal changes, providing insights into transient and steady-state frequency deviations. For enhanced interpretability, the Lissajous pattern is augmented with frequency-based color-coding, transforming the figure into a frequency map. Each point is color-tagged based on its instantaneous frequency, derived through time-frequency decomposition (e.g., Hilbert transform or STFT).

Warm colours (red/orange): Represent high-frequency components (e.g., inter-harmonics, switching transients).
Cool colours (blue/green): Indicate low-frequency variations or normal operating conditions.

This color-coded frequency map enables intuitive visualization of frequency drift, transient spikes, and inter-harmonic propagation—patterns often difficult to detect via conventional spectral methods.

The instantaneous area of the Lissajous figure at time t is calculated using:

$$\:{A}_{v-i}=\underset{i(\tau\:=t-T)}{\overset{i(\tau\:=t)}{\int\:}}v\left(\tau\:\right)di\left(\tau\:\right)$$

(1)

where:

To detect changes, the area at the current time t is compared with the area at the previous time t − Δt where Δt is the time step:

$$\:{A}_{v-i}(t-\varDelta\:t)=\underset{i\left(\tau\:=t-T-\varDelta\:t\right)}{\overset{i\left(\tau\:=t-\varDelta\:t\right)}{\int\:}}v\left(\tau\:\right)di\left(\tau\:\right)$$

(2)

Frequency mapping of lissajous patterns for visualization

To visualize the frequency variations dynamically, the Lissajous figure is enhanced with frequency-based color-coding. The instantaneous frequency is determined using advanced signal processing techniques, and each point of the Lissajous figure is assigned a colour corresponding to its frequency at that moment. This creates a dynamic frequency map where:

High frequencies are represented with warmer colours (e.g., red, orange).
Low frequencies are depicted with cooler colours (e.g., blue, green).

The color-coded Lissajous figure provides an intuitive representation of frequency drift, inter-harmonics and transient events.

Frequency-Dependent Similarity Index for Lissajous Figures.

The similarity index (SI) is modified to incorporate frequency-dependent changes:

$$\:SI\left(t\right)=1-\frac{\left|{A}_{v-i}\left(t,f\right)-{A}_{v-i}\left(t-\varDelta\:t,f\right)\right|}{max\left\{{A}_{v-i}\left(t,f\right)-{A}_{v-i}\left(t-\varDelta\:t,f\right)\right\}}$$

(2)

where Δt is the time difference between successive intervals.

Under normal operating conditions, the successive areas of the Lissajous figure are nearly identical, resulting in an SI value close to 1. During a PQ disturbance, the area changes significantly, causing the SI to deviate from 1. By choosing an appropriate threshold for the SI, the disturbance can be reliably detected. While the conventional SI provides reliable detection of PQ disturbances in static conditions, its performance diminishes when frequency variations, such as inter-harmonics, frequency drift, or transient frequency deviations, are introduced. The conventional SI is derived purely from the geometric area and does not account for the influence of frequency changes on the Lissajous pattern. During events involving overlapping PQ disturbances and frequency drift, the similarity index may yield values close to the threshold, leading to false positives or missed detections. Modern microgrids experience rapid changes in frequency due to distributed energy sources and varying load profiles, which are not adequately captured by the area-based SI. To address the aforementioned challenges, a frequency-dependent Similarity Index (FDSI) is proposed. This new formulation incorporates instantaneous frequency variations into the calculation of the Lissajous figure’s area, enabling the detection of events influenced by frequency-related disturbances. By dynamically modulating the Lissajous figure and integrating frequency-dependent attributes, the proposed FDSI overcomes the limitations of the conventional method, providing improved sensitivity and robustness in detecting PQ disturbances under dynamic grid conditions. This frequency-weighted SI enhances detection sensitivity for frequency-related disturbances. By combining dynamic modulation and frequency mapping of Lissajous figures, the method provides precise detection and localization of PQ disturbances. This approach is particularly effective for identifying events involving frequency drift or inter-harmonics ensuring robust monitoring of power quality in modern microgrids. This innovative Lissajous-based frequency-dependent analysis empowers utilities to track, visualize, and mitigate frequency-related PQ issues, enhancing grid stability and reliability.