## 2 PQD Signal Visualization Processing Based on Lissajous Trajectory
### Sequence Visualization via Lissajous Trajectory  
In mathematics, Lissajous curves refer to the trajectories of a family of curves synthesized by two sinusoidal vibrations in mutually perpendicular directions. Due to their diversity in image generation, this concept is often applied in the field of signal and image processing [15]. In recent years, the application scope of Lissajous curves has gradually expanded in power system engineering, such as for identifying fault locations in transmission lines [16] or monitoring the health status of power systems [17]. However, the two synthesis components selected in the construction of Lissajous curves in existing studies are both voltage waveforms and current waveforms, which are not applicable to the PQD classification issue targeted in this paper. This is because the research object of this topic only targets the single physical quantity of voltage and its single-phase data. To the best of our knowledge, no prior study has applied Lissajous synthesis technology to PQD classification research. To fully extract the sequence feature information of single-phase voltage signals, this paper proposes a new Lissajous curve concept called the Lissajous trajectory. This concept is motivated by the observation that the LT not only offers comprehensive and precise temporal sequence feature extraction capabilities for PQD signals but also has rapid visualization conversion efficiency. The LT is constructed by synchronously synthesizing curves at each time point, using an ideal reference signal and the target disturbance signal as the vertical and horizontal coordinates in a Cartesian coordinate system, respectively, resulting in the LT representation corresponding to the target PQD signal. When no disturbance occurs, the LT corresponding to a normal signal forms a standard circular trajectory. Once a disturbance occurs, the shape of the corresponding LT of the voltage signal will change, and the specific shape after the change will depend on the type of PQDs. Specifically, an ideal reference signal, \(U_{\text{ISP}}\), with an amplitude of 1 and a phase different from the target signal, is constructed, which differs from the target signal by a phase, \(\varphi\), where \(\varphi\) is set to \(90^{\circ}\) (the lead or lag of phase angle only change the direction of \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Label** &
\begin{tabular}{c} **Type of** \\ **PQDs** \\ \end{tabular} & \multicolumn{2}{c}{**Mathematical Model**} & \multicolumn{2}{c}{**Parameters**} \\ \hline C0 & Normal & \(f(t)=\sin(\omega t)\) & \(\omega=2\pi t_{b}f_{c}=5\text{GHz}\) \\ C1 & Sag & \(f(t)=\{1-a\varepsilon(t-t_{1})-\varepsilon(t-t_{2})\}\)\(\sin(\omega t)\) & \(0.1\leq a\leq 0.9T\leq t_{b}-t_{1}\leq 9T\) \\ C2 & Swell & \(f(t)=\{1+a\varepsilon(t-t_{1})-\varepsilon(t-t_{2})\}\)\(\sin(\omega t)\) & \(0.1\leq a\leq 0.8T\leq t_{b}-t_{1}\leq 9T\) \\ C3 & Interruption & \(f(t)=\{1-a\varepsilon(t-t_{1})-\varepsilon(t-t_{2})\}\)\(\sin(\omega t)\) & \(0.9\leq a\leq 1.5T\leq t_{b}-t_{1}\leq 9T\) \\ C4 & Harmonic & \(f(t)=\sin(\omega t)+\frac{1}{a\varepsilon(t-t_{2})\}\)\(\sin(\omega t)\) & \(0.05\leq a_{i}\leq 0.15\) \\ C5 & Oscillatory & \(0.1<a\leq 0.8T\leq t_{b}-t_{1}<3T\) \\  & transient & \(f(t)=\sin(\omega t)+a\varepsilon(t-t_{1})-\varepsilon(t-t_{2})\) & \(0.008\leq T\leq t_{b}-t_{1}<3T\) \\ C6 & Pulse & \(f(t)=\sin(\omega t)+a\varepsilon(t-t_{1})-\varepsilon(t-t_{2})]\) & \(1\leq a\leq 3.1\leq a_{b}=t_{1}=3 ms\) \\ C7 & Flicker & \(0.1<a\leq 0.2\), \(0.1\leq b\leq 0.5\) \\ C8 & Notch & \(f(t)=\sin(\omega t)-\text{sgn}(\sin(\omega t))\times\{\sum\limits_{n=0}^{b}k( \varepsilon(t-t_{n}+0.02n))-\varepsilon(t-(t-0.02n))\}\) & \(0.01T\leq t_{b}-t_{1}\leq 0.06T\) \\ C9 & Spike & \(f(t)=\sin(\omega t)+\text{sgn}(\sin(\omega t))\times\{\sum\limits_{n=0}^{b}k( \varepsilon(t-(t_{b}+0.02n))-\varepsilon(t-(t-t_{2}+0.02n))\}\) & \(0.1\leq b\leq 0.40\leq t_{b}\leq 0.5T\) \\  & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mathematical models of PQDs.