## 2 The UKF Detection of PQDs
### Analysis of Anti-Noise Detection Performance
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Label** & **Events** & **Mathematical Equations** & **Parameters Limits** \\ \hline C1 & Swell & \(y(t)=\{1+\beta[u(t-t_{1})-u(t-t_{2})]\ \text{sin}(\omega t)\}\) & \(\beta=0.1\sim 0.9\), \\  & & \(T<t_{2}-t_{1}<8T\) \\ \hline C2 & Sag & \(y(t)=\{1-\beta[u(t-t_{1})-u(t-t_{2})]\ \text{sin}(\omega t)\}\) & \(\beta=0.1\sim 0.9\), \\  & & \(T<t_{2}-t_{1}<8T\) \\ \hline C3 & Interruption & \(y(t)=\{1-\beta[u(t-t_{1})-u(t-t_{2})]\ \text{sin}(\omega t)\}\) & \(\beta=0.9\sim 1\), \\  & & \(T<t_{2}-t_{1}<8T\) \\ \hline C4 & Fluctuations & \(y(t)=\{1+a_{\text{f}}\text{sin}(\beta_{cut})\}\sin(\omega t)\) & \(a_{t}=0.1\sim 0.3\), \\  & & \(\beta_{t}=0.1\sim 0.5\) \\ \hline C5 & Pulse & \(y(t)=\text{sin}(\omega t)+\beta[u(t-t_{1})-u(t-t_{2})]\) & \(\beta=1\sim 3\), \\  & & \(1\,\text{ms}\leq t_{2}-t_{1}\leq 3\) ms \\ \hline C6 & Oscillatory & \(y(t)=\text{sin}(\omega t)+\lambda e^{-(t-t_{1})/\tau}\sin\mu\omega(t-t_{1})[u(t -t_{1})-u(t-t_{2})]\) & \(\lambda=0.1\sim 0.8\), \\  & & \(\mu=10\sim 40\), \\  & & \(0.5T<t_{2}-t_{1}<3T\) \\ \hline C7 & Harmonics & \(y(t)=\text{sin}(\omega t)+\sum_{i=3,5=t-1}^{2+1}b_{\text{f}}\sin(\omega t(t-t_{ 1}))\) & \(1\leq n\leq 3\), \\  & & \(i=3,5,7\) \\ \hline C8 & Harmonics + Swell & \(y(t)=\{1+\beta[u(t-t_{1})-u(t-t_{2})]\ \text{sin}(\omega t)+\sum_{i=3,5=t-1}^{2+1}b_{\text{f}}\sin(i \omega(t-t_{3}))\}\) & \(\begin{array}{c}\beta=0.1\sim 0.9\text{,}\\ T<t_{2}-t_{1}<8T\text{,}\\ \end{array}\) \\ \hline C9 & Harmonics + Sag & \(y(t)=\{1-\beta[u(t-t_{1})-u(t-t_{2})]\ \text{sin}(\omega t)+\sum_{i=3,5=t-1}^{2+1}b_{\text{f}}\sin(i \omega(t-t_{3}))\}\) & \(\begin{array}{c}\beta=0.1\sim 0.9\text{,}\\ T<t_{2}-t_{1}<8T\text{,}\\ \end{array}\) \\ \hline C10 & Oscillatory + Sag & \(y(t)=\{1-\beta[u(t-t_{1})-u(t-t_{2})]\ \text{sin}(\omega t)\}\) & \(\begin{array}{c}\beta=0.1\sim 0.9\text{,}\\ T<t_{2}-t_{1}<8T\text{,}\\ \end{array}\) \\  & & \(\begin{array}{c}\lambda=0.1\sim 0.8\text{,}\\ \mu=10\sim 40\text{,}\\ 0.5T<t_{4}-t_{3}<3T\end{array}\) \\ \hline \end{tabular}
\end{table} 
Table 2: The mathematical models of fourteen kinds of PQDs events. \begin{table}
\begin{tabular}{c c c c} \hline
**Label** & **Events** & **Mathematical Equations** & **Parameters Limits** \\ \hline C11 & Oscillatory + Interruption & \(\begin{array}{l}y(t)=\{1-\beta[u(t-t_{1})-u(t-t_{2})]\ \}\sin(\omega t)\\ +\lambda e^{-(t_{0})/t}\sin i\omega e(t-t_{3})[u(t-t_{3})-u(t-t_{4})]\\ \end{array}\) & \(\begin{array}{l}\beta=0.9\sim 1.\\ T\lesssim t_{2}-t_{1}\lesssim 8T,\\ \lambda=0.1\sim 0.8,\\ n=10\sim 40.\\ 0.5T\lesssim t_{4}-t_{5}\lesssim 3T\end{array}\) \\ \hline C12 & Harmonics + Oscillatory & \(\begin{array}{l}y(t)=\sin\omega t+\lambda e^{-(t_{0}-t_{1})/t}\sin i\omega e (t-t_{1})[u(t-t_{1})-u(t-t_{2})]\\ +\sum_{t=0,t=1}^{2t+1}b\sin(i\omega(t-t_{3}))\\ \end{array}\) & \(\begin{array}{l}\lambda=0.1\sim 0.8,\\ \mu=10\sim 40,\\ 0.5T<t_{2}-t_{1}<3T,\end{array}\) \\ \hline C13 & Harmonic + Swell & \(\begin{array}{l}y(t)=\{1+a_{t}\sin(\omega t)\}(1+\beta[u(t-t_{1})-u(t-t_{2})] )\sin(\omega t)\\ +\sum_{t=0,t=1}^{2t+1}b\sin(i\omega(t-t_{3}))\\ \end{array}\) & \(\begin{array}{l}1\leq\pi\leq 3,i=3,5.7,\\ \beta=0.1\sim 0.9,\\ T<t_{2}-t_{1}<8T,\\ \beta=0.1\sim 0.9\end{array}\) & \(\begin{array}{l}\phi=0.1\sim 0.9,\\ T<t_{2}-t_{1}<8T,\\ \lambda=0.1\sim 0.8,\\ n=10\sim 40.\\ 0.5T<t_{2}-t_{1}<3T,\\ \end{array}\) \\ \hline C14 & \(\begin{array}{l}\mbox{Harmonic + }\\ \mbox{Oscillatory + Sag }\\ +\sum_{t=3}^{2t+1}b\sin(i\omega(t-t_{3}))\\ \end{array}\) & \(\begin{array}{l}y(t)=(1-\beta[u(t-t_{1})-u(t-t_{2})])\sin(\omega t)\\ +\lambda e^{-(t_{0}-t_{1})/t}\sin i\omega(t-t_{3})[u(t-t_{3})-u(t-t_{4})]\\ +\sum_{t=3}^{2t+1}b\sin(i\omega(t-t_{5}))\\ \end{array}\) & \(\begin{array}{l}\lambda=0.1\sim 0.8,\\ n=10\sim 40,\\ n=10\sim 40,\\ 0.5T<t_{2}-t_{1}<3T,\\ 1\leq n\leq 3,i=3.5,7\end{array}\) \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._