## 2 PQD Feature Detection System Model Using WPTM-Based Metrics  
### Signal Model  
The occurrence of a single PQD event in a power grid can be a major cause of internal (or external) grid failure or severely damage various precision instruments, devices, equipment, and factories using the grid energy. Table 1 lists the several major PQD events that may often occur within the grid, such as sag, swell, harmonic, interruption, and notch. A power signal delivered through a power cable can be represented as  
\[x(t)=s(t)+n(t)=s_{0}(t)+q(t)+n(t) \tag{1}\]  
where \(s(t)\) (\(=s_{0}(t)\) [pure power source] \(+q(t)\) [distortion component]) is a PQD event signal and \(n(t)\) is an additive noise signal. In the absence of any PQD event, that is, \(q(t)=0\), \(s(t)\) represents the standard (pure) sinusoidal electric power signal, that is, \(s(t)=s_{0}(t)=A\sin(\omega t)\), where the peak value of the power signal \(A=220\sqrt{2}\) V and its radian frequency \(\omega=2\pi f\) rad/s (\(f=1/T\) [=electric signal period] = 60 or 50 Hz). However, when a PQD event occurs, that is, \(q(t)\
eq 0\), \(s(t)=s_{0}(t)+q(t)\), which becomes one of the PQD event signal models listed in Table 1[9]. In these signal models, \(\tau_{1}\) and \(\tau_{2}\) indicate the starting and ending edge points of the PQD events, respectively. The signal model parameters, such as \(\tau_{1}\), \(\tau_{2}\), and \(\alpha\) (amplitude ratio), as well as the power cable noise component \(n(t)\), can be intrinsically random. \begin{table}
\begin{tabular}{c c c} \hline \hline
**PQD** & **PQD Event Signal Models** & **Parameters** \\ \hline Sag & \(\begin{array}{c}s(t)=s_{0}(t)+q(t)\\ q(t)=-\alpha A(u(t-\tau_{1})-u(t-\tau_{2}))\sin(\omega t),\\ \tau_{1}<\tau_{2},\quad u(t)=1,\quad t\geq 0\end{array}\) & \(\begin{array}{c}0.1<\alpha\leq 0.9\\ 0.5T\leq\tau_{2}-\tau_{1}\leq 30T\end{array}\) \\ \hline Swell & \(\begin{array}{c}s(t)=s_{0}(t)+q(t)\\ q(t)=\alpha A(u(t-\tau_{1})-u(t-\tau_{2}))\sin(\omega t),\\ \tau_{1}=\tau_{2}-\tau_{1}<30T\end{array}\) \\ \hline Harmonics & \(\begin{array}{c}s(t)=s_{0}(t)+q(t)\\ q(t)=A(u(t-\tau_{1})-u(t-\tau_{2}))\sin(\omega t)[\alpha_{3}\sin(3\omega t)+ \alpha_{5}\sin(5\omega t)],\\ \end{array}\) & \(\begin{array}{c}0.1<\alpha\leq 0.8\\ 0.5T\leq\tau_{2}-\tau_{1}\leq 30T\end{array}\) \\ \hline Interruption & \(\begin{array}{c}s(t)=s_{0}(t)+q(t)\\ q(t)=-aA(u(t-\tau_{1})-u(t-\tau_{2}))\sin(\omega t)\end{array}\) & \(\begin{array}{c}0.1<\alpha\leq 0.4\\ 0.01T\leq\tau_{2}-\tau_{1}\leq 0.05T\end{array}\) \\ \hline Notch & \(\begin{array}{c}s(t)=s_{0}(t)+q(t)\\ q(t)=-\alpha sign(\sin(\omega t))A(u(t-\tau_{1})-u(t-\tau_{2}))\end{array}\) & Steady state \\ \hline Swell or & \(\begin{array}{c}s(t)=s_{0}(t)+q(t)\\ \text{Sag with}\\ harmonics \end{array}\) & \(\begin{array}{c}s(t)=s_{0}(t)+q(t)\\ q(t)=\pm\alpha A(u(t-\tau_{1})-u(t-\tau_{2}))\end{array}\) & \(\begin{array}{c}0.1<\alpha\leq 0.9\\ \sin(5\omega t)[\alpha_{3}\sin(3\omega t)+\alpha_{5}\sin(5\omega t)],\\ \end{array}\) & \(\begin{array}{c}0.5T\leq\tau_{2}-\tau_{1}\leq 30T\end{array}\leq\alpha_{3}\leq 0.20\leq \alpha_{5}\leq\\ 0.02\) \\ \hline Oscillatory & \(\begin{array}{c}(t)=s_{0}(t)+q(t)\\ q(t)=A\Big{(}\alpha*e^{-\frac{(t-\tau_{1})}{\tau}}*u(t-\tau_{1})-u(t-\tau_{2}) \Big{)}\end{array}\) & \(\begin{array}{c}0.1<\alpha\leq 0.8\\ 0.5T\leq\tau_{2}-\tau_{1}\leq 3T\end{array}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Power quality disturbance (PQD) event signal models and parameters. The additive noise \(n(t)\) in (1) is the sum of the Gaussian noise component \(n_{a}(t)\) and impulsive noise component \(n_{i}(t)\)[2]:  
\[n(t)=n_{a}(t)+n_{i}(t), \tag{2}\]  
where the impulsive noise component \(n_{i}(t)\), which mainly results from the power switching ON/OFF or man-made EMI, may severely hinder the PQD signal detection (or localization) [2]. In case of Gaussian noise, the typical magnitude of noise is less than 1% (i.e., SNR = 40 dB) of the voltage magnitude [9].