## 3 Numerical Results and Discussions
### Single- and Double-Edge-Based Detection  
We evaluate the performance of the proposed scheme in terms of both double- and single-edge-based detection. In Table 4, the single-edge-based detection performance of both the proposed algorithm (denoted as \"Prop\") and conventional WC-Det algorithm (\"Conv\") is compared over either Gaussian noise channels (AWGN) or impulsive noise channels (IMPN). As the simulated PQD event, a swell signal with 120% \(pu\) and \(\varphi_{m}=\pi\) (for fixed PQD) or \(\varphi_{m}\sim U[0,\ 2\pi)\) (for random PQD) is used. As indicated in Table 4, the proposed scheme has approximately 9-10% higher DR and 2-5% lower FER for random PQD over AWGN, as compared to WC-Det. In contrast, the proposed scheme has approximately 14-15% higher DR and approximately 47% lower FER for random PQD over IMPN, as compared to WC-Det. This indicates that the proposed scheme has an improved FER and DR compared to WC-Det, especially for random PQD. For fixed PQD, the proposed scheme is slightly better (8% lower over IMPN) in terms of FER than WC-Det while maintaining the same DR. Comparing MV and RMS, especially for random PQD, while MV is relatively better than RMS in terms of DR, RMS is relatively better than MV in terms of FER. While assuming single-edge-based PQD detection, Table 4 shows that the proposed (MV or RMS) scheme has a similar DR for fixed PQD but a slightly improved DR for random PQD, as compared to the existing scheme (WC-Det). In terms of FER (for both fixed PQD and random PQD), the proposed scheme is superior to the existing scheme, especially over IMPN. In most previous studies, including [PERSON]'s papers [1; 28], only single-edge-based PQD detection has been used. However, for precise PQD localization, double edge-based detection identifying both (start and end) edges of a PQD signal is necessary. In the simulation, we set the allowable limit of error (ALE) time of double-edge-based detection to be within \(\pm 7.5\times 10^{-5}\) s (\(\pm 15\) samples). This means that the PQD detection position (time) can deviate as much as the ALE time from the actual position of PQD occurrence. The accuracy of PQD detection position (time) decreases as the ALE time increases. Table 5 compares the DR of double-edge-based PQD over AWGN or IMPN channel, where SNR = 35 dB or 40 dB is assumed, between the proposed and existing schemes (WC-Det and W-Den). For the simulated PQD, we generate either (a) a sag signal with 80% \(pu\) or (b) a swell signal with 120% \(pu\). Table 5 shows that, in case of fixed PQD, there is no large difference between the existing and proposed schemes in terms of DR. However, in case of random PQD, the proposed scheme has much more improved DR over AWGN or IMPN than the existing wavelet-based schemes (WC-Det and W-Den). Specifically, for signal (a) over AWGN with SNR = 35 dB and 40 dB, the proposed MV/RMS exhibits 5.9%/5.4% and 8.8%/8.7% (or 34.5%/34% and 16.8%/16.7%) better DR, respectively, than WC-Det (or W-Den). For the signal (b) over AWGN with SNR = 35 dB and 40 dB, the proposed MV/RMS exhibits 7.7%/6.9% and 9.1%/8.9% (or 34.6%/33.8% and 23.6%/23.4%) better DR, respectively, than WC-Det (or W-Den). When considering random PQD over IMPN, the proposed scheme is more dominant than the conventional wavelet-based schemes. For instance, for the signal (a) over IMPN with SNR = 35 dB and 40 dB, the proposed MV/RMS exhibits 26.9%/26.4% and 13.5%/13.2% (or 48.9%/48.4% and 30.9%/30.6%) better DR, respectively, than WC-Det (or W-Den). For the signal (b) over IMPN with SNR = 35 dB and 40 dB, the proposed MV/RMS exhibits 27.4%/25.4% and 13.4%/13% (or 52.6%/50.6% and 32.2%/31.8%) better DR, respectively, than WC-Det (or W-Den). In Table 5, the proposed scheme is also compared with the S-transform-based scheme [3; 27] as a non-wavelet-based detection. We confirm that the S-transform exhibits much lower DR (even worse than those of WC-Det and W-Den), especially in the case of random PQD, as compared to the proposed scheme.