## 9 Conclusions  
A protection algorithm supported by the \(S\)-transform and the WDF is introduced in this manuscript for the identification and classification of transmission line faults. The merit of the  
Figure 12: FI at the time of inductive load switching.  
Figure 13: Fault index for the L-G fault (**a**) computed using the proposed algorithm and (**b**) computed using the algorithm reported in [24].  
S-transform having the minimum influence of noise and the WDF analyzing the data over a specified window width are combined in this approach. The proposed S-index calculated from the S-transform depending on the decomposition of the current signal and the W-index obtained from the WDF of the current signal over a quarter cycle are combined to obtain the FI. This FI is utilized to detect different fault conditions by comparing the peak values with the threshold. Faults' categorization is obtained using the number of fault phases in an event. The classification of the LL fault and the LL-G fault is achieved using the proposed GFI, which is based on the WD of the zero sequence current over the quarter cycle. The proposed technique is performed by testing the algorithm in different case studies such as the variation of the fault impedance, the fault incidence angle, the location, the reverse power flow, the presence of noise, and the fault on the hybrid PTL, which has sections of underground cable and overhead line. The algorithm is also analyzed for operating events such as switching off the capacitor bank and inductive load. This algorithm can obtain efficient protection of the transmission link using the data recorded at one end, eliminating the requirement of the communication channel and GPS. It is observed that the developed technique is efficient in the identification and classification of the faults on the power transmission line in the period of a quarter cycle. Furthermore, it is less complex than previous algorithms and easy to implement.  
Conceptualization, A.K., O.P.M. and M.K.G.; methodology, A.K. and O.P.M.; software, A.K. and O.P.M.; validation, A.K., O.P.M. and B.K.; formal analysis, A.K., O.P.M., and B.K.; investigation, A.K. and O.P.M.; resources, A.K., O.P.M. and M.K.G.; data curation, A.K., O.P.M., and B.K.; writing, original draft preparation, A.K. and O.P.M.; writing, review and editing, A.K. and O.P.M.; visualization, B.K., M.K.G., H.H.A. and P.S.; supervision, O.P.M., M.K.G., H.H.A. and P.S. All authors read and agreed to the published version of the manuscript.  
The authors declare no conflict of interest.  
The abbreviations used in this article are detailed below:  
\begin{tabular}{l l} CSAE & Convolutional sparse auto-encoder \\ FDOST & Fast discrete orthogonal Stockwell transform \\ FI & Fault index \\ GFI & Ground fault index \\ GFTH & Ground fault index threshold value \\ GPS & Global Positioning System \\ GSS & Grid substations \\ HT & Hilbert transform \\ L-G & Phase-to-ground fault \\ LL & Two phase fault \\ LL-G & Two phases-to-ground fault \\ LLL-G & Three phases-to-ground fault \\ MATLAB & Matrix Laboratory \\ PTL & Power transmission line \\ SAV & Sum absolute values \\ SNR & Signal-to-noise ratio \\ S-transform & Stockwell transform \\ ST & Stockwell transform \\ TH & Fault threshold \\ UG & Underground \\ WD & Wigner distribution \\ WDF & Wigner distribution function \\ WPT & Wavelet packet transform \\ WT & Wavelet transform \\ \end{tabular}