## 7 Fault Location Method Based on Similarity of Voltage Signals
### Obsevance of the First Traveling Wave
Figure 14: Protective relaying signal for high-voltage direct current (HVdc) link under faulty conditions. Morphological Gradient Based Fault Estimation in Multi-Terminal High-Voltage Direct Current (MT-HVdc) Systems  
Mathematical morphology is a promising approach to highlight the features of the wavefronts involved in the estimation of fault location and classification. Mathematical morphology involves little computational efforts and smaller data windows compared to wavelet transform [93; 94; 95; 96]. Spatial structures analyzed with theoretical and technical knowledge of mathematical morphology [83]. It is based on set theory, integral geometry, and lattice algebra. It is a non-linear technique in contrast to linear Fourier and wavelet transforms. It deals with the waveforms in time domain rather than in the frequency or time frequency domain. Mathematical morphology is a highly effective tool for fault identification and classification because of its robustness features involved in the preservation of waveforms or shapes suppressed by noise [97; 98]. Appropriate structure of a set is extracted from the main function of mathematical morphological operators. Interaction between the set and another set, known as structuring element (SE), is used for extraction. An a-priori knowledge about the shape of signal is employed to predefine the shape of structuring element. A pair of dual transformation is formed based on dilation and erosion [96]. Arithmetical difference between dilation and erosion of a signal of structuring element is known as morphological gradient (MG). It is an effective technique for edge detection. So, it can be employed to extract features of transient behavior (composed of sharp peaks and valleys) of MT-HVdc system under faulty conditions [98; 99]. Figure 15: Protection scheme for bipolar high-voltage direct current (HVdc) transmission system. Suppose that signal and structuring element are denoted by \(x\) and \(y\), respectively. Length of structuring element is assumed to be considerably shorter than the length of signal. Dilation, erosion and morphological gradient are defined as, respectively:  
\[f\oplus g = \max_{s}\left\{f(x+s)+g(s)\big{|}(x+s)\in\ D_{f},\ s\in D_{g}\right\} \tag{84}\] \[f\ominus g = \max_{s}\left\{f(x+s)-g(s)\big{|}(x+s)\in\ D_{f},\ s\in D_{g}\right\}\] (85) \[MG(f)=(f\oplus g)-(f\ominus g) \tag{86}\]  
where \(\oplus\) is dilation operator and \(\ominus\) is erosion operator. \(f(n)\) is assumed as processing signal whose domain is \(D_{f}=1,\ 2,\ \ldots,\ M-1\). \(g(n)\) is assumed to be a one-dimensional structuring element whose domain is \(D_{g}=1,\ 2,\ \ldots,\ M-1\). \(MG(f)\) is morphological gradient of signal \(f(n)\). In case of fault in any line of MT-HVdc system, travelling waves are generated which start propagating towards the converter stations in both directions where fault sensing equipment is deployed. Voltage and current at converter stations are the summation of backward and forward travelling waves [97; 100]. In a lossless transmission line, voltage and current values are:  
\[|x_{1}|>M_{1} \tag{80}\]0  
\[|x_{1}|>M_{1} \tag{80}\]1  
where \(z=\sqrt{\frac{k}{k}}\) and \(v=1/\sqrt{LC}\). \(f_{1}\) and \(f_{2}\) are forward and backward projected travelling waves respectively. \(x\) is the distance from the fault location. \(z\) is the characteristics impedance of transmission line and \(v\) is the propagation velocity. \(f_{1}\) and \(f_{2}\) are evaluated by the values of voltage and current at starting instant of fault. Values of voltage and current at the instant of initiation of fault are dependent on fault location, fault resistance and other line parameters. Traveling waves are partly transmitted and reflected back towards the fault point until they are attenuated to a negligible value.