# Fault Sensing Using Fractal Dimension and Wavelet
## 4 The Developed Maximum Wavelet Coefficient (DMWC) Recognition Algorithm  
In this section, we develop the DMWC recognition algorithm for the single-phase grounded faults (DMWCSP) and the two-phase grounded faults (DMWCTP). Similar to the previous IFBD recognition algorithm, the three phase currents were chosen as the original signals of the online power cable for the DMWC recognition algorithm. In the fault state, the three phase components are asymmetric and interlocked, so the calculation and analyses are complex. After the K-transform below, we obtained the \(\alpha\) modulus, the \(\beta\) modulus, and the 0 modulus of the three phases. \[\left\{ \begin{array}{l}i_{0}=\frac{1}{3}\left(i_{A}+i_{B}+i_{C}\right)\\ i_{\alpha}=\frac{1}{3}\left(i_{A}-i_{B}\right)\\ i_{\beta}=\frac{1}{3}\left(i_{A}-i_{C}\right)\end{array} \right. \tag{9}\]  
where \(i_{A}\), \(i_{B}\), and \(i_{C}\) are the three phase currents of the online power cable. The modulus components of 0, \(\beta\), and \(\alpha\) for the different faults are shown in Table 3. It can be seen that the different classes of faults correspond to different patterns of the modulus \(\alpha\), modulus \(\beta\), and the modulus 0. The DMWC algorithm of DMWCSP and DMWCTP are developed below. Based on the experiments, the wavelet Db3 were used to analyze the modulus components of \(\alpha\), \(\beta\), and 0 for the different faults. The maximum wavelet coefficients \(I_{0}\), \(I_{\alpha}\), and \(I_{\beta}\) are listed in Table 4 for the 10 types of short circuit faults. The flow chart of the calculation of the maximum wavelet coefficients [31] is shown in Figure 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type of Fault** & **Boundary Condition** & **0 Modulus** & **\(\alpha\) Modulus** & \(\beta\) **Modulus** \\ \hline Grounded fault of phase \(A\) & \(i_{b}=i_{c}=0\) & \(i_{a}\) & \(i_{a}\) & \(i_{a}\) \\ Grounded fault of phase \(B\) & \(i_{a}=i_{c}=0\) & \(i_{b}\) & \(-i_{b}\) & 0 \\ Grounded fault of phase \(C\) & \(i_{a}=i_{b}=0\) & \(i_{c}\) & 0 & \(-i_{c}\) \\ Phase-between fault of \(A\) and \(B\) & \(i_{a}+i_{b}=0,i_{c}=0\) & 0 & \(2i_{a}\) & \(i_{a}\) \\ Phase-between fault of \(A\) and \(C\) & \(i_{a}+i_{c}=0,i_{b}=0\) & 0 & \(-i_{b}\) & \(i_{b}\) \\ Phase-between fault of \(B\) and \(C\) & \(i_{c}+i_{b}=0,i_{a}=0\) & 0 & \(i_{a}\) & \(2i_{a}\) \\ Grounded faults of \(A\) and \(B\) & \(i_{c}=0\) & \(i_{a}+i_{b}\) & \(i_{a}-i_{b}\) & \(i_{a}\) \\ Grounded faults of \(A\) and \(C\) & \(i_{b}=0\) & \(i_{a}+i_{c}\) & \(i_{a}\) & \(i_{a}-i_{c}\) \\ Grounded faults of \(B\) and \(C\) & \(i_{a}=0\) & \(i_{b}+i_{c}\) & \(-i_{b}\) & \(-i_{c}\) \\ Grounded fault of \(A\), \(B\), and \(C\) & \(i_{a}+i_{b}+i_{c}=0\) & 0 & \(i_{a}-i_{b}\) & \(i_{a}-i_{c}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Modulus components of \(\alpha\), \(\beta\), and 0 for the different faults. The proposed DMWCSP algorithm is described below. 1. Calculate the modulus components \(i_{0}\), \(i_{a}\), and \(i_{\beta}\) of the three phase currents \(i_{A}\), \(i_{B}\), and \(i_{C}\) by using the K-transform. 2. Calculate the maximum wavelet coefficients \(I_{a}\) and \(I_{\beta}\) of the modulus components \(i_{a}\) and \(i_{\beta}\) by using the Wavelet transform. 3. If \(I_{a}<0.01\), then the fault is classified as the grounded fault of \(C\). Otherwise, go to the next step. 4. If \(I_{\beta}<0.01\), then the fault is classified as the grounded fault of \(B\). Otherwise, go to the next step. 5. The fault is classified as the grounded fault of \(A\). Similar to the proposed DMWCSP algorithm, the proposed DMWCTP algorithm was described below to support DMWC recognition algorithm. 1. Calculate the modulus components \(i_{0}\), \(i_{a}\), and \(i_{\beta}\) of the three phase currents \(i_{A}\), \(i_{B}\), and \(i_{C}\) by using the K-transform. 2. Calculate the maximum wavelet coefficients \(I_{0}\) of the modulus components \(i_{0}\) by using the Wavelet transform. 3.