# Fault Location Method of Multi-Terminal Transmission Line Based on Fault Branch Judgment Matrix
## 3 Traveling Wave Head Detection Algorithm Based on CEEMDAN-TEO
### Selection of IMFs  
CEEMDAN's improved decomposition of the target signal results in IMF components derived from its highest to lowest instantaneous frequencies. Due to the existence of spurious components, in order to extract sensitive elements that can characterize the fault of the target signal, this paper makes additional improvements at the principle level by using Pearson coefficients to calculate the correlation degree between two variables and combining it with ranking entropy to achieve the accurate selection of IMF components.  
For the time series \(y(t)\) and \(x(t)\), the correlation coefficient is given in Equation (12).  
\[\rho_{xy}=\frac{cov(x,y)}{\delta_{x}\delta_{y}} \tag{12}\]where \(cov(x,y)\) is the covariance of \(x(t)\) and \(y(t)\), \(\delta_{x}\) and \(\delta_{x}\) and \(\delta_{y}\) are the standard deviations of \(x(t)\) and \(y(t)\).  
The ranking entropy is a measure of the complexity of a time series, and the method is simple and fast to calculate and robustly and accurately capture the moment of sudden change of the fault signal.  
By defining a time series as \(\{M(i),i=1,2,\ldots,N\}\), matrix M is obtained using phase space reconstruction given in Equation (13):  
\[\begin{bmatrix}m(1)&m(1+\tau)&\cdots&m(1+(q-1)\tau)\\ m(2)&m(2+\tau)&\cdots&m(2+(q-1)\tau)\\ m(j)&m(j+\tau)&\cdots&m(j+(q-1)\tau)\\ \vdots&\vdots&\cdots&\vdots\\ m(n)&m(n+\tau)&\cdots&m(n+(q-1)\tau)\end{bmatrix} \tag{13}\]  
where \(\tau\) represents the delay time, \(q\) represents the embedding dimension, n denotes the number of reconstructed components, and \(m(j)\) is the jth row component of the reconstruction matrix. The jth reconstructed row component of the matrix M is arranged in ascending order according to the principle from smallest to largest. The corresponding position index of each element in the vector is obtained \(j_{1},j_{2},\ldots,j_{q}\), that is.  
\[m(j+(j_{1}-1)\tau)\leq x(j+(j_{2}-1)\tau)\leq\cdots\leq x\big{(}j+(j_{q}-1)t \big{)} \tag{14}\]  
If there are equal values in the components of the phase space reconstruction matrix, they are arranged according to the original ordinal numbers. For each row of the q-dimensional phase space reconstruction matrix, a sequence of notations can be obtained \(\big{\{}j_{1},j_{2},\ldots,j_{q}\big{\}}\). There are a total of q! permutations possible, and the probability of generating each sequence of notation is derived \(P_{1},P_{2},\ldots,P_{n}\). At this point, the permutation entropy PE of the sequence _M(i)_ can be calculated according to the formula of information entropy, i.e.,  
\[PE=-\sum_{j=1}^{n}P_{j}ln\big{(}P_{j}\big{)} \tag{15}\]  
In Equation (15), \(P_{j}\) is the probability of producing the jth sequence. When \(P_{j}=1/q!\), \(PE\) has the maximum value in (15). In general, the \(PE\) value is normalized to \(PE=PE/ln(q!)\), and the alignment entropy of this time series _M(i)_ is the normalized \(PE\) value.  
Based on the above analysis, the correlation-alignment entropy function is constructed and given in Equation (16).  
\[V=Max\big{(}\rho_{xy}*PE\big{)} \tag{16}\]  
The improved CEEMDAN algorithm decomposes the target signal to obtain several IMF components and then processes them, i.e., calculates the correlation coefficients and the alignment entropy values of these components, respectively, and then substitutes into the correlation-alignment entropy function constructed by Equation (16). This allows the IMF-sensitive components that can best characterize the fault features of the target signal to be selected by this function. This method can improve the accuracy of fault feature recognition and the extraction of weak fault feature signals under solid noise interference.