# Novel Faulted-Section Location Method for Active Distribution Networks of New-Type Power Systems
## 3 Improved Matrix Algorithm
### Improved Matrix Algorithm Verification  
The improved matrix algorithm cannot accurately locate multiple faults with T-connected sections. This section proposes a verification method based on the traditional matrix algorithm and superposition principle. According to \(I^{\prime}=\left|\,L^{\prime}D^{\prime}\,\right|\), the matrix \(I^{\star}\) can be derived from the matrix \(L^{\prime}\) and the matrix \(D^{\prime}\). Determine whether the calculated matrix \(I^{\star}\) is the same as the matrix \(I\) brought by equipment status information. If they are the same, the positioning result is correct. Otherwise, further fault location is performed in the next section. Each positioning result of the improved matrix algorithm is verified. The calculation steps of the verification method are as follows:  
1. Write the feeder section status matrix \(L^{\prime}\) according to the positioning result.
2. When analyzing one of the power supplies, the correlation matrix \(D^{\prime}\) considering the causality between the faulted section and the alarm of the switch, has the same structure as the matrix shown in Equation (2). Matrix \(D^{\prime}\) is defined as follows: \[D^{\prime}_{ij}=\left\{\begin{array}{rl}1,\text{ When section $i$ fail, node $j$ alarms in the positive direction}\\ 0,\text{ When section $i$ fail, node $j$ does not send an alarm}\\ -1,\text{ When section $i$ fail, node $j$ alarms in the opposite direction}\end{array}\right.\] (18)  
When analyzing one of the power supplies, the nodes nearest to the other power supplies are regarded as disconnected. Delete the rows corresponding to the disconnected nodes in the network description matrix \(D\) to attain the network description matrix \(D_{i}\) in the square matrix form. When describing the same system, the network description matrix and the correlation matrix contain the same line topology. Through calculation, we find that the two matrices of the same system are inverse matrices to each other. Therefore, the correlation matrix of the system can be calculated using the following equation:  
\[D^{\prime}_{i}=D^{-1}_{i} \tag{19}\]  
Each power supply is analyzed separately to obtain \(m_{s}-m_{l}+1\) correlation matrices \(D_{i}^{\prime}\).
3. Define the following logical operation. Matrix \(K\) and matrix \(M\) are two matrices of the same dimension: \[K=\parallel M\parallel\] (20)  
\[K_{ij}=\left\{\begin{array}{rl}1,\ M_{ij}>0\\ 0,\ M_{ij}=0\\ -1,\ M_{ij}<0\end{array}\right. \tag{21}\]  
Same as the traditional matrix algorithm principle, using the equation:  
\[I_{i}=L^{\prime}D^{\prime}_{i} \tag{22}\]  
to obtain \(m_{s}-m_{l}+1\) fault information matrix \(I_{i}\).
4. Each matrix \(I_{i}\) must be supplemented with \"0\" at the nodes it does not contain to obtain a matrix \(I_{i}^{\star}\) of \(1\times m_{s}\). Then, through the equation: \[I^{\star}=I^{\star}_{1}+I^{\star}_{2}+\ldots+I^{\star}_{m_{S}-m_{l}+1}\] (23)  
obtain the fault information matrix \(I^{\star}\). Determine whether the matrix \(I^{\star}\) and the matrix \(I\) obtained from the equipment status information are the same.  
The flow chart of the verification is shown in Figure 3.