## 2 A Proposed ASPC Approach for OCRs in a DG-Integrated Distribution Network
### Coordination Constraints of OCRs
#### 2.3.3. Calculating the Min-Max Confidence Interval of Fault Currents in the DN
* Solving the power flow problem in case of a ring-type distribution network Figure 4 shows a single-line diagram of meshed distribution network. A new branch is connected from bus 5 to bus 6, hence, the equivalent currents at all buses are calculated as the following. Equation (19) can be re-formulated by:  
\[[\Delta V]=[BV]=\left[ \begin{array}{c}V_{1}\\ V_{1}\\ V_{1}\\ V_{1}\\ V_{1}\\ 0\end{array} \right]-\left[ \begin{array}{c}V_{2}\\ V_{3}\\ V_{4}\\ V_{5}\\ V_{6}\\ 0\end{array} \right]=\left[ \begin{array}{cccccc}Z_{12}&0&0&0&0&0\\ Z_{12}&Z_{23}&0&0&0&0\\ Z_{12}&Z_{23}&Z_{34}&0&0&0\\ Z_{12}&Z_{23}&Z_{34}&Z_{45}&0&0\\ Z_{12}&Z_{23}&0&0&Z_{36}&0\\ 0&0&Z_{34}&Z_{45}&-Z_{36}&Z_{56}\end{array} \right]\left[ \begin{array}{c}B_{1} \\ B_{2}\\ B_{3}\\ B_{4}\\ B_{5}\\ B_{6}\end{array} \right] \tag{24}\]  
\[\left[ \begin{array}{c}\Delta V\\ 0\end{array} \right]=[Z_{ncv}]\left[ \begin{array}{c}BC\\ B_{ncv}\end{array} \right] \tag{25}\]  
Generally, a new branch \(B_{k}\) is built on a distribution network to create a ring topology; specifically, the new branch is connected from the bus \(i\) to the bus \(j\). The initial \([Z]\) matrix can be re-calculated by the following steps: (i) adding a new row to the initial \([Z]\) matrix and each element of the added new row is calculated by subtracting the \(j\)-th row from the \(i\)-th row; (ii) an element of the new row that belongs to a diagonal of \([Z]\) matrix is also an intersectional position of the new row and the new column of the \([Z]\) matrix; and it will be an impedance of the new branch \(B_{k}\). In addition, Equation (14) is also re-written as follows:  
\[\left[ \begin{array}{c}B_{1}\\ B_{2}\\ B_{3}\\ B_{4}\\ B_{5}\\ B_{6}\end{array} \right]=\left[ \begin{array}{cccccc}1&1&1&1&1&0\\ 0&1&1&1&1&0\\ 0&0&1&1&0&1\\ 0&0&0&1&0&1\\ 0&0&0&0&1&-1\\ 0&0&0&0&0&1\end{array} \right]\left[ \begin{array}{c}I_{2}\\ I_{3}\\ I_{4}\\ I_{5}\\ I_{6}\\ B_{6}\end{array} \right] \tag{26}\]  
\[\left[ \begin{array}{c}BC\\ B_{ncv}\end{array} \right]=\left[C_{new}\right]\left[ \begin{array}{c}I\\ B_{new}\end{array} \right] \tag{27}\]  
A new branch \(B_{k}\) is built in a meshed distribution network, which is connected from the bus \(i\) to the bus \(j\). The \([C]\) matrix is then re-calculated by the following steps: (i) copying the elements of the column related to the \(i\)-th bus to the \(k\)-th column (called a new column) and subtracting the elements of the column related to the \(j\)-th bus from this new column; and (ii) adding the value (+1) to the \(k\)-th row and \(k\)-th column in the \([C]\) matrix. Figure 4: A typical single-line diagram of meshed distribution network. Substituting two Equations (27) and (25) into Equation (22), it shows that:  
\[\left[ \begin{array}{c}\Delta V_{new}\\ 0\end{array} \right]=[Z_{new}][C_{new}]\Big{[} \begin{array}{c}I\\ B_{new}\end{array} \Big{]}=\left[ \begin{array}{c}AM^{T}\\ MN\end{array} \right]\left[ \begin{array}{c}I\\ B_{new}\end{array} \right] \tag{28}\]  
Then, the Kron reduction method is applied to Equation (28), it becomes:  
\[[\Delta V_{new}]=\Big{[}A-M^{T}N^{-1}M\Big{]}[I]=[PF_{new}][I] \tag{29}\]  
As a result, Equation (29) with the \([PF_{new}]\) matrix can be solved by the LU factorization method as mentioned in Equation (23). (iv) An Adaptable Fault Analysis Technique  
According to the work in [6], a simplified and automated fault-current calculation approach can be appropriately modified for this study. The total short-circuit current observed by the \(r\)-th OCR for a faulted location in the DG-contained DN is calculated by Equation (30). The configuration of DG-based distribution network is changeable due to two plug-and-play and peer-to-peer characteristics of DGs, so a real-time management system (RTMS) using the SCADA function is deployed to observe the operating states of DGs as well as the working status of protective devices like OCRs and other devices such as CBs, tie switches, load break switches, or reclosers in the DN.