## 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
In general, Equation (14) can be re-written as the following:  
\[[BC]=[C][I] \tag{18}\]  
Figure 3: A typical single-line diagram of DN with the integration of distributed generators. where a \([C]\) matrix is an upper triangular matrix filled by \"0\" or \"1\". Next, the relationship of branch-currents matrix \([BC]\) and bus-voltages matrix \([BV]\) is shown in Equation (19):  
\[[\Delta V]=[BV]=\left[ \begin{array}{c}V_{1}\\ V_{1}\\ V_{1}\\ V_{1}\\ V_{1}\end{array} \right]-\left[ \begin{array}{c}V_{2}\\ V_{3}\\ V_{4}\\ V_{5}\\ V_{6}\end{array} \right]=\left[ \begin{array}{ccccc}Z_{12}&0&0&0&0\\ Z_{12}&Z_{23}&0&0&0\\ Z_{12}&Z_{23}&Z_{34}&0&0\\ Z_{12}&Z_{23}&Z_{34}&Z_{45}&0\\ Z_{12}&Z_{23}&0&0&Z_{36}\end{array} \right]\left[ \begin{array}{c}B_{1}\\ B_{2}\\ B_{3}\\ B_{4}\\ B_{5}\end{array} \right] \tag{19}\]  
\[V_{j}=V_{i}-B_{i}Z_{\rm ij}\to B_{i}Z_{\rm ij}=V_{i}-V_{j},\ \ \forall i,j=1,2,\ldots,N;\ i\
eq j \tag{20}\]  
which \(V_{i}\) is the \(i\)-th bus voltage; \(V_{j}\) is the \(j\)-th bus-voltage; and \(Z_{\rm ij}\) is the line impedance between bus \(i\) and bus \(j\). Equation (19) can be generally re-written as follows. \[[\Delta V]=[BV]=[Z][BC] \tag{21}\]  
where \([\Delta V]\), or called \([BV]\), is the matrix of voltage drop between the bus \(i\) and the bus \(j\) on the DN; and \([Z]\) is a lower triangular matrix of line impedances in the system. In summary, when the min-max confidence interval of load power at the \(l\)-th load bus, \([P_{load\_l\_min},P_{load\_l\_max}]\), has been already calculated/forecasted, the min-max confidence interval of load current, \([I_{load\_l\_min},I_{load\_l\_max}]\), will be then appropriately determined. Next, the min-max confidence interval of equivalent current injected into the \(i\)-th bus, is also promptly defined. After that, the \([BC_{\rm min}]\) and \([BC_{\rm max}]\) matrices are calculated by Equation (18), with \([BC_{\rm min}]=[B_{1\_{\rm min}}B_{2\_{\rm min}}\ \ldots\ B_{\rm{i\_{min}}}]^{T}\) and \([BC_{\rm max}]=[B_{1\_{\rm max}}B_{2\_{\rm max}}\ \ldots\ B_{\rm{i\_{max}}}]^{T}\). Finally, the \([\Delta V_{\rm min}]\) and \([\Delta V_{\rm max}]\) voltage-drop matrices will be determined by Equation (21). As a result, the min-max confidence interval of bus voltage at any \(i\)-th bus on the DN, \([V_{i\_{\rm min}},V_{i\_{\rm max}}]\), can be effectively calculated, specifically for \([V_{1}]-[B_{i\_{\rm min}}]=[V_{i\_{\rm max}}]\) and \([V_{1}]-[B_{i\_{\rm max}}]=[V_{i\_{\rm min}}]\) in this study. It is worth noting that \(V_{1}\) is the nominal voltage value of bus 1 (also called the slack bus) in the distribution network. * Solving the power flow problem in case of a radial distribution network To obtain the \([\Delta V]\) matrix, Equation (18) is substituted to Equation (21) as the following: \[[\Delta V]=[Z][C][I]=[PF][I]\] (22)  
The \([PF]\) matrix in Equation (22) can be determined by a Lower-Upper (LU) factorization solution. The \([PF]\) matrix can be factorized by two \([Z]\) and \([C]\) matrices, where \([C]\) is the upper triangular matrix and \([Z]\) is the lower triangular matrix. The use of LU-decomposition and forward/backward algorithms allows skipping the calculation of Jacobian matrices or admittance matrices. Therefore, the computation time of power flows can be significantly reduced. As a result, the LU-decomposition-based power-flow analysis can be adaptable to the online power-flow analysis for a DG-integrated distribution system, as referred to Equation (23). \[\left\{ \begin{array}{l}I_{i}^{k}=I_{i}^{real}\Big{(}V_{i}^{k}\Big{)}+jI_{i} ^{imag}\Big{(}V_{i}^{k}\Big{)}=\left(\frac{P_{i}+|Q_{i}}{V_{i}^{k}}\right)^{*} \\ \left[\Delta V_{i}^{k+1}\right]=[PF]\Big{[}I_{i}^{k}\Big{]}\\ \left[PF\right]=[Z][C]\\ \left[V_{i}^{k+1}\right]=[V_{1}]-\left[\Delta V_{i}^{k+1}\right]\end{array} \right. \tag{23}\]  
where \(k\) is the iteration number of the current-injection-based power-flow analysis method.