## 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
The Current-Injection-Based Power Flow Analysis  
After the min-max confidence thresholds of load current at each load bus have been appropriately determined, they will be used for calculating voltage-fluctuating thresholds at all buses of the DG-based distribution network through the current-injection-based power-flow (CIBPF) analysis [83; 84; 85; 86]. Then, the calculated voltage-fluctuating thresholds will be input to an adaptable fault analysis technique as mentioned in Section of An Adaptable Fault Analysis Technique. The modified (CIBPF) analysis method uses two matrices which are (i) branch currents (BC) matrix and (ii) bus voltages (BV) matrix. By considering the \(i\)-th bus in the distribution network, the power injected into this \(i\)-th bus can be expressed by Equation (12). \[S_{i}=(P_{i}+jQ_{i})=(P_{G,i}-P_{L,i})+j(Q_{G,i}-Q_{L,i}),\forall i=1,2,\dots,N \tag{12}\]  
where \(P_{G,i}\) and \(Q_{G,i}\) are the active and reactive power of a generation source at the \(i\)-th bus, respectively; and \(P_{L,i}\) and \(Q_{L,i}\) are the active and reactive power of a load connected to the \(i\)-th bus, respectively. It is noted that a distribution network is assumed to have \(N\) buses. The injection of equivalent current to the \(i\)-th bus at the \(k\)-th iteration of the power flow analysis is calculated by Equation (13) as below:  
\[I_{i}^{k}=I_{i}^{real}\Big{(}V_{i}^{k}\Big{)}+jl_{i}^{imag}\Big{(}V_{i}^{k} \Big{)}=\Bigg{(}\frac{P_{i}+jQ_{i}}{V_{i}^{k}}\Bigg{)}^{*},\forall i=1,2,\dots,N \tag{13}\]  
where \(V_{i}^{k}\) and \(I_{i}^{k}\) are the bus voltage and the equivalent current injected to the \(i\)-th bus at the \(k\)-th iteration, respectively; \(I_{i}^{real}\) and \(I_{i}^{imag}\) are real and imaginary components of the equivalent current injected to the \(i\)-th bus, which are considered as a function of \(V_{i}^{k}\). By considering a DG-integrated distribution network as in Figure 3, the injected-to-bus power can be properly converted to the injected-to-bus current by conducting Equation (13). According to [84; 86], the relationship between the \([BC]\) matrix and the \([BV]\) matrix can be obtained through Kirchhoff laws. The \([BC]\) matrix is determined by the injected-to-bus currents, as shown in Equation (14). \[\left[ \begin{array}{c}B_{1}\\ B_{2}\\ B_{3}\\ B_{4}\\ B_{5}\end{array} \right]=\left[ \begin{array}{c}1\;1\;1\;1\;1\\ 0\;1\;1\;1\;1\\ 0\;0\;1\;1\;0\\ 0\;0\;0\;1\;0\\ 0\;0\;0\;0\;1\\ \end{array} \right]\left[ \begin{array}{c}I_{2}\\ I_{3}\\ I_{4}\\ I_{5}\\ I_{6}\end{array} \right] \tag{14}\]  
\[\left\{ \begin{array}{l}I_{6}=I_{\text{load}6}-I_{\text{DC}2}\\ I_{5}=I_{\text{load}5}-I_{\text{DC}1}\\ I_{4}=I_{\text{load}4}\\ I_{3}=I_{\text{load}3}\\ I_{2}=I_{\text{load}2}\end{array} \right. \tag{15}\]\[I_{load\_J}\in\left\{\frac{P_{load\_J\_min}}{V_{nom}};\frac{P_{load\_J\_max}}{V_{nom }}\right\};\;\;l=1\ldots N_{load} \tag{16}\]  
\[I_{DG\_d}=\left(\frac{P_{DG\_d}+jQ_{DG\_d}}{V_{i}}\right)^{*};\;\;d=1\ldots N_{DG} \tag{17}\]  
which \(B_{1},\,B_{2},\,B_{3},\,\,B_{4}\) and \(B_{5}\) are the branch currents on the distribution network; \(I_{2},\,I_{3},\,\,I_{4},\,\,I_{5}\) and \(I_{6}\) are the equivalent currents injected into the bus; \(I_{load\_J}\) is the load current at a \(l\)-th load bus on the DN, with \(l=1,\ldots,N_{load}\); \(N_{load}\) is the total number of load buses; \(I_{DG\_d}\) is the source current of the \(d\)-th DG injected to the bus, with \(d=1,\ldots,N_{DG}\); \(N_{DG}\) is the total number of distributed generators penetrated to the DN; \(P_{load\_J\_min}\) and \(P_{load\_J\_max}\) are the minimum and maximum thresholds of load power at the \(l\)-th load bus, respectively, as calculated in Section of Forecasting a Min-Max Confidence Interval of Load Power at Each Load Bus; \(V_{nom}\) is the nominal bus-voltage value of the distribution network; \(P_{DG\_d}\) and \(Q_{DC\_d}\) represent the active and reactive power of the \(d\)-th DG, respectively; and \(V_{i}\) is the \(i\)-th bus voltage on the distribution system.