## 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
\right],V_{i}(0)=V_{i\_\text{max}},\forall i\in N\]  
where \(V_{f,i,k}\) is the fault voltage of the \(i\)-th bus for a \(k\) faulted location in the DN; \(V_{i}(0)\) is the min/max pre-fault bus-voltage at the \(i\)-th bus; the values \(I_{f,grid-r}^{\min}\) and \(I_{f,grid-r}^{\max}\) are equal the values \(I_{f,grid-i}^{\min}\) and \(I_{f,grid-i}^{\max}\) respectively, because each bus \(i\) has already installed the \(r\)-th OCR in the system; \(\left[Z_{\text{bus}}\right]_{N\times N}\) is the Thevenin impedance matrix with a size of \(N\times N\); and \(\left[Z_{f,k}\right]_{N\times N}\) is the fault impedance at the \(k\) faulted location; normally, the faulted location is considered at the buses of a distribution system. For the asymmetrical faults, the positive- negative-, and zero-sequence Thevenin impedance matrices \(\left[Z_{bus}^{012}\right]_{N\times N}\) need to be calculated. The subscripts 0, 1, 2 are represented as zero-, positive-, and negative-sequence components, respectively. Hence, Equation (32) can be re-written as the following. \[\left\{ \begin{array}{l}V_{f,i,k}^{012}=V_{i}^{012}(0)-\left(\left[Z_{bus}^{0 12}\right]_{N\times N}+\left[Z_{f,k}\right]_{N\times N}\right)\left[\begin{array} []{l}I_{f,grid-1,k}^{012,\min}\\ \ldots\\ I_{f,grid-i,k}^{012,\min}\\ \ldots\\ I_{f,grid-N,k}^{012,\min}\end{array}\right],V_{i}^{012}(0)=V_{i\_\text{min}}^{0 12},\forall i\in N\\ \\ V_{f,i,k}^{012}=V_{i}^{012}(0)-\left(\left[Z_{bus}^{012}\right]_{N\times N}+ \left[Z_{f,k}\right]_{N\times N}\right)\left[\begin{array}{l}I_{f,grid-1,k}^{ 012,\max}\\ \ldots\\ I_{f,grid-i,k}^{012,\max}\\ I_{f,grid-N,k}^{012,\max}\end{array}\right],V_{i}^{012}(0)=V_{i\_\text{max}}^{0 12}\forall i\in N\end{array} \right. \tag{33}\]  
* Determining fault-current division coefficients of DGs in the DN In general, the min-max confidence thresholds of grid fault current seen by the \(r\)-th relay for a given fault, \(I_{f,grid-r}^{\min}\) and \(I_{f,grid-r}^{\max}\), can be used as the basis for parameter settings of the \(r\)-th relay in the DN. A fault-current division coefficient of the \(i\)-th RBDG impacting the operation of the \(r\)-th relay is expressed by: \[\left\{ \begin{array}{l}k_{ri}^{\min}=\frac{I_{f,DGI-r}^{\min}}{I_{f,DGI}^{ \min}}=\left(\frac{V_{i\_\min}}{Z_{ir}}\right)/I_{f,DGI}^{\max},\forall i\
eq r \\ k_{ri}^{\max}=\frac{I_{f,DGI-r}^{\max}}{I_{f,DGI}^{\min}}=\left(\frac{V_{i\_ \max}}{Z_{ir}}\right)/I_{f,DGI}^{\min},\forall i\
eq r\\ k_{ri}=1\text{ for }i=r\end{array} \right.\] (34) where \(k_{ri}^{\min}\) and \(k_{ri}^{\max}\) are the minimum and maximum fault-current division coefficients of the \(i\)-th RBDG with respect to the \(r\)-th relay, respectively; it is assumed that the \(i\)-th RBDG is connected to the \(i\)-th bus in the DN, so \(Z_{ir}\) is the impedance from the bus \(i\) to the bus \(r\) that is determined from the \(\left[Z_{\text{bus}}\right]_{N\times N}\) matrix. In the case of \(i=r\), the \(k_{ri}\) coefficient is unit; and the value \(k_{ri}\) is in an available range of [0-1]. On the other hand, the fault-current division coefficient \(k_{ri}\) of the \(i\)-th IBDG impacting the operation of the \(r\)-th relay can be expressed by Equation (35):  
\[\left\{ \begin{array}{l}k_{i-downstream\_r}=\frac{Z_{i-upstream\_r}}{Z_{i- downstream\_r}+Z_{i-upstream\_r}}\\ k_{i-upstream\_r}=\frac{Z_{i-downstream\_r}}{Z_{i-downstream\_r}+Z_{i-upstream \_r}}\\ k_{ri}=1\text{ for a radial topology}\end{array} \right. \tag{35}\]  
where the subscripts '\(downstream\_r\)' and '\(upstream\_r\)' represent the fault current direction of the \(i\)-th IBDG flowing into the downstream and upstream sites of the \(r\)-th relay respectively, with regard to a given fault location in the DN. When a fault occurs in the distribution system, the IBDG will limit its fault current contribution and operate as a fault current source. Therefore, the coefficient \(k_{ri}\) equals to 1 for a radial topology of DN.