## 2 A Proposed ASPC Approach for OCRs in a DG-Integrated Distribution Network
### Objective Function of Overcurrent Protection Coordination  
The objective is to find the optimal value of \(TDS_{i}\) and three \(A\), \(B\), and \(C\) coefficients of each OCR, while \(I_{pu\_i}\) and \(I_{f\_ik}\) are properly selected from fault analysis results of the DG-based distribution network. The coordination problem of OCRs is to minimize the total operating time of all relays working as the primary protection but still satisfying certain constraints and maintaining the reliability of the protection system. The objective function is expressed by Equation (2) as follows:  
\[\text{min}Z=\sum_{i=1}^{n}T_{F51\_ik,pri} \tag{2}\]where \(n\) is the total number of relays; and \(T_{F51\_ik,pri}\) is the operating time of the \(i\)-th OC relay at the \(k\) faulted location working as the primary protection. This study proposes to only consider one faulted point/location for each protected line/zone, normally at the remote end of the protected line/zone where the minimum fault current can be calculated as the basis to select a pick-up current threshold for the primary OC relay of this protected line/zone in the distribution network, as referred to Section 2.3.4.  
On the other hand, it is most common to use the OCR pairs, e.g., one primary relay and one backup relay for each OCR pair, to detect and clear a given fault in the protected line/zone. However, to improve the scalability and selectivity of an overcurrent relay coordination system in the DN having DGs, this study proposes to use the OCR triples, i.e., one for the primary relay and two other consecutive ones for the backup relays. The total tripping time of the OCR triple is formulated as follows:  
* For the upward direction of fault current to the relay \(R_{i}\): \[2*\text{CTI}^{\text{min}}\leq t_{F51,(upward-upward-i)k,backup}+t_{F51,(upward -i)k,backup}+t_{F51,ik,pri}\leq 2*\text{CTI}^{\text{max}}\] (3)
* For the backward direction of fault current to the relay \(R_{i}\): \[2*\text{CTI}^{\text{min}}\leq t_{F51,(backward-backward-i)k,backup}+t_{F51,(backward-i)k,backup}+t_{F51,ik,pri} \leq 2*\text{CTI}^{\text{max}}\] (4)  
where CTI is a coordination time interval between two adjacent relays, commonly \(\text{CTI}\in\left\{\text{CTT}^{\text{min}},\text{CTI}^{\text{max}}\right\} =\{0.2,0.5\}\)[57,58]; \(t_{F51,ik,pri}\) is the operation time of the \(i\)-th relay \(R_{i}\) when it works as primary protection for a \(k\) faulted location in the protected line/zone; \(t_{F51,(upward-i)k,backup}\) is the backup operating time of the upward adjacent relay of the relay \(R_{i}\); \(t_{F51,(upward-upward-i)k,backup}\) is the backup operating time of the last upward adjacent relay of the relay \(R_{i}\) in the OCR triple; \(t_{F51,(backward-i)k,backup}\) is the backup operating time of the backward adjacent relay of the relay \(R_{i}\); and \(t_{F51,(backward-backward-i)k,backup}\) is the backup operating time of the last backward adjacent relay of the relay \(R_{i}\) in the OCR triple.  
In general, the optimization of the objective function is subjected to the following constraint conditions, (i) relay characteristic constraints; and (ii) protection coordination constraints. The former contains the relay operating time, TDS multiplier, coefficients of the inverse-time curve, and the pick-up current of OC relays. The latter considers the operation time difference of each pair of primary and backup OCRs as well as the total tripping time of each OCR triple.  
Figure 1: Illustration of protection characteristic curves (**a**) 50P/G–definite-time overcurrent function and (**b**) 51P/G–inverse-time overcurrent function.