# Optimal Coordination of Over-Current Relays in Microgrids Using Unsupervised Learning Techniques
## 2 Coordination Protection Problem Formulation
### Constraints  
The constraints of the optimal DOCRs coordination problem are given by Equations (2)-(9).  
\[t_{if}-t_{if}\geq CTI \tag{2}\]  
\[t_{if}=\frac{A.TMS_{i}}{PSM_{if}^{b}-1} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Reference** & **Adaptive** & **PSM Constraint** & **Limits Setting Groups** & **K-Means** & **SOM** & **Hierarchical** \\ \hline
[15; 23; 24] & X & & & & & \\
[14] & X & X & & & & \\
[10] & X & & X & & & \\
[16] & X & & & X & X & \\
[17] & X & & X & & X & \\
**[Proposed]** & X & X & X & X & X & X \\ \hline \hline \end{tabular}
\end{table}
Table 1: Optimal coordination of Directional over-current relays (DOCRs) in microgrids (knowledge gap).  
\[PSM_{if}=\frac{I_{fi}}{ipickup_{i}} \tag{4}\]  
\[t_{imin}\leq t_{if}\leq t_{imax} \tag{5}\]  
\[TMS_{imin}\leq TMS_{i}\leq TMS_{imax} \tag{6}\]  
\[ipickup_{imin}\leq ipickup_{i}\leq ipickup_{imax} \tag{7}\]  
\[PSM_{imin}\leq PSM_{if}\leq PSM_{imax} \tag{8}\]  
\[SG_{imin}\leq SG_{i}\leq SG_{imax} \tag{9}\]  
Equation (2) illustrates the coordination time between backup and primary DOCRs. In this case, \(t_{if}\) is the operation time of the backup relay \(j\) when fault \(f\) occurs, and \(t_{if}\) is the operation time of the primary relay \(i\), for the same fault. The coordination time interval \(CTI\) is the period of time that is allowed for the backup protection to operate. When a fault takes place, both the backup and primary DOCRs identify the fault occurrence. The backup DOCR is in charge of tripping the fault in case the primary DOCR misses to isolate the fault. The ORCs that are considered in this study present a normal inverse characteristic, as indicated by Equation (3). In this case, \(A\) and \(B\) are constant parameters of the curve, \(TMS_{i}\) is the time multiplying setting of relay \(i\), and \(PSM_{if}\) is the ratio between the fault current \(I_{fi}\) and the pick up current \(ipickup_{i}\) given by Equation (4). Equation (5) indicates the operating time limits of the DOCRs. In this case, \(t_{imin}\) and \(t_{imax}\) are the minimum and maximum operating time of relay \(i\), respectively. Equation (6) represents the minimum and maximum limits of \(TMS\) for relay \(i\) that is given by \(TMS_{imin}\) and \(TMS_{imax}\), respectively. Similarly, Equation (7) represents lower and upper limits of the pickup current \(ipickup_{i}\), denoted as \(ipickup_{imin}\) and \(ipickup_{imax}\), respectively.  
Equation (8) represents the lower and upper limits of the \(PSM\) denoted as \(PSM_{imin}\) and \(PSM_{imax}\), respectively. PSM is the ratio between the fault current seen by the relay and the pickup current. The standard characteristic curves of commercial relays are generally defined in a region where the minimum and maximum values are 1.1 and 20 times the PSM, respectively. Nonetheless, the presence of DG units in microgrids considerably increases the short circuit levels. Consequently, for different faults, the maximum value that is defined for the PSM might be exceeded, affecting the sensitivity of the protections and causing a loss of the coordination protection scheme. In order to avoid this, a new constraint regarding the PSM was introduced in [14]. In this case, the maximum value of the \(PSM\) is the highest current level of the IEC normally inverse curve programmed in the industrial protective relay before the definite time region of the curve.  
Equation (9) represents the lower and upper limits of setting groups (\(SG\)), denoted as \(SG_{imin}\) and \(SG_{imax}\), respectively. This consideration is not taken into account in traditional protection coordination approaches. In this case, \(SG_{imax}\) corresponds to the highest number of \(SG\) programmed in the industrial protective relay, which may range from two up to eight, depending on the manufacturer.