## 3 Coordination with Linear Programming  
OCRs are currently the most widely used protection devices [13; 22], this device generally being used as backup protection. However, in some situations, it may be the only protection. The OCR should act as primary protection by removing faults that occur in the area under its supervision. Only in the event that the primary protection fails to operate should the backup protection initiate tripping. A typical electrical distribution network may consist of hundreds of equipment protection relays. Each relay in the system must be coordinated with the adjacent equipment protection relay. If backup protection is not well coordinated, coordination failure may occur, and therefore, coordination of backup protections is one of the main concerns in the network protection system [13; 23]. Generally, protection coordination can be performed by topology [24; 25], by optimization methods [26; 27], or by expert methods [28]. Topological analysis is used for relay tuning in multiterminal networks, graph theory, and functional approximation techniques are employed to provide the best solution that does not necessarily have to be the optimal one. In optimization methods, some researchers [26; 29] use non-linear programming techniques to determine the optimal relay settings, subject to constraints due to coordination and limits of the relay settings themselves. In reference [30], the big M method is proposed to find the optimal value of the OCR time multiplier setting (TMS) in which the stated values of the plug setting are assumed to be known and fixed. The problem of OCR coordination in the distribution system with DG presence can thus be defined as an optimization problem with constraints. The objective is to minimize the operating time of the relay closest to the place where the fault has occurred. The constraints imposed are due to limits on the operating time of the relay, coordination criteria, and relay characteristics. In this work, the OCR coordination problem in radial distribution systems is formulated as a linear programming (LP) problem with constraints, where the sum of the operating times of the relays in the system for different minimum fault points:  
\[\min\,\frac{\alpha}{\left(k_{ij}\right)^{\beta}-1}\,\sum_{i=1}^{n}\left(\sum _{j=1}^{m}\text{TMS}_{i,j}-\text{TMS}_{i+1,j+1}\right), \tag{1}\]  
where \(\alpha\) and \(\beta\) are OCR shape constants according to Table 1, \(k_{ij}\) is the ratio between the short-circuit current, \(I_{CC}\), and the relay setting current, \(I_{nr}\), regarding relay \(i\) with fault in section \(j\), and TMS is the time multiplier setting. The proposed optimization problem is subjected to the following group of constraints:  
* Coordination criterion--The protection coordination criterion establishes the minimum time that must elapse between the operation of the primary protection and the operation of the backup protection. The fault is detected simultaneously by the primary protection, PP, and the secondary protection, PS. To avoid erroneous operation, the PS will only have to operate in case the PP fails. If we define \(R_{i}\) as the primary fault protection at a certain point, \(j\), and \(R_{i+1}\) as the secondary or backup protection for the same fault. The constraint condition for coordination criteria is: \[t_{i+1,j}-t_{i,j}>\Delta t,\] (2) \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type of Curve** & \(\alpha\) & \(\beta\) \\ \hline Standard inverse time & 0.14 & 0.02 \\ Very inverse & 13.5 & 1 \\ Extremely inverse & 80 & 2 \\ Long inverse time & 120 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Form constants for the exponential equation IEC [31]. where, \(t_{i+1,j}\) is the operation time of the back-up, PS, operation for fault at point \(j\), \(t_{ij}\) is the operation time of PP operation for the same fault, and \(\Delta t\) is the coordination time interval (CTI). * Relay operating time limits--The time taken by a relay to detect and isolate a fault produced in its zone of influence must be bounded.