Objective Function

The main variable for the OCR coordination issue in Equation (4) is the TMS which manages the OT of the relay. In this section, the operational time for OCRs and the coordination problem, as described in Section 2, is formulated as an objective function. This objective function (OF) minimizes the overall OT for the main relay and backup relay. The operational time, *t*, for the total number of relays, *x*, and total number fault locations, *y*, is formulated as follows [29,36]:

$$OF = \sum\_{j=1}^{x} \sum\_{k=1}^{y} t\_{j,k} \tag{5}$$

where *j* is the relay number (*j* = 1, 2, ... ., *x*), *k* is the fault location number (*k* = 1, 2, ... , *y*), and *tj*,*<sup>k</sup>* is the operational time for *j* relay at *k* fault. In this work, the NSTCC is used to calculate the total tripping time, OT, where the coefficient A is controllable at both the maximum and minimum fault currents. Equation (5) can be rewritten as follows:

$$\underset{\mathbf{A}}{\text{arg min }} \sum\_{j=1}^{x} \sum\_{k=1}^{y} t\_{j,k} \tag{6}$$

There are various constraints taken into consideration during application of the OF in Equations (5) and (6) as shown below:

• Coordination Criteria and Selectivity

The selectivity constraint for OCR coordination aims to add operational time delays between the primary and backup OCRs, to minimize power outages on the network based on the location of the fault. The backup OCR will not work except when the main OCR is nonoperational. The formulation of the criteria for selectivity can be done based on CTI as constraints of inequality:

$$t\_b - t\_p \ge \text{CTI} \tag{7}$$

where *tp* represents the OT for primary relays and *tb* represents the OT for backup relays. Generally, the CTI (in seconds) is between 0.2–0.5 to guarantee selectivity [29–37]. The value of CTI is dependent on various parameters like relay type and circuit breaker speed. This study works with a CTI from 0.2 to 0.5.

• Relay Setting, Operating Time Bounds:

To keep the limitations of operational time, the constraints should be presented for the minimal and maximal OCR operational time. Nevertheless, the protective relays should have quick operation taking the minimum possible time; if the OCR operation takes more time, there will be damage to the equipment and an unstable power system. The minimal and maximal operation time bounds are shown below:

$$TMS\_{\min} \le TMS\_j \le TMS\_{\max} \tag{8}$$

$$OT\_{\min} \le OT\_{\bar{j}} \le OT\_{\max} \tag{9}$$

where *TMSmin* and *TMSmax* are the minimal and maximal *TMS* values of relay *j* and *OTmin* and *OTmax* represent the minimal and maximal time of operation needed for the relay *j* [28,30]. The OCRs' operation must be within the protection scheme's normal operation time. As a result, the PSM needs to be set in the domain of the minimal and maximal values in the minimum and maximum fault currents in the relay, even with light overloads.

• Proposed Setting of Coefficient Bounds:

In this study, the characteristic coefficient in Equation (4) is considered as a decision variable, as presented in Equation (6). In previous studies, the inverse curve requires more than one variable coefficient to shift it upwards and downwards [20,45], which leads to greater OCR tripping times with a slight shift in steepness and increase in the number of constraints. This causes miscoordination between primary and backup relays. For development purposes, the NSTCC is formulated in Equation (4) with just a single variable coefficient, which is *A*. The NSTCC tends to shift the curve downwards by changing *Ai* values; this leads to a reduction in OCR tripping times and the reduction of constraints. As a result, the OCR coordination performance is guaranteed. The following equation shows the maximum and minimum bounds of the variable coefficient *A*:

$$A\_{\rm min} \le A\_{\rm j} \le A\_{\rm max} \tag{10}$$

where *Amin* and *Amax* represent the minimal and maximal variable coefficient needed for relay *j*. It has been chosen to be between 2 and 6.5 in this study.
