**3. The Proposed Methodology: Nonstandard Time Current Characteristics**

This section aims to introduce optimal OCR coordination based on a new nonstandard time–current characteristic (NSTCC) with dynamic coefficients to reduce the tripping time associated with the value of fault currents. The proposed NSTCC scheme will be compared to the traditional OCR scheme (inverse definite minimum time (IDMT)) [29,36]. Further, there are different applications, such as the thermal stress issues occurring in the equipment (such as transformers and cables), that can use the proposed NSTCC. The next equation represents the proposed NSTCC, and the logarithmic function therein [19] is the basis of this equation.

$$t = \left(A - 1.35 \times \log\_{\varepsilon} \left(\frac{Isc}{Ip}\right)\right) \times \text{TMS} \tag{4}$$

To ensure OCR coordination selectivity, the grading time should be kept constant and free from the network's location of the fault or the current level of the fault. Equation (4) represents the NSTCC for all relays through the use of logarithmic [19,36] and variable coefficients (A) with a range between 2 and 6.5; the time of grading will not be affected by the degree and point of fault. This will make the selectivity of the protection system better and independent of the fault location or current. Moreover, it was difficult for the normal inverse curves to detect the minimum fault. The NSTCC offers ample area for the detection and coordination of the OCRs in the minimum fault, as illustrated in Figure 4, to ensure selectivity without missing the tripping time. The following section describes an optimization task for determining the TMS based on Equation (4) that reduces the OT to a minimum. Therefore, coordination on the basis of nonstandard tripping characteristics will result in an optimal time of grading in relation to the time of tripping.

The effect of adding renewable energy sources connected to the DN on the PSM and miscoordination problems that appear between OCRs during the maximum and minimum faults is shown in Figure 3b. Generally, the ratio between short-circuit current and the pickup current (Isc/Ip) is presented as a PSM. This section presents the importance of using NSTCC in coordinating the OCRs as illustrated in Figure 4. The fault's location near or at the end of the protected zone is responsible for obtaining the OCR coordination task. The fault's location near the protected line (maximum fault current) is covered by the F1 point, while the end of the protected zone (minimum fault current) is related to the F2 point. The two scenes were selected to attain the required CTI, cover on time attributes, and raise the OCR OT because of the addition of sources of renewable energy as shown in Figures 3 and 4.

It can be noticed that in Figure 4, the curve of the standard time–current characteristic (STCC) has high values of fault currents at both maximum and minimum fault currents. This curve represents the inverse definite minimum time (IDMT) overcurrent relay. As seen in Figure 4, the OTs of the STCC at the minimum and maximum fault current are *t*<sup>6</sup> and *t*8, which are unchangeable values. Then, two variables' coefficients A and B for the maximum and minimum faults are required to reduce the tripping time effectively to control the two sides of the curve of the STCC. The researchers in the available literature such as [20,44] used this approach to reduce the operating time; however, this leads to increasing the number of constraints which is another disadvantage. In addition, the authors in ref. [25] have proposed the curve of the nonstandard scheme (NSS), which is represented by the black curve in Figure 4; they have used Equation (4) with constant coefficient A equal to 5.8. Yet, due to the curve being constant as seen in Figure 5, the fault currents at minimum faults, especially in islanding mode, are still slightly high-valued and there is a delay time that will lead to miscoordination problems between OCRs at *t*<sup>7</sup> as seen in Figure 4. In this work, this gap can be filled by using the NSTCC which reduces the tripping time compared with the STCC at maximum and minimum faults and nonstandard curve in ref. [25] by making the coefficient A a variable needed to have optimal value to achieve the best reduction in the total OT for relays. The coefficient A in the proposed NSTCC is controllable at both the maximum and minimum fault currents as shown in Figure 4; the blue curves represent the NSTCC and they illustrate how just the one variable coefficient A can control at both ends of the curve and reduce the OCR tripping time. It can be seen that the NSTCC can decrease the OT from *t*<sup>5</sup> to *t*<sup>3</sup> and from *t*<sup>2</sup> to *t*<sup>1</sup> at minimum and maximum faults.

**Figure 5.** Proposed Nonstandard Time–Current Characteristics with variable coefficient A (NSTCC) and Nonstandard Scheme with constant coefficient A (NSS).

For the first point, the minimum tripping time (maximum fault current) must be guaranteed by each OCR in the DN. A lower OT is provided by using the NSTCC curve which is represented by the red dotted line compared to NSS which is demonstrated by the black line, as shown in Figure 5. With the existence of the DG units at the grid at the minimum fault current (islanding current) for the second point, the OCR OT will be increased more than the distribution grid in absence of the DGs. The avoidance of any miscoordination problem or nonoperational cases can be achieved by applying the proposed NSTCC curve, which minimizes the OCR tripping time as illustrated in Figure 5.
