# 6-Bus System (System I)

I. Introduction:

$$\bullet$$ In this test case, both near-end and far-end $$3\phi$$ fault locations are considered, as shown in the figure given below.
$$\bullet$$ This network consists of 6 buses, 7 branches and 14 directional overcurrent relays (DOCRs).
$$\bullet$$ The objective function of this system is mathematically expressed as:
$$min \ Z=\sum^m_{p=1} T^{near}_{pr,p}+\sum^n_{q=1} T^{far}_{pr,q} \text{ ...... (1)}$$
where $$T^{near}_{pr,p}$$ and $$T^{far}_{pr,q}$$ are respectively the operating time of the primary relay at the near-end $$3\phi$$ fault (at the $$p$$th location) and the far-end $$3\phi$$ fault (at the $$q$$th location), which can be calculated as follows:
$$T^{near}_{pr,p}=TMS_p \left[\frac{\displaystyle \beta}{\displaystyle \left(\frac{\displaystyle a_p }{\displaystyle PS_p \times b_p}\right)^{\alpha}-1}\right] \text{ ...... (2)}$$
$$T^{far}_{pr,q}=TMS_q \left[\frac{\displaystyle \beta}{\displaystyle \left(\frac{\displaystyle c_p }{\displaystyle PS_q \times d_p}\right)^{\alpha}-1}\right] \text{ ...... (3)}$$
where:
$$\alpha$$ and $$\beta$$ are time-current characteristic curve (TCCC) constants. For IDMT-based DOCRs, $$\alpha$$ and $$\beta$$ are equal to 0.14 and 0.02, respectively.
$$\bullet$$ Note that the constants of $$T^{far}_{pr}$$ in Eq.(3) are calculated in sequence of $$p$$ instead of $$q$$.
$$\bullet$$ Also, these $$p$$ and $$q$$ notations are $$i$$ and $$j$$ in the cited references. This replacement is essential to prevent any confusing with other systems, because the notations $$i$$ and $$j$$ are assigned for primary and backup relays, respectively.
$$\bullet$$ Based on the observation of Birla in [2], 10 constraints are relaxed [3].
$$\bullet$$ The selectivity constraint is:
$$T_{jk}-T_{ik}\geq CTI \text{ ...... (4)}$$
where $$T_{jk}$$ and $$T_{ik}$$ are respectively the operating times of the $$j$$th backup and $$i$$th primary relays for a $$3\phi$$ fault happens at the $$k$$th location. They can be computed by the following equations:
$$T_{jk} = TMS_j \left[\frac{\displaystyle \beta}{\displaystyle \left(\frac{\displaystyle e_p }{\displaystyle PS_j \times f_p}\right)^{\alpha}-1}\right] \text{ ...... (5)}$$
$$T_{ik} = TMS_i \left[\frac{\displaystyle \beta}{\displaystyle \left(\frac{\displaystyle g_p }{\displaystyle PS_i \times h_p}\right)^{\alpha}-1}\right] \text{ ...... (6)}$$
$$\bullet$$ It is considered, for this test system, that the primary operating time of each relay $$(T_i)$$ should be bounded between lower and upper limits as:
$$T^{min}(=0.05) \leq T \leq T^{max}(=1.0) \text{ ...... (7)}$$
$$\bullet$$ Both plug-setting ($$PS$$) and time-multiplier setting ($$TMS$$) are considered continuous independent variables. The CT rations ($$CTRs$$) for the relays $$R_1$$ to $$R_6$$, the listed primary/backup (P/B) relay pairs, and the corresponding $$3\phi$$ fault currents are available as $$\{a, b, c, d, e, f, g, h\}$$ constants in the given below (click on them for bigger size):

II. Single-Line Diagram:

$$\bullet$$ This single-line diagram was drawn by Ali R. Alroomi in Sept. 2015 and all the necessary data were coded in MATLAB m-files.

III. Files:

$$\bullet$$ High Resolution Images (JPEG and TIFF Formats) [Download]
$$\bullet$$ Results Tester (MATLAB, m-file Format) [Download]

IV. References (Some selected papers that use this system):

[1] D. Birla, R. P. Maheshwari, and H. O. Gupta, "A New Nonlinear Directional Overcurrent Relay Coordination Technique, and Banes and Boons of Near-End Faults Based Approach," IEEE Transactions on Power Delivery, vol. 21, no. 3, pp. 1176-1182, Jul. 2006.
[2] D. Birla, R. P. Maheshwari, H. O. Gupta, K. Deep, and M. Thakur, "Application of Random Search Technique in Directional Overcurrent Relay Coordination,"
International Journal of Emerging Electric Power Systems, vol. 7, no. 1, pp. 1-14, Sept. 2006.
[3] R. Thangaraj, M. Pant, and K. Deep, "Optimal Coordination of Over-Current Relays Using Modified Differential Evolution Algorithms,"
Engineering Applications of Artificial Intelligence, vol. 23, no. 5, pp. 820–829, Aug. 2010.
[4] J. Moirangthem, Krishnanand K.R., S.S. Dash, and R. Ramaswami, "Adaptive Differential Evolution Algorithm for Solving Non-Linear Coordination Problem of Directional Overcurrent Relays,"
IET Generation, Transmission & Distribution, vol. 7, no. 4, pp. 329-336, 2012.
[5] M. Singh, B. Panigrahi, and A. Abhyankar, "Optimal Coordination of Directional Over-Current Relays Using Teaching Learning-Based Optimization (TLBO) Algorithm,"
International Journal of Electrical Power & Energy Systems, vol. 50, pp. 33–41, 2013.
[6] T. R. Chelliah, R. Thangaraj, S. Allamsetty, and M. Pant, "Coordination of Directional Overcurrent Relays Using Opposition Based Chaotic Differential Evolution Algorithm,"
International Journal of Electrical Power & Energy Systems, vol. 55, pp. 341–350, Feb. 2014.
[7] Rafael Corrêaa, Ghendy Cardoso Jr., Olinto C.B. de Araújo, and Lenois Mariotto, "Onlin
e Coordination of Directional Overcurrent Relays Using Binary Integer Programming," Electric Power Systems Research, vol. 127, pp. 118-125, Oct. 2015.