# 3-Bus System (System II)

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 3 buses, 3 branches and 6 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.
$$\bullet$$ The number of selectivity constraints depends on the network, where only 8 selectivity constraints out of 12 are considered. This will result in a total number of 26 constraints.
$$\bullet$$ 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] K. Deep, D. Birla, R. P. Maheshwari, H. O. Gupta, and M. Thakur, "A Population Based Heuristic Algorithm for Optimal Relay Operating Times," World Journal of Modeling and Simulation, vol. 2, no. 3, pp. 167-176, Mar. 2006.
[2] J.C. Bansal and K. Deep, "Optimization of Directional Overcurrent Relay Times by Particle Swarm Optimization," in 2008 IEEE Swarm Intelligence Symposium (SIS 2008), Sept. 2008, pp. 1-7.
[3] K. Deep and J.C. Bansal, "Optimization of Directional Overcurrent Relay Times Using Laplace Crossover Particle Swarm Optimization (LXPSO)," in 2009 World Congress on Nature Biologically Inspired Computing (NaBIC 2009), Dec. 2009, pp. 288-293.
[4] 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.
[5] 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.
[6] 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.
[7] 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.