Complexity
Volume 2018 (2018), Article ID 7039790, 15 pages
https://doi.org/10.1155/2018/7039790

Research Article

Improved Solutions for the Optimal Coordination of DOCRs Using Firefly Algorithm

Muhammad Sulaiman, Waseem, Shakoor Muhammad, and Asfandyar Khan

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan

Correspondence should be addressed to Muhammad Sulaiman; sulaiman513@yahoo.co.uk

Received 9 October 2017; Revised 24 December 2017; Accepted 11 January 2018; Published 11 February 2018

Academic Editor: Arturo Buscarino

Copyright © 2018 Muhammad Sulaiman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Nature-inspired optimization techniques are useful tools in electrical engineering problems to minimize or maximize an objective function. In this paper, we use the firefly algorithm to improve the optimal solution for the problem of directional overcurrent relays (DOCRs). It is a complex and highly nonlinear constrained optimization problem. In this problem, we have two types of design variables, which are variables for plug settings (PSs) and the time dial settings (TDSs) for each relay in the circuit. The objective function is to minimize the total operating time of all the basic relays to avoid unnecessary delays. We have considered four models in this paper which are IEEE (3-bus, 4-bus, 6-bus, and 8-bus) models. From the numerical results, it is obvious that the firefly algorithm with certain parameter settings performs better than the other state-of-the-art algorithms.

1. Introduction

Power systems in electrical engineering function at contrary voltage levels ranging from 415 V to 400 KV or even more. Power systems consist of three parts: production, supply, and usage of the power generated. Moreover, supply lines, which transmit the power, are not insulated. These lines undergo irregularities more often than other domains of the system due to several causes like production of overcurrent due to lightening. These irregularities disrupt the provision of power and often result in the impairment of the other equipment attached to the power system. Thus, it is necessary to take protective measures in case this happens. To overcome this difficulty, overcurrent relays (OCRs) are commonly used as a safety strategy in the power system to avoid disruption in power supply and other technical faults. Hence, directional OCRs are easy to use and are an economic option for the safety of the attached subtransmission power systems and secondary layer of backup protection in transmission systems [1].

During the design of a power system, the coordination of these DOCRs is an important aspect to keep in mind. DOCRs are logical elements and are placed at both ends of a line, which helps the breaker in case any fault arises. Moreover, DOCRs are supplied in electrical power systems to disable the faulted portions of the power systems. The problem of DOCRs is to decide which portions of the system are to function during a fault so that the faulty part is detached and does not stop the whole system, with certain constraints like adequate coordination tolerance and without extra disruptions. This procedure is mainly based on the networking of the system, nature of relays, and other protection measures as in [2]. We have two types of design variables: variables for plug settings (PSs) and the time dial settings (TDSs) for each relay in the circuit. After optimizing the design settings, the faulty lines are isolated, thus ensuring continuous supply of power to the remaining parts of the system.

In this article, the problem of DOCRs in a power system is handled with the firefly algorithm (FA) using the standard benchmark IEEE (3-, 4-, 6-, and 8-bus, resp.) systems [1, 3–5], as presented in Figures 8, 9, 10, and 11. The two types of decision variables are named PS and TDS. The objective of this problem is to minimize the total operating time taken by primary relays. Primary relays are expected to isolate the faulty lines satisfying the constraints on design variables.

The rest of the paper is organized as follows. In Section 2, we briefly recall different algorithms used in the literature to solve the problem of DOCRs. In Section 3, we elaborate the mathematical formulation of the problem. The firefly algorithm (FA) is recalled in Section 4. Parameter settings and statistical and graphical results are discussed in Section 5. Section 6 concludes the present study.

2. Literature Review

Initially, a trial-and-error approach was used by researchers, which consumed a large number of iterations to reach an optimal relay setting. Thus, many scientists and researchers adopted the setting of DOCRs, depending on experience. In particular, although [1] uses linear programming, however, it also uses a change of variables to convert the nonlinear problem to a linear problem, without any approximation. The problem is solved linearly in terms of the TDS and PS variables, and then after the linear solution is reached, the roots of these nonlinear expressions (TDS and PS) must be found to reach the solution for TDS and PS. In [1], the application of optimization techniques was first reported for this problem. A literature review on this problem can be found in [2]. Previously, the problem of DOCRs was handled with different techniques, like curve fitting, graph theoretical approach, and classical optimization techniques. Furthermore, a version of sparse dual revised simplex algorithm was used to solve the problem assuming nonlinear PS settings to optimize the remaining variables as for TDS settings [6]. Other techniques in linear programming, to solve the problem of DOCRs, used by [7–10] have optimized the TDS and PS settings using the simplex algorithm and the Rosenbrock hill-climbing algorithm to solve the problem.

Nature-inspired algorithms (NIAs) and artificial intelligence (AI) based techniques have been gaining the attention of researchers recently [12, 13]. The NIAs which have been used to tackle the problem of DOCRs include, but are not limited to, particle swarm optimization (PSO) [14, 15], genetic algorithm (GA) [4, 16], modified evolutionary programming, and evolutionary programming. Expert systems [17–20] and fuzzy logic [21] are applications of AI algorithms to tackle the problem of DOCRs. A random search algorithm (RST-2) was applied to tackle the problem of DOCRs using IEEE 6-, 4-, and 3-bus models, respectively [2].

Firefly algorithm (FA) is a well-known and reliable nature-inspired algorithm for solving complex and highly nonlinear constrained optimization problems, and as far as the authors know FA is not implemented for optimization of DOCR settings using IEEE 3-, 4-, 6-, and 8-bus systems which are presented in this paper. We have implemented FA to solve the problem of DOCR settings and the outcomes of our simulations are compared with other state-of-the-art algorithms.

3. Problem Formulation

Determining the direction of flow of current in an electrical circuit is the job of a protection system placed in a circuit. In case a fault arises, the protection system passes the signals to the associated switch. DOCRs are useful tools to handle the situation when there are fault currents flowing in both directions. These relays operate when current flow is detected in the opposite direction. Loop systems and parallel feeders include DOCRs, because these feeders have current flowing from both ends.

There are two units of a DOCR; the first unit, which is known as the instantaneous current settings, is activated when the current is above a defined threshold. On the other hand, when the current is below a defined limit, time-delay unit is operated. This unit is activated when a fault happens with a time delay. These are two types of settings associated with the delay unit, which are named as follows:

(i)Time dialing settings (TDSs)(ii)Plug settings (PSs) (e.g., tap setting)

The TDS points out the delayed relay, whenever the value of fault current either reaches or exceeds the picking current value. PS represents a value of the picking current for a relay [2].

3.1. Problem Formulation and Explanation

The mathematical formulation of the problem is the same as given in [4]. We have tackled the problem with the firefly algorithm. The relay working time is a nonlinear relation of the dependent variables TDS and PS and the fault current denoted by observed by the corresponding relay. So, the equation for the relay operating time is given by

and the variables TDS and PS are to be determined. , , and take the values 0.14, 0.02, and 1.0, respectively, as per [IEEE Std. (1997)]. These constants define the behavior of DOCR in terms of operating time. The value of CT keeps a level of current which is suitable for a relay. And defines the number of windings in a current transformer (CT). Each relay is attached to a “CT,” which makes a known value. The fault current represented by is continuously measured by instruments. The line in the system is directly proportional to the number of constraints. Details of lines involved in different systems are furnished in Table 6. It is worth noting that, in real power systems, several types of relays operate besides DOCRs. The involvement of other types of relays along with DOCRs further complicates the problem by imposing more constraints as shown in Table 6.

3.2. The Objective and Limitations of the Problem

DOCRs need to be coordinated optimally, where optimization techniques, like firefly algorithm, minimize the corresponding objective function according to boundary and coordination limitations. The primary relay is used to clear the fault. The close-in fault is a fault which is close to the relay while a fault occurring at a far end is called a far-bus fault. By convention, the objective function is the summation of the operation time of all initial relays which clears any close-in and far-bus faults. The objective function is given by

In (2), represent close-in fault relays, while denotes the number of far-bus fault relays. Also, shows the working time of the primary close-in fault for the relay. is the working time of the primary far-bus fault for the relay.

The following are the constraints.

(1) Limits on Decision Variables TDSs.

where varies from 1 to . The lowest value for is which is 0.05 and the highest value is which is 1.1.

(2) Limits on Decision Variables PSs.

where varies from 1 to . is the lowest value which is 1.25 and is the highest value of which is 1.50.

(3) Bounds on Initial Working Times. According to this constraint, each term of the objective is bounded in the interval .

(4) Constraints on the Selection of Pairs of the Relays.

where is the working time of backup relay, is the working time of primary relay, and CTI is the coordinating time interval.

3.3. The IEEE 3-Bus System

In this problem, the values of and are taken as 6 which is twice the number of lines involved. There are 12 decision variables, namely, to and to . Figure 8 represents the model for the 3-bus system. The objective (objftn) to be minimized is given in

where

The rest of the constants , , , and are given in Table 7; see the Appendix.

Descriptions of different constraints for the model are given below.

Limits on Variables TDSs., where starts from 1 and varies to 6 ().

Limits on Variables PSs.  , where starts from 1 and varies to 6 ().

Bounds on Initial Working Time. According to this constraint, each term of the objective is bounded in the interval .

Constraints on the Selection of Pairs of the Relays.

where is the working time of the backup relay and is the working time of the primary relay. The value of CTI is 0.3. Here,

Table 8 represents the values of constants , and .

3.4. The IEEE 4-Bus System

In this problem, the values of and are taken as 8 which is twice the number of lines involved. There are 16 decision variables, namely, to and to . Figure 9 represents the model for the 4-bus system. The value of CTI for this system is 0.3. There are 9 selectivity constraints. The mathematical form for this model will be as in (6) with . The rest of the constants , , , and and for the 4-bus system are given in Tables 9 and 10; see the Appendix.

3.5. The IEEE 6-Bus System

This problem of coordination of DOCRs is named the IEEE 6-bus system; both and take a value equal to 14. Basically, this problem is comprised of 28 design variables (two for each relay), that is, to and to . Figure 10 represents the model for the 6-bus system. CTI takes the value 0.2. For the current sample of the 6-bus system, there are 38 selectivity constraints for all faults detected by the corresponding relay. Considering the observations given in [2], 10 constraints are relaxed. The mathematical form for this model will be as in (6). The rest of the constants , , , and and for the 6-bus system are given in Tables 11 and 12; see the Appendix.

3.6. The IEEE 8-Bus System

The last model investigated in this study is the 8-bus system as shown in Figure 11. The mathematical form of this model will be as in (6). It consists of 8 buses, 14 relays, 7 lines, 2 transformers, and 2 generators. The main characteristics of this system are given in Tables 13 and 14. Therefore, this 8-bus system has 28 design variables and 40 constraints [3].

4. Firefly Algorithm

The firefly algorithm was introduced by Yang in [11], and it simulates the flashing phenomena of fireflies. To simulate the idea, the following assumptions are made [9, 11].

All fireflies are of the same gender. Brighter fireflies have greater attraction than others with less brightness. The attraction will decrease with the increase in distance between fireflies. If none of the fireflies is brighter than a specific one, then it will move randomly.

The landscape of the given objective function affects the brightness of a firefly [11]. For a maximization problem, the relation of brightness and the objective function can be represented as . The second assumption can be represented mathematically as in (10) and (11).

where is the actual intensity of light and is called the light absorption coefficient. The attractiveness can be related as follows:

where represents the distance between two fireflies and is their actual attractiveness at . The distance between two fireflies at positions and can be calculated as follows:

where is the th component of the spatial coordinate of the th firefly. The movement of a firefly which is attracted to another more attractive (brighter) firefly is determined by

Here, is a random parameter in the interval . And rand is a random number drawn from a uniform Gaussian distribution in . A pseudocode for the firefly algorithm is given in Algorithm 1.

(1) Objective function ,  

(2) Initialize a population of fireflies ,  

(3) Calculate the light intensity at by

(4) Define light absorption coefficient

(5) While (< MaxGeneration)

(6) for : all fireflies

(7) for : all fireflies

(8) Calculate the distance between and using Cartesian distance equation

(9) if  

(10) Attractiveness varies with distance via

(11) Move firefly towards in all dimensions

(12) end if

(14) Evaluate new solutions and update light intensity

(15) end for

(16) end for

(17) Rank the fireflies and find the current best

(18) end while

Algorithm 1: Pseudocode for firefly algorithm [11].

5. Simulation Settings and Discussion on Results

For a fair comparison, the population size is fixed. We have executed FA a fixed number of times to get better values for the decision variables. Based on the analysis given in [2], the size of the population is fixed as 50, the constant CR is taken as 0.5, and the scaling factor (F) is set as 0.5 for differential evolution. The stopping criterion is either that the algorithm completes 1000 generations or that we get an optimal value which satisfies the following condition: . Here, is the objective value. Constraints are handled using the penalty function approach. All experiments are repeated 30 times and a global optimum solution is recorded for each algorithm.

The best solutions obtained by FA and other versions of DE for the IEEE 3-bus model in terms of best decision features, minimum objective function value, and number of function evaluations taken to complete each simulation are given in Table 1 and Figures 1 and 2. Here, it is evident that, in terms of the objective function value, DE gave the worse objective value (4.8422) and all the other versions of the DE algorithm gave almost similar values. On the other hand, FA gave us an objective value of 4.6557, which is much better than other algorithms. However, if we compare the number of function evaluations (NFE), then the performance of DE and its variants is worse as compared to the FA algorithm as it took only 25200 NFE to complete the simulations. Thus, FA is significantly better than all the other algorithms under consideration.

Table 1: Best design variables, objective function values (F), and total number of function evaluations (NFE) for IEEE 3-bus model by DE, modified DE, and firefly algorithms.

DE
MDE-1
MDE-2
MDE-3
MDE-4
MDE-5
FA

0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500

0.2194
0.2178
0.1979
0.1988
0.1976
0.1976
0.3512

0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500

0.2135
0.2090
0.2094
0.2090
0.2090
0.2090
0.3518

0.19498
0.1812
0.1847
0.1812
0.1812
0.1812
0.1290

0.1953
0.1807
0.1827
0.1807
0.1806
0.1806
0.1222

1.250
1.250
1.250
1.250
1.250
1.250
1.3583

1.250
1.250
1.4999
1.4849
1.4999
1.500
1.4107

1.2500
1.250
1.250
1.250
1.250
1.250
1.3950

1.4605
1.4999
1.4999
1.4999
1.4999
1.500
1.2500

1.250
1.500
1.4318
1.4998
1.4999
1.500
1.3109

1.250
1.4999
1.4619
1.4999
1.4999
1.500
1.2789

F
4.8422
4.80707
4.7873
4.7822
4.7806
4.7806
4.6557

NFE
78360
72350
73350
97550
69270
38250
25200

Figure 1: Best results obtained, for the IEEE 3-bus model, by different techniques are compared with the firefly algorithm.

Figure 2: The number of function evaluations done by different algorithms to solve the 3-bus model.

The simulation results of the IEEE 4-bus model are given in Table 2 and Figures 3 and 4. It is observed for the IEEE 4-bus model that DE algorithm and all the variants of the DE algorithm performed in a similar manner in terms of objective function values with MDE-4 giving a slightly better value than other variant algorithms of DE. But FA outperformed the algorithms under consideration in terms of the number of function evaluations (NFE). Better performance was shown by FA, which took 35500 NFE to converge to a solution 3.5624.

Table 2: Best design variables, objective function values (F), and number of function evaluations (NFE) for IEEE 4-bus model by DE, modified DE, and firefly algorithms.

DE
MDE-1
MDE-2
MDE-3
MDE-4
MDE-5
FA

0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500

0.2248
0.2121
0.2123
0.2121
0.2121
0.2121
0.0920

0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500

0.1515
0.1515
0.1515
0.1515
0.1515
0.1515
0.0687

0.1264
0.1264
0.1264
0.1264
0.1262
0.1264
0.0725

0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500

0.1337
0.1338
0.1371
0.1338
0.1337
0.1337
0.0871

0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500

1.2734
1.2733
1.2733
1.2733
1.250
1.2734
1.3964

1.250
1.4998
1.4959
1.500
1.500
1.4999
1.3885

1.2500
1.2500
1.2500
1.250
1.250
1.2500
1.3102

1.4997
1.4996
1.4997
1.4995
1.500
1.4999
1.4686

1.4997
1.500
1.500
1.4997
1.500
1.500
1.4236

1.250
1.2500
1.250
1.250
1.250
1.2500
1.3177

1.500
1.4997
1.4274
1.4995
1.4998
1.500
1.3697

1.250
1.250
1.250
1.250
1.250
1.250
1.3729

F
3.6774
3.6694
3.6734
3.6692
3.6674
3.6694
3.5624

NFE
95400
43400
67200
99700
55100
35330
35500

Figure 3: Best results obtained, for the IEEE 4-bus model, by different techniques are compared with the firefly algorithm.

Figure 4: The number of function evaluations done by different algorithms to solve the 4-bus model.

Furthermore, the results for the IEEE 6-bus model as in Table 3 and Figures 5 and 6, in terms of the best objective function value, are again slightly different from each other with MDE-4 and MDE-5 giving slightly improved solutions than other variant algorithms of DE. On the other hand, FA gave us an objective value of 10.3232, which is much better than other results in the table. However, in terms of NFE, the worst convergence rate was shown by DE while FA took 49980 NFE to get the best solution presented in the table.

Table 3: Best design variables, objective function values (F), and total number of function evaluations (NFE) for IEEE 6-bus model by DE, modified DE, and firefly algorithms.

DE
MDE-1
MDE-2
MDE-3
MDE-4
MDE-5
FA

0.1173
0.1171
0.1149
0.1034
0.1144
0.1024
0.7976

0.2082
0.1866
0.2037
0.1863
0.1864
0.1863
0.4522

0.0997
0.0965
0.0982
0.0961
0.0947
0.0946
0.1524

0.1125
0.1119
0.10367
0.1125
0.1006
0.1067
0.4740

0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.5339

0.0580
0.0500
0.0500
0.0500
0.0500
0.0500
0.2565

0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.08948

0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.3796

0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.3313

0.0719
0.0706
0.0575
0.0703
0.0701
0.0563
0.0980

0.0649
0.0649
0.0667
0.0649
0.0649
0.0650
0.3816

0.0617
0.0617
0.0566
0.0509
0.0509
0.0553
0.5236

0.0500
0.0500
0.0635
0.0500
0.0500
0.0500
0.8287

0.0856
0.0860
0.0859
0.0857
0.0709
0.0709
0.3320

1.2505
1.2515
1.2635
1.4995
1.2602
1.4991
1.3909

1.2500
1.4959
1.2993
1.4999
1.4987
1.4999
1.3483

1.2512
1.2525
1.2622
1.2575
1.2761
1.2771
1.2674

1.2515
1.2632
1.4322
1.2508
1.4992
1.3650
1.4622

1.2500
1.2500
1.2500
1.2500
1.2500
1.2500
1.3329

1.2500
1.3822
1.3885
1.3810
1.3814
1.3818
1.3563

1.2500
1.2500
1.2508
1.2500
1.2500
1.2500
1.4024

1.2500
1.2501
1.2500
1.2500
1.2505
1.2500
1.317

1.2502
1.2500
1.2514
1.2500
1.2500
1.2500
1.3291

1.2502
1.2501
1.4970
1.2521
1.2500
1.4996
1.3830

1.4998
1.4999
1.4759
1.4998
1.4999
1.4998
1.2923

1.2575
1.2529
1.4700
1.4997
1.5000
1.3931
1.3817

1.4805
1.4664
1.2728
1.4647
1.4615
1.4613
1.4035

1.2557
1.2500
1.2624
1.2540
1.4979
1.4974
1.2921

F
10.6272
10.5067
10.6238
10.4370
10.3812
10.3514
10.3232

NFE
212190
72960
18180
101580
100860
106200
49980

Figure 5: Best results obtained, for the IEEE 6-bus model, by different techniques are compared with the firefly algorithm.

Figure 6: The number of function evaluations done by different algorithms to solve the 6-bus model.

The last problem in this study is the 8-bus model. The results for the IEEE 8-bus model as in Table 4 and Figure 7, in terms of the best objective function values, are again in favor of FA. It is observed that, according to the quoted results obtained by the electromagnetic field optimization (EFO) algorithm, differential evolution (DE) algorithm, particle swarm optimization (PSO) algorithm, black hole (BH) algorithm, electromagnetism-like mechanism (EM), biogeography-based optimization (BBO), and harmony search (HS) [3] are slightly different from each other with EM, HS, BH, and PSO giving the worst results, while EFO, DE, and BBO gave improved solutions. On the other hand, FA gave us an objective value of 6.6463 which is much better than other results in the table.

Table 4: Comparison of the best values obtained for IEEE 8-bus systems [3].

Relays
EFO
DE
PSO
BH
EM
BBO
HS
FA

Number
TDS
(A)
TDS
(A)
TDS
(A)
TDS
(A)
TDS
(A)
TDS
(A)
TDS
(A)
TDS
(A)

1
0.197
237.11
0.074
480.00
0.077
480.00
0.290
228.31
0.430
163.31
0.164
335.22
0.251
241.57
0.0501
480.00

2
0.348
293.32
0.248
468.14
0.050
120.00
0.544
240.79
0.635
303.26
0.449
241.13
0.412
385.67
0.050
468.36

3
0.285
190.79
0.209
253.80
0.271
104.36
0.475
173.52
0.622
209.59
0.312
277.86
0.434
146.58
0.053
104.36

4
0.212
120.01
0.079
480.00
0.110
318.24
0.295
269.15
0.631
149.17
0.311
120.14
0.199
293.48
0.050
150.64

5
0.076
235.93
0.050
284.54
0.050
297.61
0.208
192.64
0.306
147.02
0.200
125.73
0.156
200.00
0.050
125

6
0.329
193.40
0.154
462.12
0.309
120.00
0.420
203.05
0.608
216.38
0.386
120.13
0.371
228.30
0.050
230.08

7
0.273
283.04
0.230
318.07
0.320
320.00
0.448
196.24
0.566
209.12
0.430
156.06
0.450
208.13
0.050
156.65

8
0.303
237.31
0.151
432.90
0.219
480.00
0.363
288.15
0.546
272.14
0.286
251.25
0.425
285.95
0.0817
251.48

9
0.098
207.89
0.050
320.00
0.050
278.30
0.237
127.96
0.348
116.53
0.226
98.89
0.179
177.36
0.050
187.36

10
0.174
225.25
0.050
480.00
0.050
268.71
0.264
257.61
0.663
146.32
0.198
301.97
0.246
215.11
0.050
220.30

11
0.263
250.91
0.226
121.51
0.200
120.00
0.367
249.57
0.515
363.46
0.290
300.12
0.348
254.75
0.050
255.30

12
0.343
319.59
0.221
472.95
0.345
127.16
0.496
237.60
0.723
240.59
0.529
138.33
0.432
404.57
0.0817
416.00

13
0.202
233.39
0.070
480.00
0.152
451.17
0.278
239.78
0.488
141.82
0.162
334.18
0.285
206.76
0.0534
230.56

14
0.311
230.26
0.217
320.00
0.309
154.82
0.397
216.33
0.455
260.14
0.328
263.29
0.428
223.33
0.058
230.34

F

7.611

6.654

10.421

11.401

15.913

7.547

11.760

6.6463

Figure 7: Best results obtained, for the IEEE 8-bus model, by different techniques are compared with the firefly algorithm [3].

Figure 8: Model for the IEEE 3-bus system.

Figure 9: Model for the IEEE 4-bus system.

Figure 10: Model for the IEEE 6-bus system.

Figure 11: Model for the IEEE 8-bus system [3].

Moreover, results obtained by FA are compared to previously quoted results of DE and MDE versions, RST-2, GA, SOMA, SOM-GA, LX-POL, and LX-PM [2, 4, 14]. The experimental outcomes are recorded in Tables 1–5 in terms of the best objective value and total number of function evaluations. Graphical analyses are presented, for each system, in shape of bar graphs given in Figures 1–7. Concise data is recorded in Table 5 which compares FA to other state-of-the-art algorithms. All experimental results suggest that FA can tackle all the three problems efficiently and optimally. It is noted that FA consumes a smaller number of function evaluations as compared to other methods.

Table 5: Comparison of the best values obtained for IEEE (3-, 4-, and 6-bus) systems.

Algorithms
3-bus model
4-bus model
6-bus model

FA
4.6557
3.5624
10.3232

DE
4.8421
3.6774
10.6272

MDE-1
4.8069
3.6694
10.5067

MDE-2
4.7872
3.6734
10.6238

MDE-3
4.7822
3.6692
10.4370

MDE-4
4.7806
3.6674
10.3812

MDE-5
4.7806
3.6694
10.3514

RST-2
4.8354
3.7050
10.6192

GA
5.0761
3.8587
13.7996

SOMA
8.0101
3.7892
26.1495

SOM-GA
4.7898
3.6745
10.3578

LX-POL
4.8265
3.5749
10.6028

LX-PM
4.8286
3.5830
10.6219

Table 6: The complexity of the DOCR problems as the bus size increases.

Number
IEEE 3-bus
IEEE 4-bus
IEEE 6-bus
IEEE 8-bus

Number of lines
3
4
7
7

Number of DOCRs (relays)
6
8
14
14

Number of decision variables
12
16
28
28

Number of selectivity constraints
8
9
38
40

Number of restricted constraints
24
32
104
104

Table 7: Values of constants , and for Model 1.

9.46
2.06

100.63
2.06

26.91
2.06

14.08
2.06

8.81
2.23

136.23
2.23

37.68
2.23

12.07
2.23

17.93
0.8

19.2
0.8

14.35
0.8

25.9
0.8

Table 8: Values of constants , and for Model I.

5
14.08
0.8
1
14.08
2.06

6
12.07
0.8
3
12.07
2.23

4
25.9
2.23
5
25.9
0.8

2
14.35
0.8
6
14.35
2.06

5
9.46
0.8
1
9.46
2.06

6
8.81
0.8
3
8.81
2.23

2
19.2
2.06
6
19.2
0.8

4
17.93
2.23
5
17.93
0.8

Table 9: Values of constants , and for Model II.

20.32
0.48

23.75
0.48

88.85
0.48

12.48
0.48

13.61
1.1789

31.92
1.1789

116.81
1.1789

10.38
1.1789

116.7
1.5259

12.07
1.5259

16.67
1.5259

31.92
1.5259

71.7
1.2018

11
1.2018

19.27
1.2018

18.91
1.2018

Table 10: Values of constants , and for Model II.

5
20.32
1.5259
1
20.32
0.48

5
12.48
1.5259
1
12.48
0.48

7
13.61
1.2018
3
13.61
1.1789

7
10.38
1.2018
3
10.38
1.1789

1
1.16
0.48
4
116.81
1.1789

2
12.07
0.48
6
12.07
1.1789

2
16.67
0.48
6
16.67
1.5259

4
11
1.1789
8
11
1.2018

4
19.27
1.1789
8
19.27
1.2018

Table 11: Values of constants , and for Model III.

2.5311
0.2585

5.9495
0.2585

2.7376
0.2585

5.3752
0.2585

2.9723
0.4863

6.6641
0.4863

4.1477
0.4863

4.5897
0.4863

1.9545
0.7138

6.2345
0.7138

2.7678
0.7138

4.2573
0.7138

3.8423
1.746

6.3694
1.746

5.618
1.746

4.1783
1.746

4.6538
1.0424

3.87
1.0424

3.5261
1.0424

5.2696
1.0424

2.584
0.7729

6.1144
0.7729

3.8006
0.7729

3.9005
0.7729

2.4143
0.5879

2.9011
0.5879

5.3541
0.5879

4.335
0.5879

Table 12: Values of constants , and for Model III.

p

q

8
4.0909
1.746
1
5.3752
0.2585

11
1.2886
0.7729
1
5.3752
0.2585

8
2.9323
1.746
1
2.5311
0.2585

3
0.6213
0.4863
2
2.7376
0.2585

3
1.6658
0.4863
2
5.9495
0.2585

10
0.0923
1.0424
3
4.5897
0.4863

10
2.561
1.0424
3
2.9723
0.4863

13
1.4995
0.5879
3
4.5897
0.4863

1
0.8869
0.2585
4
4.1477
0.4863

1
1.5243
0.2585
4
6.6641
0.4863

12
2.5444
0.7729
5
4.2573
0.7138

12
1.4549
0.7729
5
1.9545
0.7138

14
1.7142
0.5879
5
4.2573
0.7138

3
1.4658
0.4863
6
6.2345
0.7138

3
1.1231
0.2585
6
6.2345
0.7138

11
2.1436
0.7729
7
4.1783
1.746

2
2.0355
0.2585
7
4.1783
1.746

11
1.9712
0.7729
7
3.8423
1.746

2
1.8718
0.2585
7
3.8423
1.746

13
1.8321
0.5879
9
5.2696
1.0424

4
3.4386
0.4863
9
5.2696
1.0424

13
1.618
0.5879
9
4.6538
1.0424

4
3.0368
0.4863
9
4.6538
1.0424

14
2.0871
0.5879
11
3.9005
0.7729

6
1.8138
0.7138
11
3.9005
0.7729

14
1.4744
0.5879
11
2.584
0.7729

6
1.1099
0.7138
11
2.584
0.7729

8
3.3286
1.746
12
3.8006
0.7729

2
0.4734
0.2585
12
3.8006
0.7729

8
4.5736
1.746
12
6.1144
0.7729

2
1.5432
0.2585
12
6.1144
0.7729

12
2.7269
0.7729
13
4.335
0.5879

6
1.6085
0.7138
13
4.335
0.5879

12
1.836
0.7729
13
2.4143
0.5879

10
2.026
1.0424
14
2.9011
0.5879

4
0.8757
0.4863
14
2.9011
0.5879

10
2.7784
1.0424
14
5.3541
0.5879

4
2.5823
0.4863
14
5.3541
0.5879

Table 13: Primary/backup relays and fault currents for the 8-bus test system.

Primary relay
(A) (primary)
Backup relay
(A) (backup)

1
2666.3
6
2666.3

2
5374.8
1
804.7

2
5374.8
7
1531.5

3
3325.6
2
3325.6

4
2217.1
3
2217.1

5
1334.3
4
1334.3

6
4975
5
403.6

6
4975
14
1533

7
4247.6
5
403.6

7
4247.6
13
805.5

8
4973.2
7
1531.5

8
4973.2
9
403.2

9
1420.9
10
1420.9

10
2313.5
11
2313.5

11
3474.3
12
3474.3

12
5377
13
805.5

12
5377
14
1533

13
2475.7
8
2475.7

14
4246.4
1
804.7

14
4246.4
9
403.2

Table 14: The main characteristics of the 8-bus test system.

(A)

Relay
CT ratio
Min
Max

1
1200/5
120
480

2
1200/5
120
480

3
800/5
80
320

4
1200/5
120
480

5
1200/5
120
480

6
1200/5
120
480

7
800/5
80
320

8
1200/5
120
480

9
800/5
80
320

10
1200/5
120
480

11
1200/5
120
480

12
1200/5
120
480

13
1200/5
120
480

14
800/5
80
320

6. Conclusion

In electrical engineering, the problem of optimal coordination of several directional overcurrent relays is NP-hard with many constraints. The mathematical nature of the problem is highly complex, with several constraints imposed on the objective function. The problem of DOCRs requires suitable and robust optimization algorithms to tackle it. Experimental results obtained by FA are compared to previously quoted results of DE, RST-2, GA, SOMA, SOM-GA, LXPOL, LXPM, EFO, PSO, BH, EM, BBO, and HS. In order to summarize the achievements of the research, we present the following key aspects:

(i)In this paper, we implemented the firefly algorithm (FA) to improve the solutions for four optimization problems, namely, the IEEE (3-, 4-, 6-, and 8-bus) systems.(ii)To tackle the DOCR problems, the use of a suitable and efficient metaheuristic is needed.(iii)The outcome of our simulations shows that FA can efficiently minimize all the four models of the problem.(iv)The efficiency of FA can be observed from the minimum function evaluations required by the algorithm to reach the optimum as compared to other state-of-the-art algorithms.(v)The outcome suggests that FA is overall better and more efficient than the other optimization techniques considered in this research.(vi)In the future, one can extend this work to solve problems of higher buses and complex power systems. Moreover, extensive statistical analysis and parameters tuning can further highlight and improve the efficiency of FA.

Appendix

Tables 6–14 for IEEE 3-, 4-, 6-, and 8-bus systems.

Nomenclature

AI: | Artificial intelligence
CT: | Current transformer
DOCR: | Directional overcurrent relay
FA: | Firefly algorithm
GA: | Genetic algorithm
NIA: | Nature-inspired algorithm
PS: | Plug setting
PSO: | Particle swarm optimization
TDS: | Time dial setting.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding this publication.

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