# Optimal Overcurrent Relay Coordination Using Optimized Objective Function

Authors: Seyed Hadi Mousavi Motlagh; Kazem Mazlumi
Journal: ISRN Power Engineering
 (2014)

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## Abstract

A novel strategy for directional overcurrent relays (DOCRs) coordination is proposed. In the proposed method, the objective function is improved during the optimization process and objective function coefficients are changed in optimization problem. The proposed objective function is more flexible than the old objective functions because  various coefficients of objective function are set by optimization algorithm. The optimization problem is solved using hybrid genetic algorithm and particle swarm optimization algorithm (HGAPSOA). This method is applied to 6-bus and 30-bus sample networks.


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## Body

## 1. Introduction

Protection of distribution networks is one of the most important issues in power systems. Overcurrent relay is one of the most commonly used protective relays in these systems. There are two types of settings for these kinds of relays: current and time settings. A proper relay setting plays a crucial role in reducing undesired effects of faults on the power systems [1, 2]. Overcurrent relays commonly have plug setting (PS) ranging from 50 to 200% in steps of 25%. The PS shows the current setting of the overcurrent relays. For a relay installed on a line, PS is defined by two parameters: the minimum fault current and the maximum load current. However, the most important variable in the optimal coordination of overcurrent relays is the time multiplier setting (TMS) [3].So far, some researches have been carried out on coordination of overcurrent relays [3–7]. Due to the difficulty of nonlinear optimal programming techniques, the usual optimal coordination of overcurrent relays is generally carried out by linear programming techniques, including simplex, two-phase simplex, and dual simplex methods [3]. In these methods, the discrimination time of the main and backup relays (Δtmb) are considered as constraints and then the optimal coordination problem is solved using both objective function and constraints. In [8], a fast method for optimization of the TMSs and current settings by evolutionary algorithm and linear programming has been proposed. In [4], an online technique to estimate the setting of DOCRs is introduced. This technique is based on estimation of parameters of a proper equivalent circuit of the grid. Relay coordination which is very constrained discrete optimization problem is hardly solved by traditional optimization techniques [5]. In [9], the pickup current and the TMS of the relays have been considered the optimization parameters for optimal coordination of directional overcurrent relays. These optimization techniques are started by an initial conjecture and are possibly trapped in a local optimum [3]. The intelligent optimization techniques have come up in such a way that can adjust the settings of the relays without the mentioned problems. In these methods, the constraints are considered a part of objective function. In [6], a developed method based on genetic algorithm (GA) for optimal coordination has been proposed; this method has not considered the main principle of the coordination. So there may be some miscoordination in the settings of the relays. As a result of the miscoordination, when a fault occurs in front of the main relays, some backup relays may operate faster than the main one. This causes more lines of network to go out due to the occurring faults in some parts of the network. Obviously, in this case, the number of interrupted customers increases and consequently the power quality declines.In [10], continuous genetic algorithm has been used in a ring fed distribution system for optimal coordination of overcurrent relays. In [11], an objective function has been proposed in which the problem of miscoordination has almost been solved. The objective function of [11] can prevent the appearance of inappropriate results which make miscoordination. Suitable parameter initializing has been considered in the objective function of the paper that weights both the time and the delay time of the backup relays. The main shortcoming of the objective function of [11] is that some of the coefficients of objective function are set by try and error. It is proved by experience that using such coefficients cannot guarantee accessing to the smallest operation time of the relays. In some cases, using inappropriate parameters in objective function may result in some miscoordination.In this paper, a new method for directional overcurrent relay coordination is proposed in which not only the miscoordination is omitted but also operation times of the relays are the smallest. This is because, in this novel algorithm, the coefficients of the previous proposed objective function are considered a part of the optimization problem. So the coefficients of the objective function are not constant and they are not set by try and error. In the proposed method, the coefficients are calculated based on the optimization technique. The optimization problem of this paper is solved using a hybrid genetic and particle swarm optimization algorithm. This method is applied to 6-bus and 30-bus sample networks. The results of this method are compared with the results of the existing ones. The simulation results and the comparisons demonstrate the effectiveness and the advantage of the proposed algorithm.


## 2. Hybrid GA and PSO (HGAPSO) in Relay Coordination Application

By incorporating the GA into the PSO, the novel HGAPSO algorithm is obtained [12]. The HGAPSOA can provide better results [13]. In this algorithm, the initial population of GA is assigned by the solution of PSO. The total numbers of the iterations are equivalently shared by GA and PSO. First half of the iterations are carried out by PSO and the solutions are given as initial population of GA [14]. In this section, we briefly discuss the application of HGAPSOA in overcurrent relay coordination. According to Figure 1, at first, the user initializes the information such as algorithm parameters (population size, iteration numbers, mutation, and crossover range), relay data, and network data. The HGAPSOA basics in coordination of the DOCRs are described in 4 steps.Figure 1
Flowchart of the proposed method.Step
1: Initial Generation. A number of random particles are produced. The variables of HGAPSOA are the TMSs of the DOCRs and the coefficients of the objective function. In our problem, the number of the variables is equal to the number of the relays and the number of the objective function coefficients. The TMSs are randomly generated between 0.05 and 1.Step
2: Evaluate the Objective Function. In this step, the value of the objective function is calculated for every gene and discerns between good and bad chromosomes.Step
3: PSO Operators. In PSO box, update the local best position experienced by every particle denoted as pbest and update the best position experienced by whole particles denoted as gbest.Step
4: GA Operators. The best particles are selected as the initial population of GA. In this step, the objective function evaluation is compared with its initial best chromosome (minimum value in the TMS vector). If the evaluated value is less than the initial best chromosome, set the best chromosome equal to the evaluated value. Then, update the position of the genes by using crossover and mutation operators.Finally, if any of the following stopping criteria (in this study maximum number of iterations), then go to the following module. This module is used to transfer the optimal relay TMSs calculated from HGAPSOA through the interface system to each relay.


## 3. Problem Statement

To prevent overcurrent relays miscoordination and to find the optimum results of the objective function of [11], the novel objective function is formulated as follows:
(1)O·F=α1×∑(ti)M+α2×∑(Δtmb-β×(Δtmb-|Δtmb|))N,
where α1, α2, and β are weighting coefficients; M and N are even numbers; ti is the operating time of ith overcurrent relay when a fault occurs next to the relay; Δtmb is the discrimination time between the main and backup overcurrent relays. Δtmb is obtained by
(2)Δtmb=tb-tm-CTI,
where tb and tm are the operating time of the backup and main relays when a fault occurs next to the main relay. The value of coordination time interval (CTI) is mainly chosen based on the practical limitations, which consist of the relay over-travel time, the breaker operating time, and the safety margin for the relay error [15]. Generally, the suitable CTI is selected between 0.2 and 0.5 second. In this study, CTI is considered to be 0.3 seconds. According to the proposed objective function of [11], the weighting coefficients (β,α1,α2) are set by try and error and also the coefficients of (M,N) are constant and cannot guarantee accessing to the smallest operation time of the relays.The first term of (1) is the sum of overcurrent relays’ operating time and the second term is the coordination constraint. To describe the role of the second term of the above object function, consider thatΔtmb must be positive; then the relative expression (Δtmb-β·(Δtmb-|Δtmb|) is equal toΔtmb. Exactly the mentioned equation, ifΔtmb is negative is, as follows:
(3)(1+2β)(Δtmb).If the coefficientβ is a large number, the value of (3) is greater than the Δtmb, and the object function results in a large value and is not selected for the next iterations in the optimization algorithm. Consequently, the output results contain no miscoordination.


## 4. New Method

In this paper, the five coefficients of (1), that is, β,α1,α2, M, and N, beside TMSs, are incorporated as a part of the optimization parameters. The new objective function is shown in (1).To understand the importance of these coefficients the following notes are helpful.(i)
To assure minimizing the main relay operating time, a great value must be assigned toα1 and M.(ii)
To assure minimizing the backup relay operation time, a great value must be assigned toα2 and N.(iii)
To prevent miscoordination and have faster and more accurate convergence, choose a large value forβ and N.According to the above notes, it is obvious that determining these five coefficients as well as the TMS of the relays with an optimization algorithm will result in a better relay coordination. Therefore, the method presented in this paper optimizes the objective function coefficients as well as the TMSs of the relays to have a tradeoff between the fast operating time and preventing miscoordination. The algorithm applied to the relays coordination optimization problem for the proposed technique is provided in Figure1. In the first step of the algorithm, the initial values ​​are randomly guessed. The initial values consist of both relay’s TMS and the coefficients of the objective function. The relay settings are packed into a packet; each packet is a probable result for the optimization problem; and therefore each member of a packet represents a TMS of a relay. In addition, there are five extra coefficients that should be optimized which belong to the objective function (α1, α2, β, M, and  N). In the second step, the packet is evaluated by the objective function of (1). The objective function determines the value of each packet by receiving the TMS of all relays. After evaluation process, the best packet is chosen from the least objective function output point of view. In the next step, finishing criteria of algorithm are checked to be the limited iterations and yield a prespecified fitness value. If one of them is satisfied, then the algorithm terminates and goes to the next stage; otherwise it returns to the second step and starts the next iteration.


## 5. Mathematical Overcurrent Relay Models

There are many mathematical models for the overcurrent relays. In this study, the mathematical model of overcurrent relays is considered to be the standard inverse type. In this mathematical model, the operating time of the overcurrent relay is expressed as follows [16]:
(4)t=k×TMS(isc/i0)n-1+L,
where I0 is relay current setting; Isc is short-circuit current; T is the relay operating time; and TMS is the time multiplier setting of the relay. TMS varies from 0.05 to 1. From the IEC curves, for standard inverse type relays, the parameters of (4) are assumed to be L=0, k=0.14, and n=0.02.


## 6. Comparison of the Results and Discussion

As described in Section4, the values of the five coefficients are variable; so the output value of the objective function will not be an appropriate criterion for comparing results of the proposed algorithm in [11]. Therefore, in order to gain this aim, two indicators are defined as follows.(i)
Summation of TMSs. If, in a set of results of relay coordination, there is not any miscoordination between each two relays, it can be said that the better coordination has smaller summation of TMSs and, therewith, relays have lower operating times.(ii)
Summation of Difference between Each Time Operation (∑Δtm,b). For comparing two sets of the coordination results, for example, result A and result B, which do not have any miscoordination, if value of (5) is true, then the set of result A is better than B. Consider
(5)∑(m,b)=1n(Δt(m,b)A-Δt(m,b)B)<0.In (5), the number of pairs of main/backup (m,b) relays is equal to n. In this paper, the proposed method is applied to two different networks, namely, case study 1 and case study 2 (case study 1 consists of 6 buses and case study 2 is the IEEE 30-bus system) which are selected.


### 6.1. Case Study 1: Network and Protection Information

Case study 1 is 6-bus network shown in Figure2. This network consists of 7 lines, 6 buses, 2 generators, 2 transformers, and 14 overcurrent relays [17].Figure 2
Case study 1 network.All the information about this network including short-circuit current of backup and main relays and relevant information relative to main/backup relays have been provided in [11] and also are shown in Table 1.Table 1
Main/backup pair information.
Main relay
Backup relay
Primary relay SC current
Backup relay current
8
9
4961
410
8
7
4961
1520
2
7
5362
1528
2
1
5362
804
3
2
3334
3334
4
3
2234
2234
5
4
1352
1352
6
5
4965
411
6
14
4965
1522
14
1
4232
794
14
9
4232
407
1
6
2682
2682
9
10
1443
1443
10
11
2334
2334
11
12
3480
3480
12
14
5365
1529
12
13
5365
805
13
8
2490
2490
7
5
4232
407
7
13
4232
794Table2 shows the maximum load current passing through the relays. The way of the relay current settings is described as follows [11]:
(6)I0=1.2×Ilmax⁡.Table 2
I
l
max
⁡ information.
Relay number
Maximum load current
Relay number
Maximum load current
1
416
8
458
2
666
9
450
3
500
10
458
4
666
11
541
5
458
12
458
6
458
13
500
7
541
14
666All control parameters of algorithm are listed in Table3. The number of the variables in optimization problem is 19. Fourteen of the variables are related to the TMS of the relays and the remaining five variables are the coefficients of the objective function, that is, β,α1,α2,M,N.Table 3
The input parameters of the optimization algorithm.
Mutate range
0.3
Crossover range
0.7
Number of variables
19
Number of population
100
Maximum iteration
200
Maximum subiteration GA
10
Maximum subiteration PSO
10The results of the proposed method by GAPSO algorithm and the best results of [11] are compared in Table 4. The best results of [11] are shown in the second column of Table 4. The third column of Table 4 represents feasible results for coordination of the relays using the proposed method. From Table 4, all Δtmb values are small and positive and the largest Δtmb is 0.511 sec. The results mean there is not any miscoordination in the results. Table 4 shows two superiority indicators, that is, ∑TMS and ∑Δtmb, optimized using the proposed algorithm. The first superiority indicator (∑TMS) for the results of [11] is 2.75 and for the proposed method is 1.741; and the second superiority indicator (∑Δtmb) for the results of [11] is 6.699 sec and for the proposed method is 1.55 sec. The superiority indicators show that the results would be more optimized when the coefficients are optimized by the proposed algorithm. The advantage of this method is revealed when this new method is compared with the result of [11] in which the parameters are set manually in Table 4.Table 4
The coordination results.
Coordination output
Reference [11]
GAPSO
TMS
1
0.15
0.066
TMS
2
0.35
0.1833
TMS
3
0.25
0.146
TMS
4
0.1
0.069
TMS
5
0.1
0.08
TMS
6
0.25
0.13
TMS
7
0.2
0.12
TMS
8
0.25
0.12
TMS
9
0.05
0.095
TMS
10
0.15
0.135
TMS
11
0.25
0.174
TMS
12
0.4
0.273
TMS
13
0.1
0.05
TMS
14
0.15
0.1
t
1
0.5894
0.27
t
2
1.2173
0.66
t
3
0.9621
0.58
t
4
0.6636
0.46
t
5
0.7660
0.61
t
6
0.7623
0.4
t
7
0.7059
0.43
t
8
0.7625
0.37
t
9
0.3479
0.66
t
10
0.6883
0.64
t
11
0.9835
0.71
t
12
1.1847
0.82
t
13
0.4665
0.24
t
14
0.5944
0.43
Δ
t
89
0
0
Δ
t
87
0.4673
0.296
Δ
t
27
0.0028
0
Δ
t
21
0.6290
0
Δ
t
32
0.2657
0
Δ
t
43
0.2057
0
Δ
t
54
0.1885
0
Δ
t
65
0
0
Δ
t
6,14
0.4825
0.418
Δ
t
14,1
1.3181
0.258
Δ
t
14,9
0
0
Δ
t
16
0.0532
0
Δ
t
9,10
0.3173
0
Δ
t
10,11
0.2195
0
Δ
t
11,12
0.0492
0
Δ
t
12,14
0.0489
0
Δ
t
12,13
0.8135
0.065
Δ
t
13,8
0.2292
0
Δ
t
7,5
0
0
Δ
t
7,13
1.4086
0.511
∑
Δ
t
6.699
1.55
∑
TMS
2.75
1.741With attend to Table4, more relays operating time that are earn from proposed method, are smaller than [11]. For example, t2 in the proposed method is 0.66 seconds and in [11] is 1.2173 seconds or t12 in proposed method is 0.82 seconds and in [11] is 1.1847 sec. Also, the values of the five parameters of objective function are shown in Table 5.Table 5
The object function parameters.
α
1
25
α
2
0.1
β
15.722
M
100
N
2


### 6.2. Case Study 2: Network and Protection Information

For the other test case, 30-bus IEEE network is considered. This network has 86 OC relays. It consists of 30 buses (132- and 33-kV buses), 37 lines, 6 generators, 4 transformers, and 86 OC relays [18]. The distribution section of network that will be studied here is shown in Figure 3 [19]. The generator, transmission lines, and transformer information are given from [20].Figure 3
IEEE 30-bus system.In this paper, four different cases are simulated for a suitable comparison. Three cases are simulated according to the method of [11] in which the objective function coefficients are set manually, and the last case is simulated according to the method proposed in this paper, in which the objective function coefficients are not set before simulation and the optimization results show the values of the coefficients. All cases are evaluated by HGAPSO algorithm and all control parameters of algorithm are listed in Table 6. The parameters are considered the same as 4 test cases. A number of variables in new method are considered 91, 86 variables for relays number and 5 variables for (β,α1,α2,M,N) and for method proposed in [11] are considered 86.Table 6
Algorithm parameters information.
Mutate range
0.3
Crossover range
0.8
Number of variables
91
Number of population
200
Maximum iteration
500
Maximum subiteration GA
10
Maximum subiteration PSO
10Firstly, the results of the proposed method by GAPSOA are shown in Tables7 and 8. The first column of Table 7 shows relay number, the second column (T) shows relay operating time for a fault close to the circuit breaker of each relay, and the third column indicates the TMS of the relays. From Table 8, all of the values of Δtmb are small and positive numbers. The positive values of Δtmb show that there is no miscoordination in the results. For example, Δt4,1 refers to the relay pairs (4 and 1) that are obtained at 0.3529 seconds. The values of the five coefficients of the objective function β, α1, α2, M, and N are obtained as 585.5, 15, 5, 12.5, and 6, respectively. When a fault occurs at downstream, operating time of the downstream protective relays would not be affected [21]. In current grading, the pickup current was chosen by considering the maximum possible load current due to normal overloading or contingency conditions. The loads of the buses 11 and 26 are the static types and there is no need to install overcurrent relays on these buses. So TMSs of relays 57 and 79 that are connected to these buses are considered to be 0.05 [22] and other TMSs were calculated.Table 7
Coordination output results.
Relay number
T
TMS
1
0.3092
0.117
2
0.3092
0.117
3
0.3467
0.119
4
0.2095
0.068
5
0.7561
0.102
6
0.5671
0.227
7
0.248
0.091
8
0.7271
0.189
9
0.5679
0.107
10
0.6679
0.245
11
0.67
0.241
12
0.423
0.160
13
0.621
0.195
14
0.9412
0.25
15
0.9268
0.170
16
0.2343
0.074
17
0.5493
0.139
18
0.5749
0.190
19
0.3947
0.105
20
0.5669
0.163
21
0.2716
0.059
22
0.5094
0.090
23
0.5468
0.139
24
0.7716
0.142
25
0.7482
0.217
26
0.6391
0.239
27
0.6033
0.203
28
0.6351
0.184
29
0.6351
0.184
30
0.6358
0.183
31
0.4274
0.088
32
0.5374
0.118
33
0.252
0.090
34
0.2694
0.089
35
0.3322
0.084
36
0.8262
0.174
37
0.4902
0.106
38
0.1732
0.05
39
0.1731
0.061
40
0.1627
0.068
41
0.6102
0.123
42
0.2748
0.115
43
0.656
0.25
44
0.2103
0.076
45
0.2249
0.084
46
0.397
0.081
47
0.1756
0.056
48
0.5501
0.179
49
0.7792
0.155
50
0.1803
0.061
51
0.654
0.211
52
0.6578
0.166
53
0.6173
0.129
54
0.5556
0.093
55
1.0366
0.224
56
0.5839
0.143
58
0.6095
0.176
59
0.2618
0.065
60
0.4635
0.129
61
0.2769
0.056
62
0.3095
0.061
63
0.3954
0.066
64
0.5534
0.128
65
0.3576
0.094
66
0.5278
0.099
67
0.5855
0.099
68
0.7435
0.161
69
0.5109
0.095
70
0.6543
0.113
71
0.2269
0.078
72
0.477
0.143
73
0.2072
0.070
74
0.6521
0.204
75
0.3362
0.104
76
0.5226
0.115
77
0.2853
0.073
78
0.4494
0.116
80
0.4341
0.137
81
0.3934
0.111
82
0.8588
0.154
83
0.9303
0.202
84
0.5091
0.077
85
0.3488
0.144
86
0.3566
0.079Table 8
Time difference for each pair of M/B relays.
Δ
t
4,1
0.3529
Δ
t
18,16
0.0021
Δ
t
3,44
0
Δ
t
15,57
0
Δ
t
56,71
0.0122
Δ
t
6,1
0
Δ
t
19,16
0.1591
Δ
t
1,45
0.0337
Δ
t
55,57
0
Δ
t
58,71
0.0025
Δ
t
7,1
0.3146
Δ
t
20,16
0
Δ
t
3,45
0
Δ
t
14,58
0.1002
Δ
t
26,72
0
Δ
t
45,1
0.3418
Δ
t
21,18
1.0402
Δ
t
1,46
0
Δ
t
55,58
0
Δ
t
27,72
0
Δ
t
4,2
0.3529
Δ
t
23,19
0
Δ
t
2,46
0
Δ
t
8,59
0
Δ
t
28,72
0
Δ
t
6,2
0
Δ
t
32,19
0.0027
Δ
t
6,47
0
Δ
t
47,59
0.5927
Δ
t
30,72
0
Δ
t
7,2
0.3146
Δ
t
64,19
0
Δ
t
7,47
0.3238
Δ
t
48,59
0.1788
Δ
t
56,72
0
Δ
t
44,2
0.3565
Δ
t
22,20
0.5459
Δ
t
44,47
0.3147
Δ
t
18,60
0
Δ
t
58,72
0
Δ
t
5,3
0
Δ
t
23,21
0.0023
Δ
t
45,47
0.3
Δ
t
19,60
0.1665
Δ
t
26,73
0.0003
Δ
t
8,4
0
Δ
t
32,21
0
Δ
t
46,48
0
Δ
t
20,60
0.0008
Δ
t
27,73
0
Δ
t
16,4
0.56
Δ
t
62,21
0.2384
Δ
t
4,49
0
Δ
t
59,60
0.3069
Δ
t
28,73
0
Δ
t
48,4
0.1792
Δ
t
70,22
0
Δ
t
7,49
0
Δ
t
17,61
0
Δ
t
29,73
0
Δ
t
8,5
0
Δ
t
24,23
0.3009
Δ
t
44,49
0
Δ
t
19,61
0
Δ
t
56,73
0.0081
Δ
t
16,5
0.5385
Δ
t
25,24
0
Δ
t
45,49
0
Δ
t
20,61
0
Δ
t
58,73
0
Δ
t
47,5
0.5848
Δ
t
69,25
0
Δ
t
4,50
0.3518
Δ
t
59,61
0
Δ
t
71,74
0.2501
Δ
t
9,6
0.6558
Δ
t
68,26
0.4949
Δ
t
6,50
0
Δ
t
17,62
0
Δ
t
72,74
0
Δ
t
10,7
0.015
Δ
t
65,27
0.2957
Δ
t
44,50
0.3118
Δ
t
18,62
0.0312
Δ
t
23,75
0.0048
Δ
t
11,7
0
Δ
t
31,28
0
Δ
t
45,50
0.2971
Δ
t
20,62
0.0022
Δ
t
62,75
0.2319
Δ
t
12,7
0.2883
Δ
t
72,28
0
Δ
t
16,51
0.3311
Δ
t
59,62
0.3093
Δ
t
64,75
0
Δ
t
13,7
0.0488
Δ
t
31,29
0
Δ
t
47,51
0.3843
Δ
t
17,63
0
Δ
t
73,76
0.4701
Δ
t
43,7
0.0184
Δ
t
71,29
0
Δ
t
48,51
0
Δ
t
18,63
0.0059
Δ
t
74,76
0
Δ
t
51,7
0.0165
Δ
t
33,30
0
Δ
t
49,52
0
Δ
t
19,63
0.1605
Δ
t
75,77
0
Δ
t
10,8
0.0076
Δ
t
74,30
0
Δ
t
11,53
0
Δ
t
59,63
0.3125
Δ
t
76,78
0
Δ
t
11,8
0
Δ
t
33,31
0
Δ
t
12,53
0.2759
Δ
t
61,64
0
Δ
t
77,78
0
Δ
t
12,8
0.2671
Δ
t
73,31
0.0514
Δ
t
13,53
0.0489
Δ
t
63,65
0
Δ
t
37,79
0
Δ
t
13,8
0.0494
Δ
t
34,32
0.6332
Δ
t
43,53
0.0171
Δ
t
32,66
0.0072
Δ
t
78,79
0.2437
Δ
t
43,8
0.0161
Δ
t
35,33
0
Δ
t
50,53
0.5228
Δ
t
62,66
0.2319
Δ
t
36,80
0
Δ
t
50,8
0.5093
Δ
t
77,33
0.0416
Δ
t
51,53
0.0163
Δ
t
64,66
0
Δ
t
78,80
0.3633
Δ
t
53,9
0
Δ
t
35,34
0.2067
Δ
t
10,54
0
Δ
t
66,67
0
Δ
t
85,81
0
Δ
t
52,10
0.0881
Δ
t
76,34
0
Δ
t
12,54
0
Δ
t
67,68
0
Δ
t
86,81
0
Δ
t
42,11
0.3288
Δ
t
36,35
0
Δ
t
13,54
0
Δ
t
27,69
0.0239
Δ
t
40,82
0
Δ
t
14,12
0.0006
Δ
t
37,35
0.2988
Δ
t
43,54
0
Δ
t
28,69
0
Δ
t
80,82
0
Δ
t
15,12
0
Δ
t
39,37
0.2264
Δ
t
50,54
0
Δ
t
29,69
0
Δ
t
81,82
0
Δ
t
26,13
0.005
Δ
t
40,37
0.2523
Δ
t
51,54
0
Δ
t
30,69
0.0006
Δ
t
39,83
0
Δ
t
27,13
0.027
Δ
t
81,37
0
Δ
t
10,55
0
Δ
t
56,69
0.0331
Δ
t
80,83
0
Δ
t
28,13
0.0001
Δ
t
39,38
0.3152
Δ
t
11,55
0
Δ
t
58,69
0.0202
Δ
t
81,83
0
Δ
t
29,13
0.0001
Δ
t
40,38
0.3684
Δ
t
13,55
0
Δ
t
26,70
0.0155
Δ
t
82,84
0
Δ
t
30,13
0
Δ
t
80,38
0.0055
Δ
t
43,55
0
Δ
t
28,70
0
Δ
t
54,85
0
Δ
t
58,13
0.0262
Δ
t
41,39
0
Δ
t
50,55
0
Δ
t
29,70
0
Δ
t
10,86
0
Δ
t
26,15
0
Δ
t
84,40
0
Δ
t
51,55
0
Δ
t
30,70
0.0003
Δ
t
11,86
0.0116
Δ
t
27,15
0
Δ
t
83,41
0
Δ
t
10,56
0.0082
Δ
t
56,70
0.0368
Δ
t
12,86
0.2807
Δ
t
28,15
0
Δ
t
38,42
0.1969
Δ
t
11,56
0
Δ
t
58,70
0.0215
Δ
t
13,86
0.0363
Δ
t
29,15
0
Δ
t
86,42
0
Δ
t
12,56
0.2863
Δ
t
26,71
0
Δ
t
50,86
0.5105
Δ
t
30,15
0
Δ
t
38,43
0.5089
Δ
t
43,56
0.0169
Δ
t
27,71
0.0031
Δ
t
51,86
0
Δ
t
56,15
0
Δ
t
85,43
0.3278
Δ
t
50,56
0.5088
Δ
t
29,71
0.0142
Δ
t
17,16
0.0001
Δ
t
2,44
0.0333
Δ
t
51,56
0.0146
Δ
t
30,71
0.006In Table9, the results of the simulations for the four cases are illustrated. In this table, the first row shows the four different cases; and the second and the third rows show the results of the first index (∑Δt) and the second index (∑TMS) for these cases. The three first cases are related to the method of [11], where in these cases the coefficients of the objective function are set before optimization process. The values of the coefficients are shown in the first column of the table. The last case shows the results of the proposed method. The values of Δt in all cases are positive and have not any miscoordination.Table 9
Comparison between the proposed method and the method of [11].
Case1
Case2
Case3
Case4
β = 1
β = 10
β = 10
β = 585.5
α
1 = 1
α
1 = 10
α
1 = 1
α
1 = 15
α
2 = 100
α
2 = 1
α
2 = 10
α
2 = 5
M = 2
M = 2
M = 2
M = 12.5
N
= 2
N
= 2
N
= 2
N
= 6
∑
Δ
t
29.8211
24.115
26.9
22.19
∑
TMS
15.15
14.55
15.6
11.32The advantage of the proposed method is revealed when the results of the proposed method (Case 4) are compared with the best results of the traditional method of [11]. According to Table 9, the best ∑TMS and ∑Δt of method [11] (between three cases: 1 and 2 and 3) are 14.55 and 24.115, respectively, whereas ∑TMS and ∑Δt of the proposed method are 11.32 and 22.19, respectively. This means that the results of Case 4, related to the new proposed method, show the best coordination. So, considering the coefficients of objective function as a part of the optimization problem, results are better compared with conventional method according to the results of Tables 4 and 9.


## 6.1. Case Study 1: Network and Protection Information

Case study 1 is 6-bus network shown in Figure2. This network consists of 7 lines, 6 buses, 2 generators, 2 transformers, and 14 overcurrent relays [17].Figure 2
Case study 1 network.All the information about this network including short-circuit current of backup and main relays and relevant information relative to main/backup relays have been provided in [11] and also are shown in Table 1.Table 1
Main/backup pair information.
Main relay
Backup relay
Primary relay SC current
Backup relay current
8
9
4961
410
8
7
4961
1520
2
7
5362
1528
2
1
5362
804
3
2
3334
3334
4
3
2234
2234
5
4
1352
1352
6
5
4965
411
6
14
4965
1522
14
1
4232
794
14
9
4232
407
1
6
2682
2682
9
10
1443
1443
10
11
2334
2334
11
12
3480
3480
12
14
5365
1529
12
13
5365
805
13
8
2490
2490
7
5
4232
407
7
13
4232
794Table2 shows the maximum load current passing through the relays. The way of the relay current settings is described as follows [11]:
(6)I0=1.2×Ilmax⁡.Table 2
I
l
max
⁡ information.
Relay number
Maximum load current
Relay number
Maximum load current
1
416
8
458
2
666
9
450
3
500
10
458
4
666
11
541
5
458
12
458
6
458
13
500
7
541
14
666All control parameters of algorithm are listed in Table3. The number of the variables in optimization problem is 19. Fourteen of the variables are related to the TMS of the relays and the remaining five variables are the coefficients of the objective function, that is, β,α1,α2,M,N.Table 3
The input parameters of the optimization algorithm.
Mutate range
0.3
Crossover range
0.7
Number of variables
19
Number of population
100
Maximum iteration
200
Maximum subiteration GA
10
Maximum subiteration PSO
10The results of the proposed method by GAPSO algorithm and the best results of [11] are compared in Table 4. The best results of [11] are shown in the second column of Table 4. The third column of Table 4 represents feasible results for coordination of the relays using the proposed method. From Table 4, all Δtmb values are small and positive and the largest Δtmb is 0.511 sec. The results mean there is not any miscoordination in the results. Table 4 shows two superiority indicators, that is, ∑TMS and ∑Δtmb, optimized using the proposed algorithm. The first superiority indicator (∑TMS) for the results of [11] is 2.75 and for the proposed method is 1.741; and the second superiority indicator (∑Δtmb) for the results of [11] is 6.699 sec and for the proposed method is 1.55 sec. The superiority indicators show that the results would be more optimized when the coefficients are optimized by the proposed algorithm. The advantage of this method is revealed when this new method is compared with the result of [11] in which the parameters are set manually in Table 4.Table 4
The coordination results.
Coordination output
Reference [11]
GAPSO
TMS
1
0.15
0.066
TMS
2
0.35
0.1833
TMS
3
0.25
0.146
TMS
4
0.1
0.069
TMS
5
0.1
0.08
TMS
6
0.25
0.13
TMS
7
0.2
0.12
TMS
8
0.25
0.12
TMS
9
0.05
0.095
TMS
10
0.15
0.135
TMS
11
0.25
0.174
TMS
12
0.4
0.273
TMS
13
0.1
0.05
TMS
14
0.15
0.1
t
1
0.5894
0.27
t
2
1.2173
0.66
t
3
0.9621
0.58
t
4
0.6636
0.46
t
5
0.7660
0.61
t
6
0.7623
0.4
t
7
0.7059
0.43
t
8
0.7625
0.37
t
9
0.3479
0.66
t
10
0.6883
0.64
t
11
0.9835
0.71
t
12
1.1847
0.82
t
13
0.4665
0.24
t
14
0.5944
0.43
Δ
t
89
0
0
Δ
t
87
0.4673
0.296
Δ
t
27
0.0028
0
Δ
t
21
0.6290
0
Δ
t
32
0.2657
0
Δ
t
43
0.2057
0
Δ
t
54
0.1885
0
Δ
t
65
0
0
Δ
t
6,14
0.4825
0.418
Δ
t
14,1
1.3181
0.258
Δ
t
14,9
0
0
Δ
t
16
0.0532
0
Δ
t
9,10
0.3173
0
Δ
t
10,11
0.2195
0
Δ
t
11,12
0.0492
0
Δ
t
12,14
0.0489
0
Δ
t
12,13
0.8135
0.065
Δ
t
13,8
0.2292
0
Δ
t
7,5
0
0
Δ
t
7,13
1.4086
0.511
∑
Δ
t
6.699
1.55
∑
TMS
2.75
1.741With attend to Table4, more relays operating time that are earn from proposed method, are smaller than [11]. For example, t2 in the proposed method is 0.66 seconds and in [11] is 1.2173 seconds or t12 in proposed method is 0.82 seconds and in [11] is 1.1847 sec. Also, the values of the five parameters of objective function are shown in Table 5.Table 5
The object function parameters.
α
1
25
α
2
0.1
β
15.722
M
100
N
2


## 6.2. Case Study 2: Network and Protection Information

For the other test case, 30-bus IEEE network is considered. This network has 86 OC relays. It consists of 30 buses (132- and 33-kV buses), 37 lines, 6 generators, 4 transformers, and 86 OC relays [18]. The distribution section of network that will be studied here is shown in Figure 3 [19]. The generator, transmission lines, and transformer information are given from [20].Figure 3
IEEE 30-bus system.In this paper, four different cases are simulated for a suitable comparison. Three cases are simulated according to the method of [11] in which the objective function coefficients are set manually, and the last case is simulated according to the method proposed in this paper, in which the objective function coefficients are not set before simulation and the optimization results show the values of the coefficients. All cases are evaluated by HGAPSO algorithm and all control parameters of algorithm are listed in Table 6. The parameters are considered the same as 4 test cases. A number of variables in new method are considered 91, 86 variables for relays number and 5 variables for (β,α1,α2,M,N) and for method proposed in [11] are considered 86.Table 6
Algorithm parameters information.
Mutate range
0.3
Crossover range
0.8
Number of variables
91
Number of population
200
Maximum iteration
500
Maximum subiteration GA
10
Maximum subiteration PSO
10Firstly, the results of the proposed method by GAPSOA are shown in Tables7 and 8. The first column of Table 7 shows relay number, the second column (T) shows relay operating time for a fault close to the circuit breaker of each relay, and the third column indicates the TMS of the relays. From Table 8, all of the values of Δtmb are small and positive numbers. The positive values of Δtmb show that there is no miscoordination in the results. For example, Δt4,1 refers to the relay pairs (4 and 1) that are obtained at 0.3529 seconds. The values of the five coefficients of the objective function β, α1, α2, M, and N are obtained as 585.5, 15, 5, 12.5, and 6, respectively. When a fault occurs at downstream, operating time of the downstream protective relays would not be affected [21]. In current grading, the pickup current was chosen by considering the maximum possible load current due to normal overloading or contingency conditions. The loads of the buses 11 and 26 are the static types and there is no need to install overcurrent relays on these buses. So TMSs of relays 57 and 79 that are connected to these buses are considered to be 0.05 [22] and other TMSs were calculated.Table 7
Coordination output results.
Relay number
T
TMS
1
0.3092
0.117
2
0.3092
0.117
3
0.3467
0.119
4
0.2095
0.068
5
0.7561
0.102
6
0.5671
0.227
7
0.248
0.091
8
0.7271
0.189
9
0.5679
0.107
10
0.6679
0.245
11
0.67
0.241
12
0.423
0.160
13
0.621
0.195
14
0.9412
0.25
15
0.9268
0.170
16
0.2343
0.074
17
0.5493
0.139
18
0.5749
0.190
19
0.3947
0.105
20
0.5669
0.163
21
0.2716
0.059
22
0.5094
0.090
23
0.5468
0.139
24
0.7716
0.142
25
0.7482
0.217
26
0.6391
0.239
27
0.6033
0.203
28
0.6351
0.184
29
0.6351
0.184
30
0.6358
0.183
31
0.4274
0.088
32
0.5374
0.118
33
0.252
0.090
34
0.2694
0.089
35
0.3322
0.084
36
0.8262
0.174
37
0.4902
0.106
38
0.1732
0.05
39
0.1731
0.061
40
0.1627
0.068
41
0.6102
0.123
42
0.2748
0.115
43
0.656
0.25
44
0.2103
0.076
45
0.2249
0.084
46
0.397
0.081
47
0.1756
0.056
48
0.5501
0.179
49
0.7792
0.155
50
0.1803
0.061
51
0.654
0.211
52
0.6578
0.166
53
0.6173
0.129
54
0.5556
0.093
55
1.0366
0.224
56
0.5839
0.143
58
0.6095
0.176
59
0.2618
0.065
60
0.4635
0.129
61
0.2769
0.056
62
0.3095
0.061
63
0.3954
0.066
64
0.5534
0.128
65
0.3576
0.094
66
0.5278
0.099
67
0.5855
0.099
68
0.7435
0.161
69
0.5109
0.095
70
0.6543
0.113
71
0.2269
0.078
72
0.477
0.143
73
0.2072
0.070
74
0.6521
0.204
75
0.3362
0.104
76
0.5226
0.115
77
0.2853
0.073
78
0.4494
0.116
80
0.4341
0.137
81
0.3934
0.111
82
0.8588
0.154
83
0.9303
0.202
84
0.5091
0.077
85
0.3488
0.144
86
0.3566
0.079Table 8
Time difference for each pair of M/B relays.
Δ
t
4,1
0.3529
Δ
t
18,16
0.0021
Δ
t
3,44
0
Δ
t
15,57
0
Δ
t
56,71
0.0122
Δ
t
6,1
0
Δ
t
19,16
0.1591
Δ
t
1,45
0.0337
Δ
t
55,57
0
Δ
t
58,71
0.0025
Δ
t
7,1
0.3146
Δ
t
20,16
0
Δ
t
3,45
0
Δ
t
14,58
0.1002
Δ
t
26,72
0
Δ
t
45,1
0.3418
Δ
t
21,18
1.0402
Δ
t
1,46
0
Δ
t
55,58
0
Δ
t
27,72
0
Δ
t
4,2
0.3529
Δ
t
23,19
0
Δ
t
2,46
0
Δ
t
8,59
0
Δ
t
28,72
0
Δ
t
6,2
0
Δ
t
32,19
0.0027
Δ
t
6,47
0
Δ
t
47,59
0.5927
Δ
t
30,72
0
Δ
t
7,2
0.3146
Δ
t
64,19
0
Δ
t
7,47
0.3238
Δ
t
48,59
0.1788
Δ
t
56,72
0
Δ
t
44,2
0.3565
Δ
t
22,20
0.5459
Δ
t
44,47
0.3147
Δ
t
18,60
0
Δ
t
58,72
0
Δ
t
5,3
0
Δ
t
23,21
0.0023
Δ
t
45,47
0.3
Δ
t
19,60
0.1665
Δ
t
26,73
0.0003
Δ
t
8,4
0
Δ
t
32,21
0
Δ
t
46,48
0
Δ
t
20,60
0.0008
Δ
t
27,73
0
Δ
t
16,4
0.56
Δ
t
62,21
0.2384
Δ
t
4,49
0
Δ
t
59,60
0.3069
Δ
t
28,73
0
Δ
t
48,4
0.1792
Δ
t
70,22
0
Δ
t
7,49
0
Δ
t
17,61
0
Δ
t
29,73
0
Δ
t
8,5
0
Δ
t
24,23
0.3009
Δ
t
44,49
0
Δ
t
19,61
0
Δ
t
56,73
0.0081
Δ
t
16,5
0.5385
Δ
t
25,24
0
Δ
t
45,49
0
Δ
t
20,61
0
Δ
t
58,73
0
Δ
t
47,5
0.5848
Δ
t
69,25
0
Δ
t
4,50
0.3518
Δ
t
59,61
0
Δ
t
71,74
0.2501
Δ
t
9,6
0.6558
Δ
t
68,26
0.4949
Δ
t
6,50
0
Δ
t
17,62
0
Δ
t
72,74
0
Δ
t
10,7
0.015
Δ
t
65,27
0.2957
Δ
t
44,50
0.3118
Δ
t
18,62
0.0312
Δ
t
23,75
0.0048
Δ
t
11,7
0
Δ
t
31,28
0
Δ
t
45,50
0.2971
Δ
t
20,62
0.0022
Δ
t
62,75
0.2319
Δ
t
12,7
0.2883
Δ
t
72,28
0
Δ
t
16,51
0.3311
Δ
t
59,62
0.3093
Δ
t
64,75
0
Δ
t
13,7
0.0488
Δ
t
31,29
0
Δ
t
47,51
0.3843
Δ
t
17,63
0
Δ
t
73,76
0.4701
Δ
t
43,7
0.0184
Δ
t
71,29
0
Δ
t
48,51
0
Δ
t
18,63
0.0059
Δ
t
74,76
0
Δ
t
51,7
0.0165
Δ
t
33,30
0
Δ
t
49,52
0
Δ
t
19,63
0.1605
Δ
t
75,77
0
Δ
t
10,8
0.0076
Δ
t
74,30
0
Δ
t
11,53
0
Δ
t
59,63
0.3125
Δ
t
76,78
0
Δ
t
11,8
0
Δ
t
33,31
0
Δ
t
12,53
0.2759
Δ
t
61,64
0
Δ
t
77,78
0
Δ
t
12,8
0.2671
Δ
t
73,31
0.0514
Δ
t
13,53
0.0489
Δ
t
63,65
0
Δ
t
37,79
0
Δ
t
13,8
0.0494
Δ
t
34,32
0.6332
Δ
t
43,53
0.0171
Δ
t
32,66
0.0072
Δ
t
78,79
0.2437
Δ
t
43,8
0.0161
Δ
t
35,33
0
Δ
t
50,53
0.5228
Δ
t
62,66
0.2319
Δ
t
36,80
0
Δ
t
50,8
0.5093
Δ
t
77,33
0.0416
Δ
t
51,53
0.0163
Δ
t
64,66
0
Δ
t
78,80
0.3633
Δ
t
53,9
0
Δ
t
35,34
0.2067
Δ
t
10,54
0
Δ
t
66,67
0
Δ
t
85,81
0
Δ
t
52,10
0.0881
Δ
t
76,34
0
Δ
t
12,54
0
Δ
t
67,68
0
Δ
t
86,81
0
Δ
t
42,11
0.3288
Δ
t
36,35
0
Δ
t
13,54
0
Δ
t
27,69
0.0239
Δ
t
40,82
0
Δ
t
14,12
0.0006
Δ
t
37,35
0.2988
Δ
t
43,54
0
Δ
t
28,69
0
Δ
t
80,82
0
Δ
t
15,12
0
Δ
t
39,37
0.2264
Δ
t
50,54
0
Δ
t
29,69
0
Δ
t
81,82
0
Δ
t
26,13
0.005
Δ
t
40,37
0.2523
Δ
t
51,54
0
Δ
t
30,69
0.0006
Δ
t
39,83
0
Δ
t
27,13
0.027
Δ
t
81,37
0
Δ
t
10,55
0
Δ
t
56,69
0.0331
Δ
t
80,83
0
Δ
t
28,13
0.0001
Δ
t
39,38
0.3152
Δ
t
11,55
0
Δ
t
58,69
0.0202
Δ
t
81,83
0
Δ
t
29,13
0.0001
Δ
t
40,38
0.3684
Δ
t
13,55
0
Δ
t
26,70
0.0155
Δ
t
82,84
0
Δ
t
30,13
0
Δ
t
80,38
0.0055
Δ
t
43,55
0
Δ
t
28,70
0
Δ
t
54,85
0
Δ
t
58,13
0.0262
Δ
t
41,39
0
Δ
t
50,55
0
Δ
t
29,70
0
Δ
t
10,86
0
Δ
t
26,15
0
Δ
t
84,40
0
Δ
t
51,55
0
Δ
t
30,70
0.0003
Δ
t
11,86
0.0116
Δ
t
27,15
0
Δ
t
83,41
0
Δ
t
10,56
0.0082
Δ
t
56,70
0.0368
Δ
t
12,86
0.2807
Δ
t
28,15
0
Δ
t
38,42
0.1969
Δ
t
11,56
0
Δ
t
58,70
0.0215
Δ
t
13,86
0.0363
Δ
t
29,15
0
Δ
t
86,42
0
Δ
t
12,56
0.2863
Δ
t
26,71
0
Δ
t
50,86
0.5105
Δ
t
30,15
0
Δ
t
38,43
0.5089
Δ
t
43,56
0.0169
Δ
t
27,71
0.0031
Δ
t
51,86
0
Δ
t
56,15
0
Δ
t
85,43
0.3278
Δ
t
50,56
0.5088
Δ
t
29,71
0.0142
Δ
t
17,16
0.0001
Δ
t
2,44
0.0333
Δ
t
51,56
0.0146
Δ
t
30,71
0.006In Table9, the results of the simulations for the four cases are illustrated. In this table, the first row shows the four different cases; and the second and the third rows show the results of the first index (∑Δt) and the second index (∑TMS) for these cases. The three first cases are related to the method of [11], where in these cases the coefficients of the objective function are set before optimization process. The values of the coefficients are shown in the first column of the table. The last case shows the results of the proposed method. The values of Δt in all cases are positive and have not any miscoordination.Table 9
Comparison between the proposed method and the method of [11].
Case1
Case2
Case3
Case4
β = 1
β = 10
β = 10
β = 585.5
α
1 = 1
α
1 = 10
α
1 = 1
α
1 = 15
α
2 = 100
α
2 = 1
α
2 = 10
α
2 = 5
M = 2
M = 2
M = 2
M = 12.5
N
= 2
N
= 2
N
= 2
N
= 6
∑
Δ
t
29.8211
24.115
26.9
22.19
∑
TMS
15.15
14.55
15.6
11.32The advantage of the proposed method is revealed when the results of the proposed method (Case 4) are compared with the best results of the traditional method of [11]. According to Table 9, the best ∑TMS and ∑Δt of method [11] (between three cases: 1 and 2 and 3) are 14.55 and 24.115, respectively, whereas ∑TMS and ∑Δt of the proposed method are 11.32 and 22.19, respectively. This means that the results of Case 4, related to the new proposed method, show the best coordination. So, considering the coefficients of objective function as a part of the optimization problem, results are better compared with conventional method according to the results of Tables 4 and 9.


## 7. Conclusion

In this paper, a new flexible technique for overcurrent relays coordination has been proposed. In this new technique, the coefficients of the conventional objective functions have been improved by optimization problem to obtain the minimum values for the TMSs of the relays. In the proposed technique, GAPSOA optimization method is used to solve the optimization problem. This proposed method is tested on 6-bus case study and 30-bus IEEE case study. The results of the simulation show the flexibility of the technique and the best reliability because of the smallest∑TMS and ∑Δtmb compared to the conventional coordination methods.



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*Source: 869617-2014-04-03.xml*