# Hybridized Symbiotic Organism Search Algorithm for the Optimal Operation of Directional Overcurrent Relays

Authors: Muhammad Sulaiman; Ashfaq Ahmad; Asfandyar Khan; Shakoor Muhammad
Journal: Complexity
 (2018)

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## Abstract

This paper presents the solution of directional overcurrent relay (DOCR) problems using Simulated Annealing based Symbiotic Organism Search (SASOS). The objective function of the problem is to minimize the sum of the operating times of all primary relays. The DOCR problem is nonlinear and highly constrained with two types of decision variables, namely, the time dial settings (TDS) and plug setting (PS). In this paper, three models of the problem are considered, the IEEE 3-bus, 4-bus, and 6-bus, respectively. We have applied SASOS to solve the problem and the obtained results are compared with other algorithms available in the literature.


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## Body

## 1. Introduction

Due to rapidly growing power systems, the stability and security issues are highly important for the power system researchers [1–3]. The protection systems are mainly used to detect and clear faults as fast and selective as possible [4–6]. The protection relays are used for detecting the faults in the system and to detach the faulty parts from the system in real time. Proper coordination of relays is essential to maintain the appropriate operation of the overall protection system. There are various types of relays with different operating principles. An example of these relays, which are used as a good technical tool for the protection of power systems, is directional overcurrent relay [7–9]. Such a relay is divided into two units, that is, the instantaneous unit and the time overcurrent unit. The parameters to be defined in the overcurrent unit are the time dial settings (TDS) and the plug settings (PS). Computers have made the huge calculations in DOCR problems in power systems easy [10, 11]. Different optimization algorithms are used to solve the relays coordination problems in which the objective function is to minimize activity time of all main relays. The constraints of this optimization problem are considered in the second layer of relay, which should respond, if the main layer of relay fails to operate on nearby fault. This depends on the variables, TDS, PS, and the minimized working time of relay. There is a nonlinear relationship between the operating time of overcurrent relays, TDS, and PS.In electrical engineering, the power system engineering has the longest history among all other areas. Ever since, different numerical optimization techniques have been applied to power systems engineering and played an important role [12]. Optimization problems are usually nonlinear, which have nonlinear objective functions and constraints [13, 14].Nowadays, researchers use different optimization algorithms to find the optimal solutions for the problems of relays settings and coordination. Examples of these optimization algorithms are Evolutionary Algorithm (EA) [15], Differential Evolution (DE) [16], Modified Differential Evolution (MDE) [17], Self-Adaptive Differential Evolutionary (SADE) [18], Particle Swarm Optimization (PSO) [19], Modified Particle Swarm Optimizer [20, 21], Evolutionary Particle Swarm Optimization (EPSO) [22], Box-Muller Harmony Search (BMHS) [23], Zero-One Integer Programming (ZOIP) approach [24, 25], Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [26], Seeker Algorithm [27], Chaotic Differential Evolution Algorithm (CDEA) [28], Adaptive Differential Evolution [29], Artificial Bee Colony (ABC) [30], Firefly Optimization Algorithm (FOA) [31], Modified Swarm Firefly Algorithm (MSFA) [32], and Biogeography Based Optimization (BBO) [33]. BFOA has been applied to obtain the optimal location and size of multiple distributed generators (DG) [34], optimal placement and sizing of DG [35], power system harmonics estimation [36], distribution systems reconfiguration for loss minimization [37], minimum load balancing index for distribution system [38], power system stabilizer for the suppression of oscillations [39], and optimum economic load dispatch [40].During the last few years, PSO algorithm has been applied for Optimal Power Flow (OPF) control in power systems [41], OPF problem with FACTS devices [42], economic dispatch problems [43], optimal sizing and placement of DG [44], optimal location and sizing of static synchronous series compensator [45], and optimized controller design of energy storage devices [45, 46].This paper proposes the use of a hybrid optimization technique—namely, Simulated Annealing based Symbiotic Organism Search (SASOS) [1, 47]—to find the optimal solutions for the relays settings. This algorithm is applied to different models of the DOCR problems such as the IEEE 3-bus, 4-bus, and 6-bus models. To check the efficiency of the proposed algorithm for the three cases, we have minimized the total activity time (T) for each relay.


## 2. Problem Formulation

There are two important settings in each overcurrent relay for its satisfactory operations. The time dial settings (TDS) represent the activation time of each relay and the relay operation is decided by the plug settings (PS). The plug settings (PS) depend on the maximum load current and fault current due to short circuit. The main factors which control the total operating time of the relay are TDS and PS, and the fault current is represented byIf [15, 22], where(1)T=α×TDSIf/PS×CTpri-ratingβ-γ,where If denotes the fault current at the current transformer (CT) initial terminal and CTpri-rating is the primary rating of CT. Constants α, β, and γ are assigned values 0.14, 0.02, and 1.0, respectively, and that is according to IEEE standards [26].The current, seen by the relay, denoted byIrelay, is equal to the ratio between If and CTpri-rating, which is a nonlinear equation:(2)Irelay=IfCTpri-rating.


### 2.1. Objective Function

In coordination studies [22], the main objective is to minimize the total time taken in operation of primary relays for clearing a fault. The objective function takes the following form:(3)F=∑i=1Ncl⁡Tpri-cl-ini+∑j=1NfarTpri-far-busj,where(4)Tpri-cl-ini=0.14×TDSiIfi/PSi×CTpri-ratingi0.02-1,Tpri-far-busj=0.14×TDSjIfj/PSj×CTpri-ratingj0.02-1,where Tpri-cl-in is the relay operation time to clear a near-end fault while Tpri-far-bus is its operation time in case of a far end fault. Ncl⁡ and Nfar represent the relays fixed at the ends of the front line.


### 2.2. Constraints

The objective function is bound to the three constraints related toTDS, PS, and If.Equation (5) represents the bound constraints on TDS:(5)TDSimin⁡≤TDSi≤TDSimax⁡.TDSimin⁡ and TDSimax⁡ are the lower and upper bounds for TDS, whose values are given by 0.05 and 1.1, respectively, while i varies from 1 to Ncl⁡.The second constraint is PS of the relay that takes the following form:(6)PSimin⁡≤PSi≤PSimax⁡.PSimin⁡ and PSimax⁡ are the lower and upper bounds of PS, whose values are given by 1.25 and 1.50, respectively, while i varies from 1 to Nfar.The third constraint is related to the fault current and pickup current. The operating time of relay depends on the type of relay. According to [24, 33], the operating time of relay is defined by(7)Timin⁡≤Ti≤Timax⁡.Timin⁡ and Timax⁡ are the lower and upper bounds for relay functioning time, whose values are adopted as 0.05 and 1, respectively.During the optimization procedure, the time taken by primary relays to coordinate with the backup relays is constrained as in(8)Tbackup-Tprimary≥CTI,where CTI represents the specified coordination time.T
backup and Tprimary represent the working time of primary and backup relays, which can be obtained using the following equations:(9)Tibackup=0.14×TDSxIfi/PSx×CTpri-ratingi0.02-1,Tiprimary=0.14×TDSyIfi/PSy×CTpri-ratingi0.02-1.


### 2.3. The Standard IEEE System of 3-Bus

There are six overcurrent relays in this model. According to the number of relays, the value assigned to each ofNcl⁡ and Nfar is 6 and the number of decision variables is 12, and this means TDS1 to TDS6 and the variables PS1 to PS6. Figure 1 shows the 3-bus model.Figure 1
The 3-bus model.Mathematically the objective function can be written as(10)min⁡F=∑i=16Tpri-cl-ini+∑i=16Tpri-far-busj,where(11)Tpri-cl-ini=0.14×TDSiIfi/PSi×CTpri-ratingi0.02-1,Tpri-far-busj=0.14×TDSjIfj/PSj×CTpri-ratingj0.02-1.The values of constants Ifi, CTpri-ratingi, Ifj, and CTpri-ratingj are shown in Table 6.Constraints imposed on the 3-bus system are as follows.Limits on Variables
T
D
S
i
.
T
D
S
i
min
⁡
≤
T
D
S
i
≤
T
D
S
i
max
⁡, where i varies from 1 to 6.Limits on Variables
P
S
i
.
P
S
i
min
⁡
≤
P
S
i
≤
P
S
i
max
⁡, where i varies from 1 to 6.Another constraint limits each term of objective between 0.05 and 1.Selectivity constraints on working time of backup relay and the primary relay are given in the following relation:(12)Tibackup-Tiprimary≥CTI,where CTI takes the value 0.3. Here,(13)Tbackup=0.14×TDSxIfi/PSx×CTpri-ratingi0.02-1,Tprimary=0.14×TDSyIfi/PSy×CTpri-ratingj0.02-1.The values of the constants for Tbackup and Tprimary of 3-bus model are given in Table 7.


### 2.4. The Standard IEEE System of 4-Bus

In the IEEE 4-bus model, there are 8 overcurrent relays. Accordingly, the value ofNcl⁡ and Nfar is 8, which is twice the number of transmission lines and the number of decision variables is 16. The 4-bus model is shown in Figure 2. CTI takes the value for this model as 0.3. Total selectivity constraints are 9.Figure 2
The 4-bus model.Moreover, the values of constants for the 4-bus model are given in Tables8 and 9.


### 2.5. The Standard IEEE System of 6-Bus

In this section, the IEEE 6-bus model is elaborated. Here, value ofNcl⁡ and Nfar is 14. Hence, there are 28 variables in this problem, namely, TDS1 to TDS14 and PS1 to PS14. The 6-bus model is shown in Figure 3. CTI takes the value 0.2 for this model. There are 48 selectivity constraints in this problem. Based on the observation of [48], we have relaxed 10 constraints. The values of constants Ifi, CTpri-ratingi, Ifj, and CTpri-ratingj for Model 3 are given in Tables 10 and 11. A summary of all components involved in models of IEEE 3-bus, 4-bus, and 6-bus is presented in Table 5.Figure 3
The 6-bus model.


## 2.1. Objective Function

In coordination studies [22], the main objective is to minimize the total time taken in operation of primary relays for clearing a fault. The objective function takes the following form:(3)F=∑i=1Ncl⁡Tpri-cl-ini+∑j=1NfarTpri-far-busj,where(4)Tpri-cl-ini=0.14×TDSiIfi/PSi×CTpri-ratingi0.02-1,Tpri-far-busj=0.14×TDSjIfj/PSj×CTpri-ratingj0.02-1,where Tpri-cl-in is the relay operation time to clear a near-end fault while Tpri-far-bus is its operation time in case of a far end fault. Ncl⁡ and Nfar represent the relays fixed at the ends of the front line.


## 2.2. Constraints

The objective function is bound to the three constraints related toTDS, PS, and If.Equation (5) represents the bound constraints on TDS:(5)TDSimin⁡≤TDSi≤TDSimax⁡.TDSimin⁡ and TDSimax⁡ are the lower and upper bounds for TDS, whose values are given by 0.05 and 1.1, respectively, while i varies from 1 to Ncl⁡.The second constraint is PS of the relay that takes the following form:(6)PSimin⁡≤PSi≤PSimax⁡.PSimin⁡ and PSimax⁡ are the lower and upper bounds of PS, whose values are given by 1.25 and 1.50, respectively, while i varies from 1 to Nfar.The third constraint is related to the fault current and pickup current. The operating time of relay depends on the type of relay. According to [24, 33], the operating time of relay is defined by(7)Timin⁡≤Ti≤Timax⁡.Timin⁡ and Timax⁡ are the lower and upper bounds for relay functioning time, whose values are adopted as 0.05 and 1, respectively.During the optimization procedure, the time taken by primary relays to coordinate with the backup relays is constrained as in(8)Tbackup-Tprimary≥CTI,where CTI represents the specified coordination time.T
backup and Tprimary represent the working time of primary and backup relays, which can be obtained using the following equations:(9)Tibackup=0.14×TDSxIfi/PSx×CTpri-ratingi0.02-1,Tiprimary=0.14×TDSyIfi/PSy×CTpri-ratingi0.02-1.


## 2.3. The Standard IEEE System of 3-Bus

There are six overcurrent relays in this model. According to the number of relays, the value assigned to each ofNcl⁡ and Nfar is 6 and the number of decision variables is 12, and this means TDS1 to TDS6 and the variables PS1 to PS6. Figure 1 shows the 3-bus model.Figure 1
The 3-bus model.Mathematically the objective function can be written as(10)min⁡F=∑i=16Tpri-cl-ini+∑i=16Tpri-far-busj,where(11)Tpri-cl-ini=0.14×TDSiIfi/PSi×CTpri-ratingi0.02-1,Tpri-far-busj=0.14×TDSjIfj/PSj×CTpri-ratingj0.02-1.The values of constants Ifi, CTpri-ratingi, Ifj, and CTpri-ratingj are shown in Table 6.Constraints imposed on the 3-bus system are as follows.Limits on Variables
T
D
S
i
.
T
D
S
i
min
⁡
≤
T
D
S
i
≤
T
D
S
i
max
⁡, where i varies from 1 to 6.Limits on Variables
P
S
i
.
P
S
i
min
⁡
≤
P
S
i
≤
P
S
i
max
⁡, where i varies from 1 to 6.Another constraint limits each term of objective between 0.05 and 1.Selectivity constraints on working time of backup relay and the primary relay are given in the following relation:(12)Tibackup-Tiprimary≥CTI,where CTI takes the value 0.3. Here,(13)Tbackup=0.14×TDSxIfi/PSx×CTpri-ratingi0.02-1,Tprimary=0.14×TDSyIfi/PSy×CTpri-ratingj0.02-1.The values of the constants for Tbackup and Tprimary of 3-bus model are given in Table 7.


## 2.4. The Standard IEEE System of 4-Bus

In the IEEE 4-bus model, there are 8 overcurrent relays. Accordingly, the value ofNcl⁡ and Nfar is 8, which is twice the number of transmission lines and the number of decision variables is 16. The 4-bus model is shown in Figure 2. CTI takes the value for this model as 0.3. Total selectivity constraints are 9.Figure 2
The 4-bus model.Moreover, the values of constants for the 4-bus model are given in Tables8 and 9.


## 2.5. The Standard IEEE System of 6-Bus

In this section, the IEEE 6-bus model is elaborated. Here, value ofNcl⁡ and Nfar is 14. Hence, there are 28 variables in this problem, namely, TDS1 to TDS14 and PS1 to PS14. The 6-bus model is shown in Figure 3. CTI takes the value 0.2 for this model. There are 48 selectivity constraints in this problem. Based on the observation of [48], we have relaxed 10 constraints. The values of constants Ifi, CTpri-ratingi, Ifj, and CTpri-ratingj for Model 3 are given in Tables 10 and 11. A summary of all components involved in models of IEEE 3-bus, 4-bus, and 6-bus is presented in Table 5.Figure 3
The 6-bus model.


## 3. Hybridization of Simulated Annealing (SA) and Symbiotic Organism Search (SOS)

The hybrid algorithm is named as SASOS [47]. In SASOS algorithm, the new solutions are created using the three phases of SOS algorithm by shifting the earlier solutions towards randomly picked solutions from the system [49]. The convergence rate of SOS algorithm is slow due to its strategy for locating the global optima. If the updated solution is better than previous solution, the convergence will be faster. Therefore, to further improve the convergence rate, Simulated Annealing technique is employed in the mutualism and commensalism phase of the SOS algorithm. The parasitism stage is left unchanged [47]. A pseudo code for SASOS algorithm is given in Algorithm 1. The SA technique accepts poor neighbour solutions along with the current best solutions by implementing certain probability. In each generation, probability of selecting the bad solutions is higher at the initial stages but decreases at the later stages of the search. This probability of either selecting poor solutions or good ones is calculated as in(14)Pr=exp⁡-fXi∗-fXiT,where f(Xi∗) and f(Xi) are the fitness values of ith solution and current best solutions, respectively, and here T is the control parameter. Equation (15) is used to reduce the temperature in Simulated Annealing:(15)T=δi∗T0+Tf.Algorithm 1:The pseudo code for SASOS [47].
INPUT: Set ecosize, create population of organisms Xi, i=1,2,3,…, ecosize, initialize Xi, Set stopping criteria,
Initialize SA parameters: Initial temperatureTinit, Final temperature Tfin, Cooling rate δ.
OUTPUT: Optimal schedule
(
1
)
Identify the best organism Xbest
(
2
)
While stopping criterion is not met
(
3
)
For  i=1 to ecosize
(
4
)
Mutualism Phase
(
5
)
Simulated annealing on Xi
(
6
)
Simulated annealing on Xj
(
7
)
Transform Xi and Xj
(
8
)
Commensalism Phase
(
9
)
Simulated annealing on Xi
(
10
)
Apply Eq (15) to reduce the temperature
(
11
)
Transform Xi
(
12
)
Parasitism Phase
(
13
)
Create parasite_vector
(
14
)
Update Xj
(
15
)
Transform Xj
(
16
)
Identify the best organism Xbest
(
17
)
End For
(
18
)
End While


## 4. Experimental Settings and Discussion

### 4.1. Experimental Settings

The parameters involved in SASOS are ecosize, initial temperatureTinit, final temperature Tfinal, and cooling rate δ. The ecosize is taken as 50 for all the cases. The value of initial temperature Tinit is taken as 1, while the final temperature is 10-10 and cooling rate δ is taken as 1.For a fair comparison, a total of 30 simulations were performed for each of the three cases and the current best solution through the simulation was recorded as the current best solution.


### 4.2. Discussion

Performance of the SASOS algorithm is analyzed based on the best objective values (see Figures5, 7, and 9) and total function evaluations taken by algorithms (see Figures 4, 6, and 8). The outcomes of SASOS are highlighted and compared with the values obtained by different versions of Differential Evolution and other algorithms available in the literature [17]. The results for 3-bus, 4-bus, and 6-bus systems are given in Tables 1, 2, and 3, respectively.Table 1
Optimal values of design variables, objective values, and function evaluations obtained by SASOS, DE, and modified DE: IEEE 3-bus system.
T
D
S
1
T
D
S
2
T
D
S
3
T
D
S
4
T
D
S
5
T
D
S
6
P
S
1
P
S
2
P
S
3
P
S
4
P
S
5
P
S
6
F
NFE
DE
0.05
0.2194
0.05
0.2135
0.19498
0.1953
1.25
1.25
1.2500
1.4605
1.25
1.25
4.8422
78360
MDE-1
0.05
0.2178
0.05
0.2090
0.1812
0.1807
1.25
1.25
1.25
1.4999
1.5
1.4999
4.80707
72350
MDE-2
0.05
0.1979
0.05
0.2094
0.1847
0.1827
1.25
1.4999
1.25
1.4999
1.4318
1.4619
4.7873
73350
MDE-3
0.05
0.1988
0.05
0.2090
0.1812
0.1807
1.25
1.4849
1.25
1.4999
1.4998
1.4999
4.7822
97550
MDE-4
0.05
0.1976
0.05
0.2090
0.1812
0.1806
1.25
1.4999
1.25
1.4999
1.4999
1.4999
4.7806
69270
MDE-5
0.05
0.1976
0.05
0.2090
0.1812
0.1806
1.25
1.5
1.25
1.5
1.5
1.5
4.7806
38250
SASOS
0.05
0.207161
0.05
0.30155
0.203466
0.091295
1.44387
1.25
1.252555
1.265668
1.2500
1.359382
4.7201
31500Table 2
Optimal values of design variables, objective values, and function evaluations obtained by SASOS, DE, and modified DE: IEEE 4-bus system.
T
D
S
1
T
D
S
2
T
D
S
3
T
D
S
4
T
D
S
5
T
D
S
6
T
D
S
7
T
D
S
8
P
S
1
P
S
2
P
S
3
P
S
4
P
S
5
P
S
6
P
S
7
P
S
8
F
NFE
DE
0.05
0.2248
0.05
0.1515
0.1264
0.05
0.1337
0.0500
1.2734
1.25
1.2500
1.4997
1.4997
1.25
1.5
1.25
3.6774
95400
MDE-1
0.05
0.2121
0.0500
0.1515
0.1264
0.05
0.1338
0.05
1.2733
1.4998
1.2500
1.4996
1.5
1.2500
1.4997
1.2500
3.6694
43400
MDE-2
0.05
0.2123
0.05
0.1515
0.1264
0.05
0.1371
0.05
1.2733
1.4959
1.2500
1.4997
1.5
1.25
1.4274
1.25
3.6734
67200
MDE-3
0.05
0.2121
0.05
0.1515
0.1264
0.05
0.1338
0.05
1.2733
1.5
1.25
1.4995
1.4997
1.25
1.4995
1.25
3.6692
99700
MDE-4
0.05
0.2121
0.05
0.1515
0.1262
0.05
0.1337
0.05
1.25
1.5
1.25
1.5
1.5
1.25
1.4998
1.25
3.6674
55100
MDE-5
0.05
0.2121
0.05
0.1515
0.1264
0.0500
0.1337
0.0500
1.2734
1.4999
1.2500
1.4999
1.5
1.2500
1.5
1.25
3.6694
35330
SASOS
0.0500
0.2183
0.0500
0.1430
0.1265
0.0500
0.1411
0.0500
1.2800
1.2500
1.2500
1.2517
1.2500
1.2500
1.2513
1.2500
3.5533
30000Table 3
Optimal values of design variables, objective values, and function evaluations obtained by SASOS, DE, and modified DE: IEEE 6-bus system.
DE
MDE-1
MDE-2
MDE-3
MDE-4
MDE-5
SASOS
TDS1
0.1173
0.1171
0.1149
0.1034
0.1144
0.1024
0.1425
TDS2
0.2082
0.1866
0.2037
0.1863
0.1864
0.1863
0.2076
TDS3
0.0997
0.0965
0.0982
0.0961
0.0947
0.0946
0.0622
TDS4
0.1125
0.1119
0.10367
0.1125
0.1006
0.1067
0.1010
TDS5
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0524
TDS6
0.0580
0.0500
0.0500
0.0500
0.0500
0.0500
0.0536
TDS7
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0501
TDS8
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0501
TDS9
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0539
TDS10
0.0719
0.0706
0.0575
0.0703
0.0701
0.0563
0.0506
TDS11
0.0649
0.0649
0.0667
0.0649
0.0649
0.0650
0.0500
TDS12
0.0617
0.0617
0.0566
0.0509
0.0509
0.0553
0.0502
TDS13
0.0500
0.0500
0.0635
0.0500
0.0500
0.0500
0.0500
TDS14
0.0856
0.0860
0.0859
0.0857
0.0709
0.0709
0.0824
PS1
1.2505
1.2515
1.2635
1.4995
1.2602
1.4991
1.4593
PS2
1.2500
1.4959
1.2993
1.4999
1.4987
1.4999
1.2545
PS3
1.2512
1.2525
1.2622
1.2575
1.2761
1.2771
1.2500
PS4
1.2515
1.2632
1.4322
1.2508
1.4992
1.3650
1.3348
PS5
1.2500
1.2500
1.2500
1.2500
1.2500
1.2500
1.2511
PS6
1.2500
1.3822
1.3885
1.3810
1.3814
1.3818
1.2734
PS7
1.2500
1.2500
1.2508
1.2500
1.2500
1.2500
1.2502
PS8
1.2500
1.2501
1.2500
1.2500
1.2505
1.2500
1.2737
PS9
1.2502
1.2500
1.2514
1.2500
1.2500
1.2500
1.2508
PS10
1.2502
1.2501
1.4970
1.2521
1.2500
1.4996
1.2536
PS11
1.4998
1.4999
1.4759
1.4998
1.4999
1.4998
1.5000
PS12
1.2575
1.2529
1.4700
1.4997
1.5000
1.3931
1.2501
PS13
1.4805
1.4664
1.2728
1.4647
1.4615
1.4613
1.4190
PS14
1.2557
1.2500
1.2624
1.2540
1.4979
1.4974
1.2515
F
10.6272
10.5067
10.6238
10.4370
10.3812
10.3514
9.8948
NFE
212190
72960
18180
101580
100860
106200
35000Figure 4
Comparison of number of function evaluations done, for obtaining the best results, by SASOS algorithm with DE and its modified versions for 3-bus model.Figure 5
Comparison of best objective values obtained by SASOS algorithm with DE and its modified versions for 3-bus model.Figure 6
Comparison of number of function evaluations done, for obtaining the best results, by SASOS algorithm with DE and its modified versions for 4-bus model.Figure 7
Comparison of best objective values obtained by SASOS algorithm with DE and its modified versions for 4-bus model.Figure 8
Comparison of number of function evaluations done, for obtaining the best results, by SASOS algorithm with DE and its modified versions for 6-bus model.Figure 9
Comparison of best objective values obtained by SASOS algorithm with DE and its modified versions for 6-bus model.The best solutions obtained by SASOS and other versions of DE for 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 Table1 and Figures 4 and 5. Here, it is evident that, in terms of objective function value, DE gave the worse objective value as 4.8422 and all the other versions of DE algorithm gave almost similar values. On the other hand, SASOS gave us an objective value 4.7201 which is much better than other algorithms. However, if we compare the number of function evaluations (NFE), then the performances of DE and its variants are worse as compared to SASOS algorithm as it took only 31500 NFE to complete the simulations. Thus, SASOS is significantly better than all the other algorithms under consideration.The simulation results of IEEE 4-bus model are given in Table2 and Figures 6 and 7, respectively. It is observed for the IEEE 4-bus model that DE algorithm and all the variants of 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 SASOS outperformed the algorithms under consideration in terms of the number of function evaluations (NFE). The better performance was shown by SASOS, which took 30000 NFE to converge to a solution 3.5533.Furthermore, the results for IEEE 6-bus model as in Table3 and Figures 8 and 9, in terms of best objective function value, are again slightly different to each other with MDE-4 and MDE-5 giving slightly improved solutions than other variant algorithms of DE. On the other hand, SASOS gave us an objective value 9.8948 which is much better than other results in the table. However, in terms of NFE, the worst convergence rate was shown by DE while SASOS took 35000 NFE to get the best solution presented in the table.Summarily, it can be observed from Tables1, 2, and 3 that SASOS provides better solutions to the given problems compared to DE. Also, the number of function evaluations (NFE) taken by SASOS algorithm is considerably reduced in comparison to DE algorithms. In Table 4, the objective function values obtained by SASOS are compared with the values obtained by other algorithms such as DE, RST2, SOMA, and SOMGA for all the three models, as reported in [50], LX-POL and LX-PM [51], and MDE algorithms [17]. It can be observed from Table 4 that SASOS provided better results for all the three systems among all results quoted in [17], in terms of best objective values.Table 4
Comparison of results obtained by SASOS and other algorithms for IEEE 3-, 4-, and 6-bus systems.
Algorithm
3-bus model
4-bus model
6-bus model
SASOS
4.7201
3.5533
9.8948
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
SOMGA
4.7898
3.6745
10.3578
LX-POL
4.8265
3.5749
10.6028
LX-PM
4.8286
3.5830
10.6219Table 5
The summary table of different components involved in the problem of DOCR.
Number
IEEE 3-bus
IEEE 4-bus
IEEE 6-bus
Number of lines
3
4
7
Number of DOCRs (relays)
6
8
14
Number of decision variables
12
16
28
Number of selectivity constraints
8
9
38
Number of restricted constraints
24
32
104Table 6
Values of constants for objective function of 3-bus model.
T
pri-cl-in
i
T
pri-far-bus
i
T
D
S
i
I
f
i
C
T
p
r
i
-
r
a
t
i
n
g
i
T
D
S
j
I
f
j
CTpri-ratingj
T
D
S
1
9.46
2.06
T
D
S
2
100.63
2.06
T
D
S
2
26.91
2.06
T
D
S
1
14.08
2.06
T
D
S
3
8.81
2.23
T
D
S
4
136.23
2.23
T
D
S
4
37.68
2.23
T
D
S
3
12.07
2.23
T
D
S
5
17.93
0.8
T
D
S
6
19.2
0.8
T
D
S
6
14.35
0.8
T
D
S
5
25.9
0.8Table 7
Values of constants forTbackup and Tprimary of 3-bus model.
T
backup
i
T
primary
i
x
I
f
i
C
T
p
r
i
-
r
a
t
i
n
g
i
y
I
f
j
C
T
p
r
i
-
r
a
t
i
n
g
j
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
0.8Table 8
Values of constants for objective function of 4-bus model.
T
pri-cl-in
i
Tpri-far-busi
T
D
S
i
I
f
i
C
T
p
r
i
-
r
a
t
i
n
g
i
T
D
S
j
I
f
j
C
T
p
r
i
-
r
a
t
i
n
g
j
TDS1
20.32
0.48
TDS2
23.75
0.48
TDS2
88.85
0.48
TDS1
12.48
0.48
TDS3
13.61
1.1789
T
D
S
4
31.92
1.1789
TDS4
116.81
1.1789
T
D
S
3
10.38
1.1789
TDS5
116.7
1.5259
TDS6
12.07
1.5259
TDS6
16.67
1.5259
T
D
S
5
31.92
1.5259
TDS7
71.7
1.2018
TDS8
11
1.2018
TDS8
19.27
1.2018
TDS7
18.91
1.2018Table 9
Values of constants forTbackup and Tprimary of 4-bus model.
T
backup
i
T
primary
i
x
I
f
i
C
T
p
r
i
-
r
a
t
i
n
g
i
y
I
f
j
C
T
p
r
i
-
r
a
t
i
n
g
j
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
1.2018Table 10
Values of constants for objective function of 6-bus model.
T
pri-cl-in
i
T
pri-far-bus
i
T
D
S
i
I
f
i
C
T
p
r
i
-
r
a
t
i
n
g
i
T
D
S
j
I
f
j
C
T
p
r
i
-
r
a
t
i
n
g
j
TDS1
2.5311
0.2585
TDS2
5.9495
0.2585
TDS2
2.7376
0.2585
TDS1
5.3752
0.2585
TDS3
2.9723
0.4863
TDS4
6.6641
0.4863
TDS4
4.1477
0.4863
TDS3
4.5897
0.4863
TDS5
1.9545
0.7138
TDS6
6.2345
0.7138
TDS6
2.7678
0.7138
TDS5
4.2573
0.7138
TDS7
3.8423
1.746
TDS8
6.3694
1.746
TDS8
5.618
1.746
TDS7
4.1783
1.746
TDS9
4.6538
1.0424
T
D
S
10
3.87
1.0424
TDS10
3.5261
1.0424
TDS9
5.2696
1.0424
TDS11
2.584
0.7729
TDS12
6.1144
0.7729
TDS12
3.8006
0.7729
TDS11
3.9005
0.7729
TDS13
2.4143
0.5879
TDS14
2.9011
0.5879
TDS14
5.3541
0.5879
TDS13
4.335
0.5879Table 11
Values of constants forTbackup and Tprimary of 6-bus model.
T
backup
i
T
primary
i
x
I
f
i
C
T
p
r
i
-
r
a
t
i
n
g
i
y
I
f
j
C
T
p
r
i
-
r
a
t
i
n
g
j
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


## 4.1. Experimental Settings

The parameters involved in SASOS are ecosize, initial temperatureTinit, final temperature Tfinal, and cooling rate δ. The ecosize is taken as 50 for all the cases. The value of initial temperature Tinit is taken as 1, while the final temperature is 10-10 and cooling rate δ is taken as 1.For a fair comparison, a total of 30 simulations were performed for each of the three cases and the current best solution through the simulation was recorded as the current best solution.


## 4.2. Discussion

Performance of the SASOS algorithm is analyzed based on the best objective values (see Figures5, 7, and 9) and total function evaluations taken by algorithms (see Figures 4, 6, and 8). The outcomes of SASOS are highlighted and compared with the values obtained by different versions of Differential Evolution and other algorithms available in the literature [17]. The results for 3-bus, 4-bus, and 6-bus systems are given in Tables 1, 2, and 3, respectively.Table 1
Optimal values of design variables, objective values, and function evaluations obtained by SASOS, DE, and modified DE: IEEE 3-bus system.
T
D
S
1
T
D
S
2
T
D
S
3
T
D
S
4
T
D
S
5
T
D
S
6
P
S
1
P
S
2
P
S
3
P
S
4
P
S
5
P
S
6
F
NFE
DE
0.05
0.2194
0.05
0.2135
0.19498
0.1953
1.25
1.25
1.2500
1.4605
1.25
1.25
4.8422
78360
MDE-1
0.05
0.2178
0.05
0.2090
0.1812
0.1807
1.25
1.25
1.25
1.4999
1.5
1.4999
4.80707
72350
MDE-2
0.05
0.1979
0.05
0.2094
0.1847
0.1827
1.25
1.4999
1.25
1.4999
1.4318
1.4619
4.7873
73350
MDE-3
0.05
0.1988
0.05
0.2090
0.1812
0.1807
1.25
1.4849
1.25
1.4999
1.4998
1.4999
4.7822
97550
MDE-4
0.05
0.1976
0.05
0.2090
0.1812
0.1806
1.25
1.4999
1.25
1.4999
1.4999
1.4999
4.7806
69270
MDE-5
0.05
0.1976
0.05
0.2090
0.1812
0.1806
1.25
1.5
1.25
1.5
1.5
1.5
4.7806
38250
SASOS
0.05
0.207161
0.05
0.30155
0.203466
0.091295
1.44387
1.25
1.252555
1.265668
1.2500
1.359382
4.7201
31500Table 2
Optimal values of design variables, objective values, and function evaluations obtained by SASOS, DE, and modified DE: IEEE 4-bus system.
T
D
S
1
T
D
S
2
T
D
S
3
T
D
S
4
T
D
S
5
T
D
S
6
T
D
S
7
T
D
S
8
P
S
1
P
S
2
P
S
3
P
S
4
P
S
5
P
S
6
P
S
7
P
S
8
F
NFE
DE
0.05
0.2248
0.05
0.1515
0.1264
0.05
0.1337
0.0500
1.2734
1.25
1.2500
1.4997
1.4997
1.25
1.5
1.25
3.6774
95400
MDE-1
0.05
0.2121
0.0500
0.1515
0.1264
0.05
0.1338
0.05
1.2733
1.4998
1.2500
1.4996
1.5
1.2500
1.4997
1.2500
3.6694
43400
MDE-2
0.05
0.2123
0.05
0.1515
0.1264
0.05
0.1371
0.05
1.2733
1.4959
1.2500
1.4997
1.5
1.25
1.4274
1.25
3.6734
67200
MDE-3
0.05
0.2121
0.05
0.1515
0.1264
0.05
0.1338
0.05
1.2733
1.5
1.25
1.4995
1.4997
1.25
1.4995
1.25
3.6692
99700
MDE-4
0.05
0.2121
0.05
0.1515
0.1262
0.05
0.1337
0.05
1.25
1.5
1.25
1.5
1.5
1.25
1.4998
1.25
3.6674
55100
MDE-5
0.05
0.2121
0.05
0.1515
0.1264
0.0500
0.1337
0.0500
1.2734
1.4999
1.2500
1.4999
1.5
1.2500
1.5
1.25
3.6694
35330
SASOS
0.0500
0.2183
0.0500
0.1430
0.1265
0.0500
0.1411
0.0500
1.2800
1.2500
1.2500
1.2517
1.2500
1.2500
1.2513
1.2500
3.5533
30000Table 3
Optimal values of design variables, objective values, and function evaluations obtained by SASOS, DE, and modified DE: IEEE 6-bus system.
DE
MDE-1
MDE-2
MDE-3
MDE-4
MDE-5
SASOS
TDS1
0.1173
0.1171
0.1149
0.1034
0.1144
0.1024
0.1425
TDS2
0.2082
0.1866
0.2037
0.1863
0.1864
0.1863
0.2076
TDS3
0.0997
0.0965
0.0982
0.0961
0.0947
0.0946
0.0622
TDS4
0.1125
0.1119
0.10367
0.1125
0.1006
0.1067
0.1010
TDS5
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0524
TDS6
0.0580
0.0500
0.0500
0.0500
0.0500
0.0500
0.0536
TDS7
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0501
TDS8
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0501
TDS9
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0539
TDS10
0.0719
0.0706
0.0575
0.0703
0.0701
0.0563
0.0506
TDS11
0.0649
0.0649
0.0667
0.0649
0.0649
0.0650
0.0500
TDS12
0.0617
0.0617
0.0566
0.0509
0.0509
0.0553
0.0502
TDS13
0.0500
0.0500
0.0635
0.0500
0.0500
0.0500
0.0500
TDS14
0.0856
0.0860
0.0859
0.0857
0.0709
0.0709
0.0824
PS1
1.2505
1.2515
1.2635
1.4995
1.2602
1.4991
1.4593
PS2
1.2500
1.4959
1.2993
1.4999
1.4987
1.4999
1.2545
PS3
1.2512
1.2525
1.2622
1.2575
1.2761
1.2771
1.2500
PS4
1.2515
1.2632
1.4322
1.2508
1.4992
1.3650
1.3348
PS5
1.2500
1.2500
1.2500
1.2500
1.2500
1.2500
1.2511
PS6
1.2500
1.3822
1.3885
1.3810
1.3814
1.3818
1.2734
PS7
1.2500
1.2500
1.2508
1.2500
1.2500
1.2500
1.2502
PS8
1.2500
1.2501
1.2500
1.2500
1.2505
1.2500
1.2737
PS9
1.2502
1.2500
1.2514
1.2500
1.2500
1.2500
1.2508
PS10
1.2502
1.2501
1.4970
1.2521
1.2500
1.4996
1.2536
PS11
1.4998
1.4999
1.4759
1.4998
1.4999
1.4998
1.5000
PS12
1.2575
1.2529
1.4700
1.4997
1.5000
1.3931
1.2501
PS13
1.4805
1.4664
1.2728
1.4647
1.4615
1.4613
1.4190
PS14
1.2557
1.2500
1.2624
1.2540
1.4979
1.4974
1.2515
F
10.6272
10.5067
10.6238
10.4370
10.3812
10.3514
9.8948
NFE
212190
72960
18180
101580
100860
106200
35000Figure 4
Comparison of number of function evaluations done, for obtaining the best results, by SASOS algorithm with DE and its modified versions for 3-bus model.Figure 5
Comparison of best objective values obtained by SASOS algorithm with DE and its modified versions for 3-bus model.Figure 6
Comparison of number of function evaluations done, for obtaining the best results, by SASOS algorithm with DE and its modified versions for 4-bus model.Figure 7
Comparison of best objective values obtained by SASOS algorithm with DE and its modified versions for 4-bus model.Figure 8
Comparison of number of function evaluations done, for obtaining the best results, by SASOS algorithm with DE and its modified versions for 6-bus model.Figure 9
Comparison of best objective values obtained by SASOS algorithm with DE and its modified versions for 6-bus model.The best solutions obtained by SASOS and other versions of DE for 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 Table1 and Figures 4 and 5. Here, it is evident that, in terms of objective function value, DE gave the worse objective value as 4.8422 and all the other versions of DE algorithm gave almost similar values. On the other hand, SASOS gave us an objective value 4.7201 which is much better than other algorithms. However, if we compare the number of function evaluations (NFE), then the performances of DE and its variants are worse as compared to SASOS algorithm as it took only 31500 NFE to complete the simulations. Thus, SASOS is significantly better than all the other algorithms under consideration.The simulation results of IEEE 4-bus model are given in Table2 and Figures 6 and 7, respectively. It is observed for the IEEE 4-bus model that DE algorithm and all the variants of 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 SASOS outperformed the algorithms under consideration in terms of the number of function evaluations (NFE). The better performance was shown by SASOS, which took 30000 NFE to converge to a solution 3.5533.Furthermore, the results for IEEE 6-bus model as in Table3 and Figures 8 and 9, in terms of best objective function value, are again slightly different to each other with MDE-4 and MDE-5 giving slightly improved solutions than other variant algorithms of DE. On the other hand, SASOS gave us an objective value 9.8948 which is much better than other results in the table. However, in terms of NFE, the worst convergence rate was shown by DE while SASOS took 35000 NFE to get the best solution presented in the table.Summarily, it can be observed from Tables1, 2, and 3 that SASOS provides better solutions to the given problems compared to DE. Also, the number of function evaluations (NFE) taken by SASOS algorithm is considerably reduced in comparison to DE algorithms. In Table 4, the objective function values obtained by SASOS are compared with the values obtained by other algorithms such as DE, RST2, SOMA, and SOMGA for all the three models, as reported in [50], LX-POL and LX-PM [51], and MDE algorithms [17]. It can be observed from Table 4 that SASOS provided better results for all the three systems among all results quoted in [17], in terms of best objective values.Table 4
Comparison of results obtained by SASOS and other algorithms for IEEE 3-, 4-, and 6-bus systems.
Algorithm
3-bus model
4-bus model
6-bus model
SASOS
4.7201
3.5533
9.8948
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
SOMGA
4.7898
3.6745
10.3578
LX-POL
4.8265
3.5749
10.6028
LX-PM
4.8286
3.5830
10.6219Table 5
The summary table of different components involved in the problem of DOCR.
Number
IEEE 3-bus
IEEE 4-bus
IEEE 6-bus
Number of lines
3
4
7
Number of DOCRs (relays)
6
8
14
Number of decision variables
12
16
28
Number of selectivity constraints
8
9
38
Number of restricted constraints
24
32
104Table 6
Values of constants for objective function of 3-bus model.
T
pri-cl-in
i
T
pri-far-bus
i
T
D
S
i
I
f
i
C
T
p
r
i
-
r
a
t
i
n
g
i
T
D
S
j
I
f
j
CTpri-ratingj
T
D
S
1
9.46
2.06
T
D
S
2
100.63
2.06
T
D
S
2
26.91
2.06
T
D
S
1
14.08
2.06
T
D
S
3
8.81
2.23
T
D
S
4
136.23
2.23
T
D
S
4
37.68
2.23
T
D
S
3
12.07
2.23
T
D
S
5
17.93
0.8
T
D
S
6
19.2
0.8
T
D
S
6
14.35
0.8
T
D
S
5
25.9
0.8Table 7
Values of constants forTbackup and Tprimary of 3-bus model.
T
backup
i
T
primary
i
x
I
f
i
C
T
p
r
i
-
r
a
t
i
n
g
i
y
I
f
j
C
T
p
r
i
-
r
a
t
i
n
g
j
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
0.8Table 8
Values of constants for objective function of 4-bus model.
T
pri-cl-in
i
Tpri-far-busi
T
D
S
i
I
f
i
C
T
p
r
i
-
r
a
t
i
n
g
i
T
D
S
j
I
f
j
C
T
p
r
i
-
r
a
t
i
n
g
j
TDS1
20.32
0.48
TDS2
23.75
0.48
TDS2
88.85
0.48
TDS1
12.48
0.48
TDS3
13.61
1.1789
T
D
S
4
31.92
1.1789
TDS4
116.81
1.1789
T
D
S
3
10.38
1.1789
TDS5
116.7
1.5259
TDS6
12.07
1.5259
TDS6
16.67
1.5259
T
D
S
5
31.92
1.5259
TDS7
71.7
1.2018
TDS8
11
1.2018
TDS8
19.27
1.2018
TDS7
18.91
1.2018Table 9
Values of constants forTbackup and Tprimary of 4-bus model.
T
backup
i
T
primary
i
x
I
f
i
C
T
p
r
i
-
r
a
t
i
n
g
i
y
I
f
j
C
T
p
r
i
-
r
a
t
i
n
g
j
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
1.2018Table 10
Values of constants for objective function of 6-bus model.
T
pri-cl-in
i
T
pri-far-bus
i
T
D
S
i
I
f
i
C
T
p
r
i
-
r
a
t
i
n
g
i
T
D
S
j
I
f
j
C
T
p
r
i
-
r
a
t
i
n
g
j
TDS1
2.5311
0.2585
TDS2
5.9495
0.2585
TDS2
2.7376
0.2585
TDS1
5.3752
0.2585
TDS3
2.9723
0.4863
TDS4
6.6641
0.4863
TDS4
4.1477
0.4863
TDS3
4.5897
0.4863
TDS5
1.9545
0.7138
TDS6
6.2345
0.7138
TDS6
2.7678
0.7138
TDS5
4.2573
0.7138
TDS7
3.8423
1.746
TDS8
6.3694
1.746
TDS8
5.618
1.746
TDS7
4.1783
1.746
TDS9
4.6538
1.0424
T
D
S
10
3.87
1.0424
TDS10
3.5261
1.0424
TDS9
5.2696
1.0424
TDS11
2.584
0.7729
TDS12
6.1144
0.7729
TDS12
3.8006
0.7729
TDS11
3.9005
0.7729
TDS13
2.4143
0.5879
TDS14
2.9011
0.5879
TDS14
5.3541
0.5879
TDS13
4.335
0.5879Table 11
Values of constants forTbackup and Tprimary of 6-bus model.
T
backup
i
T
primary
i
x
I
f
i
C
T
p
r
i
-
r
a
t
i
n
g
i
y
I
f
j
C
T
p
r
i
-
r
a
t
i
n
g
j
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


## 5. Conclusion

We present a conclusion of this research as follows:(i)
The problems of optimal coordination of directional overcurrent relays are highly nonlinear, NP-hard, and highly constrained in nature.(ii)
Dealing with the DOCR problems, use of efficient metaheuristic is needed.(iii)
SASOS is implemented to solve the problem of DOCRs for standard IEEE 3-, 4-, and 6-bus systems.(iv)
The outcome of our simulations shows that SASOS can efficiently minimize all the three models of the problem.(v)
The efficiency of SASOS can be observed from the minimum function evaluations required by the algorithm to reach the optimum as compared to other standard algorithms.(vi)
In 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 SASOS.



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*Source: 4605769-2018-01-23.xml*