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TRANSCRIPT
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CHAPTER – 5
SIMULATION RESULTS & DISCUSSION
5.0 INTRODUCTION
This chapter deals with testing of GSHDC algorithm for IEEE
test systems. The standard IEEE 14, 30 and 57 systems are
considered to investigate the effectiveness of the proposed
methodology. The test is carried with a 1.4-GHz Pentium-IV PC. The
GSHDC has been developed by the use of MATLAB version 7. The
simulation results are compared with other popular methodologies in
judicious way.
GSHDC Method is implemented for two Test cases:
Test-1: Suboptimal Solution obtained through IP method
Test-2 : Suboptimal Solution obtained through PSO method
Suboptimal solution is obtained for two individual objectives and
one Multi-objective:
Objective-1: Minimum Fuel Cost
Objective-2 : Minimum Power Loss
Using the OPF solutions obtained through objective-1 &2 as
parent chromosomes, population is generated for the multi-objective
OPF problem. This is referred as:
Objective-3 : Multi-Objective
GSHDC is implemented for each Test case and each objective for
three case studies that is, three IEEE Test systems.
Case-1: IEEE 14-Bus System
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Case-2 : IEEE 30-Bus System
Case-3 : IEEE 57-Bus System
In addition to above two tests, GSHDC is also implemented with
suboptimal solution obtained through modified penalty factor method
to test its effectiveness. This case is referred as Test-3.
Simulation Test results are presented as per the following tree
diagram shown in the Fig.5.1
Tree diagram can be read as follows:
Example:
1) Test-1, Objective-1, Case-1 indicates the GSHDC results for IEEE
14-Bus System for minimum fuel cost using OPF suboptimal
solution based on IP Method.
Fig: 5.1 Tree Diagram indicating various simulation test results
OPF- Simulation Test Results
Test-3: OPF suboptimal Solution Using
modified penalty factor method
TEST-1OPF suboptimal Solution Using IP Method
Objective-1GSHDC solution for
minimum power lossCase-1: 14- Bus SystemCase-2: 30- Bus SystemCase-3: 57- Bus System
Objective-2GSHDC solution for
minimum fuel costCase-1: 14- Bus SystemCase-2: 30- Bus SystemCase-3: 57- Bus System
Objective-3 GSHDC-MOGAMulti-Objective solution for minimumfuel cost & minimum power loss
Case-1: 14- Bus SystemCase-2: 30- Bus SystemCase-3: 57- Bus System
TEST-2OPF suboptimal Solution Using PSO Method
Objective-1GSHDC solution for
minimum power lossCase-1: 14- Bus SystemCase-2: 30- Bus SystemCase-3: 57- Bus System
Objective-2GSHDC solution for
minimum fuel costCase-1: 14- Bus SystemCase-2: 30- Bus SystemCase-3: 57- Bus System
Objective-3 GSHDC-MOGAMulti-Objective solution for minimumfuel cost & minimum power loss
Case-1: 14- Bus SystemCase-2: 30- Bus SystemCase-3: 57- Bus System
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2) Test-1, Objective-2, Case-3 indicates the GSHDC results for IEEE
57-Bus System for minimum power loss using OPF suboptimal
solution based on IP Method.
3) Test-2, Objective-3, Case-1 indicates the multi objective GSHDC-
MOGA results for IEEE 14-Bus System where the OPF for
minimum fuel cost and power loss using suboptimal solution
based on PSO Method.
5.1 OPF SIMULATION RESULTS - IEEE 14 BUS TEST SYSTEM
In this study, the standard IEEE 14-Bus 5 Generator test
system is considered to investigate effectiveness of the GSHDC
approach. The IEEE 14-bus system has 20 transmission lines. The
single line diagram is shown in Fig.5.2. The values of fuel cost
coefficients are given in Table 5.1. The total load demand of the
system is 259 MW and 5 -Generators should share load optimally.
Fig: 5.2 IEEE 14 – Bus Test System [101]
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Table 5.2: Generator Operating Limits
Minimum or Maximum Generation
limits of Generators are presented
in Table 5.2.
Parameter values for GA are presented in Table 5.3
Table 5.3: Parameter values Genetic Algorithm
5.1.1 Test-1 Objective-1 case-1
Testing of GSHDC Algorithm for OPF Solution using suboptimal
solution obtained by Interior Point Method-Minimum Fuel Cost
For the IEEE 14 Bus Test system initially, an OPF solution is
obtained by using IP method. Taking this as suboptimal solution, a
high density cluster for minimum fuel cost is formed in the vicinity of
suboptimal solution by GSHDC Algorithm. Finally with the help of a
well defined fitness function genetic search is carried out to find the
optimal solution. The results are furnished for the objective namely,
minimum cost. The test results include the total cost of generation,
generation schedule, generator bus voltage magnitudes and CPU
Table 5.1: Generator Fuel Cost Coefficients
Sl.NoGeneratorat bus #
i ($/h) i ($/MWhr) i ($/MWhr2)
1 1 0 20 0.04302932 2 0 20 0.25
3 3 0 40 0.014 6 0 40 0.015 8 0 40 0.01
Sl.NoGeneratorat bus #
P Gi Mn (MW)
P Gi Max
(MW)1 1 0 332.42 2 0 140
3 3 0 100
4 6 0 1005 8 0 100
Population Size 100 Mutation Probability 0.01
No. of Generations 300 Crossover Probability 0.08
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execution time. Table 5.4 provides generation schedule, cost of
generation and CPU time for the minimum fuel cost objective.
Table 5.5 provides bus voltage magnitudes for the minimum fuel cost
objective. From Table 5.4, it can be seen both cost of generation and
CPU execution time in GSHDC method as compared IP method are
superior.
Table 5.4 OPF Solution for IEEE 14-Bus System
Test-1 Objective-1 Case-1 (Generation Schedule, cost, CPU time)
Table 5.5 OPF Solution for IEEE 14-Bus System-Test-1 Objective-
1 Case-1 (Generator Bus Voltage Magnitude, power loss)
From Table 5.5, it can be seen bus voltage magnitudes at
generator buses are improved in GSHDC method. Also, the power loss
in transmission system is found to be less as compared to IP method.
Parameter Suboptimal OPF solution byIP Method
GSHDC-IPMethod
P G1 (MW) 194.33 195.01
P G2 (MW) 36.72 39.45
P G3 (MW) 28.74 27.94
P G6 (MW) 11.20 9.20
P G8 (MW) 8.50 7.84
Total Cost ofGeneration
8081.53 $/h 8043.30 $/h
CPU execution time 1.75 seconds 1.43 seconds
ParameterSuboptimal OPF solution by IP
MethodGSHDC-IP Method
V G1 1.06 1.06
V G2 1.041 1.045
V G3 1.01 1.016
V G6 1.06 1.07
V G8 1.06 1.09
Power loss (MW) 9.287 9.2523
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5.1.2 Test-1 Objective-2 case-1
Testing of GSHDC Algorithm for OPF Solution using suboptimal solution
obtained by Interior Point Method-Minimum Power loss
For the IEEE 14 Bus Test system initially, an OPF solution for
minimum power loss is obtained by using IP method. Taking this as
suboptimal solution, a high density cluster for minimum power loss in
the vicinity of suboptimal solution is formed. Finally with the help of
a well defined fitness function for minimum power loss, a genetic
search is carried out to find the optimal solution. The results are
furnished for the objective namely, minimum power loss. The test
results include the total cost of generation, generation schedule,
generator bus voltage magnitudes and CPU execution time. Table 5.6
provides generation schedule, cost of generation and CPU time for the
minimum power loss objective. Table 5.7 provides bus voltage
magnitudes for the minimum power loss objective.
Table 5.6 OPF Solution for IEEE 14-Bus System
Test-1 Objective-2 Case-1 (Generation Schedule, cost, CPU time)
From Table 5.6, it can be seen both cost of generation and CPU
execution time in GSHDC method as compared IP method are
superior.
ParameterSuboptimal OPF solution by
IP MethodGSHDC-IP Method
P G1 (MW) 194.32 193.49P G2 (MW) 40.27 40.20P G3 (MW) 27.85 28.86P G6 (MW) 10.73 10.66P G8 (MW) 6.28 6.15
Total Cost ofGeneration
8082.77 $/h 8043.80 $/h
CPU execution time 1.72 seconds 1.52 seconds
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Table 5.7 OPF Solution for IEEE 14-Bus System - Test-1
Objective-2 Case-1 (Generator Bus Voltage Magnitude, power loss)
From Table 5.7, it can be seen bus voltage magnitudes at
generator buses are improved in GSHDC method. Also, the power loss
in transmission system is found to be less as compared to IP method.
Comparison of Bus voltage magnitudes in both the methods indicates
that there is no significant difference.
5.1.3 Test-1 Objective-3 case-1
Testing of MOGA-GSHDC Algorithm for OPF Solution, using two
high density core points of two individual high density clusters for
minimum fuel cost and minimum Power loss,
Now, for the IEEE 14 Bus Test system, a multi objective OPF
solution is obtained using core points available in two high density
clusters that is, for minimum fuel cost and minimum power loss by
using IP method. Table 5.8 (a) provides member ship function values
of the non-dominant OPF solutions which are the core points of each
of high density clusters.
ParameterSuboptimal OPF solution by
IP MethodGSHDC-IP Method
V G1 1.06 1.06V G2 1.045 1.047
V G3 1.01 1.010V G6 1.07 1.072V G8 1.09 1.09
Power loss (MW) 9.2469 9.1643
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Table 5.8 (a) OPF Solution for IEEE 14-Bus System - Test-1
Objective-3 Case-1
f 1,max=8063.70 f 1,min = 8043.30 f 2,max = 9.5041 f 2,min = 9.1645
f 1,max - f 1,min = 20.40
f 2,max - f 2,min = 0.3396
Membership function Values: Membership function values for the items
in 2nd row are calculated as per the following.
μ 1 = (8063.70- 8043.60)/ 20.40 = 0.9852
μ 2 = (9.5041 - 9.3725)/ 0.3396 = 0.3875
∑ μ 1 +∑ μ 2 = 8.1591 + 7.363=15.5221
μ D = (0.9852+ 0.3875) / (15.5221) = 0.08843
Multi-Objective OPF Solution-Decision Making
From the Table 5.8, it is observed the μ D has maximum value in
7th row. Accordingly the corresponding values of f 1 and f 2 are taken as
the multi objective OPF solution for the objectives minimum fuel cost
and minimum power loss respectively.
Minimum Fuel Cost Minimum Power Loss
Sl.
No.
Total fuel
cost forminimumgeneration
cost
Member
shipfunction
value
Total fuelcost forminimumpower loss
Member
shipfunction
value
Decisionmaking
f 1 μ 1 f 2 μ 2 μ D 01 8043.30 1.0 9.3706 0.3931 0.0897402 8043.60 0.9852 9.3725 0.3875 0.0884303 8043.80 0.9754 9.5041 0.0 0.0616704 8044.10 0.9607 9.2523 0.7414 0.1096505 8044.40 0.9460 9.2737 0.6784 0.1046506 8045.10 0.9117 9.3039 0.5895 0.0967107 8046.35 0.8504 9.1900 0.9249 0.11437
08 8047.23 0.8073 9.2069 0.9010 0.1100509 8055.43 0.4053 9.2469 0.7573 0.0748910 8057.23 0.3171 9.1645 1.0 0.0848511 8063.70 0.0 9.1679 0.9899 0.06377
8.1591 7.363
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The values of f 1 and f 2 are:
f 1 - Minimum Fuel Cost: 8046.35 $/h.
f 2 - Minimum Power Loss - 9.1900 MW.
Table 5.8 (b) provides generation schedule, cost of generation and
CPU time, bus voltage magnitudes for the MOGA-IP OPF solution for
IEEE 14- Bus System.
Table 5.8 (b) OPF Solution for IEEE 14-Bus System - Test-1
Objective-3 Case-1
5.1.4 Test-2 Objective-1 case-1
Testing of GSHDC-PSO Algorithm for OPF Solution using suboptimal
solution obtained by Particle Swarm Optimization Method
For the IEEE 14 Bus Test system initially, an OPF solution is
obtained by using PSO method. Taking this as suboptimal solution, a
high density cluster for minimum fuel cost is formed in the vicinity of
suboptimal solution by GSHDC-PSO Algorithm. Finally with the help
of a well defined fitness function genetic search is carried out to find
the optimal solution. The results are furnished for the objective
namely, minimum cost. The test results include the total cost of
generation, generation schedule, generator bus voltage magnitudes
and CPU execution time. Table 5.9 provides generation schedule, cost
Parameter MOGA-IP OPF Result Parameter MOGA-IP OPF ResultP G1 (MW) 195.49 V G1 1.06P G2 (MW) 40.70 V G2 1.023P G3 (MW) 29.29 V G3 1.02P G6 (MW) 11.22 V G6 1.072P G8 (MW) 5.83 V G8 1.09
Total Cost ofGeneration
8046.35 $/hPower loss(MW)
9.1900 CPU executiontime
1.83 seconds
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of generation and CPU time for the min. cost objective. Table 5.10
provides bus voltage magnitudes for the min. cost objective.
From Table 5.9, it can be seen both cost of generation and CPU
execution time in GSHDC method as compared PSO method are
superior. From Table 5.10, it can be seen bus voltage magnitudes at
generator buses are improved in GSHDC method. Also, the power loss
in transmission system is found to be less as compared to PSO
method.
Table 5.10 OPF Solution for IEEE 14-Bus System - Test-2
Objective-1 Case-1 (Generator Bus Voltage Magnitude, power loss)
5.1.5 Test-2 Objective-2 case-1
Testing of GSHDC Algorithm for OPF Solution using suboptimal solution
obtained by Interior Point Method-Minimum Power loss
Table 5.9 OPF Solution for IEEE 14-Bus System
Test-2 Objective-1 Case-1 (Generation Schedule, cost, CPU time)
ParameterSuboptimal OPF solution by PSOMethod
GSHDC-PSOMethod
P G1 (MW) 195.45 193.36
P G2 (MW) 36.93 40.86
P G3 (MW) 29.51 25.51P G6 (MW) 6.64 7.99P G8 (MW) 11.06 10.67
Total Cost of Generation 8079.40 $/h 8038.80 $/hCPU execution time 6.00 seconds 1.43 seconds
Parameter Suboptimal OPF solution by PSOMethod
GSHDC-PSOMethod
V G1 1.06 1.06
V G2 1.042 1.045V G3 1.012 1.018
V G6 1.05 1.09
V G8 1.062 1.09Power loss (MW) 9.257 9.1995
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For the IEEE 14 Bus Test system initially, an OPF solution for
minimum power loss is obtained by using IP method. Taking this as
suboptimal solution, a high density cluster for minimum power loss in
the vicinity of suboptimal solution is formed. Finally with the help of
a well defined fitness function for minimum power loss, a genetic
search is carried out to find the optimal solution. The results are
furnished for the objective namely, minimum power loss. The test
results include the total cost of generation, generation schedule,
generator bus voltage magnitudes and CPU execution time. Table 5.11
provides generation schedule, cost of generation and CPU time for the
minimum power loss objective. Table 5.12 provides bus voltage
magnitudes for the minimum power loss objective. From Table 5.11, it
can be seen both cost of generation and CPU execution time in
GSHDC method as compared PSO method are superior.
Table 5.11 OPF Solution for IEEE 14-Bus System
Test-2 Objective-2 Case-1 (Generation Schedule, cost, CPU time)
ParameterSuboptimal OPF solution by
PSO MethodGSHDC-PSO Method
P G1 (MW) 195.32 193.35
P G2 (MW) 39.27 39.80
P G3 (MW) 28.85 27.86P G6 (MW) 09.73 11.66
P G8 (MW) 5.28 5.80
Total Cost of Generation 8072.77 $/h 8042.10 $/h
CPU execution time 6.72 seconds 2.41 seconds
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Table 5.12 OPF Solution for IEEE 14-Bus System - Test-2
Objective-2 Case-1(Generator Bus Voltage Magnitude, power loss)
From Table 5.12, it can be seen bus voltage magnitudes at
generator buses are improved in GSHDC method. Also, the power loss
in transmission system is found to be less as compared to PSO
method. Comparison of Bus voltage magnitudes in both the methods
indicates that there is no significant difference.
5.1.6 Test-2 Objective-3 case-1
Testing of MOGA-GSHDC Algorithm for OPF Solution, using two
high density core points of two individual high density clusters for
minimum fuel cost and minimum Power loss.
Now, for the IEEE 14 Bus Test system, a multi objective OPF
solution is obtained using core points available in two high density
clusters that is, for minimum fuel cost and minimum power loss by
using PSO method. Table 5.13 (a) provides member ship function
values of the non-dominant OPF solutions which are the core points of
each of high density clusters.
Parameter Suboptimal OPF solution byPSO Method
GSHDC-PSO Method
V G1 1.06 1.06
V G2 1.05 1.047
V G3 1.02 1.010
V G6 1.065 1.072
V G8 1.09 1.09
Power loss (MW) 9.2567 9.1587
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Table 5.13 (a) OPF Solution for IEEE 14-Bus System - Test-2
Objective-3 Case-1
Minimum Fuel Cost Minimum Power Loss
Sl.No.
Total fuel costfor minimumgeneration cost
Member
shipfunction
value
TotalPower loss
Member
shipfunction
value
Decisionmaking
f 1 μ 1 f 2 μ 2 μ D
01 8038.80 1.0 9.3506 0.2212 0.0847202 8039.60 0.9282 9.3625 0.1729 0.076403 8041.80 0.8564 9.4051 0.0 0.05941804 8042.10 0.8421 9.2423 0.6607 0.093049
05 8042.40 0.8277 9.2747 0.5292 0.08518
06 8043.10 0.7942 9.3139 0.3701 0.07591807 8044.35 0.7344 9.1800 0.9131 0.114307
08 8046.23 0.6445 9.1881 0.8807 0.1058209 8048.13 0.5536 9.1981 0.8044 0.09422
10 8053.42 0.3004 9.1587 1.0 0.08987
11 8059.70 0.0 9.1609 0.9910 0.06268
7.4815 6.9308
f 1,max=8059.70 f 1,min = 8038.80 f 2,max = 9.4051 f 2,min = 9.1587
f 1,max - f 1,min = 20.90 f 2,max - f 2,min = 0.2464
Membership function Values: Membership function values for 2nd row
are calculated as per the following.
μ 1 = (8059.70- 8039.60)/ 20.90 = 0.9282
μ 2 = (9.4051- 9.3625)/ 0.2464= 0.1729
μ D = (0.9282+ 0.1729) / (7.4815+ 6.9308) = 0.0764
Multi-Objective OPF Solution-Decision Making
From the Table 5.13, it is observed theμ
D has maximum value
in 7th row. Accordingly the corresponding values of f 1 and f 2 are taken
as the multi objective OPF solution for the objectives minimum fuel
cost and minimum power loss respectively.
The values of f 1 and f 2 are:
f 1 - Minimum Fuel Cost: 8044.35 $/h.
f 2 - Minimum Power Loss - 9.1800 MW.
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Table 5.13 (b) OPF Solution for IEEE 14-Bus System - Test-2
Objective-3 Case-1
Table 5.13 (b) provides generation schedule, cost of generation
and CPU time, bus voltage magnitudes for the MOGA-PSO OPF
solution for IEEE 14- Bus System. MOGA-PSO results when compared
to MOGA-IP results, it can be seen OPF results are better through
former method.
5.2 OPF SIMULATION RESULTS - IEEE 30 BUS TEST SYSTEM
In this study, the standard IEEE 30-Bus 6 Generator test
system is considered to investigate effectiveness of the GSHDC
approach. The IEEE 30-bus system has 41 transmission lines. The
single line diagram is shown in Fig.5.2. The total load demand of the
system is 283.40 MW and 6 -Generators should share load optimally.
The values of fuel cost coefficients are given in Table 5.14. Minimum
or Maximum Generation limits of Generators are presented in
Table 5.15. The parameters values for GA are parented in Table: 5.16
ParameterMOGA-PSO OPFResult
ParameterMOGA-PSO OPFResult
P G1 (MW) 194.49 V G1 1.06P G2 (MW) 41.70 V G2 1.043P G3 (MW) 29.89 V G3 1.015P G6 (MW) 12.00 V G6 1.042P G8 (MW) 5.12 V G8 1.012
Total Cost ofGeneration
8044.35 $/h
Power loss (MW) 9.1800 CPU executiontime
1.92 seconds
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Fig 5.3 IEEE 30-Bus Test System [101]
Table 5.15: Generator Operating Limits
Table 5.14: Generator Fuel Cost Coefficients
Sl.NoGenerator atbus #
i ($/h) I ($/MWhr) i ($/MWhr2)
1 1 0 2.0 0.022 2 0 1.75 0.01753 5 0 1.0 0.06254 8 0 3.25 0.00835 11 0 3.0 0.0256 13 0 3.0 0.025
Sl.NoGenerator atbus #
P Gi Mn (MW) P Gi Max (MW)
1 1 50 200
2 2 20 803 5 15 504 8 10 355 11 10 306 13 12 40
Table 5.16: Parameter values Genetic Algorithm
Population Size 100 Mutation Probability 0.01
No. of Generations 300 Crossover Probability 0.08
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5.2.1 Test-1 Objective-1 case-2
Testing of GSHDC Algorithm for OPF Solution using suboptimal
solution obtained by Interior Point Method-Minimum Fuel Cost.
For the IEEE 30 Bus Test system initially, an OPF solution is
obtained by using IP method. Taking this as suboptimal solution, a
high density cluster for minimum fuel cost is formed in the vicinity of
suboptimal solution by GSHDC Algorithm. Finally with the help of a
well defined fitness function genetic search is carried out to find the
optimal solution. The results are furnished for the objective namely,
minimum cost. The test results include the total cost of generation,generation schedule, generator bus voltage magnitudes and CPU
execution time. Table 5.17 provides generation schedule, cost of
generation and CPU time for the min. cost objective. Table 5.18
provides bus voltage magnitudes for the min. cost objective.
Table 5.18 OPF Solution for IEEE 30-Bus System - Test-1
Objective-1 Case-2 (Generator Bus Voltage Magnitude, power loss)
Table 5.17 OPF Solution for IEEE 30-Bus System
Test-1 Objective-1 Case-2 (Generation Schedule, cost, CPU time)
Parameter Suboptimal OPF solution byIP Method
GSHDC-IPMethod
P G1 (MW) 175.76 175.42
P G2 (MW) 48.81 48.85
P G5 (MW) 21.54 21.71P G8 (MW) 24.71 23.68P G11 (MW) 12.35 12.71P G13 (MW) 12 11.62
Total Cost of Generation 810.61 $/h 806.7008CPU execution time 1.91 seconds 1.70 seconds
Parameter Suboptimal OPF solution by IP Method GSHDC-IP MethodV G1 1.019 1.05V G2 1.03 1.041V G5 1.00 1.013V G8 1.00 1.07V G11 1.00 1.09V G13 1.00 1.02power loss(MW)
11.43 10.5920
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From Table 5.17, it can be seen both cost of generation and CPU
execution time in GSHDC method as compared IP method are
superior. From Table 5.18, it can be seen bus voltage magnitudes at
generator buses are improved in GSHDC method. Also, the power loss
in transmission system is found to be less as compared to IP method.
5.2.2 Test-1 Objective-2 case-2
Testing of GSHDC Algorithm for OPF Solution using suboptimal
solution obtained by Interior Point Method-Minimum Power loss.
For the IEEE 30 Bus Test system initially, an OPF solution for
minimum power loss is obtained by using IP method. Taking this as
suboptimal solution, a high density cluster for minimum power loss in
the vicinity of suboptimal solution is formed. Finally with the help of
a well defined fitness function for minimum power loss, a genetic
search is carried out to find the optimal solution. The results are
furnished for the objective namely, minimum power loss. The test
results include the total cost of generation, generation schedule,
generator bus voltage magnitudes and CPU execution time. Table 5.19
provides generation schedule, cost of generation and CPU time for the
minimum power loss objective. Table 5.20 provides bus voltage
magnitudes for the minimum power loss objective.
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From Table 5.19, it can be seen both cost of generation and CPU
execution time in GSHDC method as compared to IP method are
superior. From Table 5.20, it can be seen bus voltage magnitudes at
generator buses are improved in GSHDC method. Also, the power loss
in transmission system is found to be less as compared to IP method.
Comparison of Bus voltage magnitudes in both the methods indicates
that there is no significant difference.
5.2.3 Test-1 Objective-3 case-2
Testing of MOGA-GSHDC Algorithm for OPF Solution, using two
high density core points of two individual high density clusters for
minimum fuel cost and minimum Power loss,
Table 5.19 OPF Solution for IEEE 30-Bus System
Test-1 Objective-2 Case-2 (Generation Schedule, cost, CPU time)
ParameterSuboptimal OPF solution by IP
MethodGSHDC-IP Method
P G1 (MW) 175.43 175.44
P G2 (MW) 47.81 48.86
P G5 (MW) 25.54 23.10P G8 (MW) 25.71 23.67P G11 (MW) 12.56 11.56P G13 (MW) 12 11.32
Total Cost ofGeneration
812.00 $/h 806.8495
CPU execution time 3.54 seconds 2.74
Table 5.20 OPF Solution for IEEE 30-Bus System - Test-1
Objective-2 Case-2 (Generator Bus Voltage Magnitude, power loss)
ParameterSuboptimal OPF solution byIP Method
GSHDC-IP Method
VG1 1.012 1.019
VG2 1.000 1.000
VG5 1.000 1.000VG8 1.000 1.000VG11 1.000 1.000VG13 1.000 1.000
power loss (MW) 10.830 10.558
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Now, for the IEEE 30 Bus Test system, a multi objective OPF
solution is obtained using core points available in two high density
clusters that is, for minimum fuel cost and minimum power loss by
using IP method. Table 5.21(a) provides member ship function values
of the non-dominant OPF solutions which are the core points of each
of high density clusters.
f 1,max=806.8495 f 1,min =806.7008 f 2,max =10.7330 f 2,min = 10.5580
f 1,max - f 1,min = 0.1487 f 2,max - f 2,min = 0.1750
Membership function Values: Membership function values for 2nd row
are calculated as per the following.
μ 1 = (806.8495- 806.7031)/ 0.1487 = 0.9845
μ 2 = (10.7330- 10.7109)/ 0.1750= 0.1262
∑ μ 1 + ∑ μ 2 =7.4636+5.675 =13.1386
μ D = (0. 9845+0.12620)/( 13.1386) = 0.08453
Table 5.21 (a) OPF Solution for IEEE 30-Bus System - Test-1
Objective-3 Case-2
Minimum Fuel Cost Minimum Power Loss
Sl.No.
Total fuel cost forminimumgeneration
cost
Member shipfunction
value
TotalPower loss
Member shipfunction value
Decisionmaking
f 1 μ 1 f 2 μ 2 μ D
01 806.7008 1.0 10.6934 0.2262 0.093332
02 806.7031 0.9845 10.7109 0.1262 0.084537
03 806.7073 0.9562 10.7330 0.0 0.07277704 806.7135 0.9145 10.6296 0.5908 0.114570
05 806.7228 0.8520 10.6301 0.5880 0.109600
06 806.7289 0.8110 10.6571 0.4337 0.094736
07 806.7332 0.7821 10.6157 0.6702 0.110536
08 806.7555 0.6321 10.6226 0.6308 0.096121
09 806.7860 0.4270 10.6274 0.6034 0.076425
10 806.8340 0.1042 10.5580 1.0 0.084044
11 806.8495 0.0 10.5920 0.8057 0.061323
∑ μ 1 =7.4636 ∑ μ 2=5.675
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Multi-Objective OPF Solution-Decision Making
From the Table 5.21, it is observed the μ D has maximum value
in 4th row. Accordingly the corresponding values of f 1 and f 2 are taken
as the multi objective OPF solution for the objectives minimum fuel
cost and minimum power loss respectively.
The values of f 1 and f 2 are:
f 1 - Minimum Fuel Cost: 806.7135 $/h.
f 2 - Minimum Power Loss- 10.6296 MW.
Table 5.21 (b) provides generation schedule, cost of generation
and CPU time, bus voltage magnitudes for the MOGA-IP OPF solution
for IEEE 14- Bus System.
Table 5.21 (b) OPF Solution for IEEE 30-Bus System - Test-1
Objective-3 Case-2
5.2.4 Test-2 Objective-1 case-2
For the IEEE 30 Bus Test system initially, an OPF solution is
obtained by using PSO method. Taking this as suboptimal solution, a
high density cluster for minimum fuel cost is formed in the vicinity of
suboptimal solution by GSHDC Algorithm. Finally with the help of a
well defined fitness function genetic search is carried out to find the
optimal solution. The results are furnished for the objective namely,
minimum cost. The test results include the total cost of generation,
ParameterMOGA-IP OPFResult
ParameterMOGA-IP OPFResult
P G1 (MW) 176.43 V G1 1.019P G2 (MW) 48.81 V G2 1.020P G5 (MW) 25.54 V G3 1.003P G8 (MW) 23.71 V G6 1.023P G11 (MW) 11.56 V G8 1.011P G13 (MW) 12.00 V G9 1.000
Total Cost of Generation 806.7135 Power loss(MW)
10.6296 CPU execution time 3.1 sec
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generation schedule, generator bus voltage magnitudes and CPU
execution time. Table 5.22 provides generation schedule, cost of
generation and CPU time for the min. cost objective. Table 5.23
provides bus voltage magnitudes for the min. cost objective.
Table 5.22 OPF Solution for IEEE 30-Bus System
Test-2 Objective-1 Case-2 (Generation Schedule, cost, CPU time)
From Table 5.22, it can be seen both cost of generation and CPU
execution time in GSHDC method as compared to PSO method are
superior.
Table 5.23 OPF Solution for IEEE 30-Bus System - Test-2
Objective-1 Case-2 (Generator Bus Voltage Magnitude, power loss)
From Table 5.23, it can be seen bus voltage magnitudes at
generator buses are improved in GSHDC method. Also, the power loss
in transmission system is found to be less as compared to PSO
method.
ParameterSuboptimal OPF solution byPSO Method
GSHDC-PSOMethod
P G1 (MW) 167.76 150.45P G2 (MW) 47.77 59.28P G5 (MW) 22.54 23.11
P G8 (MW) 23.71 30.20P G11 (MW) 14.56 15.00P G13 (MW) 12 14.08
Total Cost of Generation 807.961 $/h 798.9925CPU execution time 3.57 seconds 2.54 sec
ParameterSuboptimal OPF solution by PSO
MethodGSHDC-PSO
MethodV G1 1.02 1.016
V G2 1.04 1.000
V G5 1.00 1.000V G8 1.00 1.000
V G11
1.00 1.000V G13 1.00 1.000Power loss (MW) 11.11 8.7190
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5.2.5 Test-2 Objective-2 case-2
Testing of GSHDC Algorithm for OPF Solution using suboptimal
solution obtained by Interior Point Method-Minimum Power loss.
For the IEEE 30 Bus Test system initially, an OPF solution for
minimum power loss is obtained by using PSO method. Taking this
as suboptimal solution, a high density cluster for minimum power loss
in the vicinity of suboptimal solution is formed. Finally with the help
of a well defined fitness function for minimum power loss, a genetic
search is carried out to find the optimal solution. The results are
furnished for the objective namely, minimum power loss. The test
results include the total cost of generation, generation schedule,
generator bus voltage magnitudes and CPU execution time. Table 5.24
provides generation schedule, cost of generation and CPU time for the
minimum power loss objective. Table 5.25 provides bus voltage
magnitudes for the minimum power loss objective.
Table 5.24 OPF Solution for IEEE 30-Bus System - Test-2
Objective-2 Case-2 (Generation Schedule, cost, CPU time)
From Table 5.24, it can be seen both cost of generation and CPU
execution time in GSHDC method as compared PSO method are
superior.
ParameterSuboptimal OPF solution by
PSO MethodGSHDC-PSO Method
P G1 (MW) 174.20 150.18P G2 (MW) 47.90 58.80P G5 (MW) 24.44 23.17P G8 (MW) 26.12 31.62
P G11 (MW) 13.27 14.76
P G13 (MW) 12 13.53 Total Cost of
Generation807.56 $/h 799.1345
CPU execution time 3.12 seconds 2.76 sec
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From Table 5.25, it can be seen bus voltage magnitudes at
generator buses are improved in GSHDC method. Also, the power loss
in transmission system is found to be less as compared to PSO
method. Comparison of Bus voltage magnitudes in both the methods
indicates that there is no significant difference.
Table 5.26 (a) OPF Solution for IEEE 30-Bus System - Test-2
Objective-3 Case-2
Table 5.25 OPF Solution for IEEE 30-Bus System - Test-2
Objective-2 Case-2 (Generator Bus Voltage Magnitude,
power loss)
Parameter Suboptimal OPF solution byPSO Method
GSHDC-PSO Method
V G1 1.022 1.016V G2 1.034 1.000
V G5 1.00 1.000V G8 1.00 1.000V G11 1.00 1.000V G13 1.00 1.000
Power loss (MW) 10.47 8.6699
Minimum Fuel Cost Minimum Power Loss
Sl.No.
Total fuel cost forminimumgeneration cost
Member shipfunction value
TotalPowerloss
Member shipfunction value
Decisionmaking
f 1 μ 1 f 2 μ 2 μ D
01 798.9925 1.0000 8.7190 0.0924 0.10688
02 798.9951 0.9579 8.7223 0.0314 0.09679
03 799.0021 0.8446 8.7240 0.0000 0.08264
04 799.0044 0.8074 8.7184 0.1035 0.0891205 799.0066 0.7718 8.7185 0.1016 0.08545
06 799.0076 0.7556 8.7189 0.0942 0.08315
07 799.0089 0.7346 8.7090 0.2741 0.09869
08 799.0100 0.7168 8.7113 0.2347 0.09310
09 799.0138 0.6553 8.7175 0.1201 0.07587
10 799.0171 0.5970 8.6699 1.0000 0.15626
11 799.0543 0.0000 8.7063 0.3271 0.03200
∑ μ 1=7.841 ∑ μ 2=2.3791
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5.2.6 Test-2 Objective-3 case-2
Testing of MOGA-GSHDC Algorithm for OPF Solution, using two
high density core points of two individual high density clusters for
minimum fuel cost and minimum Power loss,
Now, for the IEEE 30 Bus Test system, a multi objective OPF
solution is obtained using core points available in two high density
clusters that is, for minimum fuel cost and minimum power loss by
using PSO method. Table 5.26 (a) provides member ship function
values of the non-dominant OPF solutions which are the core points of
each of high density clusters.
f 1,max=799.0543 f 1,min = 798.9925 f 2,max = 8.7240 f 2,min = 8.6699
f 1,max - f 1,min = 0.0618 f 2,max - f 2,min = 0.0541
Membership function Values: Membership function values for 2nd row
are calculated as per the following.
μ 1 = (799.0543- 798.9951)/ 0.0618 = 0.9579
μ 2 = (8.7240- 8.7223)/ 0.0541 = 0.0314
∑ μ 1 + ∑ μ 2 =7.841+ 2.3791 =10.22
μ D = (0.9579+ 0.0314) / (10.22) = 0.09679
Multi -Objective OPF Solution-Decision Making
From the Table 5.26 (a) it is observed the μ D has maximum
value in 10th row. Accordingly the corresponding values of f 1 and f 2
are taken as the multi objective OPF solution for the objectives
minimum fuel cost and minimum power loss respectively.
The values of f 1 and f 2 are:
f 1 - Minimum Fuel Cost: 799.0171 $/h.
f 2 - Minimum Power Loss - 8.6699 MW.
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Table 5.26 (b) OPF Solution for IEEE 30-Bus System - Test-2
Objective-3 Case-2
Table 5.26 (b) provides generation schedule, cost of generation
and CPU time, bus voltage magnitudes for the MOGA-PSO OPF
solution for IEEE 30- Bus System. MOGA-PSO results when compared
to MOGA-IP results, it can be seen OPF results are better through
former method.
5.3 SIMULATION RESULTS - IEEE 57 BUS TEST SYSTEM
In this study, the standard IEEE 57-Bus 7 Generator test
system is considered to investigate effectiveness of the GSHDC
approach. The IEEE 57-bus system has 80 transmission lines. The
single line diagram is shown in Fig. 5.4. The total load demand of the
system is 259MW and 7-Generators should share load optimally. The
values of fuel cost coefficients are given in Table 5.27. Generator
active power limits are presented in Table 5.28. Table 5.29 provides
Parameter values of Genetic Algorithm.
Parameter MOGA-PSO OPFResult
Parameter MOGA-PSO OPFResult
P G1 (MW) 152.23 V G1 1.016P G2 (MW) 59.10 V G2 1.001P G5 (MW) 24.17 V G3 1.020P G8 (MW) 30.62 V G6 1.010P G11 (MW) 15.70 V G8 1.010P G13 (MW) 13.23 V G9 1.000
Total Cost ofGeneration
799.0171 Power loss(MW)
8.6699 CPU execution time 3.2 sec
Table 5.27: Generator Fuel Cost Coefficients
Sl.No Generatorat bus #
i ($/h) i ($/MWhr) i ($/MWhr2)
1 1 0 20 0.0775
2 2 0 40 0.01
3 3 0 20 0.25
4 6 0 40 0.01
5 8 0 20 0.0222
6 9 0 40 0.017 12 0 20 0.022
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5.3.1 Test-1 Objective-1 case-3
Testing of GSHDC-IP Algorithm for OPF Solution using suboptimal
solution obtained by Interior Point Method-Minimum Fuel Cost.
For the IEEE 57 Bus Test system initially, an OPF solution is
obtained by using IP method. Taking this as suboptimal solution, a
high density cluster for minimum fuel cost is formed in the vicinity of
suboptimal solution by GSHDC-IP Algorithm. Finally with the help of
a well defined fitness function genetic search is carried out to find the
optimal solution. The results are furnished for the objective namely,
minimum cost. The test results include the total cost of generation,
generation schedule, generator bus voltage magnitudes and CPU
execution time. Table 5.30 provides generation schedule, cost of
generation and CPU time for the minimum cost objective.
Table 5.28: Generator Operating Limits
Sl.No Generator at bus # P Gi Mn (MW) P Gi Max (MW)
1 1 0 577.88
2 2 0 100
3 3 0 1404 6 0 1005 8 0 3506 9 0 1007 12 0 410
Table 5.29: Parameter values Genetic Algorithm
No. of Generations 300 Crossover Probability 0.8
Population Size 100 Mutation Probability 0.01
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Fig: 5.4 IEEE 57 – Bus Test System [101]
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Table 5.30 OPF Solution for IEEE 57-Bus System
Test-1 Objective-1 Case-3 (Generation Schedule, cost, CPU time)
From Table 5.30, it can be seen both cost of generation and CPU
execution time in GSHDC method as compared to IP method are
superior.
From Table 5.31, it can be seen bus voltage magnitudes at
generator buses are improved in GSHDC-IP method. Also, the power
loss in transmission system is found to be less as compared to IP
method.
5.3.2 Test-1 Objective-2 case-3
Testing of GSHDC-IP Algorithm for OPF Solution using suboptimal
solution obtained by Interior Point Method-Minimum Power loss
Parameter IP Method GSHDC -IP Method
P G1 (MW) 146.63 144.89P G2 (MW) 97.79 93.08
P G3 (MW) 47.07 45.19
P G6 (MW) 72.86 68.15
P G8 (MW) 489.80 476.03P G9 (MW) 97.63 95.90
P G12 (MW) 361.52 365.97
Total Cost ofGeneration
42,737.79 $/h 41,873.00 $/h
CPU execution time 3.17 sec 2.89 sec
Table 5.31 OPF Solution for IEEE 57-Bus System
Test-1 Objective-1 Case-3 (Generator Bus Voltage
Magnitude, power loss)
Parameter Suboptimal OPF solution by IP Method GSHDC-IP MethodV G1 1.040 1.050V G2 1.008 1.010
V G3 0.985 1.003V G6 0.980 1.026V G8 1.044 1.050V G9 0.980 1.044V G12 0.992 1.015Power loss (MW) 18.0692 17.4038
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For the IEEE 57 Bus Test system initially, an OPF solution for
minimum power loss is obtained by using IP method. Taking this as
suboptimal solution, a high density cluster for minimum power loss in
the vicinity of suboptimal solution is formed. Finally with the help of
a well defined fitness function for minimum power loss, a genetic
search is carried out to find the optimal solution. The results are
furnished for the objective namely, minimum power loss. The test
results include the total cost of generation, generation schedule,
generator bus voltage magnitudes and CPU execution time. Table 5.32
provides generation schedule, cost of generation and CPU time for the
minimum power loss objective. Table 5.33 provides bus voltage
magnitudes for the minimum power loss objective.
From Table 5.32, it can be seen both cost of generation and CPU
execution time in GSHDC method as compared to IP method are
superior.
Table 5.32 OPF Solution for IEEE 57-Bus System
Test-1 Objective-2 Case-3 (Generation Schedule, cost,CPU time)
Parameter IP Method GSHDC-IP MethodP G1 (MW) 142.63 144.78P G2 (MW) 87.79 92.83P G3 (MW) 45.07 45.29
P G6 (MW) 72.86 68.11
P G8 (MW) 459.80 457.30
P G9 (MW) 97.63 95.62
P G12 (MW) 361.52 366.27
Total Cost of Generation 42,354.90 $/h 41,956 $/h
CPU execution time 3.23 sec 2.98 sec
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From Table 5.33, it can be seen bus voltage magnitudes at
generator buses are improved in GSHDC-IP method. Also, the power
loss in transmission system is found to be less as compared to IP
method. Comparison of Bus voltage magnitudes in both the methods
indicates that there is no significant difference.
5.3.3 Test-1 Objective-3 case-3
Testing of MOGA-GSHDC Algorithm for OPF Solution, using two
high density core points of two individual high density clusters for
minimum fuel cost and minimum Power loss,
Now, for the IEEE 57 Bus Test system, a multi objective OPF
solution is obtained using core points available in two high density
clusters that is, for minimum fuel cost and minimum power loss by
using IP method. Table 5.34 provides weightage factors and member
ship function values of the non-dominant OPF solutions which are the
core points of each of high density clusters.
Table 5.33 OPF Solution for IEEE 57-Bus System Test-1
Objective-2 Case-3 (Generator Bus Voltage Magnitude,
power loss)
Parameter Suboptimal OPF solution by IPMethod GSHDC-IPMethod
V G1 1.009 1.04
V G2 1.008 1.01V G3 1.003 0.985V G6 1.026 0.980
V G8 1.044 1.005
V G9 1.044 0.980V G12 0.992 1.015
power loss (MW) 17.116 16.998
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Table 5.34 (a) OPF Solution for IEEE 57-Bus System - Test-1
Objective-3 Case-3
f 1,max=41,907.00 f 1,min = 41,873.00 f 2,max = 17.4038 f 2,min = 16.9980
f 1,max - f 1,min = 34.00 f 2,max - f 2,min = 0.4058
Membership function Values: Membership function values for 2nd row
are calculated as per the following.
μ 1 = (41,907.00- 41,874.00)/ 34.00 = 0.9705
μ 2 = (17.4038- 17.3735)/ 0.4058= 0.07466
μ D = (0.9705+ 0.07466) / (6.6172+ 4.92006) = 0.090589
Multi -Objective OPF Solution-Decision Making
From the Table 5.34, it is observed the μ D has maximum
value in 4th row. Accordingly the corresponding values of f 1 and f 2 are
taken as the multi objective OPF solution for the objectives minimum
fuel cost and minimum power loss respectively.
The values of f 1 and f 2 are:
f 1 - Minimum Fuel Cost: 41,877.00 $/h.
f 2 - Minimum Power Loss- 17.2410 MW.
Minimum Fuel Cost Minimum Power Loss
Sl.No. Total fuel costfor minimumgeneration cost
Member shipfunction
value
TotalPowerloss
Membershipfunction value
Decisionmaking
f 1 μ 1 f 2 μ 2 μ D
01 41,873.00 1.0000 17.3512 0.12962 0.09790002 41,874.00 0.9705 17.3735 0.07466 0.09058903 41,876.00 0.9117 17.4038 0.00000 0.079020
04 41,877.00 0.8823 17.2410 0.40118 0.111246
05 41,881.00 0.7647 17.2827 0.29842 0.092146
06 41,883.00 0.7058 17.29461 0.26909 0.084499
07 41,885.00 0.6470 17.1231 0.69172 0.116034
08 41,889.00 0.5294 17.1686 0.57959 0.09612209 41,903.00 0.1176 17.1871 0.53400 0.056477
10 41,90400 0.0882 16.9980 1.00000 0.094320
11 41,907.00 0.0000 17.0183 0.94997 0.082339
∑ μ 1=6.6172 ∑ μ 2=4.92006
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Table 5.34 (b) provi2des generation schedule, cost of generation
and CPU time, bus voltage magnitudes for the MOGA-IP OPF solution
for IEEE 57- Bus System.
Table 5.34 (b) OPF Solution for IEEE 57-Bus System - Test-1
Objective-3 Case-3
5.3.4 Test-2 Objective-1 case-3
Testing of GSHDC-PSO Algorithm for OPF Solution using
suboptimal solution obtained by Particle Swarm Optimization Method
For the IEEE 57 Bus Test system initially, an OPF solution is
obtained by using PSO method. Taking this as suboptimal solution, a
high density cluster for minimum fuel cost is formed in the vicinity of
suboptimal solution by GSHDC-PSO Algorithm. Finally with the help
of a well defined fitness function genetic search is carried out to find
the optimal solution. The results are furnished for the objective
namely, minimum cost. The test results include the total cost of
generation, generation schedule, generator bus voltage magnitudes
and CPU execution time. Table 5.35 provides generation schedule,
cost of generation and CPU time for the min. fuel cost objective. Table
5.36 provides bus voltage magnitudes for the min. fuel cost objective.
Parameter MOGA-IP OPF Result Parameter MOGA- IP OPF ResultP G1 (MW) 145.00 V G1 1.04P G2 (MW) 93.25 V G2 1.005P G3 (MW) 46.45 V G3 1.001P G6 (MW) 69.25 V G6 1.001P G8 (MW) 461.34 V G8 1.004P G9 (MW) 96.62 V G9 1.0032P G12 (MW) 367.85 V G12 1.016
Total Cost ofGeneration
41,877.00 $/h Power loss(MW)
17.2410 CPU execution time 3.02 sec
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From Table 5.35, it can be seen both cost of generation and CPU
execution time in GSHDC-PSO method as compared to PSO method
are superior.
Table 5.36 OPF Solution for IEEE 57-Bus System
Test-2 Objective-1 Case-3 (Generator Bus Voltage Magnitude,
power loss)
From Table 5.36, it can be seen bus voltage magnitudes at
generator buses are improved in GSHDC method. Also, the power loss
in transmission system is found to be less as compared to IP method.
5.3.5 Test-2 Objective-2 case-3
Testing of GSHDC-PSO Algorithm for OPF Solution using
suboptimal solution obtained by Particle Swarm Optimization Method -
Minimum Power loss.
Table 5.35 OPF Solution for IEEE 57-Bus System
Test-2 Objective-1 Case-3 (Generation Schedule, cost, CPU time)
Parameter PSO Method GSHDC-PSO Method
P G1 (MW) 145.43 140.24
P G2 (MW) 95.56 81.60
P G3 (MW) 46.12 48.32P G6 (MW) 69.78 68.72P G8 (MW) 479.80 476.83P G9 (MW) 96.63 84.05P G12 (MW) 363.52 367.69
Total Cost of Generation 42,145.79 $/h 41,327.00 $/hCPU execution time 3.45 sec 2.98 se
ParameterSuboptimal OPF solutionby PSO Method
GSHDC-PSO Method
V G1 1.002 1.050V G2 1.009 1.015
V G3 0.995 1.025V G6 0.995 1.030V G8 1.046 1.050V G9 0.980 1.050V G12 1.000 1.030Power loss (MW) 17.956 16.4471
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For the IEEE 57 Bus Test system initially, an OPF solution for
minimum power loss is obtained by using PSO method. Taking this
as suboptimal solution, a high density cluster for minimum power loss
in the vicinity of suboptimal solution is formed. Finally with the help
of a well defined fitness function for minimum power loss, a genetic
search is carried out to find the optimal solution. The results are
furnished for the objective namely, minimum power loss. The test
results include the total cost of generation, generation schedule,
generator bus voltage magnitudes and CPU execution time. Table 5.37
provides generation schedule, cost of generation and CPU time for the
minimum power loss objective. Table 5.38 provides bus voltage
magnitudes for the minimum power loss objective.
From Table 5.37, it can be seen both cost of generation and CPU
execution time in GSHDC method as compared to PSO method are
superior.
Table 5.37 OPF Solution for IEEE 57-Bus System Test-2
Objective-2 Case-3 (Generation Schedule, cost, CPU time)
Parameter PSO MethodGSHDC-PSOMethod
P G1 (MW) 140.43 140.24
P G2 (MW) 85.55 81.60
P G3 (MW) 47.12 48.32
P G6 (MW) 70.70 68.72
P G8 (MW) 460.80 476.83
P G9 (MW) 97.65 84.05
P G12 (MW) 360.77 367.69
Total Cost of Generation 42,244.79 $/h 41,346.00 $/hCPU execution time 3.4 sec 3.02 sec
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Table 5.38 OPF Solution for IEEE 57-Bus System Test-2
Objective-2 Case-3 (Generator Bus Voltage Magnitude, power loss)
From Table 5.38, it can be seen bus voltage magnitudes at
generator buses are improved in GSHDC-PSO method. Also, the power
loss in transmission system is found to be less as compared to PSO
method. Comparison of Bus voltage magnitudes in both the methods
indicates that there is no significant difference.
5.3.6 Test-2 Objective-3 case-3
Testing of MOGA-GSHDC Algorithm for OPF Solution, using two
high density core points of two individual high density clusters for
minimum fuel cost and minimum Power loss.
Now, for the IEEE 57 Bus Test system, a multi objective OPF
solution is obtained using core points available in two high density
clusters that is, for minimum fuel cost and minimum power loss by
using PSO method. Table 5.39 (a) provides member ship function
values of the non-dominant OPF solutions which are the core points of
each of high density clusters.
ParameterSuboptimal OPF solution by PSO
MethodGSHDC-PSO
MethodV G1 1.009 1.009V G2 1.008 1.008
V G3 1.003 1.003V G6 1.026 1.026V G8 1.044 1.044V G9 1.044 1.044V G12 0.992 0.992
Power loss (MW) 17.0692 16.0692
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Table 5.39 (a) OPF Solution for IEEE 57-Bus System - Test-2
Objective-3 Case-3
f 1,max=41,349.00 f 1,min = 41,327.00 f 2,max = 16.6601 f 2,min = 16.0692
f 1,max - f 1,min = 22.00
f 2,max - f 2,min = 0.5909
Membership function Values: Membership function values for 2nd row
are calculated as per the following.
μ 1 = (41,349.00- 41,328.00)/ 22.00 = 0.9545
μ 2 = (16.6601- 16.0692)/ 0.5909= 0.17363
μ D = (0.9545+ 0.17363) / (5.7268+ 4.4531) = 0.11081
Multi -Objective OPF Solution-Decision Making
From the Table 5.39(a), it is observed the μ D has maximum
value in 1st row. Accordingly the corresponding values of f 1 and f 2 are
taken as the multi objective OPF solution for the objectives minimum
fuel cost and minimum power loss respectively.
The values of f 1 and f 2 are:
f 1 -Minimum Fuel Cost: 41,327.00 $/h.
f 2 - Minimum Power Loss - 16.5312 MW.
Minimum Fuel Cost Minimum Power Loss
Sl.No. Total fuel costfor minimumgeneration cost
Member shipfunctionvalue
TotalPowerloss
Member shipfunctionvalue.
Decisionmaking
f 1 μ 1 f 2 μ 2 μ D
01 41,327.00 1.0 16.5312 0.21814 0.11966
02 41,328.00 0.9545 16.5575 0.17363 0.1108103 41,330.00 0.8636 16.6601 0.0 0.0848304 41,331.00 0.8181 16.5010 0.26925 0.1068105 41,335.00 0.6363 16.5027 0.26637 0.0867106 41,338.00 0.5000 16.5261 0.22677 0.0713907 41,340.00 0.4090 16.4471 0.36046 0.0755808 41,342.00 0.3181 16.2431 0.6041 0.0905909 41,346.00 0.1363 16.2886 0.6287 0.0751410 41,347.00 0.0909 16.0692 1.0 0.1071611 41,349.00 0.0 16.1183 0.7057 0.06932
∑ μ 1=5.7268 ∑ μ 2=4.4531
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Table 5.39 (b) provides generation schedule, cost of generation
and CPU time, bus voltage magnitudes for the MOGA-PSO OPF
solution for IEEE 57 Bus System.
Table 5.39 (b) OPF Solution for IEEE 57-Bus System - Test-2
Objective-3 Case-3
Table 5.39 (b) provides generation schedule, cost of generation
and CPU time, bus voltage magnitudes for the MOGA-PSO OPF
solution for IEEE 30- Bus System. MOGA-PSO results when compared
to MOGA-IP results, it can be seen OPF results are better through
former method.
5.4 SUMMARY OF RESULTS
GSHDC Method is implemented for two Test cases:
Test-1: Suboptimal Solution obtained through IP method
Test-2: Suboptimal Solution obtained through PSO method
Suboptimal solution is obtained for two individual objectives
and Multi-objective:
Objective-1: Minimum Fuel Cost
Objective-2: Minimum Power Loss
Objective-3: Multi-Objective
ParameterMOGA-PSO OPFResult
ParameterMOGA-PSO OPFResult
P G1 (MW) 141.43 V G1 1.009P G2 (MW) 87.55 V G2 1.009P G3 (MW) 47.12 V G3 1.004P G6 (MW) 69.43 V G6 1.028P G8 (MW) 462.85 V G8 1.044
P G9 (MW) 98.45 V G9 1.044P G12 (MW) 362.65 V G12 0.992
Total Cost ofGeneration
41,327.00 Power loss(MW)
16.5312 CPU execution time 4.2 sec
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GSHDC is implemented for each Test case and each objective for
three case studies that is, three IEEE Test systems.
Case-1: IEEE 14-Bus System
Case-2: IEEE 30-Bus System
Case-3: IEEE 57-Bus System
Simulation results for all the Test cases, Objectives as well as
for different case studies is furnished in earlier sections. This section
presents summary of all results obtained.
Table 5.40 presents summary of GSHDC results for the case 14
bus system.
Table 5.40: Summary of Results – Case-1: IEEE 14 - Bus System
Table 5.41 presents summary of GSHDC results for the case 30
bus system.
Table 5.41 Summary of Results – Case-2: IEEE 30 - Bus System
.
ParameterIPMethod
GSHDC-IPMethod
PSOMethod
GSHDC-PSOMethod
MOGA-GSHDC(IP Based)
MOGA-GSHDC(PSO Based)
Fuel Cost($/h)Objective-1
8081.53 8043.30 8079.40 8038.80 8046.35 8044.35
Power Loss(MW)Objective-2
9.2469 9.1643 9.2567 9.1587 9.190 9.180
ParameterIPMethod
GSHDC-IPMethod
PSOMethod
GSHDC-PSOMethod
MOGA-GSHDC(IP Based)
MOGA-GSHDC(PSO Based)
Fuel Cost($/h)Objective-1
810.61 806.7008 807.961 798.9925 806.7135 799.0171
Power Loss(MW)Objective-2
10.830 10.558 10.47 8.6699 10.6296 8.6699
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Table 5.42 presents summary of GSHDC results for the case 57
bus system.
Table 5.42 Summary of Results – Case-3: 57 - Bus System
When compared to GSHDC-IP, the results of GSHDC-PSO are better in
all the three cases. Though, GSHDC-PSO is giving best results, for the
single objective of minimum fuel cost and the single objective of
minimum losses, individually, the MOGA-GSHDC (PSO based) is
giving a better compromised OPF solution including both fuel cost
and losses.
5.5 OPF SIMULATION RESULTS - IEEE 14 BUS TEST SYSTEM-
MODIFIED PENALTY FACTOR METHOD
In addition to suboptimal solutions obtained through IP and
PSO methods, a modified penalty factor method presented in Section
5.4 is used to obtain suboptimal solution or a core point in High
Density Cluster. This section presents results for this case.
The GSHDC -penalty factor performance is evaluated on the
standard IEEE 30-bus test system [27]. The system consists of 41-
lines, 6-generators, 4-Tap-hanging transformers and shunt capacitor
banks located at 9-buses. The test is carried with a 1.4-GHz Pentium-
IV PC. The GSHDC -penalty factor has been developed by the use of
Parameter IPMethod
GSHDC-IP Method
PSOMethod
GSHDC-PSO
Method
MOGA-GSHDC
(IP Based)
MOGA-GSHDC
(PSOBased)
Fuel Cost($/h)
42,739.79 41,873.00 42,145.79 41,327.00 41,877.00 41327.00
PowerLoss (MW)
17.116 16.998 17.0692 16.0692 17.2410 16.5312
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MATLAB version 7. The parameter settings to execute GSHDC-penalty
factor are probability of crossover=0.5, probability of Mutation= 0.7,
the population size is 20. The study is carried out for a total system
load of 283.4 MW. The power mismatch tolerance is 0.0001 p.u. and
other parameters are presented in Table 5.43.
Table 5.43: Test-3 Objective-1 Case 2
Power Generation Limits and Generator cost parameters of IEEE
30 Bus System (Base MVA 100)
Table-5.44 Test-3 Objective-1 Case 2
Test results of GSHDC-penalty factor and EGA method [103]
The performance of GSHDC is -penalty factor compared with the
results of EGA [103] method and is tabulated in Table-5.44. For a
given system load, the total generation in the system by GSHDC-
penalty factor method is found slightly higher compared to that of EGA
Bus P min P max Q min Q max
1 0.5 2 -0.2 2 0 200 37.5
2 0.2 0.8 -0.2 1 0 175 175
5 0.15 0.5 -0.15 0.8 0 100 625
8 0.1 0.35 -0.15 0.6 0 325 83.4
11 0.1 0.3 -0.1 0.5 0 300 250
13 0.12 0.4 -0.15 0.6 0 300 250
GEN.NO
BUSNO
BUS VOLTAGESACTIVE POWERGENERATION
COST OFGENERATION
EGA[103]
GSHDC-penaltyfactor
EGA[103]
GSHDC-penaltyfactor
EGA[103]
GSHDC-penaltyfactor
1 1 1.050 1.0600176.20
177.216 468.84 468.3056
2 2 1.038 1.0430 48.75 48.3660 126.89 127.3034
3 5 1.012 1.0100 21.44 21.203 50.19 49.3009
4 8 1.020 1.0100 21.95 21.977 75.35 77.2442
5 11 1.087 1.082 12.42 12.182 41.13 40.61776 13 1.067 1.0710 12.02 12.00 39.67 39.600
TOTAL 292.79 292.944 802.06 802.3709
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[103] method. The % high values are presented in Table-5.45. The
numerical difference can be ignored. The EGA [103] for an IEEE30-
Bus system is carried out with a computer having the same
configuration as mentioned above. Now, the comparison is made in
terms of generation cost and CPU time. The GSHDC -penalty factor
method gave less cost of generation. The GSHDC-penalty factor
method has completed objective-1 study in 8 seconds and objective-1
and objective -2 together in 12 seconds in contrast to 85 seconds that
is taken by EGA method. The authors of EGA method in their
conclusions have mentioned the high execution time of their method.
This proves the GSHDC-penalty factor method is quite acceptable for
large size power systems and for on-line studies.
Table-5.45: Test-3 Objective-1 & Objective-2 Case 2
Generation Schedule of GSHDC-penalty factor Compared to EGA [103]
Method
Total ActivePower GenerationObjective-1
TransmissionLossesObjective-2
Total costCPU
Time
MW
%
H i g h
c
o m p a r e d t o
E
G A m e t h o d
MW
%
H i g h
c
o m p a r e d t o
E
G A m e t h o d
$/h
% H i g h
c o m p a r e d t o
E G A m e t h o d
Sec
EGA[103] 292.79 ---- 9.39 ---- 802.06 ---- 85
GSHDC-penaltyfactor ( Objective-1 Total fuel Costminimum)
292.94 0.028 9.54 0.84 802.370 0.038 8
GSHDC-penaltyfactor Objective-2(Total lossminimum)
292.78 ----- 9.38 ------ 802.510 12
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Next, the performance of GSHDC-penalty factor is compared with the
other methods and is tabulated in Table-5.46. For a given system
load and total generation, the results of GSHDC -penalty factor
method is found better as compared to other existing methods.
However, Test-1 (sub optimal solution by IP method) and Test-2 (sub
optimal solution by PSO method) are much superior. Hence, Test-3
case (sub optimal solution by modified penalty factor method) is not
considered and not studied for other case studies like 14, and 57 bus
systems.
Table-5.46 Test-3 Objective-1 & Objective-2 Case 2
Generation Schedule of GSHDC-penalty factor Compared with
Evolutionary methods
5.6 COMPARISON OF GSHDC-IP & GSHDC-PSO OPF RSULTS
WITH OTHER METHODS.
The simulation results of GSHDC-IP (with suboptimal solution
obtained through IP) method and GSHDC-PSO (with suboptimal
solution obtained through PSO) method have been presented in
earlier sections for two objectives (minimum fuel cost and minimum
power loss) and multi-objective for different case studies 14,30, and
OPF Method Total ActivePower Generationin MW
TransmissionLosses in MW
Total cost in$/h
CPU Time in Sec
GSHDC-penalty factor 292.8722 9.47 802.3709 8
EGA[103] 292.79 9.39 802.06 85
GAOPF[26]L.Lai
293.0372 9.6372 802.4484 315
EPOPF[25]Yuryevich
292.7682 9.3683 802.62 51.4
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57 bus systems. It can be observed, if single objective is the criteria,
GSHDC-PSO gives the best results. However, simulation results
indicate the multi-objective results are not far deviating from the best
results obtained from the single objective case studies.
This section presents comparative results of GSHDC-PSO with
the existing methodologies. A typical case study of IEEE 30-Bus
system is taken for the performance evaluation of the proposed
GSHDC-PSO. The comparison results are presented in Table 5.47.
Table-5.47 COMPARISON OF GSHDC-PSO OPF RESULTS WITH
OTHER METHODS.
As seen in Table 5.47, the results of GSHDC-PSO method are
found better as compared to other existing methods. Further, the
results obtained through MOGA-GSHDC (PSO based) are comparable
with those of GSHDC-PSO and better than other methods. Losses as
well as CPU time using GSHDC-PSO are much improved. Though the
single objective (of minimum fuel cost) GSHDC-PSO is giving best
minimum fuel cost, but the MOGA-GSHDC (PSO based) is giving a
better compromised OPF solution between losses and cost.
OPF Method
Total ActivePowerGenerationin MW
TransmissionLossesin MW
Total costin $/h
CPU Timein Sec
GSHDC-PSO 292.12 8.7190 798.9925 2.54
MOGA-GSHDC(PSO based)
292.12 8.7185 799.0021 8.475
EGA[103] 292.79 9.39 802.06 85IGAOPF[102] L.Lai 292.54 9.14 800.805 315
EPOPF[25] Yuryevich 292.7682 9.3683 802.62 51.4
AGA[105] Liladhur.G 297.45 14.05 801.17 433
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5.7 CONCLUSIONS
A novel method for the solution of Optimal Power Flow is
proposed in this chapter. The limitations of analytical and intelligent
methods have been overcome by the proposed methods namely,
GSHDC-IP Method, GSHDC-PSO Method, MOGA-GSHDC (IP based)
and MOGA-GSHDC (PSO based).
In this chapter, testing of GSHDC-IP Algorithm, for OPF problem
using suboptimal solution obtained by Interior Point Method is carried
out to obtain solution individually for minimum fuel cost and
minimum power loss. In addition testing of MOGA-GSHDC (IP based)
Algorithm has been carried out to obtain multi objective solution
simultaneously for minimum fuel cost and minimum power loss. The
testing of these Algorithms has been done for the well-known standard
IEEE test cases such as 14-bus system, 30-bus system and 57-bus
system.
Similarly, testing of GSHDC-PSO Algorithm for OPF problem
using suboptimal solution obtained by PSO Method is carried out to
obtain solution individually for minimum fuel cost and minimum
power loss. In addition testing of MOGA-GSHDC (IP based) Algorithm
has been carried out to obtain multi objective solution simultaneously
for minimum fuel cost and minimum power loss.
When compared to GSHDC-IP, the results of GSHDC-PSO are
better in all the three cases. Though, GSHDC-PSO is giving best
results, for the single objective of minimum fuel cost and the single
objective of minimum losses, individually, the MOGA-GSHDC (PSO
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based) is giving a better compromised OPF solution including both
fuel cost and losses.
Further, results of GSHDC-PSO are compared with the existing
methodologies. A typical case study of IEEE 30-Bus system is taken
for the performance evaluation of the proposed GSHDC-PSO. The
results of GSHDC-PSO method are found better as compared to other
existing methods. Further, the results obtained through MOGA-
GSHDC (PSO based) are comparable with those of GSHDC-PSO and
better than other methods. Losses as well as CPU time using GSHDC-
PSO are much improved. Though the single objective (of minimum
fuel cost) GSHDC-PSO is giving best minimum fuel cost, but the
MOGA-GSHDC (PSO based) is giving a better compromised OPF
solution between losses and cost.