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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


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


minimum power lossCase-1: 14- Bus SystemCase-2: 30- Bus SystemCase-3: 57- Bus System


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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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.



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


8082.77 $/h 8043.80 $/h



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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)




Comparison of Bus voltage magnitudes in both the methods indicates

that there is no significant difference.


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.


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


Testing of GSHDC-PSO Algorithm for OPF Solution using suboptimal

solution obtained by Particle Swarm Optimization Method


obtained by using PSO method. Taking this as suboptimal solution, a


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


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.


execution time in GSHDC method as compared PSO method are

superior. From Table 5.10, it can be seen bus voltage magnitudes at


in transmission system is found to be less as compared to PSO

method.




Testing of GSHDC Algorithm for OPF Solution using suboptimal solution

obtained by Interior Point Method-Minimum Power loss



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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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.




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



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Objective-2 Case-1(Generator Bus Voltage Magnitude, power loss)




method. Comparison of Bus voltage magnitudes in both the methods

indicates that there is no significant difference.




minimum fuel cost and minimum Power loss.




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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Objective-3 Case-1


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 - 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


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.





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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




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


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


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]



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





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solution obtained by Interior Point Method-Minimum Fuel Cost.







minimum cost. The test results include the total cost of generation,generation schedule, generator bus voltage magnitudes and CPU


generation and CPU time for the min. cost objective. Table 5.18






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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execution time in GSHDC method as compared IP method are






solution obtained by Interior Point Method-Minimum Power loss.














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execution time in GSHDC method as compared to IP method are




Comparison of Bus voltage magnitudes in both the methods indicates

that there is no significant difference.







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


812.00 $/h 806.8495

CPU execution time 3.54 seconds 2.74



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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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




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


Objective-3 Case-2


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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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.



f 2 - Minimum Power Loss- 10.6296 MW.


and CPU time, bus voltage magnitudes for the MOGA-IP OPF solution

for IEEE 14- Bus System.


Objective-3 Case-2









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 and CPU time for the min. cost objective. Table 5.23





execution time in GSHDC method as compared to PSO method are

superior.






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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solution obtained by Interior Point Method-Minimum Power loss.


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









Objective-2 Case-2 (Generation Schedule, cost, CPU time)


execution time in GSHDC method as compared PSO method are

superior.


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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Objective-3 Case-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


Sl.No.

Total fuel cost forminimumgeneration cost


TotalPowerloss


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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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.





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Objective-3 Case-2





former method.

5.3 SIMULATION RESULTS - IEEE 57 BUS TEST SYSTEM




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


799.0171 Power loss(MW)



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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Testing of GSHDC-IP Algorithm for OPF Solution using suboptimal

solution obtained by Interior Point Method-Minimum Fuel Cost.




suboptimal solution by GSHDC-IP Algorithm. Finally with the help of

a well defined fitness function genetic search is carried out to find the





generation and CPU time for the minimum cost objective.


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





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Fig: 5.4 IEEE 57 – Bus Test System [101]


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superior.


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.


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


42,737.79 $/h 41,873.00 $/h

CPU execution time 3.17 sec 2.89 sec


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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superior.


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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generator buses are improved in GSHDC-IP method. Also, the power

loss in transmission system is found to be less as compared to IP










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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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




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


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.


f 1 - Minimum Fuel Cost: 41,877.00 $/h.

f 2 - Minimum Power Loss- 17.2410 MW.


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.


Objective-3 Case-3


Testing of GSHDC-PSO Algorithm for OPF Solution using

suboptimal solution obtained by Particle Swarm Optimization Method




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


41,877.00 $/h Power loss(MW)



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execution time in GSHDC-PSO method as compared to PSO method

are superior.


Test-2 Objective-1 Case-3 (Generator Bus Voltage Magnitude,

power loss)





Testing of GSHDC-PSO Algorithm for OPF Solution using

suboptimal solution obtained by Particle Swarm Optimization Method -

Minimum Power loss.



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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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









execution time in GSHDC method as compared to PSO method are

superior.


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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generator buses are improved in GSHDC-PSO method. Also, the power

loss in transmission system is found to be less as compared to PSO






minimum fuel cost and minimum Power loss.







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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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



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


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.


f 1 -Minimum Fuel Cost: 41,327.00 $/h.



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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solution for IEEE 57 Bus System.


Objective-3 Case-3





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



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


41,327.00 Power loss(MW)



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GSHDC is implemented for each Test case and each objective for

three case studies that is, three IEEE Test systems.




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


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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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.

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