v 3.3.4
Minimum Shutdown Time and Minimum Running Time 50
3.3.5 Spinning Reserve Contribution
51 3.4
Gas Turbine 54
3.5 System Operational Requirements
55 3.5.1
System Total Demand 55
3.5.2 System Total Spinning Reserve Requirements
58 3.5.3
Transmission Losses 58
3.6 System Operational Limitations
59 3.6.1
Station Generation Limit Constraints 60
3.6.2 Group Import and Export Constraints
60 3.7
Objective Function 62
3.8 Procedures of Problem Solving
64 3.9
Summary 65
4. RESULTS AND DISCUSSION
66
4.1 Introduction
66 4.2
Economic Dispatch Problems: Dynamic in Nature 67
4.3 Nonlinear Function Optimization
70 4.3.1 Unconstrained Parameter Optimization
71 4.3.2 Equality Constrained Optimization
72 4.3.3 Inequality Constraints Optimization
73 4.4
PSO Algorithm 75
4.5 Mathematical Formulation
78 4.5.1
Basic Model 80
4.5.2 Additional Constraints
81 4.6
DED Solution for 10 Generators and 6 Periods Case 83
4.6.1 Solution with Dantzig-Wolfe Decomposition Method
83 4.6.2
Solution with PSO Method 87
4.6.3 Comparison of Dantzig-Wolfe Decomposition and PSO Method 95
4.7 Complex DED Solution with PSO
96 4.7.1
Solution for 20 Generators and 24 Periods Case 96
4.7.2 Solution for 100 Generators and 5 Periods Case
133 4.8
Summary 149
5. CONCLUSION AND RECOMMENDATIONS FOR FUTURE WORK 151
5.1 Conclusion
151 5.2
Attainment of Research Objectives 153
5.3 Significance of Research Outcomes
155 5.4
Recommendations for Future Work 156
REFERENCES 158
APPENDICES 167
APPENDIX A 167
APPENDIX B 210
vi
LIST OF TABLES
TABLE TITLE
PAGE
2.1 Some of the available PSO versions Coelho and Lee, 2008
23 4.1
Optimal case solution Matlab code Dantzig-Wolfe algorithm 86
4.2 Design variables for the optimal dispatch
88 4.3
Suboptimal dispatch case period by period dispatch PSO algorithm 91 4.4
Infeasible dispatch case neglecting loading rates constraints 92
4.5 Violation of loading rates constraints in the infeasible case solution
93 4.6
Optimal dispatch case PSO algorithm 94
4.7 Comparison between the Dantzig-Wolfe and PSO methods for DED
98 4.8
Results for 20 generator and 6 period, Optimal Solution 107
4.9 Results for 20 generator and 6 period, Sub-optimal Solution
108 4.10
Results for 20 generator and 12 period, Optimal Solution 109
4.11 Results for 20 generator and 12 period, Sub-optimal Solution
112 4.12
Results for 20 generator and 18 period, Optimal Solution 115
4.13 Results for 20 generator and 18 period, Sub-optimal Solution
119 4.14
Results for 20 generator and 24 period, Optimal Solution 123
4.15 Results for 20 generator and 24 period, Sub-optimal Solution
128 4.16
Comparison between the two methods for 100-generator System 138
4.17 Results for 100 generator and 5 period, Optimal Solution
139 4.18
Results for 100 generator and 5 period, Sub-Optimal Solution 144
vii
LIST OF FIGURES
FIGURE TITLE
PAGE
1.1 Power system economic operation function
3 2.1
Flying pattern of a flock of birds 34
2.2 Swimming pattern of a school of fish
34 3.1
Schematic of a thermal power plant 38
3.2 Linear cost model
40 3.3
Nonlinear and piecewise linear model 42
3.4 An exponential start-up cost function
46 3.5
Typical demandsupply unit variations with time 48
3.6 Loading rate
49 3.7
Deloading rate 50
3.8 Cost as a function of spinning reserve
53 3.9
Gas turbine plant 54
3.10 Typical daily demand curve
56 3.11
Demand curve approximation with a linear curve 57
3.12 Group constraints
61 3.13
Flowchart of problem solving procedure 64
4.1 DED problem – objective function
68 4.2
DED problem – constraint function 69
4.3 Flow chart of PSO
77 4.4
Power generation limits of different generators 79
viii 4.5
Demand variation with time 79
4.6 Spinning reserve contribution of a generator Ab Ghani, 1989
81 4.7
Convergence of total cost to optimal and feasible solution with PSO 94 4.8
Spinning reserve contribution 97
4.9 Power generation limits of different generators
100 4.10
Cost of generating powers for different generators 101
4.11 a Cost solution of 20 generators and 6 periods
105 b Cost solution of 20 generators and 12 periods
105 c Cost solution of 20 generators and 18 periods
106 d Cost solution of 20 generators and 24 periods
106 4.12
Power generation limits of different generators 134
4.13 Cost per unit of different generators
135 4.14
Cost solution of 100 generators and 5 periods 138
ix
LIST OF APPENDICES
APPENDIX TITLE
PAGE APPENDIX A
167
A1 Dantzig-Wolfe Code
167 A2
PSO - Basic Model 171
A3 PSO - 10 Generators and 6 Periods – Sub-Optimal Case
175 A4
PSO - 10 Generators and 6 Periods – Infeasible Case 178
A5 PSO - 10 Generators and 6 Periods – Optimal Case
181 A6
PSO - 20 Generators and 24 Periods Case 186
A7 PSO - 100 Generators and 5 Periods Case
198
APPENDIX B 210
B1 Test Data for 10 generators and 6 period’s case study.
210 B2
Input Data for 20 generators and 24 period’s case study 211
B3 Test Data for 100 generators and 5 period’s case study
213
x
LIST OF ABBREVIATIONS
ABC Artificial Bee Colony
ACO Ant colony optimization
AIWF Adaptive Inertia Weight Factor
APSO Adaptive PSO
CEED Combined Economic Emission Dispatch
CEED Cost and Emission Economic Dispatch
CF-IW Constriction Factor and Inertia Weight
CO2 Carbon Dioxide
CPSO Canonical PSO or Chaotic PSO
DED Dynamic Economic Dispatch
D-W Dantzig-Wolfe
ECD POZ Economic Cost Dispatch with Prohibited Operating Zone
ECD-MF Economic Cost Dispatch with Multiple Fuel
ECD-VPL Economic Cost Dispatch – Valve Point Loading
ECD-VPL-MF Economic Cost Dispatch – Valve Point Loading-Multiple Fuel
ED-POZ Economic Dispatch with Prohibited Operating Zone
EPSO Enhanced PSO
GA Genetic Algorithms
G_best Global best
GCPSO Global vision of PSO with Constriction Factor
GWPSO Global vision of PSO with Inertia Weight
xi HDE
Hybrid Differential Evolution HGPSO
hybrid gradient search PSO LCPSO
Local vision of PSO with Constriction Factor LWPSO
Local vision of PSO with Inertia Weight MAED
Multi-Area Economic Dispatch MAEED
Multi-Area Emission Economic Dispatch MIPSO
Multi-Pass Iteration PSO MOPSO
Multi-Objective Particle Swarm Optimization MPSO
Modified Particle Swarm Optimization NO2
Nitrogen Dioxide NPSO
New PSO OPF
Optimal Power Flow P_best
Personal best POZ
Prohibited Operating Zones PQCF
Piecewise Quadratic Cost Function PSO
Particle Swarm Optimization PSO-CF-IW
PSO with Inertia Weight and Constriction Factor PSO-LVIW
Particle Swarm Optimization - Linearly Varying Inertia Weight PSO-LVIW
Particle Swarm Optimization-Linearly Decreasing Inertia Weight RRL
Ramp Rate Limits SO2
Sulphur Dioxide SPLS
Synchronous Particle Local Search SQP
Sequential Quadratic Programming UC
Unit Commitment VURMFIPSO
Velocity Update Relaxation Momentum Factor-Induced PSO
xii
LIST OF PUBLICATIONS
The following publications have been achieved by this research work:
Journals: 1. Mohd Ruddin Ab Ghani, Saif Tahseen Hussein, M.T. Mohamad,