CONCLUSION AND RECOMMENDATIONS FOR FUTURE WORK 151

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,