Optimal Long-Term Operation of Electric Power Systems

This book deals with a very important problem in power system planning for countries in which hydrogeneration accounts for the greatest part of the system power production. During the past thirty years many techniques have been developed to cope with the long-term operation of hydro reser­ voirs. These techniques have been discussed in a number of publications, but they have not until now been documented in book form. This book is intended as the foundation for a special graduate course dealing with aspects of electrical engineering, operational research, water resource research, and applied mathematics. It may also be used for self­ study by practicing personnel involved in the planning and operation of hydroelectric power systems for utilities, consulting groups, and government regulatory agencies. The book consists of eight chapters. Chapter 1 reviews the historical developments in the field, discusses briefly all techniques used to solve the problem, and summarizes the modeling of hydroplants for long-term operation studies. At the end of the chapter we present in detail an outline of the book.

1. Introduction.- 1.1. A Historical Survey.- 1.2. Hydro Plant Modeling for Long-Term Operation.- 1.2.1. Pumped Storage Plants.- 1.2.2. Run-of-River Plants.- 1.2.3. Storage Plants.- 1.2.4. Reservoir Models.- 1.2.5. Operational Constraints.- 1.3. Outline of the Book.- References.- 2. Mathematical Optimization Techniques.- 2.1. Introduction.- 2.2. A Review of Matrices.- 2.2.1. Vectors.- 2.2.2. Matrices.- 2.2.3. Quadratic Forms and Definiteness.- 2.3. Discrete Variational Calculus.- 2.3.1. Unconstrained Discrete Optimization.- 2.3.2. Constrained Discrete Optimization.- 2.4. Discrete Maximum Principle.- 2.4.1. Stochastic Discrete Maximum Principle.- 2.5. Dynamic Programming.- 2.6. Functional Analysis Optimization Technique.- 2.6.1. Norms and Inner Products.- 2.6.2. Hilbert Space.- 2.6.3. Hilbert Space of Random Variables.- 2.6.4. A Minimum Norm Theorem.- References.- 3. Long-Term Operation of Reservoirs in Series.- 3.1. Introduction.- 3.2. Problem Formulation.- 3.2.1. The System under Study.- 3.2.2. The Objective Function.- 3.3. The Problem Solution.- 3.3.1. Turgeon Approaches.- 3.3.2. A Minimum Norm Approach.- 3.3.3. A Nonlinear Model: Minimum Norm Approach.- References.- 4. Long-Term Operation of Multichain Power Systems.- 4.1. Introduction.- 4.2. Problem Formulation.- 4.2.1. The System under Study.- 4.2.2. The Objective Function.- 4.3. The Aggregation Approach (Turgeon Approach).- 4.3.1. The Objective.- 4.3.2. The Solution by Dynamic Programming.- 4.3.3. The One-at-a-Time Method.- 4.3.4. The Aggregation-Decomposition Method.- 4.3.5. Comments.- 4.4. Discrete Maximum Principle.- 4.4.1. General Problem Formulation.- 4.4.2. Solution Algorithm.- 4.4.3. Practical Example.- 4.4.4. Comments.- 4.5. A Minimum Norm Approach, Linear Model.- 4.5.1. The Objective Function.- 4.5.2. A Minimum Norm Formulation.- 4.5.3. The Optimal Solution.- 4.5.4. Algorithm of Solution.- 4.5.5. Practical Example.- 4.6. A Minimum Norm Approach, Nonlinear Model.- 4.6.1. Modeling of the System.- 4.6.2. The Optimal Solution.- 4.6.3. Algorithm of Solution.- 4.6.4. Practical Example.- 4.6.5. Comments.- References.- 5. Modeling and Optimization of a Multireservoir Power System for Critical Water Conditions.- 5.1. Introduction.- 5.2. Problem Formulation.- 5.2.1. The System under Study.- 5.2.2. The Objective Function.- 5.2.3. A Minimum Norm Approach.- 5.2.4. The Optimal Solution.- 5.2.5. Computer Logic.- 5.2.6. Practical Example.- 5.2.7. Concluding Remarks.- 5.3. Nonlinear Storage Model.- 5.3.1. Objective Function.- 5.3.2. A Minimum Norm Formulation.- 5.3.3. The Optimal Solution.- 5.4. A Discrete Maximum Principle Approach (Linear Model).- 5.4.1. Problem Formulation.- 5.4.2. Optimal Equations.- 5.5. Optimization of Power System Operation with a Specified Monthly Generation.- 5.5.1. Problem Formulation.- 5.5.2. The Optimal Solution.- 5.5.3. Algorithm of Solution.- 5.5.4. Practical Example.- 5.5.5. Concluding Remarks.- References.- 6. Optimization of the Firm Hydro Energy Capability for Hydroelectric Systems.- 6.1. Introduction.- 6.2. Nonlinear Programming Model (Hicks et al. Approach).- 6.2.1. System Modeling and Relationships.- 6.2.2. Optimization Objective and Constraints.- 6.2.3. Method of Solution.- 6.2.4. Practical Example.- 6.2.5. Conclusion.- 6.3. A Minimum Norm Approach.- 6.3.1. Optimization Objective and Constraints.- 6.3.2. A Minimum Norm Formulation.- 6.3.3. The Optimal Solution.- 6.3.4. Algorithm of Solution.- 6.4. A Nonlinear Model (Minimum Norm Approach).- 6.4.1. Optimization Objective and Constraints.- 6.4.2. Modeling of the System.- 6.4.3. A Minimum Norm Formulation.- 6.4.4. The Optimal Solution.- References.- 7. Long-Term Optimal Operation of Hydrothermal Power Systems.- 7.1. Introduction.- 7.2. All-Thermal Power Systems.- 7.2.1. Linear Fuel-Cost Function.- 7.2.2. Quadratic Fuel-Cost Function.- 7.2.3. General Solution Method for Nonlinear Functions.- 7.2.4. Dynamic Programming Approach for Nonmonotonic Heat Rate Curves.- 7.2.5. Conclusion.- 7.3. Optimal Scheduling of Hydrothermal Power Systems.- 7.3.1. A Direct Method Approach.- 7.4. Discrete Maximum Principle.- 7.4.1. Concluding Remarks.- 7.5. Stochastic Nonlinear Programming.- 7.5.1. Optimal Stochastic Control Policy.- 7.5.2. Computer Algorithm.- 7.5.3. Practical Example.- 7.5.4. Concluding Remarks.- 7.6. Aggregation with Stochastic Dynamic Programming Approach.- 7.6.1. Aggregation of Reservoirs and Hydro Storage Characteristics.- 7.6.2. Optimization Problem Formulation.- 7.6.3. Optimization Algorithm.- 7.6.4. Practical Example.- 7.6.5. Concluding Remarks.- 7.7. Aggregation-Decomposition Approach.- 7.7.1. Spatial Decomposition.- 7.7.2. Algorithm of Solution.- 7.7.3. Stage Optimization, Optimization Procedure.- 7.7.4. Practical Example.- 7.7.5. Concluding Remarks.- 7.8. A Minimum Norm Approach, Linear Storage-Elevation Model.- 7.8.1. Problem Formulation and the Objective Function.- 7.8.2. A Minimum Norm Formulation.- 7.8.3. Optimal Solution.- 7.8.4. Algorithm of Solution.- 7.8.5. Concluding Remarks.- 7.9. A Minimum Norm Approach, Nonlinear Storage-Elevation Curve.- 7.9.1. Problem Formulation and Objective Functional.- 7.9.2. A Minimum Norm Formulation.- 7.9.3. Optimal Solution.- 7.9.4. Algorithm of Solution.- 7.9.5. Concluding Remarks.- 7.10. Nuclear, Hydrothermal Power Systems.- 7.10.1. Problem Formulation.- 7.10.2. A Minimum Norm Formulation.- 7.10.3. The Optimal Solution.- 7.10.4. Algorithm of Solution, Computer Logic.- 7.10.5. Conclusion.- 7.11. General Comments.- Appendix A: One Dimension Minimization.- Appendix B: Projection Matrix P.- Appendix C: Some Probability Characteristics of Electric Power Systems.- C.l. Approximate Evaluation of Expected Value and Variance of g(x).- C.2. Variances of Thermal Power Generation.- C.3. Probability Properties of Hydroelectric Generation.- C.4. Expected Transmission Losses.- C.5. Expected Cost of Thermal Generation.- References.- 8. Conclusion.- 8.1. Summary.- 8.2. Future Work.