Mathematical Models and Algorithms for Power System Optimization

Chapter 1: Introduction1.1 General Ideas About Modeling1.2 Ideas About the Setting of Variables and Functions1.3 Ideas About the Selection of Model Types1.4 Ideas About the Selection of an AlgorithmChapter 2: Daily Economic Dispatch Optimization With Pumped Storage Plant for a Multiarea System2.1 Introduction2.1.1 Outline of the Problem2.1.2 Basic Requirements of Pumped Storage Plant Operation2.1.3 Overview of This Chapter2.2 Basic Ideas of Developing an Optimization Model2.2.1 Way of Processing Objective Function2.2.2 Way of Processing Variables and Constraints2.2.3 Way of Processing Integer Variables for Pumped Storage Plant2.3 Formulation of the Problem2.3.1 Notations2.3.2 Basic Expression of the Optimization Model2.3.3 Basic Structure of the Constraint Matrix2.4 Preprocessing of the Optimization Calculation2.4.1 Validity Test of the Input Data2.4.2 Validity for the Rationality of the Constraints2.4.3 Forming the Virtual Cost Function for a Pumped Storage Plant2.4.4 Forming the Cost Function for an Individual Power Plant2.4.5 Forming the Constraints at Different Periods for Each Unit2.4.6 Forming the Coefficient Matrix2.5 Computation Procedure for Optimization2.5.1 Main Calculation Procedure2.5.2 Description of the Input Data2.5.3 Special Settings to Meet the Calculation Requirements2.5.4 Description of the Output Results2.6 Implementation2.6.1 Concrete Expression of Objective and Constraint Functions forSmall-Scale Systems2.6.2 Practical Scale of the Test Systems2.6.3 Analysis of Peak-Valley Difference in the Load Curve2.6.4 Constraints of Pumped Storage Plant and Related Conversion Calculation2.6.5 Optimization Calculation Results2.7 ConclusionChapter 3: Optimization of Annual Generator Maintenance Scheduling3.1 Introduction3.1.1 Description of the Problem3.1.2 Basic Requirements for Annual Generator Maintenance Scheduling3.1.3 Overview of This Chapter3.2 Basic Ideas of Developing an GMS Model3.2.1 Way of Handling Unit Maintenance Intervals3.2.2 Principles to Set Priority for Unit Maintenance3.2.3 Way of Processing the Objective Function3.2.4 Way of Processing the Variable Settings and Constraints3.3 Formulation of the GMS Problem3.3.1 Notations3.3.2 Objective Function3.3.3 Constraints3.4 Fuzzification of GMS Model3.4.1 Selection of Fuzzy Membership Function3.4.2 Selection of Fuzzy Objective Index of GMS3.4.3 Formation of Fuzzy Constraints for GMS Problem3.5 Expert System Developed for GMS3.5.1 Selecting of Time Units3.5.2 Introducing of Operation Index3.5.3 Related Rules of the Expert System3.6 Calculation Procedure of GMS for Optimization3.6.1 Search Paths and Recursive Formulas of Fuzzy Dynamic Programming3.6.2 Main Calculation Procedure3.6.3 Description of the Input data3.6.4 Description of the Output Results3.7 Implementation3.7.1 Input Data3.7.2 Part of Output Results3.8 ConclusionChapter 4: New Algorithms Related to Power Flow4.1 Introduction4.1.1 Way of Processing Variables in Traditional Power Flow Equations4.1.2 Way of Processing Variables in New Power Flow Equation4.1.3 Overview of Unconstrained Power Flow With Objective Function (Based on SA Method)4.1.4 Overview of Constrained Power Flow With Objective Function (Based on OPF Method)4.1.5 Chapter Overview4.2 Ideas of Modeling for Unconstrained Power Flow With Objective Function4.2.1 Description of Ill-Conditioned Power Flow Problem4.2.2 Outline of Simulated Annealing Method4.2.3 Way of Modifying Iteration Step Size by SA Method4.2.4 Way of Constructing a Nonlinear Quadratic Objective Function4.3 Formulation of Unconstrained Power Flow Model With Nonlinear Quadratic Objective Function4.3.1 Notation4.3.2 Formulation of Power Flow With Quadratic Function4.4 Calculation Procedure Based on SA and N-R Method4.5 Implementation of SA Method4.5.1 Initial Conditions4.5.2 Conditions and Results of Four Cases4.6 Formulation of Discrete Optimal Power Flow4.6.1 Similarities and Differences Between LF and OPF4.6.2 Description of the Problem4.6.3 Features of the Problem4.6.4 Mathematical Model4.7 Discrete OPF Algorithm4.7.1 Main Solution Procedure of Discrete OPF4.7.2 Linearization of the Problem4.7.3 Iterative Solution Procedure for Mixed-Integer Linear Programming Problem4.8 Implementation of Discrete OPF4.8.1 Concrete Formulation of X, Y, A, B, of 5-Bus System4.8.2 Conditions and Results of Four Cases for 135-Bus Large-scale System4.9 ConclusionChapter 5: Load Optimization for Power Network5.1 Introduction5.1.1 Description of Minimizing Load Curtailment5.1.2 Description of Maximizing Load Supply Capability5.1.3 Overview of This Chapter5.2 Basic Idea of Load Optimization Modeling5.2.1 Way of Processing the Objective Function5.2.2 Way of Processing the Variable Settings and Constraints5.3 Load Optimization Model5.3.1 Notations5.3.2 Model of Minimum Curtailed Load5.3.3 Model of Maximum Load Supply Capability5.3.4 The Derivation Process of LP Model for Load Curtailment Optimization5.3.5 The Derivation Process of LP Model for Load Supply Capability5.4 Calculation Procedure of Minimizing LCO5.4.1 Step One: Input Data5.4.2 Step Two: Data Preprocessing5.4.3 Step Three: Optimization Calculation5.4.4 Step Four: Result Output5.5 Implementation of Load Curtailment Optimization5.5.1 Verification of Proposed Models and Calculation Methods5.5.2 Basic Conditions of a Real-Scale System5.5.3 Results of the Real-Scale System5.6 Calculation Procedure of Maximizing Load Supply Capability5.7 Numerical Examples for Maximizing LSC5.7.1 Description of the Test System5.7.2 Results Analysis5.8 ConclusionChapter 6: Discrete Optimization for Reactive Power Planning6.1 Introduction6.1.1 Practical Method for Discrete VAR Optimization6.1.2 Overview of This Chapter6.2 Basic Ideas of Forming an Optimization Model6.2.1 Way of Processing Discreteness6.2.2 Way of Nonlinearity Processing6.2.3 Way of Processing Multiple States6.2.4 Way of Selecting of Initial Value6.2.5 Consideration to Obtain Global Optimization6.2.6 Verification of the Correctness of Discrete Solutions6.2.7 Special Dealing With Practical Problems6.2.8 Way of Processing Objective Function6.2.9 Way of Processing Transformer Tap T and Capacitor Bank C6.3 Single-State Discrete VAR Optimization6.3.1 Outline6.3.2 Mathematical Model for Single-State Discrete VAR Optimization6.3.3 Algorithm for Single-State Discrete Optimal VAR Planning6.3.4 Implementation of Single-State Discrete VAR Optimization6.3.5 Summary6.4 Multistate Discrete VAR Optimization6.4.1 Overview6.4.2 Multistate Model for Discrete VAR Optimization6.4.3 Overall Solution Procedure of Multistate VAR Optimization6.4.4 Implementation6.4.5 Summary6.5 Discrete VAR Optimization Based on Expert Rules6.5.1 Overview6.5.2 Necessity of Introducing Expert Rules6.5.3 Algorithm Based on Expert Rules for Discrete VAR Optimization6.5.4 Implemetation6.5.5 Summary6.6 Discrete VAR Optimization Based on GA6.6.1 Overview6.6.2 Necessity of Applying Artificial Intelligence Algorithms6.6.3 GA-Based Discrete VAR Optimization Model6.6.4 GA-Based Discrete VAR Optimization Algorithm6.6.5 Implementation6.6.6 Summary6.7 ConclusionChapter 7: Optimization Method for Load Frequency Feed Forward Control7.1 Introduction7.1.1 Descriptions of the Problem7.1.2 Overview of Chapter7.2 Basic Ideas of Modeling7.2.1 Formulating the Load Disturbance Model7.2.2 Way of Constructing Estimator at All Levels7.2.3 Way of Setting Up the Load Frequency Controller by the Invariance Principle7.2.4 Considerations of Transformation Methods for Linear Models7.3 Model Identification of Load Disturbance ΔPL7.3.1 Brief Descriptions of Random Sequence7.3.2 Brief Descriptions of Linear Models for Stochastic Process7.3.3 Identification of Model ΔPL7.3.4 Parameter Estimation of Model ΔPL7.4 Model for a Typical Power System7.4.1 Generator Model7.4.2 Turbogenerator Model7.4.3 Hydrogenerator Model7.4.4 Equivalent Generator Model of the Power System7.5 Hierarchical Estimation for the Power System7.5.1 Local Estimator7.5.2 Central Estimator..7.5.3 Estimation and Forecasting of Load Disturbance PL7.6 Load Frequency Controller of the Power System7.6.1 Invariance Principle7.6.2 Load Frequency Control Applying Invariance Principle7.6.3 Simulation Procedure of Tracking Control7.7 Transformation Methods of Linear Models7.7.1 Formulation of Difference Equation and Differential Equation7.7.2 Transformation Method Between Difference Equations7.7.3 Transformation Method From Differential Type Into Difference One7.8 Implementation7.8.1 Parameters From Figs. 7.4 and 7.77.8.2 Simulation Results of PL Model Identification7.8.3 Simulation Results of Local Estimator and Central Estimator7.8.4 Simulation Results of Compensation Controller7.8.5 Simulation Results of Tracking Control for Five-Unit Test System7.8.6 Results of Transformation Between Mathematical Models7.9 ConclusionChapter 8: Local Decoupling Control Method for Transient Stability of a Power System8.1 Introduction8.1.1 Description of the Problem8.1.2 Investigating the Feasibility of Control System Stability Based on Local Information8.1.3 Overview of This Chapter8.2 Basic Ideas of Solving the Problem8.2.1 Analysis of Two Scenarios in Power System Instability8.2.2 Purposes of Introducing an Observation Decoupled State Space8.2.3 Two Stage Countermeasures in Local Stability Controls8.3 Basic Concepts of Control Criterion Based on Local Control8.3.1 Simplified Model and Typical Network of the Power System8.3.2 Basic Concept of First Stage Control Criterion (Energy Equilibrium)8.3.3 Basic Concept of the Second Stage Control Criterion (Norm Reduction)8.4 Formulation and Proof of the First Stage Control Criterion (Energy Equilibrium)8.5 Formulation and Proof of the Second Stage Control Criterion (Norm Reduction)8.5.1 Structure of Mathematical Model and Its Generalized Formation of the Observation Decoupled State Space8.5.2 Proof of Topology Equivalence Between Observation Decoupled State Space and Original System State Space8.5.3 Proof of Topology Equivalence Between Different Forms of Observation Decoupled State Space and Original System State Space8.5.4 Origin of Observation Decoupled State Space (the Only Equilibrium Point in the Power System)8.5.5 Sufficient Condition for Convergence of the Second Stage Control (Norm Reduction)8.6 General Simulation Calculation Procedure in Two-Stage Control8.6.1 Simplified Assumptions and Network Diagram8.6.2 Variables of Measuring and Calculating in Online Control8.6.3 Preprocessing of Calculations8.6.4 Main Steps of Simulation Calculation Procedure8.7 Numerical Model in Simulation Calculation8.7.1 Concrete Formulations of Dynamic Equations and Network Equations8.7.2 Calculation of Equivalent Impedance in Case of Fault8.7.3 Braking Power Calculation After Braking Resistor Switched on8.7.4 Calculation of the First Stage Control Criterion (Energy Equilibrium)8.7.5 Calculation of Observation Decoupled State Vector and the Second Stage Control Criterion8.8 Implementation8.8.1 Network Structure and Parameters8.8.2 Operation Mode hIi8.8.3 Operation Mode hIIi8.8.4 Analysis of Calculation Results8.9 ConclusionChapter 9: Optimization of Electricity Market Transaction Decision Based on Market General Equilibrium9.1 Introduction9.1.1 Problem Description9.1.2 Microeconomics Equilibrium Principle9.1.3 The Overview of a Power Market9.1.4 Chapter Overview9.2 The Idea of Establishing the Model9.2.1 Consideration of the Type of Goods9.2.2 Consideration of Power System Characteristics9.2.3 Objective Function9.2.4 Constraint Conditions9.3 Equivalent Optimization Model of General Equilibrium in a Power Markets9.3.1 Model of Active Power Transaction9.3.2 Model Considering Both Active and Reactive Transactions9.3.3 Solution Algorithm9.4 Implementation9.4.1 Example Analysis Considering Active Power Transaction9.4.2 Example Analysis Considering Active and Reactive Power Transaction9.5 Conclusion

Appendix A: An Approximation Method for Mixed Integer ProgrammingAppendix B: The Differential Expressions for Transformer Tap and Shunt Capacitor UnitAppendix C: A DC Load Flow Method for Calculating Generation Angle