Mathematical Models and Algorithms for Power System Optimization

Mathematical Models and Algorithms for Power System Optimization helps readers build a thorough understanding of new technologies and world-class practices developed by the State Grid Corporation of China, the organization responsible for the world’s largest power distribution network. This reference covers three areas: power operation planning, electric grid investment and operational planning and power system control. It introduces economic dispatching, generator maintenance scheduling, power flow, optimal load flow, reactive power planning, load frequency control and transient stability, using mathematic models including optimization, dynamic, differential and difference equations.

• Provides insights on the development of new mathematical models of power system optimization
• Analyzes power systems comprehensively to create novel mathematic models and algorithms for issues related to the planning operation of power systems
• Includes research on the optimization of power systems and related practical research projects carried out since 1981

Graduate students and researchers who work in the area of novel optimization models for planning and operation of power systems

Chapter 1: Introduction
1.1 General Ideas About Modeling
1.2 Ideas About the Setting of Variables and Functions
1.3 Ideas About the Selection of Model Types
1.4 Ideas About the Selection of an Algorithm
Chapter 2: Daily Economic Dispatch Optimization With Pumped Storage Plant for a Multiarea System
2.1 Introduction
2.1.1 Outline of the Problem
2.1.2 Basic Requirements of Pumped Storage Plant Operation
2.1.3 Overview of This Chapter
2.2 Basic Ideas of Developing an Optimization Model
2.2.1 Way of Processing Objective Function
2.2.2 Way of Processing Variables and Constraints
2.2.3 Way of Processing Integer Variables for Pumped Storage Plant
2.3 Formulation of the Problem
2.3.1 Notations
2.3.2 Basic Expression of the Optimization Model
2.3.3 Basic Structure of the Constraint Matrix
2.4 Preprocessing of the Optimization Calculation
2.4.1 Validity Test of the Input Data
2.4.2 Validity for the Rationality of the Constraints
2.4.3 Forming the Virtual Cost Function for a Pumped Storage Plant
2.4.4 Forming the Cost Function for an Individual Power Plant
2.4.5 Forming the Constraints at Different Periods for Each Unit
2.4.6 Forming the Coefficient Matrix
2.5 Computation Procedure for Optimization
2.5.1 Main Calculation Procedure
2.5.2 Description of the Input Data
2.5.3 Special Settings to Meet the Calculation Requirements
2.5.4 Description of the Output Results
2.6 Implementation
2.6.1 Concrete Expression of Objective and Constraint Functions for
Small-Scale Systems
2.6.2 Practical Scale of the Test Systems
2.6.3 Analysis of Peak-Valley Difference in the Load Curve
2.6.4 Constraints of Pumped Storage Plant and Related Conversion Calculation
2.6.5 Optimization Calculation Results
2.7 Conclusion
Chapter 3: Optimization of Annual Generator Maintenance Scheduling
3.1 Introduction
3.1.1 Description of the Problem
3.1.2 Basic Requirements for Annual Generator Maintenance Scheduling
3.1.3 Overview of This Chapter
3.2 Basic Ideas of Developing an GMS Model
3.2.1 Way of Handling Unit Maintenance Intervals
3.2.2 Principles to Set Priority for Unit Maintenance
3.2.3 Way of Processing the Objective Function
3.2.4 Way of Processing the Variable Settings and Constraints
3.3 Formulation of the GMS Problem
3.3.1 Notations
3.3.2 Objective Function
3.3.3 Constraints
3.4 Fuzzification of GMS Model
3.4.1 Selection of Fuzzy Membership Function
3.4.2 Selection of Fuzzy Objective Index of GMS
3.4.3 Formation of Fuzzy Constraints for GMS Problem
3.5 Expert System Developed for GMS
3.5.1 Selecting of Time Units
3.5.2 Introducing of Operation Index
3.5.3 Related Rules of the Expert System
3.6 Calculation Procedure of GMS for Optimization
3.6.1 Search Paths and Recursive Formulas of Fuzzy Dynamic Programming
3.6.2 Main Calculation Procedure
3.6.3 Description of the Input data
3.6.4 Description of the Output Results
3.7 Implementation
3.7.1 Input Data
3.7.2 Part of Output Results
3.8 Conclusion
Chapter 4: New Algorithms Related to Power Flow
4.1 Introduction
4.1.1 Way of Processing Variables in Traditional Power Flow Equations
4.1.2 Way of Processing Variables in New Power Flow Equation
4.1.3 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 Overview
4.2 Ideas of Modeling for Unconstrained Power Flow With Objective Function
4.2.1 Description of Ill-Conditioned Power Flow Problem
4.2.2 Outline of Simulated Annealing Method
4.2.3 Way of Modifying Iteration Step Size by SA Method
4.2.4 Way of Constructing a Nonlinear Quadratic Objective Function
4.3 Formulation of Unconstrained Power Flow Model With Nonlinear Quadratic Objective Function
4.3.1 Notation
4.3.2 Formulation of Power Flow With Quadratic Function
4.4 Calculation Procedure Based on SA and N-R Method
4.5 Implementation of SA Method
4.5.1 Initial Conditions
4.5.2 Conditions and Results of Four Cases
4.6 Formulation of Discrete Optimal Power Flow
4.6.1 Similarities and Differences Between LF and OPF
4.6.2 Description of the Problem
4.6.3 Features of the Problem
4.6.4 Mathematical Model
4.7 Discrete OPF Algorithm
4.7.1 Main Solution Procedure of Discrete OPF
4.7.2 Linearization of the Problem
4.7.3 Iterative Solution Procedure for Mixed-Integer Linear Programming Problem
4.8 Implementation of Discrete OPF
4.8.1 Concrete Formulation of X, Y, A, B, of 5-Bus System
4.8.2 Conditions and Results of Four Cases for 135-Bus Large-scale System
4.9 Conclusion
Chapter 5: Load Optimization for Power Network
5.1 Introduction
5.1.1 Description of Minimizing Load Curtailment
5.1.2 Description of Maximizing Load Supply Capability
5.1.3 Overview of This Chapter
5.2 Basic Idea of Load Optimization Modeling
5.2.1 Way of Processing the Objective Function
5.2.2 Way of Processing the Variable Settings and Constraints
5.3 Load Optimization Model
5.3.1 Notations
5.3.2 Model of Minimum Curtailed Load
5.3.3 Model of Maximum Load Supply Capability
5.3.4 The Derivation Process of LP Model for Load Curtailment Optimization
5.3.5 The Derivation Process of LP Model for Load Supply Capability
5.4 Calculation Procedure of Minimizing LCO
5.4.1 Step One: Input Data
5.4.2 Step Two: Data Preprocessing
5.4.3 Step Three: Optimization Calculation
5.4.4 Step Four: Result Output
5.5 Implementation of Load Curtailment Optimization
5.5.1 Verification of Proposed Models and Calculation Methods
5.5.2 Basic Conditions of a Real-Scale System
5.5.3 Results of the Real-Scale System
5.6 Calculation Procedure of Maximizing Load Supply Capability
5.7 Numerical Examples for Maximizing LSC
5.7.1 Description of the Test System
5.7.2 Results Analysis
5.8 Conclusion
Chapter 6: Discrete Optimization for Reactive Power Planning
6.1 Introduction
6.1.1 Practical Method for Discrete VAR Optimization
6.1.2 Overview of This Chapter
6.2 Basic Ideas of Forming an Optimization Model
6.2.1 Way of Processing Discreteness
6.2.2 Way of Nonlinearity Processing
6.2.3 Way of Processing Multiple States
6.2.4 Way of Selecting of Initial Value
6.2.5 Consideration to Obtain Global Optimization
6.2.6 Verification of the Correctness of Discrete Solutions
6.2.7 Special Dealing With Practical Problems
6.2.8 Way of Processing Objective Function
6.2.9 Way of Processing Transformer Tap T and Capacitor Bank C
6.3 Single-State Discrete VAR Optimization
6.3.1 Outline
6.3.2 Mathematical Model for Single-State Discrete VAR Optimization
6.3.3 Algorithm for Single-State Discrete Optimal VAR Planning
6.3.4 Implementation of Single-State Discrete VAR Optimization
6.3.5 Summary
6.4 Multistate Discrete VAR Optimization
6.4.1 Overview
6.4.2 Multistate Model for Discrete VAR Optimization
6.4.3 Overall Solution Procedure of Multistate VAR Optimization
6.4.4 Implementation
6.4.5 Summary
6.5 Discrete VAR Optimization Based on Expert Rules
6.5.1 Overview
6.5.2 Necessity of Introducing Expert Rules
6.5.3 Algorithm Based on Expert Rules for Discrete VAR Optimization
6.5.4 Implemetation
6.5.5 Summary
6.6 Discrete VAR Optimization Based on GA
6.6.1 Overview
6.6.2 Necessity of Applying Artificial Intelligence Algorithms
6.6.3 GA-Based Discrete VAR Optimization Model
6.6.4 GA-Based Discrete VAR Optimization Algorithm
6.6.5 Implementation
6.6.6 Summary
6.7 Conclusion
Chapter 7: Optimization Method for Load Frequency Feed Forward Control
7.1 Introduction
7.1.1 Descriptions of the Problem
7.1.2 Overview of Chapter
7.2 Basic Ideas of Modeling
7.2.1 Formulating the Load Disturbance Model
7.2.2 Way of Constructing Estimator at All Levels
7.2.3 Way of Setting Up the Load Frequency Controller by the Invariance Principle
7.2.4 Considerations of Transformation Methods for Linear Models
7.3 Model Identification of Load Disturbance ΔPL
7.3.1 Brief Descriptions of Random Sequence
7.3.2 Brief Descriptions of Linear Models for Stochastic Process
7.3.3 Identification of Model ΔPL
7.3.4 Parameter Estimation of Model ΔPL
7.4 Model for a Typical Power System
7.4.1 Generator Model
7.4.2 Turbogenerator Model
7.4.3 Hydrogenerator Model
7.4.4 Equivalent Generator Model of the Power System
7.5 Hierarchical Estimation for the Power System
7.5.1 Local Estimator
7.5.2 Central Estimator..
7.5.3 Estimation and Forecasting of Load Disturbance PL
7.6 Load Frequency Controller of the Power System
7.6.1 Invariance Principle
7.6.2 Load Frequency Control Applying Invariance Principle
7.6.3 Simulation Procedure of Tracking Control
7.7 Transformation Methods of Linear Models
7.7.1 Formulation of Difference Equation and Differential Equation
7.7.2 Transformation Method Between Difference Equations
7.7.3 Transformation Method From Differential Type Into Difference One
7.8 Implementation
7.8.1 Parameters From Figs. 7.4 and 7.7
7.8.2 Simulation Results of PL Model Identification
7.8.3 Simulation Results of Local Estimator and Central Estimator
7.8.4 Simulation Results of Compensation Controller
7.8.5 Simulation Results of Tracking Control for Five-Unit Test System
7.8.6 Results of Transformation Between Mathematical Models
7.9 Conclusion
Chapter 8: Local Decoupling Control Method for Transient Stability of a Power System
8.1 Introduction
8.1.1 Description of the Problem
8.1.2 Investigating the Feasibility of Control System Stability Based on Local Information
8.1.3 Overview of This Chapter
8.2 Basic Ideas of Solving the Problem
8.2.1 Analysis of Two Scenarios in Power System Instability
8.2.2 Purposes of Introducing an Observation Decoupled State Space
8.2.3 Two Stage Countermeasures in Local Stability Controls
8.3 Basic Concepts of Control Criterion Based on Local Control
8.3.1 Simplified Model and Typical Network of the Power System
8.3.2 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 Space
8.5.2 Proof of Topology Equivalence Between Observation Decoupled State Space and Original System State Space
8.5.3 Proof of Topology Equivalence Between Different Forms of Observation Decoupled State Space and Original System State Space
8.5.4 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 Control
8.6.1 Simplified Assumptions and Network Diagram
8.6.2 Variables of Measuring and Calculating in Online Control
8.6.3 Preprocessing of Calculations
8.6.4 Main Steps of Simulation Calculation Procedure
8.7 Numerical Model in Simulation Calculation
8.7.1 Concrete Formulations of Dynamic Equations and Network Equations
8.7.2 Calculation of Equivalent Impedance in Case of Fault
8.7.3 Braking Power Calculation After Braking Resistor Switched on
8.7.4 Calculation of the First Stage Control Criterion (Energy Equilibrium)
8.7.5 Calculation of Observation Decoupled State Vector and the Second Stage Control Criterion
8.8 Implementation
8.8.1 Network Structure and Parameters
8.8.2 Operation Mode hIi
8.8.3 Operation Mode hIIi
8.8.4 Analysis of Calculation Results
8.9 Conclusion
Chapter 9: Optimization of Electricity Market Transaction Decision Based on Market General Equilibrium
9.1 Introduction
9.1.1 Problem Description
9.1.2 Microeconomics Equilibrium Principle
9.1.3 The Overview of a Power Market
9.1.4 Chapter Overview
9.2 The Idea of Establishing the Model
9.2.1 Consideration of the Type of Goods
9.2.2 Consideration of Power System Characteristics
9.2.3 Objective Function
9.2.4 Constraint Conditions
9.3 Equivalent Optimization Model of General Equilibrium in a Power Markets
9.3.1 Model of Active Power Transaction
9.3.2 Model Considering Both Active and Reactive Transactions
9.3.3 Solution Algorithm
9.4 Implementation
9.4.1 Example Analysis Considering Active Power Transaction
9.4.2 Example Analysis Considering Active and Reactive Power Transaction
9.5 Conclusion

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