Mathematical Models and Algorithms for Power System Optimization

Modeling Technology for Practical Engineering Problems

Mingtian Fan, Zuping Zhang, Chengmin Wang

• Publisher: Academic Press
• Year: 2019
• ISBN:
  9780128132326
• Topic: Technology & Engineering
• Subtopic: Power Resources
• Index: Technology & Engineering

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

Chapter 1

Introduction

Abstract

This book provides technical knowledge about power system optimization modeling, so that the advantages of optimization models can be fully learnt for the analysis of power system problems. This chapter focuses on the main aspects of how optimization models are constructed and solved. It introduces general ideas about modeling techniques, and then provides some ideas for the setting of variables and functions, the selection of model types, and the selection of algorithms, all of which provide main aspects for power system model constructions and solutions. Furthermore, the present and the future applications of AI technologies as possible solutions to optimization modeling are also discussed.

Keywords

Modeling techniques; Setting of variables and functions; Selection of model types; Selection of algorithms

Chapter Outline

• 1.1 General Ideas about Modeling
• 1.2 Ideas about the Setting of the Variable and Function
• 1.3 Ideas about the Selection of the Model Type
• 1.4 Ideas about the Selection of the Algorithm
• 1.5 Ideas about the Applications of Artificial Intelligence Technology

1.1 General Ideas about Modeling

Many practical power system problems can be represented as mathematical models and sets of rules that connect the model’s elements. Because of the mathematical model’s good reproducibility, the inherent law of practical problems can be found via numerical algorithms. Therefore, it is necessary to study the model and algorithms of power systems in depth.

An appropriate approximation is reasonable for guiding practical theory. It is generally believed that, as long as human observation of practical problems reaches 10-6 orders of magnitude, it can meet actual needs of measurement. Beyond this limit, only theorists are interested. As the famous mathematician Klein has said, approximation mathematics is the very part of mathematics applied to practical applications, whereas precision mathematics is the solid framework on which approximation mathematics is built. Approximate mathematics is not “approximate mathematics” but “precision mathematics about approximate relationships.” Therefore, the priority of modeling is to determine how to approximately solve a problem by taking advantage of the existing solvable conditions of the problem.

In modeling research of power system optimization, it is necessary to dialectically deal with the different types of variables, such as discreteness and continuous variables; the different types of models, such as linear and nonlinear; and the different types of algorithms, such as numerical or non-numerical procedures. To integrate more new elements and meet new development trends, new models should be developed using existing and newly developed methods based on the current computer technology to satisfy the actual requirements of the power system, which require a solid mathematical foundation and engineering background knowledge. The development of the power system optimization model is full of dialectic wisdom. The authors’ main ideas are as follows:

1. (1) The optimization modeling is to transfer a practical problem into a mathematical problem and obtain a feasible solution for the practical problem considering the existing conditions. It has to fully consider the compromise among conflicting goals, such as the simple model versus the complex calculation procedure, and vice verse, nonlinear model versus linearized solution, and large-scale discrete optimization model versus the current computing condition.
2. (2) Whether the developed model is solvable must consider many problems from the theoretical perspective. For example, the problem with local solutions for any optimization algorithm, which can be avoided by the method of the multi-point search in the solution space. As for the nonconvex problem, there is a possibility of converting the model from nonconvex to convex by some recent researches. In addition, for some problems that cannot be solved by mathematical formulation, some non-numerical procedures could be successfully applied.
3. (3) What the top issue of numerical calculation is that an approximate solution with engineering precision can be obtained, by which the difficulty of the optimization model could be tested under the conditions without need of an analytical solution, and the consistency between the theoretical model and actual problem could be verified. However, because any numerical calculation method has its own limitation, the complex relationship between a theoretical model and a practical problem may be expressed by computer vision technology in the near future.
4. (4) The mutual transformation of mathematical models is very helpful for solving difficult problems, because mathematical models can be transformed into each other under such certain conditions, such as discrete and continuous, accurate and approximate, differential and difference, solvable and nonsolvable, convex and nonconvex, optimal and suboptimal, etc.

Solvability discussions about mathematical models are also explained in this book, such as search scope, initial point, the limit of the variables, and the range of the equations. Some special modeling techniques for practical engineering problems are also provided in this book. Some ideas are briefly given in the following, including ideas about how to set variables and functions, ideas about how to determine model types and algorithms. These modeling techniques have been implemented in the practical problems of this book and can be effectively applied to help solve power system optimization problems.

1.2 Ideas about the Setting of the Variable and Function

1. (1) Ideas about the variable in conventional power system analysis:
   Two kinds of the basic components, single-ended and double-ended (which could also be briefly represented as node and branch), are included in power system analysis, in which the former can represent loads, generators, capacitors, reactors, and other grounded components, whereas the latter can represent lines, switches, transformers, and other branch components. The conventional calculation model of AC power flow for each node i is as follows:
   $\begin{array}{l}{P}_{\mathrm{G}i}-P_{\mathrm{L}i}-U_i\sum _{j\in i}{U}_j\left(G_{\mathit{ij}}cos{\theta }_{\mathit{ij}}+B_{\mathit{ij}}sin{\theta }_{\mathit{ij}}\right)=0\\ Q_{\mathrm{G}i}-Q_{\mathrm{L}i}-U_i\sum _{j\in i}{U}_j\left(G_{\mathit{ij}}sin{\theta }_{\mathit{ij}}-B_{\mathit{ij}}cos{\theta }_{\mathit{ij}}\right)=0\end{array}$
   
   where θ_ij = θ_i−θ_j, which is the angle difference between node i and j. The assigned values include the load (P_L, Q_L), some generations (P_G, Q_G), and some voltages and phase angles (U and θ). That is, there are only two variables and two equations for each node.
   The basic variables in the analysis and calculation for the steady-state of a power system can be classified into four types: active power, reactive power, voltage, and phase angle (namely: P, Q, U, and θ). Among them, active power and reactive power can be divided into active power generation and reactive power generation (P_G, Q_G), and active load and reactive power load (P_L, Q_L), respectively. Sometimes, the “P” and “Q” on the node can be considered as the corresponding impedance rather than the variable. In the transient calculation of a power system, besides the basic variables previously described, the power angle δ and the angular frequency or rotational speed of the generator ω = 2πf are also included (where f is the system frequency).
   When all of P, Q, U, and θ are treated as variables, with their upper and lower limits added, the conventional optimization method can be applied to search for an optimal solution. In addition, all these variables can be subscripted to indicate changes over time (such as seconds, minutes, hours, months, or years). For example, the power generation output of unit (i) in a different time period (t) can be expressed as P_Gi(t).
2. (2) The parameter and variable can be transformed from one to another:
   If the parameters of components are taken as variables with upper and lower limits, then they can be adjusted in the optimization calculation. For example, if the expression of the transformers and capacitors, are expanded as variables with limits for the capacitor bank number C and the transformer tap ratio T, then they can be optimized by way of an optimization method.
   The idea to optimize the parameters of the components is to taken component parameters as variables is to optimize the parameters of the components. However, in optimization calculation models, the device parameters are usually given as constants or transformed into impedances, but they are rarely directly handled as variables. The general way to convert device parameters to variables is to transform F(x₁, …, x_n) = 0 to F(x₁, …, x_{n − 1}, x_n(y)) = 0, where y = T or C.
3. (3) The variable and function can be switched from one to another.
   There is one way to process variables as functions. For example, the traditional AC power flow equation previously given can be written as F(x₁, …, x_n) = g, which can then be transformed to F(x₁, …, x_n) – g = 0, F(x₁, …, x_n, g) = 0, so the right hand side (the node injection power g) can be considered as a variable.

1.3 Ideas about the Selection of the Model Type

Under certain conditions, some model types can be mutually transformed into each other, suc...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Abstract
• Preface
• Chapter 1: Introduction
• Chapter 2: Daily Economic Dispatch Optimization With Pumped Storage Plant for a Multiarea System
• Chapter 3: Optimization of Annual Generator Maintenance Scheduling
• Chapter 4: New Algorithms Related to Power Flow
• Chapter 5: Load Optimization for Power Network
• Chapter 6: Discrete Optimization for Reactive Power Planning
• Chapter 7: Optimization Method for Load Frequency Feed Forward Control
• Chapter 8: Local Decoupling Control Method for Transient Stability of a Power System
• Chapter 9: Optimization of Electricity Market Transaction Decisions based on Market General Equilibrium
• 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
• References
• Index