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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley - IEEE
- Sprache: Englisch
Explore the theoretical foundations and real-world power system applications of convex programming In Mathematical Programming for Power System Operation with Applications in Python, Professor Alejandro Garces delivers a comprehensive overview of power system operations models with a focus on convex optimization models and their implementation in Python. Divided into two parts, the book begins with a theoretical analysis of convex optimization models before moving on to related applications in power systems operations. The author eschews concepts of topology and functional analysis found in more mathematically oriented books in favor of a more natural approach. Using this perspective, he presents recent applications of convex optimization in power system operations problems. Mathematical Programming for Power System Operation with Applications in Python uses Python and CVXPY as tools to solve power system optimization problems and includes models that can be solved with the presented framework. The book also includes: * A thorough introduction to power system operation, including economic and environmental dispatch, optimal power flow, and hosting capacity * Comprehensive explorations of the mathematical background of power system operation, including quadratic forms and norms and the basic theory of optimization * Practical discussions of convex functions and convex sets, including affine and linear spaces, politopes, balls, and ellipsoids * In-depth examinations of convex optimization, including global optimums, and first and second order conditions Perfect for undergraduate students with some knowledge in power systems analysis, generation, or distribution, Mathematical Programming for Power System Operation with Applications in Python is also an ideal resource for graduate students and engineers practicing in the area of power system optimization.
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Mathematical Programming for Power Systems Operation
Mathematical Programming for Power Systems Operation
From Theory to Applications in Python
Alejandro GarcésTechnological University of PereiraPereira, Colombia
This edition first published 2022
© 2022 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data
A catalogue record for this book is available from the Library of Congress
Paperback ISBN: 9781119747260; ePub ISBN: 9781119747284;
ePDF ISBN: 9781119747277; oBook ISBN: 9781119747291
Cover image: © Redlio Designs/Getty Images
Cover design by Wiley
Set in 9.5/12.5pt STIXTwoText by Integra Software Services Pvt. Ltd, Pondicherry, India
Contents
Cover
Title page
Copyright
Table of Contents
Acknowledgment
Introduction
1 Power systems operation
1.1 Mathematical programming for power systems operation
1.2 Continuous models
1.2.1 Economic and environmental dispatch
1.2.2 Hydrothermal dispatch
1.2.3 Effect of the grid constraints
1.2.4 Optimal power flow
1.2.5 Hosting capacity
1.2.6 Demand-side management
1.2.7 Energy storage management
1.2.8 State estimation and grid identification
1.3 Binary problems in power systems operation
1.3.1 Unit commitment
1.3.2 Optimal placement of distributed generation and capacitors
1.3.3 Primary feeder reconfiguration and topology identification
1.3.4 Phase balancing
1.4 Real-time implementation
1.5 Using Python
Part I Mathematical programming
2 A brief introduction to mathematical optimization
2.1 About sets and functions
2.2 Norms
2.3 Global and local optimum
2.4 Maximum and minimum values of continuous functions
2.5 The gradient method
2.6 Lagrange multipliers
2.7 The Newton’s method
2.8 Further readings
2.9 Exercises
3 Convex optimization
3.1 Convex sets
3.2 Convex functions
3.3 Convex optimization problems
3.4 Global optimum and uniqueness of the solution
3.5 Duality
3.6 Further readings
3.7 Exercises
4 Convex Programming in Python
4.1 Python for convex optimization
4.2 Linear programming
4.3 Quadratic forms
4.4 Semidefinite matrices
4.5 Solving quadratic programming problems
4.6 Complex variables
4.7 What is inside the box?
4.8 Mixed-integer programming problems
4.9 Transforming MINLP into MILP
4.10 Further readings
4.11 Exercises
5 Conic optimization
5.1 Convex cones
5.2 Second-order cone optimization
5.2.1 Duality in SOC problems
5.3 Semidefinite programming
5.3.1 Trace, determinant, and the Shur complement
5.3.2 Cone of semidefinite matrices
5.3.3 Duality in SDP
5.4 Semidefinite approximations
5.5 Polynomial optimization
5.6 Further readings
5.7 Exercises
6 Robust optimization
6.1 Stochastic vs robust optimization
6.1.1 Stochastic approach
6.1.2 Robust approach
6.2 Polyhedral uncertainty
6.3 Linear problems with norm uncertainty
6.4 Defining the uncertainty set
6.5 Further readings
6.6 Exercises
Part II Power systems operation
7 Economic dispatch of thermal units
7.1 Economic dispatch
7.2 Environmental dispatch
7.3 Effect of the grid
7.4 Loss equation
7.5 Further readings
7.6 Exercises
8 Unit commitment
8.1 Problem definition
8.2 Basic unit commitment model
8.3 Additional constraints
8.4 Effect of the grid
8.5 Further readings
8.6 Exercises
9 Hydrothermal scheduling
9.1 Short-term hydrothermal coordination
9.2 Basic hydrothermal coordination
9.3 Non-linear models
9.4 Hydraulic chains
9.5 Pumped hydroelectric storage
9.6 Further readings
9.7 Exercises
10 Optimal power flow
10.1 OPF in power distribution grids
10.1.1 A brief review of power flow analysis
10.2 Complex linearization
10.2.1 Sequential linearization
10.2.2 Exponential models of the load
10.3 Second-order cone approximation
10.4 Semidefinite approximation
10.5 Further readings
10.6 Exercises
11 Active distribution networks
11.1 Modern distribution networks
11.2 Primary feeder reconfiguration
11.3 Optimal placement of capacitors
11.4 Optimal placement of distributed generation
11.5 Hosting capacity of solar energy
11.6 Harmonics and reactive power compensation
11.7 Further readings
11.8 Exercises
12 State estimation and grid identification
12.1 Measurement units
12.2 State estimation
12.3 Topology identification
12.4
Y
bus
estimation
12.5 Load model estimation
12.6 Further readings
12.7 Exercises
13 Demand-side management
13.1 Shifting loads
13.2 Phase balancing
13.3 Energy storage management
13.4 Further readings
13.5 Exercises
A The nodal admittance matrix
B Complex linearization
C Some Python examples
C.1 Basic Python
C.2 NumPy
C.3 MatplotLib
C.4 Pandas
Bibliography
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1. Bounds of some ordered sets.
Chapter 3
Table 3.1. Summary of the main properties of convex optimization problems
Chapter 4
Table 4.1. Details of the node generated by the branch and bound problem.
Table 4.2. Parameters for a transportation problem with...
Chapter 5
Table 5.1. Constraints that can be transformed to SDP.
Chapter 6
Table 6.1. Dual norms for the most common cases
Chapter 7
Table 7.1 Cost functions and operative limits for a...
Chapter 8
Table 8.1. Logic table for operation...
Chapter 10
Table 10.1 34-bus test system taken from [83]
Table 10.2 Parameters of a 21-nodes DC power distribution grid
Chapter 11
Table 11.1 Parameters of the three-feeder test system for...
Table 11.2 IEEE 33 nodes test distribution network [102].
Chapter 13
Table 13.1. Three industrial process and price of the energy in each time
Table 13.2. Feasible permutations for the phase-balancing problem
Table 13.3. Expected generation, demand and price for 24h operation of a microgrid
Appendix C
Table C.1. Comparison of the installed power and increase....
Guide
Cover
Title page
Copyright
Table of Contents
Acknowledgments
Introduction
Begin Reading
A The nodal admittance matrix
B Complex linearization
C Some Python examples
Bibliography
Index
End User License Agreement
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Acknowledgment
Throughout the writing of this book, I have received a great deal of support and assistance from many people. I would first like to thank my friends Lucas Paul Perez at Welltec, Adrian Correa at Universidad Javeriana in Bogotá-Colombia, Ricardo Andres Bolaños at XM (the transmission system operator in Colombia), Raymundo Torres at Sintef-Norway, and Juan Carlos Bedoya at the Pacific Northwest National Laboratory (USA), who, in 2020 (during the COVID-19 pandemic), agreed to discuss some practical aspects associated to power system operation problems. The discussions during these video conferences were invaluable to improve the content of the book. I am also very grateful to my students, who are the primary motivation for writing this book. Special thanks to my former Ph.D. students, Danilo Montoya and Walter Julian Gil. Finally, I want to thank the Department of Electric Power Engineering at the Universidad Tecnológica de Pereira in Colombia and the Von Humbolt Foundation in Germany for the financial support required to continue my research about the operation and control of power systems.
Alejandro Garcés
Introduction
Electrification is the most outstanding engineering achievement in the 20th century, a well-deserved award if we consider the high complexity of generation, transmission, and distribution systems. An electric power system includes hundreds or even thousands of generation units, transformers, and transmission lines, located throughout an entire country and operated continuously 24 hours per day. Running such a complex system is a great challenge that requires using advanced mathematical techniques.
All industrial systems seek to increase their competitiveness by improving their efficiency. Electric power systems are not the exception. We can improve efficiency by introducing new technologies but also by implementing mathematical optimization models into daily operation. In every mathematical programming model, we require to perform four critical stages depicted in Figure . The first stage is an informed review of reality, identifying opportunities for improvement. This stage may include conversations with experts in order to establish the available data and the variables that are subject to be optimized. The second stage is the formulation of an optimization model as given below:
Where x is the vector of decision variables, f is the objective function and, Ω is the set of feasible solutions. Going from stage one (reality) to stage two (model) is more of an art than a science. One problem may have different models and different degrees of complexity. Practice and experience are required to master this stage, as some models are easier to solve than others. Subsequently, the third stage consists of the implementation of the mathematical model into a software. After that, the fourth stage is the analysis of results in the context of the real problem.
Figure 0.1 Stages of solving an optimization problem.
This book will focus on stages two and three, associated with power system operations models. In particular, we are interested in models with a geometric characteristic called convexity, that present several advantages, namely:
We can guarantee the global optimum and unique solution under well-defined conditions. This aspect is interesting from both theoretical and practical points of view. In general, a global optimum advisable in real operation problems.
There are efficient algorithms for solving convex problems. In addition, we can guarantee convergence of these algorithms. This is a critical aspect for operation problems where the algorithm requires to be solved in real-time.
There are commercial and open-source packages for solving convex optimization models. In particular, we are going to use CvxPy, a free Python-embedded modeling language for convex problems.
Many power system operations problems are already convex; for example, the economic and environmental dispatches, the hydrothermal coordination, and the load estimation problem. Besides, it is possible to find efficient convex approximations to non-convex problems such as the optimal power flow.
In summary, convex problems have both theoretical and practical advantages for power systems operation. This book studies both aspects. The book is oriented to bachelor and graduated students of power systems engineering. Concepts related to power systems analysis such as per-unit representation, the nodal admittance matrix, and the power flow problem are taken for granted. A previous course of linear programming is desirable but not mandatory. We do not pretend to encompass all the theory behind convex optimization; instead, we try to present particular aspects of convex optimization which are useful in power systems operation. The book is divided into two parts: In the first part, the main concepts of convex optimization are presented, including a distinct chapter about conic optimization. After that, selected applications for power systems operation are presented. Most of the solvers for convex optimization allow mixed-integer convex problems. Therefore, we include models that can be solved in this framework too. The student is recommended to do numerical experiments in order to acquire practical intuition of the problems.
All applications are presented in Python, which is a language that is becoming more important in power systems applications. Students are not expected to have previous knowledge in Python, although basic concepts about programming (in any language) are helpful. Our methodology is based on many examples and toy-models
