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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: IEEE Press Series on Power Engineering
- Sprache: Englisch
Interval Methods for Uncertain Power System Analysis In Interval Methods for Uncertain Power System Analysis, accomplished engineer Dr. Alfredo Vaccaro delivers a comprehensive discussion of the mathematical foundations of range analysis and its application to solving traditional power system operation problems in the presence of strong and correlated uncertainties. The book explores highly relevant topics in the area, from interval methods for uncertainty representation and management to a variety of application examples. The author offers readers the latest methodological breakthroughs and roadmaps to implementing the mathematics discussed within, as well as best practices commonly employed across the industry. Interval Methods for Uncertain Power System Analysis includes examinations of linear and non-linear equations, as well as: * A thorough introduction to reliable computing, including discussions of interval arithmetic and interval-based operators * Comprehensive explorations of uncertain power flow analysis, including discussions of problem formulation and sources of uncertainty in power flow analysis * In-depth examinations of uncertain optimal power flow analysis * Fulsome discussions of uncertain small signal stability analysis, including treatments of how to compute eigenvalues of uncertain matrices Perfect for engineers working in power flow and optimal power flow analyses, optimization theory, and computer aided simulation, Interval Methods for Uncertain Power System Analysis will also earn a place in the libraries of researchers and graduate students studying decision making under uncertainty in power systems operation.
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Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Title Page
Copyright
Dedication
About the Author
Preface
Acknowledgments
Acronyms
Introduction
1 Introduction to Reliable Computing
1.1 Elements of Reliable Computing
1.2 Interval Analysis
1.3 Interval-Based Operators
1.4 Interval Extensions of Elementary Functions
1.5 Solving Systems of Linear Interval Equations
1.6 Finding Zeros of Nonlinear Equations
1.7 Solution of Systems of Nonlinear Interval Equations
1.8 The Overestimation Problem
1.9 Affine Arithmetic
1.10 Integrating AA and IA
2 Uncertain Power Flow Analysis
2.1 Sources of Uncertainties in Power Flow Analysis
2.2 Solving Uncertain Linearized Power Flow Equations
2.3 Solving Uncertain Power Flow Equations
3 Uncertain Optimal Power Flow Analysis
3.1 Range Analysis-Based Solution
3.2 AA-Based Solution
4 Uncertain Markov Chain Analysis
4.1 Mathematical Preliminaries
4.2 Effects of Data Uncertainties
4.3 Matrix Notation
4.4 AA-Based Uncertain Analysis
4.5 Application Examples
5 Small-Signal Stability Analysis of Uncertain Power Systems
5.1 Problem Formulation
5.2 The Interval Eigenvalue Problem
5.3 Applications
6 Uncertain Power Components Thermal Analysis
6.1 Thermal Rating Assessment of Overhead Lines
6.2 Thermal Rating Assessment of Power Cables
References
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Bus data.
Table 2.2 Line data.
Table 2.3 Bus data.
Table 2.4 Line data.
Table 2.5 Bounds of the PF solution.
Table 2.6 Bounds of the PF solution.
Table 2.7 Deterministic PF solution.
Table 2.8 Sensitivity coefficients of the bus voltage magnitudes.
Table 2.9 Sensitivity coefficients of the bus voltage angles.
Table 2.10 Bounds of the PF solution.
Chapter 3
Table 3.1 Solution bounds of the optimal economic dispatch problem.
Chapter 4
Table 4.1 Transition rate matrix.
Table 4.2 Transition rate matrix.
Chapter 5
Table 5.1 Eigenvalues bounds.
Table 5.2 Eigenvalues bounds.
Chapter 6
Table 6.1 Simulation data.
Table 6.2 Thermophysical soil parameters.
Table 6.3 Thermophysical properties of the power cable components.
Table 6.4 Thermophysical properties of the soil.
List of Illustrations
Chapter 1
Figure 1.1 The error explosion problem.
Figure 1.2 Joint range of two partially dependent quantities in AA.
Figure 1.3 Affine approximation of the function .
Chapter 2
Figure 2.1 Example of a three-bus power system.
Figure 2.2 Joint range of the power flow solutions.
Chapter 3
Figure 3.1 Solution bounds – bus voltage magnitudes.
Figure 3.2 Solutions bounds – bus voltage angle.
Figure 3.3 Solution bounds – bus voltage magnitude.
Figure 3.4 Solution bounds – bus voltage angle.
Chapter 4
Figure 4.1 Case Study 1: Transient state probabilities bounds computed by AA...
Figure 4.2 Case Study 2: Monte-Carlo and Affine Arithmetic bounds, uncertain...
Chapter 6
Figure 6.1 Deterministic profiles of the wind speed and direction.
Figure 6.2 Deterministic profiles of the environmental temperature and solar...
Figure 6.3 Bounds of the static thermal ratings computed by AA, IA, and Mont...
Figure 6.4 Bounds of the conductor temperature for a step load change of 900...
Figure 6.5 Thermal modeling of the cell.
Figure 6.6 Equivalent thermal model of the power cables.
Figure 6.7 Schematic of the analyzed power cable system.
Figure 6.8 Time evolution of the environmental temperature.
Figure 6.9 Time evolution of the load current.
Figure 6.10 Time evolution of the hot-spot temperature.
Guide
Cover Page
Title Page
Copyright
Dedication
About the Author
Preface
Acknowledgments
Acronyms
Introduction
Table of Contents
Begin Reading
References
Index
End User License Agreement
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IEEE Press445 Hoes LanePiscataway, NJ 08854
IEEE Press Editorial BoardSarah Spurgeon, Editor in Chief
Jón Atli BenediktssonAnjan BoseJames DuncanAmin MoenessDesineni Subbaram Naidu
Behzad RazaviJim LykeHai LiBrian Johnson
Jeffrey ReedDiomidis SpinellisAdam DrobotTom RobertazziAhmet Murat Tekalp
Interval Methods for Uncertain Power System Analysis
Alfredo Vaccaro
University of Sannio
Italy
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Library of Congress Cataloging-in-Publication Data:
Names: Vaccaro, Alfredo, author. | John Wiley & Sons, publisher.
Title: Interval methods for uncertain power system analysis / Alfredo Vaccaro.
Description: Hoboken, New Jersey : Wiley-IEEE Press, [2023] | Includes index.
Identifiers: LCCN 2023014418 (print) | LCCN 2023014419 (ebook) | ISBN 9781119855040 (cloth) | ISBN 9781119855057 (adobe pdf) | ISBN 9781119855064 (epub)
Subjects: LCSH: Electric power systems. | Electric power systems–Mathematical models. | Electric power systems–Reliability. | System analysis.
Classification: LCC TK1005 .V25 2023 (print) | LCC TK1005 (ebook) | DDC 621.3101/5118–dc23/eng/20230506
LC record available at https://lccn.loc.gov/2023014418LC ebook record available at https://lccn.loc.gov/2023014419
Cover Design: WileyCover Image: © Sukpaiboonwat/Shutterstock
To my family
About the Author
Alfredo Vaccaro, PhD, is a full professor of electric power systems at the Department of Engineering of University of Sannio. He is the editor-in-chief of Smart Grids and Sustainable Energy, Springer Nature and associate editor of IEEE trans. on Smart Grids, IEEE trans. on Power Systems, and IEEE Power Engineering Letters.
Preface
This book summarizes the main results of my research activities in the field of uncertain power system analysis by interval methods.
I started working on this interesting and challenging issue in early 2000, inspired by the papers of Prof. Fernando Alvarado about the application of interval arithmetic in uncertain power flow analysis. The first contributions were focused on mitigating the effects of the “dependency problem” and “wrapping effect” in interval analysis, which can reduce the value of the results by overestimating the bounds of the power flow solutions. This overestimation problem has been observed when solving many conventional power system operation problems by “naive” interval analysis, which may lead to aberrant solutions, due to the inability of interval arithmetic to model the correlations between the uncertain variables. Consequently, each step of the algorithm introduces spurious values, causing the solution bounds to converge to overconservative values. This phenomena has been extensively studied in qualitative systems analysis and requires the use of complex and time-consuming preconditioning techniques.
This stimulated the research into alternative interval methods based on Affine Arithmetic, which is one of the main topics of this book.
In this approach, each uncertain variable is described by a first-degree polynomial composed of a central value and a number of partial deviations, each one modeling the effect of an independent source of uncertainty.
The adoption of Affine Arithmetic makes it possible to solve uncertain mathematical programming problems and obtain a reliable estimation of the solution hulls by including the effect of correlation between the uncertain variables, as well as the heterogeneity of the sources of uncertainty.
This reliable computing paradigm has been successfully deployed to solve a large number of power system operation problems in the presence of data uncertainty. The examples presented in this book cover power flow studies, optimal power flow analysis, dynamic thermal rating assessment, state estimation, and stability analysis.
The obtained results demonstrate the effectiveness of interval methods to solve the complex uncertain problems encountered when analyzing realistic power system operation scenarios, hence, making these methods one of the most promising alternatives for stochastic information management in modern power systems.
May 2023
Alfredo Vaccaro
Benevento, Italy
Acknowledgments
I wish to express my sincere gratitude to my mentor Prof. Claudio Canizares, who inspired and stimulated my research activities in the field of uncertain power system analysis.
I would also like to thank Prof. Kankar Bhattacharya for his valuable and qualified suggestions about the potential role of range analysis in market studies, and Dr. Adam J. Collin and Dr. Fabrizio De Caro for their valuable support in reviewing this book.
Alfredo Vaccaro
Acronyms
AA
affine arithmetic
DTR
dynamic thermal rating
IA
interval analysis
IM
interval mathematics
MC
Markov Chain
OPF
optimal power flow
PF
power flow
Introduction
Power system analysis is often affected by large and correlated uncertainties, which could seriously affect the validity of the obtained results. Uncertainties in modern power systems stem from both internal and external factors, including model inaccuracies, measurement errors, inconsistent data, and imprecise knowledge about some input information.
Conventional methods for uncertainty modeling in power system analysis involve using probabilistic techniques to characterize the variability in input data and sampling-based approaches to simulate the system behavior for a large number of possible operating scenarios.
However, relying on probabilistic methods has its limitations, as power system engineers may struggle to express their imprecise knowledge about certain input variables with probability distributions (due to the subjective and qualitative nature of their expertise), and, more generally, there may be a lack of reliable data for characterizing the probability parameters. Additionally, the use of probabilistic methods often requires the assumptions of normal distributions and statistical independence, e.g. in the case of weather variables, but operational experience shows that these assumptions are frequently unsupported by empirical evidence.
Recent advances in the field have expanded the range of methods for addressing uncertainty by introducing a number of nonprobabilistic techniques, such as interval analysis, fuzzy arithmetic, and evidence theory.
Nonprobabilistic techniques are often used when uncertainty arises from limitations in our understanding of the system, rather than unpredictable numerical data. In these cases, only rough estimates of values and relationships between variables are available. Consider, for example the power profiles generated by small and dispersed renewable generators, which strictly depend on the evolution of some weather variables, e.g. solar radiation for photovoltaic generators and wind speed and direction for wind turbines. Although these variables can be measured at a specific location, it is challenging to determine their distributions over a wide geographical area. Nonetheless, weather forecasts periodically provide qualitative information about the expected evolution of the weather variables on different time horizons, but these predicted profiles cannot be easily expressed as probabilities.
It follows that the availability of reliable frameworks for modeling and managing nonprobabilistic knowledge can greatly enhance the robustness and effectiveness of power system analysis.
For this purpose, interval methods have been recognized as an enabling methodology for uncertain power system analysis.
The primary benefit of these techniques is that they intrinsically keep track of the computing accuracy of each elementary mathematical operation, without needing information or assumptions about the input parameter uncertainties.
The simplest and most commonly used of these models is interval mathematics, which enables numerical computations in which each value is represented by a range of floating-point numbers without a probability structure. These intervals are processed by proper addition, subtraction, and/or multiplication operators, ensuring that each computed interval encloses the unknown value it represents.
Many analysts view interval mathematics as a subset of fuzzy theory, as interval variables can be viewed as a specific instance of fuzzy numbers. However, connecting interval mathematics to fuzzy set theory is not straightforward. Recently, fuzzy set theory and interval analysis have both been linked to a broader topological theory. Similarly, it is argued that fuzzy information granulation, rough set theory, and interval analysis are all subsets of a larger computational paradigm called granular computing, where these methodologies are complementary and symbiotic, rather than conflicting and exclusive.
This book focuses on the application of interval methods in uncertain power system analysis. In particular, after introducing the basic elements of interval computing, a set of conventional power system operation problems in the presence of data uncertainties are formalized and solved. Many numerical examples are presented and discussed in order to demonstrate the effectiveness of interval methods to reliably solve the complex problems encountered when analyzing the realistic operating scenarios of modern power systems.
1Introduction to Reliable Computing
Many power-engineering computations, especially those related to system analysis, are affected by large and complex uncertainties. These uncertainties make it difficult to compute the “exact” problem solution, and the analyst is required to identify a proper approximated solution, which is as close as possible to the “exact” one. The difference between the “exact” and the approximated solution is commonly referred to as the solution “error.”
The sources of uncertainties affecting power system analysis are multiple and heterogeneous, and can be both external and internal to the computing process. External uncertainties include measurement errors, missing data, and simplified models; while internal uncertainties are mainly related to the computing errors induced by digital processing, which frequently requires replacing rigorous mathematical models with discrete approximations (e.g. time discretization, truncation, and round-off errors).
Hence, when integrating the results of numerical analysis in power system operation tools, the impact of these uncertainties must be assessed and a formal error analysis should be considered an important part of the development process. The objective of such formal error analysis is to comprehensively assess the accuracy of all the computations involved, define the magnitude of the solution errors as a function of the values and the errors of the input data (i.e. variables and parameters).
Unfortunately, defining a formal mathematical process for specifying the accuracy of numerical computing algorithms is extremely complex, since estimating the propagation of both the external and internal errors for all the basic operations composing the computing process is often unfeasible, even for simple algorithms.
Moreover, accuracy specification requires the compliance of the input data with a set of strict prerequisites (e.g. well-conditioned matrices, no overflow, and functions with bounded derivatives). Guaranteeing or even checking these prerequisites for all the possible combinations of the input data represents another challenging issue to address.
Hence, power system analysts frequently deploy numerical algorithms without formal accuracy specifications and rigorous error analysis, checking the consistency of the obtained results on the basis of their own experience or by crude tests. This practice could hinder the integration of approximate numerical computing in modern power systems tools, which are characterized by the presence of large data uncertainties, stemming from multiple and heterogeneous sources.
To try and overcome this limitation, the power system research community started adopting reliable computing-based models, which allow the accuracy of the computed quantities to be automatically estimated as part of the process of computing them. The application of these models in numerical computations is also referred to as self-validated computing, since it can estimate “a posteriori” the error magnitude of the entire computing process (Stolfi and De Figueiredo, 1997). This feature is extremely important, especially when the data uncertainties are induced by external causes. In this case, if the output errors computed by the self-validated model become too large, i.e. overcoming a fixed acceptable threshold, then specific remedial actions can be automatically triggered in order to enhance the model accuracy (e.g. acquire more data, re-adjourn the input parameters, and use more accurate models).
1.1 Elements of Reliable Computing
Let be a continuous mathematical function, and suppose we need to compute for . For this, we should implement a discrete numerical computation , where and are discrete mathematical objects approximating the corresponding continuous variables and .
To solve this issue different reliable computing models can be adopted, including probability distributions and range-based methods.
In particular, the adoption of probability distributions can approximate, in a statistical sense, the computed result by considering each component of the vector as a real-value random variable, whose probability distribution function is frequently assumed to follow a Gaussian distribution. In this case, a reliable computing model should specify the statistical moments of each component , and the corresponding covariance matrix describing the joint Gaussian probability distribution of the random vector , given those characterizing the random input variables .
The application of this probabilistic-based reliable computing model is often limited to specific application domains, which are characterized by Gaussian uncertainties, linear mappings, and negligible truncation errors. The lack of these conditions makes the statistical characterization of the computed result extremely complex or even mathematically intractable (Stolfi and De Figueiredo, 1997).
To try and overcome this limitation, most reliable computing models approximate the computed results by ranges, rather than by probability distributions.
According to these models, the approximated solution is described by means of its range , which is a compact set containing the “exact” solutions for all the input variables lying in the range .
This important feature, which is usually referred to as the fundamental invariant of range analysis, guarantees that the range
