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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley-IEEE Press Book Series on Control Systems Theory and Applications
- Sprache: Englisch
Merging Optimization and Control in Power Systems
A novel exploration of distributed control in power systems with insightful discussions of physical and cyber restrictions
In Merging Optimization and Control in Power Systems an accomplished team of engineers deliver a comprehensive introduction to distributed optimal control in power systems. The book re-imagines control design within the framework of cyber-physical systems with restrictions in both the physical and cyber spaces, addressing operational constraints, non-smooth objective functions, rapid power fluctuations caused by renewable generations, partial control coverage, communication delays, and non-identical sampling rates.
This book bridges the gap between optimization and control in two ways. First, optimization-based feedback control is explored. The authors describe feedback controllers which automatically drive system states asymptotically to specific, desired optimal working points. Second, the book discusses feedback-based optimization. Leveraging the philosophy of feedback control, the authors envision the online solving of complicated optimization and control problems of power systems to adapt to time-varying environments.
Readers will also find:
- A thorough argument against the traditional and centralized hierarchy of power system control in favor of the merged approach described in the book
- Comprehensive explorations of the fundamental changes gripping the power system today, including the increasing penetration of renewable and distributed generation, the proliferation of electric vehicles, and increases in load demand
- Data, tables, illustrations, and case studies covering realistic power systems and experiments
- In-depth examinations of physical and cyber restrictions, as well as the robustness and adaptability of the proposed model
Perfect for postgraduate students and researchers with the prerequisite knowledge of power system analysis, operation, and dynamics, convex optimization theory, and control theory, Merging Optimization and Control in Power Systems is an advanced and timely treatment of distributed optimal controller design.
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Veröffentlichungsjahr: 2022
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Table of Contents
Cover
Title Page
Copyright
Dedication
Books in the IEEE Press Series on Control Systems Theory and Applications
Foreword
Preface
Acknowledgments
1 Introduction
1.1 Traditional Hierarchical Control Structure
1.2 Transitions and Challenges
1.3 Removing Central Coordinators: Distributed Coordination
1.4 Merging Optimization and Control
1.5 Overview of the Book
Bibliography
Notes
2 Preliminaries
2.1 Norm
2.2 Graph Theory
2.3 Convex Optimization
2.4 Projection Operator
2.5 Stability Theory
2.6 Passivity and Dissipativity Theory
2.7 Power Flow Model
2.8 Power System Dynamics
Bibliography
Notes
3 Bridging Control and Optimization in Distributed Optimal Frequency Control
3.1 Background
3.2 Power System Model
3.3 Design and Stability of Primary Frequency Control
3.4 Convergence Analysis
3.5 Case Studies
3.6 Conclusion and Notes
Bibliography
Notes
4 Physical Restrictions: Input Saturation in Secondary Frequency Control
4.1 Background
4.2 Power System Model
4.3 Control Design for Per‐Node Power Balance
4.4 Optimality and Uniqueness of Equilibrium
4.5 Stability Analysis
4.6 Case Studies
4.7 Conclusion and Notes
Bibliography
Notes
5 Physical Restrictions: Line Flow Limits in Secondary Frequency Control
5.1 Background
5.2 Power System Model
5.3 Control Design for Network Power Balance
5.4 Optimality of Equilibrium
5.5 Asymptotic Stability
5.6 Case Studies
5.7 Conclusion and Notes
Bibliography
Notes
6 Physical Restrictions: Nonsmoothness of Objective Functions in Load‐Frequency Control
6.1 Background
6.2 Notations and Preliminaries
6.3 Power System Model
6.4 Control Design
6.5 Optimality and Convergence
6.6 Case Studies
6.7 Conclusion and Notes
Bibliography
7 Cyber Restrictions: Imperfect Communication in Power Control of Microgrids
7.1 Background
7.2 Preliminaries and Model
7.3 Distributed Control Algorithms
7.4 Optimality and Convergence Analysis
7.5 Real‐Time Implementation
7.6 Numerical Results
7.7 Experimental Results
7.8 Conclusion and Notes
Bibliography
Notes
8 Cyber Restrictions: Imperfect Communication in Voltage Control of Active Distribution Networks
8.1 Background
8.2 Preliminaries and System Model
8.3 Problem Formulation
8.4 Asynchronous Voltage Control
8.5 Optimality and Convergence
8.6 Implementation
8.7 Case Studies
8.8 Conclusion and Notes
Bibliography
Notes
9 Robustness and Adaptability: Unknown Disturbances in Load‐Side Frequency Control
9.1 Background
9.2 Problem Formulation
9.3 Controller Design
9.4 Equilibrium and Stability Analysis
9.5 Robustness Analysis
9.6 Case Studies
9.7 Conclusion and Notes
Bibliography
Notes
10 Robustness and Adaptability: Partial Control Coverage in Transient Frequency Control
10.1 Background
10.2 Structure‐Preserving Model of Nonlinear Power System Dynamics
10.3 Formulation of Optimal Frequency Control
10.4 Control Design
10.5 Optimality and Stability
10.6 Implementation With Frequency Measurement
10.7 Case Studies
10.8 Conclusion and Notes
Bibliography
Notes
11 Robustness and Adaptability: Heterogeneity in Power Controls of DC Microgrids
11.1 Background
11.2 Network Model
11.3 Optimal Power Flow of DC Networks
11.4 Control Design
11.5 Implementation
11.6 Case Studies
11.7 Conclusion and Notes
Bibliography
Note
Appendix A Typical Distributed Optimization Algorithms
A.1 Consensus‐Based Algorithms
A.2 First‐Order Gradient‐Based Algorithms
A.3 Second‐Order Newton‐Based Algorithms
A.4 Zeroth‐Order Online Algorithms
Bibliography
Appendix B Optimal Power Flow of Direct Current Networks
B.1 Mathematical Model
B.2 Exactness of SOC Relaxation
B.3 Case Studies
B.4 Discussion on Line Constraints
Bibliography
Index
End User License Agreement
List of Tables
Chapter 4
Table 4.1 System parameters.
Table 4.2 Capacity limits and load disturbance.
Table 4.3 Equilibrium points.
Chapter 5
Table 5.1 System parameters.
Table 5.2 Capacity limits in network case.
Table 5.3 Equilibrium points.
Table 5.4 Simulation results with congestion.
Chapter 7
Table 7.1 System parameters.
Chapter 8
Table 8.1 Comparison between the regulations with and without active power....
Chapter 9
Table 9.1 Parameters used in the controller (9.11).
Table 9.2 Parameters used in the controller (9.11).
Chapter 10
Table 10.1 Capacity limits of generators.
Table 10.2 Equilibrium points.
Chapter 11
Table 11.1 System parameters.
Table 11.2 Comparision with centralized optimization.
Appendix B
Table B.1 Results of 16‐bus system.
Table B.2 Results of power injections and nodal voltages.
Table B.3 Exactness of RLS1 and comparison of two models.
List of Illustrations
Chapter 1
Figure 1.1 Time‐scale decomposition of controls in a traditional power syste...
Figure 1.2 Diagram of hierarchical frequency control.
Figure 1.3 Diagram of hierarchical voltage control.
Figure 1.4 Diagram of centralized (a), decentralized (b), and distributed (c...
Figure 1.5 Conceptual diagram of optimization‐guided feedback control.
Figure 1.6 Conceptual diagram of feedback‐based optimization.
Chapter 2
Figure 2.1 An illustrative graph with five nodes.
Figure 2.2 Illustration of (non)convex sets.
Figure 2.3 (a) The convex hull of 10 points. (b) The convex hull of a star....
Figure 2.4 Illustration of the normal cone.
Figure 2.5 Illustration of the tangent cone.
Figure 2.6 Illustration of the second‐order cone. The cone is
.
Figure 2.7 Illustration of a convex function.
Figure 2.8 Illustration of function
.
Figure 2.9 The projection of a point onto a closed convex set.
Figure 2.10 The illustration of the invariance principle.
Figure 2.11 Summary of notations at a generator bus.
Figure 2.12 Block diagram of DG inverters.
Chapter 3
Figure 3.1 Schematic of a generator bus
, where
is the frequency deviatio...
Figure 3.2
is the set on which
,
is the set of equilibrium points of (3...
Figure 3.3 Single line diagram of the IEEE 68‐bus test system.
Figure 3.4 The (a) frequency and (b) voltage on bus 66 for the following cas...
Figure 3.5 The (a) new steady‐state frequency, (b) lowest frequency, and (c)...
Figure 3.6 The cost trajectory of OLC (solid line) compared with the minimum...
Chapter 4
Figure 4.1 Four‐area power system.
Figure 4.2 Closed‐loop system diagram.
Figure 4.3 Frequency dynamics in network balance case.
Figure 4.4 Tie‐line power dynamics in network balance case.
Figure 4.5 Mechanical outputs with/without capacity constraints.
Figure 4.6 Controllable loads with/without capacity constraints.
Figure 4.7 Dynamics of frequency with proposed controller and AGC.
Figure 4.8 Dynamics of mechanical power with different update rates.
Figure 4.9 Dynamics of controllable load with different update rates.
Chapter 5
Figure 5.1 Four‐area power system.
Figure 5.2 Closed‐loop system diagram.
Figure 5.3 Frequency dynamics in network balance case.
Figure 5.4 Tie‐line power dynamics in network balance case.
Figure 5.5 Mechanical outputs (a) with/(b) without capacity constraints.
Figure 5.6 Controllable loads (a) with/(b) without capacity constraints.
Figure 5.7 Frequency dynamics with proposed controller and AGC. (a) Frequenc...
Figure 5.8 Tie‐line power (a) with/(b) without capacity constraints.
Figure 5.9 Mechanical outputs with different time delays. (a) With 100 ms de...
Figure 5.10 Controllable loads with different time delays. (a) With 100 ms d...
Chapter 6
Figure 6.1 Distributed control architecture.
Figure 6.2 The IEEE 68‐bus system for test.
Figure 6.3 Comparison of the frequency dynamics under AGC and OLC.
Figure 6.4 Dynamics of
.
Figure 6.5 Dynamics of controllable loads.
Figure 6.6 Active power dynamics of line
and controllable load.
Figure 6.7 Dynamics of
when line power congestion exists. (a). Nodes near ...
Chapter 7
Figure 7.1 Local clocks versus global clock.
Figure 7.2 Simplified model of a DC MG.
Figure 7.3 Control configuration of the proposed RTASDPD algorithm.
Figure 7.4 A schematic diagram of a typical 43‐bus MG system.
Figure 7.5 Communication graph of the system.
Figure 7.6 Dynamics of frequencies. For a DC MG, its frequency implies the f...
Figure 7.7 Dynamics of voltages.
Figure 7.8 Dynamics of generations.
Figure 7.9 Dynamics of
.
Figure 7.10 Frequencies under different/varying time delays.
Figure 7.11 Generations under different/varying time delays.
Figure 7.12 Dynamics of frequencies under synchronous and asynchronous cases...
Figure 7.13 Dynamics of generations under synchronous and asynchronous cases...
Figure 7.14 Experiment platform based on dSPACE RTI 1202 controller.
Figure 7.15 Topology of the experiment system.
Figure 7.16 Dynamics of frequencies in the experiments. The computed power v...
Figure 7.17 Dynamics of power generations in the experiments. The black line...
Figure 7.18 Comparison between the RTASDPD and the SDPD under different time...
Chapter 8
Figure 8.1 Two‐step communications.
Figure 8.2 The graph of the 8‐bus distribution network.
Figure 8.3 Comparison of algorithm convergence in terms of number of average...
Figure 8.4 Algorithm convergence with non‐strongly objective function.
Figure 8.5 Convergence with different time (a) delays and (b)
. The time de...
Figure 8.6 IEEE 123‐bus system.
Figure 8.7 Active and reactive loads and solar generation through the 24 h....
Figure 8.8 Daily voltage mismatch under the SDVC and the ASDVC in the presen...
Chapter 9
Figure 9.1 The New England 39‐bus system.
Figure 9.2 Communication graph.
Figure 9.3 Diagram of the closed‐loop system.
Figure 9.4 Power variation of renewable resources in each area.
Figure 9.5 System frequencies under two controls.
Figure 9.6 Dynamics of
.
Figure 9.7 Dynamics of steady parts of controllable loads.
Figure 9.8 Variation and internal model output with load increases.
Figure 9.9 Variation and internal model output with noise.
Figure 9.10 Frequency under external noise.
Figure 9.11 Wind power in each area.
Figure 9.12 Frequency dynamics with real data.
Figure 9.13 Variation and internal model output with real wind data.
Figure 9.14 Frequency dynamics compared with (a) the droop control and (b) t...
Chapter 10
Figure 10.1 Summary of notations.
Figure 10.2 The New England 39‐bus system.
Figure 10.3 Diagram of the closed‐loop system.
Figure 10.4 Frequency dynamics with (a) distributed controller and (b) AGC....
Figure 10.5 Dynamics of frequencies with the controller in [6].
Figure 10.6 Dynamics of mechanical powers under AGC. (a) Power of controllab...
Figure 10.7 Dynamics of mechanical powers under the proposed control. (a) Po...
Figure 10.8 Dynamics of bus voltages under load changes.
Figure 10.9 Dynamics of
under load changes.
Figure 10.10 Dynamics of
under load changes.
Figure 10.11 Dynamics of frequencies under a generator tripping.
Figure 10.12 Dynamics of mechanical powers under a generator tripping.
Figure 10.13 Dynamics of frequencies under a line tripping.
Figure 10.14 Dynamics of voltage under a line tripping.
Figure 10.15 Dynamics of
under a line tripping.
Figure 10.16 Dynamics of mechanical powers under a line tripping.
Chapter 11
Figure 11.1 The proposed distributed optimal control diagram that unifies th...
Figure 11.2 Six‐microgrid system.
Figure 11.3 Simulation design combined with PSCAD and Matlab.
Figure 11.4 Generation dynamics (a) with and (b) without constraints.
Figure 11.5 Voltage dynamics (a) with and (b) without constraints.
Figure 11.6 Generation dynamics of different constraints.
Figure 11.7 Generation dynamics under different time delays.
Figure 11.8 Voltage dynamics under different time delays.
Figure 11.9 Generation dynamics of plug and play.
Figure 11.10 Voltage dynamics of plug and play.
Appendix A
Figure A.1 An illustration of the individual and global feasible regions.
Figure A.2 The framework of the zeroth‐order online algorithm.
Appendix B
Figure B.1 Summary of notations.
Figure B.2 Relationship between different feasible sets.
Figure B.3 16‐bus system with DGs.
Figure B.4 Geometrical interpretation of rank and line constraints. (a) Rank...
Guide
Cover Page
Table of Contents
Title Page
Copyright
Dedication
Foreword
Begin Reading
Appendix A Typical Distributed Optimization Algorithms
Appendix B Optimal Power Flow of Direct Current Networks
Index
End User License Agreement
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IEEE Press
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Anjan Bose
Saeid Nahavandi
Ahmet Murat Tekalp
Adam Drobot
Jeffrey Reed
Peter (Yong) Lian
Thomas Robertazzi
Merging Optimization and Control in Power Systems
Physical and Cyber Restrictions in Distributed Frequency Control and Beyond
Feng Liu
Tsinghua University
Zhaojian Wang
Shanghai Jiao Tong University
Changhong Zhao
The Chinese University of Hong Kong
Peng Yang
Tsinghua University
IEEE Press Series on Control Systems Theory and Applications
Maria Domenica Di Benedetto, Series Editor
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To our families
Books in the IEEE Press Series on Control Systems Theory and Applications
Series Editor: Maria Domenica DiBenedetto, University of l'Aquila, Italy
The series publishes monographs, edited volumes, and textbooks which are geared for control scientists and engineers, as well as those working in various areas of applied mathematics such as optimization, game theory, and operations.
Autonomous Road Vehicle Path Planning and Tracking Control
Levent Güvenç, Bilin Aksun‐Güvenç, Sheng Zhu, Şükrü Yaren Gelbal
Embedded Control for Mobile Robotic Applications
Leena Vachhani, Pranjal Vyas, and Arunkumar G. K.
Merging Optimization and Control in Power Systems: Physical and Cyber Restrictions in Distributed Frequency Control and Beyond
Feng Liu, Zhaojian Wang, Changhong Zhao, and Peng Yang.
Foreword
With the ever‐increasing penetration of distributed energy resources (DERs), such as distributed generations (DGs), electric vehicles (EVs), and customer energy storages (CESs), the centralized, hierarchical control architecture developed for the traditional power system faces unprecedented challenges arising from massive but fragmented controllable resources. On the one hand, it might be unwise to regulate millions of individual DERs dispersed across a vast land. On the other hand, such an architecture heavily depends on intensive communication and usually slows down the control response, which may not cope with the fast variation of renewable generations. These issues bring potential risks to the stability and economy of power system operation. In this regard, distributed control and distributed optimization approaches are expected to supplement the existing control structure to relieve the dependence on the control center by leveraging neighboring communications among controllable (aggregate) DERs.
Conventionally, the two approaches are investigated separately due to different goals and time scales. Specifically, the distributed control focuses on fast‐time‐scale stability problems like primary and secondary control, while the distributed optimization addresses the slow‐time‐scale optimization problems like economic dispatch and energy trading. However, our power system desires a coalescence of the two approaches to effectively organize and utilize the massive but fragmented controllable resources to facilitate stable and economic operation.
This is the first book to systematically explore the integration of optimization and control in power systems toward a distributed paradigm. Rooted in the idea of “forward and reverse engineering,” the book suggests the methodologies of “optimization‐guided control” and “feedback‐based optimization.” It envisions the distributed optimal control of power systems with appealing design frameworks, particularly when considering complicated constraints enforced by physical and cyber restrictions. This book also gives rigorous proofs with in‐depth insights and rich demonstrative examples from practical applications, which may benefit both theoretical research and industrial deployment.
The authors are all professional and reputational young researchers who have accomplished impactful works in theories and applications of distributed control and optimization in power systems, ranging from transmission and distribution grids to microgrids. I am therefore delighted to see that they summarize their recent inspiring works and shed new light on the control design of power systems at the cutting edge of transition. I believe it is an outstanding reference for graduate students, researchers, and engineers interested in power system control and optimization.
Chengshan Wang
Director of School of Electrical and
Information Engineering, Tianjin University
Academician of Chinese Academy of Engineering
Preface
As a worldwide trend, our power system is undergoing a historical transition from the traditional to a new landscape with high penetration of renewable energies and power electronics. As P. W. Anderson said about the structure of science in 1972, “more is different.” Not surprisingly, the shift of our power system follows the same principle.
With the explosion of renewable generation, millions of small‐capacity, volatile, and heterogeneous renewable generators, such as wind‐turbine generators and photovoltaics, have been encroaching on the dominion of a few large‐capacity traditional generators. Moreover, the number is rapidly ever growing every day. With the new era looming, one natural but critical question arises: Can the current, top‐down, central‐dominant control architecture established for the traditional power system still cope with the transition? Despite a broad spectrum of opinions, at least one general agreement could be reached from disagreement. That is, the future power system calls for reshaping the control paradigm to make it more scalable to the vast number of devices, more compatible with ever‐increasing heterogeneity, more efficient in fast‐varying operation environments, more adaptable to diverse operation modes, and more robust against unexpected perturbations and even failures.
Recent years have witnessed remarkable progress in distributed control theory, convex optimization, measurement and communication technologies, and advanced power electronics engineering. The cross‐fertilization among these terrific successes further creates potential opportunities to meet the challenges mentioned above. The two most common might be the following: (i) relieving the strong dependence on a central coordinator and (ii) reshaping the top‐down architecture by breaking the original control hierarchy. The former advocates a distributed control paradigm to endow the power system with higher scalability, compatibility, and robustness, while the latter suggests emerging the slow‐time‐scale optimization and fast‐time‐scale control to achieve stronger adaptability and faster response. As a consequence, many fruitful and innovative researches have been conducted to date. This book is not an account of all the leading‐edge achievements but the personal efforts of the authors alongside, focusing on how to address the physical and cyber restrictions in distributed control design for meeting the requirement of future power systems.
Particularly, in this book, we concentrate the attention in frequency control as the ever‐increasing renewables and power electronics lead to a dramatic drop of system inertia, raising great concerns with frequency stability and system security. However, this book also stretches out to part of voltage control and beyond, involving different levels/sizes of power systems, from high‐voltage transmission systems and middle/low‐voltage distribution systems to microgrids. Advanced mathematical tools, design methodologies, and illustrative examples with numerical/experimental simulations are systematically organized, aiming to provide the reader with a vision on how to emerge optimization and control in power systems under the distributed control paradigm with special considerations posed by diverse physical and cyber restrictions.
This book is organized into four parts. Part I (Chapters 1–2) presents the introduction and necessary preliminaries in optimization theory, control theory, and power system modeling. Part II (Chapters 3–6) demonstrates how to bridge optimization and control to devise optimization‐guided distributed frequency control from both “forward engineering” and “reverse engineering” perspectives. Physical restrictions arising from operational constraints and objective functions are addressed with rigorous proof. Part III (Chapters 7–8) is dedicated to designing distributed frequency and voltage control with different cyber restrictions due to imperfect communication, where the feedback‐based optimization is mostly involved in. Lastly, Part IV (Chapters 9–11) extends to the scope of robustness and adaptability of distributed optimal control, such as the robustness against unknown disturbances, flexibility to partial control coverage, and adaptability to heterogeneous control configurations.
Hence, this book is intended to be of value to graduate students and researchers who work in theory and applications of distributed control and optimization, especially their applications in power systems. This also can be used as a reference by electrical engineers to improve industry practices in power system control.
October, 2021
Feng Liu, Beijing
Zhaojian Wang, Shanghai
Changhong Zhao, Hong Kong
Peng Yang, Beijing
Acknowledgments
This book summarizes our recent research about distributed optimal control of power systems. Most works were jointly carried out in the Department of Electrical Engineering at Tsinghua University and the Department of Electrical Engineering at the California Institute of Technology.
The authors would like to thank many people for their contributions to this book sincerely. Moreover, the authors are particularly indebted to Prof. Steven H. Low from the California Institute of Technology, Prof. Shengwei Mei and Prof. Qiang Lu from Tsinghua University, Prof. Felix F. Wu from the University of California at Berkeley, and Prof. Xinping Guan from Shanghai Jiao Tong University for their long‐lasting inspiration, encouragement, and support to our research. We also thank Mr. Yifan Su, Mr. Sicheng Deng, Prof. Laijun Chen, and Prof. Ying Chen from Tsinghua University, Dr. John ZF Pang from California Institute of Technology, Prof. Peng Yi from Tongji University, and Prof. Ming Cao from the University of Groningen, who have contributed materials to this book. In addition, we acknowledge Sandra Grayson, Kimberly Monroe‐Hill, Viniprammia Premkumar, and other staffs at IEEE Wiley for their assistance and help in preparing this book.
The authors appreciate the support from the Major Smart Grid Joint Project of the National Natural Science Foundation of China and State Grid (U1766206), the Natural Science Foundation of China (51677100, 62103265), the “Chenguang Program” supported by Shanghai Education Development Foundation and Shanghai Municipal Education Commission of China (20CG11), the Direct Grant (4055128) of the Chinese University of Hong Kong, and the Hong Kong Research Grants Council through Early Career Award (24210220).
1Introduction
Modern power grids rely on hierarchical control architectures to achieve stable, secure, and economic operation, which involves various kinds of advanced measurement, communication, and control techniques [1, 2]. Under the pressures of global climate change and energy shortage, power systems have been undergoing fundamental changes. The past decade has witnessed the leaping penetration of renewable energy resources and distributed generations, the proliferation of electric vehicles, and the active participation of customers, all of which are devoted to recreating a more reliable, flexible, sustainable, and affordable power grid.
On the generation side, fossil fuels are gradually giving place to environment‐friendly renewable generations. By the end of 2020, the total installed capacity of global renewable energies reached 2802.004 GW, including 1332.885 GW of hydropower, 732.41 GW of wind energy, and 716.152 GW of solar energy [3]. The rapid growth of renewable energy shows signs of speeding up in the near future. On the consumption side, many new forms of loads have emerged and started participating in system operation with unprecedented enthusiasm. These include, just to name a few, electric vehicles, active participation of load demand [4, 5], energy storages [6, 7], and microgrids [8].
Despite the tremendous environmental and economical benefits, the ongoing changing trends on both generation and consumption sides are gravely challenging the traditional power system control technologies. Renewable energies such as photovoltaics (PVs) and wind power generations (WTGs) are intrinsically uncertain [9], leading to volatile operating conditions. Besides, most PVs and WTGs are integrated via power electronic devices with low/zero inertia and may operate in various control modes. The load‐side diversity and participation also complicate the system control problem, which requires a careful design of the interaction protocol to achieve the smart grid vision.
We have to thoroughly and carefully address all these issues to facilitate the grand ambition of the ongoing system revolution. This extremely challenging task calls for advanced future power system control technologies to handle volatile operating conditions and massive controllable participants. Unfortunately, the existing power system control paradigm that features a centralized hierarchical structure may fail to achieve this goal. Here, we envision a new paradigm that reshapes the hierarchy and merges optimization with control, which may provide a new opportunity to tackle the task. This chapter will first introduce the traditional control methods in power systems and then introduce some state‐of‐the‐art methods.
1.1 Traditional Hierarchical Control Structure
The functional operation of a power system depends on its control systems. As one of the largest and most complicated man‐made systems, the modern power system must keep the frequency strictly synchronized and the voltages around their nominal values among thousands of generators and loads spanning over continents and interconnected through tens of thousands of miles of electric wires and cables. Therefore, an appropriate control architecture appears to be highly crucial to a reliable and efficient operation of power systems, if not the most. As a matter of fact, during the past one hundred years, power system control has evolved to be a layered/hierarchical structure encompassing diverse time scales and control objectives, ranging from millisecond to years. Figure 1.1 illustrates the time scales of power system controls with different control objectives. Generally, a slow time‐scale layer is more concerned with the economy of operation during a long‐time period, while a fast time‐scale layer focuses on stability and security during a short‐time dynamic process.
The time‐scale decomposition and the hierarchical control structure in traditional power systems greatly simplify the control synthesis problem. For example, in most cases, detailed fast time‐scale dynamics can be neglected in slow time‐scale control problems and vice versa. Such decomposition works well when the time scales of the two layers are significantly different. Even if the difference is less noticeable, e.g. between seconds and minutes, it is still acceptable. Nonetheless, the recent transition of our power system shows that it might be more suitable to combine layers in different time scales, say, merging slow optimization and fast control. This idea sets the first motivation for us to write the book.
In the rest of this section, we shortly introduce the hierarchies of traditional frequency and voltage controls.
1.1.1 Hierarchical Frequency Control
In an alternating current (AC) power system, frequency reflects the active power balance across the overall system. The frequency goes down when generation is less than load and vice versa. Therefore, a power system must adopt frequency control to maintain its frequency within a small neighborhood of the nominal value, such as 50 or 60 Hz.
Figure 1.1 Time‐scale decomposition of controls in a traditional power system.
In traditional power systems, most electric power is supplied by large‐capacity synchronous generators. The huge rotating inertia of generators can serve as a buffer of kinetic energy to mitigate moderate power imbalance, limiting frequency changes instantaneously. For example, a sudden load demand increase will be naturally supported by extracting the kinetic energy from synchronous generators. Consequently, the frequency will drop. However, the kinetic energy stored in the inertia is quite limited, which is inadequate to cope with large or long‐term frequency deviation. Therefore, intentional frequency control becomes a must to maintain system frequency more effectively and flexibly.
In accord with the control hierarchy mentioned above, frequency control includes three layers with respect to three different time scales, i.e. the primary control with a typical time scale in tens of seconds, the secondary control in several minutes, and the tertiary control in several minutes to tens of minutes, as shown in Figure 1.2. The first two layers act in a fast time scale that involves physical dynamics such as excitation control and governor control, while the last layer in a slow time scale involves operational or market dynamics such as economic dispatch (ED) and electricity market clearing. It is obvious from Figure 1.2 that the hierarchy of frequency control heavily relies on a control center.
Figure 1.2 Diagram of hierarchical frequency control.
1.1.1.1 Primary Frequency Control
Primary frequency control is designed to limit frequency deviation within an acceptable range. It is usually fulfilled via automatic governor regulation of generators. Denote by the mechanical power of generator and the frequency deviation from the nominal value at bus . Let the power compensation be with , and send it to change the valve opening of the prime mover, such that the system frequency regains a state of operating equilibrium.
Obviously, the primary frequency control is indeed a proportional feedback control, or droop control. It responds fairly fast to frequency deviation since only local frequency measurement is required. However, it may not restore system frequency to the nominal value due to the proportional control strategy.
In practice, not all generators in the system need to be equipped with a governor control. Those generators, however, usually are competent to provide fast power support. Hence one can still categorize them into primary frequency control when needed.
1.1.1.2 Secondary Frequency Control
Secondary frequency control is designed to eliminate frequency deviation. It is usually fulfilled via automatic generation control (AGC). In a multi‐area power system, the area control error (ACE) is a linear combination of the deviations of system frequency and the tie‐line powers delivered to or received from its neighboring areas [10]. For the th area, the ACE is defined as
where is a constant that stands for the responsibility of this area in response to the frequency deviation, which is referred to as area frequency response coefficient (AFRC). is the power deviation of the tie‐line connecting areas and . As a matter of fact, when the ACEs of all control areas converge to zero, the system frequency restores to the nominal value. Therefore, traditional AGC uses ACEs as the feedback signals to compute the control command that reflects the total required power compensation in the area. The obtained control command is then distributed to individual generators in proportion to their participation factors.
AGC typically adopts a proportional–integral (PI) control to drive ACEs to zero. However, as a power system always works in a time‐varying environment, the ACEs do not converge to zero but rather fluctuate around zero. So does the system frequency.
1.1.1.3 Tertiary Frequency Control
Although the secondary frequency control can eliminate the frequency deviation, it is not responsible for achieving an optimal power generation allocation. Instead, this task is fulfilled by the tertiary frequency control that aims to minimize the operation cost of the system (e.g. generation cost or network loss) by reallocating the power production among generators. Thus, it is also called ED, usually performed every 5–15 min. This problem can be mathematically formulated as a constrained optimization problem, where the constraints include power balance, generation limits, line flow limits, etc. Traditionally, the ED problem is solved to generate the control commands in the control center, and then the control commands are sent to the dispatchable generators as the set points. In a deregulated power system, or power market, the ED problem is replaced by a market clearing problem. Pricing issues and strategic behaviors of participants need to be carefully considered as well.
1.1.2 Hierarchical Voltage Control
While frequency is determined by the active power balance of the overall system, voltage is more closely relevant to local reactive power supply. When the reactive power supply is inadequate, the voltage will be lower than the nominal value, and vice versa. Therefore, voltage control is usually realized by regulating reactive power generation or consumption of generators, transformers, reactive power compensators, loads, etc. Usually, the load‐based voltage regulation is adopted in an emergency only. Except for reactive power sources/loads, flexible AC transmission systems based on power electronics provide an additional option to voltage control by changing the equivalent line impedance.
Unlike frequency, voltage is more like a local indicator for reactive power adequacy than a system‐wide indicator for active power balance. In addition, since reactive power is not suitable for long‐distance transmission (for avoiding unexpected transmission loss and voltage instability), a voltage control problem is usually considered and implemented in local regions. For coordinating voltage controls in different regions to enhance system‐wide performance, a hierarchical voltage structure emerges. Similar to frequency control, voltage control also has a central‐dominant hierarchy that comprises three layers: the primary voltage control with a typical time scale in tens of seconds, the secondary voltage control in several minutes, and the tertiary voltage control in several minutes to tens of minutes. Figure 1.3 illustrates the diagram of hierarchical voltage control.
Figure 1.3 Diagram of hierarchical voltage control.
1.1.2.1 Primary Voltage Control
As the fastest control layer in the hierarchy, primary voltage control aims to stabilize the voltage rapidly when voltage deviation from the preset value exceeds a certain threshold. It is fulfilled by using local feedback control of various reactive power sources including synchronous generators, capacitors, series inductors, static Var compensator (SVC), static synchronous compensator (STATCOM), etc. For example, one can use automatic voltage regulator (AVR) to flexibly control the terminal voltage of generators (via the excitation system), often within a range of with respect to its rated value. In practice, PI controllers are extensively employed; otherwise, droop controllers or proportional controllers appear to be more helpful for a stable operation.
1.1.2.2 Secondary Voltage Control
As mentioned above, primary voltage control is built on fully local voltage deviation. Hence a controllable reactive power source or load in a subregion would provide little support to other subregions even if it has plenty of reactive power reserve. In this regard, secondary voltage control is developed. It is implemented in a control center of one region, within a time scale from tens of seconds to several minutes, or even longer up to the time constant of controlled devices. Secondary voltage control aims to coordinate the region‐wide voltage control by utilizing the reference voltage of a pilot bus that represents the voltage situation or reactive adequacy in a local region. In operation, the reference voltage deviation of the pilot bus is sent to local primary voltage controllers, attached to the primary control signals, to coordinate all the controllable voltage/Var sources and loads within this region.
Except for reactive power sources and loads, onload tap changers can also be used to facilitate secondary voltage control by changing the taps of transformers. However, it should be aware that adjusting transformer taps only changes reactive power distribution rather than generating reactive power. Therefore, we only enable it when the reactive power is sufficient in the region. Otherwise, it may worsen the reactive power shortage, even resulting in disastrous voltage collapse.
1.1.2.3 Tertiary Voltage Control
Reactive power distribution has a remarkable influence on the power loss of power transmission. In this regard, tertiary voltage control is responsible for optimizing the reactive power distribution across the overall power grid. This target is achieved by changing the reference voltages of pilot buses in each control region. In this sense, tertiary voltage control can also be regarded as a particular type of optimal power flow (OPF) problem, where only reactive power is optimized. Therefore its time scale follows the pace of ED or slower, varying from several minutes to several hours.
1.2 Transitions and Challenges
The central‐dominant hierarchical control architecture has effectively supported the operation of traditional power systems for decades. As power systems evolve into a new era, a critical question arises: Can the traditional control paradigm still apply? The key features in future power systems that may hinder the classical control paradigm include the following:
Massive entities
: A huge amount of dynamic devices are integrated into the power system from different voltage levels, including WTGs, PVs, electric vehicles, and energy storage, to name a few. The number of controllable and uncontrollable devices increases by orders of magnitudes, calling for a more scalable control scheme with an effective coordination.
Heterogeneous dynamics
: Due to the increasing diversity of control modes and physical natures of electrical devices, the dynamics in future power systems are extensively heterogeneous. It brings intricate interaction patterns and significantly complicates system‐level analysis and control design, calling for a more compatible control scheme.
Uncertain and fast‐changing environments
: The high penetration of renewable energy resources and complex loads significantly increases the uncertainties in power systems. In addition, distributed energies usually belong to individual owners, which could be switched on or off frequently. The increasing uncertainty leads to volatile operating conditions, which requires much faster control responses to retain power balance and economical operation in fast‐changing environments, calling for a more adaptive and robust control scheme.
In light of these transitions, the classical center‐dominant hierarchical control paradigm may fail to meet what the future power systems demand, and reshaping it to fit the future becomes a must. Ideally, we envision that the future paradigm should be scalable to the massive amounts of devices, compatible with ever‐increasing heterogeneity, efficient in fast‐changing operation environments, flexible to diverse operation modes, and robust against unexpected perturbations and even failures. Toward this ambition, fruitful progress has been made to combine advanced control and optimization theories with power system engineering. Two of the most promising topics among them are (i) relieving the dependence on central coordinators and (ii) reshaping the original control hierarchy. The former advocates a distributed control paradigm that endows the power system with higher scalability, compatibility, and robustness. The latter suggests merging the slow time‐scale optimization and fast time‐scale control to achieve stronger adaptability and faster response. The following two sections will briefly introduce the state of the art of these two innovative topics.
1.3 Removing Central Coordinators: Distributed Coordination
In terms of the way of coordinating, three approaches may be involved: centralized control, decentralized control, and distributed control.
Centralized control features a control center that collects information of all agents, performs a central computation to get control commands, and sends them back to each agent (Figure 1.4a). The control center can solve a complex centralized optimization problem to increase the economic efficiency of the whole system. However, the system may break down if the control center fails, which is the notorious single‐point‐of‐failure issue. In addition, privacy becomes a big problem because the control center requires information from all agents. Another problem is that this approach is not scalable as it heavily relies on detailed information of all agents in the system. Moreover, collecting the information is time‐consuming in large systems, which remarkably slows down the response.
Decentralized control needs no control center and adopts purely local control strategies to compute control commands, i.e. no communication between agents (Figure 1.4b). Thus, decentralized control usually has a rapid response, which has been widely adopted, particularly in the primary control. However, agents adopting decentralized control may conflict with each other since there is a lack of coordination.
The distributed control also needs no control center but requires communication among agents such that each agent can compute control commands locally1 (Figure 1.4c). Roughly speaking, the complexity of distributed control is between those of centralized and decentralized ones. Similar to decentralized control, the structure of distributed control is also simple and easy to apply. However, distributed control can enable coordination among agents to facilitate global objectives as the agents can exchange information. Compared with centralized control, distributed control has several potential advantages. First, each agent only needs to share limited information with a subset of the other agents, i.e. its neighbors on the communication graph. This feature consequently improves cybersecurity, better protects privacy, and reduces the expense of communication networks. Second, distributed control avoids the single‐point‐of‐failure issue and hence is more robust. Third, distributed control may be computationally superior to the centralized rival by decomposing a large‐scale global problem into a set of small‐scale local problems. Finally, distributed control can adapt to volatile operation conditions, such as fast variation of renewable generations and topology changes.
Since the distributed control structure inherits advantages both from the centralized and decentralized ones, it has been widely recognized as a promising solution to the aforementioned challenges and has inspired plenty of achievements in this field. Interestingly, it has many overlaps with another topic, distributed optimization, in both problem formulations and algorithms. The two closely related topics, sometimes, may lead to confusion. In this regard, this book adopts the following terminology. When considering distributed control, we design a controller and implement it to physical systems in a distributed manner, and hence the controller's dynamics directly couple with the physical dynamics. On the other hand, when considering distributed optimization, we construct a distributed algorithm to solve optimization problems subject to some snapshots of the physical systems while the solving process does not directly couple with the physical dynamics. In this sense, the primary and secondary frequency/voltage regulations are categorized as a control problem, while the tertiary frequency/voltage control, i.e. ED, is categorized as an optimization problem. However, we use distributed optimal control when both optimization and control are taken into account.
Figure 1.4 Diagram of centralized (a), decentralized (b), and distributed (c) control.
1.3.1 Distributed Control
Distributed control considers a group of dynamic agents as follows:
where is the state variable of agent , is the output, and is the set of the neighbors of dynamic agent on the physical network. Agent can exchange information with its neighbors on the communication network, denoted by . Note that generally .
We expect to design the following distributed controller that depends on the local output:
One of the most extensively used approaches is to adopt the consensus‐based distributed control [11], which takes the following form:
where is the weight between buses and . According to the consensus algorithm, holds for all in the steady state. In this sense, the output essentially serves as a global variable to coordinate all agents in the system. Individual agents estimate the variable locally and approach the consensus through iterative computation and information exchange. Therefore, by appropriately choosing the output , various kinds of distributed controllers can be devised.
Specifically, in power systems, the global coordination variable could be generation ratio and marginal cost. The former one is the ratio between actual generation and the maximal capacity, which implies that all generators supply the load fairly up to their maximal capability [12–17]. References [12, 13] apply a consensus algorithm to the active power control of PVs for achieving a unified utilization ratio. This method can also apply in reactive power control [14–16] and harmonic control [17]. The latter implies that all generators share the same marginal cost and hence reach the economical optimum [18, 19]. Reference [18] shows that individual distributed generators in a microgrid can maintain identical marginal costs by using a consensus‐based distributed controller while restoring the nominal frequency. This idea is further extended to direct current (DC) microgrids in [19].
Consensus‐based distributed control has a simple structure, which can achieve a fair or economical operation. The fairness is realized by maintaining equal generation ratios, while the economy is realized by holding an equal marginal generation cost. On the other side, this simple structure conversely restricts the applicability of consensus‐based distributed control when considering complicated optimization objectives or constraints such as line flow limits, which are very common in power systems. In such circumstances, we need more sophisticated designs, which will be discussed in detail in the remaining chapters of this book.
1.3.2 Distributed Optimization
Distributed optimization considers a group of agents that cooperatively solve the following separable (convex) optimization problem with constraints:
where are convex functions and is a convex set.
In distributed optimization, each agent carries out local computation on a subproblem and exchanges information with neighbors. Convergent computation gives an optimal solution to the original optimization problem. Some commonly used distributed optimization algorithms include consensus‐based algorithms, dual decomposition, alternating direction method of multipliers (ADMM), and gradient‐based algorithms, to name a few. An introduction to these algorithms can be found in Appendix A. Here, we briefly introduce the applications of the most popular consensus‐based algorithms and ADMM in power systems.
Consensus‐based algorithms are especially relevant in ED problems, where marginal costs or prices of generators serve as the global coordination variables for consensus [21–24]. In [22], an average consensus method is presented to solve the ED problem in a distributed fashion, where two stopping criteria are derived based on sign consensus. Reference [21] extends the consensus method to solve the ED problem with transmission losses. In [23] an incremental cost‐consensus algorithm is suggested with a convergence proof. In addition, literature [24] shows that a consensus‐based algorithm can track the optimal solution to the active power ED problem with generation capacity constraints. However, as mentioned in the previous part, this appears to be restrictive in dealing with complicated constraints.
Compared with consensus‐based algorithms, ADMM has been more extensively employed to deal with complicated constraints, such as OPF problems. In terms of power flow models, the appliances of ADMM roughly fall into four categories: (i) AC OPF problems [25–27], (ii) DC OPF problems [28–30], (iii) distribution flow with the second‐order cone relaxation [31–33], and (iv) linearized distribution flow [34–36]. Reference [25] proposes an asynchronous ADMM algorithm of the AC OPF problem to cope with the communication delay by extending the synchronous ADMM algorithm [26, 27]. In [28], a fully distributed accelerated ADMM algorithm is presented, where a consensus‐based push‐sum method is derived to improve the convergence. In [29], a consensus‐based ADMM approach is employed to solve DC OPF problems. Reference [30] utilizes machine learning to predict the optimal dual variable under different realizations of system loads, remarkably accelerating the convergence of ADMM. The distribution flow (DistFlow) model is considered in references [31–36]. To cope with the nonconvexity of the DistFlow model, references [31–33] apply a second‐cone relaxation while references [34–36] choose to linearize the DistFlow model by ignoring the line loss.
Note that ADMM algorithms are usually concerned with optimization problems only by assuming the dynamics of physical power systems are fast enough and negligible. One consequence is that the algorithms are not applicable in a fast‐changing environment. Otherwise, the influence of physical power system dynamics will turn to be non‐ignorable, making the assumption invalid.
1.4 Merging Optimization and Control
As analyzed above, distributed control and optimization have their own advantages and deficiencies and are applied in different layers. So a natural question is whether they could be combined to address both optimization and control issues in complicated environments. This question inspires the idea of merging primary and secondary controls in a fast time scale with ED or OPF in a slow time scale [10, 37, 38]. This leads to a cross‐layer design approach of distributed optimal control that bridges the gap between optimization and control in different time scales. As a result, we expect the merged distributed optimal control could achieve optimality automatically and rapidly while guaranteeing system stability, even under complicated operational constraints and changing environments.
Generally, this idea has two sides: optimization‐guided control and feedback‐based optimization. On the one side, we intend to design a feedback controller for power systems, which could drive the system states to the optimal solution of an optimization problem in the steady state, such as ED. At the same time, the system should be asymptotically stable. On the other side, we hope to solve optimization problems by making use of feedback control. In this case, some state variables are obtained by measuring instead of solving complicated system equations, such as power flow equations. In this section, we explain them, respectively.
1.4.1 Optimization‐Guided Control
The main idea of optimization‐guided control is to design a (dynamic) feedback controller for power systems, which drives the system to the optimal solution to an optimization problem, such as ED or OPF. The framework is illustrated in Figure 1.5.
Figure 1.5 Conceptual diagram of optimization‐guided feedback control.
In this framework, the lower layer is the fast‐time‐scale dynamics of a power system with control inputs. We formulate it into a general control system:
where is the vector of state variables, is the vector of control inputs, and is the vector of system outputs.
The upper layer is the slow‐time‐scale ED or OPF process of the power system. We formulate it into a general constrained (convex) optimization problem with adjustable parameters:
where is the vector of decision variables and is the vector of adjustable parameters.
Our goal is to design a dynamic controller:
such that the output of (1.7) at the steady state is equal to the optimal solution to (1.6). This equivalent relation can be established by noting that the solving process of optimization problem (1.6) essentially defines a dynamic process, e.g. the primal–dual gradient dynamics.
To merge the two layers above, we interconnect the controller (1.7) and the system (1.5) by letting and ,2 as shown in Figure 1.5. Then we need to appropriately design the controller (1.7) such that the closed‐loop power system guarantees the asymptotic stability with the equilibrium point identical to the solution to the optimization problem (1.6). This approach is closely related to the primal–dual gradient dynamics (or saddle‐point dynamics) [39, 40] by noting that one can construct the controller (1.7) from the primal–dual gradient dynamics of the optimization problem (1.6). That is why this is called optimization‐guided control.
The idea of optimization‐guided control can date back to [41] that presents a methodology to regulate a nonlinear dynamical system to an optimal operation point, i.e. a solution to a given constrained convex optimization problem in terms of the steady‐state operation. To this end, it constructs a dynamic extension of the Karush–Kuhn–Tucker (KKT) optimality conditions for the corresponding optimization problem. A similar method is applied in power systems [20], which exploits the pricing interpretation of the Lagrange multipliers to guarantee economically optimal operation at the steady state. This idea is further generalized as the notion of reverse and forward engineering methodology for designing optimal controllers, particularly in optimal frequency and voltage control of power systems [10, 37, 42, 43] Following and extending this idea, we will design frequency controllers considering various physical restrictions in this book.
1.4.2 Feedback‐Based Optimization
Feedback‐based optimization provides an alternative way to merge optimization and control. One of the primary motivations for feedback‐based optimization is to fast respond to power fluctuations due to time‐varying loads and volatile renewable generations, since optimized strategies based on fixed operational conditions may not be applicable in this situation. Therefore, unlike optimization‐guided control, feedback‐based optimization is usually more concerned with the operational optimality than the stability of a power system. As a result, (quasi‐) steady‐state models of a power system other than its dynamical model are adopted.
