Merging Optimization and Control in Power Systems

Feng Liu, PhD, is Associate Professor in the Department of Electrical Engineering at Tsinghua University in Beijing, China.Zhaojian Wang, PhD, is Assistant Professor in the Department of Automation at Shanghai Jiao Tong University in Shanghai, China. Changhong Zhao, PhD, is Assistant Professor in the Department of Information Engineering at the Chinese University of Hong Kong, Hong Kong SAR, China. Peng Yang is a PhD Candidate in the Department of Electrical Engineering at Tsinghua University in Beijing, China.

• Foreword xvPreface xviiAcknowledgments xix1 Introduction 11.1 Traditional Hierarchical Control Structure 21.1.1 Hierarchical Frequency Control 21.1.1.1 Primary Frequency Control 41.1.1.2 Secondary Frequency Control 51.1.1.3 Tertiary Frequency Control 51.1.2 Hierarchical Voltage Control 51.1.2.1 Primary Voltage Control 61.1.2.2 Secondary Voltage Control 71.1.2.3 Tertiary Voltage Control 71.2 Transitions and Challenges 71.3 Removing Central Coordinators: Distributed Coordination 81.3.1 Distributed Control 111.3.2 Distributed Optimization 121.4 Merging Optimization and Control 131.4.1 Optimization-Guided Control 141.4.2 Feedback-Based Optimization 161.5 Overview of the Book 17Bibliography 192 Preliminaries 232.1 Norm 232.1.1 Vector Norm 232.1.2 Matrix Norm 242.2 Graph Theory 262.2.1 Basic Concepts 262.2.2 Laplacian Matrix 262.3 Convex Optimization 282.3.1 Convex Set 282.3.1.1 Basic Concepts 282.3.1.2 Cone 302.3.2 Convex Function 312.3.2.1 Basic Concepts 312.3.2.2 Jensen’s Inequality 352.3.3 Convex Programming 352.3.4 Duality 362.3.5 Saddle Point 392.3.6 KKT Conditions 392.4 Projection Operator 412.4.1 Basic Concepts 412.4.2 Projection Operator 422.5 Stability Theory 442.5.1 Lyapunov Stability 442.5.2 Invariance Principle 462.5.3 Input–Output Stability 472.6 Passivity and Dissipativity Theory 492.6.1 Passivity 492.6.2 Dissipativity 512.7 Power Flow Model 522.7.1 Nonlinear Power Flow 532.7.1.1 Bus Injection Model (BIM) 532.7.1.2 Branch Flow Model (BFM) 542.7.2 Linear Power Flow 552.7.2.1 DC Power Flow 552.7.2.2 Linearized Branch Flow 562.8 Power System Dynamics 562.8.1 Synchronous Generator Model 572.8.2 Inverter Model 58Bibliography 603 Bridging Control and Optimization in Distributed Optimal Frequency Control 633.1 Background 643.1.1 Motivation 643.1.2 Summary 663.1.3 Organization 673.2 Power System Model 673.2.1 Generator Buses 683.2.2 Load Buses 693.2.3 Branch Flows 703.2.4 Dynamic Network Model 723.3 Design and Stability of Primary Frequency Control 743.3.1 Optimal Load Control 743.3.2 Main Results 753.3.3 Implications 793.4 Convergence Analysis 793.5 Case Studies 883.5.1 Test System 883.5.2 Simulation Results 893.6 Conclusion and Notes 92Bibliography 934 Physical Restrictions: Input Saturation in Secondary Frequency Control 974.1 Background 984.2 Power System Model 1004.3 Control Design for Per-Node Power Balance 1014.3.1 Control Goals 1024.3.2 Decentralized Optimal Controller 1034.3.3 Design Rationale 1054.3.3.1 Primal–Dual Algorithms 1054.3.3.2 Design of Controller (4.6) 1054.4 Optimality and Uniqueness of Equilibrium 1084.5 Stability Analysis 1124.6 Case Studies 1204.6.1 Test System 1204.6.2 Simulation Results 1224.6.2.1 Stability and Optimality 1224.6.2.2 Dynamic Performance 1234.6.2.3 Comparison with AGC 1244.6.2.4 Digital Implementation 1244.7 Conclusion and Notes 128Bibliography 1315 Physical Restrictions: Line Flow Limits in Secondary Frequency Control 1355.1 Background 1365.2 Power System Model 1375.3 Control Design for Network Power Balance 1385.3.1 Control Goals 1395.3.2 Distributed Optimal Controller 1415.3.3 Design Rationale 1425.3.3.1 Primal–Dual Gradient Algorithms 1425.3.3.2 Controller Design 1435.4 Optimality of Equilibrium 1445.5 Asymptotic Stability 1485.6 Case Studies 1555.6.1 Test System 1555.6.2 Simulation Results 1565.6.2.1 Stability and Optimality 1565.6.2.2 Dynamic Performance 1585.6.2.3 Comparison with AGC 1585.6.2.4 Congestion Analysis 1585.6.2.5 Time Delay Analysis 1615.7 Conclusion and Notes 165Bibliography 1656 Physical Restrictions: Nonsmoothness of Objective Functions in Load-Frequency Control 1676.1 Background 1676.2 Notations and Preliminaries 1696.3 Power System Model 1706.4 Control Design 1716.4.1 Optimal Load Frequency Control Problem 1726.4.2 Distributed Controller Design 1736.5 Optimality and Convergence 1766.5.1 Optimality 1766.5.2 Convergence 1786.6 Case Studies 1836.6.1 Test System 1836.6.2 Simulation Results 1846.7 Conclusion and Notes 187Bibliography 1887 Cyber Restrictions: Imperfect Communication in Power Control of Microgrids 1917.1 Background 1927.2 Preliminaries and Model 1937.2.1 Notations and Preliminaries 1937.2.2 Economic Dispatch Model 1947.3 Distributed Control Algorithms 1957.3.1 Synchronous Algorithm 1957.3.2 Asynchronous Algorithm 1967.4 Optimality and Convergence Analysis 1987.4.1 Virtual Global Clock 1997.4.2 Algorithm Reformulation 2007.4.3 Optimality of Equilibrium 2037.4.4 Convergence Analysis 2047.5 Real-Time Implementation 2067.5.1 Motivation and Main Idea 2067.5.2 Real-Time ASDPD 2087.5.2.1 AC MGs 2087.5.2.2 DC Microgrids 2087.5.3 Control Configuration 2107.5.4 Optimality of the Implementation 2117.6 Numerical Results 2137.6.1 Test System 2137.6.2 Non-identical Sampling Rates 2147.6.3 Random Time Delays 2177.6.4 Comparison with the Synchronous Algorithm 2177.7 Experimental Results 2197.8 Conclusion and Notes 222Bibliography 2248 Cyber Restrictions: Imperfect Communication in Voltage Control of Active Distribution Networks 2298.1 Background 2308.2 Preliminaries and System Model 2328.2.1 Note and Preliminaries 2328.2.2 System Modeling 2338.3 Problem Formulation 2348.4 Asynchronous Voltage Control 2358.5 Optimality and Convergence 2378.5.1 Algorithm Reformulation 2388.5.2 Optimality of Equilibrium 2428.5.3 Convergence Analysis 2438.6 Implementation 2458.6.1 Communication Graph 2458.6.2 Online Implementation 2468.7 Case Studies 2468.7.1 8-Bus Feeder System 2478.7.2 IEEE 123-Bus Feeder System 2508.8 Conclusion and Notes 253Bibliography 2549 Robustness and Adaptability: Unknown Disturbances in Load-Side Frequency Control 2579.1 Background 2589.2 Problem Formulation 2599.2.1 Power Network 2599.2.2 Power Imbalance 2609.2.3 Equivalent Transformation of Power Imbalance 2619.3 Controller Design 2639.3.1 Controller for Known P_in j 2639.3.2 Controller for Time-Varying Power Imbalance 2649.3.3 Closed-Loop Dynamics 2659.4 Equilibrium and Stability Analysis 2669.4.1 Equilibrium 2669.4.2 Asymptotic Stability 2699.5 Robustness Analysis 2749.5.1 Robustness Against Uncertain Parameters 2749.5.2 Robustness Against Unknown Disturbances 2759.6 Case Studies 2779.6.1 System Configuration 2779.6.2 Self-Generated Data 2799.6.3 Performance Under Unknown Disturbances 2829.6.4 Simulation with Real Data 2829.6.5 Comparison with Existing Control Methods 2849.7 Conclusion and Notes 286Bibliography 28710 Robustness and Adaptability: Partial Control Coverage in Transient Frequency Control 28910.1 Background 28910.2 Structure-Preserving Model of Nonlinear Power System Dynamics 29110.2.1 Power Network 29110.2.2 Synchronous Generators 29210.2.3 Dynamics of Voltage Phase Angles 29310.2.4 Communication Network 29410.3 Formulation of Optimal Frequency Control 29410.3.1 Optimal Power-Sharing Among Controllable Generators 29410.3.2 Equivalent Model With Virtual Load 29510.4 Control Design 29610.4.1 Controller for Controllable Generators 29610.4.2 Active Power Dynamics of Uncontrollable Generators 29710.4.3 Excitation Voltage Dynamics of Generators 29810.5 Optimality and Stability 29810.5.1 Optimality 29810.5.2 Stability 30010.6 Implementation With Frequency Measurement 30610.6.1 Estimating Μ I Using Frequency Feedback 30610.6.2 Stability Analysis 30710.7 Case Studies 31010.7.1 Test System and Data 31010.7.2 Performance Under Small Disturbances 31210.7.2.1 Equilibrium and its Optimality 31210.7.2.2 Performance of Frequency Dynamics 31310.7.3 Performance Under Large Disturbances 31610.7.3.1 Generator Tripping 31710.7.3.2 Short-Circuit Fault 31810.8 Conclusion and Notes 321Bibliography 32211 Robustness and Adaptability: Heterogeneity in Power Controls of DC Microgrids 32511.1 Background 32511.2 Network Model 32811.3 Optimal Power Flow of DC Networks 32911.3.1 OPF Model 32911.3.2 Uniqueness of Optimal Solution 33111.4 Control Design 33411.4.1 Distributed Optimization Algorithm 33411.4.2 Optimality of Equilibrium 33511.4.3 Convergence Analysis 33811.5 Implementation 34411.6 Case Studies 34611.6.1 Test System and Data 34611.6.2 Accuracy Analysis 34811.6.3 Dynamic Performance Verification 34811.6.4 Performance in Plug-n-play Operations 35211.7 Conclusion and Notes 353Bibliography 354Appendix A Typical Distributed Optimization Algorithms 357A.1 Consensus-Based Algorithms 357A.1.1 Consensus Algorithms 358A.1.2 Cutting-Plane Consensus Algorithm 359A.2 First-Order Gradient-Based Algorithms 362A.2.1 Dual Decomposition 363A.2.2 Alternating Direction Method of Multipliers 366A.2.3 Primal–Dual Gradient Algorithm 368A.2.4 Proximal Gradient Method 371A.3 Second-Order Newton-Based Algorithms 374A.3.1 Barrier Method 374A.3.2 Primal–Dual Interior-Point Method 375A.4 Zeroth-Order Online Algorithms 377Bibliography 379Appendix B Optimal Power Flow of Direct Current Networks 385B. 1 Mathematical Model 385B.. 1 Formulation 385B.1. 2 Equivalent Transformation 387B. 2 Exactness of SOC Relaxation 388B.2. 1 SOC Relaxation of OPF in DC Networks 388B.. 2 Assumptions 388B.2. 3 Exactness of the SOC Relaxation 389B.2. 4 Topological Independence 396B.2. 5 Uniqueness of the Optimal Solution 396B.2. 6 Branch Flow Model 397B. 3 Case Studies 399B.3. 1 16-Bus System 399B.3. 2 Larger-Scale Systems 401B. 4 Discussion on Line Constraints 402B.4. 1 OPF with Line Constraints 402B.4. 2 Exactness Conditions with Line Constraints 403B.4. 3 Constructing Approximate Optimal Solutions 406B.4.3. 1 Direct Construction Method 407B.4.3. 2 Slack Variable Method 408Bibliography 409Index 411