Merging Optimization and Control in Power Systems
Physical and Cyber Restrictions in Distributed Frequency Control and Beyond
Wiley-IEEE Press Book Series on Control Systems Theory and Applications
1. Auflage August 2022
432 Seiten, Hardcover
Fachbuch
ISBN:
978-1-119-82792-4
John Wiley & Sons
Weitere Versionen
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.
Foreword xv
Preface xvii
Acknowledgments xix
1 Introduction 1
1.1 Traditional Hierarchical Control Structure 2
1.1.1 Hierarchical Frequency Control 2
1.1.1.1 Primary Frequency Control 4
1.1.1.2 Secondary Frequency Control 5
1.1.1.3 Tertiary Frequency Control 5
1.1.2 Hierarchical Voltage Control 5
1.1.2.1 Primary Voltage Control 6
1.1.2.2 Secondary Voltage Control 7
1.1.2.3 Tertiary Voltage Control 7
1.2 Transitions and Challenges 7
1.3 Removing Central Coordinators: Distributed Coordination 8
1.3.1 Distributed Control 11
1.3.2 Distributed Optimization 12
1.4 Merging Optimization and Control 13
1.4.1 Optimization-Guided Control 14
1.4.2 Feedback-Based Optimization 16
1.5 Overview of the Book 17
Bibliography 19
2 Preliminaries 23
2.1 Norm 23
2.1.1 Vector Norm 23
2.1.2 Matrix Norm 24
2.2 Graph Theory 26
2.2.1 Basic Concepts 26
2.2.2 Laplacian Matrix 26
2.3 Convex Optimization 28
2.3.1 Convex Set 28
2.3.1.1 Basic Concepts 28
2.3.1.2 Cone 30
2.3.2 Convex Function 31
2.3.2.1 Basic Concepts 31
2.3.2.2 Jensen's Inequality 35
2.3.3 Convex Programming 35
2.3.4 Duality 36
2.3.5 Saddle Point 39
2.3.6 KKT Conditions 39
2.4 Projection Operator 41
2.4.1 Basic Concepts 41
2.4.2 Projection Operator 42
2.5 Stability Theory 44
2.5.1 Lyapunov Stability 44
2.5.2 Invariance Principle 46
2.5.3 Input-Output Stability 47
2.6 Passivity and Dissipativity Theory 49
2.6.1 Passivity 49
2.6.2 Dissipativity 51
2.7 Power Flow Model 52
2.7.1 Nonlinear Power Flow 53
2.7.1.1 Bus Injection Model (BIM) 53
2.7.1.2 Branch Flow Model (BFM) 54
2.7.2 Linear Power Flow 55
2.7.2.1 DC Power Flow 55
2.7.2.2 Linearized Branch Flow 56
2.8 Power System Dynamics 56
2.8.1 Synchronous Generator Model 57
2.8.2 Inverter Model 58
Bibliography 60
3 Bridging Control and Optimization in Distributed Optimal Frequency Control 63
3.1 Background 64
3.1.1 Motivation 64
3.1.2 Summary 66
3.1.3 Organization 67
3.2 Power System Model 67
3.2.1 Generator Buses 68
3.2.2 Load Buses 69
3.2.3 Branch Flows 70
3.2.4 Dynamic Network Model 72
3.3 Design and Stability of Primary Frequency Control 74
3.3.1 Optimal Load Control 74
3.3.2 Main Results 75
3.3.3 Implications 79
3.4 Convergence Analysis 79
3.5 Case Studies 88
3.5.1 Test System 88
3.5.2 Simulation Results 89
3.6 Conclusion and Notes 92
Bibliography 93
4 Physical Restrictions: Input Saturation in Secondary Frequency Control 97
4.1 Background 98
4.2 Power System Model 100
4.3 Control Design for Per-Node Power Balance 101
4.3.1 Control Goals 102
4.3.2 Decentralized Optimal Controller 103
4.3.3 Design Rationale 105
4.3.3.1 Primal-Dual Algorithms 105
4.3.3.2 Design of Controller (4.6) 105
4.4 Optimality and Uniqueness of Equilibrium 108
4.5 Stability Analysis 112
4.6 Case Studies 120
4.6.1 Test System 120
4.6.2 Simulation Results 122
4.6.2.1 Stability and Optimality 122
4.6.2.2 Dynamic Performance 123
4.6.2.3 Comparison with AGC 124
4.6.2.4 Digital Implementation 124
4.7 Conclusion and Notes 128
Bibliography 131
5 Physical Restrictions: Line Flow Limits in Secondary Frequency Control 135
5.1 Background 136
5.2 Power System Model 137
5.3 Control Design for Network Power Balance 138
5.3.1 Control Goals 139
5.3.2 Distributed Optimal Controller 141
5.3.3 Design Rationale 142
5.3.3.1 Primal-Dual Gradient Algorithms 142
5.3.3.2 Controller Design 143
5.4 Optimality of Equilibrium 144
5.5 Asymptotic Stability 148
5.6 Case Studies 155
5.6.1 Test System 155
5.6.2 Simulation Results 156
5.6.2.1 Stability and Optimality 156
5.6.2.2 Dynamic Performance 158
5.6.2.3 Comparison with AGC 158
5.6.2.4 Congestion Analysis 158
5.6.2.5 Time Delay Analysis 161
5.7 Conclusion and Notes 165
Bibliography 165
6 Physical Restrictions: Nonsmoothness of Objective Functions in Load-Frequency Control 167
6.1 Background 167
6.2 Notations and Preliminaries 169
6.3 Power System Model 170
6.4 Control Design 171
6.4.1 Optimal Load Frequency Control Problem 172
6.4.2 Distributed Controller Design 173
6.5 Optimality and Convergence 176
6.5.1 Optimality 176
6.5.2 Convergence 178
6.6 Case Studies 183
6.6.1 Test System 183
6.6.2 Simulation Results 184
6.7 Conclusion and Notes 187
Bibliography 188
7 Cyber Restrictions: Imperfect Communication in Power Control of Microgrids 191
7.1 Background 192
7.2 Preliminaries and Model 193
7.2.1 Notations and Preliminaries 193
7.2.2 Economic Dispatch Model 194
7.3 Distributed Control Algorithms 195
7.3.1 Synchronous Algorithm 195
7.3.2 Asynchronous Algorithm 196
7.4 Optimality and Convergence Analysis 198
7.4.1 Virtual Global Clock 199
7.4.2 Algorithm Reformulation 200
7.4.3 Optimality of Equilibrium 203
7.4.4 Convergence Analysis 204
7.5 Real-Time Implementation 206
7.5.1 Motivation and Main Idea 206
7.5.2 Real-Time ASDPD 208
7.5.2.1 AC MGs 208
7.5.2.2 DC Microgrids 208
7.5.3 Control Configuration 210
7.5.4 Optimality of the Implementation 211
7.6 Numerical Results 213
7.6.1 Test System 213
7.6.2 Non-identical Sampling Rates 214
7.6.3 Random Time Delays 217
7.6.4 Comparison with the Synchronous Algorithm 217
7.7 Experimental Results 219
7.8 Conclusion and Notes 222
Bibliography 224
8 Cyber Restrictions: Imperfect Communication in Voltage Control of Active Distribution Networks 229
8.1 Background 230
8.2 Preliminaries and System Model 232
8.2.1 Note and Preliminaries 232
8.2.2 System Modeling 233
8.3 Problem Formulation 234
8.4 Asynchronous Voltage Control 235
8.5 Optimality and Convergence 237
8.5.1 Algorithm Reformulation 238
8.5.2 Optimality of Equilibrium 242
8.5.3 Convergence Analysis 243
8.6 Implementation 245
8.6.1 Communication Graph 245
8.6.2 Online Implementation 246
8.7 Case Studies 246
8.7.1 8-Bus Feeder System 247
8.7.2 IEEE 123-Bus Feeder System 250
8.8 Conclusion and Notes 253
Bibliography 254
9 Robustness and Adaptability: Unknown Disturbances in Load-Side Frequency Control 257
9.1 Background 258
9.2 Problem Formulation 259
9.2.1 Power Network 259
9.2.2 Power Imbalance 260
9.2.3 Equivalent Transformation of Power Imbalance 261
9.3 Controller Design 263
9.3.1 Controller for Known P ¯_in j 263
9.3.2 Controller for Time-Varying Power Imbalance 264
9.3.3 Closed-Loop Dynamics 265
9.4 Equilibrium and Stability Analysis 266
9.4.1 Equilibrium 266
9.4.2 Asymptotic Stability 269
9.5 Robustness Analysis 274
9.5.1 Robustness Against Uncertain Parameters 274
9.5.2 Robustness Against Unknown Disturbances 275
9.6 Case Studies 277
9.6.1 System Configuration 277
9.6.2 Self-Generated Data 279
9.6.3 Performance Under Unknown Disturbances 282
9.6.4 Simulation with Real Data 282
9.6.5 Comparison with Existing Control Methods 284
9.7 Conclusion and Notes 286
Bibliography 287
10 Robustness and Adaptability: Partial Control Coverage in Transient Frequency Control 289
10.1 Background 289
10.2 Structure-Preserving Model of Nonlinear Power System Dynamics 291
10.2.1 Power Network 291
10.2.2 Synchronous Generators 292
10.2.3 Dynamics of Voltage Phase Angles 293
10.2.4 Communication Network 294
10.3 Formulation of Optimal Frequency Control 294
10.3.1 Optimal Power-Sharing Among Controllable Generators 294
10.3.2 Equivalent Model With Virtual Load 295
10.4 Control Design 296
10.4.1 Controller for Controllable Generators 296
10.4.2 Active Power Dynamics of Uncontrollable Generators 297
10.4.3 Excitation Voltage Dynamics of Generators 298
10.5 Optimality and Stability 298
10.5.1 Optimality 298
10.5.2 Stability 300
10.6 Implementation With Frequency Measurement 306
10.6.1 Estimating Mu I Using Frequency Feedback 306
10.6.2 Stability Analysis 307
10.7 Case Studies 310
10.7.1 Test System and Data 310
10.7.2 Performance Under Small Disturbances 312
10.7.2.1 Equilibrium and its Optimality 312
10.7.2.2 Performance of Frequency Dynamics 313
10.7.3 Performance Under Large Disturbances 316
10.7.3.1 Generator Tripping 317
10.7.3.2 Short-Circuit Fault 318
10.8 Conclusion and Notes 321
Bibliography 322
11 Robustness and Adaptability: Heterogeneity in Power Controls of DC Microgrids 325
11.1 Background 325
11.2 Network Model 328
11.3 Optimal Power Flow of DC Networks 329
11.3.1 OPF Model 329
11.3.2 Uniqueness of Optimal Solution 331
11.4 Control Design 334
11.4.1 Distributed Optimization Algorithm 334
11.4.2 Optimality of Equilibrium 335
11.4.3 Convergence Analysis 338
11.5 Implementation 344
11.6 Case Studies 346
11.6.1 Test System and Data 346
11.6.2 Accuracy Analysis 348
11.6.3 Dynamic Performance Verification 348
11.6.4 Performance in Plug-n-play Operations 352
11.7 Conclusion and Notes 353
Bibliography 354
Appendix A Typical Distributed Optimization Algorithms 357
A.1 Consensus-Based Algorithms 357
A.1.1 Consensus Algorithms 358
A.1.2 Cutting-Plane Consensus Algorithm 359
A.2 First-Order Gradient-Based Algorithms 362
A.2.1 Dual Decomposition 363
A.2.2 Alternating Direction Method of Multipliers 366
A.2.3 Primal-Dual Gradient Algorithm 368
A.2.4 Proximal Gradient Method 371
A.3 Second-Order Newton-Based Algorithms 374
A.3.1 Barrier Method 374
A.3.2 Primal-Dual Interior-Point Method 375
A.4 Zeroth-Order Online Algorithms 377
Bibliography 379
Appendix B Optimal Power Flow of Direct Current Networks 385
B. 1 Mathematical Model 385
B.. 1 Formulation 385
B.1. 2 Equivalent Transformation 387
B. 2 Exactness of SOC Relaxation 388
B.2. 1 SOC Relaxation of OPF in DC Networks 388
B.. 2 Assumptions 388
B.2. 3 Exactness of the SOC Relaxation 389
B.2. 4 Topological Independence 396
B.2. 5 Uniqueness of the Optimal Solution 396
B.2. 6 Branch Flow Model 397
B. 3 Case Studies 399
B.3. 1 16-Bus System 399
B.3. 2 Larger-Scale Systems 401
B. 4 Discussion on Line Constraints 402
B.4. 1 OPF with Line Constraints 402
B.4. 2 Exactness Conditions with Line Constraints 403
B.4. 3 Constructing Approximate Optimal Solutions 406
B.4.3. 1 Direct Construction Method 407
B.4.3. 2 Slack Variable Method 408
Bibliography 409
Index 411
Preface xvii
Acknowledgments xix
1 Introduction 1
1.1 Traditional Hierarchical Control Structure 2
1.1.1 Hierarchical Frequency Control 2
1.1.1.1 Primary Frequency Control 4
1.1.1.2 Secondary Frequency Control 5
1.1.1.3 Tertiary Frequency Control 5
1.1.2 Hierarchical Voltage Control 5
1.1.2.1 Primary Voltage Control 6
1.1.2.2 Secondary Voltage Control 7
1.1.2.3 Tertiary Voltage Control 7
1.2 Transitions and Challenges 7
1.3 Removing Central Coordinators: Distributed Coordination 8
1.3.1 Distributed Control 11
1.3.2 Distributed Optimization 12
1.4 Merging Optimization and Control 13
1.4.1 Optimization-Guided Control 14
1.4.2 Feedback-Based Optimization 16
1.5 Overview of the Book 17
Bibliography 19
2 Preliminaries 23
2.1 Norm 23
2.1.1 Vector Norm 23
2.1.2 Matrix Norm 24
2.2 Graph Theory 26
2.2.1 Basic Concepts 26
2.2.2 Laplacian Matrix 26
2.3 Convex Optimization 28
2.3.1 Convex Set 28
2.3.1.1 Basic Concepts 28
2.3.1.2 Cone 30
2.3.2 Convex Function 31
2.3.2.1 Basic Concepts 31
2.3.2.2 Jensen's Inequality 35
2.3.3 Convex Programming 35
2.3.4 Duality 36
2.3.5 Saddle Point 39
2.3.6 KKT Conditions 39
2.4 Projection Operator 41
2.4.1 Basic Concepts 41
2.4.2 Projection Operator 42
2.5 Stability Theory 44
2.5.1 Lyapunov Stability 44
2.5.2 Invariance Principle 46
2.5.3 Input-Output Stability 47
2.6 Passivity and Dissipativity Theory 49
2.6.1 Passivity 49
2.6.2 Dissipativity 51
2.7 Power Flow Model 52
2.7.1 Nonlinear Power Flow 53
2.7.1.1 Bus Injection Model (BIM) 53
2.7.1.2 Branch Flow Model (BFM) 54
2.7.2 Linear Power Flow 55
2.7.2.1 DC Power Flow 55
2.7.2.2 Linearized Branch Flow 56
2.8 Power System Dynamics 56
2.8.1 Synchronous Generator Model 57
2.8.2 Inverter Model 58
Bibliography 60
3 Bridging Control and Optimization in Distributed Optimal Frequency Control 63
3.1 Background 64
3.1.1 Motivation 64
3.1.2 Summary 66
3.1.3 Organization 67
3.2 Power System Model 67
3.2.1 Generator Buses 68
3.2.2 Load Buses 69
3.2.3 Branch Flows 70
3.2.4 Dynamic Network Model 72
3.3 Design and Stability of Primary Frequency Control 74
3.3.1 Optimal Load Control 74
3.3.2 Main Results 75
3.3.3 Implications 79
3.4 Convergence Analysis 79
3.5 Case Studies 88
3.5.1 Test System 88
3.5.2 Simulation Results 89
3.6 Conclusion and Notes 92
Bibliography 93
4 Physical Restrictions: Input Saturation in Secondary Frequency Control 97
4.1 Background 98
4.2 Power System Model 100
4.3 Control Design for Per-Node Power Balance 101
4.3.1 Control Goals 102
4.3.2 Decentralized Optimal Controller 103
4.3.3 Design Rationale 105
4.3.3.1 Primal-Dual Algorithms 105
4.3.3.2 Design of Controller (4.6) 105
4.4 Optimality and Uniqueness of Equilibrium 108
4.5 Stability Analysis 112
4.6 Case Studies 120
4.6.1 Test System 120
4.6.2 Simulation Results 122
4.6.2.1 Stability and Optimality 122
4.6.2.2 Dynamic Performance 123
4.6.2.3 Comparison with AGC 124
4.6.2.4 Digital Implementation 124
4.7 Conclusion and Notes 128
Bibliography 131
5 Physical Restrictions: Line Flow Limits in Secondary Frequency Control 135
5.1 Background 136
5.2 Power System Model 137
5.3 Control Design for Network Power Balance 138
5.3.1 Control Goals 139
5.3.2 Distributed Optimal Controller 141
5.3.3 Design Rationale 142
5.3.3.1 Primal-Dual Gradient Algorithms 142
5.3.3.2 Controller Design 143
5.4 Optimality of Equilibrium 144
5.5 Asymptotic Stability 148
5.6 Case Studies 155
5.6.1 Test System 155
5.6.2 Simulation Results 156
5.6.2.1 Stability and Optimality 156
5.6.2.2 Dynamic Performance 158
5.6.2.3 Comparison with AGC 158
5.6.2.4 Congestion Analysis 158
5.6.2.5 Time Delay Analysis 161
5.7 Conclusion and Notes 165
Bibliography 165
6 Physical Restrictions: Nonsmoothness of Objective Functions in Load-Frequency Control 167
6.1 Background 167
6.2 Notations and Preliminaries 169
6.3 Power System Model 170
6.4 Control Design 171
6.4.1 Optimal Load Frequency Control Problem 172
6.4.2 Distributed Controller Design 173
6.5 Optimality and Convergence 176
6.5.1 Optimality 176
6.5.2 Convergence 178
6.6 Case Studies 183
6.6.1 Test System 183
6.6.2 Simulation Results 184
6.7 Conclusion and Notes 187
Bibliography 188
7 Cyber Restrictions: Imperfect Communication in Power Control of Microgrids 191
7.1 Background 192
7.2 Preliminaries and Model 193
7.2.1 Notations and Preliminaries 193
7.2.2 Economic Dispatch Model 194
7.3 Distributed Control Algorithms 195
7.3.1 Synchronous Algorithm 195
7.3.2 Asynchronous Algorithm 196
7.4 Optimality and Convergence Analysis 198
7.4.1 Virtual Global Clock 199
7.4.2 Algorithm Reformulation 200
7.4.3 Optimality of Equilibrium 203
7.4.4 Convergence Analysis 204
7.5 Real-Time Implementation 206
7.5.1 Motivation and Main Idea 206
7.5.2 Real-Time ASDPD 208
7.5.2.1 AC MGs 208
7.5.2.2 DC Microgrids 208
7.5.3 Control Configuration 210
7.5.4 Optimality of the Implementation 211
7.6 Numerical Results 213
7.6.1 Test System 213
7.6.2 Non-identical Sampling Rates 214
7.6.3 Random Time Delays 217
7.6.4 Comparison with the Synchronous Algorithm 217
7.7 Experimental Results 219
7.8 Conclusion and Notes 222
Bibliography 224
8 Cyber Restrictions: Imperfect Communication in Voltage Control of Active Distribution Networks 229
8.1 Background 230
8.2 Preliminaries and System Model 232
8.2.1 Note and Preliminaries 232
8.2.2 System Modeling 233
8.3 Problem Formulation 234
8.4 Asynchronous Voltage Control 235
8.5 Optimality and Convergence 237
8.5.1 Algorithm Reformulation 238
8.5.2 Optimality of Equilibrium 242
8.5.3 Convergence Analysis 243
8.6 Implementation 245
8.6.1 Communication Graph 245
8.6.2 Online Implementation 246
8.7 Case Studies 246
8.7.1 8-Bus Feeder System 247
8.7.2 IEEE 123-Bus Feeder System 250
8.8 Conclusion and Notes 253
Bibliography 254
9 Robustness and Adaptability: Unknown Disturbances in Load-Side Frequency Control 257
9.1 Background 258
9.2 Problem Formulation 259
9.2.1 Power Network 259
9.2.2 Power Imbalance 260
9.2.3 Equivalent Transformation of Power Imbalance 261
9.3 Controller Design 263
9.3.1 Controller for Known P ¯_in j 263
9.3.2 Controller for Time-Varying Power Imbalance 264
9.3.3 Closed-Loop Dynamics 265
9.4 Equilibrium and Stability Analysis 266
9.4.1 Equilibrium 266
9.4.2 Asymptotic Stability 269
9.5 Robustness Analysis 274
9.5.1 Robustness Against Uncertain Parameters 274
9.5.2 Robustness Against Unknown Disturbances 275
9.6 Case Studies 277
9.6.1 System Configuration 277
9.6.2 Self-Generated Data 279
9.6.3 Performance Under Unknown Disturbances 282
9.6.4 Simulation with Real Data 282
9.6.5 Comparison with Existing Control Methods 284
9.7 Conclusion and Notes 286
Bibliography 287
10 Robustness and Adaptability: Partial Control Coverage in Transient Frequency Control 289
10.1 Background 289
10.2 Structure-Preserving Model of Nonlinear Power System Dynamics 291
10.2.1 Power Network 291
10.2.2 Synchronous Generators 292
10.2.3 Dynamics of Voltage Phase Angles 293
10.2.4 Communication Network 294
10.3 Formulation of Optimal Frequency Control 294
10.3.1 Optimal Power-Sharing Among Controllable Generators 294
10.3.2 Equivalent Model With Virtual Load 295
10.4 Control Design 296
10.4.1 Controller for Controllable Generators 296
10.4.2 Active Power Dynamics of Uncontrollable Generators 297
10.4.3 Excitation Voltage Dynamics of Generators 298
10.5 Optimality and Stability 298
10.5.1 Optimality 298
10.5.2 Stability 300
10.6 Implementation With Frequency Measurement 306
10.6.1 Estimating Mu I Using Frequency Feedback 306
10.6.2 Stability Analysis 307
10.7 Case Studies 310
10.7.1 Test System and Data 310
10.7.2 Performance Under Small Disturbances 312
10.7.2.1 Equilibrium and its Optimality 312
10.7.2.2 Performance of Frequency Dynamics 313
10.7.3 Performance Under Large Disturbances 316
10.7.3.1 Generator Tripping 317
10.7.3.2 Short-Circuit Fault 318
10.8 Conclusion and Notes 321
Bibliography 322
11 Robustness and Adaptability: Heterogeneity in Power Controls of DC Microgrids 325
11.1 Background 325
11.2 Network Model 328
11.3 Optimal Power Flow of DC Networks 329
11.3.1 OPF Model 329
11.3.2 Uniqueness of Optimal Solution 331
11.4 Control Design 334
11.4.1 Distributed Optimization Algorithm 334
11.4.2 Optimality of Equilibrium 335
11.4.3 Convergence Analysis 338
11.5 Implementation 344
11.6 Case Studies 346
11.6.1 Test System and Data 346
11.6.2 Accuracy Analysis 348
11.6.3 Dynamic Performance Verification 348
11.6.4 Performance in Plug-n-play Operations 352
11.7 Conclusion and Notes 353
Bibliography 354
Appendix A Typical Distributed Optimization Algorithms 357
A.1 Consensus-Based Algorithms 357
A.1.1 Consensus Algorithms 358
A.1.2 Cutting-Plane Consensus Algorithm 359
A.2 First-Order Gradient-Based Algorithms 362
A.2.1 Dual Decomposition 363
A.2.2 Alternating Direction Method of Multipliers 366
A.2.3 Primal-Dual Gradient Algorithm 368
A.2.4 Proximal Gradient Method 371
A.3 Second-Order Newton-Based Algorithms 374
A.3.1 Barrier Method 374
A.3.2 Primal-Dual Interior-Point Method 375
A.4 Zeroth-Order Online Algorithms 377
Bibliography 379
Appendix B Optimal Power Flow of Direct Current Networks 385
B. 1 Mathematical Model 385
B.. 1 Formulation 385
B.1. 2 Equivalent Transformation 387
B. 2 Exactness of SOC Relaxation 388
B.2. 1 SOC Relaxation of OPF in DC Networks 388
B.. 2 Assumptions 388
B.2. 3 Exactness of the SOC Relaxation 389
B.2. 4 Topological Independence 396
B.2. 5 Uniqueness of the Optimal Solution 396
B.2. 6 Branch Flow Model 397
B. 3 Case Studies 399
B.3. 1 16-Bus System 399
B.3. 2 Larger-Scale Systems 401
B. 4 Discussion on Line Constraints 402
B.4. 1 OPF with Line Constraints 402
B.4. 2 Exactness Conditions with Line Constraints 403
B.4. 3 Constructing Approximate Optimal Solutions 406
B.4.3. 1 Direct Construction Method 407
B.4.3. 2 Slack Variable Method 408
Bibliography 409
Index 411
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.
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.
