{"product_id":"merging-optimization-and-control-in-power-systems-9781119827924","title":"Merging Optimization and Control in Power Systems","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eForeword xv\u003c\/p\u003e \u003cp\u003ePreface xvii\u003c\/p\u003e \u003cp\u003eAcknowledgments xix\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Introduction 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Traditional Hierarchical Control Structure 2\u003c\/p\u003e \u003cp\u003e1.1.1 Hierarchical Frequency Control 2\u003c\/p\u003e \u003cp\u003e1.1.1.1 Primary Frequency Control 4\u003c\/p\u003e \u003cp\u003e1.1.1.2 Secondary Frequency Control 5\u003c\/p\u003e \u003cp\u003e1.1.1.3 Tertiary Frequency Control 5\u003c\/p\u003e \u003cp\u003e1.1.2 Hierarchical Voltage Control 5\u003c\/p\u003e \u003cp\u003e1.1.2.1 Primary Voltage Control 6\u003c\/p\u003e \u003cp\u003e1.1.2.2 Secondary Voltage Control 7\u003c\/p\u003e \u003cp\u003e1.1.2.3 Tertiary Voltage Control 7\u003c\/p\u003e \u003cp\u003e1.2 Transitions and Challenges 7\u003c\/p\u003e \u003cp\u003e1.3 Removing Central Coordinators: Distributed Coordination 8\u003c\/p\u003e \u003cp\u003e1.3.1 Distributed Control 11\u003c\/p\u003e \u003cp\u003e1.3.2 Distributed Optimization 12\u003c\/p\u003e \u003cp\u003e1.4 Merging Optimization and Control 13\u003c\/p\u003e \u003cp\u003e1.4.1 Optimization-Guided Control 14\u003c\/p\u003e \u003cp\u003e1.4.2 Feedback-Based Optimization 16\u003c\/p\u003e \u003cp\u003e1.5 Overview of the Book 17\u003c\/p\u003e \u003cp\u003eBibliography 19\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Preliminaries 23\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Norm 23\u003c\/p\u003e \u003cp\u003e2.1.1 Vector Norm 23\u003c\/p\u003e \u003cp\u003e2.1.2 Matrix Norm 24\u003c\/p\u003e \u003cp\u003e2.2 Graph Theory 26\u003c\/p\u003e \u003cp\u003e2.2.1 Basic Concepts 26\u003c\/p\u003e \u003cp\u003e2.2.2 Laplacian Matrix 26\u003c\/p\u003e \u003cp\u003e2.3 Convex Optimization 28\u003c\/p\u003e \u003cp\u003e2.3.1 Convex Set 28\u003c\/p\u003e \u003cp\u003e2.3.1.1 Basic Concepts 28\u003c\/p\u003e \u003cp\u003e2.3.1.2 Cone 30\u003c\/p\u003e \u003cp\u003e2.3.2 Convex Function 31\u003c\/p\u003e \u003cp\u003e2.3.2.1 Basic Concepts 31\u003c\/p\u003e \u003cp\u003e2.3.2.2 Jensen’s Inequality 35\u003c\/p\u003e \u003cp\u003e2.3.3 Convex Programming 35\u003c\/p\u003e \u003cp\u003e2.3.4 Duality 36\u003c\/p\u003e \u003cp\u003e2.3.5 Saddle Point 39\u003c\/p\u003e \u003cp\u003e2.3.6 KKT Conditions 39\u003c\/p\u003e \u003cp\u003e2.4 Projection Operator 41\u003c\/p\u003e \u003cp\u003e2.4.1 Basic Concepts 41\u003c\/p\u003e \u003cp\u003e2.4.2 Projection Operator 42\u003c\/p\u003e \u003cp\u003e2.5 Stability Theory 44\u003c\/p\u003e \u003cp\u003e2.5.1 Lyapunov Stability 44\u003c\/p\u003e \u003cp\u003e2.5.2 Invariance Principle 46\u003c\/p\u003e \u003cp\u003e2.5.3 Input–Output Stability 47\u003c\/p\u003e \u003cp\u003e2.6 Passivity and Dissipativity Theory 49\u003c\/p\u003e \u003cp\u003e2.6.1 Passivity 49\u003c\/p\u003e \u003cp\u003e2.6.2 Dissipativity 51\u003c\/p\u003e \u003cp\u003e2.7 Power Flow Model 52\u003c\/p\u003e \u003cp\u003e2.7.1 Nonlinear Power Flow 53\u003c\/p\u003e \u003cp\u003e2.7.1.1 Bus Injection Model (BIM) 53\u003c\/p\u003e \u003cp\u003e2.7.1.2 Branch Flow Model (BFM) 54\u003c\/p\u003e \u003cp\u003e2.7.2 Linear Power Flow 55\u003c\/p\u003e \u003cp\u003e2.7.2.1 DC Power Flow 55\u003c\/p\u003e \u003cp\u003e2.7.2.2 Linearized Branch Flow 56\u003c\/p\u003e \u003cp\u003e2.8 Power System Dynamics 56\u003c\/p\u003e \u003cp\u003e2.8.1 Synchronous Generator Model 57\u003c\/p\u003e \u003cp\u003e2.8.2 Inverter Model 58\u003c\/p\u003e \u003cp\u003eBibliography 60\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Bridging Control and Optimization in Distributed Optimal Frequency Control 63\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Background 64\u003c\/p\u003e \u003cp\u003e3.1.1 Motivation 64\u003c\/p\u003e \u003cp\u003e3.1.2 Summary 66\u003c\/p\u003e \u003cp\u003e3.1.3 Organization 67\u003c\/p\u003e \u003cp\u003e3.2 Power System Model 67\u003c\/p\u003e \u003cp\u003e3.2.1 Generator Buses 68\u003c\/p\u003e \u003cp\u003e3.2.2 Load Buses 69\u003c\/p\u003e \u003cp\u003e3.2.3 Branch Flows 70\u003c\/p\u003e \u003cp\u003e3.2.4 Dynamic Network Model 72\u003c\/p\u003e \u003cp\u003e3.3 Design and Stability of Primary Frequency Control 74\u003c\/p\u003e \u003cp\u003e3.3.1 Optimal Load Control 74\u003c\/p\u003e \u003cp\u003e3.3.2 Main Results 75\u003c\/p\u003e \u003cp\u003e3.3.3 Implications 79\u003c\/p\u003e \u003cp\u003e3.4 Convergence Analysis 79\u003c\/p\u003e \u003cp\u003e3.5 Case Studies 88\u003c\/p\u003e \u003cp\u003e3.5.1 Test System 88\u003c\/p\u003e \u003cp\u003e3.5.2 Simulation Results 89\u003c\/p\u003e \u003cp\u003e3.6 Conclusion and Notes 92\u003c\/p\u003e \u003cp\u003eBibliography 93\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Physical Restrictions: Input Saturation in Secondary Frequency Control 97\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Background 98\u003c\/p\u003e \u003cp\u003e4.2 Power System Model 100\u003c\/p\u003e \u003cp\u003e4.3 Control Design for Per-Node Power Balance 101\u003c\/p\u003e \u003cp\u003e4.3.1 Control Goals 102\u003c\/p\u003e \u003cp\u003e4.3.2 Decentralized Optimal Controller 103\u003c\/p\u003e \u003cp\u003e4.3.3 Design Rationale 105\u003c\/p\u003e \u003cp\u003e4.3.3.1 Primal–Dual Algorithms 105\u003c\/p\u003e \u003cp\u003e4.3.3.2 Design of Controller (4.6) 105\u003c\/p\u003e \u003cp\u003e4.4 Optimality and Uniqueness of Equilibrium 108\u003c\/p\u003e \u003cp\u003e4.5 Stability Analysis 112\u003c\/p\u003e \u003cp\u003e4.6 Case Studies 120\u003c\/p\u003e \u003cp\u003e4.6.1 Test System 120\u003c\/p\u003e \u003cp\u003e4.6.2 Simulation Results 122\u003c\/p\u003e \u003cp\u003e4.6.2.1 Stability and Optimality 122\u003c\/p\u003e \u003cp\u003e4.6.2.2 Dynamic Performance 123\u003c\/p\u003e \u003cp\u003e4.6.2.3 Comparison with AGC 124\u003c\/p\u003e \u003cp\u003e4.6.2.4 Digital Implementation 124\u003c\/p\u003e \u003cp\u003e4.7 Conclusion and Notes 128\u003c\/p\u003e \u003cp\u003eBibliography 131\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Physical Restrictions: Line Flow Limits in Secondary Frequency Control 135\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Background 136\u003c\/p\u003e \u003cp\u003e5.2 Power System Model 137\u003c\/p\u003e \u003cp\u003e5.3 Control Design for Network Power Balance 138\u003c\/p\u003e \u003cp\u003e5.3.1 Control Goals 139\u003c\/p\u003e \u003cp\u003e5.3.2 Distributed Optimal Controller 141\u003c\/p\u003e \u003cp\u003e5.3.3 Design Rationale 142\u003c\/p\u003e \u003cp\u003e5.3.3.1 Primal–Dual Gradient Algorithms 142\u003c\/p\u003e \u003cp\u003e5.3.3.2 Controller Design 143\u003c\/p\u003e \u003cp\u003e5.4 Optimality of Equilibrium 144\u003c\/p\u003e \u003cp\u003e5.5 Asymptotic Stability 148\u003c\/p\u003e \u003cp\u003e5.6 Case Studies 155\u003c\/p\u003e \u003cp\u003e5.6.1 Test System 155\u003c\/p\u003e \u003cp\u003e5.6.2 Simulation Results 156\u003c\/p\u003e \u003cp\u003e5.6.2.1 Stability and Optimality 156\u003c\/p\u003e \u003cp\u003e5.6.2.2 Dynamic Performance 158\u003c\/p\u003e \u003cp\u003e5.6.2.3 Comparison with AGC 158\u003c\/p\u003e \u003cp\u003e5.6.2.4 Congestion Analysis 158\u003c\/p\u003e \u003cp\u003e5.6.2.5 Time Delay Analysis 161\u003c\/p\u003e \u003cp\u003e5.7 Conclusion and Notes 165\u003c\/p\u003e \u003cp\u003eBibliography 165\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Physical Restrictions: Nonsmoothness of Objective Functions in Load-Frequency Control 167\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Background 167\u003c\/p\u003e \u003cp\u003e6.2 Notations and Preliminaries 169\u003c\/p\u003e \u003cp\u003e6.3 Power System Model 170\u003c\/p\u003e \u003cp\u003e6.4 Control Design 171\u003c\/p\u003e \u003cp\u003e6.4.1 Optimal Load Frequency Control Problem 172\u003c\/p\u003e \u003cp\u003e6.4.2 Distributed Controller Design 173\u003c\/p\u003e \u003cp\u003e6.5 Optimality and Convergence 176\u003c\/p\u003e \u003cp\u003e6.5.1 Optimality 176\u003c\/p\u003e \u003cp\u003e6.5.2 Convergence 178\u003c\/p\u003e \u003cp\u003e6.6 Case Studies 183\u003c\/p\u003e \u003cp\u003e6.6.1 Test System 183\u003c\/p\u003e \u003cp\u003e6.6.2 Simulation Results 184\u003c\/p\u003e \u003cp\u003e6.7 Conclusion and Notes 187\u003c\/p\u003e \u003cp\u003eBibliography 188\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Cyber Restrictions: Imperfect Communication in Power Control of Microgrids 191\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Background 192\u003c\/p\u003e \u003cp\u003e7.2 Preliminaries and Model 193\u003c\/p\u003e \u003cp\u003e7.2.1 Notations and Preliminaries 193\u003c\/p\u003e \u003cp\u003e7.2.2 Economic Dispatch Model 194\u003c\/p\u003e \u003cp\u003e7.3 Distributed Control Algorithms 195\u003c\/p\u003e \u003cp\u003e7.3.1 Synchronous Algorithm 195\u003c\/p\u003e \u003cp\u003e7.3.2 Asynchronous Algorithm 196\u003c\/p\u003e \u003cp\u003e7.4 Optimality and Convergence Analysis 198\u003c\/p\u003e \u003cp\u003e7.4.1 Virtual Global Clock 199\u003c\/p\u003e \u003cp\u003e7.4.2 Algorithm Reformulation 200\u003c\/p\u003e \u003cp\u003e7.4.3 Optimality of Equilibrium 203\u003c\/p\u003e \u003cp\u003e7.4.4 Convergence Analysis 204\u003c\/p\u003e \u003cp\u003e7.5 Real-Time Implementation 206\u003c\/p\u003e \u003cp\u003e7.5.1 Motivation and Main Idea 206\u003c\/p\u003e \u003cp\u003e7.5.2 Real-Time ASDPD 208\u003c\/p\u003e \u003cp\u003e7.5.2.1 AC MGs 208\u003c\/p\u003e \u003cp\u003e7.5.2.2 DC Microgrids 208\u003c\/p\u003e \u003cp\u003e7.5.3 Control Configuration 210\u003c\/p\u003e \u003cp\u003e7.5.4 Optimality of the Implementation 211\u003c\/p\u003e \u003cp\u003e7.6 Numerical Results 213\u003c\/p\u003e \u003cp\u003e7.6.1 Test System 213\u003c\/p\u003e \u003cp\u003e7.6.2 Non-identical Sampling Rates 214\u003c\/p\u003e \u003cp\u003e7.6.3 Random Time Delays 217\u003c\/p\u003e \u003cp\u003e7.6.4 Comparison with the Synchronous Algorithm 217\u003c\/p\u003e \u003cp\u003e7.7 Experimental Results 219\u003c\/p\u003e \u003cp\u003e7.8 Conclusion and Notes 222\u003c\/p\u003e \u003cp\u003eBibliography 224\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Cyber Restrictions: Imperfect Communication in Voltage Control of Active Distribution Networks 229\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Background 230\u003c\/p\u003e \u003cp\u003e8.2 Preliminaries and System Model 232\u003c\/p\u003e \u003cp\u003e8.2.1 Note and Preliminaries 232\u003c\/p\u003e \u003cp\u003e8.2.2 System Modeling 233\u003c\/p\u003e \u003cp\u003e8.3 Problem Formulation 234\u003c\/p\u003e \u003cp\u003e8.4 Asynchronous Voltage Control 235\u003c\/p\u003e \u003cp\u003e8.5 Optimality and Convergence 237\u003c\/p\u003e \u003cp\u003e8.5.1 Algorithm Reformulation 238\u003c\/p\u003e \u003cp\u003e8.5.2 Optimality of Equilibrium 242\u003c\/p\u003e \u003cp\u003e8.5.3 Convergence Analysis 243\u003c\/p\u003e \u003cp\u003e8.6 Implementation 245\u003c\/p\u003e \u003cp\u003e8.6.1 Communication Graph 245\u003c\/p\u003e \u003cp\u003e8.6.2 Online Implementation 246\u003c\/p\u003e \u003cp\u003e8.7 Case Studies 246\u003c\/p\u003e \u003cp\u003e8.7.1 8-Bus Feeder System 247\u003c\/p\u003e \u003cp\u003e8.7.2 IEEE 123-Bus Feeder System 250\u003c\/p\u003e \u003cp\u003e8.8 Conclusion and Notes 253\u003c\/p\u003e \u003cp\u003eBibliography 254\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Robustness and Adaptability: Unknown Disturbances in Load-Side Frequency Control 257\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Background 258\u003c\/p\u003e \u003cp\u003e9.2 Problem Formulation 259\u003c\/p\u003e \u003cp\u003e9.2.1 Power Network 259\u003c\/p\u003e \u003cp\u003e9.2.2 Power Imbalance 260\u003c\/p\u003e \u003cp\u003e9.2.3 Equivalent Transformation of Power Imbalance 261\u003c\/p\u003e \u003cp\u003e9.3 Controller Design 263\u003c\/p\u003e \u003cp\u003e9.3.1 Controller for Known P \u003csup\u003e_in\u003c\/sup\u003e \u003csub\u003ej\u003c\/sub\u003e 263\u003c\/p\u003e \u003cp\u003e9.3.2 Controller for Time-Varying Power Imbalance 264\u003c\/p\u003e \u003cp\u003e9.3.3 Closed-Loop Dynamics 265\u003c\/p\u003e \u003cp\u003e9.4 Equilibrium and Stability Analysis 266\u003c\/p\u003e \u003cp\u003e9.4.1 Equilibrium 266\u003c\/p\u003e \u003cp\u003e9.4.2 Asymptotic Stability 269\u003c\/p\u003e \u003cp\u003e9.5 Robustness Analysis 274\u003c\/p\u003e \u003cp\u003e9.5.1 Robustness Against Uncertain Parameters 274\u003c\/p\u003e \u003cp\u003e9.5.2 Robustness Against Unknown Disturbances 275\u003c\/p\u003e \u003cp\u003e9.6 Case Studies 277\u003c\/p\u003e \u003cp\u003e9.6.1 System Configuration 277\u003c\/p\u003e \u003cp\u003e9.6.2 Self-Generated Data 279\u003c\/p\u003e \u003cp\u003e9.6.3 Performance Under Unknown Disturbances 282\u003c\/p\u003e \u003cp\u003e9.6.4 Simulation with Real Data 282\u003c\/p\u003e \u003cp\u003e9.6.5 Comparison with Existing Control Methods 284\u003c\/p\u003e \u003cp\u003e9.7 Conclusion and Notes 286\u003c\/p\u003e \u003cp\u003eBibliography 287\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Robustness and Adaptability: Partial Control Coverage in Transient Frequency Control 289\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Background 289\u003c\/p\u003e \u003cp\u003e10.2 Structure-Preserving Model of Nonlinear Power System Dynamics 291\u003c\/p\u003e \u003cp\u003e10.2.1 Power Network 291\u003c\/p\u003e \u003cp\u003e10.2.2 Synchronous Generators 292\u003c\/p\u003e \u003cp\u003e10.2.3 Dynamics of Voltage Phase Angles 293\u003c\/p\u003e \u003cp\u003e10.2.4 Communication Network 294\u003c\/p\u003e \u003cp\u003e10.3 Formulation of Optimal Frequency Control 294\u003c\/p\u003e \u003cp\u003e10.3.1 Optimal Power-Sharing Among Controllable Generators 294\u003c\/p\u003e \u003cp\u003e10.3.2 Equivalent Model With Virtual Load 295\u003c\/p\u003e \u003cp\u003e10.4 Control Design 296\u003c\/p\u003e \u003cp\u003e10.4.1 Controller for Controllable Generators 296\u003c\/p\u003e \u003cp\u003e10.4.2 Active Power Dynamics of Uncontrollable Generators 297\u003c\/p\u003e \u003cp\u003e10.4.3 Excitation Voltage Dynamics of Generators 298\u003c\/p\u003e \u003cp\u003e10.5 Optimality and Stability 298\u003c\/p\u003e \u003cp\u003e10.5.1 Optimality 298\u003c\/p\u003e \u003cp\u003e10.5.2 Stability 300\u003c\/p\u003e \u003cp\u003e10.6 Implementation With Frequency Measurement 306\u003c\/p\u003e \u003cp\u003e10.6.1 Estimating Μ I Using Frequency Feedback 306\u003c\/p\u003e \u003cp\u003e10.6.2 Stability Analysis 307\u003c\/p\u003e \u003cp\u003e10.7 Case Studies 310\u003c\/p\u003e \u003cp\u003e10.7.1 Test System and Data 310\u003c\/p\u003e \u003cp\u003e10.7.2 Performance Under Small Disturbances 312\u003c\/p\u003e \u003cp\u003e10.7.2.1 Equilibrium and its Optimality 312\u003c\/p\u003e \u003cp\u003e10.7.2.2 Performance of Frequency Dynamics 313\u003c\/p\u003e \u003cp\u003e10.7.3 Performance Under Large Disturbances 316\u003c\/p\u003e \u003cp\u003e10.7.3.1 Generator Tripping 317\u003c\/p\u003e \u003cp\u003e10.7.3.2 Short-Circuit Fault 318\u003c\/p\u003e \u003cp\u003e10.8 Conclusion and Notes 321\u003c\/p\u003e \u003cp\u003eBibliography 322\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Robustness and Adaptability: Heterogeneity in Power Controls of DC Microgrids 325\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Background 325\u003c\/p\u003e \u003cp\u003e11.2 Network Model 328\u003c\/p\u003e \u003cp\u003e11.3 Optimal Power Flow of DC Networks 329\u003c\/p\u003e \u003cp\u003e11.3.1 OPF Model 329\u003c\/p\u003e \u003cp\u003e11.3.2 Uniqueness of Optimal Solution 331\u003c\/p\u003e \u003cp\u003e11.4 Control Design 334\u003c\/p\u003e \u003cp\u003e11.4.1 Distributed Optimization Algorithm 334\u003c\/p\u003e \u003cp\u003e11.4.2 Optimality of Equilibrium 335\u003c\/p\u003e \u003cp\u003e11.4.3 Convergence Analysis 338\u003c\/p\u003e \u003cp\u003e11.5 Implementation 344\u003c\/p\u003e \u003cp\u003e11.6 Case Studies 346\u003c\/p\u003e \u003cp\u003e11.6.1 Test System and Data 346\u003c\/p\u003e \u003cp\u003e11.6.2 Accuracy Analysis 348\u003c\/p\u003e \u003cp\u003e11.6.3 Dynamic Performance Verification 348\u003c\/p\u003e \u003cp\u003e11.6.4 Performance in Plug-n-play Operations 352\u003c\/p\u003e \u003cp\u003e11.7 Conclusion and Notes 353\u003c\/p\u003e \u003cp\u003eBibliography 354\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix A Typical Distributed Optimization Algorithms 357\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eA.1 Consensus-Based Algorithms 357\u003c\/p\u003e \u003cp\u003eA.1.1 Consensus Algorithms 358\u003c\/p\u003e \u003cp\u003eA.1.2 Cutting-Plane Consensus Algorithm 359\u003c\/p\u003e \u003cp\u003eA.2 First-Order Gradient-Based Algorithms 362\u003c\/p\u003e \u003cp\u003eA.2.1 Dual Decomposition 363\u003c\/p\u003e \u003cp\u003eA.2.2 Alternating Direction Method of Multipliers 366\u003c\/p\u003e \u003cp\u003eA.2.3 Primal–Dual Gradient Algorithm 368\u003c\/p\u003e \u003cp\u003eA.2.4 Proximal Gradient Method 371\u003c\/p\u003e \u003cp\u003eA.3 Second-Order Newton-Based Algorithms 374\u003c\/p\u003e \u003cp\u003eA.3.1 Barrier Method 374\u003c\/p\u003e \u003cp\u003eA.3.2 Primal–Dual Interior-Point Method 375\u003c\/p\u003e \u003cp\u003eA.4 Zeroth-Order Online Algorithms 377\u003c\/p\u003e \u003cp\u003eBibliography 379\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix B Optimal Power Flow of Direct Current Networks 385\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eB. 1 Mathematical Model 385\u003c\/p\u003e \u003cp\u003eB.. 1 Formulation 385\u003c\/p\u003e \u003cp\u003eB.1. 2 Equivalent Transformation 387\u003c\/p\u003e \u003cp\u003eB. 2 Exactness of SOC Relaxation 388\u003c\/p\u003e \u003cp\u003eB.2. 1 SOC Relaxation of OPF in DC Networks 388\u003c\/p\u003e \u003cp\u003eB.. 2 Assumptions 388\u003c\/p\u003e \u003cp\u003eB.2. 3 Exactness of the SOC Relaxation 389\u003c\/p\u003e \u003cp\u003eB.2. 4 Topological Independence 396\u003c\/p\u003e \u003cp\u003eB.2. 5 Uniqueness of the Optimal Solution 396\u003c\/p\u003e \u003cp\u003eB.2. 6 Branch Flow Model 397\u003c\/p\u003e \u003cp\u003eB. 3 Case Studies 399\u003c\/p\u003e \u003cp\u003eB.3. 1 16-Bus System 399\u003c\/p\u003e \u003cp\u003eB.3. 2 Larger-Scale Systems 401\u003c\/p\u003e \u003cp\u003eB. 4 Discussion on Line Constraints 402\u003c\/p\u003e \u003cp\u003eB.4. 1 OPF with Line Constraints 402\u003c\/p\u003e \u003cp\u003eB.4. 2 Exactness Conditions with Line Constraints 403\u003c\/p\u003e \u003cp\u003eB.4. 3 Constructing Approximate Optimal Solutions 406\u003c\/p\u003e \u003cp\u003eB.4.3. 1 Direct Construction Method 407\u003c\/p\u003e \u003cp\u003eB.4.3. 2 Slack Variable Method 408\u003c\/p\u003e \u003cp\u003eBibliography 409\u003c\/p\u003e \u003cp\u003eIndex 411\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default 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