Preface -- Acknowledgments -- 1. Introduction and Overview -- 1.1 Introduction -- 1.2 Trends of Operating Environment -- 1.3 Online TSA -- 1.4 Need for New Tools -- 1.5 Direct Methods: Limitations and Challenges -- 1.6 Purposes of This Book -- 2. System Modeling and Stability Problems -- 2.1 Introduction -- 2.2 Power System Stability Problem -- 2.3 Model Structures and Parameters -- 2.4 Measurement-Based Modeling -- 2.5 Power System Stability Problems -- 2.6 Approaches for Stability Analysis -- 2.7 Concluding Remarks -- 3. Lyapunov Stability and Stability Regions of Nonlinear Dynamical Systems -- 3.1 Introduction -- 3.2 Equilibrium Points and Lyapunov Stability -- 3.3 Lyapunov Function Theory -- 3.4 Stable and Unstable Manifolds -- 3.5 Stability Regions -- 3.6 Local Characterizations of Stability Boundary -- 3.7 Global Characterization of Stability Boundary -- 3.8 Algorithm to Determine the Stability Boundary -- 3.9 Conclusion -- 4. Quasi-Stability Regions: Analysis and Characterization -- 4.1 Introduction -- 4.2 Quasi-Stability Region -- 4.3 Characterization of Quasi-Stability Regions -- 4.4 Conclusions -- 5. Energy Function Theory and Direct Methods -- 5.1 Introduction -- 5.2 Energy Functions -- 5.3 Energy Function Theory -- 5.4 Estimating Stability Region Using Energy Functions -- 5.5 Optimal Schemes for Estimating Stability Regions -- 5.6 Quasi-Stability Region and Energy Function -- 5.7 Conclusion -- 6. Constructing Analytical Energy Functions for Transient Stability Models -- 6.1 Introduction -- 6.2 Energy Functions for Lossless Network-Reduction Models -- 6.3 Energy Functions for Lossless Structure-Preserving Models -- 6.4 Nonexistence of Energy Functions for Lossy Models -- 6.5 Existence of Local Energy Functions -- 6.6 Concluding Remarks -- 7. Construction of Numerical Energy Functions for Lossy Transient Stability Models -- 7.1 Introduction -- 7.2 A Two-Step Procedure -- 7.3 First Integral-Based Procedure -- 7.4 Ill-Conditioned Numerical Problems -- 7.5 Numerical Evaluations of Approximation Schemes.

7.6 Multistep Trapezoidal Scheme -- 7.7 On the Corrected Numerical Energy Functions -- 7.8 Concluding Remarks -- 8. Direct Methods for Stability Analysis: An Introduction -- 8.1 Introduction -- 8.2 A Simple System -- 8.3 Closest UEP Method -- 8.4 Controlling UEP Method -- 8.5 PEBS Method -- 8.6 Concluding Remarks -- 9. Foundation of the Closest UEP Method -- 9.1 Introduction -- 9.2 A Structure-Preserving Model -- 9.3 Closest UEP -- 9.4 Characterization of the Closest UEP -- 9.5 Closest UEP Method -- 9.6 Improved Closest UEP Method -- 9.7 Robustness of the Closest UEP -- 9.8 Numerical Studies -- 9.9 Conclusions -- 10. Foundations of the Potential Energy Boundary Surface Method -- 10.1 Introduction -- 10.2 Procedure of the PEBS Method -- 10.3 Original Model and Artifi cial Model -- 10.4 Generalized Gradient Systems -- 10.5 A Class of Second-Order Dynamical Systems -- 10.6 Relation between the Original Model and the Artifi cial Model -- 10.7 Analysis of the PEBS Method -- 10.8 Concluding Remarks -- 11. Controlling UEP Method: Theory -- 11.1 Introduction -- 11.2 The Controlling UEP -- 11.3 Existence and Uniqueness -- 11.4 The Controlling UEP Method -- 11.5 Analysis of the Controlling UEP Method -- 11.6 Numerical Examples -- 11.7 Dynamic and Geometric Characterizations -- 11.8 Concluding Remarks -- 12. Controlling UEP Method: Computations -- 12.1 Introduction -- 12.2 Computational Challenges -- 12.3 Constrained Nonlinear Equations for Equilibrium Points -- 12.4 Numerical Techniques for Computing Equilibrium Points -- 12.5 Convergence Regions of Equilibrium Points -- 12.6 Conceptual Methods for Computing the Controlling UEP -- 12.7 Numerical Studies -- 12.8 Concluding Remarks -- 13. Foundations of Controlling UEP Methods for Network-Preserving Transient Stability Models -- 13.1 Introduction -- 13.2 System Models -- 13.3 Stability Regions -- 13.4 Singular Perturbation Approach -- 13.5 Energy Functions for Network-Preserving Models -- 13.6 Controlling UEP for DAE Systems.

13.7 Controlling UEP Method for DAE Systems -- 13.8 Numerical Studies -- 13.9 Concluding Remarks -- 14. Network-Reduction BCU Method and Its Theoretical Foundation -- 14.1 Introduction -- 14.2 Reduced-State System -- 14.3 Analytical Results -- 14.4 Static and Dynamic Relationships -- 14.5 Dynamic Property (D3) -- 14.6 A Conceptual Network-Reduction BCU Method -- 14.7 Concluding Remarks -- 15. Numerical Network-Reduction BCU Method -- 15.1 Introduction -- 15.2 Computing Exit Points -- 15.3 Stability-Boundary-Following Procedure -- 15.4 A Safeguard Scheme -- 15.5 Illustrative Examples -- 15.6 Numerical Illustrations -- 15.7 IEEE Test System -- 15.8 Concluding Remarks -- 16. Network-Preserving BCU Method and Its Theoretical Foundation -- 16.1 Introduction -- 16.2 Reduced-State Model -- 16.3 Static and Dynamic Properties -- 16.4 Analytical Results -- 16.5 Overall Static and Dynamic Relationships -- 16.6 Dynamic Property (D3) -- 16.7 Conceptual Network-Preserving BCU Method -- 16.8 Concluding Remarks -- 17. Numerical Network-Preserving BCU Method -- 17.1 Introduction -- 17.2 Computational Considerations -- 17.3 Numerical Scheme to Detect Exit Points -- 17.4 Computing the MGP -- 17.5 Computation of Equilibrium Points -- 17.6 Numerical Examples -- 17.7 Large Test Systems -- 17.8 Concluding Remarks -- 18. Numerical Studies of BCU Methods from Stability Boundary Perspectives -- 18.1 Introduction -- 18.2 Stability Boundary of Network-Reduction Models -- 18.3 Network-Preserving Model -- 18.4 One Dynamic Property of the Controlling UEP -- 18.5 Concluding Remarks -- 19. Study of the Transversality Conditions of the BCU Method -- 19.1 Introduction -- 19.2 A Parametric Study -- 19.3 Analytical Investigation of the Boundary Property -- 19.4 The Two-Machine Infi nite Bus (TMIB) System -- 19.5 Numerical Studies -- 19.6 Concluding Remarks -- 20. The BCU-Exit Point Method -- 20.1 Introduction -- 20.2 Boundary Property -- 20.3 Computation of the BCU-Exit Point -- 20.4 BCU-Exit Point and Critical Energy.

20.5 BCU-Exit Point Method -- 20.6 Concluding Remarks -- 21. Group Properties of Contingencies in Power Systems -- 21.1 Introduction -- 21.2 Groups of Coherent Contingencies -- 21.3 Identifi cation of a Group of Coherent Contingencies -- 21.4 Static Group Properties -- 21.5 Dynamic Group Properties -- 21.6 Concluding Remarks -- 22. Group-Based BCU-Exit Method -- 22.1 Introduction -- 22.2 Group-Based Verifi cation Scheme -- 22.3 Linear and Nonlinear Relationships -- 22.4 Group-Based BCU-Exit Point Method -- 22.5 Numerical Studies -- 22.6 Concluding Remarks -- 23. Group-Based BCU-CUEP Methods -- 23.1 Introduction -- 23.2 Exact Method for Computing the Controlling UEP -- 23.3 Group-Based BCU-CUEP Method -- 23.4 Numerical Studies -- 23.5 Concluding Remarks -- 24. Group-Based BCU Method -- 24.1 Introduction -- 24.2 Group-Based BCU Method for Accurate Critical Energy -- 24.3 Group-Based BCU Method for CUEPs -- 24.4 Numerical Studies -- 24.5 Concluding Remarks -- 25. Perspectives and Future Directions -- 25.1 Current Developments -- 25.2 Online Dynamic Contingency Screening -- 25.3 Further Improvements -- 25.4 Phasor Measurement Unit (PMU)-Assisted Online ATC Determination -- 25.5 Emerging Applications -- 25.6 Concluding Remarks -- Appendix -- A1.1 Mathematical Preliminaries -- A1.2 Proofs of Theorems in Chapter 9 -- A1.3 Proofs of Theorems in Chapter 10 -- Bibliography -- Index.