Direct methods for stability analysis of electric power systems : theoretical foundation BCU methodologies and applications 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 This book describes the BCU method (Boundary of Stability Region Based Controlling Unstable Equilibrium Point method) 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 Widely accepted around the world the BCU method is the only direct method used in the power industry. Direct Methods for Stability Analysis of Electric Power Systems presents a comprehensive theoretical foundation of the method and its numerical implementation. This book provides graduate students researchers and practitioners with theoretical foundations of direct methods energy functions and the BCU method as well as the group based BCU method and its applications. Numerical studies on industrial models and data are also included 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