• Declaration of Authorship (2)
• Abstract (4)
• Acknowledgements (5)
• List of Figures (10)
• List of Tables (13)
• Abbreviations
• Symbols
• 1 Introduction (17)
  • 1.1 Definitions (18)
  • 1.2 Transient Stability Analysis: Background (19)
  • 1.3 Transient Stability Study Methods (21)
    • 1.3.1 Time Domain Methods (21)
    • 1.3.2 Probabilistic Methods (22)
    • 1.3.3 Expert System Methods (22)
    • 1.3.4 Database or Pattern Recognition Methods (22)
    • 1.3.5 Equal Area Criterion: An Elementary View of Transient Stability (23)
    • 1.3.6 Transient Energy Function Method (27)
      • 1.3.6.1 Gorev's Energy Criteria (29)
      • 1.3.6.2 Magnusson's Method (31)
      • 1.3.6.3 Aylett's Method (31)
      • 1.3.6.4 Tavora and Smith's Method (32)
      • 1.3.6.5 Attay et al.'s Work (33)
  • 1.4 Current Trends in the Direct Methods (34)
  • 1.5 Dynamic Security Assessment in Restructured Power Systems (36)
  • 1.6 Thesis Outline (38)
• 2 Nonlinear Systems and Stability Theory (40)
  • 2.1 Second-Order Systems (47)
    • 2.1.1 Qualitative Behavior of Linear Systems (50)
      • 2.1.1.1 Real and Distinct Eigenvalues (51)
      • 2.1.1.2 Complex Eigenvalues (54)
      • 2.1.1.3 Nonzero Multiple Eigenvalues (55)
      • 2.1.1.4 One or Both Eigenvalues Equal to Zero (57)
    • 2.1.2 Behavior of Nonlinear Systems about the Equilibrium Point (58)
    • 2.1.3 Multiple Equilibria (61)
    • 2.1.4 Qualitative Behavior Near Equilibrium Points (63)
  • 2.2 Stability in the Sense of Lyapunov (67)
  • 2.3 Domain of Attraction (75)
    • 2.3.1 The Closest Unstable Equilibrium Point Method (78)
    • 2.3.2 Controlling Unstable Equilibrium Point (80)
    • 2.3.3 Potential Energy Boundary Surface (82)
      • 2.3.3.1 Theoretical Foundation for the PEBS (84)
    • 2.3.4 Accuracy of PEBS Method vs Controlling UEP Method (85)
• 3 Transient Stability Assessment Using Transient Energy Function (89)
  • 3.1 The Classical Power System Model (89)
  • 3.2 System Conditions at Fault-Clearing Instant (93)
  • 3.3 Motion Equations Relative to the Center of Inertia (97)
  • 3.4 Derivation of Transient Energy Function (98)
  • 3.5 Corrected Kinetic Energy (103)
  • 3.6 Transient Stability Assessment and Concept of Controlling UEP (107)
  • 3.7 Determination of the Controlling UEP (116)
    • 3.7.1 Controlling UEP Determination Using MOD Method (117)
      • 3.7.1.1 Obtaining Candidate UEPs (120)
      • 3.7.1.2 Approximating the UEPs for the Candidate MODs (123)
    • 3.7.2 Determination of the Controlling UEP Using the BCU Method (127)
  • 3.8 Calculation of Exact Controlling UEP (130)
    • 3.8.1 Calculation of the Jacobian Matrix for the Mismatch Function (133)
    • 3.8.2 Calculation of the Hessian Matrix (138)
    • 3.8.3 Case Study: Calculating the Exact Controlling UEP (142)
      • 3.8.3.1 IEEE 17-Generator 162-Bus System (143)
      • 3.8.3.2 IEEE 50-Generator 145-Bus System (145)
• 4 Relieving Transient Stability Constraint (148)
  • 4.1 Energy Margin and the its Functional Dependence (149)
    • 4.1.1 Sensitivity of Reduced Admittance Matrix (151)
    • 4.1.2 Sensitivity of Controlling UEP and Predisturbance SEP (154)
    • 4.1.3 Sensitivity of Angle and Speed at Fault Clearing Instant (160)
    • 4.1.4 Sensitivity of the Other Dependent Variables (163)
  • 4.2 Sensitivity of Static Operational Parameters (165)
    • 4.2.1 Complex Power Equations and the Jacobian Matrix (165)
    • 4.2.2 Sensitivity of Voltage at Load Controlled Buses (168)
    • 4.2.3 Sensitivity of Reactive Power of Generators and the Generation of Slack Bus (170)
    • 4.2.4 Sensitivity of Generators' Internal Voltage (172)
    • 4.2.5 Sensitivity of Transmission Lines' Flow (174)
  • 4.3 Relieving Transient Stability Constraint in Electricity Markets (176)
  • 4.4 Allocating the Cost of Transient Stability Relief in Bilateral Electricity Markets (183)
    • 4.4.1 Nodal Method (187)
    • 4.4.2 Proposed Method (187)
    • 4.4.3 Shapley Value (190)
    • 4.4.4 Case Study (192)
      • 4.4.4.1 Single Constraint (193)
      • 4.4.4.2 Multiple Constraints (195)
  • 4.5 Incorporating the Static Constraints in Transient Stability Relief (199)
    • 4.5.1 3-Machine System (200)
    • 4.5.2 New England 10-Machine System (202)
• 5 Conclusions (208)
  • 5.1 Concluding Remarks (208)
  • 5.2 Contributions (209)
  • 5.3 Future Research (210)
• A Numerical Methods for Solving the Differential Equations (211)
  • A.1 Euler Method (212)
  • A.2 Modified Euler Method (213)
  • A.3 Runge-Kutta Methods (214)
    • A.3.1 Second-Order R-K Method (215)
    • A.3.2 Fourth-Order R-K Method (216)
• B Line Search Methods in Unconstrained Optimization (217)
  • B.1 Introduction: Overview of methods (217)
  • B.2 Search Direction for Line Search Methods (219)
    • B.2.1 Steepest Descent Direction (219)
    • B.2.2 Newton Direction (220)
  • B.3 Step Length (222)
  • B.4 Termination Conditions in Line Search Methods (223)
    • B.4.1 Wolfe Conditions and Strong Wolfe Conditions (223)
    • B.4.2 Goldstein Condition (226)
  • B.5 Interpolation (228)
    • B.5.1 Quadratic Interpolation (229)
    • B.5.2 Cubic Interpolation (230)
  • B.6 Implementing Line Search Algorithm (231)
• C Transient Stability Test Systems (235)
  • C.1 Western System Coordinating Council Power System (235)
  • C.2 17-Generator 162-Bus System (237)
  • C.3 50-Generator 145-Bus System (256)
• Bibliography (280)