There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

The problem of defining and classifying power system stability has been addressed by several previous CIGRE and IEEE Task Force reports. These earlier efforts, however, do not completely reflect current industry needs, experiences and understanding. In particular, the definitions are not precise and the classifications do not encompass all practical instability scenarios. This report developed by a Task Force, set up jointly by the CIGRE Study Committee 38 and the IEEE Power System Dynamic Performance Committee, addresses the issue of stability definition and classification in power systems from a fundamental viewpoint and closely examines the practical ramifications. The report aims to define power system stability more precisely, provide a systematic basis for its classification, and discuss linkages to related issues such as power system reliability and security.

Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

1 Introduction 2 Electromagnetic Transients 3 Synchronous Machine Modeling 4 Synchronous Machine Control Models 5 Single-Machine Dynamic Models 6 Multimachine Dynamic Models 7 Multimachine Simulation 8 Small-Signal Stability 9 Energy Function Methods Appendix A: Integral Manifolds for Model Bibliography Index

Power System Dynamics and Stability

1 Power System Stability in Single Machine System.- 1.1 Introduction.- 1.2 Statement of the Stability Problem.- 1.3 Mathematical Formulation of the Problem.- 1.4 Modeling Issues.- 1.5 Motivation Through Single Machine Infinite Bus System.- 1.6 Chapter Outline.- 2 Energy Functions for Classical Models.- 2.1 Introduction.- 2.2 Internal Node Representation.- 2.3 Energy Functions for Internal Node Models.- 2.4 Individual Machine and other Energy Functions.- 2.5 Structure Preserving Energy Functions.- 2.6 Alternative Form of the Structure Preserving Energy Function.- 2.7 Positive Definiteness of the Energy Integral.- 2.8 Tsolas-Araposthasis-Varaiya Model.- 3 Reduced Order Energy Functions.- 3.1 Introduction.- 3.2 Individual Machine and Group Energy Function.- 3.3 Simplified Form of the Individual Machine Energy Function.- 3.4 Cutset Energy Function.- 3.5 Example of Cutset Energy Function.- 3.6 Extended Equal Area Criterion (EEAC).- 3.7 The Quasi Unstable Equilibrium Point (QUEP) Method.- 3.8 Decomposition-Aggregation Method.- 3.9 Time Scale Energies.- 4 Energy Functions with Detailed Models of Synchronous Machines and Its Control.- 4.1 Introduction.- 4.2 Single Machine System With Flux Decay Model.- 4.3 Multi-Machine Systems With Flux Decay Model (Method of Parameter Variations).- 4.4 Lyapunov Functions for Multi-Machine Systems With Flux Decay Model.- 4.5 Multi-Machine Systems With Flux Decay Models and AVR.- 4.6 Energy Functions With Detailed Models.- 4.7 Lyapunov Function for Multi-Machine Systems With Flux Decay and Nonlinear Voltage Dependent Loads.- 5 Region of Stability in Power Systems.- 5.1 Introduction.- 5.2 Characterization of the Stability Boundary.- 5.3 Region of Stability.- 5.4 Method of Hyperplanes and Hypersurfaces.- 5.5 Potential Energy Boundary Surface (PEBS) Method.- 5.6 Hybrid Method Using the Gradient System.- 6 Practical Applications of the Energy Function Method.- 6.1 Introduction.- 6.2 The Controlling u.e.p. Method.- 6.3 Modifications to the Controlling u.e.p. Method.- 6.4 Potential Energy Boundary Surface (PEBS) Method.- 6.5 Mode of Instability (MOI) Method.- 6.6 Dynamic Security Assessment.- 7 Future Research Issues.- Appendix A 10 Machine 39 Bus System Data.- References.

Energy function analysis for power system stability

The development of trajectory sensitivity analysis for hybrid systems, such as power systems, is presented in the paper. A hybrid system model which has a differential-algebraic-discrete (DAD) structure is proposed. This model forms the basis for the subsequent sensitivity analysis. Crucial to the analysis is the development of jump conditions describing the behavior of sensitivities at discrete events, such as switching and state resetting. The efficient computation of sensitivities is discussed. A number of examples are presented to illustrate various aspects of the theory. It is shown that trajectory sensitivities provide insights into system behavior which cannot be obtained from traditional simulation.

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Trajectory sensitivity analysis of hybrid systems

In this paper, the traditional automatic generation control (AGC) two-area system is modified to take into account the effect of bilateral contracts on the dynamics. The concept of distribution companies (DISCO) participation matrix to simulate these bilateral contracts is introduced and reflected in the two-area block diagram. Trajectory sensitivities are used to obtain optimal parameters of the system using a gradient Newton algorithm.

/pdf/simulation-and-optimization-in-an-agc-system-after-3sbh1k98e2.pdf

Simulation and Optimization in an AGC System after Deregulation

The relationship is presented between a detailed power system dynamic model and a standard load-flow model. The linearized dynamic model is examined to show how the load-flow Jacobian appears in the system dynamic-state Jacobian for evaluating steady-state stability. Two special cases are given for the situation when singularity of the load-flow Jacobian implies singularity of the system dynamic-state Jacobian. The standard load-flow Jacobian can provide information about the existence of a steady-state equilibrium point for a specified level of loading or interchange. There are two very special cases when the determinant of the standard load-flow Jacobian implies something about the steady-state stability of a dynamic model. Both of these cases involve very drastic assumptions about the synchronous machines and their control systems. The load level which produces a singular load-flow Jacobian should be considered an optimistic upper bound on maximum loadability. >

M.A. Pai

Papers

Power System Dynamics and Stability

Energy function analysis for power system stability

Trajectory sensitivity analysis of hybrid systems

Simulation and Optimization in an AGC System after Deregulation

Power system steady-state stability and the load-flow Jacobian