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The controlling unstable equilibrium point (CUEP) method developed for the traditional network-reduction power-system (transient-stability) model requires some modification for direct stability analysis of the network-preserving power system model. This is mainly due to the peculiar nonlinear jump behaviors occurring in network-preserving power system model. An extended CUEP method suitable for direct stability analysis of network-preserving models is developed. The extended CUEP method examines both external jump behaviors and stability regions of network-preserving models. It is shown that the target point of the fault-clearing point (i.e., the initial point of the post-fault system), instead of the fault-clearing point as in classical case, plays a crucial role in direct stability analysis of network-preserving power systems models. A theoretical foundation for the extended CUEP method is developed. A variation of the extended CUEP method is presented. Index Terms--Controlling unstable equilibrium point (CUEP) method, direct methods, direct stability analysis, transient-stability analysis.

Some results of higher dimensional space periodic solution of differential and algebraic systems are presented. The structure preserving models are considered for power systems with loads influenced by periodic disturbance. The existence and uniqueness of periodic solutions for multimachine power systems with nonlinear loads are given. The existent conditions of periodic solution for multimachine power systems with constraints are obtained. Some examples are given to illustrate the feasibility by higher dimensional space periodic solution theory. Index Terms--Differential and algebraic systems (DAS), existence, periodic perturbation, periodic solution, uniqueness.

The paper provides a unified exposition of various definitions of reactive (or inactive) power and of various compensation methods. In particular, it presents a comparative Hilbert space analysis of definitions used in power system studies and in electric drives (the so called instantaneous reactive power). The paper casts several compensation strategies (such as compensation without energy storage and compensation with linear shunt components) in a common framework, and utilizes the concept of orthogonal projections on suitable subspaces as the main analytical and computational tool. Various notions are illustrated on two examples. Index Terms--Frequency domain, instantaneous reactive power, reactive power.

A power system is employed to illustrate how we can apply the singular perturbation theory to decompose a full system into two subsystems, and slow and fast subsystems, respectively. Then we can study the qualitative properties of their solutions and finally obtain the stability region and the analytical expression of the approximate stability boundary of the operation point of the full system by numerical simulation and computing the local quadratic approximation of one-dimensional stable manifold at saddle point. Furthermore, we consider the effects of changing of the parameters on the size of the stability region. Index Terms--Electrical power systems, geometric singular perturbation, stability boundary, stability regions.

This paper reports the occurrence of a rare phenomenon in dynamical systems when four different attractors namely a stable equilibrium, a stable limit cycle and two strange attractors coexist in a fundamental power system model. The paper shows that power system operation could get trapped into sustained chaotic oscillations after a large disturbance even when there exists a viable stable equilibrium point in the model. Index Terms - Bifurcations, chaos, nonlinear dynamics, power systems.

Torque per unit velocity is a practical method to study the subsynchronous resonance instability of electric power systems. The torque per unit velocity method is justified as an eigenvalue perturbation technique and extended to power systems with thyristor switching devices and multiple torsional modes. A new method to estimate the damping of the swing mode is proposed.

The accuracy of stability assessment provided by the transient energy function (TEF) method depends on the determination of the controlling unstable equilibrium point (UEP). The technique that currently determines the controlling UEP in the TEF method is based on the so-called exit point method and has also been recently labeled the BCU method. The exit point method consists of two basic steps. First, the exit point is approximated by the point [[Theta].sup.egsa], where the first maximum of the potential energy along the fault-on trajectory is encountered. Second, the minimum gradient point [[Theta].sup.mgp] along the trajectory from [[Theta].sup.egsa] is computed. The controlling UEP is then obtained by solving a system of nonlinear algebraic equations with [[Theta].sup.mgp] as an initial guess. It has been observed that this method lacks robustness in the sense that the following two problems may occur. 1) There may be no detection of the minimum gradient point [[Theta].sup.mgp] and hence, no determination of the controlling UEP, 2) if [[Theta].sup.mgp] is found, then based on the definition of the controlling UEP, it may not be in the domain of convergence of the controlling UEP for the particular solving algorithm used. Hence, another equilibrium point, possibly a stable equilibrium point, not the controlling UEP will be located. This results in a flawed transient stability assessment. The result of this research has been the development of a new numerical technique for determining the controlling UEP. With an initial starting point that is close to the exit point this technique efficiently produces a sequence of points. An analytical foundation for this method is given which shows that under certain assumptions this sequence will converge to the controlling UEP. Hence this new technique exhibits a substantial improvement over the exit point method because of the following reasons: (1) the technique does not attempt to detect the point [[Theta].sup.mgp], (2) the technique can produce a point that is close to the controlling UEP thus avoiding a domain of convergence problem. The analytical foundation is provided for the unloaded gradient system, but an application of the technique to the IEEE 50-generator system shows that satisfactory stability assessment is also obtained for more general systems, for which the exit point method fails.

Bifurcations occurring in power system models exhibiting voltage collapse have been the subject of several recent studies. Although such models have been shown to admit a variety of bifurcation phenomena, the view that voltage collapse is triggered by possibly the simplest of these, namely by the (static) saddle node bifurcation of the nominal equilibrium, has been the dominant one. The authors have recently shown that voltage collapse can occur 'prior' to the saddle node bifurcation. In the present paper, a new dynamical mechanism for voltage collapse is determined: the boundary crisis of a strange attractor or synonymously a chaotic blue sky bifurcation. This determination is reached for an example power system model akin to one studied in several recent papers. More generally, blue sky bifurcations (not necessarily chaotic) are identified as important mechanisms deserving further consideration in the study of voltage collapse.

This paper is concerned with modeling and analysis of an interconnected electric power system for frequency ranges in which phasor dynamics of transmission lines and loads may be important to include. In most studies, only phasor dynamics of generators are taken into account. Dynamics of all other system components (transmission lines, loads, and generator stator windings) are assumed stable and instantaneous (static). While several papers have examined phasor dynamics of the transmission lines, particularly when these are equipped with FACTS devices, no systematic investigation has been carried out concerning validity of static load models in this case. It is shown in this paper that problems may arise in particular when a static constant power load model is used at the same time that phasor dynamics of transmission lines are included. Standard singular perturbation-based arguments for neglecting load dynamics are shown not to be applicable in this case. More generally, the paper raises a general concern about consistency of electric power system models in frequency ranges where phasor dynamics of the devices typically assumed to be static must be taken into consideration. Index Terms--Electric power systems, flexible AC transmission systems (FACTS) control, load modeling, modeling, phasor dynamics.

As the computational and analytical methods for voltage stability assessment become more mature, voltage control becomes a primary issue. Power systems are large, nonlinear, and dynamic. During the last three decades, the application of differential geometry in the area of nonlinear control has produced significant results for the controllability analysis. This paper is concerned with the controllability problem of power systems using the differential geometric methods. The modeled control devices include the mechanical power input to generators, VAr compensation devices, and tap settings of the on-load tap changers. Conceptually, the main result of this research is the characterization and construction of a complete controllability region, within which a power system can be steered from one state to another by use of piecewise constant controls. The results are obtained for two cases: 1) unbounded piecewise constant controls and 2) bounded piecewise constant controls. A complete controllability region identifies the limitations of the available controls of a power system. The proposed method is believed to be significant since it is a new and systematic approach to the analysis of power system controllability using a nonlinear control system model. Index Terms - Complete controllability, controlled dynamical systems, foliation, nonlinear dynamics, power systems.