John Zaborszky

Local bifurcations and feasibility regions in differential-algebraic systems

The dynamics of a large class of physical systems such as the general power system can be represented by parameter-dependent differential-algebraic models of the form x/spl dot/=f and 0=g. Typically, such constrained models have singularities. This paper analyzes the generic local bifurcations including those which are directly related to the singularity. The notion of a feasibility region is introduced and analyzed. It consists of all equilibrium states that can be reached quasistatically from the current operating point without loss of local stability. It is shown that generically loss of stability at the feasibility boundary is caused by one of three different local bifurcations, namely the saddle-node and Hopf bifurcations and a new bifurcation called the singularity induced bifurcation which is analyzed precisely here for the first time. The latter results when an equilibrium point is at the singular surface. Under certain transversality conditions, the change in the eigenstructure of the system Jacobian at the equilibrium is established and the local dynamical structure of the trajectories near this bifurcation point is analyzed.

On the phase portrait of a class of large nonlinear dynamic systems such as the power system

Complete results are presented on the phase portrait of a class of large nonlinear dynamic systems that includes the power system. The connection between the constant energy surface and the stability boundary of the power system is explored. This gives valuable insight into classical stability tests, which are based on the constant energy surface. An approach to stability monitoring of the power system derived from these results is outlined. >

Voltage dynamics: study of a generator with voltage control, transmission, and matched MW load

A comprehensive analysis of the dynamic behavior for a rudimentary but representative model of the power system is carried out for the case when control gain and load are varied as parameters. The voltage dynamics model is subject to algebraic constraints in the form of load flow equations and is studied as a differential-algebraic system in state and parameter spaces. Singularities in the state-space (noncausal points) and bifurcations in the parameter space are the principal and interacting structural elements. A rich structure of bifurcations emerges which is analyzed. The mathematical analysis is facilitated by singularly rescaling time, which transforms the differential algebraic system into a smooth dynamic system. In the state space, the characteristics of stability boundaries are observed and a description of the regions of attraction of all equilibria are given. The loosely understood term of voltage collapse is classified into well-defined types on both the dynamic and parameteric sides. >

On the controllability of a class of discrete bilinear systems

The subject of this paper is the controllability of a class of time invariant discrete bilinear systems. Bilinear systems are classified into two categories: homogeneous and inhomogeneous. Necessity as well as the sufficiency results are obtained by means of decomposing the bilinear system into a linear system and a multiplicative feedback. Byproducts of the decomposition are the notions of multiplicative feedback compensation and the multiplicative feedback with bias compensation which may prove to be alternatives for the classical linear feedback compensation.

Fast time-varying phasor analysis in the balanced three-phase large electric power system

Traditional phasor representation of sinusoidal signals, the standard analytical tool for power system stability analysis, is limited by the quasistationary assumption on the speeds of the phasor states. This paper provides a rigorous formulation of a time-varying phasor representation for the balanced three-phase large power system with no restrictions on the speeds. Power balance equations become a set of differential equations in the phasor dynamic states and singularly perturbed behavior of the resulting dynamics is explored. >

A Clustered Dynamic Model for a class of LInear Autonomous Systems Unsing Simple Enumerative Sorting

Simple, computationally efficient, enumerative techniques are introduced for establishing a clustered semidecoupled dynamic model for a class of autonomous linear systems exemplified by the electromechanical dynamics of the electric power system. Such a model consists of intercluster (usually slow) dynamics and intracluster dynamics (usually fast) which are respectively weakly coupled. The process starts by selecting the point in frequency where intracluster and intercluster dynamics tend to separate. The sorting technique then identifies the number of clusters, the width of separation in frequency, the specifics of the dynamic components and the coupling between them. No matrix manipulations like finding eigenvalues or inverses are required.

A new state space for emergency control in the interconnected power system

A new state space is introduced for the large interconnected power system in which the state of the system can be estimated piecewise by local computations from conventional local measurements. The new state space is defined, its mathematical equivalence to the conventional state space is proven, and initial illustration is provided of its usefulness.

Papers: Local feedback stabilization of large interconnected power systems in emergencies

On-line stabilization of the large interconnected power system poses problems which are unusual among large systems. The system can lose stability in 2-3 s or sooner-yet, even a simplified model of it consists of hundreds or thousands of state variables. Hence, a full state identification or estimation and a dynamic control utilizing the complete system dynamics is computationally out of the question. In fact, of necessity, any centralized action must be quite limited. This leaves the challenging task of using local information to achieve system-wide stabilization. Control is also limited by practical considerations to short pulses of positive or negative power of fixed magnitude. A nonlinear transformation which maps the conventional state space of the power system into a new observation decoupled state space is introduced and shown to be a homeomorphism. This state space has the property that its components can be measured piecemeal in local groups and its origin is the unique stable equilibrium point of the power system when one exists. It is then shown that a feedback control, based on such local information, is capable of assuring system-wide approach to the stable equilibrium point. The existence of a stability region for this local feedback control is also shown.

Global voltage dynamics: study of a generator with voltage control, transmission, and matched MW load

A rudimentary but representative model for voltage dynamics of a power system is studied. A comprehensive analysis of the different types of dynamic behavior across both state and parameter space is given with control gains and load varied. The model is subject to algebraic constraints in the form of load flow equations. A singular transformation changes the resulting singular model into a smooth dynamical system, facilitating the analysis. Singular manifolds in the state space and bifurcation manifolds in the parameter space turn out to be the principal and interacting determinants for the problem. Both local and global bifurcations play vital roles. Some of the bifurcations are directly connected to the singularity. The maximum practical load is increasing with control gain, but the region of transient stability, which is a halfspace at low gains, gradually shrinks down to a point at the Hopf bifurcation. The characteristics of stability boundaries are observed, and the loosely understood term of voltage collapse is classified into well-defined types. Simulations are used to illustrate the mathematical results.<<ETX>>

Paper: Control of reactive power and voltage in emergencies

A method is introduced which assures quite feasible computation of control actions to be taken to remedy voltage problems resulting in a section of a system because of a disturbance. The process is devised so as to conform to the natural structural features of the problem. A monitoring process is proposed. If a viability crisis is detected, first a cluster of interacting control tools are selected to match the cluster of violations. Then an effective viabilizing algorithm is carried out to restore viable voltage values in a manner least disruptive to the system. Note that the pinpointing of the cluster of violated and nonviolated network elements and the controls which are effective can in itself be very valuable information for the operator. The viabilizing algorithm only covers the affected clusters and it does not require on line system wide sensitivity computation. Consequently it is of small dimension and converges fast. Use of this approach should substantially improve both, cut and try approach by the operator on line and straightforward, computationally demanding computer implementations such as optimal load flow or linear programming approaches.