A. Bergen

A Structure Preserving Model for Power System Stability Analysis

A new model for the study of power system stability via Lyapunov functions is proposed. The key feature of the model is an assumption of frequency-dependent load power, rather than the usual impedance loads which are subsequently absorbed into a reduced network. The original network topology is explicitly represented. This approach has the important advantage of rigorously accounting for real power loads in the Lyapunov functions. This compares favorably with existing methods involving approximations to allow for the significant transfer conductances in reduced network models. The preservation of network topology can be exploited in stability analysis, with the concepts of critical and vulnerable cutsets playing central roles in dynamic and transient stability evaluation respectively. Of fundamental importance is the feature that the Lyapunov functions give a true representation of the spatial distribution of stored energy in the system

Stability analysis of multimachine power networks with linear frequency dependent loads

A complete stability analysis of a new power system model is presented. The essential feature of the model is the assumption of frequency dependent loads. This facilitates a dynamic representation directly in terms of the network structure. Consequently, concepts and results from circuit theory can play a strong role in the stability analysis of the model. The multivariable Popov criterion is used to obtain general Lure-Postnikov type Lyapunov functions which rigorously allow for the presence of real loads. This has not been possible with the previously used model.

A class of new multistep integration algorithms for the computation of power system dynamical response

The development of a class of efficient numerical integration schemes for computing power system dynamic response is presented. These schemes are derived by making detailed use of the structural properties of the differential-algebraic system representation of the multimachine power system. The nonlinear differential-algebraic system is split into a nonstiff part with long time constants coupled to a stiff part with a sparse Jacobian matrix whose longest time constant is shorter than that of the first part. These two parts are linear in their respective states, i.e. the system is semilinear. With the nonstiff part removed, a smaller set of stiff equations with a smaller conditioning number than the original system is obtained. Consequently, longer stepsizes can be used so as to reduce the computation time. The proposed multistep integration schemes exploit the sparsity, stiffness and semilinearity properties. Numerical results indicate that these schemes operate with good accuracy at stepsizes as large as 100 times those necessary to ensure numerical stability for conventional schemes.

Lyapunov function for multimachine power systems with generator flux decay and voltage dependent loads

This paper considers the direct stability analysis of a power system in which both active and reactive power flows are considered. Flux decay action is included in the generator model. The load is modelled with complex load power as a function of both bus voltage magnitude and frequency. The network model is structure preserving. The stability analysis is based on the use of an energy-type Lyapunov function. The results substantially generalize those given earlier by Bergen and Hill1 where bus voltages were assumed to be constant, and flux decay was neglected.

On input-output stability of nonlinear feedback systems

It is proved that if the input of a nonlinear feedback sytem and its first derivative are bounded, satisfaction of the V. M. Popov Theorem implies that the output is also bounded.

The parabola test for absolute stability

In applying the Popov stability test, a certain straight line is drawn; the Popov locus must lie on one side of this line. Thus geometric considerations alone indicate that the interesting sectors of absolute stability for conditionally stable systems cannot be found directly. However, the straight line of the Popov test may be replaced by a certain parabola and the conditionally stable sectors of absolute stability may then be discovered. The test has its best application for conditionally stable systems but can be used whenever the Popov test can be used. In fact, the Popov straight line may be obtained as a limiting form of the parabola. The test is ordinarily weaker than the Popov test, but nonlinear sectors having a nonzero lower bound may be found more directly.

On third-order systems and Aizerman's conjecture

The Aizerman conjecture is satisfied for a nonlinear feedback system with a third-order linear plant if a simple condition is satisfied. The plant need not be stable or minimum phase.

On instability of feedback systems with a single nonlinear time-varying gain

Encirclement of the critical disk implies instability of a feedback system described by input-output relations just as in the case of systems with a finite-dimensional state representation.

Computation of regions of transient stability of multimachine power systems

A major difficulty in applying Lyapunov theory to the problem of specifying transient stability regions of n -machine power systems is computational complexity, which increases markedly with n . This note outlines a method, requiring only a nominal amount of computation, to determine such regions.

An efficient algorithm for simulation on transients in large power systems

The simulation of the transient response of a large interconnected power system involves the solution of a very large system of differential-algebraic equations under a great variety of initial conditions and disturbances. The demands imposed on a digital transient stability program to i) study larger power system interconnections, ii) provide a more detailed representation of the power system components, and iii) permit the simulation of longer time periods, have the effect of increasing the computing time. The importance of, and the need for, efficient computational schemes is apparent. The method presented in this paper makes detailed use of the structural properties of the differential-algebraic system representation. The nonlinear differential-algebraic system is split into a nonstiff part with long time constants coupled to a stiff part with a sparse Jacobian matrix whose longest time constant is shorter than that of the first part. These two parts are linear in their respective states, i.e., the system is semilinear. With the nonstiff part removed, a smaller set of stiff equations with a smaller conditioning number than the original system is obtained. Consequently, longer stepsizes can be used so as to reduce the computation time. The proposed multistep integration schemes exploit the stiffness and semilinearity properties. Numerical results on a small test problem indicate that these schemes operate with good accuracy at stepsizes as large as 100 times those necessary to ensure numerical stability by more conventional schemes.