Leon Y. Bahar

Static bifurcations in electric power networks: Loss of steady-state stability and voltage collapse

This paper presents an analysis of static stability in electric power systems. The study is based on a model consisting of the classical swing equation characterization for generators and constant admittance, PV bus and/or PQ bus load representations which, in general, leads to a semi-explicit (or constrained) system of differential equations. A precise definition of static stability is given and basic concepts of static bifurcation theory are used to show that this definition does include conventional notions of steady-state stability and voltage collapse, but it provides a basis for rigorous analysis. Static bifurcations of the load flow equations are analyzed using the Liapunov-Schmidt reduction and Taylor series expansion of the resulting reduced bifurcation equation. These procedures have been implemented using symbolic computation (in MASYMA). It is shown that static bifurcations of the load flow equations are associated with either divergence-type instability or loss of causality. Causality issues are found to be an important factor in understanding voltage collapse and play a central role in organizing global power system dynamics when loads other than constant admittance are present.

STATE SPACE AFPROACH TO THERMOELASTICITY

The method of the matrix exponential, which constitutes the basis of the state space approach of modern control theory, is applied to the nondimensional equations of coupled thermoelasticity. The disadvantages of using the thermoelastic potential are thus avoided. The results obtained can be used to generate solutions in the Laplace-transform domain to a broad class of problems in thermoelasticity. Applications to problems pertaining to a half-space and a layer are presented.

A state space approach to elasticity

Abstract The two-dimensional, plane stress problem of linear elasticity is analyzed within a state space framework. The medium considered is homogeneous and isotropic. Vlasovs mixed formulation of elasticity is used throughout. The field equations are derived in closed form, thus avoiding Vlasovs intermediate infinite series solution. Finally, all the properties of the transfer matrix are shown to follow directly from embedding the problem into a state space setting.

A direct construction of first integrals for certain non-linear dynamical systems

Abstract A direct, constructive approach to the problem of finding first integrals of certain non-linear, second order ordinary differential equations is presented. The idea is motivated by the construction of the energy integral for the equations of motion of the corresponding conservative systems. Although the method developed for the class of equations studied herein is elementary, it yields the same results as the more advanced group-theoretical methods, such as the use of symmetries] in the context of Noethers theorem. The approach reveals some interesting features when it is specialized to the case of linear equations. Finally, a two-dimensional example is considered by extending the methodology developed for scalar equations to their vector counterparts. It is shown that, as a consequence, a first integral which is independent of the energy integral exists for a particular Hamiltonian of the Contopoulos type.

Extension of Noether's theorem to constrained non-conservative dynamical systems

Abstract A method based on a differential variational principle is developed in order to extend Noethers theorem to constrained non-conservative dynamical systems. The result is applied to generate constants of the motion for a generic example of a non-linear, dissipative dynamical system with time-varying coefficients represented by the Emden equation. The converse of Noethers theorem, whereby the symmetries of the system are determined from the knowledge of the Lagrangian and a first integral is also considered for both the Emden equation, and that of the damped harmonic oscillator. It is further shown that the presence of ideal constraints (whether holonomic or non-holonomic) does not affect the statement of Noethers theorem. The constraints affect the Jacobi energy integral, however, because they enter into consideration through real work instead of virtual work. It is shown that the Jacobi integral is conserved provided that: (a) the Lagrangian is explicitly independent of time, (b) the real power of the generalized forces not derivable from a potential vanish, (c) the holonomic constraints are explicitly independent of time, (d) the non-holonomic constraints are linear and homogeneous in the generalized velocities.

Energy-like Lyapunov functions for power system stability analysis

In this paper, a concept of strong stability of an equilibrium point of an electric power system is introduced. It is shown that almost all stable equilibria of the standard transient stability model are strongly stable and that strong stability is a necessary and sufficient condition for the existence of a local energy-like Lyapunov function for all small perturbations of the nominal system. Such a Lyapunov function is explicitly constructed. A complete local analysis of the stability of power system equilibria in the presence of transfer conductances is given.

CONNECTION BETWEEN THE THERMOELASTIC POTENTIAL AND THE STATE SPACE FORMULATION OF THERMOELASTICITY

The connection between the thermoelastic potential and the state space approach to thermoelasticity is established. It is shown that a formulation based on a state vector that consists of the Laplace transforms of the thermoelastic potential and its three spatial derivatives has advantages owing to the simpler nature of the matrix involved.

COUPLED THERMOELASTICITY OF A LAYERED MEDIUM

The transfer matrix method recently developed by the authors is applied to a layered medium. The components of the state vector are taken as the temperature, the displacement, the heat flux, and the strain. Interface conditions are automatically satisfied by multiplying the initial state vector by a layer matrix, and then by a point matrix that accounts for the finite discontinuity of the strain across the interface. The missing components of the initial state vector are determined from the corresponding boundary conditions. Global and local response is then determined by continued matrix multiplication.

The generalized Lagrange formulation for nonlinear RLC networks

Based on the concept of generalized Euler-Lagrange equations, this paper develops a Lagrange formulation of RLC networks of considerably broad scope. It is shown that the generalized Lagrange equations along with a set of compatibility constraint equations represents a set of governing differential equations of order equal to the order of complexity of the network. In this method the generalized coordinates include capacitor charges and inductor fluxes and the generalized velocities are comprised of an independent set of capacitor voltages and inductor currents. The generalized Hamilton equations are also developed and the connection with the Brayton-Moser equations is established.

On the use of quasi-velocities in impulsive motion

Abstract The differential variational principle of Jourdain (JVP) is extended to cover the dynamics of impulsive motion, formulated in terms of quasi-velocities, instead of time rates of change of true or Lagrangian generalized coordinates. This enlarges the scope of the JVP, because the extension of its range to encompass the use of quasi-velocities (which includes true velocities as a special case), enables the analyst to solve a wider range of problems. It should be pointed out that the mathematical foundation of the JVP, which is based on the assumption that the variation of the position vectors as well as the time is zero (in true Lagrangian as well as quasi-coordinates), is ideally suited to the physics of impulsive motion, where finite changes in the velocity are accompanied by negligible changes in configuration and time.