A method for computing minimum voltage stability margins of power systems

A new and fast method for computing the minimum voltage stability margin (VSM) of power systems is presented. The computation of the VSM is required for power systems planning and operation. Usually, it is assumed that loads increase along a predefined direction (e.g. with constant power factor, followed by a proportional MW generation increase) up to the system's maximum loading point is reached. Situations may occur where variations from the predefined load increase direction, for example an unexpected load increase at some bus or area may result in smaller VSM, taking the system to an insecure state. The computation of the minimum VSM (mVSM) allows forecasting the load increase worst scenario. The information regarding the mVSM and the corresponding load increase direction for which it occurs, along with the usual VSM, allows operators to take measures like preventive control actions to move the system to securer operating points. Simulation results for IEEE test systems and for realistic systems are shown. Also, the proposed method is compared with other methods found in the literature.     

The Hamilton-Jacobi method for arbitrary rheo-linear dynamical systems

Summary.  The Hamilton-Jacobi method is briefly summarized and then applied to arbitrary rheo-linear systems with a single degree of freedom. Various means of finding a complete solution of the Hamilton-Jacobi equation and applying the Jacobi theorem to solve the canonical differential equations are discussed. Basic to the procedure is the separation of variables in the Hamilton-Jacobi equation which leads to a Riccati equation which must be solved for the particular rheonomic differential equation. Five different cases of complete solution are illustrated by a simple rheonomic example. As a direct application of the method, a number of canonical systems corresponding to many “named” equations are solved. They are the: Airy, Bessel, Chebyshev, Error function, Euler, Gegenbauer, Hermite, Hypergeometric, Kelvin, Kummer, Laguerre, Legendre, Jacobi, Mathieu, Spherical Bessel, Weber-Hermite and Whittaker equations. Finally, the conservation law is given for a forced and damped oscillator. Received April 18, 2002; revised December 12, 2002 Published online: June 12, 2003     

Stable numerical integration of dynamical systems subject to equality state-space constraints

Numerical simulation of constrained dynamical systems is known to exhibit stability problems even when the unconstrained system can be simulated in a stable manner. We show that not the constraints themselves, but the transformation of the continuous set of equations to a discrete set of equations is the true source of the stability problem. A new theory is presented that allows for stable numerical integration of constrained dynamical systems. The derived numerical methods are robust with respect to errors in the initial conditions and stable with respect to errors made during the integration process. As a consequence, perturbations in the initial conditions are allowed. The new theory is extended to the case of constrained mechanical systems. Some numerical results obtained when implementing the numerical method here developed are shown.     

High-order integration of smooth dynamical systems: Theory and numerical experiments

This paper describes a new class of algorithms for integrating linear second-order equations and those containing smooth non-linearities. The algorithms are based on a combination of ideas from standard Newmark integration methods and extrapolation techniques. For the algorithm to work, the underlying Newmark method must be stable, second-order accurate, and produce asymptotic error expansions for response quantities containing only even-ordered terms. It is proved that setting the Newmark parameter γ equal to 1/2 gives a desirable asymptotic expansion, irrespective of the setting for β. Numerical experiments are conducted for one linear and two non-linear applications.     

Wedge simulation method for calculating the reliability of linear dynamical systems

In this paper the problem of calculating the probability of failure of linear dynamical systems subjected to random excitations is considered. The failure probability can be described as a union of failure events each of which is described by a linear limit state function. While the failure probability due to a union of non-interacting limit state functions can be evaluated without difficulty, the interaction among the limit state functions makes the calculation of the failure probability a difficult and challenging task. A novel robust reliability methodology, referred to as Wedge-Simulation-Method, is proposed to calculate the probability that the response of a linear system subjected to Gaussian random excitation exceeds specified target thresholds. A numerical example is given to demonstrate the efficiency of the proposed method which is found to be enormously more efficient than Monte Carlo Simulations.     

Pseudospectral method for assessing stability robustness for linear time-periodic delayed dynamical systems

The article presents a pseudospectral approach to assess the stability robustness of linear time-periodic delay systems, where periodic functions potentially present discontinuities and the delays may also periodically vary in time. The considered systems are subject to linear real-valued time-periodic uncertainties affecting the coefficient matrices, and the presented method is able to fully exploit structure and potential interdependencies among the uncertainties. The assessment of robustness relies on the computation of the pseudospectral radius of the monodromy operator, namely, the largest Floquet multiplier that the system can attain within a given range of perturbations. Instrumental to the adopted novel approach, a solver for the computation of Floquet multipliers is introduced, which results into the solution of a generalized eigenvalue problem which is linear w.r.t. (samples of) the original system matrices. We provide numerical simulations for popular applications modeled by time-periodic delay systems, such as the inverted pendulum subject to an act-and-wait controller, a single-degree-of-freedom milling model and a turning operation with spindle speed variation.     

A dynamical formalism for unrooted systems of rigid bodies

Summary This paper presents a general formalism for the dynamics ofunrooted systems (=systems without kinematical coupling to a Galilean frame). Starting point are Lagrange's equations in ordered form. A set of points, themain points — introduced by O. Fischer hundred years ago- and a fictious body, theaugmented generalized body, permit to separate the translation of the mass center from the motion in the other coordinates. The basic idea is to break up the virtual velocity of a particle into component velocities that can be readily expressed in algebraic form. This allows, after some tensor-algebraic manipulations, an elegant representation ofall inertia coefficents as a linear function ofone kind of tensor, namely thebasic kinetic tensor. Newglobal inertia tensors (GITs), a generalization of those introduced by M. Fayet, are defined. They allow to separate geometry from affine geometry. GITs of order zero are shown to be ageneralization of the reduced mass. A recursive method is presented for efficent formulation of GITs of order one and two. It is also briefly indicated how rooted systems can be interpreted as a special case of rooted ones. A new formula to compute generalized forces due to a nonhomogeneous Newtonian force field is proposed. Results for rooted trees are reviewed as far as they are necessary for the purposes of this paper. The whole method translates conveniently into efficent computer codes.     

An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials

A new meshless method for computing the dynamic stress intensity factors (SIFs) in continuously non-homogeneous solids under a transient dynamic load is presented. The method is based on the local boundary integral equation (LBIE) formulation and the moving least squares (MLS) approximation. The analyzed domain is divided into small subdomains, in which a weak solution is assumed to exist. Nodal points are randomly spread in the analyzed domain and each one is surrounded by a circle centered at the collocation point. The boundary-domain integral formulation with elastostatic fundamental solutions for homogeneous solids in Laplace-transformed domain is used to obtain the weak solution for subdomains. On the boundary of the subdomains, both the displacement and the traction vectors are unknown generally. If modified elastostatic fundamental solutions vanishing on the boundary of the subdomain are employed, the traction vector is eliminated from the local boundary integral equations for all interior nodal points. The spatial variation of the displacements is approximated by the MLS scheme.     

A numerical method for unilateral problems

Variational inequalities connected with Signorini's problem have appeared as a natural generalization of the minimum potential-energy theorem for bodies with unilateral constraints. In this paper, we describe numerical experience on the use of variational inequalities and Pade approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity, like those describing the equilibrium configuration of an elastic membrane stretched over an elastic obstacle. These problems have the peculiar feature of being alternatively formulated as nonlinear boundary value problems without constraints for which the technique of Pade approximants can be successfully employed. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution.     

A semi‐analytical locally transversal linearization method for non‐linear dynamical systems

An numeric‐analytical, implicit and local linearization methodology, called the locally transversal linearization (LTL), is developed in the present paper for analyses and simulations of non‐linear oscillators. The LTL principle is based on deriving the locally linearized equations in such a way that the tangent space of the linearized equations transversally intersects that of the given non‐linear dynamical system at that particular point in the state space where the solution vector is sought. For purposes of numerical implementation, two different numerical schemes, namely LTL‐1 and LTL‐2 schemes, based on the LTL methodology are presented. Both LTL‐1 and LTL‐2 procedures finally reduce the given set of non‐linear ordinary differential equations (ODEs) to a set of transcendental algebraic equations valid over a short interval of time or over a short segment of the evolving trajectories as projected on the phase space. While in the LTL‐1 scheme the desired solution vector at a forward time point enters the linearized differential equations as an unknown parameter, in the LTL‐2 scheme a set of unknown residues enters the linearized system as parameters. A limited set of examples involving a few well‐known single‐degree‐of‐freedom (SDOF) non‐linear oscillators indicate that the LTL methodology is capable of accurately predicting many complicated non‐linear response patterns, including limit cycles, quasi‐periodic orbits and even strange attractors. Copyright © 2001 John Wiley & Sons, Ltd.     

Direct control method for improving stability and reliability of nonlinear stochastic dynamical systems

Optimal control for improving the stability and reliability of nonlinear stochastic dynamical systems is of great significance for enhancing system performances. However, it has not been adequately investigated because the evaluation indicators for stability (e.g. maximal Lyapunov exponent) and for reliability (e.g. mean first-passage time) cannot be explicitly expressed as the functions of system states. Here, a unified procedure is established to derive optimal control strategies for improving system stability and reliability, in which a physical intuition-inspired separation technique is adopted to split feedback control forces into conservative components and dissipative components, the stochastic averaging is then utilized to express the evaluation indicators of performances of controlled system, the optimal control strategies are finally derived by minimizing the performance indexes constituted by the sigmoid function of maximal Lyapunov exponent (for stability-based control)/the reciprocal of mean first-passage time (for reliability-based control), and the mean value of quadratic form of control force. The unified procedure converts the original functional extreme problem of optimal control into an extremum value problem of multivariable function which can be solved by optimization algorithms. A numerical example is worked out to illustrate the efficacy of the optimal control strategies for enhancing system performance.     

Finding stable attractors of a class of dissipative dynamical systems by numerical parameter switching

In this article we prove numerically, via computer graphic simulations and specific examples, that switching the control parameter of a dynamical system belonging to a class of dissipative continuous dynamical systems, one can obtain a stable attractor. In this purpose, while a fixed step-size numerical method approximates the solution of the mathematical model, the parameter control is switched every few integration steps, the switching scheme being time periodic. The switch occurs within a considered set of admissible parameter values. Moreover, we show via numerical experiments that the obtained synthesized attractor belongs to the class of all admissible attractors for the considered system and matches to the averaged attractor obtained with the control parameter replaced with the averaged switched parameter values. This switched strategy may force the system to evolve along on a stable attractor whatever the parameter values and introduces a convex structure inside of the attractor set via a bijection between the set of parameter control values and the attractors set. The algorithm besides its utility in systems stabilization, when some desired parameter control cannot be directly accessed, may serve as a model for the dynamics encountered in reality or in experiments, e.g. three species food chain models, electronic circuits, etc. This method, compared, for example, to the OGY algorithm where only small perturbations of parameter control can be issued, allows relatively large parameter perturbations. Also, it does not allow to stabilize an unstable orbit but, using an appropriate parameter switching algorithm, it allows to reach an already existing attractor. The present work extends the results we obtained previously and is applied to Lorenz, Rössler and Chen systems.     

A new method for superconductor magnetization curve computing

A mathematical method using geometrical transformations (reflections and translations) on an arbitrary chosen curve — called the profile function — which describes the magnetic field distribution inside the superconductor sample is presented. This profile function must be strict monotone increasing and positive. By this method we have calculated the expression of ‘jump’ portion that connects the ascending and descending field branches and non-primitive minor loops. Also we propose a new model (called critical field model) based on a bounded profile function at some value B_cr; explicit expressions for computing all the branches of the hysteresis loop are derived. The comparison between the values of critical current density computed from the vertical width of the hysteresis loops ΔM and that deduced from the chosen profile function shows a good agreement for B_a > B_p. Finally, we present an exact relation between ΔM and J_c and its derivatives for an arbitrary positive J_c function.     

A simple method based for computing crack shapes

The growth of cracks from small naturally occurring material discontinuities plays a major role in the operational lifespan of aircraft structures. Calculating the life of an airframe structure requires the determination of the crack path(s) which for complex real life geometries can often be highly complex. This paper presents a simple method based on finite element analysis for estimating the crack growth path. The analysis is based on an element removal approach and uses the major principal stress as the failure criteria. Evolution strategies are derived from the biological process of evolution. Three examples are presented demonstrating the utility of the proposed technique.     

A direct violation correction method in numerical simulation of constrained multibody systems

 A new direct violation correction method for constrained multibody systems is presented. It can correct the value of state variables of the systems directly so as to satisfy the constraint equations of motion. During the integration of the dynamic equations of constrained multibody systems, this method can efficiently control the violations of constraint equations within any given accuracy at each time-step. Compared to conventional indirect methods, especially Baumgarte's Constraint Violation Stabilization Method, this method has clear physical meaning, less calculation and obvious correction effect. Besides, this method has minor effect on the form of the dynamic equations of systems, so it is stable and highly accurate. A numerical example is provided to demonstrate the effectiveness of this method. Received: 17 December 1999     

Semi-implicit numerical simulations of geometrically nonlinear beam,plate, and shell dynamical systems

The following article serves three purposes: (i) it presents a simple semi-implicit numerical formulation for nonlinear structural dynamics problems, which is computationally inexpensive and simple to use in nonlinear dynamics and chaos simulations; (ii) it serves as an introduction to numerical studies of nonlinear structural dynamics for engineering students; and (iii) it formulates a nonlinear structural dynamical system for studies of nonlinear dynamics and chaos. Numerical formulations along with results are presented for nonlinear oscillators, beams, Föppl–von Kármán plates, and thin shallow shells.     

A nonlinear dynamical systems framework for structural diagnosis and prognosis

This work aims to establish a nonlinear dynamics framework for diagnosis and prognosis in structural dynamic systems. The objective is to develop an analytically sound means for extracting features, which can be used to characterize damage, from modal-based input-output data in complex hybrid structures with heterogeneous materials and many components. Although systems like this are complex in nature, the premise of the work here is that damage initiates and evolves in the same phenomenological way regardless of the physical system according to nonlinear dynamic processes. That is, bifurcations occur in healthy systems as a result of damage. By projecting a priori the equations of motion of high-dimensional structural dynamic systems onto lower dimensional center, or so-called ‘damage’, manifolds, it is demonstrated that model reduction near bifurcations might be a useful way to identify certain features in the input-output data that are helpful in identifying damage. Normal forms describing local co-dimension one and two bifurcations (e.g. transcritical, subcritical pitchfork, and asymmetric pitchfork bifurcations) are assumed to govern the initiation and evolution of damage in a low-order model. Real-world complications in damage prognosis involving spatial bifurcations, global bifurcation phenomena, and the sensitivity of damage to small changes in initial conditions are also briefly discussed.     

Hitting times for random dynamical systems

The purpose of this article is to study the hitting times for random dynamical systems. For general systems we give a lower bound in terms of the local dimension. For fast mixing systems we obtain an equality. Moreover, under a power law decay of correlations we obtain lower and upper bounds of the hitting times for absolutely continuous stationary measures.     

Stability conditions for non-conservative dynamical systems

The present paper treats dynamic instability problems of non-conservative elastic systems. Starting from general equations of motion, the equations of the perturbed motion are derived. The boundedness of the perturbed motions is studied and sufficient conditions for instability and a necessary condition for stability are deduced. These conditions may determine the instability of non-conservative systems and they are expressed in terms of the properties of generalized tangent damping and stiffness matrices of the systems. Thus, they can easily be incorporated with finite element computations of arbitrary structures.