The authors present a method of finding a continuum of power flow solutions starting at some base load and leading to the steady-state voltage stability limit (critical point) of the system. A salient feature of the so-called continuation power flow is that it remains well-conditioned at and around the critical point. As a consequence, divergence due to ill-conditioning is not encountered at the critical point, even when single-precision computation is used. Intermediate results of the process are used to develop a voltage stability index and identify areas of the system most prone to voltage collapse. Examples are given where the voltage stability of a system is analyzed using several different scenarios of load increase. >

The continuation power flow: a tool for steady state voltage stability analysis

https://www3.nd.edu/~powers/ame.60636/maas1992.pdf

Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space

http://inis.jinr.ru/sl/m_mathematics/mn_numerical methods/mnd_numerical calculus/allgower e.l., georg k. introduction to numerical continuation methods (colorado, 1990)(397s).pdf

Numerical Continuation Methods

In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.

/pdf/reduced-basis-approximation-and-a-posteriori-error-1csv7ta4m6.pdf

Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations

From the Publisher:
Introduction to Numerical Continuation Methods continues to be useful for researchers and graduate students in mathematics, sciences, engineering, economics, and business looking for an introduction to computational methods for solving a large variety of nonlinear systems of equations. A background in elementary analysis and linear algebra is adequate preparation for reading this book; some knowledge from a first course in numerical analysis may also be helpful.

/pdf/introduction-to-numerical-continuation-methods-ily9kpadk5.pdf

Introduction to Numerical Continuation Methods

The design of a package of continuation procedures called PITCON to handle the following tasks is described' (1) follow numerically any a priori specified curve on an equilibrium manifold; (2) on such a curve determine the exact location of target points where a given variable has a specified value; and (3) on such a curve identify and compute exactly the (simple) limit points where stability may be lost. The process is based on the local parametenzatmn wh]ch uses an estimate of the curvature to control the chome of parameter varmble. Categorms and Subject Descriptors. G 1.5 [Numerical Analysis]Roots of Nonlinear Equations-tteratwe methods, systems of equattons General Terms" Algorithms, Design Addltmnal

A locally parameterized continuation process

Contihuation methods are considered here in the broad sense as algorithms for the computational analysis of specified parts of the solution field of equations of the form $Fx = b$ , where $F:R^m \to R^n $ is a given mapping and $m > n$. Such problems arise, for instance, in structural mechanics and then usually $m - n$ of the variables $x_i $ are designated as parameters. For the case $m = n + 1$ an existence theory for the regular curves of the solution field is developed here. Then approximate solutions are considered and shown to be solutions of certain perturbed problems. These results are used to prove that for the continuation methods with Eider-predictor and Newton-corrector a particular steplength algorithm is guaranteed to trace any regular solution of the field. Some numerical aspects of the procedure are discussed and a numerical example is included to illustrate the effectiveness of the approach.

https://www.jstor.org/stable/info/2156609

Solution Fields of Nonlinear Equations and Continuation Methods

Numerical analysis of continuation methods for nonlinear structural problems

Algorithm 596: a program for a locally parameterized

The continuation methods considered here are algorithms for the computational analysis of the regular parts of the solution field of equations of the form $Fx = b,F:D \subset R^{n + 1} \to R^n $, for given $b \in R^n $. While these methods are similar in structure to those used for ODE-solvers, their errors are independent of the history of the process and are solely determined by the termination criterion of the corrector at the current step. This suggests the use of a posteriors estimates of the convergence radii of the corrector. It is proved here that such estimates cannot be obtained from the sequence of corrector iterates alone but that they require some global information about F. However, it is shown that a finite sequence of corrector iterates does allow for the computation of effective estimates of the convergence quality of certain types of correctors. This is used for the design of various step-algorithms for continuation processes; two of them are based on a Newton-corrector while the third o...

Werner C. Rheinboldt

Papers

A locally parameterized continuation process

Solution Fields of Nonlinear Equations and Continuation Methods

Numerical analysis of continuation methods for nonlinear structural problems

Algorithm 596: a program for a locally parameterized

On Steplength Algorithms for a Class of Continuation Methods