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### Open Access Journals

JIMO

Motivated by the observation that some
reformulation based extragradient methods for general
monotone variational inequalities in real Hilbert space may not
generate a solution of the original problem, we propose an
iterative method with line searches and prove its convergence for
general pseudomonotone (monotone) variational inequality problems.

JIMO

In this paper, we present a new approach for finding a stable
solution of a system of nonlinear equations arising from dynamical
systems. We introduce the concept of
stability functions and use this idea to construct
stability solution models of several
typical small signal stability problems in dynamical systems.
Each model consists of a system of constrained
semismooth equations. The advantage of the new models is twofold.
Firstly, the stability requirement of dynamical systems is
controlled by nonlinear inequalities. Secondly, the semismoothness
property of the stability functions makes the models solvable by
efficient numerical methods. We introduce smoothing functions for
the stability functions and present a smoothing Newton method
for solving the problems. Global and local quadratic convergence of
the algorithm is established. Numerical examples from dynamical
systems are also given to illustrate the efficiency of the new
approach.

JIMO

In this paper, we investigate the reformulation
of the steady state security region problem for electrical power
systems. Firstly, a simple security region problem with one
changeable parameter
is reformulated into a system of semismooth equations, which is composed by
the normal power flow equations and an additional piecewise smooth
equation. Then the semismooth Newton method and the smoothing Newton
method can be applied to solve the problem. Preliminary numerical
results show that the method is promising. Finally, by using the
smoothing technique, a more complicated security region problem, the
Euclidean security region problem, is reformulated as an equality
constrained optimization problem. These works provide a possibility
to implement on-line calculation of the security region of
electrical power systems.

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