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 Energy and Power Engineering (EPE) , 2016, DOI: 10.4236/epe.2016.82009 Abstract: This paper is a further study of two papers [1] and [2], which were related to Ill-Conditioned Load Flow Problems and were published by IEEE Trans. PAS. The authors of this paper have some different opinions, for example, the 11-bus system is not an ill-conditioned system. In addition, a new approach to solve Load Flow Problems, E-ψtc, is introduced. It is an explicit method; solving linear equations is not needed. It can handle very tough and very large systems. The advantage of this method has been fully proved by two examples. The authors give this new method a detailed description of how to use it to solve Load Flow Problems and successfully apply it to the 43-bus and the 11-bus systems. The authors also propose a strategy to test the reliability, and by solving gradient equations, this new method can answer if the solution exists or not.
 Computer Science , 2014, Abstract: In this paper, pseudo-transient continuation method has been modified and implemented in power system long-term stability analysis. This method is a middle ground between integration and steady state calculation, thus is a good compromise between accuracy and efficiency. Pseudo-transient continuation method can be applied in the long-term stability model directly to accelerate simulation speed and can also be implemented in the QSS model to overcome numerical difficulties. Numerical examples show that pseudo-transient continuation method can provide correct approximations for the long-term stability model in terms of trajectories and stability assessment.
 Gerard Awanou Mathematics , 2013, Abstract: We present two numerical methods for the fully nonlinear elliptic Monge-Ampere equation. The first is a pseudo transient continuation method and the second is a pure pseudo time marching method. The methods are proven to converge to a strictly convex solution of a natural discrete variational formulation with $C^1$ conforming approximations. The assumption of existence of a strictly convex solution to the discrete problem is proven for smooth solutions of the continuous problem and supported by numerical evidence for non smooth solutions.
 Mathematics , 2006, Abstract: We bound the condition number of the Jacobian in pseudo arclength continuation problems, and we quantify the effect of this condition number on the linear system solution in a Newton GMRES solve. In pseudo arclength continuation one repeatedly solves systems of nonlinear equations $F(u(s),\lambda(s))=0$ for a real-valued function $u$ and a real parameter $\lambda$, given different values of the arclength $s$. It is known that the Jacobian $F_x$ of $F$ with respect to $x=(u,\lambda)$ is nonsingular, if the path contains only regular points and simple fold singularities. We introduce a new characterization of simple folds in terms of the singular value decomposition, and we use it to derive a new bound for the norm of $F_x^{-1}$. We also show that the convergence rate of GMRES in a Newton step for $F(u(s),\lambda(s))=0$ is essentially the same as that of the original problem $G(u,\lambda)=0$. In particular we prove that the bounds on the degrees of the minimal polynomials of the Jacobians $F_x$ and $G_u$ differ by at most 2. We illustrate the effectiveness of our bounds with an example from radiative transfer theory.
 Computer Science , 2013, Abstract: Pseudo-arclength continuation is a well-established method for generating a numerical curve approximating the solution of an underdetermined system of nonlinear equations. It is an inherently sequential predictor-corrector method in which new approximate solutions are extrapolated from previously converged results and then iteratively refined. Convergence of the iterative corrections is guaranteed only for sufficiently small prediction steps. In high-dimensional systems, corrector steps are extremely costly to compute and the prediction step-length must be adapted carefully to avoid failed steps or unnecessarily slow progress. We describe a parallel method for adapting the step-length employing several predictor-corrector sequences of different step lengths computed concurrently. In addition, the algorithm permits intermediate results of unconverged correction sequences to seed new predictions. This strategy results in an aggressive optimization of the step length at the cost of redundancy in the concurrent computation. We present two examples of convoluted solution curves of high-dimensional systems showing that speed-up by a factor of two can be attained on a multi-core CPU while a factor of three is attainable on a small cluster.
 N. S. Hoang Mathematics , 2014, Abstract: A general class of functionally-fitted explicit pseudo two-step Runge-Kutta-Nystr\"{o}m (FEPTRKN) methods for solving second-order initial value problems has been studied. These methods can be considered generalized explicit pseudo two-step Runge-Kutta-Nystr\"{o}m (EPTRKN) methods. We proved that an $s$-stage FEPTRKN method has step order $p = s$ and stage order $r = s$ for any set of distinct collocation parameters $(c_i)_{i=1}^s$. Supperconvergence for the accuracy orders of these methods can be obtained if the collocation parameters $(c_i)_{i=1}^s$ satisfy some orthogonality conditions. We proved that an $s$-stage FEPTRKN method can attain accuracy order $p = s + 3$. Numerical experiments have shown that the new FEPTRKN methods work better than do EPTRKN methods on problems whose solutions can be well approximated by the functions in bases on which these FEPTRKN methods are developed.