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We study a simple model of phase relaxation which consists of a parabolic PDE for temperature and an ODE with a small parameter " and double obstacles for phase variable. The model replaces sharp by di use interfaces and gives rise to superheating e ects. A semi-explicit time discretization with uniform time-step is combined with continuous piecewise linear nite elements for both and , over a xed quasi-uniform mesh of size h. At each time step, an inexpensive nodewise algebraic correction is performed to update , followed by the solution of a linear positive de nite symmetric system for by a preconditioned conjugate gradient method. A priori estimates for both and are derived in L 2-based Sobolev spaces provided the stability constraint " is enforced. Asymptotic behavior of the fully discrete model is examined as "; ; h # 0 independently, which leads to a rate of convergence of order O((+ h)" ?1=2), provided a natural compatibility condition on the initial data is satis ed. Numerical experiments illustrate the performance of the proposed method for the natural choice h ".
We examine the e ect of adaptively generated re ned meshes on the P 1 ?P 1 nite element method with semi-explicit time stepping of Part I, which applies to a phase relaxation model with small parameter " > 0. A typical mesh is highly graded in the so-called re ned region, which exhibits a local meshsize proportional to the time step , and is coarse in the remaining parabolic region where the meshsize is of order p. Three admissibility tests guarantee mesh quality and, upon failure, lead to remeshing and so to incompatible consecutive meshes. The most severe test checks whether the transition region, where phase changes take place, belongs to the re ned region. The other two tests monitor equidistribution of pointwise interpolation errors. The resulting adaptive scheme is shown to be stable in various Sobolev norms and to converge with a rate of order O(= p ") in the natural energy spaces. Several numerical experiments illustrate the scheme's e ciency and enhanced performance as compared with those of Part I.
Summary
The application of power electronics is an important trend of power system transformation. This paper introduces the world's first flexible interconnection in 220‐kV urban power grid for the first time. This paper presents the key technologies for this project. Firstly, this paper introduces the concept of flexible interconnection, in which the flexible loop operation mode is proposed. Then, this paper introduces the pilot project of partition flexible interconnection in Beijing, China. The project is to build a new flexible DC interconnection based on back‐to‐back modular multilevel converter technology between 2 subtransmission areas in Beijing power supply grid, in order to enhance stability and security. Finally, the project is comprehensively evaluated based on real project data, in which transient voltage stability, static voltage security, total supply capability, short‐circuit current, and reliability are analyzed. The results have shown the effectiveness of the proposed flexible interconnection in large‐scale urban power grid. This project demonstrates a promising application of advanced power electronics in urban power grid.
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