Abstract
The finite element orthogonal collocation method is widely used in the discretization of differential algebraic equations (DAEs), while the discrete strategy significantly affects the accuracy and efficiency of the results. In this work, a finite element meshing method with error estimation on noncollocation point is proposed and several cases were studied. Firstly, the simultaneous strategy based on the finite element is used to transform the differential and algebraic optimization problems (DAOPs) into large scale nonlinear programming problems. Then, the state variables of the reaction process are obtained by simulating with fixed control variables. The noncollocation points are introduced to compute the error estimates of the state variables at noncollocation points. Finally, in order to improve the computational accuracy with less finite element, moving finite element strategy was used for dynamically adjusting the length of finite element appropriately to satisfy the set margin of error. The proposed strategy is applied to two classical control problems and a large scale reverse osmosis seawater desalination process. Computing result shows that the proposed strategy can effectively reduce the computing effort with satisfied accuracy for dynamic optimization problems.
Highlights
The direct transcription method is an important method to solve the problem of optimal control
This numerical experiment is based on Intel (R) Core (TM) i3-2350M 2.3 GHz processor with 4 GB RAM; the nonlinear programming solver IPOPT [17, 18] is developed by Carnegie Mellon University
We proposed a mesh-partitioning strategy based on the direct transcription method to solve the optimal control problem
Summary
The direct transcription method is an important method to solve the problem of optimal control. The discrete method uses the finite element orthogonal collocation, and generally the number of finite elements is empirically selected and the length of each finite element is divided This results in low discretization accuracy for state and control variables, and to guarantee the satisfactory accuracy for some problems, the calculation time is too long to accept. With the need of discrete differential algebraic equations, moving finite element is becoming the popular and practical technique for chemical process. Betts and Kolmanovsky proposed a refinement procedure for nonlinear programming for discrete processes and estimating the discretization error for state variables [1]. Paiva and Fontes studied the adaptive mesh refinement algorithms which allow a nonuniform node collocation and apply a time mesh refinement strategy based on the local error into practical problems [3]. The method is applied to the chemical reaction of simple ODE equation [12, 13] and large scale reverse osmosis seawater desalination process, which proves its validity and feasibility
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