On Improving the Convergence of Radau IIA Methods Applied to Index 2 DAEs

Aubry, Anne; Chartier, Philippe

doi:10.1137/s0036142995296539

Cited by 8 publications

(9 citation statements)

References 6 publications

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“…is small enough, for example for the variables presented in Figure the AAEs are (a) 0.1489 mol/min, (b) 1.11 × 10 –4 and (c) 5.16 × 10 –3 MJ/min, these differences between the two models are explained by the precision of the discretization method. Orthogonal collocation with Radau point has an error of O ( h 2s–1 ), where s is the number of collocation points, for the algebraic variables of an index one DAE, whereas for the algebraic variables of an index two DAE, the error is O ( h s ); the precision in the differential variables is the same for index one and index two DAEs. , …”

Section: Resultsmentioning

confidence: 99%

“…This type of discretization is chosen because: (1) it is widely used to solve index two problems, (2) it is a numerical method with L-stability, which means that when the integration step tends to zero the stability region tends to infinity, (3) it is capable of stabilizing high index DAE systems. The previous properties hold only if the initial conditions are consistent, otherwise the numerical method can fail at the first steps of integration. , However, the method is less accurate in the algebraic variables for index two problems. , …”

Section: Eo-nmpc Problem Formulationmentioning

confidence: 99%

“…23,31 However, the method is less accurate in the algebraic variables for index two problems. 28,33 Applying this method, the differential equations become a weighting of the variables using the derivative of an orthogonal polynomial (Lagrange's polynomial) (eq18). These polynomials are represented in eq 19.…”

Section: Eo-nmpc Problem Formulationmentioning

confidence: 99%

See 2 more Smart Citations

Economic Oriented NMPC for an Extractive Distillation Column Using an Index Hybrid DAE Model Based on Fundamental Principles

Santamaria

Gómez

2015

Ind. Eng. Chem. Res.

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Nonlinear model predictive control (NMPC) is an advanced control strategy that uses a rigorous dynamic model of the process, usually described by differential-algebraic equations (DAE), to predict its future behavior. The economic optimization of the process can be included directly in the objectives of the NMPC controller, which make it attractive for application in highly nonlinear and energy-intensive processes that are subject to fluctuating operational conditions. Currently, the implementation of NMPC controllers with large-scale dynamic models is limited by the computational difficulty of solving the associated dynamic optimization problem at each period of time. In this work, we propose an index two DAE model based on fundamental principles that uses its equivalent reduced index one model to define consistent initial conditions (index hybrid DAE) with the aim of representing the dynamics of a distillation column. We demonstrate that this model describes the same physical behavior of its equivalent index one reduced model and that it decreases significantly the computational complexity of the online optimization associated with the NMPC controller. Finally, we use the index hybrid DAE model in the economic oriented NMPC (EO-NMPC) of an extractive distillation column considering different disturbances over the inlet conditions. Also, we compare the EO-NMPC with a classic PI (proportional and integral) controller to show that the first one has a better dynamic performance and improves the economic profit of the process.

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Section: Resultsmentioning

confidence: 99%

Section: Eo-nmpc Problem Formulationmentioning

confidence: 99%

See 1 more Smart Citation

Economic Oriented NMPC for an Extractive Distillation Column Using an Index Hybrid DAE Model Based on Fundamental Principles

Santamaria

Gómez

2015

Ind. Eng. Chem. Res.

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“…Orthogonal collocation with Radau point has an error of O(h 2s−1 ), where s is the number of collocation points, for the algebraic variables of an index one DAE, while for the algebraic variables of an index two DAE the error is O(h s ); the precision in the differential variables is the same for index one and index two DAEs. 28,30 Table 4 shows the size of the models after discretization using orthogonal collocation on finite elements with the parameters listed in Table 3 to solve the optimal control problem at each instant of time in the NMPC scheme. The hybrid model has fewer equations than the index one reduced DAE system because the dummy variables and equations used in the index reduction process (e.g., eq 12) are included in the mathematical programming transcription.…”

Section: Resultsmentioning

confidence: 99%

Index Hybrid Differential–Algebraic Equations Model Based on Fundamental Principles for Nonlinear Model Predictive Control of a Flash Separation Drum

Santamaria

Gómez

2015

Ind. Eng. Chem. Res.

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Model predictive control (MPC) has been widely used in the chemical process industry; however, the industrial applications use a simplified model to describe the time evolution of the system in order to make the dynamic optimization problem easier to solve. Models based on fundamental principles describe in detail the dynamics of the system, but they define a problem of differential−algebraic equations (DAE) that can present high-index difficulties related to the definition of consistent initial conditions. Index reduction techniques have been applied to solve this kind of problem. This study presents an alternative to using high-index DAE models in MPC. The problem is discretized using orthogonal collocation on finite elements, and the original high-index model is used in the numerical integration, but its equivalent reduced index one model is used to define efficiently consistent initial conditions. The set point tracking problem and the disturbance rejection problem for a flash separation drum using nonlinear model predictive control are considered. It is shown that the proposed model does not present a practical difference with respect to the index one reduced model and that it is more efficient in computational terms.

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“…Finally, IRK schemes allow naturally the use of adaptative time stepping, which is used in conjunction with an error estimator that would further improve their efficiency. Finally, methods of improving the time order of convergence of the pressure could be advantageously introduced to simulate with a higher accuracy the interaction between a structure and an incompressible flow.…”

Section: Discussionmentioning

confidence: 99%

High‐order implicit Runge–Kutta time integrators for fluid‐structure interactions

Cori

Étienne

Garon

et al. 2015

Numerical Methods in Fluids

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SummaryThis paper presents an approach to develop high‐order, temporally accurate, finite element approximations of fluid‐structure interaction (FSI) problems. The proposed numerical method uses an implicit monolithic formulation in which the same implicit Runge–Kutta (IRK) temporal integrator is used for the incompressible flow, the structural equations undergoing large displacements, and the coupling terms at the fluid‐solid interface. In this context of stiff interaction problems, the fully implicit one‐step approach presented is an original alternative to traditional multistep or explicit one‐step finite element approaches. The numerical scheme takes advantage of an arbitrary Lagrangian–Eulerian formulation of the equations designed to satisfy the geometric conservation law and to guarantee that the high‐order temporal accuracy of the IRK time integrators observed on fixed meshes is preserved on arbitrary Lagrangian–Eulerian deforming meshes. A thorough review of the literature reveals that in most previous works, high‐order time accuracy (higher than second order) is seldom achieved for FSI problems. We present thorough time‐step refinement studies for a rigid oscillating‐airfoil on deforming meshes to confirm the time accuracy on the extracted aerodynamics reactions of IRK time integrators up to fifth order. Efficiency of the proposed approach is then tested on a stiff FSI problem of flow‐induced vibrations of a flexible strip. The time‐step refinement studies indicate the following: stability of the proposed approach is always observed even with large time step and spurious oscillations on the structure are avoided without added damping. While higher order IRK schemes require more memory than classical schemes (implicit Euler), they are faster for a given level of temporal accuracy in two dimensions. Copyright © 2015 John Wiley & Sons, Ltd.

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On Improving the Convergence of Radau IIA Methods Applied to Index 2 DAEs

Abstract: This paper presents a simple new technique to improve the order behavior of Runge-Kutta methods when applied to index 2 differential-algebraic equations. It is then shown how this can be incorporated into a more efficient version of the code radau5 developed by E. Hairer and G. Wanner.

Cited by 8 publications

References 6 publications

Economic Oriented NMPC for an Extractive Distillation Column Using an Index Hybrid DAE Model Based on Fundamental Principles

Economic Oriented NMPC for an Extractive Distillation Column Using an Index Hybrid DAE Model Based on Fundamental Principles

Index Hybrid Differential–Algebraic Equations Model Based on Fundamental Principles for Nonlinear Model Predictive Control of a Flash Separation Drum

High‐order implicit Runge–Kutta time integrators for fluid‐structure interactions

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