MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems

In the present paper an optimal control problem governed by the heat equation is considered, where continuous as well as discrete controls are involved. To obtain the discrete controls the branch-and-bound method is utilized, where in each node a relaxed control constrained optimal control problem has to be solved involving only continuous controls. However, the solutions to many relaxed optimal control problems have to be computed numerically. For that reason model-order reduction is applied to speed-up the branch-and-bound method. In this work the method of proper orthogonal decomposition (POD) is used. A posteriori error estimation in each node ensures that the calculated solutions are sufficiently accurate. Numerical experiments illustrate the efficiency of the proposed strategy.

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“…Analogously to the continuous problem (P ), a solution to (14) is given by the eigenvectors to the largest eigenvalues of the operator…”

Section: The Pod Methodsmentioning

confidence: 99%

“…For partial differential equations (PDEs) we refer to the note [20]. In particular, mixed-integer problems for hyperbolic PDEs are considered, e.g., for problems for gas transportation systems [14] and traffic flow [15,16].…”

Section: Introductionmentioning

confidence: 99%

POD-Based Mixed-Integer Optimal Control of the Heat Equation

Bachmann

Beermann

et al. 2019

J Sci Comput

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“…The single nomination requires routing 544.32 Nm 3 /h units of flow from the entries to the exits. This network has already been used in the papers [9,10]. …”

Section: Gaslib-24mentioning

confidence: 99%

GasLib—A Library of Gas Network Instances

Schmidt

Aßmann

Burlacu

et al. 2017

Data

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102

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Abstract:The development of mathematical simulation and optimization models and algorithms for solving gas transport problems is an active field of research. In order to test and compare these models and algorithms, gas network instances together with demand data are needed. The goal of GasLib is to provide a set of publicly available gas network instances that can be used by researchers in the field of gas transport. The advantages are that researchers save time by using these instances and that different models and algorithms can be compared on the same specified test sets. The library instances are encoded in an XML (extensible markup language) format. In this paper, we explain this format and present the instances that are available in the library.Data Set: http://gaslib.zib.de Data Set License: CC BY 3.0 Keywords: gas transport; networks; problem instances; mixed-integer nonlinear optimization; GasLib MSC: 90-08; 90C90; 90B10 SummaryThe mathematical simulation and optimization of gas transport through pipeline systems is an important field of research with a large practical impact. Over the last decades, many different mathematical models on different levels of accuracy for different components of gas networks have been developed. On the basis of these models, several simulation and optimization algorithms have been proposed. We refer to [1][2][3] and the references therein for more information. With GasLib, we provide a set of network instances that can be used to test and compare such models and the algorithms for solving them.

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“…On the one hand, the modeling of valves regulating gas flows is huge in the applied literature; as far as this paper is concerned, we refer to []. Other useful sources of information are [], namely for optimization problems. On the other hand, a comprehensive mathematical treatment of this subject through system seems missing and, to the best of our knowledge, our previous paper was the first one toward this direction.…”

Section: Introductionmentioning

confidence: 99%

Coherence and chattering of a one‐way valve

Corli

Rosini

2019

Z Angew Math Mech

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In this paper we analyze the effects of a one-way valve on the isothermal gas flow through a pipe. The valve keeps the flow at a constant value * > 0, if possible; otherwise it is closed. First, for fixed * , we define a Riemann solver and characterize the coherence of its initial data; coherence is a necessary condition for the construction of solutions to a general initial-value problem based on a wave-front tracking scheme. We also give an example of an invariant and coherent domain where the valve can be either open or closed. Second, for suitable compact sets of initial data we make precise the range of values * that guarantee the coherence. At last, in the case of a real valve with finite reaction time, we show the chattering (rapid switch on and off) of the valve in correspondence with incoherent initial data. K E Y W O R D Schattering, coupling conditions, gas flow, Riemann problem, systems of conservation laws, valve 2 0 1 0 A

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MIP-based instantaneous control of mixed-integer PDE-constrained gas transport problems

Cited by 41 publications

References 43 publications

POD-Based Mixed-Integer Optimal Control of the Heat Equation

POD-Based Mixed-Integer Optimal Control of the Heat Equation

GasLib—A Library of Gas Network Instances

Coherence and chattering of a one‐way valve

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