Towards simulation based mixed‐integer optimization with differential equations

Gugat, Martin; Leugering, Günter; Martín, Alexander; Schmidt, Martin; Sirvent, Mathias; Wintergerst, David

doi:10.1002/net.21812

Cited by 21 publications

(26 citation statements)

References 32 publications

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“…We finally state that significant parts of this chapter are published in an international journal or are submitted for publication; see our articles Gugat et al [77] and Schmidt et al [141]. The author of this thesis made significant contributions for conceiving, developing, and elaborating all the presented contents that follow.…”

Section: Decomposition Methods For Mixed-integer Nonlinear Programsmentioning

confidence: 94%

Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

Sirvent

2020

Operations Research Proceedings

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List of Figures List of Tables 1.3 Scientific Contribution and Results challenge is to find formulations and corresponding algorithms that are able to tackle transient gas transport optimization problems. Note that the first challenge seeks for general algorithms that are then used for stationary gas transport optimization, whereas the second challenge is tailored to the application of transient gas transport optimization. Our roadmap to cope with the challenges is inspired by two guidelines. We want to decompose our problems and we want to use mixed-integer linear programs. The simple reason for the first point is that decomposing problems into smaller ones has a successful history and is common sense for solving big problems. Second, mixed-integer linear programs underwent an enormous amount of research in the last decades. Consequently, fast, stable, robust, and reliable solvers as Gurobi, Cplex, or Scip are available and we use these solvers, in our case Gurobi, as a workhorse; see Gurobi [80], Cplex [36], and Maher et al. [110], respectively. 1.3 Scientific Contribution and Results For the first challenge, we consider the case where the constraint functions in (1.1c) are not given analytically. We build up a new hierarchy of assumptions that restricts the knowledge about the nonlinear functions. Depending on the assumptions, we develop three new global decomposition algorithms that decompose the difficult constraints (1.1c), i.e., we rely on our first guideline. Specifically, we build upon relaxation strategies that are motivated by developments in the field of (mixed-integer) (non)linear programming and Lipschitz optimization. As a result, mixed-integer linear programs according to our second guideline are obtained. We prove correct and finite terminations and clarify the limitations of our approaches. Note that the presented framework is a general approach for problems in the form of (1.1). In particular, we consider stationary gas transport optimization, i.e., Problem (1.1) with the Euler equations in the form of ODEs in (1.1c). Moreover, we show promising numerical results for the real-world gas network of Greece. Additionally, we consider the second challenge of transient gas transport optimization, i.e., Problem (1.1) with PDEs (1.1c). We use an instantaneous control algorithm and adapt it to the Euler equations for the first time. Specifically, we decompose the problem into single time steps and present new discretization techniques that ensure the mixed-integer linear program property. In other words, we build upon our guidelines again. Finally, we

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Section: Decomposition Methods For Mixed-integer Nonlinear Programsmentioning

confidence: 94%

Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

Sirvent

2020

Operations Research Proceedings

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“…Moreover, all nomination data have been converted from heat power to volumetric units by utilizing a conversion with a single calorific value, thus neglecting the gas mixing within the network. This instance has been used in the papers [17][18][19]. …”

Section: Gaslib-134mentioning

confidence: 99%

GasLib—A Library of Gas Network Instances

Schmidt

Aßmann

Burlacu

et al. 2017

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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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“…We refer to [5] [6] [7] and refrain in the sequel from discussing issues of valves and compressors. The combined discrete and continuous optimization will be the subject of a forthcoming publication.…”

Section: Optimal Control Problems and Outlinementioning

confidence: 99%

Domain Decomposition of an Optimal Control Problem for Semi-Linear Elliptic Equations on Metric Graphs with Application to Gas Networks

Leugering¹

2017

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We consider optimal control problems for the flow of gas in a pipe network. The equations of motions are taken to be represented by a semi-linear model derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a given network and introduce a time discretization thereof. We then study the well-posedness of the corresponding time-discrete optimal control problem. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a non-overlapping domain decomposition of the semi-linear elliptic optimal control problem on the graph into local problems on a small part of the network, ultimately on a single edge.

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Towards simulation based mixed‐integer optimization with differential equations

Cited by 21 publications

References 32 publications

Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

Incorporating Differential Equations into Mixed-Integer Programming for Gas Transport Optimization

GasLib—A Library of Gas Network Instances

Domain Decomposition of an Optimal Control Problem for Semi-Linear Elliptic Equations on Metric Graphs with Application to Gas Networks

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