Chapter 11: What does “feasible” mean?

Joormann, Imke; Schmidt, Martin; Steinbach, Marc C.; Willert, Bernhard M.

doi:10.1137/1.9781611973693.ch11

Cited by 7 publications

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“…(5.11b) Figure 5.1 illustrates the classification of this transient pipe model compared to our stationary pipe model used in Chapter 4. Note that similar figures to classify gas models are used in Joormann et al [93].…”

Section: Derivationmentioning

confidence: 99%

“…15 illustrates the classification of this stationary pipe model compared to the transient pipe model used in Chapter 5. Note that similar figures to classify gas models are used in Joormann et al[93]. The remaining balance of moments (4.33) describes the pressure loss in a pipe due to ram pressure and frictional forces with a pressure function p = p(x) ∈ R >0 along the pipe, i.e., x ∈ [0,L a ].…”

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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: Derivationmentioning

confidence: 99%

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

Sirvent

2020

Operations Research Proceedings

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“…The detailed models are necessary to achieve a good understanding of the system state, but in many practical optimization applications only the stationary algebraic equations-or even further simplifications like piecewise linearizations as in [15,16,32]-are used in order to reduce the high computational effort of evaluating the state of the system with the more sophisticated models. However, it is then unclear how good the true state is approximated by these simplified models and error bounds are typically not available in this context; see the chapter [22] in [23] for a more detailed discussion of this issue. Recently, in [37], a detailed error and perturbation analysis has been developed for several components in the model hierarchy and it has been shown how the more detailed model components can be used to estimate the error obtained in the simplified models.…”

Section: Introductionmentioning

confidence: 99%

Model and Discretization Error Adaptivity Within Stationary Gas Transport Optimization

2018

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The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that a prescribed model and discretization error tolerance is satisfied in the end. We prove convergence of the adaptively controlled optimization method and illustrate the new approach with numerical examples.

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High detail stationary optimization models for gas networks: validation and results

Schmidt

Steinbach

Willert³

2015

Optim Eng

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Due to strict regulatory rules in combination with complex nonlinear physics, major gas network operators in Germany and Europe face hard planning problems that call for optimization. In part 1 of this paper we have developed a suitable model hierarchy for that purpose. Here we consider the more practical aspects of modeling. We validate individual model components against a trusted simulation tool, give a structural overview of the model hierarchy, and use its large variety of approximations to devise robust and efficient solution techniques. An extensive computational study demonstrates the suitability of our models and techniques for previously unsolvable problems in gas network planning.

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Chapter 11: What does “feasible” mean?

Cited by 7 publications

References 0 publications

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

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

Model and Discretization Error Adaptivity Within Stationary Gas Transport Optimization

High detail stationary optimization models for gas networks: validation and results

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