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We are interested in the simulation and optimisation of gas transport in networks. The gas flow through pipes can be modelled on the basis of the (isothermal) Euler equations. Further network components are described by purely algebraic equations. Depending on the data and the resulting network dynamics, models of different fidelity can be used in different regions of the network. Using adjoint techniques, we derive model and discretisation error estimators. Here, we apply a first-discretise approach. Based on the time-dependent structure of the considered problems, the adjoint systems feature a special structure and therefore allow for an efficient solution. A strategy that controls model and discretisation errors to maintain the accuracy of the solution is presented. We provide (technical) details of our implementation and give numerical results.

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... It extends an adaptive multilevel stochastic collocation method recently developed in [22] for elliptic partial differential equations with random data to systems of hyperbolic balance laws with uncertain initial and boundary conditions. We have been developing in-house software tools for fast and reliable transient simulation and continuous optimization of large-scale gas networks over the last decade [7,8,9,10,11]. Exemplarily, here we will investigate the important task of safely driving a stationary running system into a newly desired system defined by uncertain gas nominations at delivery points of the network. ...

... As a rule of thumb, the most complex nonlinear Euler equations (M 1 ) should be used when needed and the simplest algebraic model (M 3 ) should be taken whenever possible without loosing too much accuracy. In a series of papers, we have developed a posteriori error estimates and an overall control strategy to reduce model and discretization errors up to a user-given tolerance [7,8,9,10,11]. A brief introduction will be given next. ...

... A detailed description which would go beyond the scope of our paper is given in [7, Sect. 2.2], see also [9,10]. Polynomial reconstructions in space and time of appropriate orders are used to compute η x,j and η t,j , respectively. ...

In this paper, we are concerned with the quantification of uncertainties that arise from intra-day oscillations in the demand for natural gas transported through large-scale networks. The short-term transient dynamics of the gas flow is modelled by a hierarchy of hyperbolic systems of balance laws based on the isentropic Euler equations. We extend a novel adaptive strategy for solving elliptic PDEs with random data, recently proposed and analysed by Lang, Scheichl, and Silvester [J. Comput. Phys., 419:109692, 2020], to uncertain gas transport problems. Sample-dependent adaptive meshes and a model refinement in the physical space is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space. A single-level approach which balances the discretization errors of the physical and stochastic approximations and a multilevel approach which additionally minimizes the computational costs are considered. Two examples taken from a public gas library demonstrate the reliability of the error control of expectations calculated from random quantities of interest, and the further use of stochastic interpolants to, e.g., approximate probability density functions of minimum and maximum pressure values at the exits of the network.

... There, adjoint calculus is applied to measure the influence of both errors separately on a given quantity of interest. Recently, we have published an algorithm to adaptively control model and discretization errors in simulations for gas supply networks [6,7]. This is considered to be the first step towards an efficient optimization framework with reliable error estimates. ...

... We want to use the more complex models in the pipes only when necessary and to refine the discretizations only where needed. Using the solution of adjoint equations as done in [3,4,2,6,7], one may deduce model and discretization error estimators to measure the influence of the model and the discretization on a user-defined target functional M . With u being the exact solution of the (most complex) model equations and u h being the approximate (numerical) solution for some choice of models, the error in the target functional can be approximated by ...

... Concerning the underlying adjoint equations, these error estimators are currently implemented in a first-discretize manner, which will be briefly described in the following. A more detailed description with some hints on the implementation can be found in [7]. Let t j ( j = 0, . . . ...

We are interested in the simulation and optimization of gas and water transport in networks. Those networks consist of pipes
and various other components like compressor/pumping stations and valves. The flow through the pipes can be described by different
models based on the Euler equations, including hyperbolic systems of partial differential equations. For the other components,
algebraic or ordinary differential equations are used. Depending on the data, different models can be used in different regions
of the network. We present a strategy that adaptively applies the models and discretizations, using adjoint-based error estimators
to maintain the accuracy of the solution. Finally, we give numerical examples for both types of networks.

... Also in the context of gas networks, the adjoint approach is already established. It has been used to make decisions in hierarchical models [5], error estimations [6] and for optimizing gas networks [18,13] using a finite volume discretization. ...

... (55) 5 With W −1 L T 2 W = L T 2 -this holds for our diagonal matrix W The integrals resulting from partial integration have to vanish as well. This gives rise to the adjoint boundary and initial conditions. ...

The stable operation of gas networks is an important optimization target. While for this task commonly finite volume methods are used, we introduce a new finite difference approach. With a summation by part formulation for the spatial discretization, we get well-defined fluxes between the pipes. This allows a simple and explicit formulation of the coupling conditions at the node. From that, we derive the adjoint equations for the network simply and transparently. The resulting direct and adjoint equations are numerically efficient and easy to implement. The approach is demonstrated by the optimization of two sample gas networks.

... The more general aspects of our network simulation tool are described in Chapter 3. More details about the adaptive control of model and discretization errors concerning the theoretical as well as implementational aspects can be found in [20,[22][23][24] and are not part of this work. Concerning the relevance of models based on PDEs in the context of gas and water supply networks, one has to consider that coarse discretizations often lead to similar results as quasi-stationary models without significant additional computational effort. ...

... Here, adjoint-based error estimators with respect to a given target functional are used to adaptively refine or coarsen the discretizations, and additionally, also different models may be applied in each pipe. For more details, we refer to [20,[22][23][24]. ...

In this work, we consider the simulation and optimization of gas and water supply networks. In particular, we are dealing with the daily operation of both types of networks. First, we describe a mathematical model for gas and water supply networks, which consists of partial and ordinary differential equations as well as algebraic equations. Supplemented with initial and boundary conditions, these can be solved by applying appropriate discretization schemes. For this purpose, we develop a software framework for problems on networks, which allows the application of different schemes for each type of equations and which takes care of the correct coupling. A matter of particular importance is the treatment of the underlying hyperbolic partial differential equations. Here, we apply an implicit box scheme as well as explicit central schemes. For both methods, we give a profound stability analysis in a well-known setting. The next step after the solution of simulation tasks is the computation of sensitivity information. Here, we follow an adjoint approach. The resulting gradient information can directly be applied in three state-of-the-art optimization tools, for which we provide interfaces in our software. In practice, the optimization tasks in the daily operation of gas and water supply networks also involve discrete decisions for switching certain network elements on or off. To handle mixed integer problems, we follow two approaches. First, we present a heuristic penalization strategy, and secondly, we develop an adaptive linearization technique for the treatment of the underlying nonlinear problems with mixed integer linear programming methods. Finally, numerical results for a real-life gas and water supply network are presented as well as for a representative test network provided by one of our industry partners.

... 6). For further details on the computation of the error estimators and the adaptive strategy, we refer to Domschke et al. (2018Domschke et al. ( , 2011bDomschke et al. ( , c, 2015 3 ...

We are concerned with the simulation and optimization of large-scale gas pipeline systems in an error-controlled environment. The gas flow dynamics is locally approximated by sufficiently accurate physical models taken from a hierarchy of decreasing complexity and varying over time. Feasible work regions of compressor stations consisting of several turbo compressors are included by semiconvex approximations of aggregated characteristic fields. A discrete adjoint approach within a first-discretize-then-optimize strategy is proposed and a sequential quadratic programming with an active set strategy is applied to solve the nonlinear constrained optimization problems resulting from a validation of nominations. The method proposed here accelerates the computation of near-term forecasts of sudden changes in the gas management and allows for an economic control of intra-day gas flow schedules in large networks. Case studies for real gas pipeline systems show the remarkable performance of the new method.

... 17,18 Also in the context of gas networks, the adjoint approach is already established. It has been used to make decisions in hierarchical models, 19 error estimations 20 and for optimizing gas networks 21,22 using a finite volume discretization. ...

The stable operation of gas networks is an important optimization target. While for this task commonly finite volume methods are used, we introduce a new finite difference approach. With a summation by part formulation for the spatial discretization, we get well-defined fluxes between the pipes. This allows a simple and explicit formulation of the coupling conditions at the node. From that, we derive the adjoint equations for the network simply and transparently. The resulting direct and adjoint equations are numerically efficient and easy to implement.

... The hierarchy of models for process co-simulation is a vast research field in itself. This idea has to our knowledge never been implemented specifically for reactive transport, but it has been proposed e.g. for particular problem settings in fluid dynamics and elastomechanics (Altmann, 2013;Altmann and Heiland, 2015) as well as in the broader context of theoretical model reduction and error control (Domschke et al., 2011). This, however, is a fertile interdisciplinary research task, and it is not difficult to foresee that significant progress in this area will soon be required to facilitate and fully leverage the pow-erful machine-learning algorithms already available in order to speed up any complex, multiscale numerical simulations. ...

The computational costs associated with coupled reactive transport simulations are mostly due to the chemical subsystem: replacing it with a pre-trained statistical surrogate is a promising strategy to achieve decisive speedups at the price of small accuracy losses and thus to extend the scale of problems which can be handled. We introduce a hierarchical coupling scheme in which “full-physics” equation-based geochemical simulations are partially replaced by surrogates. Errors in mass balance resulting from multivariate surrogate predictions effectively assess the accuracy of multivariate regressions at runtime: inaccurate surrogate predictions are rejected and the more expensive equation-based simulations are run instead. Gradient boosting regressors such as XGBoost, not requiring data standardization and being able to handle Tweedie distributions, proved to be a suitable emulator. Finally, we devise a surrogate approach based on geochemical knowledge, which overcomes the issue of robustness when encountering previously unseen data and which can serve as a basis for further development of hybrid physics–AI modelling.

... The hierarchy of models for process co-simulation is a vast research field on itself. This idea has to our 465 knowledge never been implemented specifically for reactive transport, but has been proposed, e.g., for particular problem settings in fluid dynamics and elastomechanics (Altmann, 2013;Altmann and Heiland, 2015) and in the broader context of theoretical model reduction and error control (Domschke et al., 2011). This is however a fertile interdisciplinary research task and it is not difficult to foresee that significant progress in this area will soon be required to facilitate and fully leverage the powerful machine learning algorithms already available, in order to speedup any complex, multiscale numerical simulations. ...

The computational costs associated with coupled reactive transport simulations are mostly due to the chemical subsystem: replacing it with a pre-trained statistical surrogate is a promising strategy to achieve decisive speedups at the price of small accuracy losses and thus to extend the scale of problems which can be handled. We introduce a hierarchical coupling scheme in which full physics, equation-based geochemical simulations are partially replaced by surrogates. Errors on mass balance resulting from multivariate surrogate predictions effectively assess the accuracy of multivariate regressions at runtime: inaccurate surrogate predictions are rejected and the more expensive equation-based simulations are run instead. Gradient boosting regressors such as xgboost, not requiring data standardization and being able to handle Tweedie distributions, proved to be a suitable emulator. Finally, we devise a surrogate approach based on geochemical knowledge, which overcomes the issue of robustness when encountering previously unseen data, and which can serve as basis for further development of hybrid physics-AI modelling.

... In general, those decisions are made for certain time blocks [T k−1 , T k ] within the simulation time [0, T ] and accordingly the output functional (10) is locally evaluated (time integrals over [T k−1 , T k ] instead of [0, T ]) as well as the error estimates. For the details on the computation of the error estimators and the adaptive strategy, we refer to [4,6,7,8]. ...

We are concerned with the simulation and optimization of large-scale gas pipeline systems in an error-controlled environment. The gas flow dynamics is locally approximated by sufficiently accurate physical models taken from a hierarchy of decreasing complexity and varying over time. Feasible work regions of compressor stations consisting of several turbo compressors are included by semiconvex approximations of aggregated characteristic fields. A discrete adjoint approach within a first-discretize-then-optimize strategy is proposed and a sequential quadratic programming with an active set strategy is applied to solve the nonlinear constrained optimization problems resulting from a validation of nominations. The method proposed here accelerates the computation of near-term forecasts of sudden changes in the gas management and allows for an economic control of intra-day gas flow schedules in large networks. Case studies for real gas pipeline systems show the remarkable performance of the new method.

... Since more accurate models are computationally more expensive, an appropriate use of a hierarchy of models is desirable. In a sequence of papers [11,12,13], we have developed adaptive strategies to automatically control the model selection, mainly depending on the dynamics of the gas flow. Generally, simplified models can be applied in regions with low activity, while sophisticated models have to be used in regions, where the dynamical behaviour has to be resolved in more detail. ...

This paper is concerned with coupling conditions at junctions for transport models which differ in their fidelity to describe transient flow in gas pipelines. It also includes the integration of compressors between two pipes with possibly different models. A hierarchy of three one-dimensional gas transport models is built through the 3x3 polytropic Euler equations, the 2x2 isentropic Euler equations and a simplified version of it for small velocities. To ensure entropy preservation, we make use of the novel entropy-preserving coupling conditions recently proposed by Lang and Mindt [Netw. Heterog. Media, 13:177-190, 2018] and require the equality of the total enthalpy at the junction and that the specific entropy for pipes with outgoing flow equals the convex combination of all entropies that belong to pipes with incoming flow. We prove the existence and uniqueness of solutions to generalised Riemann problems at a junction in the neighbourhood of constant coupling functions and stationary states which belong to the subsonic region. This provides the basis for the well-posedness of certain Cauchy problems for initial data with sufficiently small total variation.

... although the gravity is neglected in the asymptotic analysis of [4]. If then a stationary model is assumed, i.e., the time-derivatives ∂ ∂t are set to zero, the gravity is neglected, and the compressibility factor z is set to be constant, the algebraic model in [7] is attained for the continuity and the impulse equations: ...

The presented work contains both a theoretical and a statistical error analysis for the Euler equations in purely algebraic form, also called the Weymouth equations or the temperature dependent algebraic model. These equations are obtained by performing several simplifications of the full Euler equations, which model the gas flow through a pipeline. The statistical analysis is performed using both a Monte Carlo Simulation and the Univariate Reduced Quadrature Method and is used to illustrate and confirm the obtained theoretical results.

... It is well known that extensive useful information is contained in the derivatives of function which one wishes to optimise. With the growth and development of derivative-based optimisation methods, it becomes evident that large-scale problems can be solved efficiently, but only if there is accurate derivative information at hand (Liu et al., 2005(Liu et al., , 2007Mu and Zhang, 2006;Carnevale et al., 2008;Domschke et al., 2011). However, many mathematical difficulties arise when dealing with realistic applications. ...

Management decisions involving the locations of industrial pollution sources and remediation often depend on optimisation techniques to obtain an effective air quality control strategy. This paper presents the implementation of derivative-free optimisation method to determine feasible air pollution control policy. The gaseous pollutants and fine particulate pollutants in the air will be numerically investigated separately. The locations of industrial pollutant sources and pollutant emission reduction percentage for five chemical plants are chosen as the decision variables. Lastly, we provide four cases to investigate how to effectively control air quality.

In this paper, we are concerned with the quantification of uncertainties that arise from intra-day oscillations in the demand for natural gas transported through large-scale networks. The short-term transient dynamics of the gas flow is modelled by a hierarchy of hyperbolic systems of balance laws based on the isentropic Euler equations. We extend a novel adaptive strategy for solving elliptic PDEs with random data, recently proposed and analysed by Lang, Scheichl, and Silvester [J. Comput. Phys., 419:109692, 2020], to uncertain gas transport problems. Sample-dependent adaptive meshes and a model refinement in the physical space is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space. A single-level approach which balances the discretization errors of the physical and stochastic approximations and a multilevel approach which additionally minimizes the computational costs are considered. Two examples taken from a public gas library demonstrate the reliability of the error control of expectations calculated from random quantities of interest, and the further use of stochastic interpolants to, e.g., approximate probability density functions of minimum and maximum pressure values at the exits of the network.

Proceeding from balanced truncation-based parametric reduced order models (BT-pROM) a matrix interpolation strategy is presented that allows the cheap evaluation of reduced order models at new parameter sets. The method extends the framework of model order reduction (MOR) for high-order parameter dependent linear time invariant systems in descriptor form by Geuss (2013) by treating not only permutations and rotations but also distortions of reduced order basis vectors. The applicability of the interpolation strategy and different variants is shown on BT-pROMs for gas transport in pipeline-networks.

We consider the simulation and optimisation of transport processes through gas and water supply networks. Using a consistent modeling of the network, adjoint equations for the whole system including initial, coupling and boundary conditions can be derived. These are suitable to compute gradients for optimization tasks but can also be used to estimate the accuracy of models and the discretization with respect to a given cost functional. We show the applicability of an adaptive algorithm that automatically steers the discretization and models while maintaining a given accuracy in an optimisation framework. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this work, the simulation and optimization of transport processes through gas and water supply networks is considered. Those networks mainly consist of pipes as well as other components like valves, tanks and compressor/pumping stations. These components are modeled via algebraic equations or ODEs while the flow of gas/water through pipelines is described by a hierarchy of models starting from a hyperbolic system of PDEs down to algebraic equations. We present a consistent modeling of the network and derive adjoint equations for the whole system including initial, coupling and boundary conditions. These equations are suitable to compute gradients for optimization tasks but can also be used to estimate the accuracy of models and the discretization with respect to a given cost functional. With these error estimators we present an algorithm that automatically steers the discretization and the models used to maintain a given accuracy. We show numerical experiments for the simulation algorithm as well as the applicability in an optimization framework.

Contemporary design in engineering and industry relies heavily on computer simulation and efficient algorithms to reduce the cost and to maximize the performance and sustainability as well as profits and energy efficiency. Solving an optimization problem correctly and efficiently requires not only the right choice of optimization algorithms and simulation methods, but also the proper implementation and insight into the problem of interest. This book consists of ten self-contained, detailed case studies of real-world optimization problems, selected from a wide range of applications and contributed from worldwide experts who are working in these exciting areas.
Optimization topics and applications include gas and water supply networks, oil field production optimization, microwave engineering, aerodynamic shape design, environmental emergence modelling, structural engineering, waveform design for radar and communication systems, parameter estimation in laser experiment and measurement, engineering materials and network scheduling. These case studies have been solved using a wide range of optimization techniques, including particle swarm optimization, genetic algorithms, artificial bee colony, harmony search, adaptive error control, derivative-free pattern search, surrogate-based optimization, variable-fidelity modelling, as well as various other methods and approaches. This book is a practical guide to help graduates and researchers to carry out optimization for real-world applications. More advanced readers will also find it a helpful reference and aide memoire.

In this paper, we study the problem of technical transient gas network optimization, which can be considered a minimum cost flow problem with a nonlinear objective function and additional nonlinear constraints on the network arcs. Applying an implicit box scheme to the isothermal Euler equation, we derive a mixed-integer nonlinear program. This is solved by means of a combination of (i) a novel mixed-integer linear programming approach based on piecewise linearization and (ii) a classical sequential quadratic program applied for given combinatorial constraints. Numerical experiments show that better approximations to the optimal control problem can be obtained by using solutions of the sequential quadratic programming algorithm to improve the mixed-integer linear program. Moreover, iteratively applying these two techniques improves the results even further.

This article surveys a general approach to error control and adaptive mesh
design in Galerkin finite element methods that is based on duality principles
as used in optimal control. Most of the existing work on a
posteriori error
analysis deals with error estimation in global norms like the ‘energy norm’
or the L2 norm, involving usually unknown ‘stability constants’. However, in
most applications, the error in a global norm does not provide useful bounds
for the errors in the quantities of real physical interest. Further, their sensitivity
to local error sources is not properly represented by global stability constants.
These deficiencies are overcome by employing duality techniques, as
is common in a
priori error analysis of finite element methods, and replacing
the global stability constants by computationally obtained local sensitivity
factors. Combining this with Galerkin orthogonality, a
posteriori estimates
can be derived directly for the error in the target quantity. In these estimates
local residuals of the computed solution are multiplied by weights which
measure the dependence of the error on the local residuals. Those, in turn,
can be controlled by locally refining or coarsening the computational mesh.
The weights are obtained by approximately solving a linear adjoint problem.
The resulting a
posteriori error estimates provide the basis of a feedback process
for successively constructing economical meshes and corresponding error
bounds tailored to the particular goal of the computation. This approach,
called the ‘dual-weighted-residual method’, is introduced initially within an
abstract functional analytic setting, and is then developed in detail for several
model situations featuring the characteristic properties of elliptic, parabolic
and hyperbolic problems. After having discussed the basic properties
of duality-based adaptivity, we demonstrate the potential of this approach by
presenting a selection of results obtained for practical test cases. These include
problems from viscous fluid flow, chemically reactive flow, elasto-plasticity, radiative
transfer, and optimal control. Throughout the paper, open theoretical
and practical problems are stated together with references to the relevant literature.

We investigate coupling conditions for gas transport in networks where the governing equations are the isothermal Euler equations. We discuss intersections of pipes by considering solutions to Riemann problem. We introduce additional assumptions to obtain a solution near the intersection, and we present numerical results for sample networks.

We are interested in the simulation and optimization of gas transport in networks. Those networks consist of pipes and various other components like compressor stations and valves. The gas flow through the pipes can be modelled by different equations based on the Euler equations. For the other components, purely algebraic equations are used. Depending on the data, different models for the gas flow can be used in different regions of the network. We use adjoint techniques to specify model and discretization error estimators and present a strategy that adaptively applies the different models while maintaining the accuracy of the solution.

Natural gas is the third most important energy source in the world. Presently, the consumption of natural gas is increasing the most in comparison to other non-renewable energy sources. Therefore, optimization of gas transport in networks poses a very important industrial problem. In this thesis we consider the problem of time-dependent optimization in gas networks, also called Transient Technical Optimization (TTO). A gas network consists of a set of pipes to transport the gas from the suppliers to the consumers. Due to friction with the pipe walls gas pressure gets lost. This pressure loss is compensated by so called compressors. The aim of TTO is to minimize the fuel consumption of the compressors, where the demands of consumers have to be satisfied. Transient optimization of gas transmission is one of the great research challenges in this area. We formulate a mixed integer approach for the problem of TTO which concentrates on time-dependent and discrete aspects. Thereby, the nonlinearities resulting from physical constraints are approximated using SOS (Special Ordered Set) conditions. A branch-and-cut algorithm is developed which guarantees global optimality in dependence on the approximation accuracy. Concerning the nonlinearities, we discuss the quality of approximation grids by calculating approximation errors. The SOS conditions are implicitly handled via a branching scheme, supported by adequate preprocessing techniques. A heuristic approach based on simulated annealing yields an upper bound in our branch-and-cut framework. To improve the lower bound, we incorporate two separation algorithms. The first one results from theoretical studies of the so called switching polytopes which are defined by runtime conditions and switching processes of compressors. Linking of different SOS conditions gives a second separation strategy. We present theoretical investigations of the SOS 2 and SOS 3 polytope. These polytopes arise from the modeling of SOS Type 2 and SOS Type 3 conditions using additional binary variables. The results do not have practical relevance for our solution algorithm, but we characterize facet-defining inequalities providing complete linear descriptions of these polytopes. We evaluate the developed branch-and-cut algorithm using three test networks provided by our project partner E.ON Ruhrgas AG. Two are of artificial nature, as they were developed for test purposes. They contain all important elements of a gas network, but are rather small. The third network characterizes the major part of the Ruhrgas AG network in Western Germany. We test instances from three up to 24 coupled time steps.

We investigate the concept of dual-weighted residuals for measuring model errors in the numerical solution of nonlinear partial differential equations. The method is first derived in the case where only model errors arise and then extended to handle simultaneously model and discretization errors. We next present an adaptive model/mesh refinement procedure where both sources of error are equilibrated. Various test cases involving Poisson equations and convection diffusion-reaction equations with complex diffusion models (oscillating diffusion coefficient, nonlinear diffusion, multicomponent diffusion matrix) confirm the reliability of the analysis and the efficiency of the proposed methodology.

We consider a multiscale network of natural gas pipelines. Different arcs of the network are to be modeled by possibly different models depending on the requisite qualitative detail required: an isothermal Euler system of equations; linearized model derived from the isothermal Euler system or a steady-state model of gas flow also referred to as an algebraic model. At the vertices (or joints) of the network coupling conditions are defined. An analysis of the well posedness of the hierarchial coupling conditions is presented. The analytical results are tested numerically on different network configurations including a real-world network based on the Canadian mainline gas network. Copyright © 2007 John Wiley & Sons, Ltd.

Abstract We are interested in simulation,and optimization,of gas transport in networks. Different regions of the network,may,be modelled,by different equations: There are three models,based on the Euler equations that describe the gas flow in pipelines qualitatively different: a nonlinear model, a semilinear model and a stationary also called algebraic model. For the whole network, adequate initial and boundary values as well as coupling,conditions at the junctions are needed.,Using adjoint techniques,one can specify model,error estimators for the simplified models. A strategy to adaptively apply the different models,in different regions of the network,while maintaining,the accuracy,of the solution is presented.

A gas network basically consists of a set of compressors and valves that are connected by pipes. The problem of gas network optimization deals with the question of how to optimize the flow of the gas and to use the compressors cost-efficiently such that all demands of the gas network are satisfied. This problem leads to a complex mixed integer nonlinear optimization problem. We describe techniques for a piece-wise linear approximation of the nonlinearities in this model resulting in a large mixed integer linear program. We study sub-polyhedra linking these piece-wise linear approximations and show that the number of vertices is computationally tractable yielding exact separation algorithms. Suitable branching strategies complementing the separation algorithms are also presented. Our computational results demonstrate the success of this approach.

We consider gas flow in pipeline networks governed by isothermal Euler equations, and introduce a new modeling of compressors in gas networks. Compressor units are modeled as pipe-to-pipe intersections with additional algebraic coupling conditions for compressor behavior. We prove existence and uniqueness of solutions with respect to these conditions, and use the results for numerical simulation and optimization of gas networks.

We investigate the stability and convergence of an implicit box scheme for subsonic flows modelled by scalar conservation
laws with dissipative and possibly stiff source terms. The scheme is proposed for solving transient gas flow problems in pipeline
networks. Such networks are operated in the subsonic flow region and are characterized by pressure losses due to dissipative
friction terms. We verify the properties stated by Kružkov’s theorem (Kružkov, Math. USSR-Sb. 10:217–243, 1970) for the approximate solution and prove its convergence to the entropy solution.

Hierarchische Modellierung der Eulerschen Flussgleichungen in der Gasdynamik

- P Bales

Bales, P. (2005) 'Hierarchische Modellierung der Eulerschen Flussgleichungen in der Gasdynamik',
Diploma thesis, Technische Universität Darmstadt, Department of Mathematics, Darmstadt.

Transiente technische optimierung

- E Sekirnjak

Sekirnjak, E. (2000) 'Transiente technische optimierung (TTO-Prototyp)', Technical report, PSI AG,
November.

Hierarchische Modellierung der Eulerschen Flussgleichungen in der Gasdynamik', Diploma thesis

- References Bales

References
Bales, P. (2005) 'Hierarchische Modellierung der Eulerschen Flussgleichungen in der Gasdynamik',
Diploma thesis, Technische Universität Darmstadt, Department of Mathematics, Darmstadt.

Hierarchical modelling and model adaptivity for gas flow on networks

- P Bales
- O Kolb
- J Lang

Bales, P., Kolb, O. and Lang, J. (2009) 'Hierarchical modelling and model adaptivity for gas flow on
networks', Computational Science -ICCS 2009, Lecture Notes in Computer Science, Vol. 5544,
pp.337-346, Springer, Berlin/Heidelberg.