# Subprojects

of TRR 154 in phase one (2014 - 2018)

## Subfield A: Integer-continuous methods

Contact:

In this subproject, adaptive methods for the global solution of nonlinear mixed-integer optimization problems with ODEs will be developed. This will be performed on the example of stationary gastransport. One goal is the global decision about optimality and feasibility of such optimization problems, respectively. The global solution of large instances is computationally hard and requires the development of new and clever combination with existing mathematical methods. One motivation for the considered optimization problems is the long-term planning of the operation of gastransport networks, i.e., the question whether a given gas amount can be transported from given entries to exits. In this context we will first deal with stationary gas flows. The basic model is formed by the Euler equations, which shall be treated with adaptive techniques. In addition, flow conservation conditions at junctions and nonlinear descriptions of compressor stations as well as integer decisions, e.g., at valves, arise. This yields a mixed-integer feasibility problem for a coupled system of differential and algebraic equations. For the solution of such problem, a simplification or approximation of the system via a coarse discretization and/or model reduction has to be performed, which will then iteratively be refined. Here, a priori error bounds will be applied. Integral decisions and non-convexities will be handled via variable and spatial branching, respectively. Of particular interest are the adequate combination of branching methods with adaptive techniques for the discretization.

A poster to A01 from 2016 can be found here.

Contact:

Johann Michael Schmidt

In this project we analyze the optimal control of hyperbolic PDE systems with state constraints for the example of unsteady hyperbolic PDE Models for gas networks.

The gas flow is described by the compressible Euler equations and the network components, e.g. junctions, compressor stations, valves and pressure regulators are modelled by appropriate node or boundary conditions.

Within the model hierarchy of the TRR 154, our approach belongs to the highly detailed descriptions and seeks to derive and validate simplified models. Furthermore it provides the analytical basis for the efficient derivative-based optimization of subnetworks, which require accurate PDE models.

Through the time-dependent control of compressor stations and valves, the pressure and velocity distribution of the transported gas in the network has to be optimized under additional constraints, in particular upper and lower bounds on the pressure. Due to the switching of network components, global existence of classical solutions cannot be guaranteed. Hence, we will consider entropy weak solutions in the class of functions with bounded variation (BV-solutions) for the hyperbolic state equation. This results in a complex optimal control problem for BV-solutions on a network of nonlinear hyperbolic equations, whereby the state and the control are subject to pointwise inequality constraints.

The main goal of this project is to provide an sensitivity and adjoint calculus as well as optimality conditions, which will form the basis of adaptive multilevel optimization methods.

A poster to A02 from 2016 can be found here.

Contact:

Fabian Rüffler

The project focuses on control theory in order to study hybrid dynamical systems composed from the model hierarchy of gas networks up to the level of nonlinear iso-thermal euler-equations on pipelines and their interaction with discrete network components such as on/off-flaps, -valves and -compressors. The main line of research concerns mathematical foundations and shall answer questions for example regarding regularity and sensitivity of solutions for hyperbolic PDEs interacting with discrete events. The main challenges are the formulation of suitable solution concepts and their mathematical abstraction including the ability to topologize the instances and to find representations of sensitivities for events using adjoint-based methods. In case of state dependend switching, zeno effects are to be considered. Besides contributions to the above theoretical foundations, we expect by our study novel, locally convergent numerical methods by embedding the developed sensitivity formulas into derivative based optimization algorithms in order to efficently solve mixed-integer optimal control problems on the level of PDE-gas-dynamics.

A poster to A03 from 2016 can be found here.

Contact:

Prof. Dr. Martin Grötschel

Dr. Benjamin Hiller

Tom Walther

The goal of this subproject is to develop algorithmic fundamentals for the efficient treatment of switching decisions in gas networks. In particular, this involves the modelling and algorithmics of the switching operations in compressor stations, since these pose a significant source of modelling and runtime complexity.

The set of feasible operating points of a compressor station is, in general, non-convex, in some circumstances even non-connected. However, it can be well approximated by the union of convex polyhedra. Hence, the treatment of such structures in MIPs and MINLPs will be the main focus of research in this subproject. While being motivated by the optimization of gas networks, the methods to be developed will be relevant for many applications of MIPs and MINLPs.

Known techniques for modelling unions of polyhedra as the feasible set of a MIP rely on the inequality description of the underlying polyhedra. In contrast to this, another approach adapted to the geometric properties of the overall set can be considered. More precisely, the goal is to find and analyze a hierarchical description of a non-convex set that provides an as good as possible polyhedral relaxation on each level. This hierarchy can then be used by suitable branching strategies in the branch-and-bound procedure for solving MINLPs.

In the long term, this subproject of SFB/TRR 154 is aiming at the development of real-time methods for obtaining combinatorial decisions. Furthermore, since the transient control of gas networks requires the successive solving of many similar MIPs/MINLPs, reoptimization techniques come into view that use known information from previous optimization problems in order to reduce running time. For these, a detailed analysis of the problem structure and a deeper understanding of the complex MIP/MINLP solving process will be an essential topic of research.

A poster to A04 from 2016 can be found here.

Contact:

Mathias Sirvent

The objective of this project is the development of mathematical algorithms to find an optimal control for mixed-integer problems on transport networks with the help of decomposition methods. For the sake of synergy inside of the TRR 154 the focus is on gas networks, but the methods should also be useful for water networks or other energy networks. The optimization problems are planned to be decomponed with respect to variables but also with respect to subsystems, with the result that we are getting a time-expansive MINLP with a hierarchic structure. On the upper level, there are integer decisions, while on the lower level the focus is on continuous variables. Eventually the continuous variables are discretized for the numerical realization. This approach investigates the whole range from totally discrete MINLPs to PDE-based MINLPs in the Banach space. While we use well-know finite-volume methods to simulate the gas equations at the beginning, we want to include methods from the sub-project C02 during the progress. The same holds for the inclusion of MINLP-Solvers from the sub-project B07. So the focus of the sub-project A05 is on the mathematical analysis of structured MINLPs in the light of hierarchic models. The methods of many classical decomposition approaches like Benders, Outer Approximation or Dantzig-Wolfe focus on a generation of cutting planes in the subproblem, which tighten the relaxed set in the masterproblem to achieve a convergence between the values of the objective functions of the masterproblem (dual bound) and the subproblem (primal bounds). In this sub-project we want the subproblem to provide disjunctions for the masterproblem as well, because such an approach enables the algorithm to find global optima for non-convex problems as well.

A poster to A05 from 2016 can be found here.

Contact:

Dr. Martin Groß

The goal of this subproject is to apply network flow theory to simplified models for gas pipelines and related transportation networks. Network flow theory has proven itself to be a powerful and valueable tool for solving complex problems in many areas of application, like traffic networks, telecommunication networks, and logistic networks. All of these networks can be designed and operated using network flow algorithms, which exhibit a very efficient runtime behaviour, making them capable of handling the large instances occuring in practice. In this subproject, we strive to employ network flow algorithms for solving problems in gas and related networks. Conventional network flow theory is insufficient to describe gas networks, since it neither accounts for pressures in nodes, nor for the non-linear dependency of the flow from the pressure in nodes. However, there are several problems in gas networks which occur in a similar form in network flow theory and are known and well understood there, for example generalized or length-bounded flows. Furthermore, non-linear relationships in other aspects like for example the flow velocity in traffic networks, have been studied and solved by network flow theory. As a base for our models we use a model based on the Weymouth equations, which can be solved using convex minimum cost flows. This flow model and its solution techniques can be generalized from the stationary to the transient case, analogous to how classic network flows can be generalized to dynamic network flows. For solving dynamic networks (approximately), there are useful techniques known from network flow theory, e.g. adaptive time-expanded networks, which have a great number of applications. These techniques are to be made useable for transient gas flows in this subproject. Gas networks usually contain a number of active elements - valves, compressors, and so on - which require integral decisions which have a significant impact on the gas flow. Therefore, these elements have to be incorporated in the network flow model. This can be handled by Branch & Bound based methods, or by adaption of techniques from network design. For Branch & Bound based techniques it is crucial to identify infeasible solutions as fast as possible. In gas networks, this means identifying partial configurations of active elements which imply infeasiblity. Therefore, analysis of infeasiblity will be a focus of the research done in this subproject. By developing efficient models and solution techniques for gas networks and characterizing infeasibility in these models, this subproject will contribute for Demonstrator 1, with a special focus on large gas networks.

A poster to A07 from 2016 can be found here.

Contact:

Susanne Beckers

The project is concerned with a posteriori error estimation in the context of mixed integer-continuous optimization. It aims at robustification of discrete decisions with respect to unavoidable discretization errors. In contrast to robustification against uncertainties that are only realized ex-post, the discretization errors can, in principle, be made arbitrarily small by increasing the computational effort.

In continuous optimization, the error in the decision can thus be made arbitrarily small by spending additional computational time. If discrete decisions are involved this is no longer true, since discrete decisions inherently depend discontinuously upon the data of the problem. Consequently, the central question of this project is to derive conditions, under which it can be ensured that a discrete decision would have been taken identically even if no discretization error occurred, i.e., the decision is robust with respect to the chosen discretization accuracy.

In the context of gas networks, this problem can be demonstrated already with a single pipe with connected compressor. If, for disabled compressor, the pressure in the pipe is close to the minimal allowed pressure, a small discretization error may lead to a wrong decision whether to activate the compressor or not. The methods to be developed within this project therefore will allow to characterize these situations, and thereby provide information whether the decision may still depend upon the discretization accuracy, or if further changes in the discretization will not have any influence on the decisions made.

## Subfield B: Model adaptation and coupling

Contact:

Pascal Mindt

The aim of this project is the development of an integrated, dynamic multiscale approach for the numerical solution of the compressible instationary Euler equations on network structures. The main components of a gas network are pipelines, compressor stations and valves to regulate the gas flow. The transport of gas is modeled by the one-dimensional Euler equations, a hyperbolic system of nonlinear partial differential equations. Algebraic equations model the behavior of compressors, valves and the gas flow at branching points. For a complete description of the network, we need to specify adequate initial, coupling and boundary conditions. The complexity of the overall system increases significantly with the size of the network. An efficient and error-controlled simulation of gas networks of practical relevance with given input/output parameters while maintaining tolerance limits is the key to a multilevel-based real-time optimization. It is the goal to run most optimization steps on coarse discretizations and only a few final steps with complex models on highly refined grids, which implies an enormous savings potential. In this project we will also develop an appropriate simulation platform.

In regions with low activity, simplified physical models based on the Euler equations can be used, which form a model hierarchy in a natural way. The degree of simplification ranges from non-linear and semi-linear partial differential equations via ordinary differential equations to algebraic relations for the description of a stationary gas flow. Here, adaptive spatial and temporal discretizations together with the models of a model hierarchy are controlled and linked together locally. This allows for an efficient simulation of the whole network over the entire time horizon with respect to a given accuracy. For this we develop a posteriori error estimators in the context of the dual weighted residual method to implement the dynamic switching from fine-scale to coarse-scale levels using adjoint calculus. The strategies that were developed for the already well understood isothermal case need to be extended to the case of temperature-dependent Euler equations and will be tested for small and medium-sized subnets in the first instance.

As a central discretization method we will extend an existing implicit box scheme that has already been used successfully for the isothermal case. Implicit methods are stable regardless of local CFL numbers and the stiffness of the friction terms. Hence, they provide the flexibility for a stable linkage of spatio, temporal and model-adaptive discretizations.

It is also the goal of this project to develop interfaces for a connection to gradient-based optimization algorithms. These will be used to check feasibility for mixed-integer optimization problems and for the calculation of feasible solutions. Here, the discrete adjoints that are computed in the context of the dual weighted residual method form a natural link.

A poster to B01 from 2016 can be found here.

[1] P. Domschke, B. Geißler, O. Kolb, J. Lang, A. Martin, and A. Morsi. Combination of nonlinear and linear optimization of transient gas networks. INFORMS Journal on Computing, 23(4):605–617, 2011.

[2] P. Domschke, O. Kolb, and J. Lang. Adjoint-based control of model and discretisation errors for gas flow in networks. Int. J. Mathematical Modelling and Numerical Optimisation, 2(2):175–193, 2011.

[3] O. Kolb, J. Lang, and P. Bales. An implicit box scheme for subsonic compressible flow with dissipative source term. Numerical Algorithms, 53(2):293–307, 2010.

Contact:

Prof. Dr. Michael Hintermüller

Dr. Nikolai Strogies

Associated: Prof. Dr. Thomas Surowiec

The project work concentrates on modeling aspects as well as the conception and analysis of robust numerical methods for solving inverse problems for switching (or hybrid) systems of partial differential equations on graphs. In this context, the graph typically represents a transportation network for a specific substance or commodity. In accordance with the focus application of this collaborative research center (CRC), the main emphasis lies on networks transporting for (natural) gas. The project pursues several research goals:

- The identification of unknown parameters (e.g. the Darcy friction factor) and the detection of anomalies (leakage).
- the optimal arrangement of sensors in the network for a robustifying the identification tasks
- the quantification of uncertainties in the identified parameters or in other statistically relevant quantities.

Within the CRC, this project will collaborate with other subprojects on (constrained) optimal control problems for hyperbolic or hybrid systems and their numerical realization, and on numerical methods for the identification of the friction coefficient. It will equip the hierarchy of models considered with the CRC with relevant physical parameters or statistical information resulting from solving various identification problems. The project participates in demonstrator D1.

A poster to B02 from 2016 can be found here.

Contact:

Jeroen J. Stolwijk

The aim of project B03 is the development of a new methodology for the coupling of widely different mathematical models in a network. Moreover, error controllers are developed on the basis of modeled errors and uncertainties using the example of gas networks. These errors and uncertainties in submodels of the complex network are balanced in the overall simulation. Therefore, measures for the errors and uncertainties should be modeled for every submodel and made comparable. This can only succeed on the basis of a detailed model hierarchy, see Figure 1. All the errors and uncertainties in the simulation and optimisation are considered as an error in the finest modeling level using a backward error analysis in the model hierarchy. The estimated error in the finest level forms the mathematical basis for the development of a robust coupling controller. The controller should allow us to control the overall error in such a way that a prescribed simulation or optimisation goal is achieved within a desired tolerance.

Image 1: Modelhierarchy for gas flow. Source: Domschke, P., Kolb, O. and Lang, J. (2011) 'Adjoint-based control of model and discretisation errors for gas flow in networks', Int. J. Mathematical Modelling and Numerical Optimisation, Vol. 2, No. 2, pp.175-193.

A poster to B03 from 2016 can be found here.

Contact:

The aim of this project consists in applying nonlinear probabilistic constraints to optimization problems in gas transportation assuming that the underlying random parameter obeys a multivariate and continuous distribution. Doing so, a robust in the sense of probability design of gas transport shall be facilitated. Stochastic optimization is the appropriate mathematical discipline to cope with uncertainty when looking for optimal decisions under random perturbations of some nominal parameters. Among different modeling options, probabilistic constraints hold a key position first of all in engineering applications. The solution of optimization problems with nonlinear probabilistic constraints with continuous multivariate distributions can be considered as new ground both from the theoretical and – at least for interesting dimension - from the numerical perspectives. Moreover, in the present project, we deal with implicit probabilistic constraints where the relation between decision and random parameter is established only by additional variables via some equation system. Although gas network problems with uncertain injection and consumption provide a very natural motivation for the research in this project, the mathematical insight to be expected has an impact on quite different applications as well, for instance, on optimization problems of power management, particularly those related with renewable energies. Beyond this, optimal control problems governed by PDEs and subjected to random state constraints promise being a potential application of implicit probabilistic constraints. In its first phase the project will investigate optimization problems arising from a simple stationary gas network model (RNET-ISO4) subject to random loads. Here the probabilistic constraint ensures the technical feasibility of loads with a specified probability. In the longer perspective, the consideration of dynamic probabilistic constraints for time-dependent decisions and of binary variables shall be pursued.

A poster to B04 from 2016 can be found here.

Contact:

Dr. Ralf Gollmer

Sabrina Nitsche

Dr. Claudia Stangl

Tobias Wollenberg

The aims of the present subproject are: (i) Deepen the structural and algorithmic understanding of steady-state stochastic gas network models, and (ii) develop algorithmic approaches to stochastic models of gas transportation with reduced time-discrete dynamics. These target topics receive their inspiration from working along the formulations in the common model catalogue of the TRR 154. Key features in this respect are incomplete or probabilistic information, for instance on (inlets and) outlets of the net or the availability of net components. The analysis proceeds step-by-step, heading for mathematical underpinning and tailored algorithms for the new stochastic models. The following subjects are addressed.

Structural Analysis, Algorithm Design, and Model Classification. Here, new insights are gathered in a broad spectrum from fundamental structural properties to design and numerical testing of algorithms for steady-state random optimization models in gas networks. In the end, approaches shall be identified which seem prosperous for problems with random dynamics, too. Here the dynamic models under investigation in Subproject C03 will be taken into account and models from uncertainty quanti cation (UQ) will be explored jointly with Subproject B02.

Algebraic Characterization of Feasibility. Here, a pivotal feasibility problem in gas networks under uncertainty is addressed, namely, whether for a given network state further (random) transportation orders are technically feasible. The latter is to be verified for balanced orders, without a priori assigning fixed gas quantities to in- and outlets. Jointly with Subproject B04 first ideas shall be extended for the case that order volumes follow Gaussian laws. Jointly with B04 and B03 related mathematical concepts from propagation of stochastic errors in coupled partial models of di ering mathematical 1 provenience shall be studied.

Explicit Flow and Pressure Profiles, Inversion of Polynomial Equations. Here, investigations are directed to basic novel insights into mappings given by piecewise multivariate polynomials of degree 2 involving absolute values. These mappings arise via elimination of pressure variables from the steadystate counterparts to the Euler equations. While the elimination as such is well-known, its consequences regarding explicit inversion are open. Also, conclusions of analytical properties of the mappings, such as monotonicity and coercivity, will be explored. The envisioned inverses lead to explicit representations of complete ow-pressure profiles depending on the in- and outlets. They are important for the analysis of models ISO-ALG from the common catalogue of TRR 154.

Decomposition of time-discrete stochastic gas network models. Investigations are related to the aim (ii) at the beginning of the present text. They shall establish an algorithmic link to transient gas ow models and algorithms. To this end, a given transient model is discretized in time and space. Up to a given moment in time, data are deterministic, and stochastic with finitely many realizations afterwards. In this way, two-stage stochastic programs arise which resemble the stochastic programming classics" to an extent enabling not just the transfer of models but also suggesting to study transferability of decomposition approaches from finite-dimensional optimization.

A poster of B05 from 2016 can be found here.

Contact:

Denis Aßmann

The goal of this research project is the development of tractable robust counterparts for global optimization problems, with a focus on gas networks. The motivation stems from the fact that for many real-life problems some parameters can only be estimated roughly. A well-known example in gas network optimization is the roughness value of the pipe that influences the friction of the gas and thereby effects the pressure loss between the endpoints of the pipe. However, the roughness depends on the contamination of the pipe and can only be measured with great effort. Another example is the real gas factor which depends on the gas mixture. Since gases with different chemical composition are mixed within the network, usually the exact gas mixture is unknown, and the real gas factor has to be estimated. Moreover, different formulas are used that describe the function for determining the friction from the pipe roughness. Finally, there are methodological uncertainties from the approximation of nonlinear functions in the context of mixed-integer linear optimization problems (MIPs). Similar situations are found in a wide range of applications. Therefore, results of this research project may be used for other optimizations problems under uncertainty, e.g. for water-network optimzation. In our robust optimization setting, continuous state variables are categorized as adjustable ("wait-and-see"), whereas binary decision variables are modeled as static or "here-and-now" variables. The robustification of the mentioned problem leads to mixed-integer linear, conic quadratic or positive semidefinite optimization problems, depending on the given uncertainty set and the occurance of the uncertain data. These different modeling options are adapted for gas-network optimization. A major goal will be the development of exact methods that use positive semidefinite subproblems. Initially, only the stationary case is considered. However, an extension to straight-forward transient models is a mid-term goal.

A poster to B06 from 2016 can be found here.

Contact:

Robert Burlacu

Dr. Lars Schewe

Goal of the project is the analysis and solution of large-scale MINLPs, especially from the application of instationary gas network optimization, using adaptive MIP models. We approximate the nonlinearities with piecewise-linear functions to construct MIP relaxations of the underlying MINLP. In addition, theoretical results linking the complexity of the relaxations to structural properties of the nonlinear functions and the linearization error shall be derived, whereby known statements of approximation theory are to be combined with techniques of polyhedral combinatorics. Furthermore the polyhedral structure of the resulting MIP relaxations shall be investigated.

The achievements shall be used to develope suitable adaptive refinement algorithms for the MINLP solution method proposed in [1]. Since in a refinement step both variables and constraints are added to the problem, it has to be considered that in the worst case a cold start of the MIP solution method is necessary. Therefore the refinement algorithms have to be developed such that they can be integrated in a Branch-and-Cut solution method.

Moreover structural results shall be used to obtain upper bounds of the complexity of solving methods for classes of nonconvex MINLPs and to enhance the method proposed in [1]. Eventually this shall lead to a method which is capable of solving nonconvex MINLPs from the field of instationary gas network optimization.

A poster to B07 from 2016 can be found here.

Contact:

Julia Grübel

Dr. Lars Schewe

The goal of this project is the analysis of the relation between (i) the equilibria of simple models of competitive natural gas markets, using complementarity problems for modeling the behavior of different players, and (ii) the solution of corresponding single-level welfare maximization problems. The understanding of this fundamental relation is a prerequisite for an analysis of the current entry-exit gas market design in Europe. Similar questions have been studied in detail in the context of electricity market modeling in the past. For natural gas markets, however, the addressed questions are much more complex and not yet well understood for adequate models of gas physics. The reasons for the high level of complexity is twofold: First, gas flow through pipeline systems is inherently nonconvex due to gas physics. This renders classical first-order optimality conditions possibly insufficient. Second, the operation of gas transport networks comprises the control of active network devices like (control) valves or compressors. These devices introduce binary aspects and thus a further type of non-convexity to the models of the underlying equilibrium problems.

As a result of the project we will obtain a first reference model that combines gas physics and a market analysis in a well-understood way. This will lay the ground for multilevel models of entry-exit natural gas markets that account for network characteristics. Beyond that, our results will enhance the understanding of binary equilibrium problems.

## Subfield C: Analysis and numerical simulation

Contact:

Christoph Huck

Björn Liljegren-Sailer

Prof. Dr. Nicole Marheineke

This subproject focuses on the development and analysis of models and methods for a stable and fast simulation of huge transient gasnetworks, which will also be used for an efficient parameter optimization and control of the network. The main aspects are the development of a numerical discretization in space and time that is adjusted to the topology of the network as well as a hierarchical modelling of several elements (pipes, compressors ect.) and subnet-structures.

For the complete network as a coupled system of nonlinear partial differential equations and algebraic equations (PDAE) we consider approximations by a spatial semi-discretization. We strive for a determination and classification of topology depending critera for the index of the time dependend differential algebraic system. Topology- and controldepending spatial discretizations will be determinded, that lead to DAEs of index 1, in order to diminish the influence of perturbations for the DAE system best possible. Moreover we want to establish a perturbationanalysis as well as existence and uniquness results for die PDAE-model. Here, the time and pressure/flow-depending control-states that can change the variable structure (dynamic as well as static) for certain points in time and for certain states of the network will be a major challange.

As a method, we focus on a Galerkin-Approach in space followed by a discretization in time of the resulting DAE with implicit or semi-implicit methods respectively, such that the algebraic constraints hold for the current point in time. Continuationmethods and space-mapping techniques are used for the initialisation to guarantee good convergence behaviour. Furthermore, to satisfy the control requirements of the systems and to enable the handling of huge networks, this subproject aims at the enhancement of the simulation speed. It is planed to detect characteristic subnetstructures and derive parameter dependen transient surrogate models with suitable error estimators by applying model order reduction methods. These input-ouput models as dynamic systems of ODEs will be coupled with die complete PDAE model in one model hierarchy.

A poster to C02 from 2016 can be found here.

Contact:

David Wintergerst

In this project optimal control problems with random boundary data will be analyzed. In the context of gas transport one aim is to find a control satisfying the given demands of consumers regarding gas flow and pressure at the boundary nodes, while economic and technical constraints of the gas flow and pressure need to be complied. In- and Output of the system are assumed to be balanced. The control is continous in time and is applied at the control elements of the network (e.g. compressors, valves). The objective function is a norm of the L^2-type, measuring the difference between boundary flows and pressures, generated by the system and the desired demands. The system dynamics are modeled by several PDEs defined on the network with coupling conditions at the network nodes. Apart from control interventions the system state is influenced by the consumers' randomly driven withdrawals and by randomly driven feeds at the boundary nodes. The goal of this project is to develop a fundamental theory of optimal nodal control of hyperbolic quasi- or semilinear systems with randomly driven parameters on graphs. Hereby, risk-neutral and risk-averse objective functions are used, hence extending the PDE-modeling by an adequate modeling of stochastics. For the resulting novel optimal control problems, the well-posedness of the system dynamics are examined and existence and regularity of solutions are proven. Furthermore, optimality conditions will be derived, followed by the developement of a sensitivity analysis describing the behavior of the optimal value and the optimal solution under noisy model data including the underlying probability distribution.

A poster to C03 from 2016 can be found here.

Contact:

Thomas Kugler

The goal of the project is the construction and analysis of numerical methods for singularly perturbed hyperbolic problems with parabolic asymptitics. Problems of this kind arise, for instance, in compressible flows, in the shallow water equation, in models of acoustic of electrmagnetic wave propagation, or in more general transport processes.

Within the research of the TRR 154, the focus of the project lies on systems of balance equations with stiff relaxation modeling the friction dominated transport of gas through pipelines. Galerkin methods of high order will be considered for the numerical soution of these problems close to the parabolic limit. Of particular interest in this context is the asymptotic stability of the discretization schemes, i.e., the methods have to be capable to stably approximate the hyperbolic problem as well as the parabolic limit problem. In addition, important physical properties, e.g., mass conservation and energy dissipation, should be preserved on the discrete level.

The methods to be considered in this project are mixed and hybrid continuous and discontinuous finite element methods. Particular research directions of the project are:

- the systematic construction of hierarchical Galerkin methods of arbitrary order for single pipes and small pipe networks.
- the a-priori and a-posteriori error analysis of these methods, with an emphasis on the asymptotic regime.
- the efficient numerical realization, e.g. the fast assembly of linear systems, static elimination, or the use of predictor-corrector strategies for efficient time integration.
- the use of the hierarchic construction as a model order reduction paradigm.
- the exact computation of sensitivities and adjoints required for the systematic error control and treatment of optimization problems.
- the calibration of the computational models to independent simulations or measurements via parameter identification.

For all these research directions, the Galerkin strategy proposed for the discretization will be a key ingredient. The methods developed in this project will be incorporated in the simulation and optimization of larger networks. The asymptotic stability and energy dissipation of the schemes will help to correctly and efficiently predict the behaviour of a gas network under sudden changes of supplies or demands.

A poster to C04from 2016 can be found here.

## Subfield Z: Central projects

Contact:

Prof. Dr. Nicole Marheineke

The Integrated Graduate School offers its members a unique scientific training in the research area of the TRR 154 “Mathematical Modeling, Simulation and Optimization exemplified by gas networks.” In addition, the members develop key skills required for a successful professional career in industry and academia. In the long term, this will increase the career prospects of TRR 154 members as well as the TRR 154's appeal to outstanding national and international applicants. The members of the Graduate School are doctoral candidates funded by the TRR 154, students whose projects are closely related, as well as candidates that have been awarded short-time grants (between 6 and 12 months). Moreover, post-doctoral researchers working on related fields are also admitted.

The scientific part of the research program is defined by the contents of the TRR 154 and includes for example trans-regional summer and winter schools, regular lectures and block courses given by guest researchers as well as lecture series that at each site are specifically designed for the main research fields of the TRR 154.

The doctoral candidates reach early scientific independence by establishing their own research networks. These networks will be built by joint events of different graduate schools, by visiting international conferences, by research stays abroad or by participating in excursions. The doctoral candidates are responsible for the content and organization of parts of the activities such as, for example, organizing academic seminars in collaboration with research partners. The members organize themselves by forming a committee of Student Representatives that consists of representatives from each site. Furthermore, trainee programs of each site’s Graduate School provide additional training to improve interdisciplinary skills.

In order to ensure a specific and comprehensive supervision that allows graduates to build a research profile and complete their thesis project within a predefined time period, each member of the Graduate School is assigned two mentors that are PIs in the TRR 154. Each research plan is specified in a PhD agreement. The agreements are based on the standards of the respective Graduate School of each site. Furthermore, they are agreed upon by the doctoral students and their supervisors. They also cover publications, attended lectures and workshops as well as soft-skill courses. Regular discussions with the supervisors and mentors allow critical review of the scientific progess.

Contact:

Prof. Dr. Thorsten Koch

Oliver Kunst

The subproject Z02 aims for a central database of realistic gas network data in standardized formats, and providing interfaces to allow an easy access to the data. This common data standard and common pool of test data drive the workflow and cooperation of all subprojects.

A gas network is described by many parameters. Experiences show that it is a very hard task to generate realistic data for gas networks with no access to real data of existing networks.This subproject cooperates with the biggest German network operator, Open Grid Europe GmbH, to compile test data from real gas network data.

The generated test data will extend the existing GasLib library for stationary networks. This database relieves other subprojects from generating their own test data, and provides common test cases for them. So, different subprojects can compare their results more easily, since they use the same underlying data.

One focus of the subproject Z02 is to standardize the data formats used, and to provide interfaces to them.This interfaces ease the data access for others. The usage of the same data structure throughout the different subprojects leads to more compatible programs and an easier cooperation.

The overall goal of the subproject Z02 is to compile a selection of data sets, which can be used to run different demonstration scenarios. The data sets and their descriptions will be published, such that transparency and reproducibility of the results are assured.