# Subprojects

of TRR 154 in phase two (2018 - 2022)

## Subfield A: Integer-continuous methods

Contact:

The goal of this project is to develop methods for the global solution of optimization problems subject to ODE or PDE constraints and integer decisions. On the one hand this should be performed for instationary gas flow and on the other hand for topology planing problems. The key issue is the development of good lower and upper bounds for the solutions and a adequate handling of the integer decisions.

Contact:

We consider the analysis and consistent numerical discretization of optimization problems for transient hyperbolic PDE models of gas networks with state constraints. We plan to analyze the convergence of numerical approximations and the corresponding sensitivities and adjoints for optimization problems governed by systems of hyperbolic balance laws on networks with continuous and switching controls. Moreover, the sensitivity and adjoint calculus developed in the first phase shall be extended to more general BV solutions on networks.

Contact:

The aim of the project is the development of control theory for mixed integer-continuous (hybrid) dynamical systems comprising partial differential equations. Based on regularity and sensitivity results obtained in the first funding period the project now designs and investigates receding horizon methods that are able to take decisions based on optimality principles for the control of such systems also under uncertainties for example being applicable for the control of a gas network by valves in non stationary situations.

Contact:

As in the first phase, we consider the semilinear Euler equations. Due to the results obtained in the first phase, we are able to compute an instantaneous control for networks of moderate size for time-dependent problems on the basis of the semilinear model. For this purpose, a mixed implicit-explicit Euler approach for the time discretization was used. This strategy makes it possible to understand the friction term as a quadratic term in the flux and to treat it as a monotonous perturbation. This analysis, which was completed in the first phase in AP 1, was extended in AP 2 by a spatial decomposition, in which the complete problem on the entire graph is iteratively decomposed into analog problems on smaller subgraphs. Therefore it is now possible to decompose the time-dependent overall problem into problems containing subgraphs of suitable size with valves and compressors, whereby, due to the formulation as an instantaneous control problem, only one time step has to be optimized at a time. The spatial decomposition allows a complete parallelization of the optimal control problems in such a way that the problem considered in the current iteration is updated on a given subgraph over the connecting edges with all adjacent subgraphs (from the last iteration). The local optimality systems then correspond to the local optimal control problems in this case. The goal of the second phase is to combine this spatial decomposition with an iterative time domain decomposition. For this purpose the problem shall be decomposed into subgraphs as well as into smaller continuous time intervals. In this way semilinear problems arise on subgraphs and time horizons, which lead to large and block-structured MINLPs after complete discretization in time and space (based on AP 3 of phase 1). The size of the subgraphs and the continuous time horizons should be controlled in such a way that the resulting finite-dimensional MINLPs can be solved to global optimality in practice. The subproblems include valves, compressors and other control elements, so that coupling between the subnetworks takes place exclusively via transmission nodes. The principle of space-time decomposition has already been described for hyperbolic equations and systems in the literature. This literature contains a posteriori estimates, which allow to design the decomposition adaptively. The procedure for time domain decomposition to be developed in AP 1 can be realized in different ways. It is intended to introduce an iterative time-domain decomposition into continuous subintervals and to combine this with a non-overlapping spatial decomposition of the entire graph into sub-graphs. As a result of these decompositions in space and time, time-dependent optimal control problems remain to be solved on smaller space-time domains. For this purpose we want to follow the discretize-then-optimize paradigm and perform a full discretization of space and time. The resulting finite-dimensional MINLPs themselves are subject of the second and third work package. The underlying constraints are of block-structured diagonal form with additional lines for coupling the time steps. With this form we have numerous experiences in the linear context as well as preliminary work in the context of stationary gas network optimization. Both will be used for the investigation of the nonlinear and instationary case. In the second work package, for the fully discretized and thus finite-dimensional case, it is investigated how both the time structure and the structure of the controls can be algorithmically exploited within the framework of decomposition-based penalty methods. The same discretized MINLPs are to be replaced by MIP relaxations in the third work package with the experiences from the first phase. Subsequently, it will be investigated whether the MIP of a time step can be projected onto the following time step. If this succeeds, an iterative procedure in time can be set up. In the ideal case, this would allow to reduce the very high-dimensional MIP to be solved by the sequential solution of MIPs of the order of size of one time step

Contact:

The project studies the optimization of the capacity usage of gas networks with efficient network flow methods. Based on structural insights and algorithms for the computation of instationary flows in time-expanded networks, new (approximation) algorithms for the robust and online-optimization of gas networks are devised. Finally, the allocation of the network capacities in incentive-compatible auctions is studied.

## Subfield B: Model adaptation and coupling

Contact:

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. These methods will be used for the description of the stochastic behavior of practically relevant outputs relative to randomized parameters in hyperbolic differential equations (quantification of uncertainty), the construction of reduced order models and an adaptive multilevel optimization for gas networks. In the first project period, modelling aspects and the development of adaptive discretizations were of primary importance. Adaptive spatial and temporal discretizations are controlled and combined with models from a newly established model hierarchy such that an efficient simulation of gas networks over the whole time horizon relative to a prescribed tolerance becomes available.

In the second project period, the influence of dynamic market fluctuations, which can be described by randomized initial and boundary values, on objective functions and scopes for the optimal control of gas networks in the framework of an uncertainty quantification will be investigated. Therefore, adaptive stochastic collocation methods with multilevel-like strategies for the reduction of the variance will be used. The integrated application of multilevel methods in space, time, and model as well as stochastic components lead to a reduction of computing time if resolution hierarchies in the corresponding approximations (space, time, model, stochastics) are employed. The stochastic collocation is realised by means of anisotropic sparse Smolyak grids. The inherent sampling strategy allows for the use of reduced, structure-preserving models in order to further reduce the computing time even perspectively for large scaled networks. It is the goal to combine adaptive grid and model refinements with adaptive collocation methods to improve the multilevel methods and to achieve rigorous quality requirements for expectation and variances of solution functionals for the uncertainty quantification at reduced computing time.

Contact:

Prof. Dr. Michael Hintermüller

The objective is to provide a mathematical description of markets that are coupled with physical processes for investigating economic questions regarding the behavior of market participants or optimal capacity utilization of the transport network. The main focus lies on analyzing generalized Nash equilibrium problems that include the physical processes as well as state- and control-constraints, the efficient numerical treatment of such problems and consideration of agents that are risk-averse against uncertainties in various parameters of the mathematical model.

Contact:

The goal of the project is to develop a new methodology for the coupling of mathematical models with strongly different modeling accuracy and different scales in a network. It is planned to use a system theoretic approach as models of port-Hamiltonian (pH) systems of differential-algebraic equations. A second topic is the data-based construction of pH surrogate models, e.g., for compressor stations, that can be implemented in the model hierarchy, as well as the structured incorporation of hybrid model components, such as valves. The third topic is the development of structure preserving model reduction methods for the network components as well as the whole network, including appropriate error estimates.

Contact:

The project is devoted to the consideration of uncertainties in gas transport, mostly random loads, via chance constraints. These allow one to find optimal and robust decisions in the sense of probability. The focus of future research will be on embedding of such constraints into equilibrium problems (MPECs). Doing so, one may complete gas market models by a component taking into account robust load satisfaction. This requires both a theoretical analysis of structure and the development of appropriate algorithms.

Contact:

The aim of the project is in the extension of results from Phase 1 on characterizing nomination validity in computationally feasible fashion for strongly meshed gas networks under stochastic uncertainty. To this end, recent results from symbolic computation (comprehensive Gröbner systems) are employed. Moreover, approaches to risk averse optimization in stochastic gas networks with network constraints and different market models, for instance, with or without nodal pricing will be developed.

Contact:

The focus of this research project consists in modelling of robust optimization problems in gas networks, their theoretical analysis, as well as the development of appropriate solution approaches. Building upon the results of the first funding phase for the stationary setting, decomposition approaches will be developed for the resulting two-stage robust optimization problems with uncertainties in the demands as well as in the physical parameters. The decomposition approach envisaged here enables a generalization towards the instationary case and coupled robust-stochastic optimization tasks. Market aspects will be integrated via welfare optimization in the nodal price system.

Contact:

In this subproject we will develop techniques to solve equilibrium problems with integer constraints using MIP techniques. To this end, we will consider first mixed-integer linear, then mixed-integer nonlinear problems as subproblems. To solve the resulting problems we will study both complete descriptions as also generalized KKT theorems for mixed-integer nonlinear optimization problems.

Contact:

The main goal of this project is the development of mathematical methods for the solution of multilevel, mixed-integer, and nonlinear optimization models for gas markets. To this end, the focus is on a genuine four-level model of the entry-exit system that can be reformulated as a Bilevel model. The mathematical and algorithmic insights shall then be used to characterize market solutions in the entry-exit system and to compare them to system optima. Particular attention is paid to booking prices for entry or exit capacity.

Contact:

The goal of this project is to develop methods for the analysis of strategic interaction in gas markets based on multi-level optimization models. The foundation is a model of the entry-exit system in gas markets with a focus on the strategic booking and nomination of gas suppliers. The resulting two-level strategic interaction can be formulated as an equilibrium problem with equilibrium constraints (EPEC). In this market model every agent chooses their strategy taking into account the decisions made by the other agents simultaneously as well as future decisions. The EPEC to be analyzed thus describes a game, where every agent has to solve a two-level optimization problem, to be precise a mathematical program with equilibrium constraints (MPEC). Exploiting the specific mathematical structure of the resulting EPEC, we derive conditions, under which solutions exist and are unique, and develop tailored algorithms for the computation of market equilibria. Based on the theoretical and algorithmic results we assess the impact of strategic interaction on booking prices and market outcomes and determine how those results change for different market structures and market designs.

Contact:

The aim of the project is to advance a new algorithm to solve constrained, piecewise smooth optimization problems involving also integer variables. First, the approach for unbounded non-smooth optimization must be supplemented by the treatment of linear constraints. This requires an extensive modification of the algorithm to solve the sub-problems coming from the abs linearization and an analysis of the resulting convergence properties. Second, a penalty strategy for complementarity conditions has to be studied taking the arising non-smooth constraints into account. Here, the non-smoothness that occurs should be explicitly exploited in the numerical solution. Third, techniques for an efficient handling of integer variables are to be developed. The research in this project on optimization methods based on abs-linearization is motivated by applications in the gas market as well as by gas transport issues.

## Subfield C: Analysis and numerical simulation

Contact:

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. Aim of this project is the combination of simulation and optimization, with special focus on the control of transient compressors, the admissibility of pressures and mass flows and with the aim to overcome problems concerning the simulation due to the opening and closing of valves. As methodical way we pursue an approach of the form 1. discretize in space 2. optimize 3. discretize in time. The focus rests on new approaches for the time discretization of the resulting DAE by a least-square collocation method that has regularizing properties and is robust with respect to switching structures.

Contact:

Turnpike results provide connections between the solutions of transient and the corresponding stationary optimal control problems that are often used as models in the control of gas transport networks. In this way turnpike results give a theoretical foundation for the approximation of transient optimal controls by the solutions of stationary optimal control problems that have a simpler structure. Turnpike studies can also be considered as investigations of the structure of the transient optimal controls. In the best case the stationary optimal controls approximate the transient optimal controls exponentially fast.

Contact:

Prof. Dr. Michael Hintermüller

The main goal of this subproject is the systematic numerical approximation of systems of partial differential equations on networks, which arise in the modeling of gas transport. The focus lies on new kinds of Galerkin methods which, due to their variational structure, allow an efficient treatment of corresponding problems on a higher level, e.g., the calibration and optimal control of gas networks.

Contact:

This project studies data assimilation methods for models of compressible flows in gas networks. The basic idea of data assimilation is to include measurement data into simulations during runtime in order to make their results more precise and more reliable. This can be achieved by augmenting the original model equations with control terms at nodes and on pipes that steer the solutions towards the measured data. This gives rise to a new system called "observer". This project is going to explore how much data is needed so that convergence of the observer towards the solution of the original system can be guaranteed, how fast this convergence is and how measurement errors affect the solution.

Contact:

In this subproject we want to investigate a new approach for using transport metrics in networks, by constructing appropriate Wasserstein-type metrics. By defining suitable dissipation functionals in the nodes and edges of the network a metric structure (ideally Riemannian) can be constructed in order to formulate different transport-diffusion equation with coupling conditions in the nodes as gradient flows. The instantaneous division of flows in the nodes as well as a (partial) storage will be investigated as different cases.

Based on gradient flow formulations we want to further advance the analysis of equations on networks as well as their discretization and numerical solution. In particular we expect important results on structure-preserving model reduction (via parameter asymptotics) and structure-preserving numerical solution of transport-diffusion equations on networks. In addition we want to investigate a scaling limit (quasi-stationary situations in edges) to algebraic or ordinary differential equations in the nodes. Our goal is to derive error estimates in the respective metrics, which allow for a validation of the model reduction approaches.

In a further step the application of Wasserstein-type metrics to more complex systems on networks without gradient structure shall be investigated, in particular compressible Euler-equations with friction, which are of central importance for modelling gas networks. The simple connection between Wasserstein-type metrics and Lagrangian formulations in one dimension shall be used to simplify the analysis on the one hand, but also to construct and analyze Lagrangian numerics in a systematic way.

A final step will investigate the application to optimization problems in network models. A particular goal is to derive optimality conditions for the state variable and the constraint, i.e. the partial differential equation, in Wasserstein-type space on networks. This yields a novel representation of the adjoint variable and allows for new approaches to the numerical optimization.

The derivation of mathematical methods in this projects is mainly driven by applications to gas networks, the rather general mathematical setup allows applications to other problems however, e.g. neuronal networks or biological transport networks, but also transport problems in higher spatial dimensions with non-homogeneous boundary and interface conditions.

## Subfield Z: Central projects

Ansprechpartner:

Das Integrierte Graduiertenkolleg bietet seinen Mitgliedern eine einzigartige wissenschaftliche Ausbildung in den Themenbereichen des TRR 154 „Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken“. Darüber hinaus werden sie mit den für einen erfolgreichen Berufsweg im industriellen und akademischen Bereich notwendigen Schlüsselqualifikationen ausgestattet. Hierdurch werden sowohl die Karrierechancen der Mitglieder als auch die Attraktivität des TRR 154 für exzellente nationale und internationale Bewerberinnen und Bewerber nachhaltig erhöht. Bei den Mitgliedern des Graduiertenkollegs handelt es sich um die Doktorandinnen und Doktoranden der Ergänzungs- und Grundausstattung, die ihr Forschungsvorhaben innerhalb oder assoziiert zu den TRR 154-Projekten verfolgen, sowie um durch Kurzzeitstipendien (6–12 Monate) gewonnene Kandidatinnen und Kandidaten. Außerdem ist es offen für die Aufnahme von Postdoktorandinnen und -doktoranden verwandter Forschungsfelder.

Der fachwissenschaftliche Teil des Studienprogrammes wird durch die wissenschaftlichen Inhalte des TRR 154 festgelegt und umfasst unter anderem halbjährlich stattfindende transregionale Sommer- und Winterschulen, regelmäßige Vorträge und Blockkurse von eingeladenen Gastwissenschaftlerinnen und -wissenschaftlern sowie eigens auf die thematischen Schwerpunkte des TRR 154 zugeschnittene standortangepasste Vorlesungsreihen. Die wissenschaftliche Eigenständigkeit der Promovierenden wird durch Netzwerkbildung, wie etwa gemeinsame Veranstaltungen mit anderen Graduiertengruppen, Besuche internationaler Konferenzen, Auslandsaufenthalte und Exkursionen gefördert. Den Promovierenden wird über einen Nachwuchsring, dem Vertreter der verschiedenen Standorte angehören, die inhaltliche Ausgestaltung und Organisation eines Teils der Aktivitäten übertragen, zum Beispiel Fachkolloquien, unter anderem mit anderen Forschungsverbünden. Ergänzende Ausbildungsangebote zur Schulung fachübergreifender Kompetenzen sind durch die Nachwuchsprogramme der Graduiertenschulen der jeweiligen Standorte gewährleistet.

Zur Sicherstellung einer gezielten, umfassenden Betreuung der Promovierenden, die eine Forschungsprofilbildung und einen Abschluss des Dissertationsvorhabens in einem definierten, begrenzten Zeitraum ermöglicht, erhalten alle Mitglieder des Graduiertenkollegs eine Doppelbetreuung durch zwei im TRR 154 vertretene Projektleiterinnen oder -leiter (Mentoren). Betreuungsstandards in Form einer Promotionsvereinbarung strukturieren dabei den Arbeitsplan. Die Vereinbarungen, die auf den Standards der Graduiertenschulen der beteiligten Standorte basieren, werden individuell zwischen den betreuenden Personen und den Promovierenden geschlossen. Sie decken auch Elemente der Eigenleistung ab, wie etwa Veröffentlichungen, Vorträge und die Teilnahme an berufsvorbereitenden Kursen. Regelmäßige Zwischenevaluationen messen und dokumentieren die wissenschaftlichen Fortschritte und geben Gelegenheit zur kritischen Diskussion mit den Betreuenden.

## Subfield: Internal (short) projects

Contact:

The goal of the project is to derive novel a posteriori error estimators for approximations of discontinuous entropy solutions to hyperbolic conservation laws. These investigations are motivated by recent progress in the stability analysis of hyperbolic conservation laws based on the a-contraction methodology by Vasseur and co-workers. The a-contraction theory can handle the difficult problem of the stability of large shocks for the general case of systems of conservation laws with multiple conserved quantities. In particular, this stability analysis does not require initial data to be small in total variation which makes it more general than other available theories. It is envisioned that combining these stability results with reconstructions of numerical solutions that were extensively investigated in our group will lead to optimal order a posteriori error estimators for continuous as well as discontinuous solutions. As a long term perspective, these stability results should not only be useful for deriving a posteriori error estimators but also for proving stability of solutions with respect to problem parameters or for investigating observer-based data assimilation techniques.

The outcome of the project is that this approach is indeed viable, but whereas handling existing shocks is rather easy, handling shock formations is intricate and requires a technical analysis. Preliminary results can be found in an Oberwolfach report https://publications.mfo.de/handle/mfo/3852

Contact:

In the short term project “Coupling feedback control and state estimation under port-Hamiltonian structure”, we have investigated the structure of optimal reduced LQG controllers for linear pH systems. For the ODE case, we have proposed a particular choice of the weighting matrices in the quadratic control and estimation cost functionals. While this seems to reduce the weights to structural design parameters, it has turned out that the resulting controller combines a linear quadratic regulator with a static output feedback. As a further contribution, we have proposed a balancing-based model reduction scheme to construct reduced pH systems and controllers. With regard to the associated error bound, we have optimized the system Hamiltonian. Towards the case of networks of pH systems, we currently study the generalization of these concepts to differential algebraic equations. In this context, we also provide a new characterization of pH descriptor systems in terms of linear matrix inequalities generalizing the Kalman–Yakubovich–Popov (KYP) inequality.

Contact:

The goal of this project consists of extending an existing multi-level market model of the European entry-exit gas market system by the component of a capacity expansion of the gas network. For this purpose, we focus on a nonlinear model of gas transport and passive tree-shaped networks. Both modeling and solution techniques are developed that allow computations on networks of real-world size. Based on these computational results, we analyze to what extend economic inefficiencies due to the entry-exit system can be reduced by a specific capacity expansion of the gas network.

Contact:

This project revolves around the question of how to ensure service security when future demands are subject to uncertainty. Decisions need to be made to address the development of new infrastructures and systematic expansion of networks. Such a setting arises, for example, in the development of upcoming hydrogen networks, where demands and production capacities are still very much uncertain. We approach these uncertainties from a point of view of combinatorial online optimization, i.e., we ask for algorithms that need to make irrevocable decisions as the input data is revealed over time. In order to model hydrogen networks, we investigate potential-based flows in graphs. They not only model gas flows, but also water and power flows. The question we consider in particular is to find an order in which to build the edges of a graph such that the flow between two points is maximized after every edge insertion. Our results show that, in a setting with parallel paths between the two points, the objective function is fractionally subadditive, and that for such objectives a competitive ratio between max{2.618, M} and max{3.293√M, 2M} is possible at best. Here, M is the maximal capacity of a path between the two points in the graph.

Contact:

This subproject under the direction of Falk Hante (Humbold University Berlin) and Enrique Zuazua (FAU Erlangen-Nürnberg) study hyperbolic and parabolic dynamics on networks and their control with random batch methods. The aim is to limit the overall dynamics to subnetworks in the sense of a random batch and to calculate therefor a stochastic gradient direction. For this purpose, a convergence theory and control methods of gas networks in the sense of model predictive approaches are being developed. Regarding to the consideration of uncertainties, this approach can finally be extended with the help of methods for the simultaneous control of parametric systems.

Contact:

The subproject ist concerned with stability questions for the non stationary gas flow in gas networks, which is described by the friction dominated ISO-3 model. From the analytical point of view the model consists in a system of nonlinear parabolic partial differential equations which can be transformed into a doubly nonlinear degenerate parabolic equation. We focus on stability issues for the (weak) solution (in the parabolic Sobolev space) with respect to variations of structural parameters and initial-boundary data. In particular, we expect results for the ISO-3 model concerning the stability under variations of friction parameters and the underlying gas model (ideal or real gas).