# Subprojects

of TRR 154 in phase three (2022 - 2026)

## Subfield A: Integer-continuous methods

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This project develops and analyzes adaptive methods for solving gas transport problems, including integer decisions, to global optimality. This includes the derivation of convex relaxations of instationary problems, based on Riemann invariants or first-discretize-then-optimize models. Moreover, starting with the stationary case, the mixing of different gases is incorporated, e.g., of hydrogen into natural gas, for gas transport as well as topology optimization. Acyclicity of the flows can be exploited in both contexts.

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The project provides a detailed mathematical analysis and consistent numerical discretization of optimal control problems for networks/systems of hyperbolic balance laws with state constraints as they arise for unsteady PDE models for gas networks. The results are used for the computation of convergent discrete gradients within derivative-based optimization methods. To this end, numerical approximations for a class of adjoint and sensitivity equations are studied, using the discretize-then-optimize as well as the optimize-then-discretize approach. In a further step, the project will consider higher order schemes and derive a priori error estimators for optimal control of entropy solutions.

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The project develops control theoretical methods for dynamical systems coupling partial differential equations and logic-based integer-valued components. In particular, it studies model-predictive control strategies and optimization-based variants thereof for such hybrid systems. In the third phase, a new focus is on distributed control schemes. The challenges concern well-posedness, closed-loop performance properties and algorithms for numerical realizations of such controllers. These will be met by studying variational equilibria in dynamic mixed-integer programming. The newly developed methods will be applied to decentralized control of hydrogen and natural gas pipeline systems being interconnected by mixing.

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In this subproject we study domain decomposition approaches for optimal control problems using the example of gas transport networks. Our main goal is to couple the space-time-domain decomposition method from the second phase with machine learning and mixed-integer programming techniques. To this end, we develop an interlinked data-driven and physics informed algorithm called NeTI (Network Tearing and Interconnection) that combines mixed-integer nonlinear programming, learning of surrogate models, and graph decomposition strategies.

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The goal of this project is to provide a comprehensive study of the problem of maximizing the welfare in hydrogen or gas networks. Welfare is generated by satisfying routing requests where the ability to satisfy requests is limited by the physical capacity of the networks, e.g., in terms of bounds on the maximum pressures at the nodes. These questions are formulated as non-linear packing problems that lead to a coupling of the discrete decision whose requests have to be satisfied and fractional decisions about how the flows are distributed over time and within the network. The project further studies variants of the problem where the requests appear in random order, and/or a mechanism design variant where the welfare of satisfying a request is unknown to the network operator.

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The goal of this project is to handle uncertainties arising in the planning and operation of potential-based flow networks from a perspective of online optimization. We will conduct a competitive analysis of different models on different time scales, combining discrete decisions such as a build order on a set of edges with continuous decisions such as storage at nodes and flows in the network. The focus will lie on devising competitive online algorithms for settings ranging from the incremental development of hydrogen infrastructure in the long term to the operation of such networks when coupled to renewable energy sources in the short to medium term.

## Subfield B: Model adaptation and coupling

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The goal of this project is to develop a holistic dynamic multi-scale ansatz for the numerical solution of compressible instationary Euler equations with uncertain data on network structures. We will use these methods for uncertainty quantification and adaptive multi-level probabilistic constrained optimization on flow networks. For this, we combine adaptive stochastic collocation methods with kernel density estimators in an adjoint-based gradient method.

Contact:

Prof. Dr. Michael Hintermüller

N.N

This subproject is concerned with the coupling of an intraday gas market with the physical transport of gas through a network, subject to uncertainty. This problem is modelled as a non-cooperative equilibrium problem where each risk-averse market player makes decisions in such a way as to maximize profit while simultaneously ensuring that their collective decisions are physically feasible along the network. The goal of this project is to characterize and compute equilibria to this problem. For this, we study the existence of solutions and their sensitivity to perturbations in parameters. To develop algorithms to handle the problem computationally, stochastic approximation and feedback-type mechanisms are employed.

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The goal of this project is the simulation of and the feedback control design for systems describing the model hierarchy of gas transport networks. Here, the main tool for incorporating different scales and levels of simplification is an energy-based modeling via port-Hamiltonian systems. In particular, it is studied whether and how this particular system structure can be exploited in constructing robust simulation and control mechanisms. For this purpose, we will study the effect of using different system Hamiltonians, optimal control cost functionals and weighting matrices.

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The goal of this project is to incorporate probabilistic or chance constraints into models of gas network optimization. This allows one to take risk averse optimal decisions in the presence of uncertain parameters such as gas load. Main applications are models with hierarchical structure (e.g. bilevel gas market models) or models with static or transient gas flow. These embedding structures lead to new challenges in the theoretical analysis (e.g. differentiability, convexity, existence of solutions) as well as in the algorithmic solution (e.g. probabilistic constraints with respect to infinite random inequality systems). A major aspect of the project’s research is the transition from static to dynamic chance constraints.

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A focus of subproject B06 is on the development of solution methodologies that can be applied to a wide class of robust problems, such as discrete-continuous nonconvex and two-stage robust optimization models. Building upon the results of the first two phases, B06 will investigate an integration of robustness and stochasticity together with discrete-continuous decisions. The goal is to develop approaches that yield reduced conservatism compared to pure robustness as well as an uncertainty protection that goes beyond stochastic guarantees. To this end, B06 will research a currently very actively studied methodology that promises these benefits, namely distributionally robust optimization that has many applications. Learning from data will be integrated.

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In this subproject we model and analyse multiparameter auction problems on graph structures, motivated by the gas network paradigm. Our main goal is to characterize the structure of revenue-optimal auctions in these network-constrained, multidimensional Bayesian settings, as well as to provide rigorous approximation guarantees. To do so we bring together machinery from the fields of optimal mechanism design, algorithmic game theory, mixed-integer programming, and polyhedral combinatorics.

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The goal of this subproject is to develop mathematical methods to solve mixed-integer and nonlinear multilevel optimization problems for gas markets coupled with markets of other energy sectors such as electricity. Motivated by the two cases of cooperating or non-cooperating network operators of the different sectors, we investigate on the one hand bilevel problems with potentially multiple solutions on the lower level, for which we establish methods to assess pessimistic solutions. On the other hand, we study multi-leader-follower games and develop problem-tailored solution approaches. Finally, based on our mathematical and algorithmic developments, we characterize equilibria in coupled energy systems for different combinations of market designs in the considered sectors.

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We develop models and solution procedures which allow us to analyze strategic supply decisions of firms in gas markets. This in general results in solving equilibrium problems. Our focus in this context is on Multi-Leader-Follower-Games (MLFGs), where a group of agents in a first step (upper level) takes decisions anticipating decisions of another group of agents in a second step (lower level). Our planned analysis in the third phase is motivated by retailer-consumer relationships where several retailers first choose the details of the supply contracts offered, then consumers choose a contract and make their consumption decisions. As an important feature of the third phase we plan to include different risk aspects which are of crucial importance for consumer decisions on the lower level.

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Bilevel optimization is a wide area in mathematical optimization and plays an important role in the TRR 154, where the coupling of producers and consumers represents one prominent application and the robust protection against uncertainties another one. Many of these bilevel problems can be formulated as nonsmooth, piecewise linear optimization problems with constraints. This project aims at the development, analysis and implementation of a structure-exploiting algorithm for bilevel problems of this kind building on the quite recent approach of abs-linearization and the active signature method. This allows also to treat nonsmooth functions in a bilevel optimization.

## Subfield C: Analysis and numerical simulation

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The project aims to develop stable simulations for coupled network DAEs that allow to find feasible and optimal dynamic controls for coupled networks. We concentrate on networks for gas transport. Couplings of interest arise, for example, from the increasing demand to produce and transport hydrogen in the future. A particular focus goes to the treatment of valves and regulators described by piecewise differentiable functions. We aim to provide an optimal dynamic control for coupled gas networks using the approach to first discretize in space, then optimize the resulting DAE system and finally use our least squares collocation approach for time integration of the boundary value optimality DAE.

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The goal of this project is to prove turnpike results for optimal control problems in gas networks. We will consider nodal control since the control action takes place at compressors that are located at a small number of points in the networks. Probabilistic constraints are included since they allow to take into account the uncertainty of e.g. the customer demand. We will also scrutinize switching conditions that arise e.g in the decision to to open or close a gas valve. Since the turnpike phenomenon relates the dynamic optimal states to steady states, we will also study steady gas flows on networks with intertwined cycles.

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The goal of this project is to provide state estimates of gas flows in networks by combining data input from nodal measurements and physical understanding of the flow problem encoded in the transient Euler equations and suitable coupling conditions. To this end, we create a twin system to the original system, called “observer”, into which measurement data are fed and study under which conditions the state of the observer converges to the state of the original system. We will study the evolution of solutions along characteristics and investigate the decay of functionals measuring the difference between system state and observer state.

Contact:

Ariane Fazeny

The goal of this project is to continue the structured analysis of transport on gas networks, with a possible extension to hydrogen cases. We will study the extension from gradient flow structures to perturbed versions, including non-conservative forces, as well as the extension from 2-Wasserstein metrics to more general exponents relevant in practice. Moreover, we aim at the development of structure-preserving variational time discretization of equations on networks. Another key question we want to tackle is the convergence of structures beyond scales, in particular we want to understand the convergence of 3D models to reduced 1D equations on networks with variational arguments.

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The goal of this subproject is to reduce the computational cost of solving dynamical optimal control problems of PDEs modelling gas transport on large networks by developing a stochastic gradient descent procedure based on domain decomposition methods. The main idea is to find a descent direction based only on the sensitivity of the dynamics on a randomly determined subgraph and couple this with an analysis of the network topology and with graph clustering methods. To apply this procedure in the context of gas transport, it is planned to significantly expand the optimal control theory and adjoint methodology for doubly nonlinear parabolic equations as a reformulation of the friction dominated isothermal Euler equations.

Contact:

Prof. Dr. Michael Hintermüller

N.N

The goal of this project is the development of stochastic gradient methods for the treatment of almost sure state constraints. Such constraints arise for example in the nomination validation of gas networks under uncertain demands but also play a role in the transition towards future hydrogen networks. A focus of the project is the consideration of sequences of relaxed problems intertwined with the stochastic gradient method and a rigorous mathematical convergence analysis of the resulting methods.

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This project is concerned with modeling and numerical analysis for thermodynamically consistent inviscid gas equations for binary mixtures in pipes with wall friction. We investigate general pressure laws and the underlying structure of mixture equations that allow for an extension to networks. For such models we develop a numerical scheme that is asymptotically preserving in the high friction and low Mach number limit.

## Subfield Z: Central projects

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The Integrated Research Training Group structures the supervision of the young researchers in the TRR154 and supports their interdisciplinary scientific education, their early independence in research as well as their soft-skill training. The graduate school organizes topic-specific lectures, summer schools, excursions, as well as soft-skill courses and career-planning measures. Doctoral researchers benefit from a double supervision across different sites including mentoring. The goal of the program is to prepare the members for a successful career path in adacemia as well as in industry in the best possible way.

## Subfield: Internal (short) projects

Contact:

Elisa Strauch

This subproject aims to assess control and control strategy robustness using probability as a measure. We analyze deterministically computed controls and evaluate their performance in the face of a posteriori uncertainty in the system. We use a probability distribution to determine the likelihood of the control satisfying constraints and measure its probabilistic robustness. Our approach combines a kernel density estimator with uncertainty quantification to efficiently compute these probabilities. We also investigate controls that are a priori probabilistically robust by considering a priori uncertainty in gas transport. For the resulting optimization problems with probabilistic constraints, we apply the theory of adjoint calculus to PDEs under uncertainty.