# Transregio/CRC 154: Mathematical Modelling, Simulation and Optimization on the Example of Gas Networks

## Subproject A01: Global Methods for Stationary Gastransport

This project will develop adaptive methods for the global solution of nonlinear integer optimization problemes with ODEs, on the example of stationary gastransport. Main issues are the construction of relaxations based on a priori error bounds, the combination of adaptive discretizations and branching methods, the handling of integral decisions via branch-and-bound as well as infeasibility cuts, the development of primal heuristics with a posteriori error bounds, and model reduction techniques.

Contact: Oliver Habeck, Marc Pfetsch, Stefan Ulbrich

## Subproject A02: Adaptive Multilevel Method for the Optimal Control of Hyperbolic Equations in Gas Networks

In this project we analyze the optimal control of hyperbolic PDE systems with state constraints on the example of gas networks. Through the time-dependent control of compressors and valves, the pressure and velocity distribution of the transported gas in the network has to be optimized under constraints, e.g. such that the pressure lies within a specified tolerance range. The constraints of the resulting optimal control problem (P) consist of coupled systems of one-dimensional isothermal Euler equations describing the gas flow, node conditions and state constraints. We plan to use Moreau-Yosida regularizations to approximate (P) in order to derive optimality conditions. The main goal of this project is to provide an optimization theory, which will form the basis of adaptive multilevel methods.

Contact: Johann Michael Schmitt, Stefan Ulbrich

## Subproject B09: Strategic Booking Decisions in the Entry-Exit-System

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: Alexandra Schwartz