We study network design problems for nonlinear and nonconvex flow models under demand uncertainties. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, demand scenarios within a given uncertainty set. For solving the corresponding adjustable robust mixed-integer nonlinear optimization problem, we show that a given network design is robust feasible, i.e., it admits a feasible transport for all demand uncertainties, if and only if a finite number of worst-case demand scenarios can be routed through the network. We compute these worst-case scenarios by solving polynomially many nonlinear optimization problems. Embedding this result for robust feasibility in an adversarial approach leads to an exact algorithm that computes an optimal robust network design in a finite number of iterations. Since all of the results are valid for general potential-based flows, the approach can be applied to different utility networks such as gas, hydrogen, or water networks. We demonstrate the applicability of the method by computing robust gas networks that are protected from future demand fluctuations.

With the move towards a climate neutral usage, gas suppliers and transport companies have begun to mix a certain percentage of hydrogen into the gas networks. This poses new challenges for the modeling and optimization of gas transport. Therefore, we have developed a model for the mixture of gases on networks. The model is based on an equation of state for the mixture, the stationary isothermal Euler equations and coupling conditions for the flow and the mixture. The equation of state or pressure law which we developed is based on the change of the speed of sound in a mixture of gases. We prove that the gas flow is unique, even in the case of a mixture. This is not trivial since the mixture is changing the flow properties and it is not clear anymore if there exist different network flows with different mixing ratios.
Further, we use the model to solve stationary gas flow problems to global optimality on large networks. Therefore, the model is implemented and solved with the help of the MINLP-solver SCIP. We examined different implementations of the model and their impact on computational performance.

The usage of hydrogen gas as an energy carrier is one likely factor towards achieving a net zero future. There are many possible applications, but one important question is whether existing natural gas infrastructure can be re-used for hydrogen and the challenges this brings for the control of these future networks. In particular, we expect that a hydrogen network which uses hydrogen generated from excess renewable electricity would be more difficult to control as the patterns of injection and withdrawal would be much less regular than today. Additional challenges arise from new operating parameters required for hydrogen and not for natural gas -- such as controlling for pressure fluctuations to prevent pipe-ageing. Motivated by a need for instationary optimization methods on networks at scale, we present a specialized interior point method for gas problems. Our test problem is a line-pack optimization problem using a discretization of the 1d isothermal Euler equations, as a step towards understanding the important questions above. By incorporating a bespoke preconditioned iterative solver to tackle the linearized systems at each iteration of the interior point method, which form the key computational bottleneck in such a method, we utilize the highly stuctured nature of the problem. This arises from both the network structure itself and the time discretization. The expectation is that the method will scale well with both network size and time windows, and be generalizable to broader instationary gas network optimization problems.

We discuss how optimization methods can be used to analyze the European Gas Market. Specifically, we discuss how we can analyze the effect of market power on booking decisions and how uncertainty makes it difficult to find market equilibria. We will close with an outlook of how a switch to hydrogen will make these questions more difficult and/or interesting to analyze.