Conference Agenda

Gas transportation and storage has become one of the most relevant and important optimization problems in energy systems. This problem includes highly nonlinear and nonconvex aspects due to gas physics, and discrete aspects due to control decisions of active network elements. In this work, we study the nonconvex sets induced by gas physics for pipes and compressors. We formulate this problem as a nonconvex mixed-integer nonlinear program (MINLP) through disjunctions. Obtaining locally feasible or global solutions for this nonconvex program presents significant mathematical and computational challenges for system operators. Thus, we propose conic programming relaxations for the nonconvex MINLP formulation. The proposed relaxations are based on the convex hull representations of the nonconvex sets as follows: We give the convex hull representation of the nonconvex set for pipes and show that it is second-order cone representable. We also give a complete characterization of the extreme points of the nonconvex set for compressors and show that the convex hull of the extreme points is power cone representable. We also propose a second-order cone outer-approximation for the nonconvex set for compressors. To obtain global or near global optimal solutions, we present an algorithmic framework based on these second-order conic programming programming relaxations. We illustrate the benefit of our relaxations through extensive computational experiments.

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.

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 finally demonstrate the applicability of the method by computing robust gas networks that are protected from future demand fluctuations.