A Mixed Integer Approach for the Transient Case of Gas Network Optimization

Natural gas is the third most important energy source in the world. Presently, the consumption of natural gas is increasing the most in comparison to other non-renewable energy sources. Therefore, optimization of gas transport in networks poses a very important industrial problem. In this thesis we consider the problem of time-dependent optimization in gas networks, also called Transient Technical Optimization (TTO). A gas network consists of a set of pipes to transport the gas from the suppliers to the consumers. Due to friction with the pipe walls gas pressure gets lost. This pressure loss is compensated by so called compressors. The aim of TTO is to minimize the fuel consumption of the compressors, where the demands of consumers have to be satisfied. Transient optimization of gas transmission is one of the great research challenges in this area. We formulate a mixed integer approach for the problem of TTO which concentrates on time-dependent and discrete aspects. Thereby, the nonlinearities resulting from physical constraints are approximated using SOS (Special Ordered Set) conditions. A branch-and-cut algorithm is developed which guarantees global optimality in dependence on the approximation accuracy. Concerning the nonlinearities, we discuss the quality of approximation grids by calculating approximation errors. The SOS conditions are implicitly handled via a branching scheme, supported by adequate preprocessing techniques. A heuristic approach based on simulated annealing yields an upper bound in our branch-and-cut framework. To improve the lower bound, we incorporate two separation algorithms. The first one results from theoretical studies of the so called switching polytopes which are defined by runtime conditions and switching processes of compressors. Linking of different SOS conditions gives a second separation strategy. We present theoretical investigations of the SOS 2 and SOS 3 polytope. These polytopes arise from the modeling of SOS Type 2 and SOS Type 3 conditions using additional binary variables. The results do not have practical relevance for our solution algorithm, but we characterize facet-defining inequalities providing complete linear descriptions of these polytopes. We evaluate the developed branch-and-cut algorithm using three test networks provided by our project partner E.ON Ruhrgas AG. Two are of artificial nature, as they were developed for test purposes. They contain all important elements of a gas network, but are rather small. The third network characterizes the major part of the Ruhrgas AG network in Western Germany. We test instances from three up to 24 coupled time steps.