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Abstract

As in the first phase, we consider the semilinear Euler equations. Due to the results obtained in the first phase, we are able to compute an instantaneous control for networks of moderate size for time-dependent problems on the basis of the semilinear model. For this purpose, a mixed implicit-explicit Euler approach for the time discretization was used. This strategy makes it possible to understand the friction term as a quadratic term in the flux and to treat it as a monotonous perturbation. This analysis, which was completed in the first phase in AP 1, was extended in AP 2 by a spatial decomposition, in which the complete problem on the entire graph is iteratively decomposed into analog problems on smaller subgraphs. Therefore it is now possible to decompose the time-dependent overall problem into problems containing subgraphs of suitable size with valves and compressors, whereby, due to the formulation as an instantaneous control problem, only one time step has to be optimized at a time. The spatial decomposition allows a complete parallelization of the optimal control problems in such a way that the problem considered in the current iteration is updated on a given subgraph over the connecting edges with all adjacent subgraphs (from the last iteration). The local optimality systems then correspond to the local optimal control problems in this case. The goal of the second phase is to combine this spatial decomposition with an iterative time domain decomposition. For this purpose the problem shall be decomposed into subgraphs as well as into smaller continuous time intervals. In this way semilinear problems arise on subgraphs and time horizons, which lead to large and block-structured MINLPs after complete discretization in time and space (based on AP 3 of phase 1). The size of the subgraphs and the continuous time horizons should be controlled in such a way that the resulting finite-dimensional MINLPs can be solved to global optimality in practice. The subproblems include valves, compressors and other control elements, so that coupling between the subnetworks takes place exclusively via transmission nodes. The principle of space-time decomposition has already been described for hyperbolic equations and systems in the literature. This literature contains a posteriori estimates, which allow to design the decomposition adaptively. The procedure for time domain decomposition to be developed in AP 1 can be realized in different ways. It is intended to introduce an iterative time-domain decomposition into continuous subintervals and to combine this with a non-overlapping spatial decomposition of the entire graph into sub-graphs. As a result of these decompositions in space and time, time-dependent optimal control problems remain to be solved on smaller space-time domains. For this purpose we want to follow the discretize-then-optimize paradigm and perform a full discretization of space and time. The resulting finite-dimensional MINLPs themselves are subject of the second and third work package. The underlying constraints are of block-structured diagonal form with additional lines for coupling the time steps. With this form we have numerous experiences in the linear context as well as preliminary work in the context of stationary gas network optimization. Both will be used for the investigation of the nonlinear and instationary case. In the second work package, for the fully discretized and thus finite-dimensional case, it is investigated how both the time structure and the structure of the controls can be algorithmically exploited within the framework of decomposition-based penalty methods. The same discretized MINLPs are to be replaced by MIP relaxations in the third work package with the experiences from the first phase. Subsequently, it will be investigated whether the MIP of a time step can be projected onto the following time step. If this succeeds, an iterative procedure in time can be set up. In the ideal case, this would allow to reduce the very high-dimensional MIP to be solved by the sequential solution of MIPs of the order of size of one time step.