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- On the identification of the friction coefficient in a semilinear system for gas transport through a network (2017)
- An identification problem for the friction parameter in a semilinear system of balance laws, describing the transport of gas through a passive network of pipelines, is considered. The existence of broad solutions to the state system is proven and sensitivity results for the corresponding solution operator are obtained. The existence of solutions to the output least squares formulation of the identification problem, based on noisy measurements over time at fixed spatial positions is established. Finally, numerical experiments validate the theoretical findings.

- Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization (2017)
- The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that a prescribed model and discretization error tolerance is satisfied in the end. We prove convergence of the adaptively controlled optimization method and illustrate the new approach with numerical examples.

- Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part (2017)
- In this paper we extend the implicit-explicit (IMEX) methods of Peer type recently developed in [Lang, Hundsdorfer, J. Comp. Phys., 337:203–215, 2017] to a broader class of two-step methods that allow the construction of super- convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes combine the necessary stability of implicit and low computational costs of ex- plicit methods to efficiently solve systems of ordinary differential equations with both stiff and non-stiff parts included in the source term. To construct super- convergent IMEX-Peer methods with favourable stability properties, we derive necessary and sufficient conditions on the coefficient matrices and apply an extrapolation approach based on already computed stage values. Optimised super-convergent IMEX-Peer methods of order s + 1 for s = 2, 3, 4 stages are given as result of a search algorithm carefully designed to balance the size of the stability regions and the extrapolation errors. Numerical experiments and a comparison to other IMEX-Peer methods are included.

- Uniqueness of Market Equilibria on Networks with Transport Costs (2017)
- We study market equilibria for perfect competition in capacitated transport networks. Using piecewise linear and symmetric transport costs, we prove uniqueness of the resulting equilibria under mild assumptions.

- A Global Optimization Approach for Instationary Gas Transport in Pipeline Networks (2017)
- We consider the optimization of a gas network that is described by a coupled system of parabolic partial differential equations. Energy and mass balance at network junctions and active elements, like compressors or valves, are modeled as additional algebraic equations. The resulting optimal control problem is discretized in space and time by a particular finite volume method, which can be shown to be well-posed under rather general assumptions. This first-discretize-then-optimize procedure yields a mixed-integer nonlinear problem (MINLP) that can be solved to global optimality. For the numerical solution of the MINLP, we consider a relaxation approach allowing to solve the problem globally by a sequence of mixed-integer problems (MIPs) with any required accuracy. The relaxation is based on piecewise linearization of the nonlinear constraints modeling the gas dynamics on the pipelines. Due to the particular discretization of the state equations, only univariate nonlinearities have to be approximated. This substantially facilitates the numerical treatment of the nonlinear constraints. To illustrate the efficiency of the proposed approach, we present numerical tests for typical benchmark problems.

- Nonconvex Equilibrium Models for Gas Market Analysis: Failure of Standard Techniques and Alternative Modeling Approaches (2017)
- This paper provides a first approach to assess gas market interaction on a network with nonconvex flow models. In the simplest possible setup that adequately reflects gas transport and market interaction, we elaborate on the relation of the solution of a simultaneous competitive gas market game, its corresponding mixed nonlinear complementarity problem (MNCP), and a first-best benchmark. We provide conditions under which the solution of the simultaneous game is also the solution of the corresponding MNCP. However, equilibria cannot be determined by the MNCP as the transmission system operator's (TSO’s) first-order conditions are insufficient, which goes back to nonconvexities of the gas flow model. This also implies that the welfare maximization problem may have multiple solutions that sometimes do not even coincide with any of the market equilibria. Our analysis shows that, even in the absence of strategic firms, market interaction fails to implement desirable outcomes from a welfare perspective due to the TSO’s incentive structure. We conclude that the technical environment calls for a market design that commits the TSO to a welfare objective through regulation and propose a design where the market solution corresponds to a welfare maximum and vice versa.

- Optimal Boundary Control of Hyperbolic Balance Laws with State Constraints (2017)
- In this paper we analyze the optimal control of initial-boundary value problems for entropy solutions of scalar hyperbolic balance laws with pointwise state constraints. Hereby, we suppose that the initial and the boundary data switch between different C¹-functions at certain switching points, where the C¹ -functions and the switching points are considered as the control. For a class of cost functionals, we prove first order necessary optimality conditions for the corresponding optimal control problem with state contraints. Furthermore, we use a Moreau-Yosida type regularization to approximate the optimal control problem with state constraints. We derive optimality conditions for the regularized problems and finally prove convergence to the solution of the optimal control problem with state constraints.

- Improving branching for disjunctive polyhedral models using approximate convex decompositions (2017)
- Disjunctive sets arise in a variety of optimization models and much esearch has been devoted to obtain strong relaxations for them. This paper focuses on the evaluation of the relaxation during the branch-and-bound search process. We argue that the branching possibilities (\ie binary variables) of the usual formulations are unsuitable to obtain strong bounds early in the search process as they do not capture the overall shape of the the entire disjunctive set. To analyze and exploit the shape of the disjunctive set we propose to compute a hierarchy of approximate convex decompositions and show how to extend the known formulations to obtain improved branching behavior.

- Joint model of probabilistic/robust (probust) constraints with application to gas network optimization (2017)
- Optimization tasks under uncertain conditions abound in many real-life applications. Whereas solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually used in case solutions are sought that are feasible for all realizations of uncertainties within some pre-defined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we introduce a class of joint probabilistic/robust constraints as its appears in optimization problems under uncertainty. Focussing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound, with high probability.

- Numerical optimal control of instationary gas transport with control and state constraints (2017)
- We consider the optimal control of a nonlinear hyperbolic system of balance laws on a one-dimensional network which arises in the context of gas transport in pipeline systems. State constraints, which are required for the safe operation of the system, are incorporated by a barrier method. We discuss the well-posedness of the governing system of partial differential-algebraic equations and investigate the existence of minimizers. For the numerical solution, we then consider the approximation of the state equation by mixed finite elements in space and a particular linear implicit time integration scheme that can be interpreted as a discontinuous Galerkin approximation. We establish well- posedness of this discretization scheme and prove the existence of minimizers for the corresponding discretized optimal control problem and discuss its numerical solution by a projected Gauß-Newton method. The efficient realization of the Jacobian and Hessian of the quadratic approximations that have to be minimized in every iteration of the Gauß-Newton method can be obtained via the solution of discretized sensitivity and adjoint equations. These are obtained by formal differentiation and transposition of the Galerkin methods employed for the discretization of the state equations. All approximations obtained after discretization can thus be interpreted as functions on the continuous level and, since the functional analytic setting is not changed by the Galerkin discretization, we observe mesh independence of the resulting fully discrete methods. For illustration of our theoretical results and to demonstrate the efficiency of the proposed method, we present numerical results for two test problems that model typical situations that may arise in the daily operation of gas networks.