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- Transient Modeling and Simulation of Gas Pipe Networks with Characteristic Diagram Models for Compressors (2017)
- One challenge for the simulation and optimization of real gas pipe networks is the treatment of compressors. Their behavior is usually described by characteristic diagrams reflecting the connection of the volumetric flow and the enthalpy change or shaft torque. Such models are commonly used for an optimal control of compressors and compressor stations [4, 7] using stationary models for the gas flow through the pipes. For transient simulations of gas networks, simplified compressor models have been studied in [1–3]. Here, we present a transient simulation of gas pipe networks with characteristic diagram models of compressors using a stable network formulation as (partial) differential-algebraic system.

- Neumann boundary feedback stabilization for a nonlinear wave equation: A strict H2-Lyapunov function (2017)
- For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict H2-Lyapunov function and show that the boundary feedback constant can be chosen such that the H2-Lyapunov function and hence also the H2-norm of the difference between the non-stationary and the stationary state decays exponentially with time.

- Domain Decomposition of an Optimal Control Problem for Semi-Linear Elliptic Equations on Metric Graphs with Application to Gas Networks (2017)
- We consider optimal control problems for the flow of gas in a pipe network. The equations of motions are taken to be represented by a semi-linear model derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a given network and introduce a time discretization thereof. We then study the well-posedness of the corresponding time-discrete optimal control problem. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a non-overlapping domain decomposition of the semi-linear elliptic optimal control problem on the graph into local problems on a small part of the network, ultimately on a single edge.

- On the relaxation approximation for 2 × 2 hyperbolic balance laws (2017)
- The relaxation approximation for systems of conservation laws has been studied intensively for example by [17, 5, 19, 24]. In this paper the corresponding relaxation approximation for 2x2 systems of balance laws is studied. Our driving example is gas flow in pipelines described by the isothermal Euler equations. We are interested in the limiting behavior as the relaxation parameter tends to zero. We give conditions where the relaxation converges to the states of the original system and counterexamples for cases where the steady states depend on the space variable.

- Uniqueness and Multiplicity of Market Equilibria on DC Power Flow Networks (2017)
- We consider uniqueness and multiplicity of market equilibria in a short-run setup where traded quantities of electricity are transported through a capacitated network in which power flows have to satisfy the classical lossless DC approximation. The firms face fluctuating demand and decide on their production, which is constrained by given capacities. Today, uniqueness of such market outcomes are especially important in more complicated multilevel models for measuring market (in)efficiency. Thus, our findings are important prerequisites for such studies. We show that market equilibria are unique on tree networks under mild assumptions and we also present a priori conditions under which equilibria are unique on cycle networks. On general networks, uniqueness fails to hold and we present simple examples for which multiple equilibria exist. However, we prove a posteriori criteria for the uniqueness of a given solution and characterize situations in which multiple solutions exist.

- Optimality Conditions for Switching Operator Differential Equations (2017)
- We consider optimal switching of hybrid abstract evolution equations. The framework includes switching semilinear partial differential equations of parabolic or hyperbolic type, discontinuous state resets, switching costs and allows switching of the principle parts of the equations. We present adjoint-based formulae for the gradient of the cost functional with respect to position and number of switching time points that lead to first order necessary conditions. Moreover, we discuss an alternate-direction approach for implementing descent methods. As an application we consider optimal open/close-switching of valves and on/off-switching control of compressors in a gas network modelled by a graph with simplified euler equations on edges and suitable coupling conditions at nodes.

- Boundary feedback stabilization of the isothermal Euler-equations with uncertain boundary data (2017)
- In a gas transport system, the customer behavior is uncertain. Motivated by this situation, we consider a boundary stabilization problem for the flow through a gas pipeline, where the outflow at one end of the pipe %that is governed by the customer's behavior is uncertain. The control action is located at the other end of the pipe. The feedback law is a classical Neumann velocity feedback with a feedback parameter $k>0$. We show that as long as the $H^1$-norm of the function that describes the noise in the customer's behavior decays exponentially with a rate that is sufficiently large, the velocity of the gas can be stabilized exponentially fast in the sense that a suitably chosen Lyapunov function decays exponentially. For the exponential stability it is sufficient that the feedback parameter $k$ is sufficiently large and the stationary state to which the system is stabilized is sufficiently small. The stability result is local, that is it holds for initial states that are sufficiently close to the stationary state. This result is an example for the exponential boundary feedback stabilization of a quasilinear hyperbolic system with uncertain boundary data. The analysis is based upon the choice of a suitably Lyapunov function. The decay of this Lyapunov function implies that also the $L^2$-norm of the difference of the system state and the stationary state decays exponentially.