RWTH Aachen University
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- Dynamic Network Flows (1)
- Dynamic Robust Flow (1)
- Gas Dynamics (1)
- Gas Market (1)
- Hyperbolic Balance Laws, Stabilization, Exact Controllability, Modeling of Gas Flow, Finite-Volume Schemes, Optimal control, Uncertainty (1)
- Isothermal Euler Equations (1)
- Keller-Segel (1)
- Nodal Control (1)
- Uncertain Travel Times (1)
- a posteriori error analysis (1)
We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent γ∈[1,3] and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the L∞(0,T;H1(Ω)) norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behaviour of the error estimator under changes of γ.
The European gas market is governed by rules that are agreed on by the European Union. We present a mathematical market model that
takes into account this structure, where the technical system operator (TSO)
offers certain transportation capacities that can be booked and later nominated within the previously chosen bookings. The TSO also fixes booking fees and defines an operational control of the gas pipeline system in order to deliver the gas according to the nominations. Since the gas
flow is governed by a system of partial differential equations, to realize this control structure partial differential equations (PDEs) should be involved in the model.
While the four level gas market model has been discussed previously, in this
paper we take into account the
flow model by PDEs in the discussion of the model and in the reduction to a single level problem, where we also state the corresponding necessary optimality conditions.
We present a positive and a negative stabilization result for a semilinear
model of gas flow in pipelines. For feedback boundary conditions we obtain an
unconditional stabilization result in the absence and conditional instability in
the presence of the source term. We also obtain unconditional instability for the
corresponding quasilinear model given by the isothermal Euler equations
In this article we survey recent progress on mathematical results on gas flow in pipe
networks with a special focus on questions of control and stabilization. We briefly present
the modeling of gas flow and coupling conditions for flow through vertices of a network. Our
main focus is on gas models for spatially one-dimensional flow governed by hyperbolic balance
laws. We survey results on classical solutions as well as weak solutions. We present results
on well–posedness, controllability, feedback stabilization, the inclusion of uncertainty in the
models and numerical methods.
We study dynamic network flows with uncertain input data under a robust optimization perspective. In the dynamic maximum flow problem, the goal is to maximize the flow reaching the sink within a given time horizon T, while flow requires a certain travel time to traverse an arc. In our setting, we account for uncertain travel times of flow. We investigate maximum flows over time under the assumption that at most Γ travel times may be prolonged simultaneously due to delay. We develop and study a mathematical model for this problem. As the dynamic robust flow problem generalizes the static version, it is NP-hard to compute an optimal flow. However, our dynamic version is considerably more complex than the static version. We show that it is NP-hard to verify feasibility of a given candidate solution. Furthermore, we investigate temporally repeated flows and show that in contrast to the non-robust case (i.e., without uncertainties) they no longer provide optimal solutions for the robust problem, but rather yield a worst case optimality gap of at least T. We finally show that for infinite delays, the optimality gap is at most O(k log T), where k is a newly introduced instance characteristic. The results obtained in this paper yield a first step towards understanding robust dynamic flow problems with uncertain travel times.