C03
Refine
Year of publication
Language
- English (91)
Keywords
- optimal control (6)
- Gas Networks (4)
- Nodal Control (3)
- Probabilistic Constraints (3)
- Turnpike (3)
- Turnpike phenomenon (3)
- isothermal Euler equations (3)
- Kernel Density Estimation (2)
- Kernel Density Estimator (2)
- Optimal Control (2)
Document Type
- Article (55)
- Preprint (32)
- Conference Proceeding (2)
- Doctoral Thesis (1)
- Habilitation (1)
In this paper, we study hydrogen-natural gas mixtures transported through pipeline networks. The flow is modeled by the isothermal Euler equations with a pressure law involving a non-constant, composition-dependent compressibility factor. For a broad class of such compressibility models, we prove the existence of steady-state solutions on networks containing compressor stations. The analysis is based on an implicit representation of the pressure profiles and a continuity argument that overcomes the discontinuous dependence of the gas composition on the flow direction. Numerical examples illustrate the influence of different compressibility models on the resulting states.
The transition to renewable energy and the increasing role of hydrogen as a future energy carrier pose major challenges for the operation and optimization of gas transport networks. A rigorous mathematical understanding of the topic is necessary as efficient and reliable operation requires advanced methods to deal with uncertainty, nonlinear dynamics, and complex network topologies. At the same time, optimal control theory – in particular the turnpike phenomenon – provides powerful tools to simplify long-term optimization problems and to connect dynamic models with their stationary counterparts.
This habilitation thesis focuses on establishing new results in three related areas. For optimization problems with probabilistic constraints, an approach for approximating probabilities based on kernel density estimation is presented and analyzed, yielding necessary and sufficient conditions for the convergence of solutions of approximated stochastic optimization problems. In the modeling of gas transport networks, the existence and uniqueness results of a mixing model are presented. Furthermore, a finite-time turnpike result for an optimal control problem governed by the wave equation as well as an integral turnpike result for an optimal control problem governed by the transport equation under uncertainty is established.
On the application side, the problem of optimal placement of compressor stations in stationary gas networks is analyzed for both deterministic and random gas demand. Results on the optimal number of compressor stations and their locations on gas networks are presented. Further, the turnpike phenomenon is applied to an optimal control and design problem for gas networks, allowing steady-state models to replace the gas dynamics in the long-term planning. For the corresponding stationary problem, a fast and efficient algorithm for identifying the optimal network topology with low control cost is provided.
We consider the pipeline flflow of blended gas. The flow is governed by a coupled system where for each component we have the isothermal Euler equations with an additional velocity coupling term that couples the velocities of the different components. Our motivation is hydrogen blending in natural gas pipelines, which will play a role in the transition to renewable energies. We show that with suitable boundary conditions the velocities of the gas components synchronize exponentially fast, as long as the L2-norm of the synchronization error is outside of a certain interval where the size of the interval is determined by the order of the interaction terms. This indicates that in some cases for a mixture of ncomponents it is justifified to use a flux model where it is assumed that all components flow with the same velocity. For the proofs we use an appropriately chosen Lyapunov function which is based upon the idea of relative energy.
In this paper we consider the boundary feedback stabilization of a quasi-linear hyperbolic system of balance laws. At one end of the space interval, there is a reflecting boundary condition. At the other end a stabilizing feedback law with a varying time-delay is prescribed. We present sufficient conditions for the exponential stability of the system. We show that exponential stabilization is possible if the product of the length of the interval and an upper bound for the source term is sufficiently small. We also show that if the product of the length of the interval and a lower bound for the source term is sufficiently large, the system is unstable. Our analysis is based on Lyapunov functions with weights that are given by hyperbolic functions that generalize the well-known exponential weights.
Compared with previous contributions, we obtain conditions that can be verified more easily in terms of the system parameters. Our results show that for sufficiently short space intervals, and also with varying time-delay, exponential stabilization is possible with appropriately chosen feedback gains that depend on the maximal value of the time-delay and the maximal absolute value of its derivative.
We study the existence, stability and optimal control of classical solutions of networked strictly hyperbolic systems of balance laws. Such networks arise, for instance, in the modeling of the gas dynamics in a network of gas pipelines, traffic networks, or networks of water channels.
It is assumed that characteristic speeds are nonzero and do not change sign. Node conditions are stated in a general way by requiring that the boundary traces of all states adjacent to a vertex satisfy an algebraic equation. With a suitable assumption, this equation can be solved for outgoing characteristic variables as a function of the incoming ones.
We prove the unique existence of a classical solution for arbitrarily large initial and boundary values on a possibly small time horizon. We derive continuity and differentiability properties of the solution operator of the quasilinear problem w.r.t. the control term located in the node conditions. Then we analyze an optimal control problem for the networked system, where the control is of boundary type. We prove the existence of optimal controls and the differentiability of the objective functional w.r.t. the controls.
We prove an existence result for the steady state flow of gas mixtures
on networks. The basis of the model are the physical principles of the isothermal
Euler equation, coupling conditions for the flow and pressure, and the mixing of
incoming flow at nodes. The state equation is based on a convex combination of
the ideal gas equations of state for natural gas and hydrogen. We analyze mathematical
properties of the model allowing us to prove the existence of solutions in
particular for tree-shaped networks and networks with exactly one cycle. Numerical
examples illustrate the results and explore the applicability of our approach
to different network topologies.
In this paper, topological derivatives are defined and employed for gas transport networks
governed by nonlinear hyperbolic systems of PDEs. The concept of topological derivatives of a shape functional is introduced for optimum design and control of gas networks. First, the dynamic model for the network is considered. The cost for the control problem includes the deviations of the pressure at the inflow and outflow nodes. For dynamic control problems of gas networks when the turnpike property occurs, the synthesis of control and optimum design of the network can be simplified. That is, the design of the network can be performed for optimal control of the steady-state network model. The cost of design is defined by the optimal control cost for the steady-state network model. The topological derivative of the design cost, given by the optimal control cost with respect to the nucleation of a small cycle, is
determined. Tree-structured networks can be decomposed into single network junctions. The topological derivative of the design cost is systematically evaluated at each junction of the decomposed network. This allows for the identification of internal nodes with negative topological derivatives, where replacing the node with a small cycle leads to an improved design cost. As the set of network junctions is finite, the iterative procedure is convergent. This design procedure is applied to representative examples and it can be generalized to arbitrary network graphs. A key feature of such modeling approach is the availability of exact steady-state solutions, enabling a fully analytical topological analysis of the design cost without numerical approximations.
Uncertainty plays a crucial role in modeling and optimization of complex systems across various applications. In this paper, uncertain gas transport through pipeline networks is considered and a novel strategy to measure the robustness of deterministically computed compressor and valve controls, the probabilistic robustness, is presented.
\noindent Initially, an optimal control for a deterministic gas network problem is computed such that the total control cost is minimized with respect to box constraints for the pressure. Subsequently, the model is perturbed by uncertain gas demands. The probability, that the uncertain gas pressures - based on the a priori deterministic optimal control - satisfy the box constraints, is evaluated. Moreover, buffer zones are introduced in order to tighten the pressure bounds in the deterministic scenario. Optimal controls for the deterministic scenario with buffer zones are also applied to the uncertain scenario, allowing to analyze the impact of the buffer zones on the probabilistic robustness of the optimal controls.
\noindent For the computation of the probability, we apply a kernel density estimator based on samples of the uncertain pressure at chosen locations. In order to reduce the computational effort of generating the samples, we combine the kernel density estimator approach with a stochastic collocation method which approximates the pressure at the chosen locations in the stochastic space. Finally, we discuss generalizations of the probabilistic robustness check and we present numerical results for a gas network taken from the public gas library.
Optimal Neumann Control of the Wave Equation with L1-Control Cost: The Finite-Time Turnpike Property
(2024)
The finite-time turnpike property describes a situation where the optimal state reaches a steady state after finite time. The steady state is a solution of a static optimal control problem. We study an optimal control problem where the objective functional is the sum of an 𝐿1-norm control cost with a weight 𝛾>0 and a differentiable tracking term. We consider a vibrating string with homogeneous Dirichlet conditions at one end and Neumann control action at the other end. The tracking term is defined by the squared 𝐿2-norm of a non-collocated Neumann-observation. We show that the problem has a unique solution and that due to the non-smoothness of the 𝐿1-norm for sufficiently large T the optimal state reaches the desired state after a finite time that is equal to the minimal time where exact controllability holds. We also study the effect of smoothing of the control cost on the structure of the optimal control and show that the finite-time turnpike property also holds in the smoothing limit in a strong 𝐿2-sense.
Lyapunov functions with exponential weights have been used successfully as a powerful tool for the stability analysis of hyperbolic systems of balance laws. In this paper we extend the class of weight functions to a family of hyperbolic functions and study the advantages in the analysis of 2 × 2 systems of balance laws. We present cases connected with the study of the limit of stabilizability, where the new weights provide Lyapunov functions that show exponential stability for a larger set of problem parameters than classical exponential weights.
Moreover, we show that sufficiently large time-delays influence the limit of stabilizability in the sense that the parameter set, for which the system can be stabilized becomes substantially smaller.
We also demonstrate that the hyperbolic weights are useful in the analysis of the boundary feedback stability of systems of balance laws that are governed by quasilinear hyperbolic partial differential equations.
