Humboldt-Universität zu Berlin
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- Gas networks (2)
- Mixed-integer nonlinear optimization (2)
- spheric-radial decomposition (2)
- Abs-smooth Algorithmic Differentiation (1)
- Active Signature Method (1)
- Closed-loop stability (1)
- Controlling Regulators and Compressors (1)
- Convergence (1)
- DAE (1)
- Euler-Gleichungen, isotherme Euler-Gleichungen, Modellhierarchie, Netzelemente (1)
We extend the framework of sequential action control to systems of partial differential equations which can be posed as abstract linear control problems in a Hilbert space. We follow a late-lumping approach and show that the control action can be explicitly obtained from variational principles using adjoint information. Moreover, we analyze the closed-loop system obtained from the SAC feedback for quadratic stage costs. We apply this theory prototypically to an unstable heat equation and verify the results numerically.
The inverse problem of identifying the friction coefficient in an isothermal semilinear Euler system is considered. Adopting a Bayesian approach, the goal is to identify the distribution of the quantity of interest based on a finite number of noisy measurements of the pressure at the boundaries of the domain. First well-posedness of the underlying non-linear PDE system is shown using semigroup theory, and then Lipschitz continuity of the solution operator with respect to the
friction coefficient is established. Based on the Lipschitz property, well-posedness of the resulting Bayesian inverse problem for the identification of the friction coefficient is inferred. Numerical tests for scalar and distributed parameters are performed to validate the theoretical results.
Identification of the friction function 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.
Joint model of probabilistic/robust (probust) constraints applied 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 deal with a class of joint probabilistic/robust constraints as its appears in
optimization problems under uncertainty. Focusing 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.
Differential-algebraic equations with higher index give rise to essentially ill-posed problems. The least-squares collocation by discretizing the pre-image space is not much more computationally expensive than
standard collocation methods used in the numerical solution of ordinary differential equations and index-1 differential-algebraic equations. This approach has displayed excellent convergence properties in numerical experiments, however, theoretically, till now convergence could be established merely for regular linear differential-algebraic equations with constant coefficients. We present now an estimate of the instability threshold which serves as the basic key for proving convergence for general regular linear DAEs.
Mit dieser Übersicht wollen wir eine Zusammenstellung von unterschiedlichen Modellen zur Beschreibung des Gasflusses in Netzwerken bereitstellen, um den Einstieg in das Thema zu erleichtern. Besonderes Augenmerk wird dabei auf die der Modellierung inneliegende hierarchische Struktur und die detaillierte Beschreibung einzelner Bauteile wie Ventile und Kompressoren
gelegt. Daneben finden sich ebenfalls Netzmodellklassen, die auf rein algebraische Relationen aufbauen. Am Ende geben wir einen kurzen Überblick über grundlegende numerische Verfahren und Konzepte zur Behandlung von hyperbolischen Bilanzgleichungen. Wir erheben keinen Anspruch auf Vollständigkeit und verweisen an vielen Stellen auf die bestehende Literatur.
Die Idee eines Modellkataloges ist uns im Rahmen der Antragstellung zum SFB/Transregio 154 „Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken“ gekommen. Wir möchten an dieser Stelle die Förderung durch die DFG dankend erwähnen.
Evolutionary quasi-variational inequality (QVI) problems of dissipative and non-dissipative nature with pointwise constraints on the gradient are studied. A semi-discretization in time is employed for the study of the problems and the derivation of a numerical solution scheme, respectively. Convergence of the discretization procedure is proven and properties of the original infinite dimensional problem, such as existence, extra regularity and non-decrease in time, are derived. The proposed numerical solver reduces to a finite number of gradient-constrained convex optimization problems which can be solved rather efficiently. The paper ends with a report on numerical tests obtained by a variable splitting algorithm involving different nonlinearities and types of constraints.
Higher-order Runge-Kutta (RK) time discretization methods for the optimal control of scalar conservation laws are analyzed and numerically tested. The hyperbolic nature of the state system introduces specific requirements on discretization schemes such that the discrete adjoint states associated with the control problem converge as well. Moreover, conditions on the RK-coefficients are derived that coincide with those characterizing strong stability preserving Runge-Kutta methods. As a consequence, the optimal order for the adjoint state is limited, e.g., to two even in the case where the conservation law is discretized by a third-order method. Finally, numerical tests for controlling Burgers equation validate the theoretical results.
Gas Network Benchmark Models
(2017)
The simulation of gas transportation networks becomes increasingly more important as its use-cases broaden to more complex applications. Classically, the purpose of the gas network was the transportation of predominantly natural gas from a supplier to the consumer for long-term scheduled volumes. With the rise of renewable energy sources, gas-fired power plants are often chosen to compensate for the fluctuating nature of the renewables, due to their on-demand power generation capability. Such an only short-term plannable supply and demand setting requires sophisticated simulations of the gas network prior to the dispatch to ensure the supply of all customers for a range of possible scenarios and to prevent damages to the gas network. In this work we describe the modeling of gas networks and present benchmark systems to test implementations and compare new or extended models.
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.
The directional differentiability of the solution mapping for a class of variational inequali-
ties of the second kind inspired by applications in fluid mechanics and moving free boundary
problems is investigated. The result is particularly relevant for the model predictive control
or optimal control of such variational inequalities in that it can be used to derive stationarity
conditions and efficient numerical methods.
Using a standard first-order optimality condition for nonsmooth optimization prob-
lems, a general framework for a descent method is developed. This setting is applied to
a class of mathematical programs with equilibrium constraints in function space from
which a new algorithm is derived. Global convergence of the algorithm is demonstrated
in function space and the results are then illustrated by numerical experiments.
We address the problem of optimally placing sensor networks for convection-diffusion
processes where the convective part is perturbed. The problem is formulated as an optimal control
problem where the integral Riccati equation is a constraint and the design variables are sensor
locations. The objective functional involves a term associated to the trace of the solution to the
Riccati equation and a term given by a constrained optimization problem for the directional derivative
of the previous quantity over a set of admissible perturbations. The paper addresses the existence
of the derivative with respect to the convective part of the solution to the Riccati equation, the
well-posedness of the optimization problem and finalizes with a range of numerical tests.
This paper is concerned with the distributed optimal control of a time-discrete Cahn–
Hilliard/Navier–Stokes system with variable densities. It focuses on the double-obstacle potential
which yields an optimal control problem for a family of coupled systems in each time instant of a
variational inequality of fourth order and the Navier–Stokes equation. By proposing a suitable time-
discretization, energy estimates are proved and the existence of solutions to the primal system and of
optimal controls is established for the original problem as well as for a family of regularized problems.
The latter correspond to Moreau–Yosida type approximations of the double-obstacle potential. The
consistency of these approximations is shown and first order optimality conditions for the regularized
problems are derived. Through a limit process with respect to the regularization parameter, a
stationarity system for the original problem is established. The resulting system corresponds to a
function space version of C-stationarity which is a special notion of stationarity for MPECs.
In this paper we study the structure of solutions of the one dimensional weighted total
variation regularisation problem, motivated by its application in signal recovery tasks. We study
in depth the relationship between the weight function and the creation of new discontinuities in
the solution. A partial semigroup property relating the weight function and the solution is shown
and analytic solutions for simply data functions are computed. We prove that the weighted total
variation minimisation problem is well-posed even in the case of vanishing weight function, despite
the lack of coercivity. This is based on the fact that the total variation of the solution is bounded
by the total variation of the data, a result that it also shown here. Finally the relationship to the
corresponding weighted fidelity problem is explored, showing that the two problems can produce
completely different solutions even for very simple data functions.
In this paper, optimal control problems subject to a nonlinear scalar conservation law are
studied. Such optimal control problems are challenging both at the continuous and at the discrete
level since the control-to-state operator poses difficulties as it is, e.g., not differentiable. Therefore
discretization of the underlying optimal control problem should be designed with care. Here the
discretize-then-optimize approach is employed where first the full discretization of the objective
function as well as the underlying PDE is considered. Then, the derivative of the reduced objective
is obtained by using an adjoint calculus. In this paper total variation diminishing Runge-Kutta
(TVD-RK) methods for the time discretization of such problems are studied. TVD-RK methods,
also called strong stability preserving (SSP), are originally designed to preserve total variation of
the discrete solution. It is proven in this paper that providing an SSP state scheme, is enough to
ensure stability of the discrete adjoint. However requiring SSP for both discrete state and adjoint is
too strong. Also approximation properties that the discrete adjoint inherits from the discretization
of the state equation are studied. Moreover order conditions are derived. In addition, optimal
choices with respect to CFL constant are discussed and numerical experiments are presented.
We present a novel mathematical algorithm to assist gas network operators in managing uncertainty,
while increasing reliability of transmission and supply. As a result, we solve an optimization problem
with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise
in the presence of uncertain parameters having partially stochastic and partially non-stochastic character.
The application that drives this new approach is a stationary network with uncertain demand
(which are stochastic due to the possibility of fitting statistical distributions based on historical measurements)
and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of
attainable measurements).
We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable
network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is
allowed (shaping a 'maximal' uncertainty set) around nominal roughness coefficients, such that random demands in
a stationary gas network can be satisfied at given high probability level for no matter which realization of
true roughness coefficients within the uncertainty set.
One ends up with a constraint, which is probabilistic with respect to the load of gas
and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in
the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions.
The numerical solution of a corresponding optimization problem is illustrated.
The results might assist the network operator with the implementation
of cost-intensive roughness measurements.
Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and
continuous probability distribution. The latter should allow for a transformation to a
distribution with independent marginals. The two-stage integrands are piecewise linear,
but neither smooth nor lie in the function spaces considered for QMC error analysis.
We show that under some weak geometric condition on the two-stage model all terms
of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial
derivatives and belong to L2 . Hence, randomly shifted lattice rules (SLR) may achieve
the optimal rate of convergence O(n−1+δ ) with δ ∈ (0, 12 ] and a constant not depending
on the dimension if the effective superposition dimension is at most two. We discuss
effective dimensions and dimension reduction for two-stage integrands. The geometric
condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming
the covariance matrix. Numerical experiments for a large scale two-stage stochastic
production planning model with normal demand show that indeed convergence rates
close to the optimal are achieved when using SLR and randomly scrambled Sobol’ point
sets accompanied with PCA for dimension reduction.