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https://opus4.kobv.de/opus4trr154/index/index/
Wed, 17 May 2017 17:15:34 +0200
Wed, 17 May 2017 17:15:34 +0200

A joint model of probabilistic/robust constraints for gas transport management in stationary networks
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/141
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 nonstochastic 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 nonstochastic 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 socalled sphericradial 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 costintensive roughness measurements.
Tatiana Gonzalez Grandon; Holger Heitsch; Rene Henrion
article
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/141
Wed, 17 May 2017 17:15:34 +0200

MIPBased Instantaneous Control of MixedInteger PDEConstrained Gas Transport Problems
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/140
We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics that are described by the system of partial differential Euler equations. This combination leads to mixedinteger optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be wellposed and affinelinear solutions of these systems are derived analytically. As a consequence, finitedimensional mixedinteger linear optimization problems are obtained for every time step that can be solved to global optimality using generalpurpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.
Martin Gugat; Günter Leugering; Alexander Martin; Martin Schmidt; Mathias Sirvent; David Wintergerst
preprint
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/140
Tue, 11 Apr 2017 18:10:58 +0200

TwoStage Stochastic Semidefinite Programming: Theory, Algorithms, and Application to AC Power Flow under Uncertainty
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/139
In real life decision problems, one almost always is confronted with uncertainty and risk. For practical optimization problems this is manifested by unknown parameters within the input data, or, an inexact knowledge about the system description itself. In case the uncertain problem data is governed by a known probability distribution, stochastic programming offers a variety of models hedging against uncertainty and risk. Most widely employed are twostage models, who admit a recourse structure: The firststage decisions are taken before the random event occurs. After its outcome, a recourse (secondstage) action is made, often but not always understood as some "compensation".
In the present thesis, the optimization problems that involve parameters which are not known with certainty are semidefinite programming problems. The constraint sets of these optimization problems are given by intersections of the cone of symmetric, positive semidefinite matrices with either affine or more general equations. Objective functions, formally, may be fairly general, although they often are linear as in the present thesis.
We consider risk neutral and risk averse twostage stochastic semidefinite programs with continuous and mixedinteger recourse, respectively. For these stochastic optimization problems we analyze their structure, derive solution methods relying on decomposition, and finally apply our results to unit commitment in alternating current (AC) power systems.
Furthermore, deterministic unit commitment in AC power transmission systems is addressed. Beside traditional unit commitment constraints, the physics of power flow are included. To gain globally optimal solutions a recent semidefinite programming (SDP) approach is used which leads to largescale semidefinite programs with discrete variables on top. As even the SDP relaxation of these programs is too large for being handled in an allatonce manner by general SDP solvers, it requires an efficient and reliable method to tackle them. To this end, an algorithm based on Benders decomposition is proposed.
With power demand (load) and infeed from renewables serving as sources of uncertainty, twostage stochastic programs are set up heading for unit commitment schedules which are both costeffective and robust with respect to data perturbations. The impact of different, risk neutral and risk averse, stochastic criteria on the shapes of the optimal stochastic solutions will be examined. To tackle the resulting twostage programs, we propose to approximate AC power flow by semidefinite relaxations. This leads to twostage stochastic mixedinteger semidefinite programs having a special structure. To solve the latter, the Lshaped method and dual decomposition have been applied and compared.
Tobias Wollenberg
doctoralthesis
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/139
Tue, 04 Apr 2017 11:40:13 +0200

Unit commitment under uncertainty in AC transmission systems via risk averse semidefinite stochastic programs
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/138
This paper addresses unit commitment under uncertainty of load and power infeed from renewables in alternating current (AC) power systems. Beside traditional unitcommitment constraints, the physics of power flow are included. To gain globally optimal solutions a recent semidefinite programming approach is used, which leads us to risk averse twostage stochastic mixed integer semidefinite programs for which a decomposition algorithm is presented.
Rüdiger Schultz; Tobias Wollenberg
article
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/138
Tue, 04 Apr 2017 11:40:12 +0200

Topology motivated discretization of hyperbolic PDAEs describing flow networks
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/137
Christoph Huck; Caren Tischendorf
preprint
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/137
Mon, 03 Apr 2017 12:38:56 +0200

Deciding Robust Feasibility and Infeasibility Using a Set Containment Approach: An Application to Stationary Passive Gas Network Operations
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/136
In this paper we study feasibility and infeasibility of nonlinear twostage
fully adjustable robust feasibility problems with an empty first stage. This
is equivalent to deciding set containment of a projection of the feasible
region and the uncertainty set. For answering this question, two very general
approaches using methods from polynomial optimization are presented  one
for showing feasibility and one for showing infeasibility. The developed
methods are approximated through sum of squares polynomials and solved using
semidefinite programs.
Deciding robust feasibility and infeasibility is important for gas network
operations, which is a \nonconvex quadratic problem with absolute values
functions. Concerning the gas network problem, different topologies are
considered. It is shown that a tree structured network can be decided exactly
using linear programming. Furthermore, a method is presented to reduce a tree
network with one additional arc to a single cycle network. In this case,
removing the absolute values and solving the problem can be decided with
linearly many polynomial optimization problems.
Lastly, the effectivity of the methods is tested on a variety of small cyclic
networks. For instances where robust feasibility or infeasibility can be
decided, level~2 or level~3 of the Lasserre relaxation hierarchy is typically
sufficient.
Denis Aßmann; Frauke Liers; Michael Stingl; Juan Vera
preprint
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/136
Sun, 19 Mar 2017 17:47:15 +0100

Probability of Feasible Loads in Passive Gas Networks with up to Three Cycles
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/135
Gas networks are of growing importance for the economy and offer interesting mathematical problems at the same time. The classical linear network ﬂow allows for approximate models that more and more have come to their limits. This has raised interest in nonlinear but, for simplicity, still steadystate models. The present paper aims at mobilizing techniques from symbolic computation and reparametrization of multivariate integrals to enable validation of stochastic nominations following Gaussian distributions in passive gas networks with more than one cycle.
Claudia Gotzes; Sabrina Nitsche; Rüdiger Schultz
preprint
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/135
Fri, 17 Mar 2017 16:31:01 +0100

A Multilevel Model of the European EntryExit Gas Market
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/134
In entryexit gas markets as they are currently implemented in Europe, network constraints do not affect market interaction beyond the technical capacities determined by the TSO that restrict the quantities individual firms can trade at the market. It is an up to now unanswered question to what extent existing network capacity remains unused in an entryexit design and to what extent feasible adjustments of the market design could alleviate inefficiencies. In this paper, we offer a fourlevel modeling framework that is capable of analyzing these issues and provide some first results on the model structure. In order to decouple gas trading from network congestion management, the TSO is required to determine technical capacities and corresponding booking fees at every entry and exit node up front. Firms book those capacities, which gives them the right to charge or discharge an amount of gas at a certain node up to this capacity in every period. Beyond these technical capacities and the resulting bookings, gas trade is unaffected by network constraints. The technical capacities have to ensure that transportation of traded quantities is always feasible. We assume that the TSO is regulated and determines technical capacities, fees, and transportation cost under a welfare objective. As a first step we moreover assume perfect competition among gas traders and show that the booking and nomination decisions can be analyzed in a single level. We prove that this aggregated model has a unique solution. We moreover show that the TSO's decisions can be subsumed in one level as well. If so, the model boils down to a mixedinteger nonlinear bilevel problem with robust aspects. In addition, we provide a firstbest benchmark that allows to assess welfare losses that occur in an entryexit system. Finally we discuss and provide guidance on how to include several important aspects into the approach, such as network and production capacity investment, uncertain data, market power, and intraday trading.
Veronika Grimm; Lars Schewe; Martin Schmidt; Gregor Zöttl
workingpaper
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/134
Fri, 17 Mar 2017 12:18:42 +0100

A System to Evaluate Gas Network Capacities: Concepts and Implementation
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/129
Since 2005, the gas market in the European Union is liberalized and
the trading of natural gas is decoupled from its transport.
The transport is done by socalled transmissions system operators (TSOs).
The market model established by the European Union views the gas
transmission network as a black box, providing shippers (gas traders
and consumers) the opportunity to transport gas from any entry to
any exit.
TSOs are required to offer maximum independent capacities at each
entry and exit such that the resulting gas flows can be realized by
the network without compromising security of supply.
Therefore, evaluating the available transport capacities is extremely
important to the TSOs.
This paper gives an overview of the toolset for evaluating gas
network capacities that has been developed within the ForNe project,
a joint research project of seven research partners initiated by
Open Grid Europe, Germany's biggest TSO.
While most of the relevant mathematics is described in the
book "Evaluating Gas Network Capacities", this article
sketches the system as a whole, describes some developments that have
taken place recently, and gives some details about the current
implementation.
Benjamin Hiller; Thorsten Koch; Lars Schewe; Robert Schwarz; Jonas Schweiger
preprint
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/129
Wed, 08 Mar 2017 15:21:39 +0100

Adaptive Refinement Strategies for the Simulation of Gas Flow in Networks using a Model Hierarchy
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/127
A model hierarchy that is based on the onedimensional isothermal Euler equations of fluid dynamics is used for the simulation and optimisation of gas flow through a pipeline network. Adaptive refinement strategies have the aim of bringing the simulation error below a prescribed tolerance while keeping the computational costs low. While spatial and temporal stepsize adaptivity is well studied in the literature, model adaptivity is a new field of research. The problem of finding an optimal refinement strategy that combines these three types of adaptivity is a generalisation of the unbounded knapsack problem. A refinement strategy that is currently used in gas flow simulation software is compared to two novel greedylike strategies. Both a theoretical experiment and a realistic gas flow simulation show that the novel strategies significantly outperform the current refinement strategy with respect to the computational cost incurred.
Pia Domschke; Aseem Dua; Jeroen J. Stolwijk; Jens Lang; Volker Mehrmann
preprint
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/127
Wed, 01 Feb 2017 13:10:27 +0100

Optimal Price Zones of Electricity Markets: A MixedInteger Multilevel Model and Global Solution Approaches
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/126
Mathematical modeling of market design issues in liberalized electricity markets often leads to mixedinteger nonlinear multilevel optimization problems for which no generalpurpose solvers exist and which are intractable in general. In this work, we consider the problem of splitting a market area into a given number of price zones such that the resulting market design yields welfareoptimal outcomes. This problem leads to a challenging multilevel model that contains a graphpartitioning problem with multicommodity flow connectivity constraints and nonlinearities due to proper economic modeling. Furthermore, it has highly symmetric solutions. We develop different problemtailored solution approaches. In particular, we present an extended KKT transformation approach as well as a generalized Benders approach that both yield globally optimal solutions. These methods, enhanced with techniques such as symmetry breaking and primal heuristics, are evaluated in detail on academic as well as on realistic instances. It turns out that our approaches lead to effective solution methods for the difficult optimization tasks presented here, where the problemspecific generalized Benders approach performs considerably better than the methods based on KKT transformation.
Veronika Grimm; Thomas Kleinert; Frauke Liers; Martin Schmidt; Gregor Zöttl
preprint
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/126
Fri, 20 Jan 2017 19:57:15 +0100

Lipschitz Continuity of the Value Function in MixedInteger Optimal Control Problems
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/125
Martin Gugat; Falk Hante
article
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/125
Mon, 09 Jan 2017 14:51:46 +0100

Error Analysis and Model Adaptivity for Flows in Gas Networks
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/124
In the simulation and optimization of gas flow in a pipeline network, a hierarchy of models is used that employs different formulations of the Euler equations. While the optimization is performed on piecewise linear models, the flow simulation is based on the simulation of one to three dimensional Euler equations including the temperature distributions. To decide which model class in the hierarchy is adequate to achieve a desired accuracy, this paper presents an error and perturbation analysis for a two level model hierarchy including the isothermal Euler equations in semilinear form and the stationary Euler equations in purely algebraic form. The focus of the work is on the effect of data uncertainty, discretization and rounding errors in the numerical simulation of these models and their interaction. Two simple discretization schemes for the semilinear model are compared with respect to their conditioning and temporal stepsizes are determined for which a wellconditioned problem is obtained. The results are based on new componentwise relative condition numbers for the solution of nonlinear systems of equations. Moreover, the model error between the semilinear and the algebraic model is computed, the maximum pipeline length is determined for which the algebraic model can be used safely, and a condition is derived for which the isothermal model is adequate.
Jeroen J. Stolwijk; Volker Mehrmann
preprint
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/124
Mon, 09 Jan 2017 14:51:45 +0100

Computing Feasible Points for MINLPs with MPECs
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/122
Nonconvex mixedinteger nonlinear optimization problems frequently appear in practice and are typically extremely hard to solve. In this paper we discuss a class of primal heuristics that are based on a reformulation of the problem as a mathematical program with equilibrium constraints. We then use different regularization schemes for this class of problems and use an iterative solution procedure for solving series of regularized problems. In the case of success, these procedures result in a feasible solution of the original mixedinteger nonlinear problem. Since we only rely on local nonlinear programming solvers the resulting method is fast and we further improve its robustness by additional algorithmic techniques. We show the strength of our method by an extensive computational study on 662 MINLPLib2 instances, where our methods are able to produce feasible solutions for 60% of all instances in at most 10s.
Lars Schewe; Martin Schmidt
preprint
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/122
Wed, 21 Dec 2016 14:19:21 +0100

Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: From modeling to industrial applications
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/121
We consider optimal control problems for the flow of gas or fresh water in pipe networks as well as drainage or sewer systems in open canals. The equations of motion are taken to be represented by the nonlinear isothermal Euler gas equations, the water hammer equations, or the St.~Venant equations for flow. We formulate model hierarchies and derive an abstract model for such network flow problems including pipes, junctions, and controllable elements such as valves, weirs, pumps, as well as compressors. We use the abstract model to give an overview of the known results and challenges concerning equilibria, wellposedness, controllability, and optimal control. A major challenge concerning the optimization is to deal with switching onoff states that are inherent to controllable devices in such applications combined with
continuous simulation and optimization of the gas flow. We formulate the corresponding mixedinteger nonlinear optimal control problems and outline a decomposition approach as a solution technique.
Falk Hante; Günter Leugering; Alexander Martin; Lars Schewe; Martin Schmidt
preprint
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/121
Fri, 09 Dec 2016 15:24:39 +0100

Are QuasiMonte Carlo algorithms efficient for twostage stochastic programs?
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/120
QuasiMonte Carlo algorithms are studied for designing discrete approximations of twostage linear stochastic programs with random righthand side and
continuous probability distribution. The latter should allow for a transformation to a
distribution with independent marginals. The twostage 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 twostage 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 twostage 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 twostage 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.
Hernan Leövey; Holger Heitsch; Werner Römisch
article
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/120
Mon, 28 Nov 2016 10:49:20 +0100

Joint dynamic probabilistic constraints with projected linear decision rules
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/118
We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto
hard constraints of waitandsee type. We establish the relation between the original (infinite
dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of linear decision rules and a generalized
linear model for the underlying stochastic process with noises that are Gaussian or truncated
Gaussian, we show that the value and gradient of the objective and constraint functions of the
approximating problems can be computed analytically.
Vincent Guigues; Rene Henrion
article
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/118
Mon, 28 Nov 2016 10:49:19 +0100

On probabilistic constraints with multivariate truncated Gaussian and lognormal distributions
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/119
Many engineering problems with uncertain data, notably arising
in power management, can be formulated as optimization problems subject to
probabilistic constraints. While dealing with such constraints under continuous distributions of the underlying random parameter remains a difficult task
in general both from the numerical and theoretical point of view, quite some
progress has been made in the special case of multivariate Gaussian distributions. These are not perfectly adequate, however, in many circumstances, in
particular not, when modeling uncertain inflows to hydro reservoirs or uncertain demands in gas networks. Interesting alternatives are offered by truncations of multivariate Gaussian distributions to polyhedra or by multivariate
lognormal distributions. The paper discusses the applicability of such distributions in the context of a simple joint linear probabilistic constraint putting
the emphasis on the numerical approximation of probabilities and their gradients (w.r.t. decisions to be optimized) as well as on the convexity of the set
of feasible decisions.
Andre Luiz Diniz; Rene Henrion
article
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/119
Mon, 28 Nov 2016 10:49:19 +0100

(Sub) Gradient formulae for probability functions of random inequality systems under Gaussian distribution
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/117
We consider probability functions of parameterdependent random inequality systems under
Gaussian distribution. As a main result, we provide an upper estimate for the Clarke subdifferential
of such probability functions without imposing compactness conditions. A constraint qualification
ensuring continuous differentiability is formulated. Explicit formulae are derived from the general
result in case of linear random inequality systems. In the case of a constant coefficient matrix an
upper estimate for even the smaller Mordukhovich subdifferential is proven.
Wim van Ackooij; Rene Henrion
article
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/117
Mon, 28 Nov 2016 10:49:18 +0100

On the quantification of nomination feasibility in stationary gas networks with random load
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/116
The paper considers the computation of the probability of feasible load constellations in a stationary gas
network with uncertain demand. More precisely, a network with a single entry and several
exits with uncertain loads is studied. Feasibility of a load constellation is understood in the sense of
an existing flow meeting these loads along with given pressure bounds in the pipes.
In a first step, feasibility of deterministic exit loads is characterized algebraically and these general
conditions are specified to networks involving at most one cycle.
This prerequisite is essential for determining probabilities in a stochastic setting when exit loads
are assumed to follow some (joint) Gaussian distribution when modeling uncertain customer demand.
The key of our approach is the application of the sphericradial decomposition of Gaussian random
vectors coupled with Quasi MonteCarlo sampling. This approach requires an efficient algorithmic
treatment of the mentioned algebraic relations moreover depending on a scalar parameter. Numerical
results are illustrated for different network examples and demonstrate a clear superiority in terms of
precision over simple generic MonteCarlo sampling. They lead to fairly accurate probability values
even for moderate sample size.
Claudia Gotzes; Holger Heitsch; Rene Henrion; Rüdiger Schultz
article
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/116
Mon, 28 Nov 2016 10:49:17 +0100

Exact penalization of terminal constraints for optimal control problems
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/115
We study optimal control problems for linear systems with prescribed initial and terminal states. We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the normminimal exact control can be computed as the solution of an optimization problem without terminal constraint but with a nonsmooth penalization of the end conditions in the objective function, if the penalty parameter is sufficiently large. We describe the application of the method for hyperbolic and parabolic systems of partial differential equations, considering the wave and heat equations as particular examples. Copyright © 2016 John Wiley & Sons, Ltd.
Martin Gugat; Enrique Zuazua
article
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/115
Fri, 11 Nov 2016 13:32:57 +0100

On the relaxation gap for PDE mixedinteger optimal control problems
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/114
Falk Hante
conferenceobject
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/114
Wed, 09 Nov 2016 17:32:13 +0100

Optimal Neumann control for the 1D wave equation: Finite horizon, infinite horizon, boundary tracking terms and the turnpike property
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/113
We consider a vibrating string that is fixed at one end with Neumann control action at the other end. We investigate the optimal control problem of steering this system from given initial data to rest, in time TT, by minimizing an objective functional that is the convex sum of the L2L2norm of the control and of a boundary Neumann tracking term.
We provide an explicit solution of this optimal control problem, showing that if the weight of the tracking term is positive, then the optimal control action is concentrated at the beginning and at the end of the time interval, and inbetween it decays exponentially. We show that the optimal control can actually be written in that case as the sum of an exponentially decaying term and of an exponentially increasing term. This implies that, if the time TT is large, then the optimal trajectory approximately consists of three arcs, where the first and the third shorttime arcs are transient arcs, and in the middle arc the optimal control and the corresponding state are exponentially close to 00. This is an example of a turnpike phenomenon for a problem of optimal boundary control. If T=+∞T=+∞ (infinite time horizon problem), then only the exponentially decaying component of the control remains, and the norms of the optimal control action and of the optimal state decay exponentially in time. In contrast to this situation, if the weight of the tracking term is zero and only the control cost is minimized, then the optimal control is distributed uniformly along the whole interval [0,T][0,T] and coincides with the control given by the Hilbert Uniqueness Method.
In addition, we establish a similarity theorem stating that, for every T>0T>0, there exists an appropriate weight λ<1λ<1 for which the optimal solutions of the corresponding finite horizon optimal control problem and of the infinite horizon optimal control problem coincide along the first part of the time interval [0,2][0,2]. We also discuss the turnpike phenomenon from the perspective of a general framework with a strongly continuous semigroup.
Martin Gugat; Emmanuel Trelat; Enrique Zuazua
article
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/113
Wed, 09 Nov 2016 17:32:12 +0100

Networks of pipelines for gas with nonconstant compressibility factor: stationary states
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/112
For the management of gas transportation networks, it is essential to know how the stationary states of the system are determined by the boundary data. The isothermal Euler equations are an accurate pdemodel for the gas ﬂow through each pipe. A compressibility factor is used to model the nonlinear relationship between density and pressure that occurs in real gas in contrast to ideal gas. The gas ﬂow through the nodes is governed by algebraic node conditions that require the conservation of mass and the continuity of the pressure. We examine networks that are described by arbitrary ﬁnite graphs and show that for suitably chosen boundary data, subsonic stationary states exist and are uniquely determined by the boundary data. Our construction of the stationary states is based upon explicit representations of the stationary states on each single pipe that can easily be evaluated numerically. We also use the monotonicity properties of these states as functions of the boundary data.
David Wintergerst; Martin Gugat; Rüdiger Schultz
preprint
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/112
Wed, 09 Nov 2016 17:32:11 +0100

PeakLoad Pricing on a Network
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/110
In this paper we analyze peakload pricing in the presence of network constraints. In our setup, firms facing fluctuating demand decide on the size and location of production facilities. They make production decisions constrained by the invested capacities, taking into account that market prices reflect scarce transmission capacities. We state general conditions for existence and uniqueness of the market equilibrium and provide a characterization of equilibrium investment and production. The presented analysis covers the cases of perfect competition and monopoly  the case of strategic firms is approximated by a conjectural variations approach. Our result is a prerequisite for analyzing regulatory policy options with computational multilevel equilibrium models, since uniqueness of the equilibrium at lower levels is of key importance when solving these models. Thus, our paper contributes to an evolving strand of literature that analyzes regulatory policy based on computational multilevel equilibrium models and aims at taking into account individual objectives of various agents, among them not only generators and customers but also, e.g., the regulator deciding on network expansion.
Veronika Grimm; Lars Schewe; Martin Schmidt; Gregor Zöttl
preprint
https://opus4.kobv.de/opus4trr154/frontdoor/index/index/docId/110
Thu, 20 Oct 2016 13:40:05 +0200