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Wed, 06 Feb 2019 17:37:45 +0100Wed, 06 Feb 2019 17:37:45 +0100On the turnpike phenomenon for optimal boundary control problems with hyperbolic systems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/250
We study problems of optimal boundary control with systems governed by linear hyperbolic partial differential equations. The objective function is quadratic and given by an integral over the finite time interval (0,T) that depends on the boundary traces of the solution. If the time horizon T is sufficiently large, the solution of the dynamic optimal boundary control problem can be approximated by the solution of a steady state optimization problem. We show that for T to infinity the approximation error converges to zero in the sense of the norm in L^2(0,1) with the rate 1/T, if the time interval (0,T) is transformed to the fixed interval (0,1). Moreover, we show that also for optimal boundary control problems with integer constraints for the controls the turnpike phenomenon occurs. In this case the steady state optimization problem also has the integer constraints. If T is sufficiently large, the integer part of each solution of the dynamic optimal boundary control problem with integer constraints is equal to the integer part of a solution of the static problem. A numerical verification is given for a control problem in gas pipeline operations.Martin Gugat; Falk Hantearticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/250Wed, 06 Feb 2019 17:37:45 +0100Bookings in the European Gas Market: Characterisation of Feasibility and Computational Complexity Results
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/245
As a consequence of the liberalisation of the European gas market in the last decades, gas trading and transport have been decoupled. At the core of this decoupling are so-called bookings and nominations. Bookings are special long-term capacity right contracts that guarantee that a specified amount of gas can be supplied or withdrawn at certain entry or exit nodes of the network. These supplies and withdrawals are nominated at the day-ahead. The special property of bookings then is that they need to be feasible, i.e., every nomination that complies with the given bookings can be transported. While checking the feasibility of a nomination can typically be done by solving a mixed-integer nonlinear feasibility problem, the verification of feasibility of a set of bookings is much harder. The reason is the robust nature of feasibility of bookings - namely that for a set of bookings to be feasible, all compliant nominations, i.e., infinitely many, need to be checked for feasibility. In this paper, we consider the question of how to verify the feasibility of given bookings for a number of special cases. For our physics model we impose a steady-state potential-based flow model and disregard controllable network elements. For this case we derive a characterisation of feasible bookings, which is then used to show that the problem is in coNP for the general case but can be solved in polynomial time for linear potential-based flow models. Moreover, we present a dynamic programming approach for deciding the feasibility of a booking in tree-shaped networks even for nonlinear flow models. It turns out that the hardness of the problem mainly depends on the combination of the chosen physics model as well as the specific network structure under consideration. Thus, we give an overview over all settings for which the hardness of the problem is known and finally present a list of open problems.Martine Labbé; Fränk Plein; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/245Thu, 06 Dec 2018 14:42:50 +0100Convergence of Finite-Dimensional Approximations for Mixed-Integer Optimization with Differential Equations
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/244
We consider a direct approach to solve mixed-integer nonlinear optimization problems with constraints depending on initial and terminal conditions of an ordinary differential equation. In order to obtain a finite-dimensional problem, the dynamics are approximated using discretization methods. In the framework of general one-step methods, we provide sufficient conditions for the convergence of this approach in the sense of the corresponding optimal values. The results are obtained by considering the discretized problem as a parametric mixed-integer nonlinear optimization problem in finite dimensions, where the maximum step size for discretizing the dynamics is the parameter. In this setting, we prove the continuity of the optimal value function under a stability assumption for the integer feasible set and second-order conditions from nonlinear optimization. We address the necessity of the conditions on the example of pipe sizing problems for gas networks.Falk M. Hante; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/244Thu, 06 Dec 2018 12:53:27 +0100Endogenous Price Zones and Investment Incentives in Electricity Markets: An Application of Multilevel Optimization with Graph Partitioning
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/243
In the course of the energy transition, load and supply centers are growing apart in many electricity markets worldwide, rendering regional price signals even more important to provide adequate locational investment incentives. In this context, the establishment of price zones in order to at least partially price grid bottlenecks is under discussion. This paper addresses the key question of how to configure price zones on a network in order to optimally govern investment and production decisions in the long run. We extend the multilevel equilibrium model from Grimm et al. (2017a) to endogenously determine welfare-maximizing price zones for a given electricity market and analyze their impact on market outcomes. This mixed-integer nonlinear model contains a graph partitioning problem on the first level to model the zoning of the network. Using a generalized Benders decomposition and a problem-tailored scenario clustering for reducing the input data size, we are able to solve the model to global optimality even for large instances. We apply the approach to the German electricity market as an example to examine the impact of optimal zoning on key performance indicators such as welfare, generation mix and locations, or electricity prices. It turns out that already a few optimally chosen zones lead to significant welfare gains.Mirjam Ambrosius; Veronika Grimm; Thomas Kleinert; Frauke Liers; Martin Schmidt; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/243Fri, 19 Oct 2018 17:54:45 +0200Structural Properties of Feasible Bookings in the European Entry-Exit Gas Market System
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/242
In this work we analyze the structural properties of the set of feasible bookings in the European entry-exit gas market system. We present formal definitions of feasible bookings and then analyze properties that are important if one wants to optimize over them. Thus, we study whether the sets of feasible nominations and bookings are bounded, convex, connected, conic, and star-shaped. The results depend on the specific model of gas flow in a network. Here, we discuss a simple linear flow model with arc capacities as well as nonlinear and mixed-integer nonlinear models of passive and active networks, respectively. It turns out that the set of feasible bookings has some unintuitive properties. For instance, we show that the set is nonconvex even though only a simple linear flow model is used.Lars Schewe; Martin Schmidt; Johannes Thüraufpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/242Wed, 03 Oct 2018 20:04:09 +0200Mixed-Integer Optimal Control for PDEs: Relaxation via Differential Inclusions and Applications to Gas Network Optimization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/241
We show that mixed-integer control problems for evolution type partial differential equations can be regarded as operator differential inclusions. This yields a relaxation result including a characterization of the optimal value for mixed-integer optimal control problems with control constraints. The theory is related to partial outer convexification and sum-up rounding methods. The results are applied to optimal valve switching control for gas pipeline operations. A numerical example illustrates the approach.Falk Hantearticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/241Wed, 19 Sep 2018 10:17:30 +0200Mixed-Integer Programming Techniques for the Connected Max-k-Cut Problem
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/239
We consider an extended version of the classical Max-k-Cut problem in which we additionally require that the parts of the graph partition are connected. For this problem we study two alternative mixed-integer linear formulations and review existing as well as develop new branch-and-cut techniques like cuts, branching rules, propagation, primal heuristics, and symmetry breaking. The main focus of this paper is an extensive numerical study in which we analyze the impact of the different techniques for various test sets. It turns out that the techniques from the existing literature are not sufficient to solve an adequate fraction of the test sets. However, our novel techniques significantly outperform the existing ones both in terms of running times and the overall number of instances that can be solved.Christopher Hojny; Imke Joormann; Hendrik Lüthen; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/239Thu, 26 Jul 2018 13:32:30 +0200Strictly and Γ-Robust Counterparts of Electricity Market Models: Perfect Competition and Nash-Cournot Equilibria
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/238
This paper mainly studies two topics: linear complementarity problems (LCPs) for modeling electricity market equilibria and optimization under uncertainty. While there have been quite some attempts to deal with uncertain LCPs in a stochastic - i.e., distributional - sense, robust LCPs have only gained attention very recently. In this paper, we consider both perfectly competitive and Nash-Cournot models of electricity markets and study their robustifications using strict robustness and the Γ-approach. For three out of the four combinations of economic competition and robustification we derive algorithmically tractable convex optimization counterparts that have a clear-cut economic interpretation. In the case of perfect competition this particularly means that the two classical welfare theorems also hold in both considered robust cases. Using the mentioned counterparts, we can also prove the existence and, in some cases, uniqueness of robust equilibria. Surprisingly, it turns out that there is no such economic sensible counterpart for the case of Γ-robustifications of Nash-Cournot models. Thus, an analogue of the welfare theorems does not hold in this case. Finally, we provide a computational case study that illustrates the different effects of the combination of economic competition and uncertainty modeling.Anja Kramer; Vanessa Krebs; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/238Fri, 13 Jul 2018 16:21:28 +0200The Impact of Potential-Based Physics Models on Pricing in Energy Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/237
Pricing of access to energy networks is an important issue in liberalized energy sectors because of the natural monopoly character of the underlying transport infrastructures. We introduce a general pricing framework for potential-based energy flows in arbitrarily structured transport networks. In different specifications of our general pricing model we discuss first- and second-best pricing results and compare different pricing outcomes of potential-free and potential-based energy flow models. Our results show that considering nonlinear laws of physics leads to significantly different pricing results on networks and that these differences can only be seen in sufficiently complex, e.g., cyclic, networks as they can be found in real-world situations.Lars Schewe; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/237Wed, 06 Jun 2018 12:59:51 +0200Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/236
We investigate the long-time behaviour of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition. The proof of the existence and uniqueness of a solution defined for all positive times is also provided in this paper.Martin Gugat; Lionel Rosier; Vincent Perrollazarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/236Tue, 05 Jun 2018 07:37:11 +0200The Cost of Not Knowing Enough: Mixed-Integer Optimization with Implicit Lipschitz Nonlinearities
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/235
It is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit constraint functions that cannot be solved for any parameter analytically. For this class of mixed-integer problems we propose a novel algorithm based on an approximation of the feasible set in the domain of the nonlinear function - in contrast to an approximation of the graph of the function considered in prior work. This method is shown to compute global optimal solutions in finite time and we also provide a worst-case iteration bound. However, first numerical experiences reveal that a lot of work is still to be done for this highly challenging class of problems and we thus finally propose some possible directions of future research.Martin Schmidt; Mathias Sirvent; Winnifried Wollnerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/235Thu, 26 Apr 2018 13:07:18 +0200Global Optimization of Multilevel Electricity Market Models Including Network Design and Graph Partitioning
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/234
We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in [25] that does not include a network design problem. The two classical discrete optimization problems of network design and graph partitioning together with nonlinearities due to economic modeling yield extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof including novel primal-dual bound tightening techniques, whereas the second is a tailored generalized Benders decomposition. For the latter, we strengthen the Benders cuts of [25] by using the structure of the newly introduced network design subproblem. We prove for both methods that they yield global optimal solutions. Afterward, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation. Finally, we present a case study that illustrates the economic effects that are captured in our model.Thomas Kleinert; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/234Wed, 07 Feb 2018 14:27:16 +0100Robust Optimal Discrete Arc Sizing for Tree-Shaped Potential Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/233
We consider the problem of discrete arc sizing for tree-shaped potential networks with respect to infinitely many demand scenarios. This means that the arc sizes need to be feasible for an infinite set of scenarios. The problem can be seen as a strictly robust counterpart of a single-scenario network design problem, which is shown to be NP-complete even on trees. In order to obtain a tractable problem, we introduce a method for generating a finite scenario set such that optimality of a sizing for this finite set implies the sizing's optimality for the originally given infinite set of scenarios. We further prove that the size of the finite scenario set is quadratically bounded above in the number of nodes of the underlying tree and that it can be computed in polynomial time. The resulting problem can then be solved as a standard mixed-integer linear optimization problem. Finally, we show the applicability of our theoretical results by computing globally optimal arc sizes for a realistic hydrogen transport network of Eastern Germany.Martin Robinius; Lars Schewe; Martin Schmidt; Detlef Stolten; Johannes Thürauf; Lara Welderpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/233Thu, 01 Feb 2018 15:28:47 +0100A Bayesian approach for parameter identification in gas networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/231
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.Soheil Hajian; Michael Hintermüller; Claudia Schillings; Nikolai Strogiespreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/231Mon, 29 Jan 2018 14:11:31 +0100Optimal Model Switching for Gas Flow in Pipe Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/232
We consider model adaptivity for gas flow in pipeline networks. For each instant in time and for each pipe in the network a model for the gas flow is to be selected from a hierarchy of models in order to maximize a performance index that balances model accuracy and computational cost for a simulation of the entire network. This combinatorial problem involving partial differential equations is posed as an optimal switching control problem for abstract semilinear evolutions. We provide a theoretical and numerical framework for solving this problem using a two stage gradient descent approach based on switching time and mode insertion gradients. A numerical study demonstrates the practicability of the approach.Fabian Rüffler; Volker Mehrmann; Falk Hantearticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/232Mon, 29 Jan 2018 14:11:31 +0100DYNAMIC BOUNDARY CONTROL GAMES WITH NETWORKS OF STRINGS
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/229
Consider a star-shaped network
of strings. Each string is governed by the wave equation.
At each boundary node of the network there is
a player that performs Dirichlet boundary control action
and in this way influences the system state.
At the central node, the states are coupled
by algebraic conditions in such a way that the energy is conserved.
We consider the corresponding antagonistic game
where each player minimizes a certain quadratic objective function
that is given by the sum of a control cost and
a tracking term for the final state.
We prove that under suitable assumptions
a unique Nash equilibrium exists
and give an explicit representation
of the equilibrium strategies.Martin Gugat; Sonja Steffensenarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/229Fri, 19 Jan 2018 11:16:58 +0100Properties of Chance Constraints in Infinite Dimensions with an Application to PDE Constrained Optimization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/227
Chance constraints represent a popular tool for finding decisions that enforce the satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in finite dimensions. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties as well as a stability result to an infinite dimensional setting. The abstract results are applied to a simple PDE constrained control problem subject to (uniform) state chance constraints.M. Hassan Farshbaf-Shaker; Rene Henrion; Dietmar Hömbergarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/227Tue, 09 Jan 2018 11:32:43 +0100Subdifferential characterization of probability functions under Gaussian distribution
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/226
Probability functions figure prominently in optimization problems
of engineering. They may be nonsmooth even if all input data are smooth.
This fact motivates the consideration of subdifferentials for such typically just
continuous functions. The aim of this paper is to provide subdifferential formulae
of such functions in the case of Gaussian distributions for possibly
infinite-dimensional decision variables and nonsmooth (locally Lipschitzian)
input data. These formulae are based on the spheric-radial decomposition of
Gaussian random vectors on the one hand and on a cone of directions of
moderate growth on the other. By successively adding additional hypotheses,
conditions are satisfied under which the probability function is locally Lipschitzian
or even differentiable.Abderrahim Hantoute; Rene Henrion; Pedro Perez-Arosarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/226Tue, 09 Jan 2018 11:32:42 +0100On the identification of the friction coefficient in a semilinear system for gas transport through a network
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/225
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.Michael Hintermüller; Nikolai Strogiespreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/225Tue, 28 Nov 2017 10:57:52 +0100Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/224
The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that a prescribed model and discretization error tolerance is satisfied in the end. We prove convergence of the adaptively controlled optimization method and illustrate the new approach with numerical examples.Volker Mehrmann; Martin Schmidt; Jeroen J. Stolwijkpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/224Mon, 27 Nov 2017 12:56:12 +0100Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/223
In this paper we extend the implicit-explicit (IMEX) methods of Peer type
recently developed in [Lang, Hundsdorfer, J. Comp. Phys., 337:203–215, 2017]
to a broader class of two-step methods that allow the construction of super-
convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes
combine the necessary stability of implicit and low computational costs of ex-
plicit methods to efficiently solve systems of ordinary differential equations with
both stiff and non-stiff parts included in the source term. To construct super-
convergent IMEX-Peer methods with favourable stability properties, we derive
necessary and sufficient conditions on the coefficient matrices and apply an
extrapolation approach based on already computed stage values. Optimised
super-convergent IMEX-Peer methods of order s + 1 for s = 2, 3, 4 stages are
given as result of a search algorithm carefully designed to balance the size of
the stability regions and the extrapolation errors. Numerical experiments and
a comparison to other IMEX-Peer methods are included.Moritz Schneider; Jens Lang; Willem Hundsdorferpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/223Sun, 26 Nov 2017 20:46:40 +0100Uniqueness of Market Equilibria on Networks with Transport Costs
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/222
We study the existence and uniqueness of equilibria for perfectly competitive markets in capacitated transport networks. The model under consideration is rather general so that it captures basic aspects of related models in, e.g., gas or electricity networks. We formulate the market equilibrium model as a mixed complementarity problem and show the equivalence to a welfare maximization problem. Using the latter we prove uniqueness of the resulting equilibrium for piecewise linear and symmetric transport costs under additional mild assumptions. Moreover, we show the necessity of these assumptions by illustrating examples that possess multiple solutions if our assumptions are violated.Vanessa Krebs; Martin Schmidtpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/222Wed, 22 Nov 2017 11:30:24 +0100Maximizing the storage capacity of gas networks: a global MINLP approach
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/221
In this paper, we study the transient optimization of gas networks, focusing in particular on maximizing the storage capacity of the network. We include nonlinear gas physics and active elements such as valves and compressors, which due to their switching lead to discrete decisions. The former is described by a model derived from the Euler equations that is given by a coupled system of nonlinear parabolic partial differential equations (PDEs). We tackle the resulting mathematical optimization problem by a first-discretize-then-optimize approach. To this end, we introduce a new discretization of the underlying system of parabolic PDEs and prove well-posedness for the resulting nonlinear discretized system. Endowed with this discretization, we model the problem of maximizing the storage capacity as a non-convex mixed-integer nonlinear problem (MINLP). For the numerical solution of the MINLP, we algorithmically extend a well-known relaxation approach that has already been used very successfully in the field of stationary gas network optimization. This method allows us to solve the problem to global optimality by iteratively solving a series of mixed-integer problems (MIPs). Finally, we present two case studies that illustrate the applicability of our approach.Robert Burlacu; Herbert Egger; Martin Groß; Alexander Martin; Marc Pfetsch; Lars Schewe; Mathias Sirvent; Martin Skutellapreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/221Tue, 21 Nov 2017 15:12:15 +0100Nonconvex Equilibrium Models for Gas Market Analysis: Failure of Standard Techniques and Alternative Modeling Approaches
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/220
This paper provides a first approach to assess gas market interaction on a network with nonconvex flow models. In the simplest possible setup that adequately reflects gas transport and market interaction, we elaborate on the relation of the solution of a simultaneous competitive gas market game, its corresponding mixed nonlinear complementarity problem (MNCP), and a first-best benchmark. We provide conditions under which the solution of the simultaneous game is also the solution of the corresponding MNCP. However, equilibria cannot be determined by the MNCP as the transmission system operator's (TSO’s) first-order conditions are insufficient, which goes back to nonconvexities of the gas flow model. This also implies that the welfare maximization problem may have multiple solutions that sometimes do not even coincide with any of the market equilibria. Our analysis shows that, even in the absence of strategic firms, market interaction fails to implement desirable outcomes from a welfare perspective due to the TSO’s incentive structure. We conclude that the technical environment calls for a market design that commits the TSO to a welfare objective through regulation and propose a design where the market solution corresponds to a welfare maximum and vice versa.Veronika Grimm; Julia Grübel; Lars Schewe; Martin Schmidt; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/220Tue, 21 Nov 2017 13:42:30 +0100Optimal Boundary Control of Hyperbolic Balance Laws with State Constraints
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/219
In this paper we analyze the optimal control of initial-boundary value problems
for entropy solutions of scalar hyperbolic balance laws with pointwise state constraints. Hereby,
we suppose that the initial and the boundary data switch between different C¹-functions at certain
switching points, where the C¹ -functions and the switching points are considered as the control. For
a class of cost functionals, we prove first order necessary optimality conditions for the corresponding
optimal control problem with state contraints. Furthermore, we use a Moreau-Yosida type regularization to approximate the optimal control problem with state constraints. We derive optimality
conditions for the regularized problems and finally prove convergence to the solution of the optimal
control problem with state constraints.Johann Michael Schmitt; Stefan Ulbrichpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/219Tue, 21 Nov 2017 13:19:55 +0100