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Mon, 07 Aug 2017 11:55:45 +0200Mon, 07 Aug 2017 11:55:45 +0200General Bounds for Incremental Maximization
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/156
We propose a theoretical framework to capture incremental s
olutions to cardinality con-
strained maximization problems. The defining characterist
ic of our framework is that the
cardinality/support of the solution is bounded by a value
k
∈
N
that grows over time, and
we allow the solution to be extended one element at a time. We i
nvestigate the best-possible
competitive ratio of such an incremental solution, i.e., th
e worst ratio over all
k
between the
incremental solution after
k
steps and an optimum solution of cardinality
k
. We define a
large class of problems that contains many important cardin
ality constrained maximization
problems like maximum matching, knapsack, and packing/cov
ering problems. We provide a
general 2
.
618-competitive incremental algorithm for this class of pr
oblems, and show that no
algorithm can have competitive ratio below 2
.
18 in general.
In the second part of the paper, we focus on the inherently inc
remental greedy algorithm
that increases the objective value as much as possible in eac
h step. This algorithm is known
to be 1
.
58-competitive for submodular objective functions, but it
has unbounded competitive
ratio for the class of incremental problems mentioned above
. We define a relaxed submod-
ularity condition for the objective function, capturing pr
oblems like maximum (weighted)
(
b
-)matching and a variant of the maximum flow problem. We show t
hat the greedy algo-
rithm has competitive ratio (exactly) 2
.
313 for the class of problems that satisfy this relaxed
submodularity condition.
Note that our upper bounds on the competitive ratios transla
te to approximation ratios
for the underlying cardinality constrained problems.Aaron Bernstein; Yann Disser; Martin Großarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/156Mon, 07 Aug 2017 11:55:45 +0200A Local-Search Algorithm for Steiner Forest
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/155
In the
Steiner Forest
problem, we are given a graph and a collection of source-sink
pairs, and the
goal is to find a subgraph of minimum total length such that all
pairs are connected. The problem is
APX-Hard and can be
2
-approximated by, e.g., the elegant primal-dual algorithm
of Agrawal, Klein, and
Ravi from 1995.
We give a local-search-based constant-factor approximati
on for the problem. Local search brings in
new techniques to an area that has for long not seen any improv
ements and might be a step towards
a combinatorial algorithm for the more general survivable n
etwork design problem. Moreover, local
search was an essential tool to tackle the dynamic MST/Stein
er Tree problem, whereas dynamic Steiner
Forest is still wide open.
It is easy to see that any constant factor local search algori
thm requires steps that add/drop many edges
together. We propose natural local moves which, at each step
, either (a) add a shortest path in the current
graph and then drop a bunch of inessential edges, or (b) add a s
et of edges to the current solution. This
second type of moves is motivated by the potential function w
e use to measure progress, combining the
cost of the solution with a penalty for each connected compon
ent. Our carefully-chosen local moves and
potential function work in tandem to eliminate bad local min
ima that arise when using more traditional
local moves.
Our analysis first considers the case where the local optimum
is a single tree, and shows optimality w.r.t.
moves that add a single edge (and drop a set of edges) is enough
to bound the locality gap. For the
general case, we show how to “project” the optimal solution o
nto the different trees of the local optimum
without incurring too much cost (and this argument uses opti
mality w.r.t. both kinds of moves), followed
by a tree-by-tree argument. We hope both the potential funct
ion, and our analysis techniques will be
useful to develop and analyze local-search algorithms in ot
her contexts.Martin Groß; Anupam Gupta; Amit Kumar; Jannik Matuschke; Daniel R. Schmidt; Melanie Schmidt; José Verschaepreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/155Mon, 07 Aug 2017 11:55:44 +0200Scheduling Maintenance Jobs in Networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/154
We investigate the problem of scheduling the maintenance
of edges in a network, motivated by the goal of minimizing outages in
transportation or telecommunication networks. We focus on maintaining
connectivity between two nodes over time; for the special case of path
networks, this is related to the problem of minimizing the busy time of
machines.
We show that the problem can be solved in polynomial time in arbitrary
networks if preemption is allowed. If preemption is restricted to integral
time points, the problem is NP-hard and in the non-preemptive case
we give strong non-approximability results. Furthermore, we give tight
bounds on the power of preemption, that is, the maximum ratio of the
values of non-preemptive and preemptive optimal solutions.
Interestingly, the preemptive and the non-preemptive problem can be
solved efficiently on paths, whereas we show that mixing both leads to a
weakly NP-hard problem that allows for a simple 2-approximation.Fidaa Abed; Lin Chen; Yann Disser; Martin Groß; Nicole Megow; Julie Meißner; Alexander T. Richter; Roman Rischkearticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/154Mon, 07 Aug 2017 11:55:43 +0200Algorithmic Results for Potential-Based Flows: Easy and Hard Cases
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/153
Potential-based flows are an extension of classical network flows in which the flow on an arc is determined by the difference of the potentials of its incident nodes. Such flows are unique and arise, for example, in energy networks. Two important algorithmic problems are to determine whether there exists a feasible flow and to maximize the flow between two designated nodes. We show that these problems can be solved for the single source and sink case by reducing the network to a single arc. However, if we additionally consider switches that allow to force the flow to 0 and decouple the potentials, these problems are NP-hard. Nevertheless, for particular series-parallel networks, one can use algorithms for the subset sum problem. Moreover, applying network presolving based on generalized series-parallel structures allows to significantly reduce the size of realistic energy networks.Martin Groß; Pfetsch Marc E.; Lars Schewe; Martin Schmidt; Martin Skutellapreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/153Fri, 04 Aug 2017 10:59:54 +0200Total variation diminishing schemes in optimal control of scalar conservation laws
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/152
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.Soheil Hajian; Michael Hintermüller; Stefan Ulbrichpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/152Fri, 28 Jul 2017 16:35:44 +0200Solving Mixed-Integer Nonlinear Programs using Adaptively Refined Mixed-Integer Linear Programs
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/151
We propose a method for solving mixed-integer nonlinear programs (MINLPs) to global optimality by discretization of occuring nonlinearities. The main idea is based on using piecewise linear functions to construct mixed-integer linear program (MIP) relaxations of the underlying MINLP. In order to find a global optimum of the given MINLP we develope an iterative algorithm which solves MIP relaxations that are adaptively refined. We are able to give convergence results for a wide range of MINLPs requiring only continuous nonlinearities with bounded domains and an oracle computing maxima of the nonlinearities on their domain. Moreover, the practicalness of our approach is shown numerically by an application from the field of gas network optimization.Robert Burlacu; Björn Geißler; Lars Schewepreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/151Fri, 21 Jul 2017 17:24:37 +0200Regular solutions of DAE hybrid systems and regularization techniques
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/149
The solvability and regularity of hybrid differential-algebraic systems (DAEs) is
studied, and classical stability estimates are extended to hybrid DAE systems. Different reasons
for non-regularity are discussed and appropriate regularization techniques are presented. This includes a generalization of Filippov regularization in the case of so-called chattering. The results are illustrated by several numerical examples.
Peter Kunkel; Volker Mehrmannpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/149Thu, 20 Jul 2017 11:08:50 +0200Port-Hamiltonian descriptor systems
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/150
The modeling framework of port-Hamiltonian systems is systematically extended to
constrained dynamical systems (descriptor systems, differential-algebraic equations). A new algebraically and geometrically defined system structure is derived. It is shown that this structure is invariant under equivalence transformations, and that it is adequate also for the modeling of high-index descriptor systems. The regularization procedure for
descriptor systems to make them suitable for simulation and control is modified to deal with the port-Hamiltonian structure. The relevance of the new structure is demonstrated with several examples.
Christopher Beattie; Volker Mehrmann; Hongguo Xu; Hans Zwartpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/150Thu, 20 Jul 2017 11:08:50 +0200On structure preserving model reduction for damped wave propagation in transport networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/148
We consider the discretization and subsequent model reduction of a system of partial differential-algebraic equations describing the propagation of pressure waves in a pipeline network. Important properties like conservation of mass, dissipation of energy, passivity, existence of steady states, and exponential stability can be preserved by an appropriate semi-
discretization in space via a mixed finite element method and also during the further dimension reduction by structure preserving Galerkin projection which is the main focus of this paper. Krylov subspace methods are employed for the construction of the reduced models and we discuss modifications needed to satisfy certain algebraic compatibility conditions; these are required to ensure the well-posedness of the reduced models and the preservation of the key properties. Our analysis is based on the underlying infinite dimensional problem and its Galerkin approximations. The proposed algorithms therefore have a direct interpretation in function spaces; in principle, they are even applicable directly to the original system of partial differential-algebraic
equations while the intermediate discretization by finite elements is only required for the actual
computations. The performance of the proposed methods is illustrated with numerical tests and the necessity for the compatibility conditions is demonstrated by examples.
Herbert Egger; Thomas Kugler; Björn Liljegren-Sailer; Nicole Marheineke; Volker Mehrmannpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/148Thu, 20 Jul 2017 11:08:49 +0200Model reduction for systems with inhomogeneous initial conditions
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/147
We consider the model reduction problem for linear time-invariant dynamical systems having nonzero
(but otherwise indeterminate) initial conditions. Building upon the observation that the full system response is decomposable as a superposition of the response map for an unforced system having nontrivial initial conditions and the response map for a forced system having null initial conditions, we develop a new approach that involves reducing these component responses independently and then combining the reduced responses into an aggregate reduced system response. This approach allows greater flexibility and offers better approximation properties than other comparable methods.
Christopher Beattie; Serkan Gugercin; Volker Mehrmannarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/147Thu, 20 Jul 2017 11:08:48 +0200Stability radii for real linear Hamiltonian systems with perturbed dissipation
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/146
We study linear dissipative Hamiltonian (DH) systems with real constant coefficients that arise in energy based modeling of dynamical systems. We analyze when such a system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much the dissipation term has to be perturbed to be on this boundary. For unstructured systems the explicit construction of
the real distance to instability (real stability radius) has been a challenging problem. We analyze this real distance under different structured perturbations to the dissipation term that preserve the DH structure and we derive explicit formulas for this distance in terms of low rank perturbations. We also show (via numerical examples) that under real
structured perturbations to the dissipation the asymptotical stability of a DH system is much more robust than for unstructured perturbations.
Christian Mehl; Volker Mehrmann; Punit Sharmaarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/146Thu, 20 Jul 2017 11:08:47 +0200A Decomposition Method for MINLPs with Lipschitz Continuous Nonlinearities
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/145
Many mixed-integer optimization problems are constrained by nonlinear functions that do not possess desirable analytical properties like convexity or factorability or cannot even be evaluated exactly. This is, e.g., the case for problems constrained by differential equations or for models that rely on black-box simulation runs. For these problem classes, we present, analyze, and test algorithms that solve mixed-integer problems with only Lipschitz continuous nonlinearities. Our theoretical results depend on the assumptions made on the (in)exactness of function evaluations and on the knowledge of Lipschitz constants. If Lipschitz constants are known, we prove finite termination at approximate globally optimal points both for the case of exact and inexact function evaluations. If only approximate Lipschitz constants are known, we prove finite termination and derive additional conditions under which infeasibility can be detected. A computational study for gas transport problems and an academic case study show the applicability of our algorithms to real-world problems and how different assumptions on the constraint functions up- or downgrade the practical performance of the methods.Martin Schmidt; Mathias Sirvent; Winnifried Wollnerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/145Mon, 17 Jul 2017 15:50:49 +0200On the existence, uniqueness and exact controllability of Lipschitz solutions of initial boundary value problems for gas networks with nonconstant compressibility factor
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/144
The flow of gas through networks of pipes
can be modeled by the isothermal Euler equations
and algebraic node conditions
that model the
flow through the vertices of
the network graph.
We prove the well-posedness of the system
for
gas with
nonconstant compressibility factor
that is given by an affine linear function.
We consider initial data and control functions
that are Lipschitz continuous
and compatible with the node and boundary conditions.
We show the existence of
semi--global Lipschitz continuous solutions of the initial boundary value problem.
The construction of the solution is based upon a
fixed point iteration along the characteristic curves.
This allows us to study the exact controllability of the system
for tree--shaped networks.
For a steering time that is sufficiently large
and stationary states that are sufficiently small
the system is locally exactly controllable
by boundary controls at all boundary nodes.
The solutions of the intial boundary value problem
on arbitrary networks satisfy a maximum principle in terms of the Riemann invariants
that states that the maximum of the absolute values is attained for the initial
or the boundary data.Martin Gugatpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/144Mon, 03 Jul 2017 15:01:43 +0200On M-stationarity conditions in MPECs and the associated qualification conditions
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/142
Depending on whether a mathematical program with equilibrium constraints
(MPEC) is considered in its original or its enhanced (via KKT conditions) form, the assumed qualification
conditions as well as the derived necessary optimality conditions may differ significantly. In this paper, we
study this issue when imposing one of the weakest possible qualification conditions, namely the calmness of
the perturbation mapping associated with the respective generalized equations in both forms of the MPEC.
It is well known that the calmness property allows one to derive the so-called M-stationarity conditions. The
restrictiveness of assumptions and the strength of conclusions in the two forms of the MPEC is also strongly
related to the qualification conditions on the “lower level”. For instance, even under the Linear Independence
Constraint Qualification (LICQ) for a lower level feasible set described by C 1 functions, the calmness properties
of the original and the enhanced perturbation mapping are drastically different. When passing to C 1,1 data, this
difference still remains true under the weaker Mangasarian-Fromovitz Constraint Qualification, whereas under
LICQ both the calmness assumption and the derived optimality conditions are fully equivalent for the original
and the enhanced form of the MPEC. After clarifying these relations, we provide a compilation of practically
relevant consequences of our analysis in the derivation of necessary optimality conditions. The obtained results
are finally applied to MPECs with structured equilibria.Lukas Adam; Rene Henrion; Jiri Outrataarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/142Mon, 03 Jul 2017 15:01:42 +0200A joint model of probabilistic/robust constraints for gas transport management in stationary networks
https://opus4.kobv.de/opus4-trr154/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 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.Tatiana Gonzalez Grandon; Holger Heitsch; Rene Henrionarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/141Wed, 17 May 2017 17:15:34 +0200MIP-Based Instantaneous Control of Mixed-Integer PDE-Constrained Gas Transport Problems
https://opus4.kobv.de/opus4-trr154/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 mixed-integer 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 well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose 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 Wintergerstpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/140Tue, 11 Apr 2017 18:10:58 +0200Two-Stage Stochastic Semidefinite Programming: Theory, Algorithms, and Application to AC Power Flow under Uncertainty
https://opus4.kobv.de/opus4-trr154/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 two-stage models, who admit a recourse structure: The first-stage decisions are taken before the random event occurs. After its outcome, a recourse (second-stage) 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 two-stage stochastic semidefinite programs with continuous and mixed-integer 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 large-scale semidefinite programs with discrete variables on top. As even the SDP relaxation of these programs is too large for being handled in an all-at-once 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 in-feed from renewables serving as sources of uncertainty, two-stage stochastic programs are set up heading for unit commitment schedules which are both cost-effective 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 two-stage programs, we propose to approximate AC power flow by semidefinite relaxations. This leads to two-stage stochastic mixed-integer semidefinite programs having a special structure. To solve the latter, the L-shaped method and dual decomposition have been applied and compared.
Tobias Wollenbergdoctoralthesishttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/139Tue, 04 Apr 2017 11:40:13 +0200Unit commitment under uncertainty in AC transmission systems via risk averse semidefinite stochastic programs
https://opus4.kobv.de/opus4-trr154/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 unit-commitment 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 two-stage stochastic mixed integer semidefinite programs for which a decomposition algorithm is presented. Rüdiger Schultz; Tobias Wollenbergarticlehttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/138Tue, 04 Apr 2017 11:40:12 +0200Topology motivated discretization of hyperbolic PDAEs describing flow networks
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/137
Christoph Huck; Caren Tischendorfpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/137Mon, 03 Apr 2017 12:38:56 +0200Deciding Robust Feasibility and Infeasibility Using a Set Containment Approach: An Application to Stationary Passive Gas Network Operations
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/136
In this paper we study feasibility and infeasibility of nonlinear two-stage
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 Verapreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/136Sun, 19 Mar 2017 17:47:15 +0100Probability of Feasible Loads in Passive Gas Networks with up to Three Cycles
https://opus4.kobv.de/opus4-trr154/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 steady-state 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 Schultzpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/135Fri, 17 Mar 2017 16:31:01 +0100A Multilevel Model of the European Entry-Exit Gas Market
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/134
In entry-exit 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 entry-exit design and to what extent feasible adjustments of the market design could alleviate inefficiencies. In this paper, we offer a four-level 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 mixed-integer nonlinear bilevel problem with robust aspects. In addition, we provide a first-best benchmark that allows to assess welfare losses that occur in an entry-exit 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 intra-day trading.
Veronika Grimm; Lars Schewe; Martin Schmidt; Gregor Zöttlworkingpaperhttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/134Fri, 17 Mar 2017 12:18:42 +0100A System to Evaluate Gas Network Capacities: Concepts and Implementation
https://opus4.kobv.de/opus4-trr154/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 so-called 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 Schweigerpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/129Wed, 08 Mar 2017 15:21:39 +0100Adaptive Refinement Strategies for the Simulation of Gas Flow in Networks using a Model Hierarchy
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/127
A model hierarchy that is based on the one-dimensional 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 greedy-like 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 Mehrmannpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/127Wed, 01 Feb 2017 13:10:27 +0100Optimal Price Zones of Electricity Markets: A Mixed-Integer Multilevel Model and Global Solution Approaches
https://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/126
Mathematical modeling of market design issues in liberalized electricity markets often leads to mixed-integer nonlinear multilevel optimization problems for which no general-purpose 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 welfare-optimal outcomes. This problem leads to a challenging multilevel model that contains a graph-partitioning problem with multi-commodity flow connectivity constraints and nonlinearities due to proper economic modeling. Furthermore, it has highly symmetric solutions. We develop different problem-tailored 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 problem-specific generalized Benders approach performs considerably better than the methods based on KKT transformation.Veronika Grimm; Thomas Kleinert; Frauke Liers; Martin Schmidt; Gregor Zöttlpreprinthttps://opus4.kobv.de/opus4-trr154/frontdoor/index/index/docId/126Fri, 20 Jan 2017 19:57:15 +0100