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Keywords
- Gas networks (4)
- Mixed-Integer Nonlinear Programming (4)
- Networks (4)
- Mixed-integer optimization (3)
- Pricing (3)
- Uniqueness (3)
- Bilevel optimization (2)
- Bookings (2)
- Combinatorial optimization (2)
- Convergence (2)
In this work, we present an exact approach for solving network design problems that is based on an iterative graph aggregation procedure. The scheme allows existing preinstalled capacities. Starting with an initial aggregation, we solve a sequence of network design master problems over increasingly fine-grained representations of the original network. In each step, a subproblem is solved that either proves optimality of the solution or gives a directive where to refine the representation of the network in the subsequent iteration. The algorithm terminates with a globally optimal solution to the original problem. Our implementation uses a standard integer programming solver for solving the master problems as well as the subproblems. The computational results on random and realistic instances confirm the profitable use of the iterative aggregation technique. The computing time often reduces drastically when our method is compared to solving the original problem from scratch.
Im Mai 2014 wurde seitens der DFG der Transregio
154 Mathematische Modellierung, Simulation und
Optimierung am Beispiel von Gasnetzwerken bewilligt.
Die Forschungsarbeiten an den beteiligten Standorten,
der Friedrich-Alexander-Universität Erlangen-Nürnberg
(Sprecheruniversität; Sprecher: Alexander Martin), der
Technischen Universität Darmstadt (stellvertretender
Sprecher: Jens Lang), der Technischen Universität Berlin,
der Humboldt Universität (stellvertretende Sprecherin:
Caren Tischendorf) sowie den Partnerinstitutionen
Weierstraß-Institut (Berlin), Konrad-Zuse-Zentrum (Berlin)
und Universität Duisburg-Essen haben im Oktober
2014 begonnen.
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, well-posedness, controllability, and optimal control. A major challenge concerning the optimization is to deal with switching on-off states that are inherent to controllable devices in such applications combined with
continuous simulation and optimization of the gas flow. We formulate the corresponding mixed-integer nonlinear optimal control problems and outline a decomposition approach as a solution technique.
Nonconvex mixed-binary 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 mixed-binary nonlinear problem. Since we rely on local nonlinear programming solvers the resulting method is fast and we further improve its reliability 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.
In this work we study polyhedra in the context of network flow problems, where the flow value on each arc lies in one of several predefined intervals. This is motivated by nonlinear
problems on transportation networks, where nonlinearities are handled by piecewise linear approximation or relaxation - a common and established approach in many applications.
Several methods for modeling piecewise linear functions are known which provide a complete description for a single network arc. However, in general this property is lost when considering multiple arcs. We show how to strengthen the formulation for specific substructures consisting of multiple arcs by linear inequalities. For the case of paths of degree-two-nodes we give a complete description of the polyhedron projected to the integer variables. Our model is based on - but not limited to - the multiple choice method; we also show how to transfer our results to a formulation based on the incremental method. Computational results show that a state-of-the-art MIP-solver greatly benefits from using our cutting planes for random and realistic network topologies.
We consider the clique problem with multiple-choice constraints (CPMC) and characterize a case where it is possible to give an efficient description of the convex hull of its feasible solutions. This case, which we call staircase compatibility, generalizes common properties in applications and allows for a linear description of the integer feasible solutions to (CPMC) with a totally unimodular constraint matrix
of polynomial size. We derive two such totally unimodular reformulations for the problem: one that is obtained by a strengthening of the compatibility constraints
and one that is based on a representation as a dual network flow problem. Furthermore, we show a natural way to derive integral solutions from fractional solutions to the problem by determining integral extreme points generating this fractional
solution. We also evaluate our reformulations from a computational point of view by applying them to two different real-world applications. The first one is
a problem in railway timetabling where we try to adapt a given timetable slightly such that energy costs from operating the trains are reduced. The second one is the piecewise linearization of non-linear flow problems on a gas network. In both cases, we are able to reduce the solution times significantly by passing to the theoretically stronger formulations of the problem.
Detailed modeling of gas transport problems leads to nonlinear
and nonconvex mixed-integer optimization or feasibility models
(MINLPs) because both the incorporation of discrete controls of the
network as well as accurate physical and technical modeling is
required in order to achieve practical solutions. Hence, ignoring
certain parts of the physics model is not valid for practice. In the
present contribution we extend an approach based on linear relaxations
of the underlying nonlinearities by tailored model reformulation
techniques yielding block-separable MINLPs. This combination of
techniques allows us to apply a penalty alternating direction method
and thus to solve highly detailed MINLPs for large-scale real-world
instances. The practical strength of the proposed method is
demonstrated by a computational study in which we apply the method to
instances from steady-state gas transport
including both pooling effects with respect to the mixing of gases of
different composition and a highly detailed compressor station model.
We propose an equilibrium model that allows to analyze the long-run impact of the electricity market design on transmission line expansion by the regulator and investment in generation capacity by private firms in liberalized electricity markets. The model incorporates investment decisions of the transmission system operator and private firms in expectation of an energy-only market and cost-based redispatch. In different specifications we consider the cases of one vs. multiple price zones (market splitting) and analyze different approaches to recover network cost—in particular lump sum, generation capacity based, and energy based fees. In order to compare the outcomes of our multilevel market model with a first best benchmark, we also solve the corresponding integrated planner problem. Using two test networks we illustrate that energy-only markets can lead to suboptimal locational decisions for generation capacity and thus imply excessive network expansion. Market splitting heals these problems only partially. These results are valid for all considered types of network tariffs, although investment slightly differs across those regimes.
We present a solution algorithm for problems from
steady-state gas transport optimization.
Due to nonlinear and nonconvex physics and engineering models as
well as discrete controllability of active network devices, these
problems lead to difficult nonconvex mixed-integer nonlinear optimization
models.
The proposed method is based on mixed-integer linear techniques using
piecewise linear relaxations of the nonlinearities and a tailored
alternating direction method.
Most other publications in the field of gas transport optimization only consider
pressure and flow as main physical quantities. In this work, we additionally
incorporate heat power supplies and demands as well as a mixing model for
different gas qualities.
We demonstrate the capabilities of our method on Germany's largest
transport networks and hereby present numerical results on the largest
instances that were ever reported in the literature for this problem
class.