24 research outputs found
Computation of Convex Hull Prices in Electricity Markets with Non-Convexities using Dantzig-Wolfe Decomposition
The presence of non-convexities in electricity markets has been an active
research area for about two decades. The -- inevitable under current marginal
cost pricing -- problem of guaranteeing that no market participant incurs
losses in the day-ahead market is addressed in current practice through
make-whole payments a.k.a. uplift. Alternative pricing rules have been studied
to deal with this problem. Among them, Convex Hull (CH) prices associated with
minimum uplift have attracted significant attention. Several US Independent
System Operators (ISOs) have considered CH prices but resorted to
approximations, mainly because determining exact CH prices is computationally
challenging, while providing little intuition about the price formation
rationale. In this paper, we describe the CH price estimation problem by
relying on Dantzig-Wolfe decomposition and Column Generation, as a tractable,
highly paralellizable, and exact method -- i.e., yielding exact, not
approximate, CH prices -- with guaranteed finite convergence. Moreover, the
approach provides intuition on the underlying price formation rationale. A test
bed of stylized examples provide an exposition of the intuition in the CH price
formation. In addition, a realistic ISO dataset is used to support scalability
and validate the proof-of-concept.Comment: 11 page
Computation of Convex Hull prices in electricity markets with non-convexities using Dantzig-Wolfe decomposition
The presence of non-convexities in electricity markets has been an active research area for about two decades. The — inevitable under current marginal cost pricing — problem
of guaranteeing that no market participant incurs losses in the day-ahead market is addressed in current practice through
make-whole payments a.k.a. uplift. Alternative pricing rules have been studied to deal with this problem. Among them, Convex Hull (CH) prices associated with minimum uplift have attracted
significant attention. Several US Independent System Operators (ISOs) have considered CH prices but resorted to approximations,
mainly because determining exact CH prices is computationally challenging, while providing little intuition about the price formation rationale. In this paper, we describe the CH price estimation problem by relying on Dantzig-Wolfe decomposition and Column Generation, as a tractable, highly paralellizable, and
exact method — i.e., yielding exact, not approximate, CH prices — with guaranteed finite convergence. Moreover, the approach
provides intuition on the underlying price formation rationale. A test bed of stylized examples provide an exposition of the intuition
in the CH price formation. In addition, a realistic ISO dataset is used to support scalability and validate the proof-of-concept.Accepted manuscrip
An exact algorithm for the Partition Coloring Problem
We study the Partition Coloring Problem (PCP), a generalization of the Vertex Coloring Problem where the vertex set is partitioned. The PCP asks to select one vertex for each subset of the partition in such a way that the chromatic number of the induced graph is minimum. We propose a new Integer Linear Programming formulation with an exponential number of variables. To solve this formulation to optimality, we design an effective Branch-and-Price algorithm. Good quality initial solutions are computed via a new metaheuristic algorithm based on adaptive large neighborhood search. Extensive computational experiments on a benchmark test of instances from the literature show that our Branch-and-Price algorithm, combined with the new metaheuristic algorithm, is able to solve for the first time to proven optimality several open instances, and compares favorably with the current state-of-the-art exact algorithm
Lagrangian-informed mixed integer programming reformulations
La programmation linéaire en nombres entiers est une approche robuste
qui permet de résoudre rapidement de grandes instances de problèmes
d'optimisation discrète. Toutefois, les problèmes gagnent constamment
en complexité et imposent parfois de fortes limites sur le temps de
calcul. Il devient alors nécessaire de développer des méthodes
spécialisées afin de résoudre approximativement ces problèmes, tout en
calculant des bornes sur leurs valeurs optimales afin de prouver la
qualité des solutions obtenues.
Nous proposons d'explorer une approche de reformulation en nombres
entiers guidée par la relaxation lagrangienne. Après l'identification
d'une forte relaxation lagrangienne, un processus systématique permet
d'obtenir une seconde formulation en nombres entiers. Cette
reformulation, plus compacte que celle de Dantzig et Wolfe, comporte
exactement les mêmes solutions entières que la formulation initiale,
mais en améliore la borne linéaire: elle devient égale à la borne
lagrangienne.
L'approche de reformulation permet d'unifier et de généraliser des
formulations et des méthodes de borne connues. De plus, elle offre
une manière simple d'obtenir des reformulations de moins grandes
tailles en contrepartie de bornes plus faibles.
Ces reformulations demeurent de grandes tailles. C'est pourquoi nous
décrivons aussi des méthodes spécialisées pour en résoudre les
relaxations linéaires.
Finalement, nous appliquons l'approche de reformulation Ă deux
problèmes de localisation. Cela nous mène à de nouvelles formulations
pour ces problèmes; certaines sont de très grandes tailles, mais nos
méthodes de résolution spécialisées les rendent pratiques.Integer linear programming is a robust and efficient approach to solve
large-scale instances of combinatorial problems. However, problems
constantly gain in complexity and sometimes impose strong constraints
on computation times. We must then develop specialised methods to
compute heuristic primal solutions to the problem and derive lower
bounds on the optimal value, and thus prove the quality of our primal
solutions.
We propose to guide a reformulation approach for mixed integer
programs with Lagrangian relaxations. After the identification of a
strong relaxation, a mechanical process leads to a second integer
formulation. This reformulation is equivalent to the initial one, but
its linear relaxation is equivalent to the strong Lagrangian dual.
We will show that the reformulation approach unifies and generalises
prior formulations and lower bounding approaches, and that it exposes
a simple mechanism to reduce the size of reformulations in return for
weaker bounds.
Nevertheless, our reformulations are large. We address this issue by
solving their linear relaxations with specialised methods.
Finally, we apply the reformulation approach to two location problems.
This yields novel formulations for both problems; some are very large
but, thanks to the aforementioned specialised methods, still
practical
Decomposition Methods and Network Design Problems
Decomposition based approaches are recalled from primal and dual point of view. The possibility of building partially disaggregated reduced master problems is investigated. This extends the idea of aggregated-versus-disaggregated formulation to a gradual choice of alternative level of aggregation. Partial aggregation is applied to the linear multicommodity minimum cost flow problem. The possibility of having only partially aggregated bundles opens a wide range of alternatives with different trade-offs between the number of iterations and the required computation for solving it. This trade-off is explored for several sets of instances and the results are compared with the ones obtained by directly solving the natural node-arc formulation.
An iterative solution process to the route assignment problem is proposed, based on the well-known Frank Wolfe algorithm. In order to provide a first feasible solution to the Frank Wolfe algorithm, a linear multicommodity min-cost flow problem is solved to optimality by using the decomposition techniques mentioned above. Solutions of this problem are useful for network orientation and design, especially in relation with public transportation systems as the Personal Rapid Transit.
A single-commodity robust network design problem is addressed. In this, an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. A set of new instances that are computationally hard for the natural flow formulation are solved by means of a new heuristic algorithm.
Finally, an efficient decomposition-based heuristic approach for a large scale stochastic unit commitment problem is presented. The addressed real-world stochastic problem employs at its core a deterministic unit commitment planning model developed by the California Independent System Operator (ISO)
Algorithms for Stochastic Integer Programs Using Fenchel Cutting Planes
This dissertation develops theory and methodology based on Fenchel cutting planes for solving stochastic integer programs (SIPs) with binary or general integer variables in the second-stage. The methodology is applied to auto-carrier loading problem under uncertainty. The motivation is that many applications can be modeled as SIPs, but this class of problems is hard to solve. In this dissertation, the underlying parameter distributions are assumed to be discrete so that the original problem can be formulated as a deterministic equivalent mixed-integer program. The developed methods are evaluated based on computational experiments using both real and randomly generated instances from the literature. We begin with studying a methodology using Fenchel cutting planes for SIPs with binary variables and implement an algorithm to improve runtime performance.
We then introduce the stochastic auto-carrier loading problem where we present a mathematical model for tactical decision making regarding the number and types of auto-carriers needed based on the uncertainty of availability of vehicles. This involves the auto-carrier loading problem for which actual dimensions of the vehicles, regulations on total height of the auto-carriers and maximum weight of the axles, and safety requirements are considered. The problem is modeled as a two-stage SIP, and computational experiments are performed using test instances based on real data.
Next, we develop theory and a methodology for Fenchel cutting planes for mixed integer programs with special structure. Integer programs have to be solved to generate a Fenchel cutting plane and this poses a challenge. Therefore, we propose a new methodology for constructing a reduced set of integer points so that the generation of Fenchel cutting planes is computationally favorable. We then present the computational results based on randomly generated instances from the literature and discuss the limitations of the methodology. We finally extend the methodology to SIPs with general integer variables in the second-stage with special structure, and study different normalizations for Fenchel cut generation and report their computational performance