On the complexity of the Unit Commitment Problem

Bendotti, Pascale; Fouilhoux, Pierre; Rottner, Cécile

doi:10.1007/s10479-018-2827-x

Cited by 44 publications

(37 citation statements)

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“…First note that π 132 ∈ G thus π = π 132 . Since both X 1 and X 2 have exactly one entry to 0, π must be such that π(e 2 ) = e 3 , where, for i ∈ {1, ..., 3} e i ∈ {0, 1} (1,3) is such that e i (1, i) = 1 and e i (1, j) = 0, ∀j = i. Since Y = e 2 , π(Y ) = e 3 ∈ X , which is a contradiction.…”

Section: Sub-symmetriesmentioning

confidence: 99%

Orbitopal fixing for the full (sub-)orbitope and application to the Unit Commitment Problem

2019

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This paper focuses on integer linear programs where solutions are binary matrices, and the corresponding symmetry group is the set of all column permutations. Orbitopal fixing, as introduced in [12], is a technique designed to break symmetries in the special case of partitioning (resp. packing) formulations involving matrices with exactly (resp. at most) one 1-entry in each row. The main result of this paper is to extend orbitopal fixing to the full orbitope, defined as the convex hull of binary matrices with lexicographically nonincreasing columns. We determine all the variables whose values are fixed in the intersection of an hypercube face with the full orbitope. Sub-symmetries arising in a given subset of matrices are also considered, thus leading to define the full sub-orbitope in the case of the sub-symmetric group. We propose a linear time orbitopal fixing algorithm handling both symmetries and sub-symmetries. We introduce a dynamic variant of this algorithm where the lexicographical order follows the branching decisions occurring along the B&B search. Experimental results for the Unit Commitment Problem are presented. A comparison with state-of-the-art techniques is considered to show the effectiveness of the proposed variants of the algorithm. DefinitionsThroughout the paper, we consider an Integer Linear Program (ILP) of the form min c(x) | x ∈ X , with X ⊆ P(m, n) and c : P(m, n) → Rwhere P(m, n) is the set of m × n binary matrices. A symmetry is defined as a permutation π of the columns {1, ..., n} such that for any solution matrix x ∈ X , matrix π(x) is also solution and has same cost, i.e., π(x) ∈ X and c(x) = c(π(x)). The symmetry group G of ILP (1) is the set of all * pascale.bendotti@edf.fr † pierre.fouilhoux@lip6.fr ‡ cecile.rottner@edf.fr such permutations. Symmetry group G partitions the solution set X into orbits, i.e., two matrices are in the same orbit if there exists a permutation in G sending one to the other. Symmetries arising in ILP can impair the solution process, in particular when symmetric solutions lead to an excessively large branch and bound (B&B) search tree (see survey [19]). Symmetry detection techniques are proposed in [15,3]. Various techniques, so called symmetry-breaking techniques, are available to handle symmetries in ILP of the form (1). The general idea is, in each orbit, to pick one solution, defined as the representative, and then restrict the solution set to the set of all representatives.A technique is said to be full-symmetry-breaking (resp. partial-symmetry-breaking) if the solution set is exactly (resp. partially) restricted to the representative set. Moreover, such a technique may introduce some specific branching rules that interfere with the B&B search. This can forbid exploiting a user-defined branching rule or, even, the default solver branching settings. A symmetrybreaking technique is said to be flexible if at any node of the B&B tree, the branching rule can be derived from any linear inequality on the variables. Such a technique can be based on specific...

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Section: Sub-symmetriesmentioning

confidence: 99%

Orbitopal fixing for the full (sub-)orbitope and application to the Unit Commitment Problem

2019

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“…The solution proposed consists in designing the spinning reserves of the dispatchable units in such a way that the reserve power can meet the demand estimation error for several levels of reliability according to the value of n r introduced in Equation (14) ( Figure 2 shows the reliability level for values of n r = 1, n r = 2 and n r = 3). This mode of operation is complementary to the ones described above since it includes additional constraints to the problem, not changing the cost functions in Equations (21) and (22). In this mode, more expensive solutions are expected to be obtained, given the severe constraints related to robustness that they must fulfill.…”

Section: Robust Operation Modementioning

confidence: 99%

“…Therefore, when the power given by the generator increases significantly, MT is the cheaper and lower-emissions solution. This fact makes cost the functions in Equations (21) and (22) have a minimum value in two different operation points.…”

Section: Example Scenario Settingsmentioning

confidence: 99%

“…Finally, it is worth pointing out the complexity of the combined UC and ED problem [21]. The schedule of the dispatchable generation units together with the management of the ESS, with the additional consideration of uncertainties in the demand forecasting generates a non-convex mathematical problem, which combines strong non-linearities and mixed integer formulation.…”

Section: Introductionmentioning

confidence: 99%

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An Evolutionary Computational Approach for the Problem of Unit Commitment and Economic Dispatch in Microgrids under Several Operation Modes

et al. 2019

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In the last decades, new types of generation technologies have emerged and have been gradually integrated into the existing power systems, moving their classical architectures to distributed systems. Despite the positive features associated to this paradigm, new problems arise such as coordination and uncertainty. In this framework, microgrids constitute an effective solution to deal with the coordination and operation of these distributed energy resources. This paper proposes a Genetic Algorithm (GA) to address the combined problem of Unit Commitment (UC) and Economic Dispatch (ED). With this end, a model of a microgrid is introduced together with all the control variables and physical constraints. To optimally operate the microgrid, three operation modes are introduced. The first two attend to optimize economical and environmental factors, while the last operation mode considers the errors induced by the uncertainties in the demand forecasting. Therefore, it achieves a robust design that guarantees the power supply for different confidence levels. Finally, the algorithm was applied to an example scenario to illustrate its performance. The achieved simulation results demonstrate the validity of the proposed approach.

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“…In a similar vein, [61] provide an extension to many units of the analysis initiated in [417] as min-up/min-down constraints are concerned; in particular, a set of new valid inequalities is introduced. By extension, the computational complexity is looked at in [59]; not surprisingly, UC is found to be NP-Hard even in simple situations. Further such investigations are related specifically to the consideration of start-up costs as in [88].…”

Section: Recent Trends In Milp Techniquesmentioning

confidence: 99%

Large-scale unit commitment under uncertainty: an updated literature survey

Ackooij¹,

et al. 2018

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The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably small time for its size. Recently, growing renewable energy shares have strongly increased the level of uncertainty in the system, making the (ideal) Unit Commitment model a large-scale, non-convex and uncertain (stochastic, robust, chanceconstrained) program. We provide a survey of the literature on methods for the Uncertain Unit Commitment problem, in all its variants. We start with a review of the main contributions on solution methods for the deterministic versions of the problem, focussing on those based on mathematical programming techniques that are more relevant for the uncertain versions of the problem. We then present and categorize the approaches to the latter, while providing entry points to the relevant literature on optimization under uncertainty. This is an updated version of the paper "Large-scale Unit Commitment under uncertainty: a literature survey" that appeared in 4OR 13(2), 115-171 (2015); this version has over 170 more citations, most of which appeared in the last three years, proving how fast the literature on uncertain Unit Commitment evolves, and therefore the interest in this subject.

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On the complexity of the Unit Commitment Problem

Cited by 44 publications

References 10 publications

Orbitopal fixing for the full (sub-)orbitope and application to the Unit Commitment Problem

Orbitopal fixing for the full (sub-)orbitope and application to the Unit Commitment Problem

An Evolutionary Computational Approach for the Problem of Unit Commitment and Economic Dispatch in Microgrids under Several Operation Modes

Large-scale unit commitment under uncertainty: an updated literature survey

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