Search

The Hydro Unit Commitment problem (HUC) specific to hydroelectric units is part of the electricity production planning problem, called Unit Commitment Problem (UCP). More specifically, the studied case is that of the HUC with a single unit, denoted 1-HUC. The unit is located between two reservoirs. The horizon is discretized in time periods. The unit operates at a finite number of points defined as pairs of the generated power and the corresponding water flow. Several constraints are considered. Each reservoir has an initial volume, as well as window resource constraints, defined by a minimum and maximum volume per time period. At each time period, there is an additional positive, negative or zero intake of water in the reservoirs. The case of a price-taker revenue maximization problem is considered. An efficient exact A* variant, so called HA*, is proposed to solve the 1-HUC accouting for window constraints, with a reduced search space and a dedicated optimistic heuristic. This variant is compared to a classical Resource Constrained Shortest Path Problem (RCSPP) algorithm and an MILP formulation solved with CPLEX. Results show that the proposed algorithm outperforms both concurrent alternatives in terms of computational time in average on a set of realistic instances, meaning that HA* exhibits a more stable behavior with more instances solved.

In this paper, we define a new variant of the knapsack problem, the Symmetric Weight Matrix Knapsack Problem (SMKP). The (SMKP) is the core structure of the Hydro Unit Commitment being a production scheduling problem relative to hydroelectric units. The (SMKP) is proven to be NP-hard. The main contribution is a polyedral study of the (SMKP) and a dedicated two-phase Branch & Cut scheme. The polyedral study focuses on inequalities with 0-1 coefficients. Necessary facet-defining conditions are described, through a new structure called pattern encoding the symmetries of the (SMKP). A special case of patterns is identified where these conditions are necessary and sufficient. A two-phase Branch & Cut scheme is defined to exploit the symmetries on the pattern inequalities. As a pre-processing step, the first phase aims to produce patterns verifying these conditions. The second phase separates the inequalities associated to the patterns produced in the first phase at the nodes of a Branch & Cut tree. Experimental results demonstrate the efficiency of the proposed scheme in particular with respect to default CPLEX.

International audience; Les systèmes physiques réels possèdent dans certains cas des non-linéarités dont il faut tenir compte lors de la résolution de problèmes d'optimisations associés. Il existe un grand nombre de possibilité pour modéliser ces problèmes, ayant une précision, et demandant des temps de calculs différents.Le Hydro Unit Commitment fait parti des problèmes non-linéaires, faisant intervenir deux non-linéarités. Plusieurs modèles sont présentés pour modéliser le HUC, et sont comparés sur un jeu d'instance varié.Le but est d'identifier les modèles pertinants, en terme de temps de calcul, précision et faisabilité, et les charactéristiques des instances qui ont un impact sur la résolution des différents modèles proposés.

A wide range of real world optimization problems involves continuous decisions and non-linearities. Each non-linear component of such problems can be modeled either linearly or non-linearly, considering or not additional integer variables. In this paper, different modeling alternatives are proposed for a real non-linear optimization problem, the single unit hydro unit commitment problem (1-HUC). The non-linearities of the 1-HUC come from the power produced at each time period. It is defined as a two-dimensional non-convex and non-concave function of the water flow and head decision variables, the latter being itself a one-dimensional convex non-linear function of the turbined volume. A common simplification is also considered, assuming that the water head is fixed and defining consequently the produced power as a one-dimensional non concave and non convex function of the water flow. Seven non-linear and linear models are described for both the 1-HUC and the fixed-head 1-HUC. These models cover multiple families of modeling alternatives, including common models of the literature as well as new models featuring less common functions. Different sets of instances are generated to evaluate the sensitivity of performance with respect to the main characteristics of the 1-HUC. Several available solvers are used for each non-linear model and the best virtual solver is retained to focus on the model capabilities rather than on the solver performance. Based on the numerical experiments, sometimes counter-intuitive recommendations are given so as to help practitioners in selecting the most adequate model and solver depending on the characteristics of the instance.

Variantes du polytope min-up/min-down

International audience; We consider integer linear programs whose solutions are binary matrices and whose (sub-)symmetry groups are symmetric groups acting on (sub-)columns. Such structured subsymmetry groups arise in important classes of combinatorial problems, e.g. graph coloring or unit commitment. For a priori known (sub-)symmetries, we propose a framework to build (sub-)symmetry-breaking inequalities for such problems, by introducing one additional variable per considered sub-symmetry group. The derived inequalities are full-symmetry-breaking and in polynomial number w.r.t. the number of sub-symmetry groups considered. The proposed framework is applied to derive such inequalities when the symmetry group is the symmetric group acting on the columns. It is also applied to derive sub-symmetry-breaking inequalities for the graph coloring problem. Experimental results give insight into how to select the right inequality subset in order to efficiently break sub-symmetries. Finally, the framework is applied to derive (sub-)symmetry breaking inequalities for Min-up/min-down Unit Commitment Problem with or without ramp constraints. We show the effectiveness of the approach by presenting an experimental comparison with state-of-the-art symmetry-breaking formulations.

The Min-up/min-down Unit Commitment Problem (MUCP), is to find a minimum-cost production plan on a discrete time horizon for a set of units producing electric power. At each time period, the total production has to meet a forecast demand. Each unit must satisfy minimum up and down time constraints. We show that the MUCP is strongly NP-hard, thus highlighting that the dynamic coupling of demands by minimum up and down time constraints represents one major source of difficulty. To cope with this difficulty, we introduce interval up-set inequalities, a new class of valid inequalities for the MUCP polytope, generalizing both min-up and extended cover inequalities from the 0-1 knapsack polytope. Facet defining cases are characterized and a Branch & Cut algorithm is devised. To deal with the symmetries impairing the solution process, we define sub-symmetries, as symmetries arising from a solution subset. We focus on integer linear programs whose (sub-)symmetry groups are symmetric groups acting on sub-columns of solution matrices. We propose a general framework to handle sub-symmetries in such problems. On this basis, two symmetry-breaking techniques are introduced. The first technique is an orbitopal fixing algorithm for the full (sub-)orbitope. The second technique is based on sub-symmetry breaking inequalities. Experimental results on MUCP instances show that the proposed techniques outperform state-of-the-art techniques. Finally we compare various Dantzig-Wolfe structures for the MUCP. We show that good quality lower bounds can be obtained by dualization of time-coupling constraints. Branch & Price results show that interval up-set inequalities are useful in this context.; Le Min-up/min-down Unit Commitment Problem (MUCP) consiste à trouver un plan de production de coût minimum pour un ensemble d’unités de production électrique sur un intervalle de temps discrétisé. A chaque pas de temps, la production totale doit satisfaire la demande prévue. Chaque unité respecte des temps minimum de marche et d’arrêt. Nous montrons que le MUCP est fortement NP-difficile, mettant ainsi en valeur l’impact du couplage dynamique des contraintes de demande sur la difficulté du problème. Pour appréhender cette difficulté, nous introduisons les inégalités interval up-set, généralisant les contraintes de min-up et les extended cover du sac à dos. Les facettes sont caractérisées, et un Branch & Cut est implémenté. Afin de briser les symétries du problème, nous définissons les sous-symétries comme des symétries apparaissant dans des sous-ensembles de solution. Nous considérons des PLNE dont les groupes de sous-symétrie sont des groupes symétriques agissant sur certaines sous-colonnes des matrices solutions. Nous proposons un cadre générique pour gérer les sous-symétries apparaissant dans ce type de problèmes. Deux techniques pour briser les sous-symétries sont proposées : la première est un algorithme de fixing orbitopal pour le full sub-orbitope, la seconde est basée sur des inégalités. Les résultats expérimentaux montrent que les techniques proposées sont plus performantes que les techniques de la littérature. Enfin, nous comparons différentes structures de décomposition pour le MUCP. Des bornes de bonne qualité sont obtenues par dualisation des contraintes dynamiques. Notre Branch&Price&Cut montre que les interval up-set sont utiles dans ce contexte.

The Min-up/min-down Unit Commitment Problem (MUCP), is to find a minimum-cost production plan on a discrete time horizon for a set of units producing electric power. At each time period, the total production has to meet a forecast demand. Each unit must satisfy minimum up and down time constraints. We show that the MUCP is strongly NP-hard, thus highlighting that the dynamic coupling of demands by minimum up and down time constraints represents one major source of difficulty. To cope with this difficulty, we introduce interval up-set inequalities, a new class of valid inequalities for the MUCP polytope, generalizing both min-up and extended cover inequalities from the 0-1 knapsack polytope. Facet defining cases are characterized and a Branch & Cut algorithm is devised. To deal with the symmetries impairing the solution process, we define sub-symmetries, as symmetries arising from a solution subset. We focus on integer linear programs whose (sub-)symmetry groups are symmetric groups acting on sub-columns of solution matrices. We propose a general framework to handle sub-symmetries in such problems. On this basis, two symmetry-breaking techniques are introduced. The first technique is an orbitopal fixing algorithm for the full (sub-)orbitope. The second technique is based on sub-symmetry breaking inequalities. Experimental results on MUCP instances show that the proposed techniques outperform state-of-the-art techniques. Finally we compare various Dantzig-Wolfe structures for the MUCP. We show that good quality lower bounds can be obtained by dualization of time-coupling constraints. Branch & Price results show that interval up-set inequalities are useful in this context.; Le Min-up/min-down Unit Commitment Problem (MUCP) consiste à trouver un plan de production de coût minimum pour un ensemble d’unités de production électrique sur un intervalle de temps discrétisé. A chaque pas de temps, la production totale doit satisfaire la demande prévue. Chaque unité respecte des temps minimum de marche et d’arrêt. Nous montrons que le MUCP est fortement NP-difficile, mettant ainsi en valeur l’impact du couplage dynamique des contraintes de demande sur la difficulté du problème. Pour appréhender cette difficulté, nous introduisons les inégalités interval up-set, généralisant les contraintes de min-up et les extended cover du sac à dos. Les facettes sont caractérisées, et un Branch & Cut est implémenté. Afin de briser les symétries du problème, nous définissons les sous-symétries comme des symétries apparaissant dans des sous-ensembles de solution. Nous considérons des PLNE dont les groupes de sous-symétrie sont des groupes symétriques agissant sur certaines sous-colonnes des matrices solutions. Nous proposons un cadre générique pour gérer les sous-symétries apparaissant dans ce type de problèmes. Deux techniques pour briser les sous-symétries sont proposées : la première est un algorithme de fixing orbitopal pour le full sub-orbitope, la seconde est basée sur des inégalités. Les résultats expérimentaux montrent que les techniques proposées sont plus performantes que les techniques de la littérature. Enfin, nous comparons différentes structures de décomposition pour le MUCP. Des bornes de bonne qualité sont obtenues par dualisation des contraintes dynamiques. Notre Branch&Price&Cut montre que les interval up-set sont utiles dans ce contexte.

Consider a time horizon and a set of possible states for a given system. The system must be in exactly one state at a time. In this paper, we generalize classical results on min-up/min-down constraints for a 2-state system to an n-state system. The minimum-time constraints enforce that if the system switches to state i at time t, then it must remain in state i for a minimum number of time steps. The minimum-time polytope is defined as the convex hull of integer solutions satisfying the minimum-time constraints. A variant of minimum-time constraints is also considered, namely the no-spike constraints. They enforce that if state i is switched on at time t, the system must remain on states j ≥ i during a minimum time. Symmetrically, they enforce that if state i is switched off at time t, the system must remain on states j < i during a minimum time. The no-spike polytope is defined as the convex hull of integer solutions satisfying the no-spike constraints. For both the minimum-time polytope and the no-spike polytope, we introduce families of valid inequalities. We prove that these inequalities lead to a complete description of linear size for each polytope.