Local consistency techniques have been introduced in logic programming in order to extend the application domain of logic programming languages. The existing languages based on these techniques consider arithmetic constraints applied to variables ranging over finite integer domains. This makes difficult a natural and concise modelling as well as an efficient solving of a class of A/P-complete combinatorial search problems dealing with sets. To overcome these problems, we propose a solution which consists in extending the notion of integer domains to that of set domains (sets of sets). We specify a set domain by an interval whose lower and upper bounds are known sets, ordered by set inclusion. We define the formal and practical framework of a new constraint logic programming language over set domains, called Conjunto. Conjunto comprises the usual set operation symbols (U, N, \), and the set inclusion relation (___). Set expressions built using the operation symbols are interpreted as relations (s U Sl = s2 .... ). In addition, Conjunto provides us with a set of constraints called graduated constraints (e.g. the set cardinality) which map sets onto arithmetic terms. This allows us to handle optimization problems by applying a cost function to the quantifiable, i.e., arithmetic, terms which are associated to set terms. The constraint solving in Conjunto is based on local consistency techniques using interval reasoning which are extended to handle set constraints. The main contribution of this paper concerns the formal definition of the language and its design and implementation as a practical language.

In CP literature combinatorial design problems such as sport scheduling, Steiner systems, error-correcting codes and more, are typically solved using Finite Domain (FD) models despite often being more naturally expressed as Finite Set (FS) models. Existing FS solvers have difficulty with such problems as they do not make strong use of the ubiquitous set cardinality information. We investigate a new approach to strengthen the propagation of FS constraints in a tractable way: extending the domain representation to more closely approximate the true domain of a set variable. We show how this approach allows us to reach a stronger level of consistency, compared to standard FS solvers, for arbitrary constraints as well as providing a mechanism for implementing certain symmetry breaking constraints. By experiments on Steiner Systems and error correcting codes, we demonstrate that our approach is not only an improvement over standard FS solvers but also an improvement on recently published results using FD 0/1 matrix models as well. 1 O(ncv √ nc) where n=num vars, c=cardinality and v=size of largest lub 2 See Sect. 6 for explanation of notation

Today, the overall goal of energy transition planning is to seek an optimal strategy for increasing the share of renewable sources in existing power networks, such that the growing power demand is satisfied at manageable short/long term investment. In this paper we address the problem of PV penetration in electricity networks, by considering both 1) the spatial issue of site selection and size, and 2) the temporal aspect of hourly load and demand satisfaction, in addition with the investment and maintenance costs to guarantee a viable and reliable solution. We propose to address this spatio-temporal optimization problem through an integrated GIS and robust optimization model, that allows handling of the ubiquitous dependencies between resource and demand time variability and the selection of optimal sites of renewable power generation. Our approach contributes to the integration of the multi-dimensional and combinatorial aspects of this problem, gathering geographical layers (regional or national scale) and temporal packing (hourly time stamp) constraints, and cost functions. This model computes the optimal geographical location and size of PV facilities allowing energy planning targets to be met at minimal cost in a reliable manner. In this paper, we illustrate our approach by studying the penetration of large-scale solar PV in the French Guiana's power system. Among the results, we show for instance that: 1) our approach performs geographical aggregation with real contextual data, i.e. balances the intermittency of RE sources by spreading out the corresponding installations (location + size) across the territory; 2) the total installed PV capacity can be doubled by removing the 35 % penetration limit on intermittent power without exceeding hourly demand; 3) the safest investment scenario is below 30 MW of new PV facilities (≈ 45 Me and 2 plants), though it is theoretically possible to install up to 45 MW (>120 Me and 11 plants). Nomenclature B iBoolean variable equal to 1 if site PS i is selected, 0 otherwise Ccap i Capital cost for implementation of a new PV power plant (e) Ccon i Connection cost for each new PV plant, transmission lines and substation (e) Clan Transmission line unit cost (e/m) Cop i Annual fixed operational cost per new PV plant (e) Csta Substation power unit cost (e/kW) Csta i Capital cost for new substation (e) Dem h Estimated hourly power demand (kWh) Dg i Minimum distance from the grid to the centroid of a candidate (m) Eint h Current hourly production from intermittent sources (kWh) * Corresponding author Email addresses: benjamin.pillot@ird.fr (Benjamin Pillot), nadeem.alkurdi@ird.fr (Nadeem Al-Kurdi), carmen.gervet@umontpellier.fr (Carmen Gervet), laurent.linguet@univ-guyane.fr (Laurent Linguet)

The computer will be the most marvellous of all tools as soon as program writing and debugging will be no longer necessary-Jean-Louis Laurière (1976) A wide range of combinatorial search problems find a natural formulation in the language of sets, multisets, strings, functions, graphs or other structured objects. Bin-packing, set partitioning, set covering, combinatorial design problems, circuits and mapping problems are some of them. They are NP-complete problems originating from areas as diverse as combinatorial mathematics, operations research or artificial intelligence. These problems deal essentially with the search for discrete structured objects. While a high-level modeling approach seems more natural, many solutions have exploited the effectiveness of finite domains or mixed integer programming solvers. In this chapter we present higher level modeling facilities utilizing constraints over structured domains.