This paper presents the development of a new advanced control method suitable for variable temperature District Heating Systems (DHS). The proposed controller determines optimal planning for the on/off status and power of the heat generators as well as for the supply temperature and differential pressure at the production plant level. Compared to existing methods, the original features of the developed solution are to fully exploit the thermal storage capacity of the network and to determine the best compromise between pumping costs and heat losses. A numerical case study based on a representative DHS is used to evaluate the method over a heating season (5 months). Results show that our method reduces production costs up to 8.3 % when compared to a more classical controller. Moreover, the observed computing time is compatible with the requirements of the receding time horizon principle, ensuring that the method is tractable on real DHS.

Given a connected graph G, a vertex v of G is said to be a branch vertex if its degree is strictly greater than 2. The Minimum Branch Vertices Spanning Tree problem (MBVST) consists in nding a spanning tree of G with the minimum number of branch vertices. This problem has been well studied in the literature and has applications specially for routing in optical networks. In this paper we propose a generalization of this problem. We introduce the notion of k-branch vertex which is a vertex with degree strictly greater than k + 2. The parameter k can be seen as the limit of the capacity of optical splitters to divide the light signal. In order to respect as far as possible this limit, we propose to search a spanning tree of G with the minimum number of k-branch vertices (k-MBVST problem). We propose a proof of NP-hardness of this new problem whatever the value of k. We also propose an ILP formulation of the k-MBVST by generalising the MBVST one. Experimental tests on random graphs show that the number of k-branch vertices increases with graph size but decreases with k as well as with the density. They also show that when k ≥ 4, the number of k-branch vertices is close to zero whatever the size and the density of the tested graph.

Given a connected edge-weighted graph G and a positive integer B, the Degree Constrained Minimum Spanning Tree problem (DCMST) consists in nding a minimum cost spanning tree of G such that the degree of each vertex in the tree is less than or equal to B. This problem, which has been extensively studied over the last few decades, has several practical applications, mainly in networks. However, some applications do not especially impose a subgraph as solution. For this purpose, a more exible so-called hierarchy structure has been proposed. Hierarchy, which can be seen as a generalization of trees, is dened as a homomorphism of a tree in a graph. In this paper, we discuss the Degree Constrained Minimum Spanning Hierarchy (DCMSH) problem which is NP-hard. An Integer Linear Program (ILP) formulation of this new problem is given. Properties of the solution are analysed, which allows us to add valid inequalities to the ILP. To evaluate the dierence of cost between trees and hierarchies, the exact solution of DCMST and DCMSH problems are compared. It appears that, in sparse random graphs, the average percentage of improvement of the cost varied from 20% to 36% when the maximal authorized degree of vertices B is equal to 2, and from 11% to 31% when B is equal to 3. The improvement increases as the graph size increases.

Abstract. Degree-bounded spanning problems are well known and are mainly used to solve capacity constrained communication (routing) problems. The degree-constrained spanning tree problems are NP-hard and the minimum cost spanning tree is not approximable nor in the case where the entire graph must be spanned neither in partial spanning problems. Most of applications (such as communications) do not need trees as solutions. Recently, a more flexible, connected, graph related structure called hierarchy was proposed to span a set of vertices in the case of constraints. The application of this structure permit the reformulation of the degree-constrained spanning problem. In this paper we present not only the advantages of the new structure but also that the new and optimal structure is approximable with the help of fast polynomial algorithm. The obtained approximation ratio is very advantageous regarding the negative approximability results with spanning trees.