Water Distribution Expansion Planning with Decomposition

In recent years, the water distribution expansion planning (WDEP) problem has become increasingly complex as the demands on water distribution systems have evolved to meet modern requirements. This paper describes an algorithm that combines the strengths of local search with global search to flexibly handle these difficult problems. The algorithm decomposes a WDEP problem into sub problems using a local search procedure called Large Neighborhood Search (LNS). Each sub problem is solved exhaustively (or partially) using global search techniques such as branchand-bound search. The utility of the approach is demonstrated on the problem described in the Battle of the Water Networks II competition. INTRODUCTION The past few decades have seen stress placed on water distribution networks around the world due to aging infrastructures, water quality concerns, increasing demand for water, diminishing supplies, and a desire to reduce the carbon footprint of water systems. To address this challenge, the 14th Water Distribution Systems Analysis Symposium has issued a Battle of the Water Networks II competition. In this paper, we describe our approach to solving the problem of expanding and controlling closed pipe water distribution systems to meet future demand, satisfy multiple objective functions, and meet robustness criteria. We generalize algorithms developed for the electric power systems domains of transmission expansion planning (Bent et al. (2012)), integrated resource planning (Bent et al. (2011)), restoration scheduling (Coffrin et al. (2012)), and vehicle routing (Bent and Van Hentenryck (2004)). In this paper we use a decomposition-based approach that separates the problem into distinct sub problems that are solved iteratively using a procedure referred to as Large Neighborhood Search (LNS) (Shaw (1998)). Each sub problem is solved within a (potentially truncated) branch-andbound search procedure that intuitively is not unlike some of the ideas developed by the simulation optimization community where each partial solution explored in the branch-and-bound search is evaluated using a water network simulation package. The key contribution of the algorithm is a technique for combining global search techniques with local search to find high quality solutions. The local search iteratively determines the sub problems to consider and the global search procedure is executed on each sub problem. Review The WDEP problem is NP-Hard due to the non-linearities present in the head-loss models of water systems (Rossman (2000)) and the discrete variables (e.g., pipe diameters, etc.) in water network expansion (Yates et al. (1984); Gupta et al. (1993)). As a result, there is a large body of solution approaches for solving the WDEP in the literature. These approaches include linear programming and non-linear programming (Kessler and Shamir (1989); Fujiwara and Khang (1990); Sherali et al. (2001); Bragalli et al. (2008)) and evolutionary algorithms (EA). The EA algorithms include genetic algorithms (Dandy et al. (1996); Wu and Simpson (2001); Reca and Martinez (2006); Ewald et al. (2008); Kadu et al. (2008)), simulated annealing (Tospornsampan et al. (2007)), ant colony optimization (Maier et al. (2003); Zecchin et al. (2005); Tong et al. (2011)), Harmony Search (Geem (2009)) and particle swarm optimization (Montalvo et al. (2008)). While these approaches have made significant contributions to the field, it is clear that there are oppurtunities to expand and enhance this literature to further the state-of-the-art. The rest of the paper is organized as follows. The next section formally defines the WDEP problem. The second section describes the algorithm used to find solutions to the WDEP problem. The third section describes to the solution the problem provided by the Battle of the Water Networks II. The fourth section discusses the results. The final section concludes the paper. PROBLEM DEFINITION The model of a water distribution network described in this paper follows the definitions provided in the EPANET software package (Rossman (2000)). The minimal set of features of a water distribution network that are required to fully define our approach are discussed in this section. Nodes The problem is described in terms of a set of nodes, N , that represent geographically located points in a water network e.g., reservoirs (RESERVOIRS), tanks (TANKS), and junctions (JUNCTIONS), such that N = RESERVOIRS ∪ TANKS ∪ JUNCTIONS. For each junction i ∈ JUNCTIONS, the function di,τ is used to define the demand for water at time τ. For each reservoir r ∈ RESERVOIRS, the function hr,τ is used to define the hydraulic head at time τ. For each tank t ∈ TANKS, e+ and e− define the maximum and minimum water storage elevation respectively. The decision variable vt is used to define the volume of the tank. vt has discrete domain [vt,vt, . . . ,vt n]. Edges The problem is also described in terms of a set of edges, E . For an edge i, j ∈ E between nodes i and j, the decision variable pi, j is used to denote the number of pipes with diameter d, where pd − i, j ≤ pi, j ≤ pd + i, j . The set of possible diameters is denoted by D. The decision variable u π i, j is used to denote the number of pumps of type π between i and j, where uπ i, j ≤ ui, j ≤ uπ + i, j . The set of possible pump types are defined by I. The decision variable vi, j is used to denote the number of valves with diameter d between i and j, where vd − i, j ≤ vi, j ≤ vd + i, j . The Boolean decision variable gi, j is used to denote the existence of a backup diesel generator for components (pumps) on edge i, j. Controls Finally, the problem is also defined by control statements that determine the status of edges. The decision variables k i, j and k − i, j denote the times when components (pumps and valves) on edge i, j are activated and deactivated respectively. The coupled decision variable κi, j and κ − i, j denotes the node attribute and value that causes edge i, j to be activated and deactivated respectively (for example, tank water levels). Solution A solution, σ, is defined as a set of variable assignments to the variables of the WDEP problem, i.e.1. ⋃