Water Distribution Network Design Using the Shuffled Frog Leaping Algorithm

min f ( D1, ........., DNP) =
∑ C( k∈NP
Dk , Lk )
(1) (2) (3)
Subject to: ∑ Qin − ∑ Qout = Qe
∑ ∆H k k∈Loop l
= 0 , ∀l ∈NL
Lk β −1 Q |Q | , β γ k k C k Dk ∀k∈NP
WATER DISTRIBUTION NETWORK DESIGN USING THE SHUFFLED FROG LEAPING ALGORITHM
Muzaffar M. Eusuff and Kevin E. Lansey Graduate Research Assistant and Associate Professor, respectively, Dept. of Civil Engin. and Eng. Mech., The University of Arizona, Tucson, AZ, 85721. Phone: 520-621-2512, Fax: 520-621-2550, E-mail: lansey@ Introduction Water distribution networks are a significant investment. As such, a large volume of research has examined the pipe design/rehabilitation problem and is summarized in other papers (e.g., Savic and Walters (1997), Lansey (2000)). This paper focuses on the application of a new optimization method to the pipe sizing problem. In recent years, the researchers have attempted to exploit expanding computer power and combined new optimization techniques with hydraulic simulation software. The computer model in this work, SFLANET, is based upon the shuffled frog leaping algorithm (SFLA), a memetic algorithm (a kind of meta-heuristic). The optimization algorithm is linked to EPANET (Rossman 2000) via the EPANET Toolkit and can be used to design large, complex pipe network systems. Here results are shown for the New York City Tunnel problem. Problem Formulation In this paper, as in most literature work, water distribution network design is formulated as a least-cost optimization problem with pipe diameters as the decision variables. The objective is to minimize the cost of piping given the pipe layout and its connectivity, nodal demands, and minimum pressure head requirements. Mathematically the formulation is:
∆H k = H u / s node, k − H d / s node, k = ω
H tiN ≥ H tmin,iN , ∀iN ∈ NN , ∀t ∈T
D k ∈{D}, ∀k ∈ NP
Байду номын сангаас
(4) (5) (6)
where D1,.....,DNP are NP available discrete pipe diameters in the set of commercial pipe sizes {D} (eq. 6). C(Dk, Lk) is the cost of pipe k with diameter Dk and length Lk. Equation 2 is conservation of mass for each node where Qin and Qout are the flow into and away from the node, respectively, and Qe is the external inflow or demand at the node. Equation 3 is the conservation of energy around loop l where ∆Hk denotes the head loss in pipe k and NL is the total number of 1
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loops in the system. The head loss in each pipe is the head difference between connected nodes (equation 4). Equation 5 limits the nodal pressure H for any node iN (where total number of nodes is NN) during any period t (where number of time periods is T) to remain equal to or above a pre-specified minimum pressure Hmin. Shuffled frog leaping algorithm (SFLA): A Memetic Meta-heuristic Shuffled frog leaping algorithm (SFLA), a memetic meta-heuristic for combinatorial optimization has been developed recently by Eusuff and Lansey (2000). The algorithm is designed to seek a global optimal solution by performing an informed heuristic search using a heuristic function. It is based on evolution of memes carried by interactive individuals and a global exchange of information within the population. SFLA has been tested on several combinatorial problems and found to be very efficient. Memetic algorithms are population-based approaches for heuristic search in optimization problems. The term memetic comes from “meme” (Dawkins 1976). Memes can be considered as the unit of cultural evolution. Ideas evolve in a manner analogous to biological evolution. Here we introduce the term memeplex to denote a group of mutually supporting memes that form an organized belief system, such as a religion. The shuffled frog leaping algorithm progresses through the time loops in the form of memetic evolution. A meme (analogous to a gene in genetic algorithms) consists of a number of memotypes. A memotype carries information about the idea in a similar manner as a chromosome of a gene. The pool of memes is defined as a Virtual Population (VP). The Virtual Population is used in modeling the meme pool that holds a diverse set of frogs with different memes. One analogy for SFLA is a population of frogs searching for food. A second is a cultural population such as an earlier cultural who are developing an idea such as pottery. A set of ideas for making pottery are progressing concurrently within different groups of craftsmen. Each idea can be thought of as converging toward some local optimal for that approach. However, each year the craftsmen may come together and interact for some economic or cultural reason. At these meetings, ideas for constructing pottery are mixed between individuals. New groups are formed pursuing the best method for pottery making and move toward the optimal method for a combined approach. By combining ideas the search for the best pottery making technique is directed toward the global optimum. This cycle continues each year as the villages and craftsmen interact. For the local search SFLA uses the particle swarm optimization (PSO) (Eberhart and Kennedy 1995) in a formulation for discrete problems. Mixing of ideas is then accomplished after the local searchs progress for some number of iterations. This interaction is intended to lead to a global search and hopefully global optimum. The basis for shuffling of ideas was derived from the shuffled complex evolution (SCE) algorithm (Duan et al. 1992). Development of SFLANET This section describes the development of SFLANET, a computer model that applies SFLA in combination with the hydraulic simulation software, EPANET, and its library functions, EPANET Toolkit. A flow chart of the SFLANET algorithm is shown in Figure 1. Encoding – Decoding of a set of design solution Memotype(s), the actual contents of a meme are treated as the decision variables for the network design problem. Each memotype represents a pipe and they are encoded as a number 2