Computational Technology Resources - CCP

Keywords: water supply networks, optimisation, genetic algorithms, fuzzy reasoning, linguistic variables, small water tanks.

Summary

In terms of mathematical simulation, a water supply and distribution network is assumed to be a system, consisting of elements (pipes, pumps) whose diameters and curves respectively are yet unknown and nodes, simulating consumption points, source points (e.g. reservoirs, tanks) and junctions. For a water distribution network to operate successfully, both pressure (or energy heads) at every node and velocities at each pipe must not exceed preset upper and/or lower bounds, for any water demand scenario (network loading) imposed to the system. Those requirements consist the system constraints.

Given the desired topographical layout of a water distribution network, design optimisation aims at defining the "best" or optimal solution out of all existing alternatives, defining proper diameters for each pipe, so as to minimize the overall system cost, taking into consideration the system constraints. Because the problem is complicated enough, especially for the improvement and renovation of old existing networks, for more than a decade genetic algorithms (GAs) have been applied for fluid (water and gas) distribution network optimisation.

Generally, when using genetic algorithms for water supply network optimisation, a penalty function is used to estimate feasibility. So, in case of constraint violation an additional penalty "cost" is added to the total objective function value, affecting any further survival and selection procedure for the string (alternative solution) involved. In this paper fuzzy reasoning for penalty functions is introduced in water supply network design optimisation using genetic algorithms, replacing penalty functions.

An original S-shaped membership function has been developed. So, obtaining a small membership function value to the set of feasible solutions simulates any deviation from constraint bounds. By modifying properly two shape parameters, fuzzy linguistic definitions, such as [AROUND], [BETWEEN], [NO_LESS_THAN], [NOMORE_THAN], [ANY_VALUE] and the qualitative hedges [MEDIUM], [STRICTLY], [TOLERANTLY], [VERY] and [TOO] can be simulated and further combinations of them formed.

Moreover, by using these linguistic parameters, an original approach to the mathematical simulation of small existing tanks, as quasi-fixed head nodes during optimisation, is introduced, for which fuzzy reasoning is appropriate. To estimate the overall membership function value to the set of feasible solutions, aggregation operations are employed.

Assuming the cost for each link is REAL_COST, the system total "real" cost for each alternative $\mathbf{D}$ , $\mathbf{D}=\{d_1, d_2, \ldots d_{NL}\}$ , is transformed to fuzzy cost, used for string fitness evaluation, by introducing aggregated membership function values, as follows :

$\displaystyle f_{fuzzy,mathbf{D}}=\frac{1}{\mu_{p,\text{all}}}\sum^{NL}_{l=1}\frac{\text{REAL\_COST}(d_l)}{\mu_l}$

(1)

where $\mu_l$ and $\mu_{p,\text{all}}$ are the total aggregated membership function values for elements (pipes) and node pressure respectively. So small membership function values affects the cost of any alternative, by increasing the fuzzy cost, and therefore making the solution less fit for selection at the GA process.

The GA applied in this paper was designed and developed from scratch [1,2]. As a result the program G_FUZNET has been developed, using Compaq $\circledR$ Visual Fortran V6.5. The program allows for up to 10 multiple loadings, a different "case-diameter" mapping for each new/replaceable link, parallel pipes, existing and replaceable links and different constraint bounds for each link/node.

The general theoretical approach is presented, together with an application of the model to the main water supply network of the city of Halkis, as case study. The paper includes also a discussion on the advantages and possibilities of linguistic variables, while pointing out sensitivity to the use of various fuzzy operators for membership functions and aggregation.

References

1: Vamvakeridou-Lyroudia L.S., "Water Supply Network Design Optimisation Using Fuzzy Reasoning and Genetic Algorithms", in Water Software Systems, Ulanicki B., Coulbeck B. and Rance J. (Editors), Research Studies Press Ltd., U.K., Vol. 1, 3-13, 2001.
2: Vamvakeridou-Lyroudia L.S., "Fuzzy Reasoning Combined with Genetic Algorithms for Water Supply Network Design Optimisation", in Evolutionary Methods for Design, Optimisation and Control, Giannakoglou K.C., Tsahalis D.T., Periaux J., Papailiou K.D., Fogarty T. (Editors), CIMNE Barcelona, Spain, 442-447, 2002.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £82 +P&P)