Genetic algorithm for optimization of water distribution systems
Introduction
Water distribution system (WDS) design belongs to a group of inherently intractable problems commonly referred to as NP-hard (Templeman, 1982, Parker and Rardin, 1988). Essentially NP-hard means that a rigorous algorithm to find an optimum design using discrete diameters is not a practical possibility. Several researchers have reported algorithms for minimising the cost through the application of mathematical techniques, such as linear, non-linear or dynamic programming. It is well known that when diameters are assumed as the decision variables (DV), the constraints are implicit functions of the DV, the feasible region is non-convex, and the objective function is multimodal. Hence, conventional optimization methods result in a local optimum which is dependent on the starting point in the search process.
The application of stochastic optimization techniques such as genetic algorithm (GA) and simulated annealing to WDS optimization is of recent origin. Simpson et al. (1994)have presented a methodology for finding the best cost alternative for pipe networks using a three operator GA comprising reproduction, crossover and mutation. An inherent problem in that the model is the large computational time in comparison to the non-linear programming techniques. Loganathan et al. (1995)proposed an outer flow search inner optimization procedure to identify lower cost design solutions. In that approach each pipe network is subjected to an outer search scheme that selects alternative flow configurations in an attempt to find an optimal flow division among pipes. For each selected set of pipe flows a linear program is used to find the associated optimal pipe diameters and energy heads.
A new GA based methodology for optimal design/augmentation of pipe networks is described in this paper. The methodology was compared with a non-linear programming (NLP) technique based on interior penalty function (IPF) with the Davidon-Fletcher-Powell (DFP) method. The NLP technique was first evaluated by application to a case study which has been previously attempted by several researchers (Loganathan et al., 1995Loganathan et al., 1990Fujiwara et al., 1987Quindry et al., 1979Alperovits and Shamir, 1977). The optimal cost obtained from the NLP technique was 0.57% higher than the solution achieved by Loganathan et al. (1995). The solutions achieved by other researchers are 1.9–18.3% higher than the solution obtained by Loganathan et al. (1995).
Further, a comparison between the results of the GA and NLP techniques for augmentation of several medium size networks showed that the GA in general provided a lower cost solution, than that obtained from the NLP technique. The hydraulic simulator ANALIS (Bassin et al., 1992) which is based on graph theory, was used in both the NLP and GA solutions to calculate pressure heads, flows and velocities in the design of branched, looped and combined systems.
Section snippets
Deterministic optimization techniques
A number of investigators have dealt with the problem of optimization of WDS by applying mathematical programming techniques.
Several researchers employed linear programming to optimise a WDS. Principal approaches include those of Alperovits and Shamir (1977), Quindry et al. (1981)and Kessler and Shamir (1989). The technique given by Alperovits and Shamir (1977)requires that a set of variables (pipe flows) be set to particular values before the linear programme can be formulated. Information
Overview of genetic algorithms
GAs are nature based stochastic computational techniques. The major advantages of these algorithms are their broad applicability, flexibility and their ability to find optimal or near optimal solutions with relatively modest computational requirements. GAs, pioneered by Holland (1975), have proven useful in a variety of search and optimization problems in engineering, science and commerce (Goldberg, 1989). The algorithms are based on the principle of the survival of the fittest which tries to
GA based pipe network optimization
GAs typically require problem system states to be represented as strings called chromosomes. For example, if eight different pipe sizes are available then a binary sub-string of three bits is used to represent the options. This process requires that the binary coding be converted to discrete pipe diameters when evaluating the cost of the network. However in the GA based methodology described in this paper it was considered unnecessary to represent the solution as a chromosome to avoid the
Comparison of GA and NLP-IPF based techniques
GA and NLP based techniques are powerful tools which have been effectively applied to water distribution system optimization problems. The effectiveness of the techniques with respect to convergence relies on the adaptation of inherent features and properties of the distribution system in the problem formulation. Both the techniques require few parameter adjustments through trial and error to obtain the best solution. The fitness function is most crucial aspect of any GA. Other important
Case study
In order to establish the efficacy of GA based algorithm in comparison with NLP technique several networks were optimized employing both the techniques.
Fig. 2 delineates network 1. This network consists of 38 pipes (30 existing and 8 new) and 23 nodes including 21 demand nodes. Water is supplied through a reservoir of 20 m height at node no. 8. Constraints on minimum nodal pressure and pipe diameter are 12 and 0.08 m respectively. The coefficients of roughness for old and new pipes are 0.7 and
Conclusions
The paper presents the applicability of genetic algorithm in the design of water distribution systems. The algorithm has been compared with a NLP technique with IPF method which was found to be fairly efficient in comparison to the techniques presented thus far in the literature. The solution set obtained from GA and NLP techniques for several medium size networks showed that GA provides a better solution in general, in comparison with that obtained with the NLP technique. The differences in
References (25)
- et al.
Optimization of water distribution system
Environmental Software
(1993) - et al.
Design of optimal water distribution systems
Water Resour. Res.
(1977) - et al.
Graph theoretic Approach to the Analysis of Water Distribution System
J. Indian Water Works Assoc.
(1992) - Brooke, A., Kendrick, D., Meeraus, A., 1988. GAMS: a user's guide. The Scientific Press, Redwood City, California,...
- et al.
Looped water distribution system optimization for single loading
J. Envir. Engrg. ASCE
(1986) - et al.
Optimal reliability based design of pumping and distribution systems
J. Hydr. Engrg., ASCE
(1990) - et al.
A modified linear programming gradient method for optimal design of looped water distribution networks
Water Resour. Res.
(1987) - Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading,...
- Holland, J.H., 1975. Adaptation in Natural and Artificial Systems. University of Michigan Press, ANN...
- et al.
Modified Hazen Williams Formula
J. Envir. Engr. Div. ASCE
(1978)
Analysis of the linear programming gradient method for optimal design of water supply networks
Water Resour. Res.
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