INTRODUCTION

A water distribution network is a system containing pipes, reservoirs, pumps, and valves of different types, which are connected to each other to provide water to consumers. It is an important component of an urban infrastructure or agricultural landscape as an irrigation project and requires significant investment. Therefore, researchers are constantly searching for new ways to create more economical and efficient designs. Although the problem of the optimal design of water distribution networks has various aspects to be considered, it is often viewed as a least-cost optimization problem with the pipe diameters acting as the primary decision variables. In most situations, the pipe layout, connectivity, and imposed minimum head constraints at the pipe junctions (nodes) are taken as fixed design targets.

The crudest optimization procedure is to evaluate all the possible parameter combinations and pick the optimum one. In practice, the experienced design engineer will rely on his personal experience to avoid analyzing every possible configuration. The result of the design is to produce a few feasible solutions (solutions that satisfy the design constraints), which can then be priced. On large systems, a number of factors limit the effectiveness of a such manual design method.Gessler (1985) suggested a simplified approach based on the enumeration of a limited number of alternatives. In his work, he took advantage of two considerations: • After a combination of pipe sizes that give a hydraulically

feasible solution has been found, there is no need to test any other pipe size combination that is significantly more expensive.

M. ČISTÝ, Z. BAJTEK

OPTIMAL DESIGN OF WATER DISTRIBUTION SYSTEMS BY A COMBINATION OF STOCHASTIC ALGORITHMS AND MATHEMATICAL PROGRAMMING

KEY WORDS

• Genetic Algorithm,• Linear Programming,• Optimal design of looped hydraulic pipe

networks

ABSTRACT

The paper demonstrates a new model for determining the minimum cost for the design of a water distribution system based on a combination of linear programming methodology and a genetic algorithms approach. The optimal design of looped hydraulic pipe networks belongs to the class of large combinatorial optimization problems that are difficult to handle using conventional operational research techniques. Although many research efforts have been made for the sake of achieving the optimal design of large loop water distribution networks, there is still some uncertainty about finding a generally reliable method. The author of the paper proposes a method in which the complementary usage of deterministic and soft computing methods will ensure a new level for the quality of outputs. The method is based on a combination of linear programming methodology and a genetic algorithms approach and is intended for use in drinking water systems and the rehabilitation of pressurized irrigation systems as well.

Milan Čistý, PhD., Assoc. ProfResearch fields: irrigation, application of soft comput-ing methods in water engineering

Zbyněk Bajtek, Eng.Research fields: design of irrigation networks, pro-gramming

Department of Land and Water Resources Management, Faculty of Civil Engineering, Slovak University of Technology,Radlinského 11, 813 68 Bratislava, Slovak Republice-mail: milan.cisty@stuba.sk

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• After an infeasible solution has been encountered, any other size combination, with all the sizes equal to or less than those, is an infeasible solution.

Alperovits and Shamir (1977) presented a linear programming gradient (LPG) in optimizing a water distribution network. A segmental length of pipe with a differential diameter was used as the decision-making variable. Kessler & Shamir (1989) used the linear programming gradient (LPG) method as an extension to this method. It consists of two stages: an LP problem is solved for a given flow distribution and then a search is conducted in the space of the flow variables. Later, Fujiwara & Khang (1990) used a two-phase decomposition method extending that of Alperovits & Shamir (1977) to non-linear modelling. Also, Eiger, et al. (1994) used the same formulation as Kessler & Shamir (1989), which leads to the determination of the lengths of one or more segments in each link with discrete diameters. Some of these methods impose a restriction on the type of hydraulic components in the network. For instance, the presence of pumps in the network increases the nonlinearity of the problem; as a result networks with pumps cannot be resolved by some of the methods. These methods also fail in resolving large looped systems.Recently, researchers have focused on stochastic optimization methods. The advantage of these methods is that they allow full consideration of a system’s nonlinearity – a problem with which the previously mentioned methods fail.Simpson, et al. (1994) used simple genetic algorithms (GA) in which each individual population is represented in a string of bits with identical lengths that encode one possible solution. The simple GA was then improved by Dandy, et al. (1996) using the concept of the variable power scaling of the fitness function, an adjacency mutation operator, and gray codes. Savic and Walters (1997) also used simple GA in conjunction with an EPANET network solver. Other evolutionary techniques have also been applied to the optimization of a water distribution system, such as ant colony optimization algorithms (Maier, et al, 2001), simulated annealing (Loganathan, et al, 1995; Cunha and Sousa, 2001) and harmony search (Geem, 2002).

MATERIALS AND METHODS

In the proposed method two algorithmic techniques will be employed – linear programming (LP) and genetic algorithms (GA). A combination of these two methods with the aim of eliminating the limitation of LP (it is not suitable for large networks with loops) is used.Genetic algorithms are search procedures inspired by the mechanics of natural genetics and natural selection. This methodology is finding

increased application to solve difficult problems of engineering, science, and commerce. The present usage of evolutionary based optimization techniques stems largely from the development of evolution strategies by Rechenberg in Germany in the 1960’s and the development of genetic algorithms by Holland in the USA soon afterwards. The basic ideas are briefly summarized below; a good introduction to the subject is given by Goldberg (1989).The first step is to represent a legal solution to the problem by a string of genes (seeking parameters) that can take on some value from a specified finite range. This string of genes, which represents the solution, is known as a chromosome. Then an initial population of legal chromosomes is constructed at random. With each generation, the fitness of each chromosome in the population is measured. The fitter chromosomes are then selected to produce offspring for the next generation, which inherit the best characteristics of both parents. This process is repeated until some form of convergence in fitness is achieved. The goal of the optimization process is to minimize or maximize the fitness.In the case of the design of a pipe network the optimization problem can be stated as follows: minimize the cost of the network components subject to satisfactory performance. The chromosome can be a binary or integer string of numbers (representing possible decisions), which defines the network’s design. Care must be taken to incorporate all reasonably possible choices for each component in the system, with an appropriate set of costs from which all the schemes can be priced. In general, the more choices and flexibility in the design options allowed at this stage, the greater the savings will be for the optimized design.A model is then set up, which incorporates all the options for the individual network components. The GA then generates trial solutions, each of which is evaluated by simulating its hydraulic performance. Any hydraulic infeasibility, for example, failure to reach a specified minimum pressure at any demand point, is noted, and a penalty cost is calculated. Operational (e.g. energy) costs can also be calculated at this point if required. The penalty costs are then combined with the predicted capital and operational costs to obtain an overall measure of the quality of the trial solution. From this quality measure the fitness of the trial solution is derived. The process can continue for many thousands of iterations, and a population of good, feasible solutions will evolve.For the purpose of clarity we will also briefly describe also the optimization procedure of the pipeline network’s optimization using linear programming. The mathematical formulation of this problem is as follows:X11 + X12 + ...+ X1n = B1X21 + X22 + ...+ X2n = B2 etc.Xm1 + Xm2 + ...+ Xmn = Bm (1)

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A11 X11 + A12 X12 + ...+ AxyXxy ≤ C1 (2)etc.

D1 X11 + D2 X11 + ...+ Dj Xmn = min (3)

The solution has to comply with the inequalities: X11 > 0; X12 > 0 etc. up to Xmn > 0 (4)

Where Xij is unknown length of selected diameter j on section i Bi – total length of section Amn – hydraulic loss in section m and diameter n Ci – allowable total loss for section Di – price of pipeline with diameter number i

When, in order to resolve the task of rehabilitating the pipeline networks, we apply linear programming, the unknown will be the lengths of the individual pipeline diameters. In conditions (1) we must mathematically express the requirement that the sum of the unknown lengths of the individual diameters in each section has to be equal to its total length. The second type of the equation in constraints (2) represents the request that the total pressure losses in a hydraulic path between a pump station and critical node (the end of the pipeline, extreme elevation inside the network) should be equal to or less than the known value. This constraint is based on the minimum network pressure requirement needed for the operation of the system. Given the minimization requirement for the investment costs, the objective function (3) sums up the products of the individual pipeline prices and their required lengths.

RESULTS

Mathematical programming methods, which undisputedly lead to global optimality, can be applied only in the case of branched networks with no loops for a single loading. In the case of looped networks, even for the simplest case of a least cost design, under one single loading for a gravity fed network (no tanks, no pumps), the debate about global optimality is still, today (after about 40 years), a research issue. In fact, for looped networks, no deterministic theoretical methods exist to reach global optimality. For this reason various heuristic methods are applied to these problems.But while using heuristic methods (e.g. GA), problem of testing it arises. Tasks which heuristic methods are capable of solving (e.g., the optimization of a looped network) are often not possible to solve with any other methods, especially not deterministic methods. So it is not possible to prove their mathematically reliability of them. For this reason it is obvious to use the so-called benchmark networks for testing. The problem which follows is that commonly used

benchmark networks are relatively small water distribution systems, and the question always remains how the developed methods will work in large (real life) systems.We conducted some experiments for this reason. For the GA method it does not matter if it resolves a looped or branched network. It is possible to optimize both of these configurations, but it is advantageous to use it for looped networks, because branched networks without loops could be absolutely reliably resolved by a linear programming method. But, for the purpose of testing we used GA for resolving a large branched network. In the next step it is also possible to resolve the same network with linear programming. Because the result from the linear programming is without doubt global minimum, we can see how close GA gets to this goal. This is competent testing of GA or another heuristic methodology on a large network. Because, as previously mentioned, there is no methodological difference for GA when it is used to resolve a branch or looped network, we can extrapolate that the precision in finding an optimal solution will be similar in the case of a looped network. From the results it could be derived that there is still a reason to look for a competent method for optimizing water distribution network, because the differences in the various hydraulic schemes were about 5-15%. The proposed method is based on a combination of linear programming methodology and a genetic algorithms approach. The main idea is that linear programming is more reliable than heuristic methods, but because it is not suitable for resolving looped networks, the GA method is used for decomposing a complex looped network into a group of branched networks - each with identical hydraulic behavior as a looped network with identical diameters. Identical hydraulic behavior means that there are identical flows in the corresponding pipes and identical pressures in the corresponding nodes in looped and branch networks. Decomposition by GA should only be done on large networks - for smaller networks, all possible branched networks could be found to which it is possible to transform the original looped network in the optimization process. The optimization models using LP are then set up for each member of this group of branched networks. After evaluating (by LP) the high number of possible branch networks which produce local minima, an optimal global solution could be easily chosen for the original looped network.The loop can be transformed to a branch layout without changing the hydraulic behavior of the network if we split the loop in the node in which flows are coming to from both sides. It is one from the demand nodes (in which water is taken from a network – e.g., a hydrant). But which node it is depends on other network conditions as the pump station location, diameters and lengths of pipes, other demand allocations, etc. Because we do not know these parameters before the diameters on network are proposed (that is the task which

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we are solving), a loop split will be done in every demand node, and then we will propose diameters in such a manner that this meeting of flows will happen in this node. In this way we can produce many alternatives of splitting the network into branched. The node which was chosen for this splitting must be doubled; the original node will be assigned to one link from which the flow is coming to the node and its twin node to another link from the opposite side of the original node. The demand in the node will be divided into these two nodes with a rational number of alternatives. The number of alternatives depends on the amount of the demand – a high demand in a node requires more alternatives. So the number of demand nodes multiplied by the number of these alternatives of demand splitting gives the number of possible branch networks to which it is possible to divide a one-loop network. A network may consist of more than one loop, of course. For this reason the following algorithm intended for the transformation of a looped network to a branched one will be run:1. The number of loops will be determined on the original

network:

L = P – J + 1 (5)

Where L is the number of loops P – the number of pipes J – the number of junctions

2. For every loop the vector of links (pipes) which it consists of will be determined. This will be done in successive steps to loop after loop and when some link has already been assigned to any previously analyzed loop (it is already part of another loop vector of pipes) it will not be assigned to a vector of links of the currently analyzed loop. This assignment of pipes to loops is graphically shown on fig. 1a (by line types).

3. To every loop a three-column matrix (tab. 1) will be assigned. Table 1 demonstrates loop 1 from fig. 1. It will consist of a column for split nodes N in which the loop can be divided and from the column of split links L (which are part of the loop vector determined in the previous step and which are connected to N). The original node N will be cloned to its twin node N-L for the sake of splitting the loop (this twin node N-L replaces the original node N on link L – fig. 1b or 1c nodes 5-4 or 5-5). For every node N one or more links L could be assigned for which we can obtain different branch networks while splitting the loop in node N. The third parameter is the ratio R in which the original demand in node N will be divided between nodes N and N-L. In table 1 only 3 ratios are considered (0.25, 0.5 and 0.75), and as can be seen for the first combination of N and L, it will produce 3 versions of this split alternative of the loop. The same situation holds for the other combinations of L and N.

4. Every alternative (row in the matrix) for every loop will be combined with each other row in the matrices for other loops in the network, which produces a group of branch networks for LP computation.

5. All the alternatives are evaluated in the case of total enumeration (for small networks), or the alternatives from step 4 are selected by a genetic algorithm when searching for the optimal solution in large networks.

Fig. 1 Schematic network with 3 loops

Tab. 1 Split matrix for loop 1 from Fig. 1Split node Split link Split ratio

1 1 0.25

1 1 0.5

1 1 0.75

1 5 0.25

… … …

2 1 0.25

… … …

3 2 0.25

… … …

4 3 0.25

… … …

4 4 0.25

… … …

5 4 0.25

… … …

5 5 0.25

… … …

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The test problem is a two-loop network with 8 pipes, 7 nodes and one reservoir (Fig. 2) which was obtained from the literature (Alperovits & Shamir, 1977). All the pipes are 1000 m long, and the Hazen-Williams coefficient is assumed to be 130 for all the pipes. The minimum nodal head requirement for all the demand nodes is 30 m (0,3 Mpa). There are 14 commercially available pipe diameters (Tab. 2). The network consists of two loops, so two three-column matrices (described in step 3 above) were determined for each of them. For the first loop it is for pipes 2, 3, 4 and 7 and for the second

5, 6 and 8. The computations show, that it was not necessary to choose a smaller ratio R for dividing the original demand in the node between the twin nodes than 0.1. This led to 1452 possible combinations of branch networks to which the original looped network was transformed. For performing linear programming computations the solver LPSOLVE was called from software originally developed by the authors of the paper. Because of the relatively small number of iterations, it was also possible to run a total enumeration of the problem. The GA was set up with integer coding, and the chromosome consists of four genes. The first gene is the indicator of the pair: split node – split link for the first loop (every possible pair is numbered); the second gene is the dividing ratio for the first loop R1 multiplied by ten. The third and fourth genes are analogous to the second loop. Roulette wheel selection is used to chose the parents for the next generation. A one-point crossover was selected because of the relatively short chromosome, and the probability of the selected pair of strings being subjected to the crossover operator was taken as pc=0.97. The mutation rate was set to be pm= 0.02. The objective function is the cost of the solution. No penalty function was used, because the LP that is in the proposed algorithm final producer of every partial solution produces in contrast to GA only feasible solutions. The results together with the results from other authors testing their methods on this benchmark network are summarized in table 4.As can be seen, the optimal solution was reached by the proposed method. In Eiger’s work a different interpretation of the Hazen-Wiliams equation was used than in other works (it is common to use the interpretation which is embedded in the Epanet network solver, which was also used in our work). This leads to the result that there are some nodes in the Eiger solution in which there is smaller pressure of 0,3 MPa than was required. Our solution gives the same results either using GA or if a total enumeration is performed.

Fig. 2 Two-looped network

Tab. 2 Cost data for the two-loop networkDiameter (mm) 25.4 50.8 76.2 102 152 203 254Cost (units) 2 5 8 11 16 23 32Diameter (mm) 305 356 406 457 508 559 610Cost (units) 50 60 90 130 170 300 550

Tab. 3 Node data for the two-loop network

NodeDemand(m3/h)

Ground level(m)

1 (Reservoir) -1,120.0 210.002 100.0 150.003 100.0 160.004 120.0 155.005 270.0 150.006 330.0 165.007 200.0 160.00

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SUMMARY AND CONCLUSIONS

Although the genetic algorithms method helps in many areas of engineering, including the design or operation of water distribution systems, it has some disadvantages:• it cannot guarantee the generation of an optimal solution,

particularly for large-scale systems • it requires extensive fine-tuning of the algorithmic parameters,

which are highly dependent on the individual problem • it can produce solutions in which the diameters can be optimally

found from an economic point of view, but the diameters could in some conditions be distributed randomly: for instance, for the branched part of networks there could be bigger diameters in some parts after smaller ones in the direction of the flow which is wrong.

For these reasons the authors of this paper propose a method in which genetic algorithms are incorporated, but the final solution is

produced by linear programming. This method is described in the paper and tested on a benchmark network successfully. The method can also be run without GA with a total enumeration (e.g., totally eliminate randomness), which of course takes more CPU time. This gives opportunity for comparing the results with and without GA. By involving GA, we get results much earlier. In the paper this testing is done on a benchmark model; large network testing will be presented in future works.

ACKNOWLEDGEMENT

This study was supported by the Scientific Grant Agency of the Ministry of Education of the Slovak Republic and the Slovak Academy of Sciences, Grants No. 1/0496/08, 1/0585/08 and by Slovak Research and Development Agency, Grant No. APVV-0443-07.

Tab.4 – Results for benchmark network

Pipe number

Alperovits and Shamir (1977)

Kessler and Shamir (1989)

Eiger et al. (1994) Geem (2002) Čistý (2008)

L [m] D [mm] L [m] D [mm] L [m] D [mm] L [m] D [mm] L [m] D [mm]

1 256.00 508.0 1000.00 457.2 1000.00 457.2 1000 457.2 1000.00 457.2

744.00 457.2

2 996.38 8.0 66.00 304.8 238.02 304.8 1000 254.0 202.00 304.8

3.62 152.4 934.00 254.0 761.98 254.0 798.00 254.0

3 1000.00 457.2 1000.00 406.4 1000.00 406.4 1000 406.4 1000.00 406.4

4 319.38 8.0 713.00 76.2 1000.00 25.4 1000 101.6 1000.00 25.4

680.62 152.4 287.00 50.8

5 1000.00 406.4 836.00 406.4 628.86 406.4 1000 406.4 693.00 406.4

164.00 355.6 371.14 355.6 307.00 355.6

6 784.94 304.8 109.00 304.8 989.05 254.0 1000 254.0 990.00 254.0

215.06 254.0 891.00 254.0 10.95 8.0 10.00 203.2

7 1000.00 152.4 819.00 254.0 921.86 254.0 1000 254.0 902.00 254.0

181.00 8.0 78.14 8.0 98.00 203.2

8 990.93 152.4 920.00 76.2 1000.00 25.4 1000 25.4 1000.00 25.4

9.07 101.6 80.00 50.8

Cost 497 525 417 500 402 352 419 000 403 454

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REFERENCES

• ALPEROVITS, E. AND SHAMIR U. (1977): Design of Optimal Water Distribution Systems. Water Resources Research, AGU, vol.13, no. 6, pp.885-900

• CUNHA, M. C. AND SOUSA, J., (2001): Hydraulic infrastructures design using simulated annealing. Journal of Infrastructure Systems, ASCE, 7(1), 32-39.

• DANDY, G. C., SIMPSON, A. R., AND MURPHY, L. J. (1996): An Improved Genetic Algorithm for Pipe Network Optimization, Water Resources Research, 32(2), 449.

• EIGER, G., SHAMIR, U. AND BEN-TAL, A. (1994): Optimal Design of Water Distribution Networks, Water Resources Research, 30(9), 2637-2646.

• FUJIWARA, O. AND KHANG, D.B. (1990): A Two-Phase Decomposition Method for Optimal Design of Looped Water Distribution Networks, Water Resources Research, 26(4), 539-549.

• GEEM, Z. W., KIM, J. H. AND LOGANATHAN, G. V. (2002): Harmony search optimization: application to pipe network design. International Journal of Modelling and Simulation, 22(2), 125-133.

• GOLDBERG, D.E. (1989): Genetic Algorithms in Search, Optimisation and Machine Learning. New York, Addison-Wesley.

• KESSLER, A. AND SHAMIR, U. (1989): Analysis of the Linear Programming Gradient Method for Optimal Design of Water Supply Networks, Water Resources Research, 25(7), 1469-1480.

• LOGANATHAN, G. V., GREENE, J. J. AND AHN, T. J. (1995): Design heuristic for globally minimum cost water-distribution systems. Journal of Water Resources Planning and Management, ASCE, 121(2), 182-192.

• MAIER, H. R., SIMPSON, A. R., FOONG, W. K., PHANG, K. Y., SEAH, H. Y., AND TAN, C. L. (2001): Ant Colony Optimization for the Design of Water Distribution Systems, Proceedings of the World Water and Environmental Resources Congress, Orlando, Florida, USA

• SAVIC, D. A. AND WALTERS, G. A. (1997): Genetic algorithms for least-cost design of water distribution networks, Journal of Water Resources Planning and Management, ASCE, 123(2), 67-77.

• SIMPSON, A.R., DANDY, G.C., AND MURPHY, L.J. (1994): Genetic Algorithms Compared to Other Techniques for Pipe Optimization, Journal of Water Resources Planning and Management, ASCE, 120(4), 423-443.

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