battle of the network models: epilogue

BATTLE OF THE NETWORK MODELS: EPILOGUE

Thomas M. Walski,1 E. Downey Brill, Jr.,2 Johannes Gessler,3 Members, ASCE, Ian C. Goulter,4 A. M. ASCE, Roland M. Jeppson,5 M. ASCE,

Kevin Lansey,6 Han-Lin Lee,7 Student Members, ASCE, Jon C. Liebman,8 Larry Mays,' Members, ASCE,

David R. Morgan,10 and Lindell Ormsbee,11

A. M. ASCE

ABSTRACT: Several models that can be used to optimally size water distribution pipes were applied to a hypothetical system. The results are summarized in this paper. The models produced solutions with costs that were within 10% of one another, although the solutions were quite different. While the models were helpful in sizing pipes, some manual calculations and a good deal of engineering judgment were required to apply them.

INTRODUCTION

In recent years a great deal of research has been conducted on optimally sizing water distribution systems, i.e., selecting the least-cost combination of pipes, pumps, and tanks. Numerous papers and models have been prepared, but to date, there have been few published applications of optimization models to sizing real water distribution systems. This can be attributed to some extent to a lack of communication between those developing the programs and engineers involved with water distribution system design. "The Battle of the Network Models" is the name given to a series of sessions held at the conference "Computers in Water Resources" at Buffalo, New York, in June 1985 to bring together researchers and practicing engineers for a critical appraisal of the current situation in pipe network optimization.

In preparation for "The Battle of the Network Models" each of the participants (or participating groups) solved a problem of sizing p u m p s , tanks, and water mains for additions to the same hypothetical in-place water system. The systems given had features and problems typical of

'Research Civ. Engr., U.S. Army Corps of Engrs., Waterways Experiment Station, Vicksburg, MS 39180.

2Prof. of Civ. Engrg. and Environmental Studies, Dept. of Civ. Engrg., Univ. of Illinois, Urbana, IL 61801.

3Assoc. Prof., Dept. of Civ. Engrg., Colorado State Univ., Ft. Collins, CO 80523. 4Assoc. Prof., Dept. of Civ. Engrg., Univ. of Manitoba, Winnipeg, Manitoba,

R3T 2N2, Canada. 5Prof. of Civ. and Environmental Engrg., Utah State Univ., Logan, UT 84322. 6Research Asst., Dept. of Civ. Engrg., Univ. of Texas, Austin, TX 78712. 7Univ. Fellow, Dept. of Civ. Engrg., Univ. of Illinois, Urbana, IL 61801. "Prof, of Environmental Engrg., Univ. of Illinois, Urbana, IL 61801. 'Assoc. Prof., Dept. of Civ. Engrg., Univ. of Texas, Austin, TX 78712. 10Civ. Engr., MacLaren Engrs., Winnipeg, Manitoba, R3L 2T4, Canada. "Asst. Prof., Dept. of Civ. Engrg., Univ. of Kentucky, Lexington, KY 40506-

0046. Note.—Discussion open until August 1, 1987. To extend the closing date one

month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on November 20, 1985. This paper is part of the Journal of Water Resources Planning and Management, Vol. 113, No. 2, March, 1987. ©ASCE, ISSN 0733-9496/ 87/0002-0191/$01.00. Paper No. 21314.

191

J. Water Resour. Plann. Manage. 1987.113:191-203.

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those found in many real systems, although in the given system smaller pipes not contributing greatly to flow and pressure distribution were removed. Each of the participants applied a different procedure generally developed from their own research for solving the problem. They reported their results at the conference in Buffalo.

Each of the participants also prepared a paper describing their solution approach for the conference proceedings (Brill, Liebman, and Lee 1985; Gessler 1985; Jeppson 1985; Lansey and Mays 1985; Morgan and Goulter 1985b; Ormsbee 1985). The first writer prepared a discussion of the literature for the conference proceedings (Walski 1985c) and prepared the example problem that was solved by the other authors.

Since the problem statement was not released until approximately two months before the conference, the problem statement and solutions could not be included in the proceedings. However, it was felt that it would be valuable to publish the problem and solutions so that the various methods could be compared. Furthermore, other approaches to solving pipe sizing problems could be benchmarked against these solutions. This paper was prepared to describe the findings of the sessions.

The problem statement presented to the session participants is given below, followed by their solutions and a discussion of the results.

PROBLEM STATEMENT

The problem focuses on the water distribution system of a hypothetical community, Anytown, U.S.A. The distribution system is shown by the pipe network in Fig. 1. The town takes its water from a river and treats it at a central plant. Three identical pumps connected in parallel take water from the clearwell at the treatment plant and pump it into the system.

FIG. 1.—-Existing Network

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1 The town originally developed around 1910 to the southeast of pipe i link 28. Most of those pipes are old cast iron with low Hazen-Williams

C-factors. There are many buried utilities in the old central city (inside dashed area in figure), which make excavation much more difficult than in the surrounding residential areas.

After World War II, the town began to grow again to the northwest and west. In particular some industries have located near node 160, and a tank was recently erected there. The utility is presently having trouble filling this tank. Nodes 65 and 165 are elevated tanks with 250,000-gallon (1,136-m3) capacity in each, while node 10 is a clearwell at the water treatment plant.

There are plans for a new industrial park to be developed to the north of town. The old central city and east side are not expected to grow significantly in the future. The city is planning to issue bonds to cover water distribution system upgrades that will meet the needs of the city through 2005.

The problem is to select: (1) New pipes, pumps, and tanks; and (2) 1 pipes that need to be cleaned and lined to meet minimum pressure re

quirements at minimum cost. Costs should include all construction costs and energy cost. Pipe will be placed in the existing right-of-way so there is no right-of-way acquisition cost. New pipes may be placed in parallel with existing pipes, which are shown as solid lines in the figure. Pipe 54 is to replace an existing 2-in. pipe and cannot be eliminated.

Pipe characteristics are shown in Table 1. The C-factors given are val-1 ues projected for the year 2005 based on extrapolating current trends.

Pipes 78 and 80 are riser pipes to the elevated tanks. All three pumps have identical pump characteristic curves as shown in Table 2. The ef-

i ficiencies given are wire-to-water efficiencies, i.e., both motor and pump efficiencies. The rated capacity of each pump is 4,000 gpm (0.252 m3/s).

Average daily water use at each node is given in Table 3 for years 1985 and 2005. The elevation of each node is also given in the table. In the case of the tanks, the elevation is that of the bottom of the tank except for the clearwellat the treatment plant for which the water level is maintained at 10 ft (3.04 m). The tanks are full at water level 255 ft (77.7 m). The utility wants to operate the tanks with water levels between elevations 225 and 250 ft (68.6 and 76.2 m).

The system must supply water at a minimum pressure of at least 40 psi (257.8 kPa) at all nodes at instantaneous peak flow, which is 1.8 times average day flow. The system must provide at least 20 psi (137.9 kPa) at all nodes while meeting fire flows. The fire flow required is 500 gpm (0.0316 m3/s) at all nodes except for: (1) 2,500 gpm (0.158 m3/s) at node 90; (2) 1,500 gpm (0.0946 m3/s) at nodes 75, 115, and 55; and (3) 1,000 gpm (0.0631 m3/s) at nodes 120 and 160. The fire flows must be met while also supplying peak day flows, which are 1.3 times average

i flow. All of the pressure requirements must be met with one pump out of service and the tank water levels at their low level during a normal day.

| The costs for pipe laying, pipe cleaning and lining, and pressure reducing valves (PRV's) are given in Table 4. Costs are given in $/ft except those for the PRV's which are given in dollars. Cost for new pipes (pipes 54, 68-76) are given in the column labeled "New" (Col. 4). Costs for

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TABLE 1.—Pipe Characteristics

Pipe no. (1) 2 4 6 8

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80

Diameter (in.) (2)

16 12 12 12 12 10 12 10 12 10 10 10 12 10 10 10 10 10 10 10 8 8 8 8

10 8

— 8

10 8 8 8 8

•— — •— — — 12 12

Length (ft) (3)

12,000 12,000 12,000 9,000 6,000 6,000 6,000 6,000 6,000 6,000 6,000 6,000 6,000 6,000 6,000 6,000 9,000 6,000 6,000 6,000 6,000 6,000 6,000 6,000 6,000 6,000 9,000 6,000 6,000 6,000 6,000

12,000 12,000 6,000 6,000 6,000 6,000 6,000

100 100

C-factor (4)

70 120 70 70 70 70 70 70 70 70 70 70 70 70

120 120 120 120 120 120 120 120 120 70

120 120 130 120 120 120 120 120 120 130 130 130 130 130 120 120

Note: 1 ft = 0.305 m; and 1 in. = 25.4 mm.

paralleling old pipes in sparse residential areas (pipes 36-66, except 48) are shown in the column labeled "Residential" (Col. 3). Costs for laying pipe in the central city (pipes 2-34, 48) are given in the column labeled "Urban" (Col. 2). Costs for cleaning and lining pipes depend on the land use. Costs for cleaning and lining pipes 2-34 and 48 are given in the column titled "Urban" (Col. 5). Costs for cleaning and lining other pipes

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I ABLE 2.—Pump Characteristics for Individual Pump

Discharge (gpm)

(D 0

2,000 4,000 6,000 8,000

Head (ft) (2)

300 292 270 230 181

Efficiency (%) (3)

0 50 65 55 40

Note: 1 gpm = 6.309 x 10"5 m3/s; and 1 ft = 0.305 m.

are given in the column labeled "Residential" (Col. 6). The final C-factor for pipes that have been cleaned and lined is 125.

The costs for pumping capacity depend on whether a new pumping station is being built or an old one is being upgraded. They are based on the rated discharge and the head of the pumping station. For new equipment the costs can be estimated by

C = 500Q°-7H0-4 (1)

TABLE 3.—Node Characteristics*

No. (1) 10 20 30 40 50 55 60 65 70 75 80 90

100 110 115 120 130 140 150 160 165 170

Average Daily Use

1985 (gpm) (2)

Tank 500 200 200 200 — 500

Tank 500 — 500

1,000 500 500 — 200 200 200 200 800

Tank 200

2005 (gpm) (3)

Tank 500 200 200 600 600 500

Tank 500 600 500

1,000 500 500 600 400 400 400 400

1,000 Tank

400

Elevation (ft) (4)

10 20 50 50 50 80 50

215 50 80 50 50 50 50 80

120 120 80

120 120 215 120

"Critical fires = 2,500 gpm at node 90; 1,500 gpm at node 75, 115, 55; and 1,000 gpm at node 120, 160; Peak day: average flow = 1.3; Instantaneous peak: average flow = 1.8.

Note: 1 gpm = 6.309 x 1Q'5 m3/s; and 1 ft = 0.305 m.

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TABLE 4.—Costs for Pipe Laying, Cleaning, and Lining ($/ft); for Pressure Reducing Valves (PRV's) ($)

Pipe diameter

(in.) 0) 6 8

10 12 14 16 18 20 24" 30"

Urban (2)

26.2 27.8 34.1 41.4 50.2 58.5 66.2 76.8

109.2 142.5

New Pipe

Residential (3)

14.2 19.8 25.1 32.4 40.2 48.5 57.2 66.8 85.5

116.1

COSTS

New (4)

12.8 17.8 22.5 29.2 36.2 43.6 51.5 60.1 77.0

105.5

Clean and Line

Urban (5)

17.0 17.0 17.0 17.0 18.2 19.8 21.6 23.5 — —

Residential (6)

12.0 12.0 12.0 13.0 14.2 15.5 17.1 20.2 — —

PRV's (7)

2,100 3,580 5,600 8,060

11,000 14,340 18,150 22,400

— —

"Added since original problem. Note: 1 ft = 0.305 m; 1 in. = 25.4 mm; 1 gal = 0.00455 m3; and 1 gpm = 6.309

x 10~5 m3/s.

where C = construction cost, in dollars; Q = rated discharge, in gpm; and H = rated head, in ft. For upgrading the mechanical equipment in existing stations, the cost can be estimated as

C = 350Q°-7H°4 (2)

For this problem, the tank cost will be treated primarily as a function of volume, since the height of the tank is already determined by the topography (Table 5). The duration of the required fire flow is 2 hrs. The cost of energy will be taken as $0.12/kWh. The present worth of energy costs will be based on an interest rate of 12% and an amortization period of 20 yrs.

Costs, of course, vary from one utility to another. The values in Table 4 are typical and are based on data from the MAPS Computer Program (USACOE 1980) and Walski (1985a).

The variation in water use throughout the day is given in Col. 2 in Table 6. For example, the value of 1.3 times average use for 9-12 a.m. means that water use is 1.3 times average use during those hours.

The system currently contains no check valves or pressure reducing

TABLE 5.—Tank Cost

Volume of tank (gal) (1) 50,000

100,000 250,000 500,000

1,000,000

Cost ($) (2)

115,000 145,000 325,000 425,000 600,000

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TABLE 8.--Actuai Water Use Pattern

Time of day (1)

12-3 a.m. 3-6 a.m. 6-9 a.m. 9-12 a.m.

12-3 p.m. 3-6 p.m. 6-9 p.m. 9-12 p.m.

Actual average use (2)

0.7 0.6 1.2 1.3 1.2

. 1.1 1.0 0.9

valves. Minor losses are considered to be accounted for in the given equivalent lengths of pipe.

To provide redundancy, every node must be connected to the remainder of the system through at least two pipes. The river is the only water source in the vicinity of the city, as supplemental wells near the new developments and interconnections with other utilities are not feasible.

The hypothetical system was developed to be typical of a small, older city with some of the small pipes removed from the model. One somewhat unrealistic feature is that the existing elevated storage capacity is somewhat low for a city of this size. In most cases such a city would have more than 500,000 gallons (2,273 m3) of elevated storage. The shape of the tanks was also not specified. Two participants assumed they were cylindrical while a third assumed they were spherical. Cylindrical tanks could be used as a reasonable approximation of most tanks. For this problem, none of the in-place pipes are to be abandoned due to breakage or leaks.

While the hypothetical system has many features of real systems, it is not construed to be an all-encompassing typical situation. Similarly, the sample data provided are typical of costs, fire flows, and pressure requirement but are not universally applicable.

SOLUTIONS

Solutions of Gessler; Lee, Brill, and Liebman; Morgan and Goulter; and Ormsbee are summarized in Tables 7 and 8 and shown in Figs. 2 (Gessler; Lee, Brill, and Liebman) and 3 (Morgan and Goulter; Ormsbee). The solution of Lansey and Mays was not included because the program based on their methodology at that time was not entirely completed, and they felt that a partial solution would not be representative of their optimization approach. Jeppson's solution was not included because he interpreted the problem somewhat differently from the others. All of the solutions featured the addition of at least one new tank, and new pipes from the pumping station. No solution, however, included cleaning of old pipe.

Gessler's solution was based on enumerating alternative pipe sizes. Lee, Brill, and Liebman relied on a combination gradient search model to identify optimal head distributions, and a linear programming model

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TABLE 7.—Summary

Researcher (D

Gessler Lee, et al.

Morgan and Goulter

Ormsbee

Diameter (in.) of Pipe Parallel to

2 (2)

16-18

12

20

4 (3)

14 14-16

8

18

6 (4)

24 18-20

20

20

12 (5)

14-16

10

16

18 (6)

6-8

20 (7)

8

22 (8)

8

26 (9)

8

30 (10)

10

32 (11)

10

34 (12)

12-14

42 (13)

14

18

44 (14)

18

48 (15)

8

50 (16)

24 10-12

18

18 aData for three pumps. 'TData for two tanks. Note: 1 in. = 25.4 mm; 1 ft = 0.305 m; 1 MG = 4,546 m3; and 1 gpm = 6.903 x 10"5 m3/s.

to select pipe sizes for each head distribution. Morgan and Goulter used a Hardy-Cross method for flow distributions, and a linear programming model for diameter for each flow distribution. Ormsbee used a Box-Complex search method.

The solutions of Lee, Brill, and Liebman have two diameters listed for each pipe link that is not at minimum diameter. This is due to the pipe-sizing algorithm which allows for continuous pipe diameters. (For example, if the continuous optimal solution is a 14.5-in (0.368-m) diameter pipe, the program would select equivalent lengths of 14-in. (0.356 m) and 16-in. (0.406 m) diameter pipe.) Another unusual feature of Lee, Brill, and Liebman's solution is that they located booster pumps at the base of each tank to pump into each of the three tanks during off-peak hours. After the conference Morgan and Goulter were able to develop a less expensive solution than the one given for them in Tables 7 and 8. They accomplished this by eliminating the booster pumping station by using a 30-in. (0.762-m) pipe for links 6 and 50 and extrapolating the costs in Table 4 to a 30-in. (0.762-m) diameter. Pipe costs in Table 4 needed to include 24- and 30-in. (0.610- and 0.762-m) diameter pipes. (Some of the participants prepared working papers describing their solutions in more detail. Copies may be obtained from the participants.)

The participants arrived at the solutions using virtually the same steps: (1) Select a location for the new tanks; (2) use a pipe-sizing optimization program; and (3) simulate operation over a 24-hr period to select tank

TABLE 8.—Summary of Costs (in $ x 106)

Description d)

Pipes Pump equipment Tanks Energy Total

Gessler (2)

4.5 0.7 0.5 6.6

12.3

Lee, et al. (3)

4.6 1.8 0.5 6.0

12.9

Morgan and Goulter (4)

3.3 2.7 0.7 6.3

13.0

Ormsbee (5)

5.7 0.0 1.8 6.3

13.8

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of Solutions

SOLUTIONS

Pipe No.:

52 (17)

10-12

6

10

54 (18)

8 6

6

20

58 (19)

16 10-12

— 10

60 (20)

14 10-12

—

-

62 (21)

_

12

10

68 (22)

12 10-12

10

10

70 (23)

12 10-12

10

6

72 (24)

6 6

8

12

74 (25)

14 10-12

8

6

76 (26)

6 6

10

—

Node (27)

150 140

170b

75 3.0

Vol. (MG) (28)

0.8 0.7

0.1b

1.0

—

Source Pump

Discharge (gpm) (29)

2,000 4,000

4,000

—

Head («) (30)

270 270

270

—

Booster Pump

Discharge (gpm) (31)

l,000a

1,000 1,000 8,000

—

Head (ft) (32)

12" 12 12 85

size and to insure the solution will work. While the solutions looked considerably different and the algorithms were completely different, the total present worth costs were similar. Most of the differences could be traced to the location of storage tanks, decisions which were made without using optimization models. For example, by locating a tank near node 90 (at node 150), Gessler was able to eliminate the need for new pipes parallel to pipes 2 and 12.

Because of the differences in tank locations, it is not possible to conclude if one approach for selecting pipe sizes is superior to the others.

/ \ GEESLERTANK

^ ^ LEE TANK

FIG. 2.—Solutions of Gessler and Lee (Brill and Liebman)

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^ ^ ORMSBEETANK

FIG. 3.—Solutions of Morgan and Goulter, and Ormsbee

Selection of one model over the others for use by a practicing engineer will be influenced more by such factors as computer time and User-friendliness of the programs which were not evaluated in this paper.

ANALYSIS

Most of the participants were able to develop a solution that would work at peak loading but would not have adequate capacity to fill tanks at off-peak loadings—a fact that would not have been recognized by optimizing the system for a single loading. This demonstrated the need for models to have the capability to handle multiple loadings and perform unsteady simulations.

While energy cost appeared to be the most significant cost item, most of this energy is lift energy (as opposed to energy used to overcome friction) and is independent of pipe size. Since the pumping station will, on the average, lift 6,400-9,800 gpm (0.404-0.619 m3/s) a height of 230 ft (70.1 m) at a wire-to-water efficiency of 0.60 and an energy cost of $0.12/kWh, the present worth lift energy cost, based on 9,800 gpm (0.619 m3/s) will be roughly $5.6 X 106. Pipe sizing will have no effect on this value, but will only affect energy use beyond this minimum energy. While the energy cost is a major item, it is not highly affected by pipe sizing, which affects only friction losses, except insofar as pipe sizes affect pump efficiency, which in turn can significantly affect costs.

On the other hand, tank construction, which is a fairly minor item in itself, can affect both the nature of the solution and the cost. One instance occurs in the area around node 170. This is an area with low

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wafer use and high elevation with pressure problems at peak use—a typical headache for a design engineer. Most of the participants used large pipes to minimize head loss to this node, e.g., Ormsbee used 10-and 20-in. (0.254- and 0.508-m) pipes to minimize head loss at a cost of $550,000. Morgan and Goulter installed a small tank to shave peaks at a cost of $100,000. At present, the models used in this session cannot size tanks or identify specific locations- where tanks can result in cost savings. Morgan and Goulter's insight saved a considerable amount of money with no decrease in service.

Some of the differences in the solutions are due to the interpretation of safety factors. For example, Ormsbee added 3,000,000 gallons (13,638 m3) of new storage to the system. This alone explains most of the differences in cost between his solution and the others. In another paper, Walski (1985b) discusses how ambiguity in performance criteria for distribution systems affects design. This ambiguity, especially with regard to reliability requirements, causes engineers to include excess capacity in systems. On the other hand, optimization models tend to remove excess capacity from designs. More precise statements of reliability requirements would make results of optimization models more meaningful.

Pipe network optimization models, when used to solve single loading problems, will result in branched rather than looped systems unless the solution is constrained to force looping. It is interesting, therefore, to note that both Lee, Brill, and Liebman's and Ormsbee's solutions contain loops. These loops apparently arise from the fact that the system must work over a wide range of loadings.

While there are differences in the solutions, all the methods used gave solutions that are within 10% of each other in total cost. The differences in designs were due primarily to differences in sizing and locating storage facilities and different approaches to providing reliability. Such differences occur in any study of distribution systems sizing due to varying interpretations of performance criteria, regardless of whether optimization models are used.

So, while the pipe network optimization programs can assist the engineer in selecting pipe sizes, a great deal of engineering judgment and experience is needed to determine a low-cost workable solution.

While it is not currently practical to prove that any of the solutions is the true, global optimum, participants felt that the solutions they presented were significant improvements over solutions they would have developed without the aid of the optimization models. In all cases, the models were unable to optimize the system in one step. Instead, the models were used to solve individual subproblems in an efficient manner. The models still required that users exercise a good deal of engineering judgment and manual calculations to solve portions of the problems. Tank location and tank sizing are examples of portions of the problem solved using engineering judgment and manual calculations, respectively.

Several important observations can be drawn from the experiences of the participants. One of the most important was that tank sizing and location can significantly affect the optimal solution. Another is that distribution sizing must not be done without careful consideration of pump

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efficiency. Large costs could be incurred if the pumps operated at an inefficient operating point.

One of the first distribution systems to be studied using optimization models was the New York tunnel system (Lai and Schaake 1969). Later, Gessler (1982), Morgan and Goulter (1985a) and Quindry, Brill, and Liebman (1981) used this system to benchmark their approaches. The New York tunnel problem had some simplifying features which made it relatively easy to optimize. The problem presented in this paper does not contain many of those simplifications and, thus, more closely resembles a real water distribution system. Some important differences are described as follows:

1. The New York system was designed for a single loading (water use distribution). Real systems must operate over a wide range of loadings. Peak flow alone cannot be used to size distribution systems, especially those with pumps, since pump stations almost never operate at peak capacity.

2. The New York tunnel system did not include pumps. Most real systems do, and pump sizing and/or operation are not known beforehand.

3. The New York tunnel problem did not include storage tank sizing and location, both of which dramatically influence pipe sizing.

4. The New York tunnel problem did not include the possibility of cleaning and lining in-place pipes as an alternative to new pipes.

5. The New York tunnel problem did not consider the fact that the unit prices of pipe are not only a function of diameter, but also of such parameters as the location in the system.

Water distribution system optimization models are more likely to become tools of practicing engineers if they can handle more complicated problems than have traditionally been used to benchmark the models. Even though the system described in this paper does not contain all of the features of real systems (e.g., multiple pressure zones, seasonal and local demand fluctuations, fiscal constraints, uncertainty of future demands and pipe roughness, and complicated staging of construction), it should serve as a challenging benchmark for optimization models that are able to consider many real-system features such as pump and tank sizing and location.

SUMMARY AND CONCLUSION

Water distribution system optimization models can help engineers in sizing water distribution systems. However, at present many portions of the problem must be solved using models in conjunction with manual calculations and engineering judgment. As model development continues, the models should be able to solve more portions of the problem and be applicable to more and more complicated systems.

There is still considerable room for engineers to interpret water system performance criteria. This creates additional difficulties in determining whether a solution is the optimal solution.

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APPENDIX.—REFERENCES

Brill, E. D., Liebman, J. C , and Lee, H. L. (1985). "Optimization of looped water distribution networks," Computer applications in water resources, H. C. Torno, Ed., ASCE, New York, N.Y., 569-571.

Gessler, J. (1982). "Optimization of pipe networks," International Symposiums on Urban Hydrology, Hyraulics and Sediment Control, Univ. of Kentucky, Lexington, Ky., 165-172.

Gessler, J. (1985). "Pipe network optimization by enumeration," Computer applications in water resources, H. C. Torno, Ed., ASCE, New York, N.Y., 572-581.

Jeppson, R. W. (1985). "Practical optimization of looped water systems," Computer applications in water resources, H. C. Torno, Ed., ASCE, New York, N.Y., 723-731.

Lai, D., and Schaake, J. (1969). "Linear programming and dynamic programming applied to water distribution network design," Hydrodynamics Lab Report 116, M.I.T., Cambridge, Mass.

Lansey, K., and Mays, L. (1985). "A methodology for optimal network design," Computer applications in water resources, H. C. Torno, Ed., ASCE, New York, N.Y., 732-738.

"Methodology for areawide planning studies," (1980). Engineer Manual 110-2-502, U.S. Army Corps of Engineers, Washington, D.C.

Morgan, D. R., and Goulter, I. C , (1985a). "Optimal urban water distribution design," Water Resource Res., 21(5), 642.

Morgan, D. R., and Goulter, I. C. (1985b). "Water distribution design with multiple demands," Computer applications in water resources, H. C. Torno, Ed., ASCE, New York, N.Y., 582-590.

Ormsbee, L. (1985). "OPNET: a nonlinear design algorithm for hydraulic networks," Computer applications in water resources, H. C. Torno, Ed., ASCE, New York, N.Y., 739-748.

Quindry, G. E., Brill, E. D., and Liebman, J. C. (1981). "Optimization of looped water distribution systems," /. Environ. Engrg. DID., 107(4), 665-680.

Walski, T. M. (1985a). "Cost of water distribution system infrastructure rehabilitation, repair and replacement," Technical Report EL-85-5, U.S. Army Corps of Engineers, Waterways Experiment Station, Vicksburg, Miss.

Walski, T. M. (1985b). "Performance criteria for water distribution system," Proceedings of the American Water Works Association Annual National Convention, Washington, D.C, 15-21.

Walski, T. M. (1985c). "State-of-the-art: pipe network optimization," Computer applications in water resources, H. C. Torno, Ed., ASCE, New York, N.Y., 559-568.

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battle of the network models: epilogue

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