Mathematical efficiency concerns in water distribution network considerations1

SWAPAN GUPTA,' EDWARD A. MCBEAN, A N D J. RICHARD COUSINS~ Deprrrttnent of Civil Engineering, Ut~iversity of Waterloo, Waterloo, Ont. , Cmada N2L 3GI

Received January 8, 1979

Revised manuscript accepted October 19, 1979

Because the governing mathematical equations for water distribution networks are nonlinear, many computerized methods of solution have been proposed as the "best" method of solving these equations. A comparison of some of the more popular methods indicates that little differ- ence exists between the methods, although a slight overall edge is available with the New- ton-Raphson technique. The exponential increase in computer model cost with increasing network size is demonstrated. The utility of network schematization models is documented, particularly as employed in design studies.

Etant donne que les equations mathematiques qui dkcrivent les reseaux de distribution d'eau sont non lineaires, beaucoup de mkthodes de solution sur ordinateur ont ete proposkes comrne Ctant la "meilleure" methode pour resoudre ces equations. Une comparaison de certaines des methodes les plus populaires denote qu'il y a trkspeu de difference entre les rnethodes, bienque la technique Newton-Raphson presente un leger avantage sur I'ensemble. L'augmentation ex- ponentielle du codt de simulation par ordinateur est demontree avec I'accroissement de la grandeur du reseau. L'utilisation de modkles de schCmatisation pour reseaux est documentee, relatif surtout B leur emploi pour les etudes d'avant projet.

Can. J. Civ. Eng., 7,78-83 (1980) [Traduit par la revue]

Introduction Sizeable expenditures continue to be made in the

design and construction of water distribution net- works. The considerable size of these expenditures has allowed small or incremental improvements in design to translate into many dollars of overall saving. In this search for incrementally improved designs during the last I + decades, computer models have been extensively employed. However, because of the nature of the equations, there are a number of alternative methods that have been com- puterized for solving the distribution network problem. Of interest in this paper is whether there are important economies of one method as compared with another that would enhance the search for improved designs. The comparisons of three of the most popular computerized methods are developed herein through application to three southern Ontario cities and subsets thereto. The case study applications encompass a wide range of network con- figurations and sizes.

Analysis Versus Design Formulation of a mathematical model of a water

distribution network requires the use of 2N + M

'Presented at the Canadian Society for Civil Engineering Annual Conference, Montreal, P.Q., June 7 and 8, 1979.

2Present address: University of Ottawa, Ottawa, Ont., Canada KlH 6S6.

3Present address: Acres Consulting Services Ltd., 5259 Dorchester Rd., Niagara Falls, Ont., Canada L2E 6W1.

variables to describe the system: a pressure head and consumption rate a t each of N nodes or con- fluences of pipes and a resistance for each of the M pipes (see Gupta et al. 1979a for more specific mathematical details). To be able to solve the N equations that can be used to describe the network, no more than N of the variables can be unknown. Frequently the equations are solved by assuming pipe sizes, pipe materials, and the consumption at each node, leaving a vector of unknown pressure heads. Note that this is a question of analysis since the network system is completely specified and the problem involves determining how the specified net- work system performs-for example, i f adequate pressures are maintained throughout the distribution system.

An alternative question to that of analysis is one of design: to determine the best set of pipes (where "best" is normally determined on the basis of minimum cost) subject to constraints stipulating a minimum acceptable system performance-(e.g., that minimum pressures are maintained throughout the network). The analysis question is therefore a subset of the design question since the design question involves requirements for both satisfactory per- formance and best system characteristics, usually "least cost. "

Although many different computer models have been suggested to address the design problem, they can be classified as being either one of two types:

(i) A formal model based on mathematical pro-

03 15- 14681801010078-06$01 .00/0 @ 1980 National Research Council of CanadaIConseil national de recherches du Canada

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GUPTA ET AL.

TABLE 1. Criteria for comparison of different program methods

Criterion Brief explanation

I. Characteristics Computer time of different methods cox storage of solution requirement

Convergence

I1 Computer model Input data finesse procedures

Model fidelity

Processing limitations

Model efficiency

Computer output

Documentation

Computational effort necessary Necessary size of computing

facility Speed and likelihood of conver-

gence of the iterative solution

Ease of data input, identification of input inconsistencies

Extent of the complexity (e.g., pumps, valves) reflected by the computer model

Size of distribution system able to be handled

Reflection of advanced tech- niques such as sparse matrix procedures

Understandability of the produced information

Extent of aid given to the user of the program

gramming principles (e.g., dynamic or linear pro- gramming) where the algorithm implicit in the model searches out the best solution. A number of these formal models have been suggested (e.g., Smith 1966; Schaake and Lai 1969; Karmeli et al. 1968; Kolhaas and Mattern 1971; Deb and Sarkar 1971; Cembrowicz and Harrington 1973; Yang 1974; Watanatada 1973).

(ii) A successive use of the analysis approach where new pipe sizes are adopted, the flows and pressures calculated, and the adequacy of the response is determined. Of the combinations at- tempted, the set of pipe sizes that best satisfies the objective function (e.g., minimum cost) and does not violate constraints is selected as the best design. The technical literature includes a number of approaches based on this methodology (e.g., Shamir 1974; Swamee et al. 1973; Swamee and Khanna 1975; Deb 1974).

More widespread utilization of the type (ii) approach has occurred than of type (i) in the design problem because of the "cussedly nonlinear" characteristics of the design question; the non- linearities introduce a significant computational difficulty in the formal models (for further details see, for example, Schaake and Lai 1969). The result is that the solution technique used in (ii) in the design stage, to solve the set of network equations, will be necessarily used many times. Therefore, any efficiency that can be gained in determining the solu- tion will improve the design process since more combinations (permutations) of pipe sizes can be

attempted and more cost-effectiveness in design attained.

Methods for Solution

For both the design problem and the subset thereof (the analysis questions) there are N equa- tions with N unknowns that must be so lved . -~n important concern is that these N equations com- prise a set of nonlinear equations requiring an iterative solution. The re~etitiveness of the iterative calculations has made t i e problem ideal for com- puterization; consequently, computers have played a fundamental role over the last 15 years in water distribution investigations.

However, the answer to the question of which of the available techniques for solving the nonlinear equations is the most cost-effective has not yet been resolved. Some of these techniques include: (i) the Hardy-Cross method; (ii) the Newton-Raphson method; (iii) the modified Newton-Raphson method; and (iv) the linear algebra method. It is important to note that each method, or algorithm, solves exactly the same set of equations. The dif- ferences between the methods arise because of the manner in which the equations are solved.

As a means of comparing the different methods of solution, consider the criteria listed in Table 1. Note that two categories of criteria are used: one that is specific t o the method or algorithm, and one that contains a much broader list-of considerations that make a computer model more or less difficult to use. In the technical literature, comparisons of different

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80 CAN. J. CIV. ENG. VOL. 7. 1980

solution methods, such as those named above, are frequently derived on the basis of computer model finesse aspects, where many of the finesse aspects are simply the result of the effort expended in the computer model development. The interest in this paper is concentrated more specifically on the charac- teristics of the different methods of solution and less on the computer model finesse.

A detailed comparison of the mathematics im- plicit in the different methods of solution is beyond the scope of this paper; for further information see Gupta et al. (1979a). However, several brief points are worthy of mention :

(a) The Hardy-Cross technique works with one equation at a time with the result that small amounts of computer storage are required. However, many iterations on individual equations may be required to achieve convergence to the solution of the non- linear equations.

(b) The Newton-Raphson technique, instead of working with individual equations as the Hardy- Cross technique does, deals with the entire network, or system of equations, simultaneously. Thus each iteration, which involves the problem of inverting a large matrix, is a major portion of the computational work (Donachie 1974) and requires large computer storage levels because of the matrix size. Therefore, although the governing equations are identical for the Newton-Raphson and Hardy-Cross techniques, a comparison of the methods on the basis of number of iterations is inappropriate-the nature of the iterations is different.

(c) The modified Newton-Raphson technique is identical to (b) except that matrix inversion is not performed at every iteration.

(d) The linear algebra technique also solves the same nonlinear equations but does so by solving linearized versions of the equations and successively updating the linear equations until convergence is attained. For further details see Epp and Fowler (1970).

Computer Model Finesse

Recall from Table 1 that the criteria for com- parison of the different computer programs were subdivided into two groups, one on the basis of the different methods or algorithms and the other on the basis of the computer model finesse. It should be noted that all of the methods briefly described above are adaptable to computer programming techniques that increase their user efficiencies. However, it is quite often the case that only the more recent algorithms such as the Newton-Raphson method receive the effort required to reduce, for example, user handling.

Case Study Application

As seen from the above, a number of alternatives exist for analyzing water distribution networks. The question as to whether it makes any difference which technique is used is multifaceted. One criterion may be minimum computer cost; however, computer costs are a function of the size of the system being examined. Furthermore, additional costs implicit in the overall cost of computer model usage must include such factors as the input data preparation costs, the availability of a computer with sufficient core storage space, the detail of model output, etc.

In order to answer strictly the economic questions introduced above, a series of case study analyses was carried out on the distribution networks of three southern Ontario communities and an artificial or synthesized distribution system. A comparison of results for the first three cities provides some indica- tion of the variations in costs realized because of the different networks that necessarily arise with dif- ferent regions being serviced; the artificial system (developed from a procedure initially suggested by Andres 1971 and slightly modified by Gupta 1978) provides a "richer" set of conclusions because of the breadth of analyses that were undertaken.

More specifically, the distribution system for city 1 consists of 68 nodes and 101 pipes fed by two pumps and a reservoir. The system for city 2 has 101 nodes and 143 pipes supplied by three reservoirs and the system for city 3 contains 130 nodes and 185 pipes supplied by three reservoirs.

Illustrated in Fig. 1 are the relative economics of the three alternative methods of solution resulting from application to the distribution system of city 1. The points on the graph indicating numbers of nodes less than 68 in number were obtained by arbitrary selections of portions of the distribution system of city 1 and therefore are indicative of smaller distri- bution networks. The execution costs on the right ordinate of Fig. 1 include the central processing unit charges, the computer core storage costs, the load- module costs, and reader and printer costs; the total costs include the execution costs and also the input data preparation costs based on a punching rate of one card per minute and labour at the rate of $10/h.

The differences in cost of utilizing the Hardy-Cross and Newton-Raphson programs are seen to be fairly minimal; the linear algebra program becomes progressively more unfavourable with increases in the number of nodes.

Given the above, a comment might then be that these results are only valid for that particular dis- tribution network and subsets thereto. As an indica- tion of the differences in costs for analyzing alterna- tive network systems (with their concomitant varia-

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GUPTA E T AL. 8 1

b AEWTON-RAPHSON PROGRAM

-LINEAR ALGEBRA PROGRAM 0 ARTIFICIAL DISTRIBUTION NETWORK ' h ClTY No3 DISTRIWTION NETWORK

0

- TOTAL COST 0

0 A

0 A EXECUTION COSTS

NUMBER OF NODES

FIG. 1. Costs of analyzing the distribution of city 1 using the three solution methods.

tions in configuration, pipe sizes, source locations, etc.), the results of applying the most efficient of the methods, the Newton-Raphson technique, are illustrated in Fig. 2. As seen from the figure the extent to which costs are a function of the distribu- tion network being analyzed is marginal.

Illustrated in Fig. 3 are the execution costs and total costs as a function of the size of the system being analyzed, using the Newton-Raphson tech- nique. Evident from the figure is the exponental increase in costs as a result of increased network distribution size: there is obviously a substantial

NUMBER OF NODES

FIG. 3. Costs of analyzing water distribution networks (artificial and real).

return to schematizing a network to a smaller size (where the simplest schematization approach is simply to neglect any pipe lower than a specified size. More complex schematization procedures are described in Gupta et al. 19796). The artificial net- work costs are seen to be comparable to the costs for solution of the real system lending credence to the exponential growth in cost characterization; this comparability in behaviour supports the suggestion that costs are quite insensitive to the network being analyzed and that substantial savings can be attained through network schematization.

If, as may be the case in a design study, the computer model is to be run a number of times, the execution cost as a proportion of the total cost will

- 0 - ClTY N a I DISTRIBUTION NETWORK

A - ClTY NO.2 DISTRIBUTION NETWORK 0 - ClTY N a 3 DISTRIBUTION NETWOAK 0

Q

A 7 A TOTAL CCGT

change dramatically because the input preparation costs will be required only once with small sub- sequent adjustments. An indication of this relative cost of computing versus total cost is provided in Table 2. As the number of computer runs increases, the relative proportion of the total costs arising from execution costs increases dramatically.

I I . I I I I I 0 2 0 40 6 0 80 100 120

NUMBER OF NODES An alternative interpretation of the effect of net- work distribution size is the number of iterations required for convergence. As seen from Fig. 4, the

FIG. 2. Costs of analyzing real networks (Newton-Raphson program).

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82 CAN. J . CIV. ENG. VOL. 7, 1980

TABLE 2. Execution costs and total costs for different numbers of computer runs

ClTY No. l DISTRIBUTION NETWORK

0 MAXIMUM PERMISSIBLE E R R . . 0.1 g l m

A MAXIMUM PERMISSIBLE ERROR 0 0001 p / m 5 0

No. of computer Execution Total Execution Ratio = -

runs costs* costs* Total

For 33 nodes 1 2.04 39.02 0.05

20 40.8 63.48 0.64 50 102.00 124.68 0.82

For 130 nodes 1 5.76 82.44 0.07

20 115.20 137.88 0.84 50 288.00 310.68 0.93

Z

COSTS

*All entries in terms of dollars.

EXECUTION I MAXIMUM ERROR . 0.1 p / m

Q- ClTY No. 3 DISTRIBUTION NETWORK

10 &CITY No.1 DISTRIBUTION NETWORK

FIG. 5. Costs of analyzinn a real network with network size

I I I I I I I

and maximum acceptable e;ror.

0

0 MAXIMUM ERROR . 0.0001 g / m - A 0 . 0 0 1 g / m 0.01 g / m 0.1 g /m

0 10 2 0 3 0 4 0 5 0 6 0 70

NUMBER OF NODES

x % I ClTY No.1 DISTRIBUTION NETWORK

2 0 4 0 6 0 8 0 100 I 2 0 140

NUMBER OF NODES

FIG. 4. Variation of number of iterations with network size (real networks).

number of iterations required for the Newton- Raphson procedure to attain convergence for the networks (and subsets thereto) of the cities 1 and 3 remains fairly constant. This result is somewhat surprising in that it was initially expected that as the

0-

NUMBER OF NODES

network size increased, there would be an increased likelihood of random errors that would cause the number of iterations required to attain convergence to increase; this initial assumption was not borne FIG. 6. Variation of number of iterations with network size

(a real network) and maximum acceptable error. out by experimentation (for further results, see Gupta 1978).

Also of interest in the analysis of distribution Conclusions systems is the cost impact of adjustment of the con- vergence criterion. As seen from Fig. 5, there is a negligible saving obtained by relaxation of the con- vergence criterion as a means of potentially de- creasing the cost of the network analysis. Further, as apparent from Fig. 6, there is only a marginal change in the number of iterations to convergence for the

A number of different techniques exist for analy- zing water distribution networks. All of the tech- niques solve the same set of equations but differ in the means by which this is accomplished. Two of the techniques, namely Newton-Raphson and Hardy- Cross, were seen to have very similar cost responses for different sizes and configurations of distribution systems, although a slight advantage was obtained by the Newton-Raphson technique. Therefore, the

Newton-Raphson technique as a function of the number of nodes being used in the analyses.

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GUPTA ET AL. 83

availability of or familiarity with a computer pro- gram may be very important (and possibly the avail- ability of adequate storage space in the computer) as the determining factor in computer model selec- tion.

It is noteworthy that there is a substantial increase in modelling cost with increases in network size, which suggests there is considerable merit in the development of schematization models that assist in representing the same network but with fewer nodes when the network is to be solved many times. For questions of analysis, where this refers to a determin- ation of how a specified network system performs, schematization has little merit. However, for ques- tions of "design," where this refers to a requirement both to attain a satisfactory performance and to find a system with the best characteristics (such as minimum cost), the return for schematization may be very evident because an extensive number of utilizations of the models will be required.

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