Keywords: demand-driven analysis, head-outflow relationship, nodal outflow function, pressure-dependent consumption, pressure-dependent modelling, water distribution systems.

Several formulae have been assumed in the literature to characterize the relationship between the nodal pressures and outflows of water distribution systems [1,2,3,4,5,6]. These formulae have been defined on the basis that the nodal demand is fully satisfied when the nodal head is greater than the desired head and zero when the nodal head is less than the minimum head. Previous work [7] has shown that the conventional demand-driven analysis approach to network modelling is unsuitable for operating conditions with insufficient pressure. It is widely accepted that water distribution systems may not be able to satisfy all the consumer demands during abnormal operating conditions because the amount of water available at a demand node depends on the pressure at that node.

Some weaknesses of previous head-outflow relationships include the absence of continuity in their derivatives at the transitions between zero and partial nodal outflow and/or between partial and full demand satisfaction. Discontinuities in functions and/or their derivatives can cause convergence difficulties in the computational solution of systems of equations. Also, some functions describing the head-outflow relationship can yield demand satisfaction ratios that either exceed, or never reach, 100%. Similarly, other functions give nodal outflows that are significantly greater than zero when the residual pressure is zero.

This paper presents a new head-outflow relationship. The proposed function has better computational properties than those in the literature and has been incorporated successfully in the system of governing equations for water distribution networks. The computational solution of the system of equations is carried out using a robust, globally convergent Newton-Raphson algorithm. Examples, including networks with pumps and pressure regulating devices, which demonstrate the superiority of the proposed function and the robustness of the overall approach, are included.

The paper includes a comparison of several nodal outflow functions. Based on the examples considered, the proposed function appears to compare favorably against both the conventional demand driven analysis approach on one hand and other nodal performance functions on the other. The results would appear to highlight the computational difficulties associated with discontinuous functions. The new nodal outflow function lends itself to direct incorporation into the system of equations for the computational solution of the latter. Moreover, the relationship proposed is a single function which covers the entire nodal performance range, i.e. zero, partial and full outflow.

The FORTRAN implementation of the solution of the system of equations, PRAAWDS (Program for the Realistic Analysis of the Availability of Water from Distribution Systems), is based on a robust, globally-convergent Newton-Raphson procedure. Evidence of the robustness includes the ability of the program to run smoothly and produce realistic, hydraulically consistent results [7] at extremely low network demand satisfaction levels. The program has an in-built procedure for selecting the initial heads and in general multiple trials are not required.

1

Germanopoulos, G., "A technical note on the inclusion of pressure dependent demand and leakage terms in water supply network models", Civil Engineering Systems, 2, 171-179, 1985. doi:10.1080/02630258508970401

2

Cullinane, M.J., Lansey, K.E. and Mays, L.W., "Optimisation-availability-based design of water-distribution networks", Journal of Hydraulic Engineering, ASCE, 118(3), 420-441, 1992. doi:10.1061/(ASCE)0733-9429(1992)118:3(420)

3

Wagner, J.M., Shamir, U. and Marks, D.H., "Water distribution reliability: simulation methods", Journal of Water Resources Planning and Management. 114(3), 276-294, 1988. doi:10.1061/(ASCE)0733-9496(1988)114:3(276)

4

Reddy, L.S. and Elango, K., "Analysis of water distribution networks with head dependent outlets", Civil Engineering Systems, 6(3), 102-110, 1989. doi:10.1080/02630258908970550

5

Fujiwara, O. and Ganesharajah, T., "Reliability assessment of water supply systems with storage and distribution networks", Water Resources Research, 29(8), 2917-2924, 1993. doi:10.1029/93WR00857

6

Gupta, R. and Bhave, P.R., "Comparison of methods for predicting deficient network performance", Journal of Water Resources Planning and Management, ASCE, 122(3), 214-217, 1996. doi:10.1061/(ASCE)0733-9496(1996)122:3(214)

7

Tanyimboh, T.T., Tahar, B. and Templeman, A.B., "Pressure-driven modelling of water distribution systems", Water Science and Technology - Water Supply, 3(1-2), 255-262, 2003.

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