A Robust, Rapidly Convergent Method That Solves the Water Distribution Equations for Pressure-Dependent Models
Publication: Journal of Water Resources Planning and Management
Volume 142, Issue 2
Abstract
In the past, pressure-dependent models (PDMs) have suffered from convergence difficulties. In this paper conditions are established for the existence and uniqueness of solutions to the PDM problem posed as two optimization problems, one based on weighted least squares (WLS) and the other based on the co-content function. A damping scheme based on Goldstein’s algorithm is used and has been found to be both reliable and robust. A critical contribution of this paper is that the Goldstein theorem conditions guarantee convergence of the new method. The new methods have been applied to a set of eight challenging case study networks, the largest of which has nearly 20,000 pipes and 18,000 nodes, and are shown to have convergence behavior that mirrors that of the global gradient algorithm on demand-dependent model problems. A line search scheme based on the WLS optimization problem is proposed as the preferred option because of its smaller computational cost. Additionally, various consumption functions, including the regularized Wagner function, are considered and four starting value schemes for the heads are proposed and compared. The wide range of challenging case study problems that the new methods quickly solve suggests that the methods proposed in this paper are likely to be suitable for a wide range of PDM problems.
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References
Ackley, J., Tanyimboh, T., Tahar, B., and Templeman, A. (2001). “Head driven analysis of water distribution systems.” Water software systems: Theory and applications, Vol. 1, Research Studies Press, England, 183–192.
Ang, W. K., and Jowitt, P. W. (2006). “Solution for water distribution systems under pressure-deficient conditions.” J. Water Resour. Plann. Manage., 175–182.
Bhave, P. R. (1981). “Node flow analysis distribution systems.” Transp. Eng. J., 107(4), 457–467.
Carpentier, P., Cohen, G., and Hamam, Y. (1985). “Water network equilibrium, variational formulation and comparison of numerical algorithms.” Proc., 7e Conf. Europeenne de Recherche Operationnelle’, EURO VII, EURO—The Association of European Operational Research Societies, Meanwood, Leeds, U.K.
Chandapillai, J. (1991). “Realistic simulation of water distribution system.” J. Transp. Eng., 258–263.
Cherry, C. (1951). “CXVII. Some general theorems for non-linear systems possessing reactance.” London, Edinburgh, Dublin Philos. Mag. J. Sci., 42(333), 1161–1177.
Cheung, P., van Zyl, J. E., and Reis, L. F. R. (2005). “Extension of EPANET for pressure driven demand modeling in water distribution system.” Proc., 8th Int. Conf. on Computing and Control in the Water Industry CCWI05 Water Management for the 21st Century, Vol. 1, Centre for Water Systems, Univ. of Exeter, Exeter, U.K.
Collins, M., Cooper, L., Helgason, R., Kennington, J., and LeBlanc, L. (1978). “Solving the pipe network analysis problem using optimization techniques.” Manage. Sci., 24(7), 747–760.
Dahlquist, G., and Bjork, A. (1974). Numerical methods, Prentice-Hall.
Deuerlein, J. (2002). “On the hydraulic system analysis of water supply networks.” Ph.D. thesis, Dept. of Civil Engineering, Geo- and Environmental Sciences, Univ. of Karlsruhe (TH), Karlsruhe, Germany (in German).
Deuerlein, J., Simpson, A., and Montalvo, I. (2012). “Preprocessing of water distribution systems to assess connectivity and solvability in the presence of flow control devices.” Proc., World Environmental and Water Resources Congress 2012, ASCE, New York, 3237–3247.
Elhay, S., and Simpson, A. R. (2011). “Dealing with zero flows in solving the non-linear equations for water distribution systems.” J. Hydraul. Eng., 1216–1224.
Elhay, S., Simpson, A. R., Deuerlein, J., Alexander, B., and Schilders, W. (2014). “A reformulated co-tree flows method competitive with the global gradient algorithm for solving the water distribution system equations.” J. Water Resour. Plann. Manage., 04014040.
Fujiwara, O., and Ganesharajah, T. (1993). “Reliability assessment of water supply systems with storage and distribution networks.” Water Resour. Res., 29(8), 2917–2924.
Fujiwara, O., and Li, J. (1998). “Reliability analysis of water distribution networks in consideration of equity, redistribution, and pressure-dependent demand.” Water Resour. Res., 34(7), 1843–1850.
Giustolisi, O., and Laucelli, D. (2011). “Water distribution network pressure-driven analysis using the enhanced global gradient algorithm (EGGA).” J. Water Resour. Plann. Manage., 498–510.
Giustolisi, O., Savic, D., and Kapelan, Z. (2008). “Pressure-driven demand and leakage simulation for water distribution networks.” J. Hydraul. Eng., 626–635.
Giustolisi, O., and Walski, T. (2012). “Demand components in water distribution network analysis.” J. Water Resour. Plann. Manage., 356–367.
Goldstein, A. (1967). Constructive real analysis, Dover, Mineola, New York.
Gratton, S., Lawless, A., and Nichols, N. (2007). “Approximate Gauss Newton methods for nonlinear least squares problems.” SIAM J. Opt., 18(1), 106–132.
Jun, L., and Guoping, Y. (2013). “Iterative methodology of pressure-dependent demand based on EPANET for pressure-deficient water distribution analysis.” J. Water Resour. Plann. Manage., 34–44.
Lippai, I., and Wright, L. (2014). “Demand constructs for risk analysis.” Proc. Eng., 89, 640–647.
Matlab [Computer software]. Natick, MA, Mathworks.
Millar, W. (1951). “CXVI. Some general theorems for non-linear systems possessing resistance.” London, Edinburgh, Dublin Philos. Mag. J. Sci., 42(333), 1150–1160.
Muranho, J., Ferreira, A., Sousa, J., Gomes, A., and Sa Marques, A. (2012). “WaterNetGen—An EPANET extension for automatic water distribution networks models generation and pipe sizing.” Water Sci. Technol.: Water Supply, 12(1), 117–123.
Parker, F. (1955). “Integrals of inverse functions.” Amer. Math. Monthly, 62(6), 439–440.
Piller, O. (1995). “Modeling the behavior of a network—Hydraulic analysis and a sampling procedure for estimating the parameters.” Ph.D. thesis, Univ. of Bordeaux, Talence, France.
Piller, O., Bremond, B., and Poulton, M. (2003). “Least action principles appropriate to pressure driven models of pipe networks.” Proc., World Water and Environmental Resources Congress 2003, ASCE, Reston, VA, 1–15.
Piller, O., and van Zyl, J. E. (2014). “Incorporating the FAVAD leakage equation into water distribution system analysis.” Proc. Eng., 89, 613–617.
Rossman, L. (2000). EPANET 2 users manual, Water Supply and Water Resources Division, National Risk Management Research Laboratory, Cincinnati.
Siew, C., and Tanyimboh, T. (2012). “Pressure-dependent EPANET extension.” Water Resour Manage., 26(6), 1477–1498.
Simpson, A. R., Elhay, S., and Alexander, B. (2012). “Forest-core partitioning algorithm for speeding up the analysis of water distribution systems.” J. Water Resour. Plann. Manage., 435–443.
Tabesh, M. (1998). “Implications of the pressure dependency of outflows on data management, mathematical modelling and reliability assessment of water distribution systems.” Ph.D. thesis, Dept. of Civil Engineering, Univ. of Liverpool, Liverpool, U.K.
Tanyimboh, T., and Templeman, A. (2004). “A new nodal outflow function for water distribution networks.” Proc., 4th Int. Conf. on Engineering Computational Technology, Civil-Comp Press, Stirlingshire, U.K.
Todini, E. (2003). “A more realistic approach to the extended period simulation of water distribution networks.” Proc., Computing and control for the water industry, C. Maksimovic, F. A. Memon, and D. Butler, eds., Taylor & Francis, London, 173–183.
Todini, E., and Pilati, S. (1988). A gradient algorithm for the analysis of pipe networks, Wiley, London, 1–20.
Tucciarelli, T., Criminisi, A., and Termini, D. (1999). “Leak analysis in pipeline systems by means of optimal valve regulation.” J. Hydraul. Eng., 277–285.
van Zyl, J., and Clayton, C. (2007). “The effect of pressure on leakage in water distribution systems.” Proc. ICE Water Manage., 160(2), 109–114.
Wagner, J., Shamir, U., and Marks, D. (1988). “Water distribution reliability: Simulation methods.” J. Water Resour. Plann. Manage., 276–294.
Wu, Z., Wang, R., Walski, T., Yang, S., Bowdler, D., and Baggett, C. (2009). “Extended global-gradient algorithm for pressure-dependent water distribution analysis.” J. Water Resour. Plann. Manage., 13–22.
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© 2015 American Society of Civil Engineers.
History
Received: Dec 16, 2014
Accepted: Jun 11, 2015
Published online: Aug 7, 2015
Discussion open until: Jan 7, 2016
Published in print: Feb 1, 2016
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