Multilinear Method for Hydraulic Analysis of Pipe Networks
Publication: Journal of Irrigation and Drainage Engineering
Volume 143, Issue 8
Abstract
Steady-state modeling of water distribution networks (WDNs) is the calculation of flow rates in pipes and nodal pressures for a given set of boundary conditions (i.e., water levels, pump curves, nodal demands, and so forth). The solution procedure is based on simultaneous solving of the energy and mass conservation equations in the network. There is a Newton-based method entitled the global gradient algorithm (GGA) for solving these equations and it is used for many commercial software programs. The GGA solves a positive definite symmetric linear system for finding head pressures and also updates the discharge of pipes at each iteration. After a predefined number of iterations, the GGA converges to the final solution quickly and it is the most efficient algorithm for WDN modeling. In this paper, for improving the convergence rate, a new method called the multilinear technique is presented that solves the nonlinear system of equations. In this method, nonlinear terms of energy equations are linearized based on maximum and minimum allowable discharge in pipes. Therefore, the set of continuity and energy equations is converted into a linear system and by solving this linear system a good initial solution is obtained. Then, this new solution is used for the linearization process in the next iteration. The process continues until convergence with the final solution with reasonable accuracy. To demonstrate the robustness and effectiveness of the multilinear algorithm, several real and hypothetical WDNs from a small to a large scale are tested. Results in benchmark and real networks show that after two iterations the multilinear algorithm converges with acceptable precision. The simulation of 1,000 hypothetical networks shows that the computational efficiency of the multilinear method in terms of time is almost half of those obtained by the GGA.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
I would like to thank Prof. Ruth Derksen, instructor of writing and rhetoric in both the English Department and in the Faculty of Applied Science at the University of British Columbia (UBC) for comments that greatly improved the manuscript.
References
Basha, H. A., and Kassab, B. G. (1996). “Analysis of water distribution systems using a perturbation method.” Appl. Math. Modell., 20(4), 290–297.
Cross, H. (1936). “Analysis of flow in networks of conduits or conductors.”, Univ. of Illinois, Engineering Experimental Station, Urbana, IL.
Giustolisi, O. (2010). “Considering actual pipe connections in WDN analysis.” J. Hydraul. Eng., 889–900.
Giustolisi, O., Laucelli, D., Berardi, L., and Savić, D. A. (2012). “A computationally efficient modeling method for large size water network analysis.” J. Hydraul. Eng., 313–326.
Giustolisi, O., and Moosavian, N. (2014). “Testing linear solvers for global gradient algorithm.” J. Hydroinf., 16(5), 1178–1193.
Guidolin, M., Kapelan, Z., and Savic, D. (2013). “Using high performance techniques to accelerate demand driven hydraulic solvers.” J. Hydroinf., 15(1), 38–54.
Isaacs, L. T., and Mills, K. G. (1980). “Linear theory methods for pipe network analysis.” J. Hydraul. Div., 106(7), 1191–1201.
KYPIPE [Computer software]. KYPipe, Cary, NC.
Li, X., Mu, C., Ma, J., and Wang, C. (2010). “Sixteenth-order method for nonlinear equations.” Appl. Math. Comput., 215(10), 3754–3758.
Liu, K. T. H. (1969). “The numerical analysis of water supply networks by digital computers.” Proc., 13th Congress of IAHR, 1, 25–42.
Martin, D. W., and Peters, G. (1963). “The application of Newton’s method to network analysis by digital computer.” J. Inst. Water Eng. Scientists, 17(2), 115–129.
MATLAB [Computer software]. MathWorks, Natick, MA.
Moosavian, N., and Jaefarzadeh, R. (2014). “Hydraulic analysis of water supply networks using a modified Hardy cross method.” Int. J. Eng., 27(9), 1331–1338.
Moosavian, N., and Lence, B. (2016). “Non-dominated sorting differential algorithms for multi-objective optimization of water distribution systems.” J. Water Res. Plann. Manage., 04016082.
Moosavian, N., and Roodsari, B. K. (2014). “Soccer league competition algorithm: A new method for solving systems of nonlinear equations.” Int. J. Intell. Sci., 4(1), 7–16.
Ormsbee, L. E. (2006). “The history of water distribution network analysis: The computer age.” 8th Annual Water Distribution Systems Analysis Symp., Cincinnati.
Rossman, L. A. (2000). “Epanet2 user’s manual.” U.S. Environmental Protection Agency, Cincinnati.
Salgado, R., and Todini, E., and O’Connell, P. E. (1987). “Comparison of the gradient method with some traditional methods for the analysis of water supply and distribution.” Computer applications in water supply, Vol. 1 (System analysis and simulation), Wiley, Hoboken, NJ.
Savić, D. A., and Banyard, J. K. (2011). Water distribution systems, ICE Publishing, London.
Todini, E. (2006). “On the convergence properties of the different pipe network algorithms.” 8th Annual Water Distribution Systems Analysis Symp., Cincinnati.
Todini, E., and Pilati, S. (1988). “A gradient method for the solution of looped pipe networks.” Int. Conf. on Computer Applications in Water Supply, System Analysis and Simulation, B. Coulbeck and C. H. Orr, eds., Vol. 1, Wiley, London, 1–20.
Todini, E., and Rossman, L. (2013). “Unified framework for deriving simultaneous equation algorithms for water distribution networks.” J. Hydraul. Eng., 511–526.
Wood, D. J. (1980). “User’s manual-computer analysis of flow in pipe networks including extended period simulations.” Univ. of Kentucky, Lexington, KY.
Wood, D. J., and Charles, C. O. A. (1972). “Hydraulic network analysis using linear theory.” J. Hydraul. Div., 98(7), 1157–1170.
Wood, D. J., and Charles, C. O. A. (1973). “Closure to discussions to Wood and Charles (1972): Hydraulic network analysis using linear theory.” J. Hydraul. Div., 99(11), 21–29.
Wood, D. J., and Funk, J. E. (1993). “Hydraulic analysis of water distribution systems.” Proc., Int. Conf. on Water Supply Systems, State of the Art and Future Trends, E. Cabrera and F. Martinez, eds., Computational Mechanics Publications, Valencia, Spain, 41, 85.
Wu, Z. Y., Wang, R. H., Walski, T. M., Yang, S. Y., Bowdler, D., and Baggett, C. C. (2009). “Extended global-gradient algorithm for pressure-dependent water distribution analysis.” J. Water Res. Plann. Manage., 13–22.
Zecchin, A., Thum, P., Simpson, A., and Tischendorf, C. (2012). “Steady-state behavior of large water distribution systems: Algebraic multigrid method for the fast solution of the linear step.” J. Water Res. Plann. Manage., 639–650.
Information & Authors
Information
Published In
Copyright
©2017 American Society of Civil Engineers.
History
Received: Oct 11, 2016
Accepted: Jan 27, 2017
Published online: May 2, 2017
Published in print: Aug 1, 2017
Discussion open until: Oct 2, 2017
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.