# American Institute of Mathematical Sciences

June  2012, 2(2): 101-120. doi: 10.3934/mcrf.2012.2.101

## Numerical approximation of an optimization problem to reduce leakage in water distribution systems

 1 Institut de Mathématiques de Bordeaux IMB UMR 5251, Institut Polytechnique de Bordeaux, Université de Bordeaux, F-33405 Talence, France, France 2 Irstea, UR REBX, F-33612 Cestas Cedex, France, France

Received  April 2011 Revised  January 2012 Published  May 2012

Leakage represents a large part of the supplied water in Water Distribution Systems (WDS). Consequently, it is important to develop some efficient strategies to manage such a phenomenon. In this paper an improved formulation of the hydraulic network equations that incorporate pressure-dependent leakage, is presented and validated. The formulation is derived from the Navier-Stokes equations and solved using an adequate splitting method. Then, this formulation is used to study a constrained optimization problem with the objective to minimize the distributed water volume reducing the leakage. The problem is described and validated for academic case studies and real networks.
Citation: Pierre Fabrie, Elodie Jaumouillé, Iraj Mortazavi, Olivier Piller. Numerical approximation of an optimization problem to reduce leakage in water distribution systems. Mathematical Control and Related Fields, 2012, 2 (2) : 101-120. doi: 10.3934/mcrf.2012.2.101
##### References:
 [1] O. Chesneau, "Un Outil d'aide à la Maîtrise des Pertes dans les Réseaux d'eau Potable: La Modélisation Dynamique de Différentes Composantes du Débit de Fuite," Ph.D thesis, University of Strasbourg, Strasbourg, 2006. [2] E. Jaumouillé, O. Piller and J. E. Van Zyl, A hydraulic model for water distribution systems incorporating both inertia and leakage,, in, (): 129. [3] Porteau software, IRSTEA (2011), accessed on March 09, 2012., Available from: \url{http://porteau.irstea.fr/}., (). [4] A. Lambert, What do we know about pressure-leakage relationships in distribution systems?, in "Proc. of IWA Congress: System Approach to Leakage Control and Water Distribution Systems Management," Brno, Czech Republic, (2001), 89-96. [5] J. E. Van Zyl and C. R. I. Clayton, The effect of pressure on leakage in water distribution systems, in "Proc. of CCWI Conference: Water Management for the 21st Century," University of Exeter, UK, 2 (2005), 131-136. [6] R. Glowinski and A. Marrocco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation dualité, d'une classe de problèmes de Dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle RAIRO Analyse Numérique, 9 (1975), 41-76. [7] M. S. Ghidaoui, On the fundamental equations of water hammer,, Urban Water Journal, 1 (): 71. [8] J. T. Beale and A. Majda, Rates of convergence for viscous splitting of the Navier-Stokes equations, Mathematics of Computation, 37 (1981), 243-259. doi: 10.1090/S0025-5718-1981-0628693-0. [9] S. Descombes and M. Massot, Operator splitting for nonlinear reaction-diffusion systems with an entropic structure: Singular perturbation and order reduction, Numerische Mathematik, 97 (2004), 667-698. doi: 10.1007/s00211-003-0496-3. [10] E. Trélat, "Contrôle Optimal. Théorie & Applications," Mathématiques Concrètes, Vuibert, Paris, 2005. [11] M. Bergounioux, "Optimisation et Contrôle des Systemes Linéaires," Dunod, 2001. [12] B. Brémond, P. Fabrie, E. Jaumouillé, I. Mortazavi and O. Piller, Numerical simulation of a hydraulic Saint-Venant type model with pressure-dependent leakage, Applied Mathematics Letters, 22 (2009), 1694-1699. doi: 10.1016/j.aml.2009.02.007. [13] P. Fabrie, G. Gancel, I. Mortazavi and O. Piller, Quality modelling of water distribution systems using sensitivity equations, Journal of Hydraulic Engineering, 136 (2010), 11 pp. doi: 10.1061/(ASCE)HY.1943-7900.0000138. [14] W. Hundsdorfer and J. G. Verwer, "Numerical Solution of Time Dependent Advection-Diffusion-Reaction Equations," Springer Series in Computational Mathematics, 33, Springer-Verlag, Berlin, 2003.

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##### References:
 [1] O. Chesneau, "Un Outil d'aide à la Maîtrise des Pertes dans les Réseaux d'eau Potable: La Modélisation Dynamique de Différentes Composantes du Débit de Fuite," Ph.D thesis, University of Strasbourg, Strasbourg, 2006. [2] E. Jaumouillé, O. Piller and J. E. Van Zyl, A hydraulic model for water distribution systems incorporating both inertia and leakage,, in, (): 129. [3] Porteau software, IRSTEA (2011), accessed on March 09, 2012., Available from: \url{http://porteau.irstea.fr/}., (). [4] A. Lambert, What do we know about pressure-leakage relationships in distribution systems?, in "Proc. of IWA Congress: System Approach to Leakage Control and Water Distribution Systems Management," Brno, Czech Republic, (2001), 89-96. [5] J. E. Van Zyl and C. R. I. Clayton, The effect of pressure on leakage in water distribution systems, in "Proc. of CCWI Conference: Water Management for the 21st Century," University of Exeter, UK, 2 (2005), 131-136. [6] R. Glowinski and A. Marrocco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation dualité, d'une classe de problèmes de Dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle RAIRO Analyse Numérique, 9 (1975), 41-76. [7] M. S. Ghidaoui, On the fundamental equations of water hammer,, Urban Water Journal, 1 (): 71. [8] J. T. Beale and A. Majda, Rates of convergence for viscous splitting of the Navier-Stokes equations, Mathematics of Computation, 37 (1981), 243-259. doi: 10.1090/S0025-5718-1981-0628693-0. [9] S. Descombes and M. Massot, Operator splitting for nonlinear reaction-diffusion systems with an entropic structure: Singular perturbation and order reduction, Numerische Mathematik, 97 (2004), 667-698. doi: 10.1007/s00211-003-0496-3. [10] E. Trélat, "Contrôle Optimal. Théorie & Applications," Mathématiques Concrètes, Vuibert, Paris, 2005. [11] M. Bergounioux, "Optimisation et Contrôle des Systemes Linéaires," Dunod, 2001. [12] B. Brémond, P. Fabrie, E. Jaumouillé, I. Mortazavi and O. Piller, Numerical simulation of a hydraulic Saint-Venant type model with pressure-dependent leakage, Applied Mathematics Letters, 22 (2009), 1694-1699. doi: 10.1016/j.aml.2009.02.007. [13] P. Fabrie, G. Gancel, I. Mortazavi and O. Piller, Quality modelling of water distribution systems using sensitivity equations, Journal of Hydraulic Engineering, 136 (2010), 11 pp. doi: 10.1061/(ASCE)HY.1943-7900.0000138. [14] W. Hundsdorfer and J. G. Verwer, "Numerical Solution of Time Dependent Advection-Diffusion-Reaction Equations," Springer Series in Computational Mathematics, 33, Springer-Verlag, Berlin, 2003.
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