Reformulated Co-Tree Flows Method Competitive with the Global Gradient Algorithm for Solving Water Distribution System Equations

Abstract

Authors may post the final draft of their work on open, unrestricted Internet sites or deposit it in an institutional repository when the draft contains a link to the bibliographic record of the published version in the ASCE Civil Engineering Database. "Final draft" means the version submitted to ASCE after peer review and prior to copyediting or other ASCE production activities; it does not include the copyedited version, the page proof, or a PDF of the published version. Journal of Water Resources Planning and ManagementCitation: Elhay, S., Simpson, A.R., Deuerlein, J., Alexander, B. and Schilders, W.H.A., (2014). "Reformulated Co-tree flows method competitive with the Global Gradient Algorithm for solving the water distribution system equations. implementation. 7In this paper a Reformulated Co-Trees Method (RCTM) is presented that simplifies the pro-8 cedure by manipulating the incidence matrix into trapezoidal form: a lower triangular block at 9 the top representing a spanning tree and rectangular block below it representing the corresponding 10 co-tree. This reordering leads to significant efficiencies which make the RCTM competitive with 11 the GGA in certain settings. 12The new method has some similarities to the Loop Flows Corrections formulation and it is is shown, by application to a set of eight case study networks with between 932 and 19,647 pipes 14 and between 848 and 17971 nodes, to be between 15% and 82% faster than the GGA in a setting, be possible for the GGA in the same computing environment. 20Unlike some alternatives to the GGA, the following features, make the RCTM attractive: (i) 21it does not require a set of initial flows which satisfy continuity, (ii) there is no need to identify 22 independent loops or the loops incidence matrix, (iii) a spanning tree and co-tree can be found from 23 the incidence matrix without the addition of virtual loops, particularly when multiple reservoirs are 24 present, (iv) it does not require the addition of a ground node and pseudo-loops for each demand 25 node and does not require the determination of cut-sets. 26In contrast to the GGA, the RCTM does not require special techniques to handle zero flow 27 problems which can occur when the head loss is modeled by the Hazen-Williams formula (a sufficient 28 condition is given).

Improving the performance of water distribution systems’ simulation on multicore systems

“…An alternative to the GGA is the loop method [11], which is also a Newton-Raphson process working with a sparse, symmetric positive definite matrix. However, the size of the matrix is in this case n l ×n l , where n l = m−n is the number of independent loops that can be found in the network, which is usually much smaller that n. Some recent papers [5,7,10,2,1] have addressed the loop method as a competitive alternative to GGA. The authors developed contributions to this method, with the corresponding implementation [5].…”

“…Recently, several papers [7,8,10,2,1] have considered the loop method for simulation of WDS, first introduced in [11], as an alternative to the more widely used Global Gradient Algorithm (GGA), implemented in Epanet. The authors presented in [5] some contributions to the loop method, with an efficient implementation that is shown to be faster than Epanet for the test networks considered.…”

Limitations of demand- and pressure-driven modeling for large deficient networks

“…However, in recent years new publications have shown that the loop method gives an efficient alternative to the hybrid approaches. Elhay et al (2014) use a tree-cotree decomposition and take advantage of the fact that loop methods are null space methods. Alvarruiz et al (2015) have shown that loops can be defined without tree-cotree decomposition to increase the sparsity of the iteration matrix.…”

Sparse Null Space Algorithms for Hydraulic Analysis of Large-Scale Water Supply Networks

“…In the numerical optimization literature, null space algorithms for saddle point problems have been studied extensively, often called reduced Hessian methods; see the review paper (Benzi et al 2005) and the references therein. Null space algorithms, as opposed to the range space method of GGA (Todini and Pilati 1988), have also been applied for hydraulic analysis of water and gas pipe networks (Nielsen 1989;Rahal 1995;Elhay et al 2014;Todini and Rossman 2012). The three references (Nielsen 1989;Rahal 1995;Elhay et al 2014) differ in the way they generate the null space bases.…”

“…Null space algorithms, as opposed to the range space method of GGA (Todini and Pilati 1988), have also been applied for hydraulic analysis of water and gas pipe networks (Nielsen 1989;Rahal 1995;Elhay et al 2014;Todini and Rossman 2012). The three references (Nielsen 1989;Rahal 1995;Elhay et al 2014) differ in the way they generate the null space bases. Nielsen (1989) proposes the use of matrix row and column permutations to compute null space bases, whereas Rahal (1995) uses graph theoretic tools to solve what he called the co-tree formulation of the problem.…”