Minimisation du Content par une méthode d'active set pour les équations d'équilibrage hydraulique conduites par la pression

International audienceA new content-based, box-constrained, active-set projected Newton method is presented that solves for the heads, the pipe flows, and the nodal outflows of a water distribution system in which nodal outflows are pressure dependent. The new method is attractive because, by comparison with the previously published weighted least-squares energy and mass residuals (EMR) damped Newton method, (1) it typically takes fewer iterations, (2) it does not require damping, (3) it takes less wall-clock time, (4) it does not require the addition of any virtual elements, and (5) it is algorithmically easier to deal with. Various pressure-outflow relationships (PORs), which model nodal outflows, were considered and two new PORs are presented. The new method is shown, by application to eight previously published case study networks with up to about 20,000 pipes and 18,000 nodes, to be up to five times faster than the EMR method and to take between 34% and 70% fewer iterations than the EMR method

Closure to "Jacobian matrix for solving water distribution system equations with the Darcy-Weisbach head-loss model" by Angus Simpson and Sylvan Elhay

Angus R. Simpson and Sylvan Elha

Formulating the water distribution system equations in terms of head and velocity

The set of equations for solving for pressures and flows in water distribution systems are non‐linear due to the head loss‐velocity relationship for each of the pipes. The solution of these non‐linear equations for the heads and flows is usually based on the Todini and Pilati method. The method is an elegant way of formulating the equations. A Newton solution method is used to solve the equations whereby the special structure of the Jacobian is exploited to minimize the computations and this leads to an extremely fast algorithm. Each iteration firstly solves for the heads and then solves for the flows. In the EPANET implementation of the Todini and Pilati algorithm an initial guess of the flows is based on an assumed velocity of 1.0 fps (0.305 m/s) in each pipe in the network. Each flow is then determined from the continuity equation by multiplying the assumed velocity by the area. Usually velocities in pipes are in the range of 0.5 to 1.5 m/s (and perhaps sometimes higher up to 3 or 4 m/s). Thus the velocities to be solved for are all of the same order of magnitude. In contrast, the range of discharges may be quite large in a system — ranging from below 10 L/s up to above 700 L/s — thus possibly three orders of magnitude of difference. As an alternative to the usual formulation of the Todini and Pilati method in terms of flows and heads, this paper recasts the Todini and Pilati formulation in terms of heads and velocities to attempt to improve the convergence properties. Results are compared for the two formulations for a range of networks from 553 to 10,354 pipes. Convergence criteria for stopping the iterative solution process are discussed. The impact of the initial guess of the velocities in each of the pipes in the network on the convergence behavior is also investigated. Statistics on mean flows and velocities in the network and the minimum and maximum velocities for each of the example networks are given and finally operation counts are also provided for these networks.Angus R. Simpson and Sylvan Elha

Closure to "Dealing with Zero Flows in Solving the Nonlinear Equations for Water Distribution Systems" by Sylvan Elhay and Angus R. Simpson

Sylvan Elhay and Angus R. Simpso

The Darcy-Weisbach Jacobian and avoiding zero flow failures in the global gradient algorithm for the water network equations

This paper considers two issues related to iteratively solving the non-linear equations governing the flows and heads in a water distribution system network. The first concerns the use of the correct Jacobian for the Global Gradient Algorithm (GGA) when the Darcy-Weisbach head loss model is used. The second relates to dealing with zero flows in the iterative solution process. A regularization procedure for the GGA with the Hazen{Williams model is demonstrated on an example network which has zero flows but for which the (full) Jacobian is invertible.Sylvan Elhay and Angus R. Simpso

A New Technique for Multidimensional Signal Compression

The problem of efficiently compressing a large number, L, of N dimensional signal vectors is considered. The approach suggested here achieves efficiencies over current pre-processing and Karhunen-Loeve techniques when both L and N are large

Dealing with zero flows in solving the nonlinear equations for water distribution systems

Three issues concerning the iterative solution of the nonlinear equations governing the flows and heads in a water distribution system network are considered. Zero flows cause a computation failure (division by zero) when the Global Gradient Algorithm of Todini and Pilati is used to solve for the steady state of a system in which the head loss is modeled by the Hazen-Williams formula. The proposed regularization technique overcomes this failure as a solution to this first issue. The second issue relates to zero flows in the Darcy-Weisbach formulation. This work explains for the first time why zero flows do not lead to a division by zero where the head loss is modeled by the Darcy-Weisbach formula. In this paper, the authors show how to handle the computation appropriately in the case of laminar flow (the only instance in which zero flows may occur). However, as is shown, a significant loss of accuracy can result if the Jacobian matrix, necessary for the solution process, becomes poorly conditioned, and so it is recommended that the regularization technique be used for the Darcy-Weisbach case also. Only a modest extra computational cost is incurred when the technique is applied. The third issue relates to a new convergence stopping criterion for the iterative process based on the infinity-norm of the vector of nodal head differences between one iteration and the next. This test is recommended because it has a more natural physical interpretation than the relative discharge stopping criterion that is currently used in standard software packages such as EPANET. In addition, it is recommended to check the infinity norms of the residuals once iteration has been stopped. The residuals test ensures that inaccurate solutions are not accepted. © 2011 American Society of Civil Engineers.Sylvan Elhay and Angus R. Simpso

Fast graph matrix partitioning algorithm for solving the water distribution system equations

In this paper a method which determines the steady-state hydraulics of a water distribution system, the Graph Matrix Partitioning Algorithm (GMPA), is presented. This method extends the technique of separating the linear and nonlinear parts of the problem and using the more time consuming nonlinear solver only on the nonlinear parts of the problem and faster linear techniques on the linear parts of the problem. The previously developed Forest-Core Partitioning Algorithm (FCPA) used this approach to separate the network graph's external forest from its looped core but did not address the fact that within the core of a network graph there may be many internal trees - nodes in series - for which a more economical linear process can be used. This extension of the separation process can significantly reduce the dimension of the nonlinear problem that must be solved: GMPA applied to eight case study networks with between 900 and 20,000 pipes show reductions to between 5% and 55% of the core dimension (after FCPA). The separation of the problem into its nonlinear and linear parts involves no approximations, such as lumping or skeletonization, and the resulting solution is precisely the solution that would have been obtained by the slower technique of solving the entire network with a nonlinear solver. The new method is applied after the network has been separated into an external forest and core by the FCPA method. The GMPA identifies all the nodes in the core which are in series (the internal forest) and then iterates alternately on the remaining core (the (nonlinear) global step) and the internal forest (the (linear) local step). In this paper, it is formally shown that the smaller set of nonlinear equations in the GMPA corresponds to the network equations of a particular topological subgraph of the original graph. Using algebraic manipulations, the size of the linearized system to be solved is reduced to the number of nodes in the core having degree greater than two. For pipe models of real world applications that are derived from GIS datasets, this can mean a dramatic reduction of the size of the nonlinear problem that has to be solved. The main contributions of the paper are (i) the derivation and presentation of formal proofs for the new method and (ii) demonstrating how significant the reduction in the dimension of the nonlinear problem can be for suitable networks. The method is illustrated on a simple example.J. Deuerlein, S. Elhay, and A. R. Simpso

Graph partitioning in the analysis of pressure dependent water distribution systems

The forest core partitioning algorithm (FCPA) and the fast graph matrix partitioning algorithm (GMPA) have been used to improve efficiency in the determination of the steady-state heads and flows of water distribution systems that have large, complex network graphs. In this paper, a single framework for the FCPA and the GMPA is used to extend their application from demand dependent models to pressure dependent models (PDMs). The PDM topological minor (TM) is characterized, important properties of its key matrices are identified, and efficient evaluation schemes for the key matrices are presented. The TM captures the network’s most important characteristics: It has exactly the same number of loops as the full network, and the flows and heads of those elements not in the TM depend linearly on those of the TM. The inverse of the TM’s Schur complement is shown to be the top, left block of the inverse of the full system Jacobian’s Schur complement, thereby providing information about the system’s essential behavior more economically than is otherwise possible. The new results are applicable to other nonlinear network problems, such as in gas, district heating, and electrical distribution.Sylvan Elhay, Jochen Deuerlein, Olivier Piller and Angus R. Simpso

Reformulated co-tree flows method competitive with the global gradient algorithm for solving water distribution system equations

Many different methods have been devised to solve the nonlinear systems of equations that model water distribution networks. Probably the most popular is Todini and Pilati's global gradient algorithm (GGA). Given the GGA's success, alternative methods have not aroused much interest. One example is the co-tree method, which requires some cumbersome steps in its implementation. In this paper, a reformulated co-trees method (RCTM) is presented that simplifies the procedure by manipulating the incidence matrix into trapezoidal form: a lower triangular block at the top representing a spanning tree and rectangular block below it representing the corresponding co-tree. This reordering leads to significant efficiencies that make the RCTM competitive with the GGA in certain settings. The new method has some similarities to the loop flows corrections formulation, and it is shown, by application to a set of eight case study networks with between 932 and 19,647 pipes and between 848 and 17,971 nodes, to be between 15 and 82% faster than the GGA in a setting, such as optimization using evolutionary algorithms, where the methods are applied hundreds of thousands, or even millions, of times to networks with the same topology. It is shown that the key matrix for the RCTM can require as little as 7% of the storage requirements of the corresponding matrix for the GGA. This can allow for the solution of larger problems by the RCTM than might be possible for the GGA in the same computing environment. Unlike some alternatives to the GGA, the following features make the RCTM attractive: (1) it does not require a set of initial flows that satisfy continuity; (2) there is no need to identify independent loops or the loops incidence matrix; (3) a spanning tree and co-tree can be found from the incidence matrix without the addition of virtual loops, particularly when multiple reservoirs are present; and (4) it does not require the addition of a ground node and pseudoloops for each demand node and does not require the determination of cut sets. In contrast with the GGA, the RCTM does not require special techniques to handle zero flow problems that can occur when the head loss is modeled by the Hazen-Williams formula (a sufficient condition is given). The paper also (1) reports a comparison of the sparsity of the key RCTM and GGA matrices for the case study networks; (2) shows mathematically why the RCTM and GGA always take the same number of iterations and produce precisely the same iterates; and (3) establishes that the loop flows corrections and the nullspace methods (previously shown by Nielsen to be equivalent) are actually identical to the RCTM.Sylvan Elhay, Angus R. Simpson, Jochen Deuerlein, Bradley Alexander and Wil H. A. Schilder