On The Convergence Properties of the Different Pipe Network Algorithms

Scope of this paper is to propose a unified insight to the problem of solving the looped water distribution systems non-linear hydraulic equations as well as to compare the convergence of the different resulting algorithms. After showing that the problem can be formulated as the solution of a system of equations, partly linear and partly non-linear, it will be demonstrated how most of the widely used algorithms for the analysis of looped pipe networks, can be formulated in terms of a gradient search approach and can be derived from the Global Gradient by simple linear transformations. For the sake of simplicity, the problem of analysing a looped pipe network will be defined on the absence of minor losses, pumps, valves and other concentrated dissipation devices, which may however be introduced without affecting the substance of the results. It will be shown how the different solution methods can be classified according to the space of the state variables used as unknowns (the pipe "flows", the nodal "heads", the mesh "residual flows"), and can be divided into two main categories: a first group incorporating the flow based algorithms and a second group based on the Nodal Heads. Using an extremely simple example, the paper shows that all the flow based algorithms have exactly the same convergence properties, in that they require the same number of non-linear Newton-Raphson steps to reach exactly the same result. Therefore, the computer time consumption only depends on the dimension of the space on which the problem is solved (the nodes, the pipes, the meshes) and on the symmetry and the sparsity of the resulting system matrix. This does not happens when the nodal head solvers are used. In this case the convergence properties of the Newton-Raphson algorithm are affected by the different concavity of the resulting functions, with the consequence of generating less convergent algorithms, which never reach the convergence properties of the flow based ones. even if corrected with relaxation coefficients. This paper was presented at the 8th Annual Water Distribution Systems Analysis Symposium which was held with the generous support of Awwa Research Foundation (AwwaRF).