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

Ezio Todini said of the Global Gradient Algorithm (GGA), which he developed with Pilati, that it is the most appropriate fast convergent and robust tool for pipe network analysis . Indeed, its speed and efficiency has seen it built into many popular water distribution simulation packages. In the face of the GGA's success, alternative methods have not aroused much interest. In this paper a Reformulated Co-Trees Method (RCTM) is presented. The new method has some similarities to the loop flows 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 17971 nodes, to be between 15% and 82% faster than the GGA in a setting, such as optimization using genetic 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 other alternatives to the GGA, the following features, make the RCTM attractive: (i) it does not require a set of initial flows which satisfy continuity, or (ii) there is no need to identify independent loops or the loops incidence matrix, (iii) a spanning tree and co-tree can be found from the matrix A1 without the addition of virtual loops, particularly when multiple reservoirs are present, (iv) the RCTM does not require the determination of cut-sets or the addition of a ground node and pseudo-loops for each demand node, (v) it may not require special techniques to handle zero flow problems (a sufficient condition is given). The paper also (i) reports a comparison of the sparsity of the key RCTM and GGA matrices for the case study networks, (ii) shows mathematically why the RCTM and GGA always take the same number of iterations and produce precisely the same iterates, (iii) establishes that the Loop-Flows Corrections and the Nullspace methods (previously shown by Nielsen to be equivalent) are actually identical to the RCTM