Modernization of Todini Global Gradient Algorithm for hydraulic analysis of networks with choked flow

The modification of well-known Global Gradient Algorithm for hydraulic network flow distribution problem is proposed. This modification is based on problem equations rewritten in “upstream” form and on modified form of linearization, and can be effectively used for piping networks with gas and multiphase gas-liquid flow with multiple choked flow.

Testing linear solvers for global gradient algorithm

Steady-state Water Distribution Network models compute pipe flows and nodal heads for assumed nodal demands, pipe hydraulic resistances, etc. The nonlinear mathematical problem is based on energy and mass conservation laws which is solved by using global linearization techniques, such as global gradient algorithm (GGA). The matrix of coefficients of the linear system inside GGA belongs to the class of sparse, symmetric and positive definite. Therefore a fast solver for the linear system is important in order to achieve the computational efficiency, especially when multiple runs are required. This work aims at testing three main strategies for the solution of linear systems inside GGA. The tests are performed on eight real networks by sampling nodal demands, considering the pressure-driven and demand-driven modelling to evaluate the robustness of solvers. The results show that there exists a robust specialized direct method which is superior to all the other alternatives. Furthermore, it is found that the number of times the linear system is solved inside the GGA does not depend on the specific solver, if a small regularization to the linear problem is applied, and that pressure-driven modelling requires a greater number which depends on the size and topology of the network and not only on the level of pressure deficiency.

Extending the global gradient algorithm to unsteady flow extended period simulations of water distribution systems

This paper introduces an extension of the Global Gradient Algorithm (GGA) to directly solve unsteady flow problems arising from the presence of variable head water storage devices, such as tanks, in Extended Period Simulations (EPS) of looped water distribution networks (WDN). Such a modification of the original algorithm was motivated by the need to overcome oscillations and instabilities reported by several users of EPANET, a worldwide available package, which uses the GGA to solve the looped WDN equations. The set of partial differential equations describing the time and space behaviour of a water distribution system is here presented. It is shown how an unsteady flow GGA can be derived by simple modifications of the original steady-state GGA. The performances of the new algorithm, referred to as EPS-GGA, are compared with the results provided by EPANET on an extremely simplified example, the solution of which is qualitatively known. As opposed to EPANET which shows significant instabilities, the EPS-GGA is stable under a wide variety of increasing integration time intervals.