Solving for the status of pressure regulating valves using Nash Equilibrium

Abstract

Computer hydraulic solvers such as EPANET allow complex water distribution system networks to be simulated by design engineers. Modeling of systems containing combinations of pressure reducing valves (PRVs), pressure sustaining valves (PSVs) and flow control valves (FCVs) can sometimes be problematical. The status of PRVs, PSVs and FCVs is usually solved for based on a heuristic approach. Starting from an assumed state of the control device, a check is performed during the next step of the iteration procedure, as to whether the original assumption of each particular assumed valve status still holds. If the assumption proves to be no longer true, a correction of the state of the control device is made. It has been observed in the past that this approach may sometimes fail to converge at all, or even worse may converge to incorrect solutions when simulating networks that include pressure regulating devices. In the majority of cases for which no correct solution was obtained, the solver converged quickly to an incorrect solution. Hence, no warning message was generated to indicate there was a problem with the solution. Some simple networks incorporating PRVs and FCVs have been considered in this research. The networks have multiple pressure regulating and flow regulating devices that are placed in series and separated by substantial lengths of pipe. Even for simple networks comprising two pressure regulating devices (one pressure regulating device and one flow regulating device) in series, current hydraulic modeling software fails to converge to the correct solution for particular configurations and settings. This paper aims to consider the circumstances in which such difficulties occur and the reasons for the failure to solve. In 2002, Deuerlein in his PhD thesis proposed a new method to ensure that the correct solution for water distribution systems containing multiple pressure and flow regulating valves is obtained. He used Nash equilibrium. The application of this approach enables all simple networks (as described above) to be correctly solved that previously could not be solved or those that converged to the wrong solution in EPANET. Details of the Nash equilibrium algorithm and the specific application to some example networks that converged to the wrong solution in EPANET are presented in this paper. Arbitrary pressure and flow controlling devices in water distribution systems can now be simulated with confidence. Convergence is assured if the system's physics are feasible. The approach has been implemented as part of the water supply network simulation and optimization model KANET.