Modeling Flow and Pressure Control in Water Distribution Systems Using the Nash Equilibrium

Pressure dependent modeling (PDM) for water distribution systems (WDSs) is now widely accepted as being much more realistic than the previously used demand driven modeling. Steady-state linkflows, $q$, outflows, $c$, and heads, $h$, of a PDM WDS with no controls of flow and pressure in the system can reliably be found as the active set method solution of a linear-equality-constrained nonlinear optimization of the system’s content. Introducing linkflow controls, such as flow control valves (FCVs) and check valves can be handled by imposing box constraints on the decision variables $q$ and $c$ in the optimization; these problems can also be found either by an ASM or an interior point method. The heads in these problems are the Lagrange multipliers in the content model, and controlling these cannot be handled simply by imposing constraints on them. In this paper, the problem of modeling pressure-control devices such as pressure-reducing valves (PRVs) is solved by finding the Nash Equilibrium of a model that treats (1) the (global) linkflow constrained content optimization; and (2) the local pressure controls, as players in a competitive, noncooperative game. While this paper details how to model FCVs and PRVs together, this modeling framework is equally applicable to pressure-sustaining valves and variable speed pumps for pressure control without essential modification. An important contribution of this proof-of-concept paper is the development of a comprehensive model that includes flow and pressure controls and which finds a solution without using heuristics. The new method is illustrated on some examples.