Synthesis of Feedback Control for Pump Operation in Water Distribution Networks

Abstract

Existing optimal scheduling packages for water distribution systems (WDS) calculate the time schedules for control elements such as pumps, valves and treatment works following the idea of open loop control. The calculations take into account the hydraulic model of WDS, operational constraints and the time dependent electrical tariff. The time horizon is typically one day or one week depending on the network storage capacity and the structure of the electrical tariff. If such a schedule is applied to a physical system the predicted and the physical reservoir levels diverge due to uncertain demands and as a result the calculations need to be repeated at some time using updated demand prediction and the current measured reservoir levels. The aim of this paper is to investigate the feasibility of synthesising continuous feedback from reservoir levels to pump and valve operation, taking into account the operational constraints and the electrical tariff. This attempt can be seen as a generalisation of a simple practical rule where a pump is controlled by the level of an associated reservoir, e.g. the rule ‘if the tank is full switch the pump OFF , if the tank is empty switch the pump ON’. The proposed methodology relies on preparing a control law in an off-line mode which is subsequently used in a real time situation. The traditional optimal scheduling problem is solved many times for typical initial reservoir levels in the system and a family of reservoir and corresponding control trajectories is generated. These trajectories are approximated by a hyper-surface which represents an optimal control law ready to use in the field. The control action will follow the sequence: log the time, measure the current reservoir levels, evaluate from the hyper-surface the corresponding operation for pumps and valves and finally apply the control action to the pumps and valves. The approach avoids time-consuming online-recalculation of pump schedules, and additionally the feedback control is much more robust with respect to uncertain demands. The limitation of the approach is dictated by the number of reservoirs and number of initial reservoir levels which need to be considered. In the worst case the computational complexity may increase exponentially with the number of reservoirs. Initially this approach was applied to a simple system containing a single reservoir fed by one pump station with a time varying tariff and subsequently to a medium-size system with four reservoirs and four pump stations.