How to use the first and second order sensitivities of steady-state solutions to water distribution systems

Overview & Synopsis

A hydraulic model can be of considerable benefit to those responsible for the design and operation of drinking water systems. In the contemporary era, it is imperative that water distribution systems are operated in a more sober manner, in a sustainable and resilient fashion, to combat the effects of global warming and to mitigate natural and anthropogenic risks. However, this may necessitate the resolution of optimisation problems through the utilisation of extensive hydraulic models, a process that could encounter intractable challenges due to the curse of dimensionality. One such example is real-time demand calibration in conjunction with research into countermeasures for critical events in a large water distribution system. Quantifying uncertainty is imperative for robust countermeasures and decisions. To address these significant challenges and potentially reduce computing time, a novel approach to deriving first and second order sensitivities was proposed at the 3rd International Joint Conference WDSA/CCWI (2024) [1]. This method involves solving a series of linear systems with the same matrix and Cholesky factors, but with different right-hand sides. This provides an efficient, low-cost way of calculating sensitivities. The objective of this study is to ascertain the potential of this innovative approach to medium and large networks. Calculating the sensitivities of many state variables with respect to many parameters is not feasible and is fortunately not useful. In hydraulic models, there are a limited number of demand patterns, and we are often interested in just a few specific outputs. The application will concentrate on two significant applications: the variation in pressure head according to local demand or the flow rate at a critical node in a large network, and the application to real-time calibration for a large network. Finally, we will discuss other potential applications for large networks and how to avoid certain pitfalls.This paper was presented at the 21st Computing and Control in the Water Industry Conference (CCWI 2025) at the University of Sheffield (1st - 3rd September 2025).