This paper presents a novel analysis on the accuracy of quadratic approximations for Hazen--Williams head loss formula, which enables the control of constraint violations in optimisation problems for water supply networks.
The two smooth polynomial approximations considered here minimise the absolute and relative errors, respectively, from the original non-smooth Hazen--Williams head loss function over a range of flows.
Since the quadratic approximations are used to formulate head loss constraints for different optimisation problems, we are interested in quantifying and controlling their absolute errors, which affect the degree of constraint violations of candidate feasible solutions.
We derive new exact analytical formulae for the absolute errors as a function of the approximation domain, pipe roughness and relative error tolerance. We assess the efficacy of the quadratic approximations in mathematical optimisation for optimal pressure regulation in an operational water supply network. We propose a strategy on how to choose the approximation domain for each pipe such that the optimisation results are sufficiently close to the exact hydraulic feasibility region. By using simulations with multiple parameters, the approximation errors are shown to be consistent with our analytical predictions.