  This article describes the methodology for formulating and solving optimal
pump scheduling problems with variable-speed pumps (VSPs) as mixed integer
linear programs (MILPs) using piece-linear approximations of the network
components. The water distribution network (WDN) is simulated with an initial
pump schedule for a defined time horizon, e.g. 24 hours, using a nonlinear
algebraic solver. Next, the network element equations including VSPs are
approximated with linear and piece-linear functions around chosen operating
point(s). Finally, a fully parameterized MILP is formulated in which the
objective is the total pumping cost. The method was used to solve a pump
scheduling problem on a a simple two variable speed pump single-tank network
that allows the reader to easily understand how the methodology works and how
it is applied in practice. The obtained results showed that the formulation is
robust and the optimizer is able to return global optimal result in a reliable
manner for a range of operating points.
