  In this paper, we propose a data-based methodology to solve a multi-period
stochastic optimal water flow (OWF) problem for water distribution networks
(WDNs). The framework explicitly considers the pump schedule and water network
head level with limited information of demand forecast errors for an extended
period simulation. The objective is to determine the optimal feedback decisions
of network-connected components, such as nominal pump schedules and tank head
levels and reserve policies, which specify device reactions to forecast errors
for accommodation of fluctuating water demand. Instead of assuming the
uncertainties across the water network are generated by a prescribed certain
distribution, we consider ambiguity sets of distributions centered at an
empirical distribution, which is based directly on a finite training data set.
We use a distance-based ambiguity set with the Wasserstein metric to quantify
the distance between the real unknown data-generating distribution and the
empirical distribution. This allows our multi-period OWF framework to trade off
system performance and inherent sampling errors in the training dataset. Case
studies on a three-tank water distribution network systematically illustrate
the tradeoff between pump operational cost, risks of constraint violation, and
out-of-sample performance.
