Water pumping in drinking water distribution networks (WDNs) can be treated as a flexible load in the power distribution network (PDN). In this article, we formulate an optimization problem to minimize the electricity costs associated with pumping subject to WDN and PDN constraints. In practice, both water and power demands are uncertain and pumps should be scheduled to ensure that pump operation does not violate either networks' constraints for nearly all possible uncertainty realizations. To address this problem, we formulate a chance-constrained (CC) optimization problem that simultaneously determines pumping schedules along with the parameters of real-time control policies that can be used to respond to water and power demand forecast errors. We use approximations and relaxations along with the scenario approach for CC programming to reformulate the optimization problem into a convex deterministic problem. We demonstrate the performance of the approach through case studies and also explore the impact of the relaxations, an approach to improve computational tractability, and tradeoffs associated with the way in which we define the cost of real-time control actions. We find that optimal scheduling and real-time control of water pumping can effectively manage water and power demand uncertainty, meaning water demand is satisfied and both the WDN and PDN operate within their limits; however, the approach is conservative leading to high reliability at high cost.

Aggregations of demand response resources can provide a variety of services to the power grid. To utilize them effectively, for both planning problems and real-time control, we must estimate their flexibility. However, flexibility estimates are uncertain because of issues such as model error and forecasting error. In this paper, we present a model of uncertain flexibility and describe the many causes of uncertainty. We conduct two case studies, one for electric vehicle aggregations and one for air conditioner aggregations, in order to show specific examples that illustrate the causes and magnitude of uncertainty. We find that uncertainty can be very large for small load aggregations, when models do not capture enough of the underlying dynamics, or when forecasts of other quantities, such as ambient conditions, are bad. Although the focus of the paper is on understanding uncertainty, we also briefly discuss how one might use knowledge of uncertainty distributions in planning problems to derive closer-to-optimal results.