Coupled water-energy operation under forecast uncertainty

Physical process and operating decision

Urban water supply coordinates treatment, transport, and distribution over an operating horizon. Each stage consumes electricity, so the schedule sets when the system serves demand and when it buys energy. Electricity prices are uncertain, so a forecast error can change both the operating cost and the selected schedule.

The study represents the full water-supply process as a stochastic scheduling problem. A probabilistic electricity-cost forecast provides the uncertain cost parameters, and the scheduling model defines the operating variables and constraints. The resulting schedule determines the decision loss used to train the forecast, so training weights a price error by its effect on the schedule.

Learning through the induced schedule

Stochastic Predict-Then-Optimize trains the forecast distribution through the optimality gap of the induced schedule. The training signal depends on how a forecast error changes the selected operating periods and their cost. This ties the forecast to the coupling between water operation and grid conditions.

Differentiable optimality gap

The stochastic scheduling problem cannot be differentiated directly. SPTO+ derives a differentiable convex upper bound on the optimality gap from Lagrangian duality. The gradient is computed from this bound, and the scheduling problem remains the source of the learning signal.

System-level evaluation

Experiments on real water-supply systems compare pointwise forecast error, operating cost, and decision regret against two-stage forecasting and optimization baselines. SPTO reports lower operating cost and lower regret.

Relation to dynamics and decisions

The learned distribution describes uncertain energy costs over the operating horizon of a water process. A constrained optimizer uses those costs to coordinate the schedule over time, and decision regret measures the consequences of forecast error. This study connects water-energy operation to the broader question of how a learned prediction affects an operating decision.

Publication

Shunyu Wu, Jingcheng Wang, Haotian Xu, Yanjiu Zhong, and Jun Rao. End-to-End Stochastic Predict-Then-Optimize for Cost-Efficient Water-Energy Resource Scheduling. IEEE Transactions on Smart Grid, 16(6), 4796-4807, 2025.