End-to-end Stochastic Predict-Then-Optimize for Cost-Efficient Water-Energy Resource Scheduling

Compared to other energy-intensive industries, urban water supply systems are unique due to the strong temporal coupling with the power grid, which intensifies grid stress during peak hours. This paper addresses the water-energy nexus by developing a water-energy resource scheduling model for wholeprocess water supply that incorporates electricity cost prediction uncertainties. Meanwhile, to minimize the scheduling cost in an end-to-end approach, we propose the Stochastic Predictthen-Optimize (SPTO), which aligns prediction distribution with the optimality gap by integrating probabilistic forecasting with stochastic optimization. Unlike traditional two-stage approaches, SPTO establishes a closed-loop feedback between forecasting and optimization, reducing cumulative errors and improving realtime adaptability. To enable end-to-end training, we derive the SPTO+, a differentiable convex upper-bound surrogate function based on Lagrangian duality, which provides surrogate gradient computation for end-to-end training. Experiments on real-world systems demonstrate that SPTO significantly reduces operational cost and decision regret, effectively coordinating water operations with dynamic grid conditions to support cost-efficient and gridresponsive scheduling.