Integrated optimal operation of power and water systems under uncertainty: An adjustable robust optimization approach

Water and power systems are deeply interdependent, yet they are typically analyzed and optimized separately. Coupling these systems leverages their mutual dependency to enhance overall efficiency and reliability, although it introduces significant modeling complexity, especially when considering the inherent uncertainty in these systems, such as water demands, power loads, and renewable generation. This study proposes Robust Optimization (RO) and adjustable robust optimization (ARO) for the integrated day-ahead scheduling of pump operations and generator dispatch under uncertainty. A novel formulation is introduced to systematically eliminate uncertain equality constraints, which hinders the direct implementation of RO and ARO models. This is done by decomposing the decision vector into dependent and independent variables and substituting the dependent variables to eliminate the equality constraints. A key advantage of the ARO framework is its ability to incorporate real-time information as uncertainty unfolds, enabling adaptive decision-making. By coupling the two systems, the model allows each system to adjust its operation not only based on internal measurements but also in response to the evolving state of the other system. Two case studies demonstrate the method's performance through a Pareto tradeoff between cost optimality and robustness. The results show that robustness can be dramatically improved, compared to deterministic optimization, increasing the constraint satisfaction rates from less than 25% to above 99% with an increase of only 2-5% in operational costs. The results also highlight the superiority of ARO over RO in achieving lower operational costs for comparable levels of robustness. These findings showcase the potential of the proposed method to generate robust, adaptive policies that support the advancement of sustainable and reliable operation of interdependent infrastructure systems.