Modeling of Time-Window Ramping and Time-Averaged Environmental Constraints In Short Term Hydrothermal Problems

This paper presents the formulation of time-window averaged (TWA) and time-window ramping (TWR) hydraulic constraints within the short-term hydrothermal scheduling problem. Conventional instantaneous hydraulic constraints enforce strict limits at every time step, which can result in an overly restrictive behavior in highly discretized horizons. The proposed formulations introduce time-window constraints, providing more flexible alternatives to instantaneous limits and enabling a more realistic representation of operational flexibility. The TWA formulation imposes bounds on average values of hydraulic variables within daily or weekly windows, while the TWR formulation restricts variations between consecutive windows or periodic time points. Both constraints can be applied to variables such as discharge, spillage, and storage. The paper also addresses upstream water level constraints, whose nonlinear relations are approximated as constraints on stored volume, in order to preserve model linearity. The proposed formulations were validated on real test cases from the Brazilian Independent System Operator, confirming their practical applicability in large-scale hydrothermal systems.