Simplified Approach to Mixed-Integer Chance-Constrained Optimization with Ensemble Streamflow Forecasts for Risk-Based Dam Operation

Abstract

We developed an integrated optimization program to design the release strategy for a dam, given a downstream flow target and considering medium-range ensemble forecasts of downstream tributary flows. A strength of the program is its ability to explicitly limit the risk of downstream flooding through chance constraints. Embedded in the program is an advanced hydrological ensemble prediction system (HEPS). Given the ensemble nature of the downstream flow forecasts (in that they comprise sets of individual scenarios), we formulated the chance constraints as discontinuous mixed-integer equations. This makes the integrated program nondeterministic polynomial-time hard (NP-hard) and thus, intractable to be solved by conventional mixed-integer programming methods based on branch-and-bound. Thus, to solve the program, we took a simplified but innovative approach—possible through exploiting certain simplifying aspects of the problem—where we solved first for the integer (binary) variables following a ranking mechanism, then the continuous variables using nonlinear programming. The results for a case study of Hume Dam and Lake Mulwala in the Murray-Darling Basin, Australia, demonstrates the efficacy of the integrated program and our simplified approach to solving it. The results show the integrated program is able to meet the optimization target when the natural flow from tributaries is low. The results also show that when that natural flow is high, the integrated program’s ability to meet the target to increase with the allowable risk of flooding. However, a higher allowable risk may lead to a higher possibility of flooding, depending on the problem specification. We recommend to run the integrated program on a continual basis because its results depend not only on the optimization for the current day, but also previous days. This study addresses a major need: the ability to use all information from an advanced HEPS, e.g., to limit risk in optimizing the operation of a mixed engineered-natural system, and with that, advances the use of ensemble hydrologic forecasts in water management.