Multi-objective Optimization of Reservoir Operation Under Uncertainty with Robust and Flexible Decision Variables Public Deposited


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  • Optimization of reservoir operation is involves various competing objectives for a scarce resource (water). To find the optimal operation of reservoirs, it is essential to consider multiple objectives simultaneously. There are various sources of uncertainty associated with the reservoir operation problem that should be considered as well. The overarching goal of this research is to develop a framework for finding flexible and reliable solutions to the reservoir operation problem with competing objectives. Because some sources of uncertainty are not well quantified, providing flexible decision variables lets the decision maker choose accordingly from a range of options knowing that all the flexible decision variables are feasible with a specified probability of failure and that are relatively optimal. To accomplish this goal, each flexible decision variable is represented by a random variable within a specific range instead of a single deterministic decision variable. An additional objective is added to the optimization problem, in order to maximize the flexibility of decision variables. The proposed methodology is tested for two mathematical test problems and the operation of the Grand Coulee reservoir, which is located on the Columbia River in the Northwestern United States. The Stochastic Collocation (SC) method is used to sample the random variables and approximate the expected values of the objectives. For the Grand Coulee reservoir, the decision variables are the daily turbine outflows. The first objective of the optimization is to minimize the forebay elevation deviation at the end of the optimization period. The second objective is to maximize the revenue from the hydropower production. The results show that the proposed methodology could find some flexible decision variables with 45% coefficient of variation. The corresponding expected objectives have less than 20% deterioration from the deterministic Pareto solutions. However, the number of function evaluations increases exponentially with the number of decision variables. Therefore, this methodology is suggested for problems with a few decision variables. For finding flexible decision variables in problems with many decision variables, a dimension reduction method called Karhunen Loeve (KL) expansion is implemented in the optimization problem. By extracting useful information from the decision variables, the decision space can be represented with merely a few random variables using a set of deterministic decision variables. The results show that three random variables are sufficient to generate decision variable realizations which have mean and variance less than 1% and 5% different from the original decision variable realizations, respectively. The proposed methodology is capable of efficiently finding flexible decision variables that lead to expected objective values close to the Pareto deterministic solutions. To force the generated decision variable realizations to stay within the feasible bounds and therefore reduce the number of constraints that need to be checked, the data is transformed to be within bounds first, and then the KL-expansion is performed. Using the transformed data decreases the computational time but the decrease in computational time is not significant. The inflow uncertainty is also considered as the only source of input uncertainty. Forecast inflow ensembles can be used as the source of inflow uncertainty. However in this study due to lack of information, historical inflows are used instead. The inflow uncertainties are represented using the KL-expansion. Robust optimization is performed by optimizing the weighted sum of the expectation and standard deviation of the objective due to uncertain inflows. The weights in the robust objective formulation can be changed based on the decision maker’s preference of robustness versus performance. Finally, the combined framework to find robust and flexible decision variables is tested on a reservoir operation problem and the results were compared to the deterministic case.
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