Inflow pattern stochastic dynamic programming and its application in reservoir operation optimization
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AbstractStochastic Dynamic Programming (SDP) has often been a useful technique for identifying the optimal operating policies for reservoirs. The stochastic nature of inflows is usually described by an order one Markov process, of which the current inflow is only conditioned on the preceding inflow, and represented by a transition probability Pr(Qt+i\Qt). An important controversy in the literature of stochastic reservoir optimizations, as well as that of stream flow modeling, concerns the appropriateness of using order one Markov process to describe the inflow sequence. If correlations among inflows beyond lag 1 are strong (the shorter the time step, the wider the correlations will span among them, e.g. daily inflows normally show wider span of correlations than weekly or monthly inflows), which implies that the occurrence of current inflow can be more precisely conditioned on more previous inflows Pr(Qt+i\Qt,Qt~ii ...,Qt-k) (higher order autoregressive models are often used in stochastic hydrological modeling under this situation). Theoretically, these previous inflows can be treated as state variables to be included in the SDP formulation. However, this addition of state variables yields a mathematical formulation that is impractical to solve due to the "curse of dimensionality" inherent in the SDP formulation. Specifically, these difficulties stem from the structural limitations of the SDP, whereby the addition of each new state variable forces exponential increases in the number of system states to be evaluated, quickly making it impractical to solve. In this study, the previous inflows Qt, Qt-i, •••> Qt-k are treated as an inflow pattern to be included in the SDP formulation as a single state variable. The number of state variables remains the same as that of the SDP, with one storage state variable and one hydrologie state variable. The computational efficiency of the SDP model is maintained, while the model captures the higher order correlation structure in the inflow sequence. The resulting model is termed inflow Pattern Stochastic Dynamic Programming (PSDP) model. The improvement of PSDP models over their classic SDP counterparts are demonstrated through their application to the Glenmore Reservoir, which is located in southwest Calgary, Canada.
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