Multi-dimensional stochastic dynamic programming algorithms considering the relevancy between streamflow and water demand

Abstract: Both Reservoir Inflow (RI) and Irrigation Water Demand (IWD) depend mainly on the basin-wide climate. Hence, various unpredictable (stochastic) weather and climate have induced significant variation in the water supply and demand processes in recent years. Meanwhile, the spatial similarity of climate conditions was also required to consider the relevancy between water supply and demand. More importantly, the smaller the basin is, the more considerable the relevancy is. However, the traditional one-dimensional stochastic dynamic programming cannot concern the relevancy between the multiple variables of water resources. In this study, a novel multi-dimensional stochastic dynamic programming was developed to deal with the relevancy and stochasticity in the streamflow and demand for the reservoir operations. Firstly, the one-dimensional distribution was selected to characterize the stochasticity of individual RI and IWD processes. Secondly, a Copula function of several variables was constructed using the marginal distribution. Specifically, the relevancy involved two aspects, i.e., the RI or IWD between the adjacent time intervals, and between RI and IWD at the same time. The stochasticity was referred to the individual RI, IWD, as well as the two-dimensional RI and IWD processes. Finally, the Copula function was integrated into the stochastic dynamic programming for the optimal dispatching model of the reservoir. A case study was set as an annual regulating reservoir responsible for agricultural irrigation. The results indicate that: 1) The Generalized Extreme Value (GEV) distribution was performed better on the RI and IWD in months. 2) There was remarkable relevancy of RI or IWD between a few adjacent time intervals. Moreover, there was a strong significance in April, May, June, July, September, and October. 3) Various types of Copula function were selected to model the dependent structure of RI/IWD variables at the adjacent time intervals. A Frank Copula was also employed to describe the negative relationship of dependent structure between RI and IWD at the same time. 4) The reservoir operation model considering the relevancy between RI and IWD was performed better than not. Specifically, the case study showed that the total water shortage was 10.37 × 108m3, when considering the relevancy and stochasticity of multiple variables, which was less than before (10.49×108 m3). Nevertheless, there was a little difference between the optimal total water shortages by the two multi-dimensional stochastic dynamic programmings. This trend was attributed to a little difference of state transition probability between with and without considering the relevancy of RI and IWD. Consequently, the newly-developed multi-dimensional stochastic dynamic programming can be widely expected to extend for the multi-dimensional stochastic optimal scheduling in fields, such as the multi-energy supplementary model considering the hydropower, wind, and light energy. Moreover, a better fitting can be suitable for the current relevancy and stochasticity in multi-dimensional processes, where the wind speed and solar radiation are stochastic variables in this case.