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This paper focused on the applying stochastic dynamic programming (SDP) to reservoir operation. Based on the two stages decision procedure, we built an operation model for reservoir operation to derive operating rules. With a case study of the China’s Three Gorges Reservoir, long-term operating rules are obtained. Based on the derived operating rules, the reservoir is simulated with the inflow from 1882 to 2005, which the mean hydropower generation is 85.71 billion kWh. It is shown that the SDP works well in the reservoir operation.

Stochastic programming is a framework for modeling optimization problems that involve uncertainty [1-2]. It has been widely used in water resources planning and management [3-9]. The classic approach is the stochastic dynamic programming (SDP), which is able to tackle not only linear but also nonlinear objective function and constraints. The SDP recurrent formulation is

where denotes the optimal benefit-to-go function from decision and g_{0}(x_{t}) is the benefit with a decision x_{t}.

However, SDP needs to discrete the state space and this makes it heavy computational burden. In order to tackle this “curse of dimensionality”, there are several types of methods have been studied. 1) Method based on aggregation and decomposition. A method was proposed in which the control problem for a system of M reservoirs in series was decomposed into M sub-problems each with two reservoirs: one a reservoir from the original problem and the other an aggregate representation of the reservoirs downstream of that reservoir [

An interval-parameter multistage stochastic programming method was proposed for supporting water resources decision making, where uncertainties expressed as random variables and interval numbers could be reflected [

The operating rules are a function between decision and observation (or prediction). Typically, single reservoir operating rules provide a release decision when inflow and current water storage have been obtained. If the operation horizon is infinite to future,

where S_{i} is the initial water storage at time period i, and R_{i} are inflow and release during time period i. and are the minimum and maximum allowable water storages during time period i, respectively. and denote the minimum and maximum reservoir releases subjected to physical constraints during time period i, respectively.

Since it is difficult to estimate accurately the inflow far behind (e.g., 100 years late), we estimate it by using periodic characterization. That is, during a cycle periodic time 1 to n,

By using a two-stage stochastic programming framework,

where is a benefit-to-go function from to end time. The means the terminal condition of optimization.

The utilization function may be a linear function (e.g., for water supply) and a nonlinear function (e.g., for hydropower generation).

The Three Gorges Reservoir (TGR) is a vital project for water resources development of China’s largest river, the Yangtze River (^{3} km long channel with a contributing

drainage area of 10^{6} km^{2}. The mean annual runoff at the dam site is 4.51 × 10^{11} m^{3}. The TGR is to date the largest multipurpose hydro-development project ever built in the world. Its benefits include flood control, power generation and navigation improvement. Streamflow records from Yichang flow gage station, located about 40 km downstream of the TGR, are used as inflow of the TGR. The interval of the time period is ten-day-long, a traditional Chinese measure of time. The streamflow series from 1882 to 2005 is used in this study.

The SDP involves two steps as follows:

1) Discretized the Inflow and Reservoir Storage: The inflow during the same time period (such as Jan), are discretized into N intervals from minimum to maximum. The probability of inflow interval i (during time period t) to inflow j (during time period t + 1) is also computed from the observed inflow series.

2) Finding the optimal solutions: By using Equation (4), the optimal solution is found and saved as optimal operating rules. The following operating rules are used in this paper.

where is the reservoir storage need to decide, and are the current reservoir storage and inflow.

The above SDP method is used to find the optimal operating rules, and it consumes about 3 minutes. Part of the operating rules is shown in

Based on the derived operating rules, the reservoir is simulated from 1882 to 2005. Finally, the mean hydropower generation is 85.71 billion kWh.

The above operating rules only use the current reservoir storage and inflow as variables for make-decision. In other words, this is a two-dimensional SDP problem. Indeed, SDP needs to discrete the state space and this makes it heavy computational burden.

In this paper, we focused on the applying stochastic programming to water resources management. We built one operation model for reservoir operation for operating rules derivation. With a case study of the China’s Three Gorges Reservoir, long-term operating rules are obtained. Based on the derived operating rules, the reservoir is simulated with the inflow from 1882 to 2005, which the mean hydropower generation is 85.71 billion kWh. It is shown that the SDP works well in the reservoir operation.

This study was supported by Program for New Century Excellent Talents in University (NCET-11-0401), the National Key Technologies Research and Development Program of China (2009BAC56B02, 2009BAC56B04) and the National Natural Science Foundation of China (50979072).