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The paper presents a two-stage approach to cope with the long-term optimal operation of cascaded hydropower systems. This approach combines progressive optimality algorithm (POA) with quadratic programming (QP) to improve the optimization results. POA is used at the first stage to generate a local optimal result, which will be selected as the initial feasible solution of QP method employed at the second stage. Around the initial solution, a rational local search range for QP method is then determined, where the nonlinear water level function and tailrace level function can be linearized nearly with high accuracy. The simplified optimization problem is formulated as a QP model with a quadratic generation function and a linear set of constraints, and solved using the available mathematic optimization software package. Simulation is performed on the long term operation of Hongshui River hydropower system which is located in southwest China and consists of 9 built hydropower plants. Results obtained from the proposed approach show a significant increase in the total energy production compared to the results from POA.

Over the past twenty years, China has put much effort on hydropower development [

Optimization of large-scale cascaded hydropower systems has been well known as a challenging practical and theoretical problem, which motivates strong demands for optimization techniques [

Among the above optimization methods, POA is usually used to solve the long-term optimal operation of cascaded hydropower plants. The significant advantages of POA are that it not only avoids resolving the nonli- near objective functions and constraints in the solution procedure, but reduces dimensionality difficulties by decomposing a multi-state decision problem into a series of non-linear programming two-stage problems. However, POA easily converges to a local optimum for complex problems with complex nonlinear relationship and spatial coupling constraints. To enhance the quality of the optimization solutions, it may be a feasible way to develop a hybrid method by incorporating POA with other global search algorithms.

The paper presents a two-stage approach to cope with the long-term optimal operation of cascaded hydropower systems. This approach combines POA with QP to improve the optimization results. POA is used at the first stage to generate a local optimal result, which will be selected as the initial feasible solution of QP method employed at the second stage. Around the initial solution, a rational local search range for QP method is then determined, where the nonlinear water level function and tailrace level function can be linearized nearly with high accuracy. The simplified optimization problem is formulated as a QP model with a quadratic generation function and a linear set of constraints, and solved using the available mathematic optimization software package. Simulation is performed on the long term operation of Hongshuihe River hydropower system which is located in southwest China and consists of 9 built hydropower plants. Results obtained from the proposed approach show a significant increase in the total energy production compared to the results from POA.

This paper uses the objective of maximizing total energy production which is a classic objective for hydropower systems [

where m = the reservoir index; M = the total number of reservoirs; t = the period index; T = the previous time index under consideration; p_{m}_{,t} = average generation of plant m in period t, in MW; Δ_{t} = time step in period t, in seconds.

Equation (1) is subjected to the following constraints:

1) Water balance:

This equation ensures the inflow and outflow balance between interconnected plants. Where V_{m}_{,t} = storage in reservoir m in period t, in m^{3}; Q_{m}_{,t} = inflow into reservoir m in period t, in m^{3}/s; q_{m}_{,t} = turbine discharge of plant

m in period t, in m^{3}/s; Ql_{m}_{,t} = spill flow of reservoir m in period t, in m^{3}/s.

number of immediate upstream plants of the mth plant; Qn_{m}_{,t} = local inflow into reservoir m in period t, in m^{3}/s; ^{3}/s;

2) Specified target demand for single reservoir:

where

3) Turbine discharge capacity for single plant:

where ^{3}/s.

4) Discharge capacity for single reservoir:

where^{3}/s.

5) Minimum and maximum forebay elevation:

where

6) Generation capacity for single plant:

where

7) System generation capacity for cascaded hydropower system:

where

The paper solves the above optimization problem using the developed two-stage approach which combines the progressive optimality algorithm (POA) with quadratic programming (QP). In this approach, the first stage determines the initial feasible solution using POA, and the second stage improves the obtained solution by using QP method. The following two sections describe the solution process at each stage, respectively.

POA has been shown to have great advantages as an effective method for solving large-scale hydropower system operations. This method has some merits, such as no need for discretizing the state variables, no resolving or linearization of nonlinear objective functions and constrains, etc. Basically, POA solves this kind of long- term hydropower system operations by repeatedly resolving a two-stage subproblem from time-step 1 to T. For each subproblem, the goal is to determine optimal state variables

Around the initial solution obtained at the first stage, a rational local search range for QP method is determined, where the nonlinear water level function and tailrace level function can be linearized nearly without losing accuracy. The simplified optimization problem is then formulated as a QP model with a quadratic generation function and a linear set of constraints, and solved using the general mathematic optimization software package.

First of all, the decision variables should be chosen to formulate the problem as a QP one. In this study, the forebay level, turbine discharge, and spill at all periods are used as the decision variables, i.e.

For each reservoir m, the storage at period t can be expressed as the linear function of the forebay level.

where k = the section index within the bounds of level.

For each plant m, the tailrace elevation

where

For each plant m, the head losses in the penstock at period t are expressed as a linear function of turbine dis- charge, shown in Equation (11). Thus the net head is obtained by subtracting head losses from the gross head, shown in Equation (12).

where

Based on the above linear relationships of decision variables, the original objective function can be reformulated as a quadratic function of decision variables.

Inserting Equation (9) into Equation (2) gives

where k1, k2 are the section indexs within the bounds of level.

For each plant m, water rate

where

By inserting

In a similar way, the lower bound of power generation can be reformulated as

For the system constraint (8), we use the new multipliers to convert the generation limitation into equivalent discharge limitation. The new system constraints are

where

Consequently, the original optimization problem is described as a QP model with a quadratic objective function (13) and linear constraints (3)-(6), (14), (16)-(18). The QP model can be easily solved using the existing standard algorithm provided in some mathematic optimization software such as MATLAB and LINGO.

The developed approach is implemented to the cascaded hydropower plants in the main stream of Hongshuihe River operated by SCPG and Guangxi Power Grid (GXPG). Hongshuihe River is located in southwest China, and flows from northwest to southeast with about 1050 km stretch, a drainage area of 190,000 km^{2} as well as annual mean precipitation of 1200 mm. The River originates from Nanpanjiang River in Yunnan Province, later known as Hongshuihe by the confluence with Beipanjiang River in Guizhou Province. There are also two main tributaries in the down reach of Hongshuihe River, Yujiang River and Liujiang River. Correspondingly, 10 cas- caded hydropower plants are under planning stage in the main stream, 9 of which have been put into operation from upstream First Reservoir of Tianshengqiao (FROTSQ) to downstream Qiaogong plant, with an installed capacity of 10,739 MW.

To test the proposed approach, two different schemes with different flow are given. The first flow scheme uses 25% frequency to determine the local inflow, and the second scheme uses 50%.

Flow scheme | Method | Total energy/10^{8} kWh | Computation time/s |
---|---|---|---|

The first scheme (25%) | Our approach | 450.96 | 31 |

POA | 445.37 | 11 | |

The second scheme (50%) | Our approach | 395.82 | 28 |

POA | 390.57 | 10 |

As can be seen from

It is easily seen from

With the rapid development of China’s hydropower, there has been a pressing need for efficient and feasible optimization methods to enhance the production benefit of large-scale hydropower systems. This study proposes a two-stage approach and implements it to the Hongshuihe hydropower system. The simulation results from a case study demonstrate that the proposed approach can enhance the energy production of cascaded hydropower plants. It is also concluded that integrating multiple typical optimization algorithms maybe a feasible way to ef- ficiently solve the complex optimization problem of large-scale hydropower system operations.

This research was supported by National Natural Science Foundation of China (No. 51209031) and the Funda- mental Research Funds for the Central Universities (No. DUT14RC(3)089), and “Xinghai Scholar” talents training plan of Dalian University of Technology.