A rapid solution adjustment strategy in solving Long-term scheduling of large cascade hydropower stations

Due to the great potential of large cascade hydropower stations on power generation, long-term scheduling of large cascade hydropower stations (LSLCHS) plays an important role in electrical power system. As more and more concentrations focused on the optimal operation of large cascade hydropower stations, the LSLCHS has been taken into a multi-dimensional, non-convex and non-linear optimization problem due to its complicated hydraulic connection relationships and varieties of complex constraints with considering its power generation, shipping and ecological characteristics. In order to solve this problem, a strategy of solution rapid adjustment regarding the principle of monotone principle is proposed accordingly. The simulation results show that the proposed method is an efficient for solving joint optimization dispatch model of cascade hydropower stations with fast convergent rate, strong robustness.


Introduction
The Jinsha River is the upper stretches of the Yangtze River. Four large hydropower stations locate downstream the river, Wudongde (under construction, 8,700 MW), Baihetan (under construction, 14,000 MW), Xiluodu (completed, 13,860 MW) and XiangjiaBa (completed, 6,400 MW). With the gradual completion of these hydropower stations, requirements of optimal operation including long-term scheduling for power production, evolution of historical operation and compensation benefit analysis are urgent. Long-term scheduling of large cascade hydropower stations (LSLCHS) is a significance task to satisfy these needs. The objective of LSLCHS is to maximize the benefit of power production by controlling the water level processes of hydropower stations while subjecting to various constraints. In comparison to the optimization of single hydropower station, LSLCHS problem has more complex constraints due to its hydraulic and electric connection, which makes this optimization problem quite difficult to solve [1,2] .
The reminder of this paper is organized as follows, Section 2 introduces the formulation of LSLCHS problem as well as the problem solving strategy. In Section 3, the proposed method is applied to solve LSLCHS in Jinsha River, and the results are analysed. Finally, conclusions are summarized in Section 4.

Problem formulation
The primary objective of LSLCHS problem is to gain the maximum generating benefit of large cascade hydropower stations. The objective formula and constraints are described as follows.

Objective function
The objective to maximize the benefit of power production is described as follows, , ,

 
(1) B represents the total benefit of power production. T is amount of periods. N denotes the number of hydropower stations. N i,t stands for output of the i-th hydropower station in period t, t Δ is the length of the interval, K i represents output coefficient which is a constant determined by the characteristic of hydropower station, Q i,t equals outflow through hydro-turbines and Q i,t is calculated by water balance equation shown in formula (2). H i,t is pure water head. p i,t is the long-term electricity price.

Constraints
(1) Water balance constraint , 1 , , V i,t is reservoir storage of the i-th hydropower station at the beginning of period t, I i,t is inflow, q i,t stands for local inflow and S i,t is deserted outflow.
(3) Water level constraint output is a function of pure water head. min , i t N is the lower output limit, which is generally called Guaranteed output.
is the initial water level and end i Z is the terminal water level of the i-th hydropower station.

A rapid solution adjustment Strategy
A rapid solution adjustment strategy based on the principle of monotone principle of joint operation of cascade hydropower stations has been proposed. The proposed method adjusts the solution from time period 0 to T station by station. Details are shown as following, (1) The solution adjustment strategy in one step The solution adjustment strategy in one step is aimed to adjust the solution in period t of hydropower station m, which could improve the total power generation of cascade hydropower stations. The adjustment contains two conditions, ,max b. Calculate the best adjustment outflow The best adjustment outflow may be the boundary of the decision space or the value which make  d. The end.

Algorithms
In order to prove the performance of the proposed method, ant colony optimization for continuous domain (ACO R ) [22] is adopt as the optimization algorithm. The rapid solution adjustment strategy was used as daemon actions in ACO R . The framework of ACO R contains three parts, solution construction, pheromone update and daemon actions..
(1) Solution Construction A Gaussian kernel function, which is made up of a series of Gaussian functions, is used as the probability density function (PDF) to generate solutions.
The Gaussian kernel function is decided by three parameters, ω is the weight of individual Gaussian function, μ is the sample value, σ is the standard deviation. K is the size of Gaussian kernel function. One Gaussian kernel function is made up of K individual Gaussian functions. Another basic conception that must be mentioned in advance is the pheromone. In ACO R , a solution archive ranked by fitness value is adopted to be the pheromone of the ant colony (shown in Fig. 1).

Case Study
In order to prove the performance of the proposed method, a case study about LSLCHS problem downstream Jinsha river has been conducted. The rapid solution adjustment strategy was used as daemon actions in ACO R , which called improved ACO R (IACO R ). Cases are named after the year of the historical daily runoff, "1989" as the wet year case, "1958" as the dry year case and "2009" as the normal year case. Simulation of this problem is based upon a Java implementation running on a personal computer, Winds7 Ultimate server Pack1 x64, Intel(R) Core(TM) i5-2320 CPU@ 3.00GHZ, RAM 4.00GB.
Comparison results in different inflow conditions are shown in Table1. From Table1, it is obvious that the performance of IACO R is better than ACO R in both inflow conditions. The gained maximum power generation by IACO R is 1.01 billion, 0.45 billion and 0.28 billion more than which gained by ACO R in wet year, normal year and dry year, respectively. On the convergence precision aspect, the performance of IACO R is better than ACO R , shown in Fig.2. In general, the rapid solution adjustment strategy has effectively improve the performance of ACO R , and it is an optional tool to solve LSLCHS problem accurately and fleetly. "Max", "Mean" and "Std." stand for maximum, average and standard deviation.

Fig.2
The maximum power generation gained by ACOR and IACOR in 100 independent simulations

Conclusion
In this paper, a rapid solution adjustment strategy is proposed to solve long-term scheduling of large cascade hydropower stations problem. To verify its performance, a case study about LSLCHS problem downstream Jinsha river has been conducted. Simulation results show that the proposed rapid solution adjustment strategy has effectively improve the performance of the original algorithm, which means that it is an effectively enhancement tool for optimization algorithms in solving LSLCHS.