Abstract

As more and more reservoirs are built, concentrated water releasing in the dry season will bring about additional flow compensation and the joint operation of cascade hydro plants is quite important. This paper discusses the water level decline strategy of four cascade hydro plants in the Jinsha River and analyses the relationship between guaranteed output and total power generation. Considering stochastic inflows, an implicit stochastic optimization method is employed and a multi-objective parallel differential evolution algorithm is proposed to extract dispatching rules. Finally, a method which combines discriminant method and dispatching rules is proposed for practical operations and achieves good performance. Compared with routine scheduling, the power generations of the proposed method are improved observably in different typical years. The average power generation increases about 3% with the same cascade minimum output.

INTRODUCTION

In recent years, as more and more large reservoirs have been put into operation, runoff has changed greatly in the dry season with the regulating function of cascade reservoirs (Schäfer et al. 2016). It is necessary to study the operation mode of these cascade hydro plants and implement integrated water management. In the dry season, the natural inflow rate is usually small and cannot supplement downstream water requirements or guaranteed output demands. So the annual regulation reservoirs need to lower their water level and increase water release. Although the increased flow improves power generation for the current time, the water head benefits for later periods are reduced. In addition, guaranteed output should also be considered, as it is an important evaluation indicator and reflects the reliability and stability of hydropower stations (Zhang et al. 2015). If the guaranteed output is too small, the water level will be kept as high as possible in the whole dry season. The water head benefits can be maximized while this is adverse for reporting generation planning to the power grid. For the operation of cascade hydro plants, the problems become more complicated. In order to maximize the comprehensive benefits, the water level retrogression opportunity, sequence and depth of each hydro plant should be decided appropriately.

The current management of hydro plants in the dry season is mainly based on rule-curves, which were designed from the time series of historical inflows (Zhou et al. 2015). Therefore, reservoir operation efficiency is limited when inflows are significantly different from their seasonal average (Ficchì et al. 2015). In addition, the rule-curves only reflect suitable single reservoir operational decisions under various historical scenarios (Zhao et al. 2011) and lose sight of the joint operation of cascade hydro plants. Other than rule-curves, various methods have been proposed for single reservoir optimal operation in stochastic condition (Xie et al. 2015; Côté & Leconte 2016). Zhao et al. (2015) evaluated the marginal utility principle for long-term hydropower scheduling. Elabd & El-Ghandour (2013) performed multi-objective optimization of Bigge Reservoir operation in dry seasons. Zhao & Zhao (2014) analyzed joint and respective effects of long- and short-term forecast uncertainties on Danjiangkou Reservoir.

For the joint operation of cascade hydro plants, there is also much research using advanced methods (Zhang et al. 2012; Ramos et al. 2015; Pereira-Cardenal et al. 2016). Li et al. (2014) developed a parallel dynamic programming algorithm to optimize the joint operation of a multi-reservoir system. Bai et al. (2015) established a multi-objective optimal operation model of cascade hydro plants with four regulation objectives and proposed an improved multi-objective algorithm. Wang et al. (2015a) proposed a multi-population ant colony optimization for continuous domains to solve long-term scheduling in the Jinsha River. In these studies, the minimum output is only restricted by the minimum permitted reservoir release (Wang et al. 2015b) or treated as a fixed value. It does not change with a different rainfall year.

Based on the above, this paper analyzes the relationship between cascade minimum output and total power generation. Then the authors establish a multi-objective optimization model for a joint water level decline strategy for four cascade hydro plants in Jinsha River. In order to strengthen the expanding ability of the population at both ends, an improved multi-objective parallel differential evolution algorithm (Zhang et al. 2013a) is proposed to solve the problem. Generally speaking, the monthly runoff cannot be forecasted accurately but the general trend of future runoff can be predicted roughly. So the historical runoff data are divided into three typical years and an implicit stochastic optimization method (Ratnayake & Harboe 2007) is employed to extract dispatching rules. Finally, a method which combines a discriminant method (Jiang et al. 2014) and dispatching rules is proposed for practical operations and achieves good performance.

STUDY METHODS

Discriminant method

The K value discriminant method is a classic method to identify a water supply or water store sequence for cascade hydro plants (Wang et al. 2014). It can be formulated as:  
formula
(1)
where K = K value of reservoir; W = total water income; ΣH = total water head of current reservoir (including the head of all downstream reservoirs); ΣV = total water storage of all upper reservoirs; F = surface area of the reservoir.

The discriminant method takes the minimum energy loss or the maximum energy increase as the basic principle (Jiang et al. 2014). The higher the K value is, the more the energy increase for impounding water and the more the energy loss for discharging water. According to the formula, the future water quantity needs to be known but it is difficult to be forecasted accurately in actual operation. So this paper simplifies the K value discriminant method.

At the beginning of the dry season, considering that the later water income is large and neglecting the influence of the total water storage of all upper reservoirs, when the water head of i-th reservoir drops , the primary loss of power generation in the later period can be expressed as:  
formula
(2)
where A = output coefficient; = lost water head; = total water income in the later period; T = number of scheduling periods. It can be seen that the power generation loss is mainly decided by .
In the dry season, the natural inflow rate is usually too small to meet the system generating discharge requirements. Assume that the output gap is N, then if the gaps are supplemented by the i-th reservoir, the lost water level can be expressed as:  
formula
(3)
where N = output gaps; Fi = surface area of the i-th reservoir; = total water head of current reservoir (including the head of all downstream reservoirs); M = number of hydro plants. So can be used to judge the optimal water supply sequence. After calculating the difference between the given system output and the output of all the hydro plants with natural inflow, the reservoir which has a higher K* value should be given priority for discharging water to compensate the output difference value.

Before deciding on the water supply sequence, the system output must be determined. So next, a multi-objective optimization model is established to give the range of cascade minimum output. And after that, in order to prevent the concentrated decline of water level before the flood season which may cause water spillage, a method which combines a simplified discriminant method and dispatching rules is proposed.

Multi-objective optimization model and method

In the dry season, minimum output maximization and power generation maximization are two contradictory objectives. So this paper establishes a multi-objective optimization model. The objective function and constraints are described as follows:  
formula
(4)
where obj1 = total power generations; obj2 = cascade minimum output; M = number of hydro plants; T = number of scheduling periods; A = output coefficient of the i-th hydro plant; = generating flow of the i-th hydro plant at period t; = hydraulic head; Δt = interval of scheduling term.

The constraints are described as follows:

  • (1)
    Reservoir storage volume, water discharge and output limits  
    formula
    (5)
  • (2)
    The continuity equation for the hydro plant network  
    formula
    (6)
where = storage; = local inflow rate; = water discharge rate. and = lower and upper storage volume limits, respectively; and = minimum and maximum water discharge limits, respectively; = maximum output limits.

In order to solve the multi-objective optimization model, this paper proposes an improved multi-objective parallel differential evolution algorithm. The basic differential evolution algorithm is introduced first. As a mature heuristic algorithm, a differential evolution algorithm (Rajesh et al. 2016) has been widely used in hydro generation scheduling (Yuan et al. 2008; Qin et al. 2010a). It includes three operators: mutation, crossover and selection. This technique starts with a randomly generated initial population and utilizes the difference vector of two objects (values) which are randomly selected from the population as a random variation source for the third object. The mutation object is generated by adding the weighted difference vector to the third object; it can be formulated as:  
formula
(7)
The mutation parameter . In order to strengthen the diversity of the parameter vector, the crossover operator mixes the mutation object with a predetermined object:  
formula
(8)

The crossover parameter . After this, the selection operator compares the trial object with the target object and chooses the better one for the next generation.

A multi-objective differential evolution (MODE) algorithm consults a non-dominated sorting genetic algorithm-II (Aboutalebi et al. 2015) and introduces the non-dominated sorting strategy and the crowding distance calculation method into differential evolution. It has been successfully applied in multi-objective scheduling problems of reservoirs (Reddy & Kumar 2007; Qin et al. 2010b).

The complicated relationship between cascade minimum output and power generation makes MODE hard to find the Pareto optimal front. So this paper proposes an improved multi-objective parallel differential evolution algorithm (IMOPDE). The individual vector is composed of water level values for various periods of reservoirs. When the number of reservoirs is too much and the length of the individual vector is too long, the effect of evolution will be affected. So partial and global optimization are implemented in parallel. For partial mutation and crossover, only one reservoir participates each time in mutation and crossover operators. In order to accelerate the computation speed, the external archive set (Zhang et al. 2013b) is employed to preserve elite solutions and guide the evolutionary process (Li et al. 2015). The original mutation strategy and multi-elite guide mutation strategy (Zhang et al. 2013b) are implemented in parallel. Finally, three selection mechanisms (power generation priority, minimum output priority and crowding distance priority) are implemented in parallel to strengthen the expanding ability of the population at both ends. For each loop iteration, their external archive sets are merged and the evolution will restart. The IMOPDE consists of the above-mentioned strategies. The procedure of parallel computing is shown as Figure 1.

Figure 1

Procedure of parallel computing.

Figure 1

Procedure of parallel computing.

STUDY AREA

As Wudongde and Baihetan hydro plants are nearing completion, four world-class hydro plants (Wudongde, Baihetan, Xiluodu and Xiangjiaba) located in the downstream of Jinsha River will form a complicated hydropower system. Based on historical inflow, this paper studies their joint water level decline strategy. The basic parameters of these hydro plants are presented in Table 1.

Table 1

Parameters of hydro plants

Parameter Wudongde Baihetan Xiluodu Xiangjiaba 
Water level range (m) [945–975] [765–825] [540–600] [370–380] 
Regulation storage (×108 m³) 26.15 111.81 64.62 9.03 
Maximum water discharge (m³/s) 50,153 50,153 50,153 50,153 
Minimum water discharge (m³/s) 1,000 1,000 1,200 1,200 
Flood control limit level (m) 952 785 560 370 
Installed capacity (MW) 10,200 16,000 13,860 6,400 
Parameter Wudongde Baihetan Xiluodu Xiangjiaba 
Water level range (m) [945–975] [765–825] [540–600] [370–380] 
Regulation storage (×108 m³) 26.15 111.81 64.62 9.03 
Maximum water discharge (m³/s) 50,153 50,153 50,153 50,153 
Minimum water discharge (m³/s) 1,000 1,000 1,200 1,200 
Flood control limit level (m) 952 785 560 370 
Installed capacity (MW) 10,200 16,000 13,860 6,400 

RESULTS AND DISCUSSION

Dispatching rules extracting

First, IMOPDE is employed to analyze the relationship between cascade minimum output and power generation in the dry season. Three typical years are calculated (dry year, normal year and wet year). The Pareto front obtained by IMOPDE is shown as Figure 2. By calculating, general ranges of cascade minimum output in three typical years are obtained. Dry year: 7,200 MW–13,800 MW. Normal year: 8,200 MW–15,000 MW. Wet year: 11,000 MW–17,000 MW. These values can be used as a reference for the actual operation. By using these output values and the discriminant method in combination, the operation mode of cascade hydro plants at an early period in the dry season can be decided.

Figure 2

Pareto front obtained by IMOPDE.

Figure 2

Pareto front obtained by IMOPDE.

The discriminant method cannot deal with the concentrated water spillage problem before the flood season. So an implicit stochastic optimization method is employed to extract dispatching rules and prevent the concentrated decline of water level at the later period in the dry season. First, we count the start time of the water level declining. Table 2 lists the general time of two schemes (the scheme for a larger power generation and the scheme for a larger minimum output) in three typical years.

Table 2

General start time of water level decline

Schemes The start time of water level decline
 
Wet year Normal year Dry year 
Wudongde Larger power generation April 1 April 1 January 10 
Larger minimum output January 1 January 1 January 1 
Baihetan Larger power generation April 10 April 20 April 20 
Larger minimum output January 20 January 20 January 20 
Xiluodu Larger power generation May 20 June 10 June 10 
Larger minimum output April 20 April 20 April 10 
Xiangjiaba Larger power generation June 1 June 20 June 20 
Larger minimum output June 1 June 20 June 20 
Schemes The start time of water level decline
 
Wet year Normal year Dry year 
Wudongde Larger power generation April 1 April 1 January 10 
Larger minimum output January 1 January 1 January 1 
Baihetan Larger power generation April 10 April 20 April 20 
Larger minimum output January 20 January 20 January 20 
Xiluodu Larger power generation May 20 June 10 June 10 
Larger minimum output April 20 April 20 April 10 
Xiangjiaba Larger power generation June 1 June 20 June 20 
Larger minimum output June 1 June 20 June 20 

From Table 2, whether the hydro plants operate with a larger minimum output or a larger power generation, Wudongde and Baihetan will lower the water level first, then Xiluodu and Xiangjiaba. The results are in accordance with the discriminant method. Wudongde and Baihetan have a larger K* value and they are close to each other. So they lower their water level almost at the same time. Only in a dry year, in order to meet the minimum water discharge limit, will Wudongde lower its water level early. The K* value of Xiluodu is greater than Xiangjiaba. So Xiangjiaba lowers its water level last.

From May, the inflow of the Jinsha River begins to increase and the cascade minimum output has already been satisfied. At this time, the discriminant method has not played its function, while the water spillage problem needs to be considered. The four reservoirs must drop their water level to flood control level for defending flood before the end of June. How to maximize the benefit of power generation is a key issue. So this paper extracts dispatching rules and draws control lines by using an implicit stochastic optimization method. According to the results of multi-objective optimization, this paper calculates multiple typical years and gets the top covering lines. Then we take an average and obtain the control lines in three typical years. In this paper, we choose five years for each typical year. The control lines are shown as Figure 3.

Figure 3

Control lines of each hydro plant.

Figure 3

Control lines of each hydro plant.

Simulation computation

By means of the discriminant method and control lines, the reservoir operating mode in the whole dry season can be determined. The schematic of the whole procedure is shown as Figure 4.

Figure 4

Schematic of the whole procedure (t = scheduling period variable; T = the total number of scheduling periods; Na = the selected cascade minimum output; N = the output with natural inflow).

Figure 4

Schematic of the whole procedure (t = scheduling period variable; T = the total number of scheduling periods; Na = the selected cascade minimum output; N = the output with natural inflow).

In order to prove its advantages, the years 1969 (Dry year: The water frequency is around 90%), 1983 (Normal year: 50%) and 2001 (Wet year: 10%) are chosen for verification. The results are compared with routine scheduling (rule-curves). The water level processes calculated by the proposed method and routine scheduling are shown as Figures 5 and 6, respectively.

Figure 5

Water-level process of the proposed method.

Figure 5

Water-level process of the proposed method.

Figure 6

Water-level process of routine scheduling.

Figure 6

Water-level process of routine scheduling.

The results of routine scheduling show that there is no obvious water supply sequence and rules. Baihetan and Xiluodu lower their water levels almost synchronously. Xiangjiaba is a seasonal regulation reservoir and the variation of water level is irregular. All the reservoirs operate with their own rule-curves and the cascaded compensation benefits cannot be reflected. By combining the discriminant method and dispatching rules, the water level decline process is quite different from that of routine scheduling. As the upstream reservoirs, Wudongde and Baihetan have a high K* value and lower their water level first. The flood control level of Baihetan is higher than the dead water level. In order to reduce the water spillage before the flood season, Baihetan will lower its water level below the flood control level in advance and impound some water at the last period. Xiluodu and Xiangjiaba lower the water level in the last several periods. In this way, the water in upstream reservoirs can utilize the downstream high hydraulic head to generate more electric power. And the cascaded compensation benefits can be brought into full play. The statistical results of routine scheduling and the proposed method are listed in Table 3.

Table 3

Statistical results of routine scheduling and proposed method

Results Routine scheduling
 
Proposed method
 
1969 1983 2001 1969 1983 2001 
Power generation of Wudongde (108 kW h) 82.54 100.61 135.84 84.57 101.25 131.31 
Power generation of Baihetan (108 kW h) 182.34 199.35 253.29 177.82 197.04 248.33 
Power generation of Xiluodu (108 kW h) 204.88 216.10 267.49 218.13 249.63 304.03 
Power generation of Xiangjiaba (108 kW h) 119.89 129.71 162.97 114.37 129.54 161.70 
Power generation of cascade (108 kW h) 589.65 645.79 819.58 594.90 677.45 845.36 
Minimum output of cascade (MW) 10,368 12,031 14,745 10,368 12,031 14,745 
Results Routine scheduling
 
Proposed method
 
1969 1983 2001 1969 1983 2001 
Power generation of Wudongde (108 kW h) 82.54 100.61 135.84 84.57 101.25 131.31 
Power generation of Baihetan (108 kW h) 182.34 199.35 253.29 177.82 197.04 248.33 
Power generation of Xiluodu (108 kW h) 204.88 216.10 267.49 218.13 249.63 304.03 
Power generation of Xiangjiaba (108 kW h) 119.89 129.71 162.97 114.37 129.54 161.70 
Power generation of cascade (108 kW h) 589.65 645.79 819.58 594.90 677.45 845.36 
Minimum output of cascade (MW) 10,368 12,031 14,745 10,368 12,031 14,745 

Compared with routine scheduling, the power generations of the proposed method are improved with the same cascade minimum output in different typical years. Although some hydro plants' power generations decrease to some degree, the power generations of Xiluodu are enhanced observably among these hydro plants. The cascaded power generations are increased as well. In 1969, the total power generations increase about 0.9%. Baihetan and Xiluodu have a larger energy storage in the last period. In 1983 and 2001, the total power generations increase about 4.9% and 3.2%, respectively. Through designing the water supply opportunity and sequence reasonably, the cascade power generations can be enhanced observably and water spillage can be avoided. The proposed method is effective for joint generation scheduling of cascade hydro plants in the dry season.

CONCLUSIONS

This paper discusses the joint operation of cascade hydro plants in the dry season and establishes a multi-objective optimization model. In order to obtain a better Pareto front, IMOPDE is proposed to solve the problem. Considering that monthly runoff cannot be forecasted accurately but the general trend of future runoff can be predicted roughly, the years are divided into three typical years and an implicit stochastic optimization method is employed to extract dispatching rules. Finally, a method which combines a discriminant method and dispatching rules is proposed for practical operation. The proposed method can be summarized as follows. 1. Apply multi-objective optimization model to give the range of cascade minimum output in each typical year. 2. Extract dispatching rules and draw control lines by using implicit stochastic optimization method according to the results of multi-objective optimization. 3. Calculate the difference between the selected system output (choose from the range of cascade minimum output) and the output of all the hydro plants with natural inflow. 4. Calculate the K* value and decide which reservoir should be given priority for discharging water to compensate the output difference value. 5. Revise water levels of each hydro plant and make them below the control lines.

Four large hydro plants in Jinsha River are taken as a case study. By employing the proposed method and reasonably scheduling the water supply opportunity and sequence, the cascade power generation and minimum output were improved observably. Compared with routine scheduling, the average power generation increases about 3% with the same cascade minimum output. The proposed method can take into account stochastic inflows and cascaded compensation benefits simultaneously, so it is meaningful and has more significance in practice compared to routine scheduling and deterministic optimal scheduling.

ACKNOWLEDGEMENTS

This work is supported by the Key Program of the Major Research Plan of the National Natural Science Foundation of China (No. 91547208), the National Natural Science Foundation of China (No. 51579107), the National Key R&D Program of China (2016YFC0402205) and the State Key Program of National Natural Science of China (No. 51239004). Special thanks are given to the anonymous reviewers and editors for their constructive comments.

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