## Abstract

In multi-objective reservoir operation, it is vital for decision-makers to select optimal scheduling schemes through efficient multi-criteria decision-making (MCDM) techniques. However, in the family of MCDM methods, it is difficult for the technique for order preference by similarity to an ideal solution (TOPSIS) to describe grey correlation, thus making decisions with less reliability. To this end, a framework supporting high-quality solutions' acquirement and optimal reservoir operation decision-making is established. The improved multi-objective particle swarm optimization (IMOPSO), a new efficient MCDM model based on TOPSIS and grey correlation analysis (GCA), and combination weighting method based on the minimum deviation (CWMMD) are included in the framework. The non-inferior solution set is efficiently obtained by IMOPSO and optimal decision information is provided for decision-makers using the MCDM model. Moreover, the CWMMD is used to determine weighting information of multiple evaluation indicators. Numerical simulations are conducted to verify the efficiency of the proposed methodology and support decision-making for multi-objective reservoir operation in Hongjiadu and Qingjiang basins. The results indicate that the proposed methodology can provide non-inferior scheduling solutions and decision-making instruction with higher reliability for multi-objective reservoir operation.

## INTRODUCTION

Complex reservoir systems, especially cascade reservoirs, are built to control flood control and increase social, economic, and ecological benefits. Optimal reservoir operation requires simultaneous optimization of multiple competing objectives, such as hydropower generation, flood control, safety requirements of upstream and downstream regions, the water supply and shipping (Jia *et al.* 2016). Due to the multiple benefits of optimization, it is unpractical to find a single scheduling scheme to balance all the objectives. Instead, solutions with trade-offs, also referred to as Pareto optimal solutions are selected by decision-makers, and thus, multi-objective reservoir operation can be defined as a multi-criteria decision-making (MCDM) problem (Zhu *et al.* 2016, 2017). As a group of satisfying solutions are provided to decision-makers, they need to locate the most ideal or comprehensive solution to operate a reservoir. To this end, it is of great theoretical and practical significance to develop an effective MCDM method to rank Pareto optimal solutions.

TOPSIS refers to the technique for order performance by similarity to ideal solution which is a classical MCDM method proposed to rank various schemes according to closeness to the ideal alternative (Hwang & Yoon 1981). Based on the concept of ideal and anti-ideal point, the most satisfying alternative obtained by the TOPSIS method is defined as the alternative that is simultaneously closest to the ideal alternative and farthest from the anti-ideal alternative. TOPSIS has been widely applied to solve MCDM problems and recommended by the United Nation Environmental Program (Abo-Sinna *et al.* 2008; Afshar *et al.* 2011). Meanwhile, improvements to TOPSIS have also received deserved attention. Yu *et al.* (2004) proposed an improved TOPSIS based on the fuzzy preference. Karimi *et al.* (2012) extended the traditional TOPSIS method to fuzzy TOPSIS method and applied it to financial risk management. Wang & Wang (2014) and Chen & Lu (2015) replaced Euclid distance with the Mahalanobis distance to modify conventional TOPSIS and combined it with fuzzy analytical hierarchy process (FAHP). Deng *et al.* (2000) attempted to improve TOPSIS from the aspect of scientific indicator or criterion weights' determination.

However, due to the limitations of cognitive ability and complexity of evaluation object, information such as attribute or preference is missing in many decision-making problems and cannot be completely represented by certain values. For an MCDM problem of multi-objective reservoir operation, it is a complicated decision-making process to evaluate and optimize the comprehensive reservoir operation schemes. In addition, as multi-objective reservoir operation is not limited to a single reservoir, cascade reservoirs are jointly operated in order to maximize overall benefit (Wang *et al.* 2014; Li & Ouyang 2015; Peng *et al.* 2015). In comparison with a single reservoir, the MCDM problem of complex cascade reservoir systems is more challenging and complex because more hydraulic constraints, uncertainty factors, and decision criteria should be considered for assessing the quality of scheduling schemes. Furthermore, the hidden interrelation and coupling relation among the indicators and criteria make the reservoir operation scheme reveal the inner grey correlation. The grey correlation also referred to as ‘grey characteristics’ between evaluation indicators cannot be described or solved by TOPSIS. To this end, grey correlation analysis (GCA) is combined with TOPSIS to efficiently show the inner correlation between scheduling schemes.

The GCA is derived from grey system theory proposed to solve the problem of incomplete information and uncertain system. It is also a family of MCDM methods for assisting the decision process in multi-criteria or attribute system. GCA describes the similarity of factors according to the closeness of variation trend among factors, and its essence is to judge the correlation degree of data according to the similarity degree of geometric relationship between data (Jiang *et al.* 2007). Therefore, GCA has become a common mathematical tool in the field of MCDM. Zeng *et al.* (2003) established a multi-objective decision model based on grey relational analysis and applied it to the optimization of normal reservoir water level. Li *et al.* (2009) applied a grey relational analysis model based on improved entropy weight to optimize the evaluation of reservoir scheduling schemes.

In this paper, we proposed an improved MCDM method based on GCA and TOPSIS dealing with optimization of scheduling scheme for multi-criteria flood control operation and multi-objective ecological and water supply operation in complex reservoir systems. The weights of indicators or multiple criteria are determined using combination weighting method based on minimum deviation (CWMMD). In recent years, single objective and multi-objective optimization techniques have been widely used to solve optimization problems including reservoir operation (Azamathulla *et al.* 2008; He *et al.* 2014; Luo *et al.* 2015; Jia *et al.* 2016; Shi *et al.* 2017; Yang *et al.* 2019a, 2019b, 2019c). As a family of multi-objective optimization methods, MOPSO and modified versions have been widely used to solve multi-objective optimization problems and received deserved attention (Dai *et al.* 2015; Li *et al.* 2016; Sheikholeslami & Navimipour 2017; Laskar *et al.* 2018; Zain *et al.* 2018). To efficiently solve the multi-objective optimization model and obtain non-inferior solutions, we proposed effective improvements based on conventional MOPSO. Thereafter, the ranking order of all the feasible scheduling schemes or alternatives can be obtained using the GCA-TOPSIS proposed. Finally, we apply the proposed methodologies to decision-making for flood control operation in Hongjiadu reservoir, and ecological and water supply operation in Qingjiang cascade reservoirs. The results show that the proposed methodology can effectively locate the ideal reservoir scheduling information for decision-makers and improve the reliability of decisions for multi-objective reservoir operation with emphasis on flood control, ecology, and water supply.

## THE IMPROVED MCDM MODEL BASED ON GCA-TOPSIS

### Definition of MCDM problem

As a large-scale water control system with multiple objectives, reservoir operation usually has multiple functions such as flood control, power generation, shipping, and water supply. Due to the presence of the multiple objectives, it is difficult to find a single optimal solution that optimizes all goals. Instead, multiple solutions exist in the form of trade-offs, also referred to as Pareto optimal solutions (Malekmohammadi *et al.* 2011). We use MCDM methods to evaluate and select the most preferred solution from the set of non-inferior solutions on the Pareto front for decision-makers (Zhu *et al.* 2017). In general, an MCDM problem consists of a set of alternatives, criteria, and an MCDM model. In terms of this study, the non-inferior solutions are viewed as alternatives. The criteria are related to the function of reservoirs and weighting information is decided by CWMMD. For the MCDM model, we proposed a decision-making method on the basis of GCA and TOPSIS.

### The description of GCA

Grey correlation analysis is a method to determine the quality grade of samples according to the correlation between the comparison sequence and the reference sequence. The reference sequence has the best correlation with the largest degree of correlation, and the quality grade of the sample can be obtained accordingly. The principle of GCA is simple and requires fewer original data with more convenient operation (Zhang & Liang 2009). These features make it easier to mine data rules. The essential steps of a GCA model include: (1) data sequence determination including reference and comparison sequences, (2) building a decision matrix, (3) calculating ideal and anti-ideal schemes, (4) calculating correlation coefficient and correlation degree. The higher the correlation degree is, the better the scheme will be.

Due to less strict quantitative requirements for data and simple calculation, GCA is widely used in fields such as social science, decision-making, and management. In practice, GCA is capable of analyzing the correlation between the same elements in different schemes; however, it is impossible to compare the close degree of different elements.

### The TOPSIS model

TOPSIS is a well-known MCDM method based on the concept of ideal and anti-ideal points, where the best alternative should be the one that is closest to the ideal alternative and farthest from the anti-ideal alternative (Hwang & Yoon 1981). This method has no strict restrictions and requirements in terms of the number of indicators, also referred to as criteria, samples, and data. It can fully make use of original data with less information loss (Lei *et al.* 2016). The main advantages of TOPSIS include: (1) its robust logical structure, (2) its powerful computation capability, and (3) its ability to consider ideal and anti-ideal points simultaneously. The main steps of TOPSIS can be elaborated as follows.

**Step (1)**: Building a normalized decision matrix and calculating the corresponding weighted matrix. The normalization and weighted methods are as follows: where is the initial value of the th indicator in the th evaluation schemes; is the weighted ; is the value with normalization; and are maximum and minimum value of the th indicator in the th evaluation schemes.

The TOPSIS also has defects including: (1) when calculating the distance between the evaluation object and ideal and anti-ideal solutions, the correlation between indicators cannot be fully considered; (2) the schemes may be equally close to the ideal and anti-ideal solutions, making it impossible to locate the best scheme.

### The GCA-TOPSIS model

Multi-objective reservoir operation including objectives such as flood control, ecological, power generation, and water supply is affected by the uncertainty of hydrology, hydraulics, and reservoir operation management. These uncertain factors have great influence on the scheduling decision, which may lead to differences between the actual scheduling process and the scheduling scheme made under certain conditions. In the decision-making process of multi-objective reservoir operation, due to uncertainty factors, insufficient statistical data, great data fluctuation, and inapparent typical distribution rule, the uncertainty of quantitative relationship between evaluation criteria will lead to obvious ‘grey characteristics’. To this end, in terms of potential defects existing in GCA and TOPSIS, we build a decision-making model coupling the GCA and TOPSIS to make full utilization of their respective strengths. The GCA contributes to reflecting the difference between the change trend in scheduling scheme sets and ideal scheme, and places emphasis on the internal connection between indicators. Meanwhile, the TOPSIS can reflect the overall similarity between alternative and ideal schemes. The combination of GCA and TOPSIS makes the decision-making for multi-objective reservoir operation more scientific and reasonable. The coupling process is elaborated as follows.

**Step (1):** Assuming that the solution set for reservoir multi-objective operation is ; the solution decision matrix is ; the weighting vector is ; the corresponding weighted solution decision matrix is .

**Step (2):** Normalize the initial decision matrix and calculate according to methods mentioned in the section ‘The TOPSIS model’.

**Step (3):** Determine the ideal and anti-ideal solutions of according to **Step (2)** in the section ‘The TOPSIS model’.

**Step (4):** Calculate the ED between each scheme and the ideal and anti-ideal solutions ( and ) in accordance with Step (3).

**Step (5):**Calculate the grey correlation coefficient (GCC) matrix between each scheme and the ideal and anti-ideal solution ( and ) according to the following equation: where and are correlation coefficient elements in the matrix; , , and are in line with the above. The empirical value of is 0.5 usually.

**Step (7):**The ED and GCD are merged after dimensionless treatment. The merge equations can be determined as follows: where and are merge equations; and are ED with dimensionless treatment; and are GCD with dimensionless treatment. The higher the value of and is, the better the feasible scheme will be. On the contrary, the lower the value of and is, the better the feasible scheme will be.

**Step (8):**The relative similarity degree is defined to reflect the closeness degree of the feasible scheme to the ideal or anti-ideal scheme in terms of situation change. The higher the value is, the better the scheme will be and vice versa. The relative similarity degree can be calculated according to the following equation: where and denote preference degree of the decision-maker. Normally, the values of and are consistent.

## THE COMBINATION WEIGHTING METHOD BASED ON MINIMUM DEVIATION

It is critical to confirm indicator weights in order to quantitatively describe the importance for evaluation indicators and improve reliability of evaluation results. To our knowledge, weighting methods can be divided into two types: subjective and objective measures. The former is to determine the indicator weight according to the subjective indicator emphasis. The latter obtains the weight from original objective information of indicators. To take advantage of the two types of weighting methods, the combination weighting method is used to achieve balance between subjective intention and objectivity of indicator.

Normally, combination weights are calculated by linear combination coefficient which is subjected to subjective experiences. Therefore, in this work, combination weighting measure based on minimum deviation is proposed to get the optimal weight combination. The analytic hierarchy process (AHP) and improved entropy method are selected as the subjective and objective weighting measure, respectively. The details of AHP and improvements for entropy method can be seen in research by Yang *et al.* (2019a).

### The improved entropy weight method

*i*are calculated according to the following equations: where

*k*is standard coefficient;

*n*denotes indicator amount;

*m*represents observed sample data of evaluation indicators; and is the th standardized sample date of the indicator

*i*.

The nature of entropy includes: (1) If the entropy value of an indicator is large, it indicates that the value difference of each scheme on this indicator is small. The corresponding value of entropy weight is also small and vice versa. (2) If all schemes have the same value of one indicator, the entropy value of this indicator is assumed to be ().

To verify the reasonability of the improved entropy weight method**,** comparison between it and conventional entropy method is displayed in Table 1.

Number | Distribution circumstance | Entropy value | Weighting results | |
---|---|---|---|---|

Conventional entropy method | Improved entropy proposed | |||

1 | (0.9999,0.9998,0.9997) | (0.167,0.333,0.500) | (0.332,0.333,0.335) | |

2 | Discrete distribution | (0.9,0.5,0.1) | (0.067,0.333,0.600) | (0.067,0.333,0.600) |

Number | Distribution circumstance | Entropy value | Weighting results | |
---|---|---|---|---|

Conventional entropy method | Improved entropy proposed | |||

1 | (0.9999,0.9998,0.9997) | (0.167,0.333,0.500) | (0.332,0.333,0.335) | |

2 | Discrete distribution | (0.9,0.5,0.1) | (0.067,0.333,0.600) | (0.067,0.333,0.600) |

From the results in Table 1, indicator weights obtained by the improved entropy method are more reasonable in terms of distribution circumstance 1. The indicator weight is distributed more evenly with small changes of . When the weights present discrete distribution, results of the improved method coincide with the conventional one as well. Therefore, the indicator weights calculated by the improved entropy method are more objective and reliable than that of the conventional one.

### Weight combination based on minimum deviation principle (CWMMD)

*r*weight determination methods are selected, and

*m*represents the quantity of evaluation indicators or criteria. Accordingly, corresponding indicator or criteria weight vector can be described as follows: The combination weighting optimization model based on minimum deviation is established and shown as follows: where

*W*denotes the minimum weight deviation between each weighting method; indicates weight distribution of all weighting methods.

*n*, , and are intermediate variables during the solving process. The simplification forms are multidimensional linear equations containing unknown numbers. The multidimensional linear equations have unique solution vector if the corresponding determinant of coefficient . Finally, the combination weights of evaluation indicators are obtained by weighted average method.

## THE IMPROVED MULTI-OBJECTIVE PARTICLE SWARM OPTIMIZATION (IMOPSO)

### The standard particle swarm optimization (PSO)

*et al.*2016; Laskar

*et al.*2018). In PSO, a particle is used to represent the potential solution of an optimization problem. The whole particle swarm flies in the feasible space to search for the global optimum. In a D-dimension hyper-space, the velocity vector and position vector are related to the th particle , where

*i*denotes the population size, and t represents current iteration number. After each iteration, local and global optima are selected and guide the updating direction of particles in the population. Normally, the velocity and position of each particle are first initialized by random vectors within feasible ranges. Then, the new velocities and positions of particles are updated in line with the following equations: where represents the personal best particle for the th particle. is the best position that the whole swarm has found so far. is the inertia weight, which was introduced (Shi & Eberhart 1998) in order to enhance the exploration and exploitation capability; and are called acceleration coefficients; and denote random numbers uniformly distributed in the range [0, 1].

In order to solve the multi-objective optimization problem, we need to convert standard PSO into multi-objective PSO. The MOPSO was first proposed to handle multi-objective optimization problems (Coello & Lechuga 2002). The MOPSO has the characteristics of less parameters, easy operation, and fast convergence. To further enhance the search capability of MOPSO, we proposed dynamic clustering and particle mutation strategies presented as follows.

### Improvement strategy

#### Dynamic clustering strategy (DCS)

The diversity of the population will gradually decrease in the evolutionary process, so it is easy to fall into the local optimum. To this end, the dynamic clustering strategy is introduced to strengthen search capability and diversity of the particle population (Yu *et al.* 2018). The specific operation of dynamic clustering strategy is as follows. Furthermore, to improve the search ability and convergence of the algorithm, the population is divided into several sub-populations in which particles are updated separately. The best position and velocity of particles in sub-populations are recorded, respectively:

Assuming that the number of sub-populations is and total

*L*particles are assigned into each sub-population.The th particle sub-population conducts non-dominated sorting first according to mutual dominance. Then, these particles are ranked in accordance with crowding distance (Raquel & Naval 2005).

- The th sub-population implementing DCS will be divided into two groups. One is the high-quality group and the other is the inferior group. The quantity of particles in the high-quality group is controlled by threshold . If the number of the particle is within the threshold , the particle will enter the high-quality group, otherwise it will be assigned to the inferior group. The is calculated according to the following equations: where is current iterations.
*T*is total iteration number. denotes the change rate of the and is times the void group searched. The design of is used to provide the existence of particles in both high-quality and inferior groups. The two types of groups are merged after particle update. The sub-population to implement DCS in the next generation is selected randomly in all sub-populations.

#### The particle update

The particles in sub-populations without implementing DCS are updated according to the strategy in conventional MOPSO. The particles in the th sub-population implementing DCS evolve according to the following method.

For the particles in the high-quality group, update method is basically the same as the standard update in MOPSO. The differences lie in that the and are replaced by personal best for the th particle in the high-quality group and the optimal particle for th particle sub-population, respectively.

- The particles in the inferior group are of relatively poor quality. However, the conventional update strategy is incapable of making full use of evolution information. To this end, besides the personal best and optimal particles for the th particle sub-population, global optimal particle is also used to provide guidance information for the evolution of the inferior particles. The specific particle velocity update equation is as follows: where represents personal best particle for the th particle in the inferior group; denotes optimal particle in the inferior group; is the best particle position in the whole population. and are the same as and .

#### The optimal particle selection

*m*, particle corresponding to the th optimization objective, respectively. and = the best and worst fitness in the EAS.

In terms of selecting the optimal particle in high-quality and inferior groups , the strategy is to compare the positions of each particle in the current group and pick the optimal one. If multiple particles exist, they will be further compared to locate the optimal particle with the largest crowding distance.

### The flow chart of the proposed methodology

In this paper, we construct a framework combining the IMOPSO with MCDM model based on GCA and TOPSIS. The flow chart of the framework is demonstrated in Figure 1.

### The numerical simulation

#### Testing function

In this section, we conduct simulation experiments to demonstrate the superiority of the proposed IMOPSO with several testing functions. The testing functions are from the ZDT and DTLZ package. The NSGAII (fast elitist non-dominated sorting genetic algorithm), MOEAD (multi-objective evolutionary algorithm based on decomposition), MOEADDE (MOEAD based on differential evolution operator), dMOPSO (multi-objective particle based on decomposition), and MOPSO (multi-objective particle swarm optimization) are selected to compare with the IMOPSO. A total of 30 independent simulations are conducted to offset randomness effect. Furthermore, the IGD and GD indicators are analyzed to evaluate the performance of IMOPSO quantitatively. The IGD is an indicator for assessing convergence and solution diversity. The GD represents the distance between the true Pareto front and Pareto front searched by IMOPSO and other rivals. The smaller the IGD and GD are, the better the convergence and distribution of the solution are. We take the ZDT1, DTLZ1, and DTLZ5 for examples. The Pareto fronts obtained by IMOPSO are shown in Figure 2. The mean and standard deviation (STD) of the IGD and GD indicators are shown in Tables 1 and 2.

Function testing | NSGAII | MOEAD | dMOPSO | MOEADDE | MOPSO | IMOPSO |
---|---|---|---|---|---|---|

The mean of IGD | ||||||

ZDT1 | 2.23 × 10^{−2} | 8.22 × 10^{−2} | 4.37 × 10^{−3} | 4.60 × 10^{−2} | 2.13 × 10^{−2} | 2.63 × 10 ^{−3} |

DTLZ1 | 6.69 × 10^{−2} | 4.11 × 10^{−1} | 8.73 × 10^{−3} | 7.77 × 10^{−2} | 6.96 × 10^{−2} | 1.14 × 10 ^{−3} |

DTLZ5 | 1.77 × 10^{−1} | 3.91 × 10^{−1} | 1.01 × 10^{−2} | 2.21 × 10^{−1} | 1.58 × 10^{−2} | 1.99 × 10 ^{−3} |

The STD of IGD | ||||||

ZDT1 | 3.06 × 10^{−2} | 5.44 × 10^{−2} | 1.70 × 10^{−3} | 8.25 × 10^{−2} | 2.75 × 10^{−3} | 2.03 × 10 ^{−5} |

DTLZ1 | 6.11 × 10^{−2} | 1.09 × 10^{−1} | 3.41 × 10^{−3} | 2.16 × 10^{−1} | 6.26 × 10^{−3} | 1.08 × 10 ^{−5} |

DTLZ5 | 1.47 × 10^{−1} | 2.00 × 10^{−1} | 4.13 × 10^{−3} | 3.54 × 10^{−1} | 1.61 × 10^{−3} | 1.88 × 10 ^{−5} |

Function testing | NSGAII | MOEAD | dMOPSO | MOEADDE | MOPSO | IMOPSO |
---|---|---|---|---|---|---|

The mean of IGD | ||||||

ZDT1 | 2.23 × 10^{−2} | 8.22 × 10^{−2} | 4.37 × 10^{−3} | 4.60 × 10^{−2} | 2.13 × 10^{−2} | 2.63 × 10 ^{−3} |

DTLZ1 | 6.69 × 10^{−2} | 4.11 × 10^{−1} | 8.73 × 10^{−3} | 7.77 × 10^{−2} | 6.96 × 10^{−2} | 1.14 × 10 ^{−3} |

DTLZ5 | 1.77 × 10^{−1} | 3.91 × 10^{−1} | 1.01 × 10^{−2} | 2.21 × 10^{−1} | 1.58 × 10^{−2} | 1.99 × 10 ^{−3} |

The STD of IGD | ||||||

ZDT1 | 3.06 × 10^{−2} | 5.44 × 10^{−2} | 1.70 × 10^{−3} | 8.25 × 10^{−2} | 2.75 × 10^{−3} | 2.03 × 10 ^{−5} |

DTLZ1 | 6.11 × 10^{−2} | 1.09 × 10^{−1} | 3.41 × 10^{−3} | 2.16 × 10^{−1} | 6.26 × 10^{−3} | 1.08 × 10 ^{−5} |

DTLZ5 | 1.47 × 10^{−1} | 2.00 × 10^{−1} | 4.13 × 10^{−3} | 3.54 × 10^{−1} | 1.61 × 10^{−3} | 1.88 × 10 ^{−5} |

From Figure 2 and Tables 2 and 3, the final solutions on the Pareto front obtained by IMOPSO are characterized by uniform distribution and strong diversity. The lower IGD and GD demonstrate that IMOPSO is closer to the true Pareto and has advantages in convergence, distribution, and diversity of solutions compared with other methods. Therefore, the IMOPSO is verified to have advantages on solution quality and stable performance compared with other methods selected.

Function testing | NSGAII | MOEAD | dMOPSO | MOEADDE | MOPSO | IMOPSO |
---|---|---|---|---|---|---|

The mean of IGD | ||||||

ZDT1 | 1.23 × 10^{−3} | 2.16 × 10^{−2} | 9.29 × 10^{−4} | 4.44 × 10^{−2} | 3.54 × 10^{−3} | 5.63 × 10 ^{−5} |

DTLZ1 | 3.68 × 10^{−3} | 4.75 × 10^{−2} | 1.81 × 10^{−4} | 3.20 × 10^{−2} | 4.50 × 10^{−3} | 2.76 × 10 ^{−5} |

DTLZ5 | 6.03 × 10^{−3} | 8.17 × 10^{−3} | 2.08 × 10^{−3} | 1.69 × 10^{−1} | 8.32 × 10^{−3} | 3.55 × 10 ^{−5} |

The STD of GD | ||||||

ZDT1 | 6.67 × 10^{−3} | 5.28 × 10^{−3} | 7.94 × 10^{−4} | 5.98 × 10^{−2} | 5.09 × 10^{−3} | 4.90 × 10 ^{−5} |

DTLZ1 | 1.33 × 10^{−2} | 1.06 × 10^{−2} | 1.58 × 10^{−4} | 6.72 × 10^{−2} | 6.51 × 10^{−3} | 1.21 × 10 ^{−6} |

DTLZ5 | 1.89 × 10^{−2} | 1.94 × 10^{−3} | 1.92 × 10^{−4} | 1.10 × 10^{−1} | 1.32 × 10^{−2} | 8.53 × 10 ^{−5} |

Function testing | NSGAII | MOEAD | dMOPSO | MOEADDE | MOPSO | IMOPSO |
---|---|---|---|---|---|---|

The mean of IGD | ||||||

ZDT1 | 1.23 × 10^{−3} | 2.16 × 10^{−2} | 9.29 × 10^{−4} | 4.44 × 10^{−2} | 3.54 × 10^{−3} | 5.63 × 10 ^{−5} |

DTLZ1 | 3.68 × 10^{−3} | 4.75 × 10^{−2} | 1.81 × 10^{−4} | 3.20 × 10^{−2} | 4.50 × 10^{−3} | 2.76 × 10 ^{−5} |

DTLZ5 | 6.03 × 10^{−3} | 8.17 × 10^{−3} | 2.08 × 10^{−3} | 1.69 × 10^{−1} | 8.32 × 10^{−3} | 3.55 × 10 ^{−5} |

The STD of GD | ||||||

ZDT1 | 6.67 × 10^{−3} | 5.28 × 10^{−3} | 7.94 × 10^{−4} | 5.98 × 10^{−2} | 5.09 × 10^{−3} | 4.90 × 10 ^{−5} |

DTLZ1 | 1.33 × 10^{−2} | 1.06 × 10^{−2} | 1.58 × 10^{−4} | 6.72 × 10^{−2} | 6.51 × 10^{−3} | 1.21 × 10 ^{−6} |

DTLZ5 | 1.89 × 10^{−2} | 1.94 × 10^{−3} | 1.92 × 10^{−4} | 1.10 × 10^{−1} | 1.32 × 10^{−2} | 8.53 × 10 ^{−5} |

#### Sensitivity analysis of parameters

In this section, simulation is conducted to analyze the sensitivity of parameters in the method proposed. The sensitivity of population size and inertia weight are analyzed, and results are shown in Tables 4 and 5 where we take the DTLZ1 testing function for instance.

Index | Statistics | The inertia weight | ||||||
---|---|---|---|---|---|---|---|---|

0.2 | 0.3 | 0.4 | 0.6 | 0.75 | 0.9 | 0.95 | ||

IGD | Mean | 1.42 × 10^{−3} | 1.37 × 10^{−3} | 1.41 × 10^{−3} | 1.33 × 10^{−3} | 1.19 × 10^{−3} | 1.18 × 10 ^{−3} | 1.20 × 10^{−3} |

STD | 2.57 × 10^{−5} | 2.30 × 10^{−5} | 1.83 × 10^{−5} | 1.39 × 10^{−5} | 1.30 × 10^{−5} | 1.24 × 10 ^{−5} | 1.31 × 10^{−5} | |

GD | Mean | 2.52 × 10^{−5} | 2.68 × 10^{−5} | 2.48 × 10^{−5} | 2.33 × 10^{−5} | 2.28 × 10^{−5} | 2.22 × 10 ^{−5} | 2.24 × 10^{−5} |

STD | 4.09 × 10^{−6} | 3.79 × 10^{−6} | 3.58 × 10^{−6} | 3.63 × 10^{−6} | 3.43 × 10^{−6} | 3.35 × 10 ^{−6} | 3.44 × 10^{−6} |

Index | Statistics | The inertia weight | ||||||
---|---|---|---|---|---|---|---|---|

0.2 | 0.3 | 0.4 | 0.6 | 0.75 | 0.9 | 0.95 | ||

IGD | Mean | 1.42 × 10^{−3} | 1.37 × 10^{−3} | 1.41 × 10^{−3} | 1.33 × 10^{−3} | 1.19 × 10^{−3} | 1.18 × 10 ^{−3} | 1.20 × 10^{−3} |

STD | 2.57 × 10^{−5} | 2.30 × 10^{−5} | 1.83 × 10^{−5} | 1.39 × 10^{−5} | 1.30 × 10^{−5} | 1.24 × 10 ^{−5} | 1.31 × 10^{−5} | |

GD | Mean | 2.52 × 10^{−5} | 2.68 × 10^{−5} | 2.48 × 10^{−5} | 2.33 × 10^{−5} | 2.28 × 10^{−5} | 2.22 × 10 ^{−5} | 2.24 × 10^{−5} |

STD | 4.09 × 10^{−6} | 3.79 × 10^{−6} | 3.58 × 10^{−6} | 3.63 × 10^{−6} | 3.43 × 10^{−6} | 3.35 × 10 ^{−6} | 3.44 × 10^{−6} |

Index | Statistics | Population size | ||||||
---|---|---|---|---|---|---|---|---|

80 | 100 | 150 | 200 | 250 | 300 | 350 | ||

IGD | Mean | 1.41 × 10^{−3} | 1.36 × 10^{−3} | 1.40 × 10^{−3} | 1.18 × 10^{−3} | 1.19 × 10^{−3} | 1.17 × 10^{−3} | 1.16 × 10^{−3} |

STD | 2.04 × 10^{−5} | 1.88 × 10^{−5} | 1.73 × 10^{−5} | 1.24 × 10^{−5} | 1.33 × 10^{−5} | 1.34 × 10^{−5} | 1.25 × 10^{−5} | |

GD | Mean | 2.39 × 10^{−5} | 2.43 × 10^{−5} | 2.40 × 10^{−5} | 2.22 × 10^{−5} | 2.25 × 10^{−5} | 2.21 × 10^{−5} | 2.21 × 10^{−5} |

STD | 3.86 × 10^{−6} | 3.58 × 10^{−6} | 3.45 × 10^{−6} | 3.35 × 10^{−6} | 3.34 × 10^{−6} | 3.35 × 10^{−6} | 3.29 × 10^{−6} |

Index | Statistics | Population size | ||||||
---|---|---|---|---|---|---|---|---|

80 | 100 | 150 | 200 | 250 | 300 | 350 | ||

IGD | Mean | 1.41 × 10^{−3} | 1.36 × 10^{−3} | 1.40 × 10^{−3} | 1.18 × 10^{−3} | 1.19 × 10^{−3} | 1.17 × 10^{−3} | 1.16 × 10^{−3} |

STD | 2.04 × 10^{−5} | 1.88 × 10^{−5} | 1.73 × 10^{−5} | 1.24 × 10^{−5} | 1.33 × 10^{−5} | 1.34 × 10^{−5} | 1.25 × 10^{−5} | |

GD | Mean | 2.39 × 10^{−5} | 2.43 × 10^{−5} | 2.40 × 10^{−5} | 2.22 × 10^{−5} | 2.25 × 10^{−5} | 2.21 × 10^{−5} | 2.21 × 10^{−5} |

STD | 3.86 × 10^{−6} | 3.58 × 10^{−6} | 3.45 × 10^{−6} | 3.35 × 10^{−6} | 3.34 × 10^{−6} | 3.35 × 10^{−6} | 3.29 × 10^{−6} |

Inspecting the results in Tables 4 and 5, it can be concluded that the best inertia weight is 0.90 and population size is 200. The IGD and GD indicators present a descending trend with the increase of inertia weight until the value of inertia weight is 0.9. Similarly, as the population size increases, the quality of solutions becomes better. Meanwhile, more computation time will cost as the complexity of algorithm is strongly associated with population size. Therefore, although the IGD and GD indicators are not their best when population size is 200, it will save a large amount of computation time compared with simulations under population sizes of 300 and 350. Finally, we defined the inertia weight and population size as 0.90 and 200, respectively.

## CASE STUDY

### Decision-making for flood control operation of Hongjiadu reservoir

In order to verify the validity and reasonability of the GCA-TOPSIS model, the flood control operation schemes of Hongjiadu reservoir were used to make the optimal decision calculation. The schemes shown in Table 6 are derived from articles by Ma *et al.* (2007) and Lu *et al.* (2015). The indicator weights are calculated and shown in Table 7 according to the proposed CWMMD.

Scheme | Description | /10^{4} kWh | /10^{8} m^{3} | /m | /10^{8} m^{3} | /m^{3} |
---|---|---|---|---|---|---|

1 | Flood pre-discharge scheduling scheme | 5,451.89 | 3.5 | 0.09 | 1.48 | 2,995 |

2 | Peak flood staggering scheme | 5,445.45 | 3.58 | 0.19 | 1.4 | 3,005 |

3 | Peak clipping scheduling scheme | 5,388.15 | 3.78 | 0.3 | 1.32 | 3,193 |

4 | Flood control schemes which adjust opening sequence and frequency of reservoir gate in Schemes 1, 2, 3 | 5,442.58 | 3.57 | 0.25 | 1.44 | 1,935 |

5 | 5,447.42 | 3.56 | 0.17 | 1.42 | 3,004 | |

6 | 5,421.96 | 3.69 | 0.22 | 1.38 | 3,106 |

Scheme | Description | /10^{4} kWh | /10^{8} m^{3} | /m | /10^{8} m^{3} | /m^{3} |
---|---|---|---|---|---|---|

1 | Flood pre-discharge scheduling scheme | 5,451.89 | 3.5 | 0.09 | 1.48 | 2,995 |

2 | Peak flood staggering scheme | 5,445.45 | 3.58 | 0.19 | 1.4 | 3,005 |

3 | Peak clipping scheduling scheme | 5,388.15 | 3.78 | 0.3 | 1.32 | 3,193 |

4 | Flood control schemes which adjust opening sequence and frequency of reservoir gate in Schemes 1, 2, 3 | 5,442.58 | 3.57 | 0.25 | 1.44 | 1,935 |

5 | 5,447.42 | 3.56 | 0.17 | 1.42 | 3,004 | |

6 | 5,421.96 | 3.69 | 0.22 | 1.38 | 3,106 |

Evaluation indicator | Indicator weight | ||
---|---|---|---|

Improved entropy method | AHP | CWMMD | |

0.1801 | 0.2520 | 0.2169 | |

0.1803 | 0.2095 | 0.1953 | |

0.2622 | 0.2423 | 0.2520 | |

0.1807 | 0.1504 | 0.1652 | |

0.1967 | 0.1458 | 0.1706 |

Evaluation indicator | Indicator weight | ||
---|---|---|---|

Improved entropy method | AHP | CWMMD | |

0.1801 | 0.2520 | 0.2169 | |

0.1803 | 0.2095 | 0.1953 | |

0.2622 | 0.2423 | 0.2520 | |

0.1807 | 0.1504 | 0.1652 | |

0.1967 | 0.1458 | 0.1706 |

In Table 6, the denotes power generation; is the water abandoned in the reservoir; is difference between the water level at the end of operation and the ideal water level; is the flood control capacity used; is the maximum reservoir discharge.

The GCC matrix between each scheme and the ideal and anti-ideal solution (and ) are shown in Tables 8 and 9.

Scheduling scheme | /10^{4} kWh | /10^{8} m^{3} | /m | /10^{8} m^{3} | /m^{3} | GCC | GCC with dimensionless treatment |
---|---|---|---|---|---|---|---|

1 | 1.000 | 1.000 | 1.000 | 0.333 | 0.372 | 0.784 | 1.000 |

2 | 0.832 | 0.636 | 0.512 | 0.500 | 0.370 | 0.581 | 0.741 |

3 | 0.333 | 0.333 | 0.333 | 1.000 | 0.333 | 0.443 | 0.566 |

4 | 0.774 | 0.667 | 0.396 | 0.400 | 1.000 | 0.635 | 0.810 |

5 | 0.877 | 0.700 | 0.568 | 0.444 | 0.370 | 0.608 | 0.775 |

6 | 0.516 | 0.424 | 0.447 | 0.571 | 0.349 | 0.462 | 0.589 |

Scheduling scheme | /10^{4} kWh | /10^{8} m^{3} | /m | /10^{8} m^{3} | /m^{3} | GCC | GCC with dimensionless treatment |
---|---|---|---|---|---|---|---|

1 | 1.000 | 1.000 | 1.000 | 0.333 | 0.372 | 0.784 | 1.000 |

2 | 0.832 | 0.636 | 0.512 | 0.500 | 0.370 | 0.581 | 0.741 |

3 | 0.333 | 0.333 | 0.333 | 1.000 | 0.333 | 0.443 | 0.566 |

4 | 0.774 | 0.667 | 0.396 | 0.400 | 1.000 | 0.635 | 0.810 |

5 | 0.877 | 0.700 | 0.568 | 0.444 | 0.370 | 0.608 | 0.775 |

6 | 0.516 | 0.424 | 0.447 | 0.571 | 0.349 | 0.462 | 0.589 |

Scheduling scheme | /10^{4} kWh | /10^{8} m^{3} | /m | /10^{8} m^{3} | /m^{3} | GCC | GCC with dimensionless treatment |
---|---|---|---|---|---|---|---|

1 | 0.333 | 0.333 | 0.333 | 1.000 | 0.761 | 0.516 | 0.579 |

2 | 0.357 | 0.412 | 0.488 | 0.500 | 0.770 | 0.494 | 0.555 |

3 | 1.000 | 1.000 | 1.000 | 0.333 | 1.000 | 0.890 | 1.000 |

4 | 0.369 | 0.400 | 0.677 | 0.667 | 0.333 | 0.496 | 0.557 |

5 | 0.350 | 0.389 | 0.447 | 0.571 | 0.769 | 0.489 | 0.550 |

6 | 0.485 | 0.609 | 0.568 | 0.444 | 0.878 | 0.590 | 0.663 |

Scheduling scheme | /10^{4} kWh | /10^{8} m^{3} | /m | /10^{8} m^{3} | /m^{3} | GCC | GCC with dimensionless treatment |
---|---|---|---|---|---|---|---|

1 | 0.333 | 0.333 | 0.333 | 1.000 | 0.761 | 0.516 | 0.579 |

2 | 0.357 | 0.412 | 0.488 | 0.500 | 0.770 | 0.494 | 0.555 |

3 | 1.000 | 1.000 | 1.000 | 0.333 | 1.000 | 0.890 | 1.000 |

4 | 0.369 | 0.400 | 0.677 | 0.667 | 0.333 | 0.496 | 0.557 |

5 | 0.350 | 0.389 | 0.447 | 0.571 | 0.769 | 0.489 | 0.550 |

6 | 0.485 | 0.609 | 0.568 | 0.444 | 0.878 | 0.590 | 0.663 |

The relative similarity degree and result of scheme decision after the scheme sorting are displayed in Table 10.

Scheduling scheme | Scheme sorting | |||
---|---|---|---|---|

1 | 0.5376 | 0.6332 | 0.5854 | ① |

2 | 0.4514 | 0.5716 | 0.5115 | ④ |

3 | 0.4630 | 0.3612 | 0.4121 | ⑥ |

4 | 0.4722 | 0.5925 | 0.5323 | ② |

5 | 0.4711 | 0.5852 | 0.5282 | ③ |

6 | 0.4200 | 0.4706 | 0.4453 | ⑤ |

Scheduling scheme | Scheme sorting | |||
---|---|---|---|---|

1 | 0.5376 | 0.6332 | 0.5854 | ① |

2 | 0.4514 | 0.5716 | 0.5115 | ④ |

3 | 0.4630 | 0.3612 | 0.4121 | ⑥ |

4 | 0.4722 | 0.5925 | 0.5323 | ② |

5 | 0.4711 | 0.5852 | 0.5282 | ③ |

6 | 0.4200 | 0.4706 | 0.4453 | ⑤ |

Inspecting Table 10, the scheduling schemes in descending order are . The sort order of schemes is consistent with that of Ma *et al.* (2007). However, the method in Ma *et al.* (2007) only consider the subjective indicator weight ignoring objective information. As a result, the decision-making results of this method are easily influenced by the knowledge and experience of decision-makers. On the contrary, we take fully objective and subjective information into consideration using the combination weighting method based on minimum deviation. Furthermore, the GCA contributes to reflect the uncertain information (grey information) and internal connection between multi-indicators. Meanwhile, the TOPSIS can describe the overall similarity between alternative and ideal schemes. The coupled GCA-TOPSIS makes the decision-making and scheduling scheme optimization process more reasonable and convincing through merged relative similarity degree.

Further comparison of the scheduling schemes shows that scheme 1 and 4 effectively utilize flood resources to increase the power generation with pre-discharge measure to control flood. Thus, the scheduling schemes 1 and 4 are optimal for the decision-maker. However, scheme 6 and 3 take more conservative measures of flood peak mitigation to hold back the flood only at the time of flood peak arrival, failing to make the best use of flood resources. Accordingly, these schemes are incompatible with the decision-maker. The decision results coincide with practical scheduling schemes for flood control in Hongjiadu reservoir.

### Decision-making for multi-objective operation of Qingjiang cascade reservoirs

As presented in the above, the GCA-TOPSIS has been verified to acquire a reasonable scheduling scheme for single reservoir flood control system. For the cascade reservoirs in the basin, the formulation, evaluation, and optimization of the scheduling scheme is essentially a multi-objective, multi-attribute, and multi-turn decision-making problem in the same way. In this section, multi-objective reservoir operation in Qingjiang river is chosen as one case. The GCA-TOPSIS model is applied to take balanced consideration of multiple scheduling objectives and find the optimal scheme sort order for the decision-maker.

#### Overview of cascade reservoirs' system in Qingjiang river

Qingjiang river is the largest tributary of the middle Yangtze river which flows through Enshi, Badong, Changyang, and Yidu located in Hubei province. A total of three reservoirs and hydropower stations with large installed capacity, regulating storage are located on the river. The Shuibuya reservoir with multi-year regulating storage and the largest installed capacity, is the most important reservoir of the cascade development in Qingjiang river basin. The Geheyan reservoir is a huge water conservancy project with major functions of power generation and flood control, and is of navigational benefit. Gaobazhou reservoir, situated on the lower reaches of the Qingjiang river, plays an important role in the reverse regulation of upstream Geheyan (Yang *et al.* 2019b, 2019c). The main characteristic parameters of cascade reservoirs are shown in Table 11. A generalization of the cascade reservoir system in Qingjiang river basin is described in Figure 3.

Characteristic water level and parameters | Cascade reservoirs in Qingjiang | ||
---|---|---|---|

Shuibuya | Geheyan | Gaobazhou | |

Normal storage water level (m) | 400 | 200 | 80 |

Flood control level (m) | 391.8 | 193.6 | 78.5 |

Dead water level (m) | 350 | 160 | 78.09 |

Total storage capacity (10^{8} m^{3}) | 45.89 | 30.18 | 4.03 |

Beneficial reservoir capacity (10^{8} m^{3}) | 23.83 | 19.75 | 0.54 |

Installed capacity (MW) | 1840 | 1212 | 270 |

Annual utilization hours of installed capacity (h) | 2,450 | 2,533 | 3,563 |

Guaranteed output (MW) | 310.0 | 241.0 | 77.3 |

Comprehensive efficiency coefficient | 8.50 | 8.50 | 8.40 |

Maximum water head (m) | 203.0 | 121.6 | 40.0 |

Minimum water head (m) | 147.0 | 80.7 | 22.3 |

Mean water head (m) | 186.6 | 112.0 | 35.4 |

Characteristic water level and parameters | Cascade reservoirs in Qingjiang | ||
---|---|---|---|

Shuibuya | Geheyan | Gaobazhou | |

Normal storage water level (m) | 400 | 200 | 80 |

Flood control level (m) | 391.8 | 193.6 | 78.5 |

Dead water level (m) | 350 | 160 | 78.09 |

Total storage capacity (10^{8} m^{3}) | 45.89 | 30.18 | 4.03 |

Beneficial reservoir capacity (10^{8} m^{3}) | 23.83 | 19.75 | 0.54 |

Installed capacity (MW) | 1840 | 1212 | 270 |

Annual utilization hours of installed capacity (h) | 2,450 | 2,533 | 3,563 |

Guaranteed output (MW) | 310.0 | 241.0 | 77.3 |

Comprehensive efficiency coefficient | 8.50 | 8.50 | 8.40 |

Maximum water head (m) | 203.0 | 121.6 | 40.0 |

Minimum water head (m) | 147.0 | 80.7 | 22.3 |

Mean water head (m) | 186.6 | 112.0 | 35.4 |

#### Parameter selection of IMOPSO

The control parameters in IMOPSO are selected as follows with contrastive analysis. The size of EAS is defined as 30. The particle population is 200 and sub-populations assigned *M* are predefined as 10. Accordingly, the number of particles in each sub-population is 20. The upper and lower bound of velocity is in the interval of [−4,4]. The inertia weight is 0.90. The maximum iterations are defined as 500.

#### The scheduling scenarios

Two scheduling scenarios are proposed with the emphasis on multi-objective ecological scheduling and water supply scheduling, respectively. The specific scenarios are described as follows.

**Scenario (1)**: Take multiple objectives including power generation, guaranteed hydropower station and ecological water spill and shortage into account. The solutions by IMOPSO are evaluated and ranked using the GCA-TOPSIS model.**Scenario (2)**: Take multiple objectives including power generation, water supply and ecological water spill and shortage into account. The solutions by IMOPSO are evaluated and ranked through the GCA-TOPSIS model.

## RESULTS AND DISCUSSION

**Scenario (1): Multi-objective ecological scheduling (MOES)**: For this scenario, the maximum power generation and guaranteed hydropower output and minimum ecological water shortage and spill are considered. The hydrologic and hydraulic constraints can be found in Zhu *et al.* (2017) and Yang *et al.* (2019b, 2019c). The details of optimization objectives are listed as follows.

*G*and = total amount of reservoir and scheduling intervals, respectively; = quantity of control cross section which is matched with reservoirs; = power output of the th hydropower station at th interval; = efficiency coefficient of unit in the th hydropower station; = the power generation flow for the th reservoir at th interval; is guaranteed hydropower output of cascade reservoirs; = reservoir water release corresponding to the th control cross section; = different level of ecological water requirements downstream of the reservoir corresponding to the th control cross section.

For the input of the MOES model, we select one typical observed inflow process. Then, the IMOPSO is used to solve the MOES model. The optimal scheme set and Pareto fronts are demonstrated in Table 12 and Figure 4.

Scheme number | Multiple objectives | ||
---|---|---|---|

Power generation (10^{8} kW·h) | Guaranteed hydropower output (10^{4} kW) | Ecological water spill and shortage (10^{8} m^{3}) | |

1 | 82.920 | 78.876 | 165.562 |

2 | 82.948 | 79.479 | 165.442 |

3 | 82.927 | 83.541 | 164.633 |

4 | 82.941 | 84.540 | 164.516 |

5 | 82.963 | 77.711 | 165.793 |

6 | 82.939 | 78.602 | 165.616 |

7 | 82.944 | 83.413 | 164.659 |

8 | 82.949 | 81.564 | 165.027 |

9 | 82.943 | 82.672 | 164.807 |

10 | 82.941 | 84.729 | 164.498 |

11 | 82.959 | 77.974 | 165.741 |

12 | 82.946 | 83.299 | 164.682 |

13 | 82.938 | 85.071 | 164.463 |

14 | 82.945 | 80.624 | 165.214 |

15 | 82.943 | 84.195 | 164.551 |

16 | 82.943 | 84.698 | 164.501 |

17 | 82.963 | 77.711 | 165.793 |

18 | 82.955 | 79.220 | 165.492 |

19 | 82.945 | 83.835 | 164.586 |

20 | 82.947 | 83.361 | 164.669 |

21 | 82.949 | 83.160 | 164.704 |

22 | 82.951 | 82.423 | 164.855 |

23 | 82.951 | 82.850 | 164.769 |

24 | 82.951 | 82.883 | 164.763 |

25 | 82.949 | 83.073 | 164.725 |

26 | 82.951 | 82.850 | 164.769 |

27 | 82.953 | 81.964 | 164.946 |

28 | 82.955 | 80.734 | 165.191 |

29 | 82.956 | 80.519 | 165.234 |

30 | 82.960 | 79.091 | 165.518 |

Scheme number | Multiple objectives | ||
---|---|---|---|

Power generation (10^{8} kW·h) | Guaranteed hydropower output (10^{4} kW) | Ecological water spill and shortage (10^{8} m^{3}) | |

1 | 82.920 | 78.876 | 165.562 |

2 | 82.948 | 79.479 | 165.442 |

3 | 82.927 | 83.541 | 164.633 |

4 | 82.941 | 84.540 | 164.516 |

5 | 82.963 | 77.711 | 165.793 |

6 | 82.939 | 78.602 | 165.616 |

7 | 82.944 | 83.413 | 164.659 |

8 | 82.949 | 81.564 | 165.027 |

9 | 82.943 | 82.672 | 164.807 |

10 | 82.941 | 84.729 | 164.498 |

11 | 82.959 | 77.974 | 165.741 |

12 | 82.946 | 83.299 | 164.682 |

13 | 82.938 | 85.071 | 164.463 |

14 | 82.945 | 80.624 | 165.214 |

15 | 82.943 | 84.195 | 164.551 |

16 | 82.943 | 84.698 | 164.501 |

17 | 82.963 | 77.711 | 165.793 |

18 | 82.955 | 79.220 | 165.492 |

19 | 82.945 | 83.835 | 164.586 |

20 | 82.947 | 83.361 | 164.669 |

21 | 82.949 | 83.160 | 164.704 |

22 | 82.951 | 82.423 | 164.855 |

23 | 82.951 | 82.850 | 164.769 |

24 | 82.951 | 82.883 | 164.763 |

25 | 82.949 | 83.073 | 164.725 |

26 | 82.951 | 82.850 | 164.769 |

27 | 82.953 | 81.964 | 164.946 |

28 | 82.955 | 80.734 | 165.191 |

29 | 82.956 | 80.519 | 165.234 |

30 | 82.960 | 79.091 | 165.518 |

Inspecting Table 12 and Figure 4, the reverse and competitive relationship between the power generation, guaranteed output is presented which implies that power generation will improve with the violation of guaranteed output benefit. Moreover, power generation of cascade reservoirs and downstream ecological benefit also shows contradictoriness and opposition. However, with the increase of guaranteed output, decrease of ecological overflow and water shortage means increase of ecological benefit, and implies the compatibility between the cascade reservoir output and ecological benefits. For the decision-maker, they should select the suitable scheme from the solution sets to implement. To this end, with the GCA-TOPSIS model, we rank solution sets in accordance with power generation tendency and ecological benefit tendency, respectively. The indicator weights calculated by CWMMD are shown in Table 13. The relative similarity degree and solution sorting results are demonstrated in Table 14.

Decision tendency | Indicator | ||
---|---|---|---|

Power generation | Guaranteed hydropower output | Ecological water spill and shortage | |

Power generation | 0.5798 | 0.1256 | 0.2946 |

Ecological benefit | 0.3059 | 0.0948 | 0.5993 |

Decision tendency | Indicator | ||
---|---|---|---|

Power generation | Guaranteed hydropower output | Ecological water spill and shortage | |

Power generation | 0.5798 | 0.1256 | 0.2946 |

Ecological benefit | 0.3059 | 0.0948 | 0.5993 |

Ranking | Power generation tendency | Ecological benefit tendency | ||
---|---|---|---|---|

Relative similarity degree | Scheme number | Relative similarity degree | Scheme number | |

1 | 0.6575 | 16 | 0.7413 | 13 |

2 | 0.6544 | 10 | 0.7288 | 10 |

3 | 0.6513 | 13 | 0.7263 | 16 |

4 | 0.6481 | 19 | 0.7254 | 4 |

5 | 0.6477 | 4 | 0.7139 | 15 |

6 | 0.6463 | 15 | 0.7064 | 3 |

7 | 0.6416 | 21 | 0.7001 | 19 |

8 | 0.6403 | 20 | 0.6818 | 7 |

9 | 0.6398 | 25 | 0.6738 | 20 |

10 | 0.6395 | 24 | 0.6717 | 12 |

11 | 0.6387 | 23 | 0.6607 | 21 |

12 | 0.6387 | 26 | 0.6536 | 25 |

13 | 0.6336 | 12 | 0.6399 | 24 |

14 | 0.6295 | 7 | 0.6376 | 23 |

15 | 0.6268 | 22 | 0.6376 | 26 |

16 | 0.6253 | 27 | 0.6331 | 9 |

17 | 0.6032 | 9 | 0.6082 | 22 |

18 | 0.5999 | 28 | 0.5733 | 27 |

19 | 0.5980 | 29 | 0.5439 | 8 |

20 | 0.5956 | 8 | 0.4767 | 28 |

21 | 0.5751 | 30 | 0.4691 | 14 |

22 | 0.5582 | 3 | 0.4593 | 29 |

23 | 0.5520 | 18 | 0.3735 | 2 |

24 | 0.5512 | 17 | 0.3613 | 1 |

25 | 0.5512 | 5 | 0.3508 | 18 |

26 | 0.5487 | 14 | 0.3440 | 30 |

27 | 0.5287 | 11 | 0.3239 | 6 |

28 | 0.5259 | 2 | 0.2673 | 11 |

29 | 0.4575 | 6 | 0.2654 | 17 |

30 | 0.3678 | 1 | 0.2654 | 5 |

Ranking | Power generation tendency | Ecological benefit tendency | ||
---|---|---|---|---|

Relative similarity degree | Scheme number | Relative similarity degree | Scheme number | |

1 | 0.6575 | 16 | 0.7413 | 13 |

2 | 0.6544 | 10 | 0.7288 | 10 |

3 | 0.6513 | 13 | 0.7263 | 16 |

4 | 0.6481 | 19 | 0.7254 | 4 |

5 | 0.6477 | 4 | 0.7139 | 15 |

6 | 0.6463 | 15 | 0.7064 | 3 |

7 | 0.6416 | 21 | 0.7001 | 19 |

8 | 0.6403 | 20 | 0.6818 | 7 |

9 | 0.6398 | 25 | 0.6738 | 20 |

10 | 0.6395 | 24 | 0.6717 | 12 |

11 | 0.6387 | 23 | 0.6607 | 21 |

12 | 0.6387 | 26 | 0.6536 | 25 |

13 | 0.6336 | 12 | 0.6399 | 24 |

14 | 0.6295 | 7 | 0.6376 | 23 |

15 | 0.6268 | 22 | 0.6376 | 26 |

16 | 0.6253 | 27 | 0.6331 | 9 |

17 | 0.6032 | 9 | 0.6082 | 22 |

18 | 0.5999 | 28 | 0.5733 | 27 |

19 | 0.5980 | 29 | 0.5439 | 8 |

20 | 0.5956 | 8 | 0.4767 | 28 |

21 | 0.5751 | 30 | 0.4691 | 14 |

22 | 0.5582 | 3 | 0.4593 | 29 |

23 | 0.5520 | 18 | 0.3735 | 2 |

24 | 0.5512 | 17 | 0.3613 | 1 |

25 | 0.5512 | 5 | 0.3508 | 18 |

26 | 0.5487 | 14 | 0.3440 | 30 |

27 | 0.5287 | 11 | 0.3239 | 6 |

28 | 0.5259 | 2 | 0.2673 | 11 |

29 | 0.4575 | 6 | 0.2654 | 17 |

30 | 0.3678 | 1 | 0.2654 | 5 |

It can be seen from Table 14 that scheme 16 and 13 are the most ideal for decision-makers when decision tendency is power generation and ecological benefit, respectively. In terms of scheme 16, the objective of power generation and guaranteed output is fully considered. Moreover, the ecological benefit obtained by this scheme outperforms most other schemes. Meanwhile, as the indicator weights calculated are 0.5798, 0.1256, and 0.2946, the decision process will place more emphasis on power generation and ecological benefit. The sorting results in Table 14 are in accordance with this principle to some extent. It should be noted that we discuss the MOES for cascade reservoirs rather than reservoir scheduling which focus on economic benefits. Therefore, scheduling schemes are not just ranked in descending order as the power generation increases. The ecological objective also has a non-negligible and significant influence on scheme evaluation and decision. On the other hand, as the decision process tends to ecological benefit, corresponding indicator weights are 0.3059, 0.0948, and 0.5993. It is evident that scheme sorting results under this situation are basically consistent with variation tendency of ecological water spill and shortage. The specific optimal scheduling schemes of each decision tendency are shown in Figures 5 and 6, respectively. The medium and worst schemes are added as comparison.

Due to the flood control task undertaken by Qingjiang cascade reservoirs in the flood season, the flood control benefit becomes the main factor. As a result, it can be seen from Figures 5 and 6 that the scheduling optimization space is strictly limited. Instead, the differences between schemes are mainly reflected in the dry season. Power generation efficiency requires the reservoir to operate at a high-water head, consequently, the variation range of flow rate is increased, and natural runoff will be changed, to a large extent, which produces obvious competition between power generation and ecological benefits. Meanwhile, guaranteed output benefit needs to maintain stability of power generation flow under the condition of certain amount of water. The ecological flow in dry season is simultaneously beneficial to hydropower output effect because the differences between schemes exist in this season. Therefore, the compatibility between ecological and guaranteed hydropower output benefits is remarkable.

Scheme 16 in Figure 5 places emphasis on the power generation with taking serious account of ecological benefit. Accordingly, it increases corresponding reservoir discharge in the dry season in order to protect the stability of guaranteed output and downstream ecological flow to a certain extent. As a result, the total power generation will be sacrificed which is shown in Table 12. In comparison, scheme 22 outperforms on total power generation and makes full use of high-water head effect; however, its guaranteed output is relatively lower than that of scheme 16 due to large variation of hydropower output. The large variation of output also implies instability of reservoir discharge which impairs the ecological benefit. In terms of scheme 13, which tends to achieve the ecological benefit, the ecological water spill and shortage in the initial stage of the flood season is larger than other schemes. However, with a small growth rate, the total amount of ecological spill and shortage is small, and the stable variation is more beneficial to ecological benefit. In the meantime, the hydropower output is also outstanding as discharge in the flood season increases.

**Scenario (2): Multi-objective water supply scheduling (MOWSS)**: In this scenario, optimization goals are adjusted so that the water supply benefit is added, and ecological benefit is described in the form of ecological flow guarantee rate. Similarly, the water supply guarantee rate is used for representing water supply benefit. Moreover, power generation benefit described above is also considered. The details are shown as follows.

- Maximum average ecological flow guarantee rate (AEFGR): where = the water supply guarantee rate, which is the ratio of volume of water supply to water demand . = the water delivered to th control section at the th interval. Similarly, = water demand for the th control section at the th interval. = ecological water supply guarantee rate which is described as the ratio of to .

Similarly, the IMOPSO is applied to solve the MOWSS model and the corresponding Pareto front of the non-inferior scheduling scheme is demonstrated in Figure 7.

It can be seen from Figure 7 that mutual competition and contradiction between the power generation and water supply and ecological flow guarantee rates is evident. Accordingly, improving power generation will bring about the violation of water supply and ecological benefits and vice versa. However, with the increase of AWSGR, the AEFGR also increases which implies the mutual compatibility and promotion between water supply and ecological benefits. This may be explained by the increase of AWSGR needs larger reservoir discharge, which is also beneficial for maintaining downstream ecological health. However, some solutions in Figure 7(d) present contradictory relation which can be attributed to the fact that the water diversion task may directly transport reservoir water to users. As a result, it will bring about an adverse effect on power generation and ecological benefits.

Furthermore, the GCA-TOPSIS model is used to evaluate and rank scheduling schemes obtained by IMOPSO. The indicator weights shown in Table 15 are calculated in the same way used in Scenario (1). The scheme evaluation and sorting results are listed in Table 16. The optimal scheduling scheme is shown in Figure 8. The medium and worst schemes are used as a comparison.

Decision tendency | Indicator | ||
---|---|---|---|

Power generation | AEFGR | AWSGR | |

Water supply | 0.2703 | 0.1043 | 0.6254 |

Decision tendency | Indicator | ||
---|---|---|---|

Power generation | AEFGR | AWSGR | |

Water supply | 0.2703 | 0.1043 | 0.6254 |

Table 16 shows that scheme 2 is the most ideal one for the decision-maker. In terms of scheme 2, the objective of AWSGR is optimum in comparison with other schemes. As the indicator weights calculated are 0.2703, 0.1043, and 0.6254, the decision process tends to place more emphasis on water supply benefit. The sorting results are in accordance with the quality of the objective of AWSGR. In other words, the scheme with the larger AWSGR is better than others. Thus, the sorting result of the optimal scheme to that of the worst scheme is basically consistent with the variation trend of water supply guarantee rate.

Ranking | Water supply tendency | |
---|---|---|

Relative similarity degree | Scheme number | |

1 | 0.6391 | 2 |

2 | 0.6372 | 3 |

3 | 0.6207 | 4 |

4 | 0.6143 | 9 |

5 | 0.6129 | 12 |

6 | 0.6002 | 15 |

7 | 0.6002 | 5 |

8 | 0.5994 | 13 |

9 | 0.5880 | 1 |

10 | 0.5799 | 7 |

11 | 0.5740 | 16 |

12 | 0.5719 | 19 |

13 | 0.5687 | 8 |

14 | 0.5639 | 14 |

15 | 0.5628 | 17 |

16 | 0.5572 | 21 |

17 | 0.5365 | 10 |

18 | 0.5225 | 18 |

19 | 0.5132 | 20 |

20 | 0.5090 | 11 |

21 | 0.5072 | 22 |

22 | 0.5003 | 23 |

23 | 0.4979 | 25 |

24 | 0.4952 | 6 |

25 | 0.4890 | 26 |

26 | 0.4471 | 27 |

27 | 0.3765 | 29 |

28 | 0.3744 | 30 |

29 | 0.3662 | 28 |

30 | 0.3404 | 24 |

Ranking | Water supply tendency | |
---|---|---|

Relative similarity degree | Scheme number | |

1 | 0.6391 | 2 |

2 | 0.6372 | 3 |

3 | 0.6207 | 4 |

4 | 0.6143 | 9 |

5 | 0.6129 | 12 |

6 | 0.6002 | 15 |

7 | 0.6002 | 5 |

8 | 0.5994 | 13 |

9 | 0.5880 | 1 |

10 | 0.5799 | 7 |

11 | 0.5740 | 16 |

12 | 0.5719 | 19 |

13 | 0.5687 | 8 |

14 | 0.5639 | 14 |

15 | 0.5628 | 17 |

16 | 0.5572 | 21 |

17 | 0.5365 | 10 |

18 | 0.5225 | 18 |

19 | 0.5132 | 20 |

20 | 0.5090 | 11 |

21 | 0.5072 | 22 |

22 | 0.5003 | 23 |

23 | 0.4979 | 25 |

24 | 0.4952 | 6 |

25 | 0.4890 | 26 |

26 | 0.4471 | 27 |

27 | 0.3765 | 29 |

28 | 0.3744 | 30 |

29 | 0.3662 | 28 |

30 | 0.3404 | 24 |

As can be seen from Figure 8, the scheduling space is limited during the flood control season, while differences are mainly reflected in the dry season. Scheme 2 increases reservoir discharge and produces water stage drawdown to guarantee the water supply task during the dry season. Meanwhile, the larger quantity of water discharge is beneficial to alleviate ecological water shortage at the dry season stage. Thus, the compatibility relationship between water supply and ecological protection is presented. On the other hand, although the hydropower output outperforms other schemes in the initial stage of the dry season, high-water head is not efficiently utilized to guarantee total power generation. In terms of scheme 24, it focuses more energy on the power generation benefit. To this end, reservoir water discharge is cut down to operate at higher-water head and concentrate drawdown before the arrival of the flood season. As a result, high-water head effect is fully used to play the power generation benefit. However, the smaller discharge in the early stage of the dry period will alleviate the benefit of AWSGR and AEFGR, and great fluctuation of reservoir discharge is harmful to maintain stability of water supply and ecological health.

In short, the decision-making method of GCA-TOPSIS has shown strong applicability in solving the MCDM problem of multi-objective reservoir ecological and water supply operation. It can quickly and reasonably select the most ideal scheme of comprehensive benefits under different decision-making scenarios for the reservoir decision-makers.

## CONCLUSION

In this paper, we propose a framework including IMOPSO for non-inferior solutions' acquirement, the MCDM model based on grey correlation analysis (GCA) and the multi-attribute method of TOPSIS. In terms of the MCDM model, the GCA contributes to reflecting the difference between the change trend in the scheduling scheme sets and ideal scheme, while the TOPSIS can reflect the overall similarity between alternative and ideal schemes. The combination of GCA and TOPSIS can take uncertainty factors and grey characteristics in the multi-objective decision process fully into account, and accurately describe the comprehensive quality of alternative scheduling schemes.

The GCA-TOPSIS is applied to MCDM, including multi-objective flood control operation of Hongjiadu reservoir, and ecological and water supply operation of Qingjiang cascade reservoirs. Based on scheduling schemes obtained by IMOPSO and combination weighting of evaluation indicators calculated by CWMMD, GCA-TOPSIS can effectively evaluate and select the most ideal scheme of comprehensive benefits under different decision-making scenarios for the reservoir decision-makers. In summary, the decision-making method of GCA-TOPSIS shows strong applicability in solving the evaluation of multi-objective flood control, ecological and water supply scheduling schemes. The rationality of the decision result is also illustrated and verified, which implies that the decision-making method GCA-TOPSIS can provide strong theoretical support for the implementation of multi-objective balanced scheduling decisions in complex reservoir systems.

## ACKNOWLEDGEMENTS

This work is supported by the Research and Extension Project of Hydraulic Science and Technology in Shanxi Province ‘Study on the Key Technology for Joint Optimal Operation of Complex Multi-Reservoir System and Water Network’ (2017DSW02), the Science and Technology Project of Yunnan Water Resources Department ‘Comprehensive Water Saving and Unconventional Water Utilization Research’, the National Key Basic Research Program of China (973 Program) (2012CB417006), the National Science Support Plan Project of China (2009BAC56B03). The first author was supported by a fellowship from the China Scholarship Council for his visit to the IIHR-Hydroscience and Engineering, University of Iowa, C. Maxwell Stanley Hydraulics Laboratory, Iowa City, USA.