The improved multi-criteria decision-making model for multi-objective operation in a complex reservoir system

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.

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.

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.

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.

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.

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).

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 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.

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.

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.

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:

1. Assuming that the number of sub-populations is and total L particles are assigned into each sub-population.

2. 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).

3. 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:

   

   (20)

   

   (21)

   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.

4. 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 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.

1. 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.

2. 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:

   

   (22)

   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 ⁠.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

1. Power generation:

   

   (25)
2. Guaranteed hydropower output:

   

   (26)
3. Ecological water spill and shortage:

   

   (27)

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.

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.

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.

1. Maximum average water supply guarantee rate (AWSGR):

   

   (28)
2. Maximum average ecological flow guarantee rate (AEFGR):

   

   (29)

   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.

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.

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.

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.

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.