Introduction

The ecological and environmental effects of hydropower projects have become a bottleneck for the sustainable development and utilization of hydropower, and it is also a long-term research hotspot in international water science1. Constructions of large hydropower plants have raised serious ecological problems, including eutrophication of backwater zones within the Three Gorges Reservoir, decrease of fish stocks, like the Chinese Carps along the Yangtze River, and reduction of fish habitats like that of the Yangtze sturgeon2.The construction of the Three Gorges Dam has deteriorated the natural ecological environment of the river, resulting in ecological security problems such as regional water quality protection, survival and reproduction of key species, and ecological restoration of lakes and wetlands in the Three Gorges Reservoir and the middle and lower reaches of the Yangtze River3. Since its construction, the ecological management of the Three Gorges Reservoir has attracted the attention of all sectors of society and scholars, and many studies have been carried out4,5,6. To coordinate the efficient utilization of water energy resources in the basin and the healthy and sustainable development of the river ecosystem, the implementation of reservoir ecological management is the most effective and convenient management measure at this stage.

As the cornerstone of freshwater fishery aquaculture in China, the " Four Major Chinese Carps" are representative species adapted to the complex ecosystem of rivers and lakes in the middle and lower reaches of the Yangtze River, and their resource dynamics are an important indication of the health status of the water ecosystem in the Yangtze River Basin. As the carrier of fish reproduction, natural spawning grounds are directly related to the early embryonic development of fish and are key habitats for the continuation of fish species. The establishment of the Three Gorges Project has changed the original spawning environment of fish in the river, which is reflected in the “flattening” of the natural inflow process due to the change of the outflow brought about by reservoir dispatching3, which has greatly changed the environmental conditions of the spawning grounds and spawning requirements of migratory fish in the Yangtze River Basin. This in turn affects the reproduction and growth of fish7. For the ecological goal of improving the conditions required for fish spawning, many researches have been carried out by scholars at home and abroad. Wen Xiaotong et al8 established a relevant model with the ecological goal of maximum membership degree of downstream river flow within the appropriate ecological flow range for fish during the scheduling period. Liu Han et al.9 proposed a set of quantitative methods for ecological scheduling targets to stimulate the reproduction of Four Major Chinese Carps by using the characteristic curve method (ROC method), and constructed a bio-hydrological response model. Lv et al.10 proposed the index-weighted usable area (WUA) to assess the suitability of fish for reproduction and established a life-cycle fish habitat reservoir operation model (ROM-FHLS). Anton et al.11 investigated the effects of changes in the frequency of hydropower generation and the closing time of floodgates on the spawning areas and stranding tendencies of fish such as European grayfish and concluded the potential impact of changes in hydropower station operating conditions on fish stocks.

As a beneficial facility, the economic benefits of a reservoir should be considered as an important indicator in the operation. Generally, an optimal operation model is established with the goal of maximizing the power generation, and the obtained operation process can fully guarantee the power generation benefits, but the relevant requirements of river ecology are not considered enough12. Most of the studies proposed operating the reservoir in an ecological-friendly manner by incorporating a constant minimum flow as a fixed constraint. However, realization of such an ecological flow may hamper socio-economic benefits, especially for diversion-type hydropower stations, because release of ecological flow could cause a significant loss of hydropower generation13. Designing specific operational modes to fit specific ecological conditions of rivers remains a key practical challenge for hydropower operation, especially in terms of balancing the various tradeoffs among multiple objectives in the context of coordinated operation of cascade hydropower plants14.Therefore, it is an important task to coordinate the downstream ecological flow demand of hydropower station with the power generation dispatching demand of hydropower station during reservoir operation, and how to evaluate the ecological satisfaction of the dispatching process is an urgent problem to be solved.In view of the ecological needs related to the subsidence period of the Three Gorges Reservoir, a multi-objective model of optimal dispatch of the Three Gorges Reservoir is proposed, considering the coordinated ecological flow demand and power generation benefits of different rising times. On this basis, taking the rising process as an important factor, multilayer entropy-weighted TOPSIS method was carried out to evaluate the model results to obtain a satisfactory solution for multi-objective scheduling. The study can provide decision-making reference for the ecological scheduling of the spawning period of the "Four Major Chinese Carps" in the Three Gorges Reservoir of the Yangtze River.

Determination of ecological demand index of the " Four Major Chinese Carps" breeding period

The spawning and reproduction activities of the " Four Major Chinese Carps " in the lower reaches of the Three Gorges Reservoir are sensitive to hydrological processes and inseparable from the changes in the hydrological situation of the basin15,16. Shen Chen et al.17 showed that water temperature conditions and upwelling process are the key hydrological factors affecting the growth and reproduction of the Four Major Chinese Carps.

(1) Water temperature conditions.

The lower water temperature threshold of Four Major Chinese Carps spawning is 18 °C18,19,20 Relevant literature shows that due to the construction of the Three Gorges Reservoir, the date when the water temperature in the lower reaches of the Three Gorges reaches 18 °C has been postponed from mid-April to early May to early to mid-May before the construction of the reservoir21, so the implementation time of ecological scheduling to adapt to the spawning and reproduction of the " Four Major Chinese Carps " should be at least later than after mid-May.

(2) Flow rise process.

The spawning activities of drifting fish such as the " Four Major Chinese Carps " are generally accompanied by the process of flow rise, which is the most important external condition to stimulate the natural reproduction of domestic fish. The longer the duration of flow rise, the longer the spawning duration of the "Four Major Chinese Carps", and the number of eggs is closely related to hydrological conditions such as water temperature, duration of flow rise, and daily rise rate22. Since 2011, the Three Gorges Reservoir has organized and implemented ecological dispatch experiments for the natural reproduction of Four Major Chinese Carps in the middle reaches of the Yangtze River, and as of 2023, a total of 17 ecological dispatch experiments have been carried out23, almost all of which were carried out after mid-to-late May, mainly in early June. The ecological scheduling experiments all achieved good results, and the fish had obvious spawning behavior, and the total spawning volume increased with the accumulation of ecological scheduling trial years24.

The factors of the stimulation degree of the flow rise process on the reproductive behavior of the Four Major Chinese Carps can be summarized as intensity, intensity change rate and time accumulation9. The intensity element can be characterized by the initial flow and peak flow, the time accumulation element can be characterized by the duration of flow rise and the total increase of flow, and the intensity change rate element can be characterized by the daily increase of flow. Table 1 summarizes the relevant indexes and suitable ranges of the relevant literature on the water requirements of the ecological environment for spawning of the Four Major Chinese Carps, and numbers the different literatures under the same index to facilitate subsequent evaluation.

Table 1 The summary of relevant articles of relevant indicators and their suitable ranges of spawning resources and environmental requirements of Four Major Chinese Carps.
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In order to better synthesize the above research results and consider the specific feasibility of the actual operation of the Three Gorges, the starting time is set as June 1, and the duration of the flow rise is divided into four flow rise schemes: 4 days, 5 days, 6 days and 7 days.Then, the multi-layer entropy weight TOPSIS method was used to evaluate the suitability of ecological dispatching for the non-inferior solution set obtained by each flow rise scheme. The constraints of each scheme index are shown in Table 2.

Table 2 Index constraints of each scheme.
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Ecological scheduling model

Objective Functions

In this paper, we focused on the impact of the discharge rate from April to June on the spawning of the Four Major Chinese Carps in the lower reaches of the Three Gorges Reservoir, and established a multi-objective operation model of the Three Gorges Reservoir including ecological and power generation targets.

Ecological objectives

Ecological flow was developed to achieve a water balance between humans and river ecosystems and to protect river health and biodiversity25,26.The ecological goal of this study mainly considers the ecological friendliness of the downstream river flow during the dispatch period. Firstly, the historical flow sequence before the construction of the Three Gorges Dam was divided into horizontal years. Based on the multi-year average flow series, it is divided into wet year, flat water year, and dry year, and the year with a corresponding frequency of more than 75% is selected as the dry year. A year with a frequency between 25% ~ 75% is defined as a flat water year; A year with a frequency of less than 25% is defined as a wet year.

Calculate the environmental flow in the river channel for each typical year. Based on the historical runoff data of the Three Gorges Reservoir site, the monthly frequency of P = 90% is the recommended minimum environmental flow (q1). P = 75% ~ 25% is the recommended suitable environment flow rate (q2 ~ q3); Frequency P = 10% is the recommended maximum ambient flow rate (q4). The results are shown in Table 3.

Table 3 Calculations of the four environmental flows in the river for each typical year. unit: m3/s.
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On the basis of Table 3, the trapezoidal membership function is used to evaluate the eco-friendliness of the flow. The traditional membership function takes a value of 0 for both the minimum and maximum parts and is graphically displayed as a trapezoid, with no distinction for exceeding the minimum or maximum. However, in the evaluation scenarios in this study, the degree of eco-friendliness of very large or very small flow is clearly different. Therefore, in this study, the method of determining ecological membership was first improved for the calculation of ecological target membership, and when the flow rate was less than the minimum environmental flow, the membership degree was set to a negative value, as shown in Fig. 1. The abscissa Q is the flow rate, and the ordinate FZ is the membership degree, which indicates the friendliness of the flow to the ecology. Flow q1, q2, q3, and q4 are characteristic values, wherein q1 and q4 are the recommended minimum and maximum environmental flow values for each time period, and q2 and q3 are the upper and lower limits of the recommended suitable environmental flow for each time period, respectively, and their graphic shapes are similar to trapezoids.

Fig. 1
figure 1

Improved graphical representation of the ecological affiliation versus flow relationship.

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Notes:Where Q has unit m3/s and FZ has no unit.

Suppose the membership degree of river discharge \(Q_{n,t}\) on the \(t\) day of the \(n\) month is \(FZ_{n,t}\), and its formula is as follows.

$$FZ_{n,t} = \left\{ {\begin{array}{*{20}c} {\frac{{Q_{n,t} - q_{1} }}{{q_{2} - q_{1} }},0 \le Q_{n,t} < q_{2} } \\ {1,q_{2} \le Q_{n,t} \le q_{3} } \\ {\frac{{q_{4} - Q_{n,t} }}{{q_{4} - q_{3} }},q_{3} \le Q_{n,t} < q_{4} } \\ {0,Q_{n,t} > q_{4} } \\ \end{array} } \right.$$
(1)

then the ecological objectives of reservoir dispatch \(f_{eco}\) can be expressed as:

$$f_{eco} = \max FZ = \frac{1}{{\sum\nolimits_{n = 4}^{6} {T_{n} } }}\sum\nolimits_{n = 4}^{6} {\sum\nolimits_{t = 1}^{{T_{n} }} {FZ_{n,t} } }$$
(2)

where, \(FZ_{n,t}\) is the membership degree of river discharge \(Q_{n,t}\) on the \(t\) day of the \(n\) month ; \(T_{n}\) is the number of days in the nth month of the scheduling period.

Power generation targets

The power generation task is the most important task of the Three Gorges Reservoir in addition to flood control, and it is also an important embodiment of the economic benefits of the Three Gorges Hydropower Station. The maximum power generation of the hydropower station is the goal during the dispatch period, and the expression is as follows:

$$f_{ene} = \max E = \max \left\{ {\sum\limits_{t = 1}^{T} {\left[ {KQ_{fd} (t)H(t)\Delta t} \right]} } \right\}$$
(3)

where, \(E\) is the total power generation of the reservoir during the dispatching period, kW·h; \(T\) is the total number of time slots in the scheduling period, the ordinal number of time slots \(t = 1,2,...,T\); \(K\) is the comprehensive output coefficient of hydropower station; \(Q_{fd} (t)\) is the power generation flow of hydropower station during the t period, m3/s; \(H(t)\) refers to the power generation head of the hydropower station during the t period, m; \(\Delta t\) is the length of the calculation period in hours, h.

Constraints

(1) Water balance constraints:

$$V(t + 1) = V(t) + 3600 \times (Q_{in} (t) - Q_{out} (t))\Delta t$$
(4)

(2) Water level constraints:

$$Z_{{}}^{\min } (t) \le Z(t) \le Z_{{}}^{\max } (t)$$
(5)

(3) Output constraints:

$$N_{{}}^{\min } (t) \le N(t) \le N_{{}}^{\max } (t)$$
(6)

(4) Boundary constraints:

$$Z(0) = Z_{start} ,Z(T) = Z_{end}$$
(7)

(5) Outbound flow constraints:

$$Q_{out}^{\min } (t) \le Q_{out} (t) \le Q_{out}^{\max } (t)$$
(8)

(6) Initial flow constraints during the flow rise period:

$$Q_{out,start} \ge 7000$$
(9)

(7) Constraint on peak flood flow during flow rise:

$$Q_{out,end} \ge 19610$$
(10)

(8) Constraints on the daily increase of flow during the flow rise period:

$$Q_{out,6.n + 1} - Q_{out,6.n} > 900,n = start,start + 1,...end - 1$$
(11)

(9) Constraints on the total increase in flow during the flow rise:

$$Q_{out,end} - Q_{out,start} > 4050$$
(12)

(10) Constraint on the duration of flow rise:

$$end - start \in \left\{ {4,5,6,7} \right\}$$
(13)

(11) Non-negative constraints: The variables are non-negative.

In the formula: \(V(t)\) is the water storage capacity of the hydropower station in time period \(t\), m3; \(Q_{in} (t)\) and \(Q_{out} (t)\) are the inflow and outflow of the hydropower station in the time period respectively, m3/s. \(Z^{\min } (t)\) and \(Z^{\max } (t)\) are the minimum and maximum allowable water level of hydropower station in time period \(t\),m. The minimum is generally the dead water level \(Z_{d}\), and the maximum is the normal storage water level \(Z_{n}\) or flood limit water level \(Z_{x}\) according to the requirements of the corresponding period,m. \(N^{\min } (t)\) and \(N^{\max } (t)\) are respectively the minimum and maximum values of hydropower station output during the time period,kW. The minimum values are generally the technical minimum output of the unit, or 0;.The maximum value can take the expected output \(N_{yx} (t)\) or specify other values according to the actual requirements of the power grid;\(Z_{start}\) and \(Z_{end}\) are the beginning and end water levels of the scheduling period of hydropower station respectively,m. \(Q_{out}^{\min } (t)\) and \(Q_{out}^{\max } (t)\) are the minimum and maximum allowable discharge of the reservoir of the hydropower station in the time period \(t\), respectively, m3/s.The minimum value is set according to other requirements except the study objective, and the maximum value is limited by the discharge capacity of the reservoir and the flood control discharge.

Constraints during the flood period are mainly the requirements for the water expansion scheme during the spawning period of Four Major Chinese Carps shown in Table 2. After the water temperature change rule of comprehensive reservoir discharge and relevant article, the start time of the ecological flow rising process is set as June 1. The union of initial flow, peak flow, daily flow increase, and total flow increase indicators obtained from each article in Table 2 is reflected in the model as constraints.

Solving algorithms

In this study, we use Dynamic Programming Based on Combination of Penalty Factors (CPF-DPSA)27 to solve the problem. The penalty function with a penalty factor of \(o_{1}\) is established for \(f_{eco}\) and added to the cumulative-value objective function. This method can transform the multi-objective problem into a single objective for solving. The specific calculation process is based on the DP algorithm with certain modifications, the state transfer equation of DP during iterative calculation is as below.

$$f_{{}}^{*} (t + 1,k) = \mathop {\max }\limits_{{j = 1\sim L_{t} }} \left( {f_{{}}^{*} (t,j) + E(t,j,k) + o_{1} FZ(t,j,k)} \right)$$
(14)

where, \(f^{*} (t + 1,k)\) is the total optimal energy generation of the reservoir during \(1\sim t\) the period when it is in \(k\) the state at the end of \(t\) period;\(E(t,j,k)\) is the power generation of time period \(t\) when time period \(t\) is in the \(j\) state at the beginning and the \(k\) state at the end of time period \(t\); \(FZ(t,j,k)\) is the ecological membership degree of time period \(t\) when time period \(t\) is in the \(j\) state at the beginning and the \(k\) state at the end of time period \(t\); \(o_{1}\) is the penalty factor and takes a non-negative value.

The value of the penalty factor directly affects the result of the optimization of the non-inferior solution set. Therefore, it is necessary to further analyze the influence range and discrete width, so as to obtain a more evenly distributed and extensive set of non-inferior solutions. The specific value process of penalty factor is divided into the following two stages.

Stage 1: Penalty factor range detection:

(1) Plot \(f_{ene} - o_{1}\) curve and examine the trend of the curve, analyze and exclude the region of no impact further get the penalty factor influence range S. Ensure that the factor value is determined within the valid range;

(2) In the effective range S, the discrete value of the factor is selected according to the gradient of the target change with the factor.In the drastic changes can choose more points, and reduce the selection points in the gentle changes. After getting the range of the penalty factor value set s, further get the set of non-inferior solutions corresponding to different penalty factors d.

Stage 2: Each element in the penalty factor value set is substituted into the comprehensive target one by one and the optimal solution is obtained by a dynamic programming algorithm. After solving the comprehensive objective function corresponding to all penalty factors, the set of non-inferior solutions is further obtained by multi-objective screening.

The multilayer entropy-weighted TOPSIS method

Entropy-weight TOPSIS evaluation and solving algorithm

The entropy-weighted TOPSIS method is further evaluated by using the TOPSIS method after the entropy-weighted method is used to weight each index4. The main steps of the entropy-weighted TOPSIS method are as follows.

1.Entropy-weight method

(1) Calculate the weight of the jth sample value under the ith indicator in relation to that indicator:

$$p_{ij} \, = \,\frac{{y_{ij} }}{{\sum\limits_{j = 1}^{n} {y_{ij} } }}$$
(15)

(2) Calculate the entropy value of the ith indicator:

$$e_{i} = - k\sum\limits_{j = 1}^{n} {p_{ij} \ln (p_{ij} )}$$
(16)

Where \(k = \frac{1}{\ln (n)}\),if \(p_{ij} \, = \,0\),then define \(p_{ij} \ln (p_{ij} ) = {0}\).

(3) Determination of the weights of the indicators:

$$\omega_{i} = \frac{{(1 - e_{i} )}}{{\sum\limits_{i = 1}^{m} {(1 - e_{i} )} }}$$
(17)

2.Comprehensive evaluation of the entropy-weight method

According to the normalized value of each evaluation indicator and the weight of each evaluation indicator, a comprehensive evaluation value is generally obtained through a weighted sum formula.

$$P_{j} = \sum\limits_{i = 1}^{N} {\omega_{i} y_{ij} }$$
(18)

Here, \(P_{j}\) is the composite evaluation score for partition \(j\). The larger \(P_{j}\) is, the better.

For example, a composite evaluation value of 0.75 or greater is considered excellent, between 0.5 and 0.75 is considered good, between 0.35 and 0.50 is considered moderate, and less than 0.35 is considered poor.

3.Entropy-weight TOPSIS comprehensive evaluation

(1) Determine the best and worst options.

The optimal scheme \(z^{ + }\) consists of the maximum value of each column element in \(Z\):

$$z^{ + } = \left( {z_{1}^{ + } ,z_{2}^{ + } ,...,z_{n}^{ + } } \right)$$
(19)

The worst scheme \(z^{ - }\) consists of the minimum value of each column element in Z:

$$z^{ - } = \left( {z_{1}^{ - } ,z_{2}^{ - } ,...,z_{n}^{ - } } \right)$$
(20)

(2) The approximate degree of each evaluation object to the optimal scheme and the worst scheme is \(D_{j}^{ + }\)、\(D_{j}^{{_{ - } }}\) respectively.

$$D_{j}^{ \pm } = \sqrt {\sum\nolimits_{i = 1}^{n} {\omega_{i} \left( {z_{i}^{ \pm } - z_{ij} } \right)^{2} } }$$
(21)

where \(\omega_{i}\) is the weight of the \(i\) th attribute, and the index weight is obtained by entropy-weighted method.

(3) The close degree \(C_{i}\) between the evaluation object \(i\) and the optimal scheme is calculated. The more \(C_{i}\) approaches 1, the better the evaluation object is.

$$C_{i} = \frac{{D_{j}^{ - } }}{{D_{j}^{ + } + D_{j}^{ - } }}$$
(22)

According to the size of \(C_{i}\), the evaluation results are given.

The multilayer entropy-weighted TOPSIS evaluation

Because of the embedded flow rise process, which can be measured by multiple indicators, as shown in Fig. 2. Since the traditional TOPSIS evaluation method cannot fully consider the level and information distribution of indicators, the hierarchical processing can be used to independently analyze and synthesize indicators at different levels, which not only preserves the internal details of indicators, but also reflects the overall situation at the comprehensive level, and improves the reliability of evaluation results. As can be seen from Fig. 2, the comprehensive evaluation of the scheduling results can be divided into four levels, starting from the innermost layer, and the evaluation score of each layer is used as the attribute value of the outer layer to participate in the new round of evaluation. The specific steps of the multi-layer entropy weight TOPSIS method to evaluate the non-inferior solution scheme are as follows.

Fig.2
figure 2

Multilevel evaluation structure of the n-day upwelling process.

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In the first step, the daily increase in flow of each day was evaluated, and the daily increase of flow d2,…,dn after the start time of spawning of the Four Major Chinese Carps was carried out for each flow rise scheme, and the suitable range given by different literatures was used to conduct a fuzzy comprehensive evaluation.

The second step is to conduct the second level of evaluation. It includes the evaluation of the initial flow during the spawning period, the peak flow, the total increase in the total volume and the daily increase in the flow. The results of the first step were used in the comprehensive evaluation of the daily increase of flow on each day of the spawning period, and the corresponding values in the representative literature selected by other applications were used for evaluation.

The third step is to conduct the third level of evaluation. This includes the evaluation of power generation, ecological affiliation, and flow rise processes. The scores of the four indicators in the second step were used as the attribute values for the evaluation of the flow rise process, and the results of the multi-objective scheduling model were used for the evaluation of other indicators.

The fourth step is a comprehensive evaluation. The entropy-weighted TOPSIS method was used to conduct a final comprehensive evaluation of each optimization scheme from three aspects: power generation, ecological membership, and flow rise scheme.

The fifth step, is sorting. After the first four steps of calculation, each scheme will get a comprehensive score, and finally the scheme will be sorted according to the score, and the optimal result of each non-inferior scheme when the flow rise time is 4d ~ 7d respectively.

Discussion

Based on the characteristic parameters of the Three Gorges Reservoir and the historical inflow data for many years, the start time of ecological water demand rise is set to June 1, and the flow rise time is set to 4d ~ 7d, a total of four schemes, with the goal of maximizing the total power generation during the scheduling period and the ecological membership degree of the downstream river environmental flow, the multi-objective dispatching model is calculated from April 1 to June 10, and the multi-objective solution algorithm based on penalty factor is used to solve the problem.Firstly, the results obtained when the two objectives are only considered for single-objective optimization (i.e., the penalty factors are 0 and 1) are calculated respectively. Secondly, the results obtained when the two objectives are considered for optimization (i.e., the penalty factors both fluctuate) are stopped until the calculation results do not change after the penalty factors change, and finally the non-inferior solution set is selected and evaluated by TOPSIS with multi-layer entropy weight to get the optimal tradeoff scheme.

Optimization results

Taking the result data of the corresponding calculation period of a typical dry year (2013) as an example, the number of non-inferior solution sets of each scheme obtained after multi-objective screening was as follows: 39 for the flow rise time of 4 days; 45 for 5d; 41 for 6d; The 48 of 7d, the Pareto front of each protocol are shown in Fig. 3 and the large images can be found in Supplementary Figure S1-S4 ,the full results tables can be found in Supplementary Table S1-S4.

Fig. 3
figure 3

Pareto frontiers for four rising water scenarios.

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Notes: The non-inferior solution sets of ecological and economic objective trade-offs of different water expansion schemes are presented.

(1) From the calculation results of the four schemes in Fig. 3, it can be seen that the total power generation E during the dispatch period has an obvious competitive relationship with the sum of the ecological membership FZ.

(2) The following is an analysis of the results of the different rise time scenarios:

4d: when the penalty factor \(o_{1}\) is 0, i.e., when single-objective optimal dispatching with the goal of maximum total power generation is carried out, the total power generation is 13.936 billion kWh; when \(o_{1}\) is an extremely large value, and the model is close to single-objective optimal dispatching with the goal of maximum eco-affiliation, the average value of eco-affiliation is 0.945; when the average value of eco-affiliation is 0.871, the total power generation is 13.519 billion kWh, and the two objective values have a good balance.

5d: when the penalty factor \(o_{1}\) is 0, the total power generation is 13.921 billion kWh; when \(o_{1}\) is extremely large, the average value of ecological affiliation is 0.916; when the average value of ecological affiliation is 0.883, the total power generation is 13.434 billion kWh, and the two target values have a good balance.

6d: when the penalty factor \(o_{1}\) is 0, the total power generation is 13.896 billion kWh; when \(o_{1}\) is a great value, the average value of ecological affiliation is 0.902; when the average value of ecological affiliation is 0.848, the total power generation is 13.603.5 billion kWh, and the two target values have a good balance.

7d :When the penalty factor \(o_{1}\) is 0, the total power generation is 13.841 billion kWh; when \(o_{1}\) is extremely large, the average value of ecological affiliation is 0.888; when the average value of ecological affiliation is 0.834, the total power generation is 13.635 billion kWh, and there is a better balance between the two target values.

(3) When the penalty factor \(o_{1}\) is 0, that is, when the single-objective optimal scheduling with the goal of maximizing the total power generation is carried out, the total power generation of the four schemes decreases with the increase of days; When the power generation weight \(o_{1}\) is the maximum and the model is close to the single-objective optimal scheduling with the goal of maximizing the ecological membership, the average ecological membership of the results obtained by the four schemes also decreases with the increase of days. It can be seen that the longer the flow rise process, the stricter the constraints on the outflow, and the smaller the target value obtained.

In order to discuss the robustness and stability of the model, the size of the parameter penalty factor \(o_{1}\) is changed in the range of minimum and maximum values, and the scores of 4 schemes from 4 to 7dare obtained as shown in the Fig. 4. It can be seen that the score is better when the parameter is small, and the performance of the model decreases when the parameter is maximum, which is related to the stricter constraints of the water expansion scheme under the influence of the parameters. In practical applications, a smaller range of parameters should be preferred to obtain the best model performance.

Fig. 4
figure 4

The score of 4 schemes under the change of penalty factor \(o_{1}\).

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Notes: The stability of the model is discussed by showing the scores of the schemes under different parameters of the model.

Evaluation results

The dispatching schemes obtained under each scheme are sorted according to the value of the objective function of power generation from large to small, and are labeled separately for the subsequent result evaluation. First, the first level of evaluation is carried out, i.e., between the different literature within the indicators of each program within a given range. Taking the d2 increase whose duration of upwelling is 6 days as an example, the evaluation results are shown in Fig. 5 and the full results table can be found in Supplementary Table S5.

Fig.5
figure 5

Results of the first level of evaluation.

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Notes: At. = Article.

For example, there are 4 increased indicators when the duration is 5 days and 5 indicators when the duration is 6 days. Therefore, compared with other indicators of the water increase scheme, the increase index needs to be evaluated based on the entropy weight TOPSIS method. Taking the rising process for 6 days as an example, after the first layer of the increase index from d2 to d6 is evaluated, the entropy-weighted TOPSIS method of the comprehensive score of the second layer of increase is evaluated, and the results are shown in Fig. 6 and the full results table can be found in Supplementary Table S6.

Fig.6
figure 6

Results of the second tier of evaluation.

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Notes:d2,d3,d4,d5 and d6 are indicators of increase when the duration of the rise is 6 days.

Furthermore, the four indicators of the initial flow, peak flow, total flow growth and daily flow increase of each scheme were evaluated based on the entropy weight TOPSIS method, and the results obtained by the flow rise process lasting 6 days were taken as an example, and the score results of the flow rise process are shown in Fig. 7 and the full results table can be found in Supplementary Table S7.

Fig.7
figure 7

Results of the third-tier evaluation.

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Finally, the fourth level of evaluation is carried out. The entropy-weighted TOPSIS method was used to comprehensively evaluate different schemes considering power generation, ecological affiliation and flow rise process. Taking the results of the flow rise process lasting 6 days as an example, the total score results are shown in Table 4 and the full results table can be found in Supplementary Table S8.

Table 4 Results of the fourth tier of evaluations.
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The scores of each scheme set corresponding to the four types of flow rise durations including 4d ~ 7d are shown in Fig. 8. The power generation at the top of each scheme number is larger, but the other two indexes ecological membership degree and flow rise process score are smaller, resulting in a smaller total score. At the bottom of the scheme number is smaller, but the other two indicators score higher, and the total score is higher, but the total score is highest in the middle of the scheme.

Fig.8
figure 8

Scheme scores of four kinds of upwelling process in 4d ~ 7d.

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In order to verify the effectiveness and superiority of the multilayer entropy-weighted TOPSIS method, the scheme scores obtained by this method and the traditional TOPSIS method for 4d ~ 7d four kinds of rising water processes are compared in the Fig. 9. It can be clearly observed that the distribution of the scheme scores obtained by using the multilayer entropy-weight TOPSIS method is more reasonable, indicating that the method has a better effect and higher accuracy in the evaluation of these kinds of schemes. This result highlights the superiority of the multilayer entropy-weight TOPSIS method in dealing with complex decision-making problems, especially when considering multidimensional influencing factors, which can identify and evaluate the advantages and disadvantages of the schemes more effectively.

Fig.9
figure 9

Comparison chart of programme scores for the two evaluation methods.

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Therefore, the scheme with the highest score in the non-inferior solution set including the 4d-7d flow rise process was selected as the preferred scheme, and the results are shown in Table 5. The optimal schemes for different flow rise times in Table 5 have good benefits in terms of power generation and ecological objectives, and the final comprehensive score is close to 1. In the 4d ~ 7d scheme, each flow rise process has good performance in the initial flow, peak flow, total flow growth and daily flow increase, which is the result of using the multi-layer entropy weight TOPSIS method to evaluate the benefits of balancing each index. The horizontal comparison shows that the score of the embedded preferred scheme with fewer rising days is slightly greater than that of the preferred scheme with longer rising days, which is speculated to be due to the more rising days, the stricter the requirements for outbound flow, and the fewer non-inferior solutions that can be selected.

Table 5 Results of the optimal scheme of each flow rise process.
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The average daily outflow process of the preferred four scenarios in Table 5 for the entire dispatch period is shown in Fig. 10, with the periods of the upwelling process in the dashed box.

Fig. 10
figure 10

The average daily outflow of four upwelling process case for the entire scheduling period.

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The upper left corner is a magnified view of the water rise process. Highlighting the water level changes during the period of water rise can be seen more clearly in the comparison of the increase of each scheme. The comparison reveals that before May, there is little difference between the optimisation results corresponding to 4d to 7d, and there is a slight change from the beginning of May, but the difference is mainly after the beginning of June, which is close to the period of high water. This is the decision-making result of the multilayer entropy-weighted TOPSIS method to synthesise the benefits of each indicator.

Conclusion

In this paper, a multi-objective optimal scheduling model of reservoirs is proposed to embed the synergistic ecological requirements and power generation benefits in response to the spawning demand of the " Four Major Chinese Carps ". The model aims to maximize the power generation and the maximum water demand of the river ecology, and the dynamic programming method based on penalty factors is used to solve the problem. The five indexes of the flow rise process are included in the model as follows: firstly, the range of the other four indicators in each literature is synthesized, and the union of the appropriate range in different kinds of literature of each index is taken as a special constraint, and then the multi-layer entropy weight TOPSIS method is used to further evaluate the rising scheme corresponding to the non-inferior set of the obtained non-inferior set with reference to the literature standard, so as to obtain the optimal scheduling scheme under different rising durations. The results show that:

(1) The maximum power generation optimized from 4 to 7 days in the flow rise time was 13.936, 13.921, 13.896, and 13.841 billion kWh, respectively. The average values of the optimized maximum ecological membership were also 0.945, 0.916, 0.902, and 0.888. The reason why both values decrease with the duration of the rising process is that the longer the duration of the rising process, the more stringent the constraints of the model, so the smaller the value of the optimization result.

(2) The multilayer entropy-weighted TOPSIS method can comprehensively consider the appropriate range given in different articles of each index, and further integrate the five indexes of the upwelling process: Initial flow, peak flow, duration of upwelling, total flow growth amount and daily flow increase. The scores of the upwelling process are brought into the next layer and evaluated together with the two objective functions, and finally the optimal scheme under different durations of upwelling is obtained.

The entropy-weighted TOPSIS evaluation in each layer has a certain distinction between schemes, the data adopt the actual operation data of the Three Gorges Reservoir, and the parameters are refined according to the simulation results of the actual operation data, within a certain suitable range. The results can provide a theoretical basis and a reference for specific schemes for the optimal operation of the Three Gorges Reservoir during the subsidence period. However, there are several points that can be improved in the future. For example, there are limitations in the accuracy of hydrological forecast and the starting time of model setting in practical application. The performance of the model is very good when the parameters are set to a small value in practical application. Further optimization of the parameters is needed to enhance the robustness and stability of the model. In addition, the computational efficiency of the model can be improved by obtaining the initial solution more efficiently and accurately, eliminating the fixed step size and parallel computation.