Introduction

The Yellow River, which originates in the Tibetan Plateau and flows through nine provinces and autonomous regions, is the mother river of the Chinese nation. Although it is the second-longest river in China, the Yellow River is more famous for its characteristics of less water and more sand, water and sand come from different sources, and the relationship between water and sand is incongruous (Wang et al. 2016). Water resources in the Yellow River basin are incredibly scarce, with the amount of water per capita and per mu being only 23% and 15% of the national average level, which leads to the sharp contradiction between living, production, and ecological water use in the area (Xiang et al. 2017). Relevant studies indicate that the runoff of the Yellow River shows a downward trend under the dual effect of climate change and human activities (Omer et al. 2020). There is no doubt that the contradiction between the supply and demand of water resources will be further aggravated by the rapid socioeconomic development and steady population growth in the Yellow River basin (Bai et al. 2019; Zhang et al. 2021a).

In the context of rigid constraints of water resources and frequent extreme weather, cascade reservoirs in the Yellow River play an indispensable role in flood control, water storage and delivery, hydropower generation, water purification, and other aspects, fundamentally ensuring the continuous flow of the river without breaking its banks for many years (Jin et al. 2021; Zhang et al. 2021b). Meanwhile, the water and sediment regulation under the joint operation of cascade reservoirs in the Yellow River effectively controls the sediment deposition, improves the available regulated storage capacity of reservoirs, and fundamentally prevents the lifting of the riverbed of the suspended river on the ground (Jin et al. 2019). Different reservoirs perform multiple tasks in operation, and each task cannot be fully met at the same time because of the apparent conflicts between the functions (Lu et al. 2022).

Under the realistic background of climate change and the reduction of runoff of the Yellow River, river basin management departments and scholars have conducted much research on the multi-objective joint operation of reservoirs (Zhang et al. 2021c). The research results show that the joint operation of cascade reservoirs can improve the comprehensive benefits of a watershed system (Lu et al. 2018), while the reasonable allocation of value-added is the basic premise to maintain the long-range feasibility of the optimal operation of cascade reservoirs (Chen et al. 2020; Wang et al. 2021). Summarize and analyzing the existing literature, it is not difficult to find that the optimization of the operation of reservoirs mainly at the minimum water shortage or the maximum power generation, while the objectives of sediment transport and ecological protection of sandy rivers mostly take the water requirement for sediment transport and ecological base flow as the media, which are transformed into constraints (Li et al. 2022; Lu et al. 2018). Therefore, relevant studies failed to quantify the benefits of sediment transport and the ecological environment, making the benefit accounting under the joint operation of reservoirs not comprehensive enough (Bai et al. 2017; Huang et al. 2019; Jin et al. 2021). More importantly, the joint operation of cascade reservoirs is of far-reaching significance to the ecological protection and social development of cities along the Yellow River. However, the existing research on benefits accounting and gains allocation under the optimized operation of reservoirs is limited to the interior of reservoirs, and there is a lack of analysis on the relationship of interest between reservoirs and cities along the river (Shen et al. 2018a, b; Shen et al. 2018a, b; Xu et al. 2018).

Sediment deposition in sediment-laden rivers will lead to various negative problems, significantly increasing the possibility of severe floods (Chen et al. 2012). Simultaneously, since China has raised the ecological protection and high-quality development of the Yellow River basin into a national strategy, the coordinated development of ecological and economic benefits of the Yellow River basin became an essential foothold for the exploitation of the Yellow River in the future (Xu and Wang 2020). Therefore, multi-dimensional benefits such as sediment reduction and ecological protection should be integrated into the existing optimized operation of reservoirs. In addition, it is essential to further explore the connection of interest between the reservoir group and city group, and reasonably distribute the value-added on this basis.

Aiming at the above problems, this paper established a multi-dimensional benefits accounting model for the joint operation of the cascade reservoirs and a value-added allocation model among multi-stakeholders to improve the existing research. In terms of benefit accounting, this paper introduces the Emergy theory to realize the unified accounting of eco-environmental and economic benefits. Based on obtaining the natural economic value, the Emergy theory converts different categories and incomparable energy or matter into solar emergy for analysis and research (Gan et al. 2023). As a quantitative research method from traditional energy analysis to emerging eco economic systems (Odum 1996), emergy theory has been widely used in water ecological compensation (Guan et al. 2019), ecological environment value assessment (Wu et al. 2019), sustainable development of the city (Li et al. 2021) and so on (Amaral et al. 2016). In terms of gain distribution, this paper establishes a double-layer value-added allocation model under different application modes of the reservoir group based on the differences between the reservoir and city stakeholders, which realizes a win–win situation for all stakeholders.

The rest of this paper is structured as follows. The emergy theory and the principles of value-added allocation are briefly described in “Theory” Section. "Methodology" Section proposes the multi-dimensional benefits evaluation model and the distribution model of value-added under the operation of the reservoir group. “Study area and data collection” Section gives the location of the study area and the source of data. In “Results and discussion” Section, the model is applied in the middle and lower reaches of the Yellow River, and the results and discussion are given. “Conclusions” Section further provides the conclusion and future research direction of this study. The research route of this paper is shown in Fig. 1.

Fig. 1
figure 1

The research flow chart of this study

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Theory

A brief introduction to emergy theory

Emergy theory was proposed by Odum (Odum 1996), a famous American ecosystem scientist. As a new evaluation theory, Emergy analysis is committed to a unified measurement standard to comprehensively analyze various natural resources in the ecosystem. Emergy theory bridges the ecosystem and the socio-economic system, which converts different types of energy or matter that cannot be compared uniformly into solar emergy. The conversion formula of solar emergy can be expressed as

$$M = \tau \times B$$
(1)

where \(\tau\) means solar transformity of \(B\) and \(B\) refers to the quality of energy or matter.

In emergy analysis, the emergy-currency ratio (ECR) connects the ecosystem and economic system and realizes the conversion between currency and emergy. ECR indicates the emergy value of a region unit currency, which can be formulated as

$$ECR = \frac{TAE}{{GNP}}$$
(2)

where \(TAE\) is the emergy input of the country (region) in a given year and \(GNP\) is the corresponding regional gross national product.

The emergy contribution rate of water resources (ECRW) is a crucial parameter of emergy analysis and is a relative index to measure the contribution of water resources to economic production and the ecological cycle. Taking social production as an example, water resources, capital, labour input, and other means of production jointly produce industrial and agricultural products or provide social services. To calculate the emergy value of water resources for specific purposes, extracting the contribution of the emergy value of water resources is essential.

$$ECRW = \frac{{EM_{EW} }}{{EM_{ET} }}$$
(3)

where \(EM_{EW}\) means the emergy input of water resources in a system (sej) and \(EM_{EI}\) means the total emergy input of the system (sej).

The energy network analysis of the sediment river system under the joint operation of reservoirs is shown in Fig. 2, and the specific steps of emergy analysis refer to the existing research (Wu et al. 2019).

Fig. 2
figure 2

Energy network diagram of the river from the perspective of multi-dimensional benefits

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Principles of value-added allocation

The optimized operation of cascade reservoirs is essentially a Kaldo-Hicks improvement process, and the profit and loss of various stakeholders are different. Therefore, the principle of “who benefits, who compensates” should be followed so that the value-added is shared between the beneficiaries and benefactors. In addition, the allocation of the benefits should follow the principle of “cost–benefit equivalence”, that is, the greater the loss of the stakeholders in the optimized operation of cascade reservoirs, the more value-added share should be allocated. Meanwhile, this paper assumes that the income distribution follows the principle of “social responsibility”, which means that for vulnerable stakeholders (lower GDP), the proportion of its value-added allocation can be appropriately increased.

Methodology

The research methods are mainly divided into two parts. The first part is the calculation of multi-dimensional benefits, and the second part is the income distribution under the joint operation of reservoirs. First, in combination with the inherent characteristics of sandy rivers, this paper mainly selects ten indicators from three dimensions to calculate the multi-dimensional benefits of sandy rivers under the operating conditions of reservoir groups. Secondly, given the individual differences of interest subjects, stakeholders are divided into two categories, as shown in Table 1.

Table 1 Dimension division and indicators selection of multidimensional benefit accounting
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Multi-dimensional benefit accounting

Unit monetary value calculation of various water use

When analyzing the emergy of some benefits of water resources, it is necessary to calculate the solar transformity of various water bodies. The methods to measure the solar transformity of surface water, groundwater, and engineering water in this paper come from existing research (Wu et al. 2019). On this basis, the unit monetary value of various water use is further calculated to quantitatively calculate the benefits of water supply and ecological environment later (Guan et al. 2020; Pulselli et al. 2011).

  1. (a)

    Industrial water, agricultural water, and domestic water

The unit value of industrial, agricultural, and domestic water calculation processes can be summarized in the following four steps.

Step 1 Calculate the ECRW of the water body.

$$ECRW_{i} = \frac{{EM_{i}^{IW} }}{{EM_{i}^{IT} }}$$
(4)

where \(EM_{i}^{IW}\) is the emergy input of water resources of the \(i{\text{ - th}}\) system in a specific year (sej). \(EM_{i}^{IT}\) is the overall input emergy of the \(i{\text{ - th}}\) system in this year (sej). \(i\) refers to agricultural, industrial, and domestic water-use systems.

Step 2 Calculate the emergy output of water resources in \(i{\text{ - th}}\) system.

$$EM_{i} = ECRW_{i} \times EM_{i}^{OT}$$
(5)

where \(EM_{i}^{OT}\) means the overall emergy output in the \(i{\text{ - th}}\) system (sej).

Step 3 Calculate the emergy value of unit water.

$$EM_{Pi} = \frac{{EM_{i} }}{{W_{i} }}$$
(6)

where \(W_{i}\) is the consumed water resource in \(i{\text{ - th}}\) system this year (m3).

Step 4 Calculate the monetary value of the unit water.

$$E_{Pi} = \frac{{EM_{Pi} }}{ECR}$$
(7)

where \(ECR\) is the emergy/currency ratio in the region where \(i{\text{ - th}}\) system is located (sej/yuan).

  1. (b)

    Ecological water use

The value of ecological water refers to the ecological protection function of water resources, including the maintenance of biodiversity, net primary productivity, carbon fixation and oxygen release, climate regulation and water purification. Therefore, the monetary value of unit ecological water is expressed as

$$EM_{PE} = \frac{{\sum\limits_{i = 1}^{5} {EM_{i} } }}{{Q_{E} }}$$
(8)
$$E_{PE} = \frac{{EM_{PE} }}{ECR}$$
(9)

where \(EM_{i}\) is the annual benefit of the representative indicators in the subsystem of the ecological environment. \(Q_{E}\) is the annual runoff of the corresponding section (m3). \(ECR\) is the emergy/currency ratio of the city where the hydrology section of the river is located (sej/yuan). \(EM_{PE}\) and \(E_{PE}\) are the emergy value and monetary value of unit water, respectively (sej/m3), (yuan/m3).

Ecological environment subsystem

Based on existing studies (Wu et al. 2019), this paper divides rivers’ ecological environmental benefit accounting indicators into the following five categories, as shown in Table 1.

  1. (1)

    Maintaining biodiversity

Water resources not only provide material needs for organisms but also create suitable living conditions. The benefits of water resources in maintaining biodiversity can be expressed as (Jonsson and Malmqvist 2003)

$$EM_{MB} = N \times R_{b} \times \tau_{b} \times ECRW$$
(10)

where \(N\) is the total biological species in a given area. \(R_{b}\) is the proportion of biological activity area to the global area (%), and \(\tau_{b}\) is the solar transformity of global species, \(\tau_{b} = 1.26 \times 10^{25}\) sej/species.

  1. (B)

    Net primary productivity

Net primary productivity mainly refers to the amount of organic carbon fixed by plants through photosynthesis to remove the part consumed by their own respiration and used for growth and reproduction.

$$EM_{NPP} = A_{npp} \times P_{npp} \times \tau_{npp}$$
(11)

where \(A_{npp}\) is the area of aquatic plants in regional water (m2). \(P_{npp}\) is the net primary productivity of the unit water area (g/m2). \(\tau_{npp}\) is the solar transformity of net primary productivity, \(\tau_{npp} = 5.78 \times 10^{7}\) sej/g.

  1. (C)

    Carbon fixation and oxygen release

Aquatic plants can absorb carbon dioxide and release oxygen, and the ecological benefits can be expressed as

$$EM_{CF} = \left( {\sigma_{{co_{2} }} \times P_{c} \times S_{c} \times \tau_{{co_{2} }} + \sigma_{{o_{2} }} \times P_{c} \times S_{c} \times \tau_{{o_{2} }} } \right) \times ECRW$$
(12)

where \(\sigma_{{co_{2} }}\) and \(\sigma_{{o_{2} }}\) are 1.47 and 1.07 respectively. \(P_{c}\) is the average productivity of vegetation (g/hm2).\(S_{c}\) is vegetation area (hm2). \(\tau_{{co_{2} }}\) and \(\tau_{{co_{2} }}\) are solar transformity of \(O_{2}\) and \(CO_{2}\) respectively, and the corresponding emergy values are 3.78 × 107sej/g and 5.11 × 107sej/g.

  1. (D)

    Climate regulation

The climate regulation function of river water resources refers to the regulation of water vapour on atmospheric temperature and humidity. The benefits of climate regulation can be transformed into the multiplication of the latent heat, amount of evaporation and solar transformity of steam(Wu et al. 2019).

$$EM_{CR} = \left( {2,507.4 - 2.39T_{t} } \right) \times W \times \tau_{z}$$
(13)

where \(T_{t}\) means the average temperature of the given area (°C). \(\tau_{z}\) is the solar transformity of steam, \(\tau_{z} = 12.20{\text{ sej/J}}\), and \(W\) is the amount of evaporated water (g).

  1. (E)

    Water purification

Water has a natural ability to purify and precipitate, which can change the concentration of various pollutants (Wu et al. 2019).

$$EM_{WP} = f \times ECRW \times \sum\limits_{p = 1}^{n} {m_{p} } \times \tau_{p}$$
(14)

where \(f\) means the self-purification ratio of the water. \(m_{p}\) is the discharge of \(p{\text{ - th}}\) pollutant (g), and \(\tau_{p}\) is the corresponding solar transformity (sej/g).

  1. (F)

    Ecological and environmental benefits are related to flow

According to formula (8) and formula (9), the eco-environmental benefits of the river system can be shown as

$$E_{PE} = \frac{{EM_{MB} + EM_{NPP} + EM_{CF} + EM_{CR} + EM_{WP} }}{{Q_{E} \times ECR}}$$
(15)
$$E_{Total} = E_{PE,v} \times Q_{v}$$
(16)

where \(E_{PE,v}\) is the eco-environmental value of unit water at the \(v{\text{ - th}}\) river section downstream (yuan/m3). \(Q_{v}\) is the ecological flow of the \(v{\text{ - th}}\) river section (m3).

Flood discharge and sand transport subsystem

  1. (1)

    Flood control

The losses caused by typical floods under different operation modes of the reservoirs can be expressed as

$$E_{FC} = \sum\limits_{j = 1}^{n} {A_{j} \times E_{j} \times \theta_{j} }$$
(17)

where \(A_{j}\) is the inundation area of \(j{\text{ - th}}\) type land under the given application mode of the reservoirs, and \(E_{j}\) is the value per unit area of \(j{\text{ - th}}\) type land use. \(\theta_{j}\) is the loss rate of inundation of \(j{\text{ - th}}\) type land. \(j\) includes arable land, housing land, etc.

  1. (B)

    Sediment reduction of the river channel

The indirect method is adopted to calculate the benefit of sediment reduction of the river channel, which is equal to the unit dredging cost multiplied by the sediment dredging volume (Liang et al. 2016).

$$E_{SR} = V_{h} \times P_{h}$$
(18)

where \(V_{h}\) is the dredging amount of river sediment, \(P_{h}\) is the unit dredging cost, \(P_{h} = 4.7\) yuan/t.

  1. (C)

    Sediment discharge of reservoir

Different operation modes of the reservoirs will also affect their own siltation, and the 7 corresponding benefits of reducing siltation are expressed as

$${E_{SD}} = \frac{{{S_d}\xi }}{{D{V_c}}}$$
(19)

where \(S_{d}\) is the sediment volume is reduced in the reservoir area (t).\(D\) is the sediment density,uniformly taken as 1.2 t/m3. \(V_{c}\) is the initial storage capacity of the reservoir (m3). \(\xi\) is the total construction cost of the reservoir (yuan).

Socioeconomic subsystem

  1. (1)

    Water supply

For the convenience of calculation, the water supply benefits under different operation modes of the reservoir are expressed as

$$E_{WS} = \sum\limits_{h = 1}^{H} {\sum\limits_{k = 1}^{3} {E_{Pk}^{h} \times W_{k}^{h} } }$$
(20)

where \(E_{Pk}^{h}\) represents the unit price of water delivery to \(k{\text{ - th}}\) water users in the city \(h\), and \(W_{k}^{h}\) represents the amount of water delivery to \(k{\text{ - th}}\) water users in the city \(h\). \(k\) includes industrial, agricultural and domestic water users.

  1. (B)

    Hydroelectric generation

The power generation benefit of reservoirs in sandy rivers also needs to consider the constraints of the sediment-carrying capacity of generator units. When the sediment concentration in the reservoir is greater than the maximum sediment discharge, the power generation benefit of the reservoir is set to 0. When the sediment concentration in storage is less than the maximum sediment discharge, the calculation formula of power generation benefit is as follows

$$E_{HG} = \left\{ \begin{gathered} \varphi N\min \left( {Q_{p} ,Q_{p\max } } \right)\Delta H\Delta t, \, Q_{s} \le Q_{s\max } \hfill \\ \, 0, \, Q_{s} > Q_{s\max } \hfill \\ \end{gathered} \right.$$
(21)

where \(\varphi\) is the network access price of the reservoir (yuan/kwh); \(Q_{p}\) is the over flow (m3/s). \(Q_{\max }\) is the maximum sediment carrying capacity of the generator set (m3/s). \(\Delta H\) is the generating head (m). \(\Delta t\) is the duration of power generation (h). \(N\) is the output coefficient of reservoir hydropower station. \(Q_{s\max }\) and \(Q_{s}\) are respectively the sediment content of water (kg/m3) and the maximum sediment volume passing through the turbine (kg/m3).

The allocation of value-added among the main stakeholders

The optimized operation of key reservoirs in sandy rivers is a Kaldo Hicks improvement process. The profit and loss of each stakeholder are different under different operation modes of reservoirs. Therefore, it is necessary to reasonably allocate the value-added after the optimized operation of reservoirs. To facilitate the dimensionality reduction analysis, this paper divides the distribution of value-added into two levels: the distribution between reservoir-group and city-group and the distribution within the group.

Allocation of gains between the reservoir group and the city group

Nash bargaining solution is an efficient tool to solve the problem of interest negotiation, which is widely used in various fields (Zhao et al. 2021). The equilibrium solution can be expressed as the optimal solution of the following Nash product form.

$$u^{*} = \mathop {\max }\limits_{u \in \Omega } \left( {u_{r} - d_{r} } \right)^{\alpha } \times \left( {u_{c} - d_{c} } \right)^{1 - \alpha }$$
(22)

where \(\Omega\) is the set of valid payment pairs,\(d_{r}\) and \(d_{c}\) are the breaking points of the reservoir group and city group. \(\alpha\) and \(1 - \alpha\) are the bargaining power of the reservoir group and city group.

Allocation of gains within a group

Multi-attribute decision-making can also solve the distribution problem (Xu et al. 2021). By reasonably selecting indicators and further weighting these indicators, the distribution proportion of each stakeholder can be obtained. Determining the weight of each indicator is the key to the decision-making problem. The product of subjective weight and objective weight is taken as the comprehensive weight in this paper, in which the best worst method (BWM) is selected for the subjective weight. Compared with other subjective methods of weighting, BWM can simplify the decision-making process and ensure the consistency of decision-making (Xu et al. 2021). The entropy weight method is adopted to determine the objective weight. Referring to the principles of value-added allocation proposed in “Principles of value-added allocation” Section, the selection of two groups of value-added allocation indicators is shown in Table 2.

Table 2 Indicators selection of intragroup gains distribution
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Study area and data collection

Study area

The study area of this paper refer to Sanmenxia reservoir (SMX), Xiaolangdi reservoir (XLD), and the lower reaches of the Yellow River. The lower reaches of the Yellow River from Taohuayu to the Yellow River estuary, with a total length of 786 km and a drainage area of 23,000 km2. The sediment deposition in the lower reaches of the Yellow River is extremely serious, and the riverbed of some sections is 4–6 m higher than the ground, which makes the Yellow River a world-famous "suspended river". Both banks of the lower Yellow River mainly rely on levees to keep out water, and floods have become the biggest hidden danger threatening the safety of the Huang-Huai-Hai Plain. Cities along the lower reaches of the Yellow River include Luoyang (LY), Jiaozuo (JZ), Zhengzhou (ZZ), Kaifeng (KF), Xinxiang (XX), Puyang (PY), Heze (HZ), Jining (JNI), Tai'an (TA), Liaocheng (LC), Dezhou (DZ), Jinan (JNA), Zibo (ZB), Binzhou (BZ) and Dongying (DY). The length of the river channel in each city is shown in Table 3.

Table 3 Channel length of cities in the lower Yellow River (unit: km)
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Located in Sanmenxia, SMX is a seasonal regulation reservoir with an effective storage capacity of 439 million m3. Due to the serious sediment deposition in the reservoir area, the regulation capacity of SMX is very limited. XLD is an incomplete annual regulation reservoir, controlling a drainage area of 69,4000 km2, accounting for 92.3% of the Yellow River drainage area. The average water level of XLD is 275 m, with a total storage capacity of 12.65 billion m3 and a regulating storage capacity of 5.1 billion m3. The total installed capacity is 1,800 MW and the average annual power generation is 5.1 billion kwh. The integrated mission of XLD focuses on flood control and sediment reduction while taking into account water supply, irrigation, and power generation. The main hydrological sections of the lower Yellow River include Huayuankou (HYK), Jiahetan (JHT), Gaocun (GC), Sunkou (SK), and Aishan (AS), Luokou (LK), and Lijin (LJ). The specific distribution of critical nodes is shown in Fig. 3.

Fig. 3
figure 3

The study area of the Yellow River

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Data collection

Taking a typical once-every-20-year flood as an example, two different joint application modes of SMX and XLD are set as mode of below the beach and mode of overflow the beach (simply named application mode1 and application mode 2). According to the multi-dimensional benefit accounting method established in this paper, the comprehensive benefit is calculated, and the value-added is further distributed to each stakeholder. Multi-dimensional benefits are calculated and distributed based on economic data and water use data in 2016. The data sources mainly include the regional water resources bulletin, statistical yearbook, national economic and social development statistical bulletin. Some data are from the Yellow River water conservancy research institute.

The ECR of each region is obtained according to the annual emergy input and the GNP of each region. Through the classified collection of agricultural, industrial and domestic economic data and water use data in each region, the unit value of various types of water use in the corresponding areas can be calculated. For the value estimation of unit ecological water, the various ecological environment value and ecological flows in each city are approximately estimated based on the monitored runoff, sediment, pollutants, biomass and other data of the main hydrological section of the lower reaches of the Yellow River and the length of the river channel in each city. Relevant data collection and calculation results are shown in Table 4.

Table 4 Total eco-environmental emergy and runoff of cities in the lower Yellow River (unit: sej, m3)
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Calculate the data in Table 4 according to the steps in “Unit monetary value calculation of various water use” Section, and the unit currency value of various types of water use in each city can be obtained as shown in Table 5. On this basis, the water supply benefit and ecological environment benefit under different operation modes of the reservoir can be further calculated.

Table 5 ECR and unit value of various types of water in each city (unit: sej/yuan, yuan/m3)
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Results and discussion

Results

This paper simulates two application modes of SMX and XLD facing typical flood conditions and calculates various benefits in turn: (1) Calculate the corresponding flood loss in combination with the inundation scope and land use type of each city; (2) Calculate the benefit of water supply by collecting the data on all kinds of water use in each city; (3) Combined with the observation data of hydrological stations and the river length, the ecological flow of each city is converted, and further ecological benefits are obtained; (4) Combined with the siltation amount monitored by each hydrological station and the channel length in each city, calculate the river siltation reduction benefit of each city; (5) Calculate the power generation and benefits of SMX and XLD; (6) Calculate the sediment discharge of the reservoir under the two application modes of SMX and XLD, and calculate the corresponding sediment discharge benefits of the reservoir. The calculation results of various benefits for stakeholders under the two application modes of SMX and XLD are shown in Table 6.

Table 6 Profit and loss of each stakeholder under the two application modes (unit: yuan)
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Comparing the comprehensive benefits of stakeholders under the two application modes, the selection preferences of each stakeholder are shown in Table 7. The preferences of stakeholders are inconsistent, but the total benefit under mode 1 is more significant than that under mode 2. Therefore, all parties will choose application mode 1 as long as the value-added can be reasonably distributed.

Table 7 Preferences of stakeholders for the two application modes
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The stakeholders are divided into the reservoir group and the city group, and the benefit distribution among groups is carried out on this basis. The asymmetric Nash bargaining model is adopted to distribute benefits between the two groups here. The breaking point of the city group is 4.24E + 08, which means that the ecological benefits of application mode 1 are reduced compared with mode 2. The breaking point of the reservoir group is 3.09 + 09, which means that the sediment discharge benefits in application mode 1 are lower than that in mode 2. The bargaining power of the main body of the reservoir group is determined by the ratio of the regulation capacity of the leading reservoir group to the average of the regulation capacity of the main reservoirs of the Yellow River. The bargaining power of the city group is given by the ratio of average water consumption in the Yellow River basin to that in the lower reaches. Based on the above, it is necessary to determine the gain to be allocated. The net benefit generated under application mode 1 is 1.04E + 09 yuan more than that under application mode 2, but it is not appropriate to directly configure this part of the gain. As can be seen from Table 7, stakeholders such as XLD, LC and TA prefer to use application mode 2. It is unfair to add the allocation amount of gain to the multidimensional benefits corresponding to application mode 2 as the final benefit of stakeholders. Therefore, this paper further adjusts the value added to net gains (1.04E + 09 yuan) plus the profit and loss difference between the two application models of stakeholders who gain more benefits in application mode 2. Then the final value-added is 4.09E + 09 yuan, and the asymmetric Nash bargaining model is established as follows

$$\left\{ \begin{gathered} u^{*} = \mathop {\max }\limits_{u \in \Omega } \left( {u_{r} - 3.09 \times 10^{9} } \right)^{0.3786} \times \left( {u_{c} - 4.24 \times 10^{8} } \right)^{0.6214} \hfill \\ u_{r} + u_{c} = 7.06{\text{E}} + 09 \hfill \\ \end{gathered} \right.$$

Solving the Nash bargaining solution above, \(u_{r} = 4.63{\text{E}} + 09\) and \(u_{c} = 2.96{\text{E}} + 09\) can be obtained. That is, the share of the reservoir group is \({1}{\text{.55E}} + 09\) yuan, and the share of the city group is \({2}{\text{.54E}} + 09\). Based on the above, the value-added is further distributed within groups. For the reservoir group, the distribution indicators selected in this paper include the variation of SD, HG, total revenue, and effective storage, the guaranteed output of the generator set. Since the stakeholders are only XLD and SMX in the reservoir group, the BWM is adopted to determine the weight of indicators. The weight of each indicator is shown in Table 8.

Table 8 Weight of each gain distribution indicator in the reservoir group
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For the city group, the entropy weight method and BWM are combined to calculate the weight of each indicator. The selection of the benefit distribution indicators and their corresponding comprehensive weights are shown in Table 9.

Table 9 Weight of each gain distribution indicator in the city group
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According to the relative value and corresponding weight of each decision-making indicator of each stakeholder, the benefit distribution within the group is carried out. The final allocation result is shown in Fig. 4. It can be seen from the figure that among all stakeholders, XLD has the highest gains allocation, accounting for about 36.16% of the total. This is because the generation of value-added under application mode 1 is based on the premise that XLD sacrifices the great benefits of sediment discharge, and XLD has a relatively large regulation capacity for flood compared with SMX, so it should enjoy more benefits. On the contrary, the share of JNI accounts for only 1.12% of the total, because the value of each evaluation indicator of JNI is lower than that of other cities. Specifically, JNI has a smaller range of changes of each benefit under the two application modes and has the shortest channel of the Yellow River in the area are the fundamental reasons for the lowest quota.

Fig. 4
figure 4

Distribution of value-added among stakeholders

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Discussion

To better allocate the value-added, it is necessary to compare the profit and loss of each stakeholder under the two scheduling modes. When SMX and XLD are in application mode 1, the profit and loss of each stakeholder are shown in Fig. 5. As seen from the figure, the profits and losses of each stakeholder vary greatly. For the city group, the benefits of flood control and river channel sedimentation reduction are negative, that is, the flood has brought substantial economic losses and led to serious siltation of the river channel. Furthermore, the negative benefit brought by river sedimentation is much greater than the direct loss brought by flood. Since flooding can ensure water supply and ecological base flow, these two benefits are positive. In general, the comprehensive benefits of cities in Henan province are far lower than those in Shandong province, resulting from the superposition of three main reasons. First, the flood has brought more inundation losses to cities in Henan province. Secondly, the flood has seriously silted the rivers in the Henan section. Finally, the water consumption of cities in Shandong province is greater than that of Henan province. Therefore, the former two bring more negative benefits to Henan province, while the third one brings fewer positive benefits to Henan province. This further makes the comprehensive benefit of DY, Shandong Province, the largest is 3.25E + 09 yuan. However, the comprehensive benefit of JZ in Henan Province is the minimum of − 3.48E + 09 yuan.

Fig. 5
figure 5

Profit and loss of each stakeholder under application mode 1

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As for the reservoirs, since SMX is in the late stage of operation, the sediment in the reservoir area can be discharged by open discharge, so its power generation and sediment discharge benefits are positive, but the benefit value is relatively small. For XLD, its power generation benefit is positive, far more significant than SMX. However, the flood caused a lot of settlement in the internal area of XLD, leading to its overall negative benefits.

When SMX and XLD are in application mode 2, the profit and loss of each stakeholder are shown in Fig. 6. Under this operation mode, the overall trend of multi-dimensional benefits is consistent with operation mode 1, whether for the city group or reservoir group. Although there are minor fluctuations in various benefits, it does not affect the order of magnitude and positive and negative conditions of multi-dimensional benefits. It is worth noting that compared with application mode 1, the flood loss of the city under application mode 2 is increased, and the deposition of some downstream channels has brought more negative benefits. However, the ecological benefits of cities have increased, and the water supply benefits are flat. For the reservoir, the increase of the benefit value of sand discharge in the reservoir under application mode 2 is more significant than the decrease of the benefit value of power generation. The city group is generally more inclined to application mode 1, while the reservoir group is more inclined to application mode 2.

Fig. 6
figure 6

Profit and loss of each stakeholder under application mode 2

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According to the subsystem divided in Table 1, the changes in stakeholders' benefits under the two modes are shown in Figs. 7, 8 and 9. The socio-economic subsystem (Fig. 7) includes the benefits of water supply in cities and the benefits of electricity generation in reservoirs. Under the two application modes, urban water supply can be guaranteed, so the benefits of water supply remain unchanged. Due to the limitation of installed capacity, the power generation benefit of SMX is basically unchanged. The benefit of XLD power generation is far greater than the economic benefit of other stakeholders, and fluctuates under the two modes. The application of mode 1 generates 1.35E + 08 yuan more than that of mode 2. Therefore, the benefits of XLD power generation in the socio-economic subsystem should be focused on in the process of flooding.

Fig. 7
figure 7

Benefit change of socio-economic under two application modes

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Fig. 8
figure 8

Benefit change of flood control and sand transport under two application modes

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Fig. 9
figure 9

Benefit change of ecological environment under two application modes

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Flood control and sand transport subsystem includes flood control, sediment reduction of the river channel, and sediment discharge of the reservoir. Under the two application modes, the benefits of flood control and the benefit of the sediment reduction of the river channel are inconsistent in the direction of change, so the city stakeholders have different preferences in the two application modes (Fig. 8). Under application mode 1, XLD lost 3.08E + 09 yuan more than in application mode 2, and SMX lost 2.82E + 06 yuan more than in application mode 2. Therefore, the reservoir stakeholders are consistent in their preference for application mode 2.

This paper mainly discusses the impact of reservoir operation on the downstream river ecological environment, so it is only directly related to the preferences of city stakeholders. The ecological environment subsystem mainly includes the benefits of the five dimensions in Table 1. As the ecological environment benefit is affected by the discharge, the application mode 2 can bring better ecological flow to the downstream river channel than application mode 1, so the city stakeholders prefer application mode 2.

Analyzing the changes in multi-dimensional benefits of stakeholders under the two application modes is to make a reasonable allocation of gains. This means the question of what to divide and how to divide. As mentioned above, when stakeholders’ preferences are inconsistent, they cannot directly add the benefits under the lower state to the quota as their final benefits, which does not meet the principle of fairness. Therefore, adjusting the gain, that is, the added value plus the income difference of individuals with poor system benefits but high individual benefits is necessary. Allocate the revised gain, and take the individual allocation plus the corresponding benefit in the lower state as the final benefit of the stakeholders, which will help the absolute benefit of the stakeholders to be between the corresponding benefits in various states.

Conclusions

The Yellow River is a world-famous river containing sand, with less water and more sand. In particular, the lower reaches of the Yellow River are facing huge flood risks. The scientific regulation of reservoirs can fundamentally ensure the safety of flood control, water supply and the ecological environment of cities along the Yellow River. Reservoirs usually have multiple competing objectives. The priority of various purposes can be weighed by establishing a multidimensional benefit calculation method. This article comprehensively introduces the multi-dimensional advantages of SMX and XLD under different joint application modes. The results show that inundation loss and sediment deposition are the biggest threats facing the lower Yellow River under typical flood conditions. Therefore, the technical innovation of water and sediment regulation should be further strengthened to reduce the adverse effects of sediment deposition in the Yellow River. At the same time, it is necessary to improve the flood control capacity of the lower reaches of the Yellow River by increasing the dam height or speeding up the construction of the planned reservoirs of the Yellow River, to reduce the losses caused by floods.

During the flooding process, reservoir stakeholders and city stakeholders have different preferences for the joint application of reservoirs, so it is necessary to redistribute the interests of stakeholders. In recent years, based on the people-oriented river management concept, the river basin management department has made every effort to ensure that there is no overbank flood in the lower reaches of the Yellow River, which is consistent with the recommendations of this study, that is, it is more reasonable to use application mode 1 in the flooding process, which also requires the downstream cities to make fair compensation for the loss of sediment deposition in the reservoir to ensure the long-term stability of the Yellow River.

The multi-dimensional benefit calculation method and value-added distribution method established in this paper not only achieve a win–win situation for all stakeholders, but also provide direction and guarantee for the optimal joint operation of reservoirs. However, this paper only compares two reservoir group application models. How to find a better reservoir application mode based on the research results of this paper will be the critical problem to be studied in the next step.