1 Introduction

Reservoir operation is significant for management of water resources and sustainable socioeconomic developments (Tan et al. 2017; Xu et al. 2021). The control of flood limited water level (FLWL) is a significantly productive method of reservoir operation.

FLWL is an important index to compromise the risk of flood and the benefit of floodwater use. It is mainly determined by reservoir regulation based on the biggest floods. However, in actual operation, most of the floods are medium or small. This means that the storage capacity, which is originally vacated for large floods, is only used to store the medium or small floods, and thus the rest of the flood control capacity is wasted. In China, the floods in most watersheds show a strong seasonal variation due to the monsoon climate, varying significantly throughout the whole flood season. If the entire flood season operates with an annually fixed FLWL, it inevitably causes a low FLWL, which would lead to waste of floodwater resources. To solve these problems, the entire flood season should be better divided into several sub-seasons according to the seasonal variations of flood. On the basis of flood season segmentation, FLWL of each stage is determined respectively. It cannot only ensure the flood control safety in the flood season but also improve the benefit to a greater extent.

The study of flood season segmentation (Jiang et al. 2019) has gone through a process of in-depth research. Advances primarily involve the techniques of statistical method, genesis analysis method (Singh et al. 2005), fuzzy clustering method (Ju et al. 2020), fractal theory (Li et al. 2018), relative frequency and directional statistics method (Cunderlik et al. 2004), and so on. Owing to the seasonality of floods, the beginning and end of the flood season have the characteristics of periodic change. The circular distribution method (Villarini 2016) is a method to analyze the phenomenon of periodic change and find its internal law. It has the characteristics of simple and convenient calculation and flexible analysis. It also takes the flood magnitude into account, and its results are more accurate than other methods. Besides, the relative frequency method is used to recheck the results of the circular distribution method. Although its result cannot be as precise as the other method down to a day, it is simple and requires less data. Using 62-year daily discharge data, we select the circular distribution method and the relative frequency method to segment the flood season in this study.

On the premise of meeting the downstream flood control safety, the efficiency of floodwater use has improved through adjustment of FLWL (Huang et al. 2017; Liu et al. 2019). There have been many studies on how to realize effective control of FLWL. The staged control of FLWL is proved to be a more efficient measure. The widely used methods (Jiang et al. 2015) mainly include fuzzy set analysis method (Ruan and Singh 2008) and multi-objective optimization method (Xie et al. 2018) etc. Researchers have made extraordinary achievements in producing suitable operating methods (Ma et al. 2020; Mo et al. 2022; Wu and Zhong 2018), whereas most of the existing studies do not separately consider FLWL at each stage. Generally, in the pre-flood season, to meet the storage requirement of an incoming flood, the decision-makers would mainly focus on flood safety control and thus prefer risk to benefit, resulting in a lower FLWL. Conversely, in the post-flood season, the decision-makers would prefer benefit to risk as the concern of water supply increases, expecting a higher FLWL. Therefore, FLWL in different stages should be considered respectively to compromise the benefit and risk.

In this paper, on the basis of flood segmentation, we establish a set of benefit and risk indexes of FLWL control following the game theory and a multi-objective cooperative decision-making model of risk and benefit. Taking Xianghongdian (XHD) reservoir as an example, we choose FLWL in the main flood season as the original design with a focus on the pre-flood and post-flood seasons according to the goal of flood control and design of flood hydrograph. Starting with the Nash negotiation theory, we obtain the optimal FLWL scheme under different preferences between risk and benefit in each stage.

2 The Division of Flood Season

2.1 Circular Distribution Method

2.1.1 Concept

Circular distribution method transforms the periodic data into linear through trigonometric function transformation. Treating the flood events as a vector, it calculates the vector eigenvalues to determine the flood season segmentation (Chen et al. 2010).

2.1.2 Computing Method

Assuming that the length of the calculation period is T, the occurrence time of the flood peak of the i-th flood sample is Di, and the magnitude is qi, the equations of the coordinate value (xi, yi) of flood occurrence time are:

$$\left( {x_{i} ,y_{i} } \right) = \begin{cases} \left( {\cos \alpha_{i} ,\sin \alpha_{i} } \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} without \, considering \, the \, magnitude \, of \, flood \hfill \\ \left( {q_{i} \cos \alpha_{i} ,q_{i} \sin \alpha_{i} } \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \, considering \, the \, magnitude \, of \, flood \hfill \\ \end{cases}$$
(1)

and

$$\left( {\overline{x},\overline{y}} \right) = \left( {\sum\limits_{i = 1}^{N} {x_{i} /N,\sum\limits_{i = 1}^{N} {y_{i} /N} } } \right)$$
(2)

where N is the sample size, and \({\alpha }_{i}={D}_{i}\frac{2\pi }{T}\) is the occurrence time of the i-th flood.

The concentration period \(\overline{\alpha }\) and concentration r are respectively:

$$\overline\alpha=\left\{\begin{array}{ll}\arctan\;\overline y/\overline x&\overline x>0,\;\overline y>0\\2\mathrm\pi+\arctan\;\overline{\mathrm y}/\overline{\mathrm x}&\overline x>0,\;\overline y<0\\\mathrm\pi+\arctan\overline{\mathrm y}/\overline{\mathrm x}&\overline x<0\\\mathrm\pi/2&\overline x=0,\;\overline y>0\\3\mathrm\pi/2&\overline x=0,\;\overline y<0\\\mathrm{Uncertainty}&\overline x=0,\;\overline y=0\end{array}\right.$$
(3)

and

$${\text{r}} = \begin{cases} \sqrt {\overline{x}^{2} + \overline{y}^{2} } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} without \, considering \, the \, magnitude \, of \, flood{\kern 1pt} {\kern 1pt} {\kern 1pt} \hfill \\ {{\sqrt {\overline{x}^{2} + \overline{y}^{2} } } \mathord{\left/ {\vphantom {{\sqrt {\overline{x}^{2} + \overline{y}^{2} } } {\overline{q}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \, considering \, the \, magnitude \, of \, flood}}} \right. \kern-\nulldelimiterspace} {\overline{q}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \, considering \, the \, magnitude \, of \, flood}} \hfill \\ \end{cases}$$
(4)

where \(\overline{q }\) is the mean flow value of N samples, and the concentration day corresponding to the concentration period \(\overline{\alpha }\) is \({D}_{i}=\overline{\alpha }\frac{T}{2\pi }\).

Concentration r describes the central tendency of αi in the circular distribution with 0 ≤ r ≤ 1. When r is closer to 1, it means that the flood occurrence time is more concentrated in a specific area, not vice-versa. s is the standard deviation of αi. The relationship between r and s is:

$$s = \sqrt { - 2\ln r}$$
(5)

where s is an index indicating the discrete trend of the flood occurrence time. Then the starting and ending dates D1 and D2 of the main flood season are, respectively:

$$D_{{1}} = \frac{{\overline{\alpha } - s}}{2\pi }T$$
(6)
$$D_{{2}} = \frac{{\overline{\alpha } + s}}{2\pi }T$$
(7)

2.2 Relative Frequency Method

The entire flood season is divided into k periods, and the amount of the maximum flood in each period is counted. The equation for the relative frequency of each period is:

$$RF_{k} = \frac{{b_{k} }}{n}$$
(8)

where RFk is the relative frequency of the k-th period, bk is the number of annual maximum floods in the k-th period, and n is the number of years of flow data.

If each period is 10 days, the length of each period of a month is different. To make each period has the same length, the period needs to be adjusted by multiplying a coefficient. For the last ten-day of a longer month (e.g., Jan. with 31 days), the last 10-day period should be multiplied by 10/11; and for the last 10-day of February, it should be multiplied by a coefficient of 10/8 (10/9 for a leap year). When the algebraic sum s of the initial frequency RFk is not equal to the total of the adjusted RFk’ s’, generally each RFk’ is multiplied by the coefficient s/s’, that is, the adjusted relative frequency is:

$$RF_{k}^{^{\prime}} = RF_{k} \cdot \frac{s}{{s^{^{\prime}} }}$$
(9)

3 Analysis of Risk and Benefit

The flood season staged operation is to take different FLWL control scheme and schedule strategy for each segmentation. To raise the FLWL for improving the benefits of water conservancy while ensuring the flood control safety for the project and the downstream, we first conduct a cooperative decision-making analysis of risk and benefit.

3.1 Risk Analysis

During reservoir operation, events that are not conducive to the dam safety, threaten the flood control safety of downstream, and affect the benefit of reservoir will happen. The occurrence probability and loss due to these events is called flood control risk (Zhou and Guo 2014). The flood control risk rate in the extreme state is referred to as the extreme risk rate. With the FLWL Z0 as the initial water level, the design flood hydrograph of each frequency is calculated for flood routing. Suppose that Zm is the highest water level of flood routing for the design flood hydrograph at a certain frequency. When Zm is exactly equal to a selected safe level, the frequency is the extreme risk rate Plim. We take the design flood level Zd as the selected safe level. The extreme risk rate can be expressed by the following equation:

$$P_{\lim } = P\left\{ {Z_{m} \ge Z_{d} } \right\}$$
(10)

3.2 Benefit Analysis

Under the premise of ensuring the flood control safety, the FLWL can be raised as much as possible. Therefore, the utilization benefit of hydropower generation and water supply are improved accordingly.

3.2.1 Water Supply Benefit

As the FLWL rises, the added water supply capacity mainly comes from the increased storage of abandoned water. In dry years, there is no water to store after the flood season, and the benefits of FLWL adjustment can be clearly reflected. On the contrary, in wet years, the reservoir can be reserved to the normal water level even if FLWL is not raised. Raising the FLWL will only bring about the hydropower generation benefit caused by the increased water head. Therefore, the multi-year average water supply, which can be increased by raising FLWL, should be deducted from wet years. The added water supply volume can be obtained as:

$$\Delta V_{2} = \Delta V_{1} \times \left( {1 - \lambda } \right)$$
(11)

where ∆V2 is the increased water supply storage, and ∆V1 is the increased water storage capacity. According to the curve between water level and capacity, the increased water storage under different FLWL can be obtained. λ is the proportion of wet years.

The increased water supply benefit brought about by the adjustment of the FLWL is:

$$B_{\Delta } = \Delta V_{2} \times \delta$$
(12)

where \({B}_{\Delta }\) is the increased water supply benefit, and \(\delta\) is the water supply benefit of per cubic meter of water.

3.2.2 Hydropower Generation Benefit

Added water capacity and increased water head cause the improvement of hydropower generation benefit. The benefit brought by the increased water capacity is calculated by the average water consumption rate of per unit electric energy, which is obtained by counting the average annual hydropower generation water consumption and annual average hydropower generation in recent years. The additional electricity generated by the increased water capacity is expressed as:

$$E_{1} = \Delta V_{2} /\varphi$$
(13)

where E1 is the additional electricity, ∆V2 is the added water supply storage, and \(\varphi\) is the average water consumption rate of per unit electricity.

The benefit generated by increased water head is expressed by the following equation:

$$E_{2} = 9.81 \cdot \eta \cdot Q_{T} \cdot \Delta H \cdot \Delta T$$
(14)

where E2 is the additional electricity generated by increased water head, η is the efficiency of the hydropower station, η ∈ (0,1), QT is the maximum hydropower generation flow, ΔH is the increased water head caused by the adjustment of the FLWL, and ΔT is the calculation time length, here refers to the time length for benefit analysis.

4 Design of Multi-objective Cooperative Decision-making Model

4.1 Establishment of Multi-objective Cooperative Decision-making Model

In the efficient use of reservoirs, flood season operation is a typical multi-objective decision-making problem. Assuming that the decision-making system has n alternative schemes to choose from, the pros and cons of n schemes are judged according to the characteristic values of m indexes. Then the eigenvalue matrix of m indexes of each scheme is:

$$X{ = }\left[ {\begin{array}{*{20}c} {x_{11} } & {x_{12} } & \cdots & {x_{1m} } \\ {x_{21} } & {x_{22} } & \cdots & {x_{2m} } \\ \vdots & \vdots & \ddots & \vdots \\ {x_{n1} } & {x_{n2} } & \cdots & {x_{nm} } \\ \end{array} } \right] = \left\{ {x_{ij} } \right\}$$
(15)

where xij is the eigenvalue of the j-th index of the i-th scheme, 1 ≤ i ≤ n and 1 ≤ j ≤ m.

In the actual decision-making, indexes can be divided into three categories: one is benefit index, the second is the risk index, and the third is an intermediate index. Due to the different units among these indexes, each index must be normalized.

For the benefit index, the equation of the relative membership degree of the evaluation index is (Fang et al. 2019):

$$r_{ij\max } = \frac{{x_{ij} - x_{i\min } }}{{x_{i\max } - x_{i\min } }}$$
(16)

For the risk index, the equation is:

$$r_{ij\min } = \frac{{x_{i\max } - x_{ij} }}{{x_{i\max } - x_{i\min } }}$$
(17)

`For the intermediate index, the equation is

$$r_{ijmid} = \left\{ \begin{array}{clc} 1 - \frac{{x_{i}^{^{\prime}} - x_{ij} }}{{x_{i}^{*} }} & x_{ij} < x_{i}^{^{\prime}} \\ 1 & \quad x_{i}^{^{\prime}} \le x_{ij} \le x_{i}^{^{\prime\prime}} \\ 1 - \frac{{x_{ij} - x_{i}^{^{\prime\prime}} }}{{x_{i}^{*} }} & x_{ij} > x_{i}^{^{\prime\prime}} \hfill \\ \end{array} \right.$$
(18)

where \(\left[{x}_{i},{x}_{i}\right]\) is the best stable interval of xij. In Eqs. (16) and (17), ximax and ximin are respectively the largest and smallest eigenvalues in the j-th (1 ≤ j ≤ m) index set, and \(x_{i}^{*} = \max \{ x_{i}^{\prime } - x_{i\min } ,x_{i\max } - x_{i}^{\prime \prime } \}\).

Then, we obtain the relative superior membership degree matrix R:

$${\text{R = }}\left[ {\begin{array}{*{20}c} {r_{11} } & {r_{12} } & \cdots & {r_{1m} } \\ {r_{21} } & {r_{22} } & \cdots & {r_{2m} } \\ \vdots & \vdots & \ddots & \vdots \\ {r_{n1} } & {r_{n2} } & \cdots & {r_{nm} } \\ \end{array} } \right] = \left\{ {r_{ij} } \right\}$$
(19)

Therefore, m objectives can be divided into p benefit indexes and (m-p) risk indexes. Then the problem can be described as:

$$\begin{gathered} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \max F\left( {u_{1} \left( x \right),u_{2} \left( x \right)} \right) \hfill \\ s.t.\;g_{k} \left( x \right) \ge 0,k = 1,2, \cdots ,T \hfill \\ \end{gathered}$$
(20)

where F(x) is the objective function, and T is the number of constraint conditions. The multi-attribute utility functions u1(x) and u2(x) are:

$$u_{1}^{j} \left( x \right) = \omega_{1} r_{1j} + \omega_{2} r_{2j} + \cdots + \omega_{p} r_{pj}$$
(21)
$$u_{2}^{j} \left( x \right) = \omega_{p + 1} r_{{\left( {p + 1} \right)j}} + \omega_{p + 2} r_{{\left( {p + 2} \right)j}} + \cdots + \omega_{m} r_{mj}$$
(22)

A decision maker can thus obtain the utility winning function by assigning weights to the multiple indexes. Based on the above, we construct a multi-objective cooperative decision-making model for reservoir operation in flood seasons.

4.2 Solution of Multi-objective Cooperative Decision-making Model

We consider the problem of cooperative decision-making of two index sets, benefit index set DM1 and risk index set DM2.. Let N = {1,2} be the sequence number of index sets. For any i ∈ N, there are ki indexes in the i-th decision indexes set.

$$f_{i} \left( {x_{1} ,x_{2} } \right) = \left( {f_{i1} \left( {x_{1} ,x_{2} } \right),f_{i2} \left( {x_{1} ,x_{2} } \right), \cdots ,f_{{ik_{i} }} \left( {x_{1} ,x_{2} } \right)} \right) \in G_{i}$$
(23)

where Gi \(\subset R^{{k_{i} }}\) is the reachable target set, xi ∈ Ωi \(\subset R^{{m_{i} }}\) is the decision vector, \(R^{{k_{i} }}\) and \(R^{{m_{i} }}\) are ki-dimensional and mi-dimensional real vector spaces, respectively. The winning function of the index set is represented by the utility function. The utility function of the i-th indicator set is:

$$u_{i} \left( {x_{1} ,x_{2} } \right) = u_{i} \left( {f_{i} \left( {x_{1} ,x_{2} } \right)} \right)$$
(24)

A pair of utility functions is mapped into a subset E of the two-dimensional Euclidean space R2, and \({E}\subset {{R}^{2}}\) represents the joint reachable utility set.

$$E = \left\{ {\left. {\left( {u_{1} \left( {f_{1} } \right),u_{2} \left( {f_{2} } \right)} \right) \in R^{2} } \right|} \right.\left. {f_{i} \left( {x_{1} ,x_{2} } \right) \in G_{i} ,x_{i} \in \Omega_{i} ,i = 1,2} \right\}$$
(25)

If E is a closed bounded convex set, d = (d1,d2) ∈ E is the current point. The relationship between E and d is called a multi-objective negotiation decision-making problem. If the decision maker assigns d1 > d2, it means that the index set DM1 has greater initiative in the process of negotiation, and the final negotiation result will also be biased towards the index set DM1 and vice versa. Suppose that h(E,d) ∈ E is the solution of (E,d), let h*(E,d) = u*, for all u ∈ E, u > d and u ≠ u*,

$$\left( {u_{1}^{*} - d_{1} } \right)\left( {u_{2}^{*} - d_{2} } \right) > \left( {u_{1} - d_{1} } \right)\left( {u_{2} - d_{2} } \right)$$
(26)

At this time, h*(E,d) = u* becomes the Nash negotiation solution to the multi-objective problem.

If no feasible scheme is exactly the same as the Nash negotiation solution, the fuzzy pattern recognition theory is used to determine the closest, which is the optimal.

5 Application

5.1 Overview of the Study Reservoir

XHD reservoir on the West source of the Pi River Basin, which is one of the upper tributaries of the Huai River in China, is selected as the study object. The river basin is an area affected by the subtropical monsoon climate and frequently subjected to rainstorms and floods. The average annual precipitation in the Pi River basin is 1334 mm. XHD reservoir is a large size, 1 year regulating reservoir with a watershed area of 1400 km2. The Pi River basin and the location of XHD reservoir are shown as Fig. 1.

Fig. 1
figure 1

Maps of the Pi River basin and XHD reservoir location

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The annual average runoff of the reservoir is 33.1 m3/s. The main task of the reservoir is flood control, and the comprehensive utilization of water supply and hydropower generation are taken into account. The design flood control level is 139.10 m, the corresponding storage capacity is 2.228 billion m3, and the check flood control level is 143.60 m. The maximum hydropower generation flow of the power station is 200 m3/s.

The flood season is from May 1st to October 1st, and the designed FLWL is 125.0 m. There is 500 million m3 of flood control capacity above 125.0 m that is responsible for downstream flood control and peak reduction tasks. When the reservoir water level exceeds 132.60 m, the discharge tunnel is fully opened, with a maximum discharge volume of 618 m3/ s.

5.2 Segmentation of Flood Season

The restored daily inflow of the reservoir during the flood season of years 1956–2017 was used as the basic data for calculation. That is, the sample size N = 62 years, and the length of the calculation period T = 154 days.

5.2.1 Segmentation of Flood Season by Circular Distribution Method

The samples were taken according to different time intervals t (t = 1, 3, 5, 7, 10, 15) to stage the flood season. Firstly, the maximum discharge qLt (L = 1, 2,…, n) of t days is selected from the L-th-year daily discharge in the flood season. Then, various indexes of flood situation are calculated by Eqs. (1)–(7). Table 1 and Fig. 1 denote the results of segmentation of different sampling intervals.

Table 1 The results of flood season segmentation by circular distribution method
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The segmentation is obtained mainly based on the results of the maximum t-day discharge (t = 1, 3, 5) considering the flood magnitude. Table 1 indicated that the calculated main flood season is from June 10 to August 17.

Figure 2 shows the occurrence time of the maximum discharge, which are expressed in angle. The distribution of data points considering and ignoring the magnitude of the flood is expressed by the coordinate value of the occurrence time multiplied by the magnitude. Obviously, the points considering the magnitude are concentrated in the period from June 10 to August 17.

Fig. 2
figure 2

The vector distribution of maximum flood occurrence time by the circular distribution method

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Based on the above, combining the actual situation of the reservoir, the main flood season of the reservoir is gained as June 10—August 17.

5.2.2 Segmentation of Flood Season by the Relative Frequency Method

The whole flood season is divided into 15 periods (October 1 is classified as the late September). In each period, the occurrence times of daily maximum discharge over the years 1956–2017 are counted to calculate the relative frequency. The results are shown in the following table.

It can be seen from the Table 2 that the relatively high frequency is mainly concentrated in the first ten days of June to the middle of August, which is consistent with the flood season segmentation results by the circular distribution method.

Table 2 The results of flood season segmentation by relative frequency method
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5.3 Analysis of Risk and Benefit

5.3.1 Risk Analysis

Several typical flood hydrographs are selected in the pre-flood season and post-flood season severally, and the flood hydrographs with different design frequencies are derived. Considering the flood situation in the reservoir area, the values of frequency P are selected as 0.01%, 0.10%, 0.20%, 1%, 2%, 3.30%, 5%, 10% and 20% respectively. For the selection of the dynamic domain of FLWL, considering that the original FLWL of the reservoir is 125.0 m, and the normal water level is 130.0 m, the FLWLs start at 125.0 m and increase at a step of 0.5 m to a maximum of 130.0 m.

The flood hydrograph that is most unfavorable for the reservoir flood control is selected. According to the design results of the reservoir, in the pre-flood season, the design flood peak is much smaller than that in the main flood season, but the flood volume is large and the peaks are dense. The flood in 1974 with peak discharge of 3,938.84 m3/s is selected as the typical. In the post-flood season, the design flood is lighter than that in the main flood season. While compared with the pre-flood season, it has a high peak, large volume and many fluctuations. The flood in 2005 with peak discharge of 7,942.69 m3/s is chosen as the typical.

Then, taking Zi of each scheme as the starting water level, the time interval is 3 h, flood regulation is carried out according to the predetermined dispatching rules. The relationship between the highest flood regulation water level Zm(Zi) and the corresponding frequency P(Zi), i = 1, 2, …, n is obtained.

Taking the designed flood control level (139.10 m) as the selected safe level, the frequency when the flood regulation upper water level starting from each FLWL is exactly equal to the selected safe level is obtained. The results of different schemes and corresponding extreme risk rates are displayed in Table 3.

Table 3 Extreme risk rate under different flood limited water levels in the flood season
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Figure 3 presents the relationship curve between different FLWLs Zi and the corresponding extreme risk rate Plim.

Fig. 3
figure 3

Extreme risk rate under different flood limited water levels

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5.3.2 Benefit Analysis

XHD reservoir supplies water mainly including industrial water, agricultural water, domestic water and ecological water, of which 70% is used for agricultural water. According to relevant research, the reservoir water supply benefit generated by per cubic metre water is 35.40 yuan (Zheng 2020). The hydropower generation operation is subject to the flood control and irrigation operation. According to the actual operation data of the hydropower station (Yu 2010), the average water consumption rate per unit electric energy was 9.786 m3.

In the pre-flood season, to meet the storage requirements of the incoming flood, the reservoir needs to empty its water in advance. The benefit of raising the FLWL only considers the hydropower generation, which positively influenced by the increase of average water head. Equation (14) is used to calculate the increased hydropower generation benefit.

In the post-flood season, the reservoir begins to store water to improve the utilization benefit. Therefore, the benefits include not only the hydropower generation benefit caused by increased water head and volume, but also the water supply benefit. According to the measured hydrological data of the watershed over the years (Hu 2015), the proportion of wet years that can meet the water supply needs without raising the FLWL is 25%. Deducting the proportion of these years, the actual increased water supply benefit by raising the FLWL can be obtained. Then Eq. (12) is used to calculate the water supply benefit. Next, Using Eqs. (13) and (14), the hydropower generation benefit caused by increased water head and volume can be calculated. Table 4 presents the benefit results in the pre-flood season and the post-flood season, respectively.

Table 4 Hydropower generation benefit and water supply benefit of different schemes in the pre-flood season and post-flood season
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5.4 Nash Negotiation Solution

The value of each index is standardized by Eqs. (16) and (17). Considering that the reservoir focuses on flood control and water supply, supplemented by hydropower generation, the weight of the two benefit indexes, namely water supply and hydropower generation, is (0.6, 0.4). The utility payoff function values u1 and u2 are calculated by Eqs. (21) and (22). Table 5 shows the solution of payoff function values of each scheme.

Table 5 DM1 and DM2 payoff function values of each FLWL scheme
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Firstly, it is assumed that the winning function values of the two index sets are both 0, that is, the current point d = (0,0). In order to obtain the joint reachable utility set E of the two index sets, the winning function values (u1,u2) of each flood season are marked, as shown in Fig. 4 below. For u1 and u2, the maximum value of the product should appear in the range where the values of the two variables are closest, corresponding to the lines of AB and BC in Fig. 4. Then the problem of Nash negotiation is transformed into the following nonlinear programming problems. The product value is optimized during the two lines respectively, and the objective function is as follows.

Fig. 4
figure 4

Joint reachable utility set in pre-flood season and post-flood season

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For pre-flood season:

$$\begin{array}{c} \max \; u_{1} \cdot u_{2} \\ \begin{aligned} s.t. \; & 0 \le u_{2} \le 1 \\ & 0.927u_{1} + u_{2} \le 1.285 \\ & 1.581u_{1} + u_{2} \le 1.722 \end{aligned}\end{array}$$

For post-flood season:

$$\begin{array}{c} \max \; u_{1} \cdot u_{2} \\ \begin{aligned} s.t. \; & 0 \le u_{1} \le 1, \; 0 \le u_{2} \le 1 \\ & 0.842u_{1} + u_{2} \le 1.252 \\ & 1.493u_{1} + u_{2} \le 1.683 \end{aligned} \end{array}$$

Based on the above, the Nash negotiation solution u1 in the pre-flood season is 0.918 and u2 is 0.669. In the post-flood season, u1 is 0.658 and u2 is 0.874. Since the two negotiation solutions are not equal to u1 and u2 of each scheme, it is necessary to adopt the fuzzy pattern recognition model and grid schedule theory to calculate the membership degree and decide the optimal scheme. Table 6 presents the membership degree of each scheme, and then the optimal water level can be determined.

Table 6 Membership degree of each scheme in the pre-flood season and post-flood season
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The membership degree is the largest when the water level in the pre-flood season is 129.0 m and in the post-flood season is 128.5 m. When d = (0,0), the FLWLs of the optimal scheme are 129.0 m and 128.5 m for the pre-flood season and the post-flood season, respectively.

The water level at different sub-seasons of the flood season should be considered for specific situations. The flood control safety is mainly considered in the pre-flood season, so that the decision preference is targeting on risk assessment. In the post-flood season, to ensure water supply, decision preference will focus on benefit assessment. A decision-maker would change the preference by changing the current point. Table 7 presents the specific results. It can be deduced that the more d2 increases, the lower the FLWL, and the better the flood control safety can be guaranteed. On the contrary, the more d1 increases, the higher the FLWL, and the more utilization benefits can be improved.

Table 7 Optimal flood limited water level under different decision preference
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6 Conclusions

  1. 1.

    We use the circular distribution method and relative frequency method to calculate the flood segmentation. The main flood season of XHD reservoir is gained as June 10—August 17. The results of circular distribution method are verified by relative frequency method, and the verification results are consistent.

  2. 2.

    We establish the index sets of risk and benefit respectively and use the Nash negotiation theorem to solve the utility payoff function of the two index sets, and, based on the game theory, we establish a multi-objective cooperative decision-making model.

  3. 3.

    Taking XHD reservoir as an example, we obtain an optimal FLWL control scheme for sub-seasons. The resulting FLWLs of the optimal scheme are 129.0 m, and 128.5 m in the pre-flood season and the post-flood season, respectively in the condition that the risk and benefit are equally valued. By adjusting the current point to reflect decision-makers' preference, the decision-makers' expectations for risk and benefit in different sub-seasons can be realized through this study.

In the future, we should deeply analyze the characteristics of rainstorm and flood in combination with the hydrological and meteorological characteristics of the basin where the reservoir is located. Based on the real-time rainfall and flow data, the FLWL of each stage more in line with the reality could be achieved.