## Abstract

There are a lot of uncertainties in the water resources system, which makes the water allocation plan very risky. In order to analyze the risks of water resources allocation under uncertain conditions, a new methodology called the stochastic simulation-based risk assessment approach is developed in this paper. First, the main hydrological stochastic variable is fitted by a proper probability distribution. Second, suitable two-stage stochastic programming is constructed to obtain the expected benefit and optimized water allocation targets. Third, the Monte Carlo method is used to obtain a suitable stochastic sample of the hydrological variable. Fourth, a pre-allocated water optimization model is proposed to obtain optimized actual benefit. The methodology can give a way for risk analysis of water allocation plans obtained by uncertain optimization models, which provides reliable assistance to water managers in decision-making. The proposed methodology is applied to the Zhanghe Irrigation District and the risk of the water allocation plan obtained under the randomness of annual inflow is assessed. In addition, three different division methods of the annual inflow are applied in the first step, namely three levels, five levels and seven levels, respectively. From the results, the risk of the water allocation scheme obtained by the TSP model is 0.372–0.411 and decreases with the increase of the number of hydrological levels. Considering both the risk and model complexity, seven hydrological levels are recommended when using the TSP model to optimize water allocation under stochastic uncertainty.

## HIGHLIGHTS

A stochastic simulation-based risk assessment method is proposed to analyze risks in water allocation plans obtained by uncertain models.

The suitable number of hydrological levels is revealed for using TSP model to optimize water allocation under stochastic uncertainty.

### Graphical Abstract

## INTRODUCTION

As one of the necessary natural resources, water plays an important role in human life and socio-economic development. However, because of population growth and economic development, limited water resources are no longer enough to meet increasing water demand (Mapani *et al.* 2016). The resulting water shortage has become a restrictive factor affecting social–economic development in some regions (Kang & Cao 2012). So, various methods should be taken to develop sound management plans to improve water utilization efficiency (Wheater & Gober 2015). Optimization methods can be divided into two categories: deterministic methods and uncertain methods. Deterministic optimization methods do not take into account the various uncertainties that exist in the water resources system. However, uncertain optimization methods can handle one or more uncertainties, such as randomness, imprecision and ambiguity. Among all the uncertainties, the randomness of runoff and rainfall is the most important uncertainty factor affecting water resources allocation.

Given that deterministic optimization methods have difficulties in handling the stochastic uncertainty that exists in water management systems, various uncertain optimization methods have been developed over the past decades. Stochastic dynamic programming was employed to optimize the operation of reservoirs, crop irrigation and so on (Alaya *et al.* 2003; Cervellera *et al.* 2006; Davidsen *et al.* 2015). Then, chance constraint programming was also developed for collaborative management of water quantity and quality (Birhanu *et al.* 2014; Grosso *et al.* 2014; Kaviani *et al.* 2015). In addition, two-stage stochastic programming and multi-stage stochastic programming were successively developed and applied to optimize urban water supply, hydropower scheduling and agricultural planting (Wang & Huang 2011; Fu *et al.* 2018; Guo *et al.* 2019). Among all the above methods, two-stage stochastic programming (TSP) is considered to be a proper, sample method to optimize water resources allocation under stochastic uncertainty. For example, Ortiz-Partida *et al.* (2019) formulated a TSP model to optimize a monthly set of single reservoir releases under the randomness of the streamflow. Khosrojerdi *et al.* (2019) coupled the TSP model and interval-parameter programming to optimize water resources allocation between multi-reservoir and multi-user in Iran.

In TSP, a pre-decision is made to maximize the ‘benefits’ before the stochastic events have occurred; then, a second decision is undertaken to minimize ‘penalties’ due to the target not being met after the value of the random variable is known (Huang & Loucks 2000). However, the method is optimized using expectation as an evaluation criterion in the second stage, which indicates that it is a kind of expected risk method. So, conditional value-at-risk has been employed and integrated into uncertain optimization models to reduce the risk (Shao *et al.* 2011; Syme 2014; Hu *et al.* 2016). The risk of a water resource allocation scheme obtained by an uncertain method is lower than that obtained by a deterministic method. Uncertain methods still have some certain risks, making the water resources allocation scheme unable to achieve the expected results. To provide reliable assistance to water managers in decision-making, risk analysis should be carried out for the water allocation scheme obtained by uncertain optimization models like the TSP model. So far, many studies have been conducted to analyze the risks in water management plans obtained by deterministic optimization models (Gu *et al.* 2009; Lambourne & Bowmer 2013). However, there are few studies on risk assessment of water resource management plans obtained by uncertain methods like TSP model.

In this paper, a new methodology named the stochastic risk assessment method is proposed to carry out risk analysis for water allocation schemes obtained by the TSP model. The methodology mainly includes four contents: (1) uncertain optimization – a proper TSP model is constructed to obtain the expected benefit and optimized water allocation targets after probability distribution analysis of the hydrological random variable; (2) stochastic simulation – a suitable stochastic sample of the random variable is obtained through the Monte Carlo method; (3) deterministic optimization – a pre-allocated water optimization model is formulated to obtain the actual benefit based on the optimized allocation targets as input; (4) risk assessment - the risk is calculated through statistical analysis. The developed methodology is used for a water allocation plan obtained under uncertainty in the Zhanghe Irrigation District.

## RESEARCH METHODOLOGY

To carry out risk analysis of a water allocation scheme obtained by the TSP model under uncertainty, a stochastic simulation-based risk assessment (SSRA) method is developed (see Figure 1). The methodology mainly includes four contents: create a suitable two-stage stochastic programming model, carry out stochastic simulation, create a proper pre-allocated water optimization model, and carry out risk assessment, which will be described separately as follows.

### Two-stage stochastic programming model

*ξ*is the selected main hydrological random variable in the water allocation system. After fitting the suitable probability distribution of the hydrological random variable, divide

*ξ*into

*Nr*levels and let

*ξ*take on value

*ξ*with probability for . Thus, following the simplified method by Huang & Loucks (2000), the TSP model for water allocation is presented as follows:where represents the expected net benefit of the system (yuan), yuan represents the legal currency of China;

_{k}*i*represents a water user;

*Nu*represents the number of water users participating in the allocation; represents the water allocation target for user

*i*(m

^{3}), which is the decision variable in the first stage; represents the net benefit of water allocated for user

*i*(yuan/m

^{3}); represents the water shortage for user

*i*under level

*k*(m

^{3}), which is the decision variable in the second stage; represents the penalty of water shortage for user

*i*(yuan/m

^{3}); represents the available water of the system under level

*k*(m

^{3}); represents the maximum water demand of user

*i.*After model–solving, let represent the optimized water allocation target for user

*i*.

### Stochastic simulation

The Monte Carlo method is a well-known method focused on stochastic modeling (Golasowski *et al.* 2015) and is used for stochastic simulation of the hydrological random variable. There are three steps to achieve the goal. Firstly, 10,000 stochastic samples with 1,000 simulated values in each sample can be obtained after fitting a suitable probability distribution of the random variable. Each simulated value is a possible value of the random variable. After that, probability distribution analysis is carried out for each sample to obtain its distribution parameters. Finally, the sample with the most suitable distribution parameters is chosen as the optimized simulation sample and the simulated values in the sample are used as the inputs for the following calculation.

### Pre-allocated water optimization model

*j*(yuan); represents the optimized water allocation target to user

*i*obtained by Equation (1) (m

^{3}); represents the water shortage for user

*i*under a simulated value

*j*(m

^{3}), decision variable; represents the available water of the system under a simulated value

*j*(m

^{3}).

### Risk assessment

*Nr*represent the number of times that is less than . The risk of the water allocation scheme (

*R*) obtained by the TSP model can be calculated by Equation (3):where represents the total number of the simulated values.

## Case study

### Problem description

Zhanghe Irrigation District (ZID) is located in the middle reaches of the Yangtze River. It can be divided into seven subregions with some small and medium reservoirs in each subregion (see Figure 2). In addition, Zhanghe Reservoir, which is a large reservoir, is the main irrigation water source for three main crops (semi-late rice, winter rape, and cotton) in the ZID. In order to utilize the water resources reasonably and efficiently in the ZID, the manager is responsible for making a water allocation scheme to optimize water allocation between the above main crops in each subregion every year beforehand. Considering that the annual inflow of Zhanghe reservoir (AIZR) is a stochastic variable, the TSP method can be used to obtain the water allocation scheme. However, the actual benefit from the above water allocation scheme may less than the expected benefit in some years. Thus, there is a need to carry out a risk analysis of the water allocation scheme obtained by the TSP method.

### Problem solution

The SSRA method is applied to solve the above problem. The main steps are as follows: analyze the probability distribution of the AIZR; set the hydrological level; create the TSP model; simulate the AIZR; create the PWO model; solve the model and perform statistical analysis.

#### Probability distribution analysis

In this study, the Pearson type III distribution is selected to fit the historical series of the AIZR, which has the length of 54 years (from 1963 to 2016). The Pearson type III distribution has been widely used to fit hydrological variables, such as runoff and precipitation in China. Following Wilks (1993), the normal quantile–quantile diagram of residuals (Q–Q diagram) is applied to assess the degree-of-fit. Figure 3(a) demonstrates the Q–Q diagram between the AIZR historical series and the best-fitted Pearson type III distribution, which indicates that the degree-of-fit is satisfactorily good.

#### Hydrological level setting

Based on the fitted probability distribution, the AIZR can be divided into several levels by selected percentiles as in Chen *et al.* (2021). In order to analyze the impact of the number of levels on the risk of the TSP model, three scenarios of hydrological level setting are applied as follow: (a) two percentiles (25th and 75th) are applied to divide the AIZR random variable into three levels. The probabilities are 0.25, 0.5 and 0.25, respectively; (b) four percentiles (12.5th, 37.5th, 62.5th, and 87.5th) are applied to divide the AIZR random variable into five levels. The probabilities are 0.125, 0.25, 0.25, 0.25 and 0.125, respectively; (c) six percentiles (12.5th, 25th, 37.5th, 62.5th, 75th and 87.5th) are applied to divide the AIZR random variable into seven levels. The probabilities are 0.125, 0.125, 0.125, 0.25, 0.125, 0.125 and 0.125, respectively. Under each inflow level, the inflow is an interval number and the expected inflow is calculated according to the method proposed by Chen *et al.* (2021), which is shown in Table 1.

#### Creation of the TSP model

*t*in subregion

*r*(m

^{3}), decision variable; represents the net benefit of crop

*t*in subregion

*r*(yuan/m

^{3}); represents the water shortage for crop

*t*in subregion

*r*under inflow level

*k*(m

^{3}), decision variable; represents the penalty of water shortage for crop

*t*in subregion

*r*(yuan/ m

^{3}); represents the available irrigation water supplied by Zhanghe Reservoir under inflow level

*k*(m

^{3}); represents the irrigation water allocated for subregion

*r*under inflow level

*k*(m

^{3}); represents the internal available irrigation water supplied by reservoirs in subregion

*r*(m

^{3});

*η*represents the water use efficiency in subregion

_{dr}*r*;

*η*represents the conveyance efficiency of the canal between subregion

_{cr}*r*and Zhanghe Reservoir; represents the maximum water demand of crop

*t*in subregion

*r*(m

^{3}).

#### Stochastic simulation

Based on the best-fitted probability distribution, an optimized stochastic sample with 1,000 simulated inflows is obtained by the method described in section 2.2. Then, the normal quantile–quantile plot of residuals is applied to show the degree-of-fit of the optimized stochastic sample, which is shown in Figure 3(b). It demonstrates satisfactorily good agreements between the best-fitted probability distribution and optimized stochastic sample.

#### Creation of the PWO model

*j*represents a simulated inflow; represents the actual net benefit of the ZID under a simulated inflow

*j*(yuan); represents the optimized water allocation target obtained by Equation (4); represents the pre-allocated water not satisfied for crop

*t*in subregion

*r*under a simulated inflow

*j*(m

^{3}), decision variable; represents the available water supplied by Zhanghe Reservoir under a simulated inflow

*j*(m

^{3}).

The PWO model is optimized under each simulated inflow within the optimized stochastic sample.

#### Parameter determination

In order to run the above two models, some parameters should be determined. Table 2 demonstrates the maximum demands, benefits, and penalties for three crops. The maximum water demands are calculated by multiplying the planting area and crop irrigation quotas. The benefit and penalty for each crop are approximated from socio-economic indicators, for example crop price, potential crop yield, and irrigation benefit coefficient. In addition, the available irrigation water supplied by Zhanghe Reservoir under each inflow level is obtained through expected inflow minus domestic and industrial water consumption and water loss. Table 3 shows the internal available irrigation water (), the water use efficiency (*η _{dr}*) and conveyance efficiency (

*η*) for each subregion.

_{cr}### Results analysis

#### Results of model optimization

Just as presented in section 3.2, the AIZR is converted into discrete variables under three scenarios (three-level, five-level and seven-level), respectively. For each scenario, the TSP model is first applied to obtain the expected benefit () and optimized target (). Then, the PWO model is used to calculate the actual benefit () under all 1,000 simulated inflows based on the optimized target as input. By statistical analysis of 1,000 optimization results, the frequency of each actual benefit can be calculated and is shown in Figure 4.

It can be seen that the frequency distributions of the actual benefits are similar under the three scenarios. In each plot, the frequency first increases and then decreases with the actual benefit, and finally increases to the maximum. However, when the AIZR is divided into three levels, the first peak is obtained at 400 million yuan (see Figure 4(a)). when the AIZR is divided into five levels or seven levels, the first peaks are both obtained at 450 million yuan (see Figures 4(b) and 4(c)).

#### Risks of optimized allocation schemes

According to the expected benefit and the distribution of the actual benefits, the risks of the water allocation scheme under the three scenarios are obtained by Equation (3) and the results are shown in Figure 5(a). The larger the risk, the less likely it is to reach the expected benefit. It is demonstrated that the risk of water allocation by the TSP method in the ZID is 0.372–0.411. In addition, the risk is the largest when the AIZR is divided into three levels. However, the risk is the smallest when the AIZR is divided into seven levels. The above results indicate that the more the number of dividing levels of the AIZR, the smaller the risk of the TSP method.

In addition, the difference between the risk under five levels and seven levels is much smaller than that between the risk under three levels and five levels. It is expected that the difference between the risk under seven levels and nine levels may be smaller than that between the risk under five levels and seven levels. The more the number of dividing levels of the random variable, the more the decision variables of the TSP model and the more difficult it is to solve the model. Therefore, it is recommended to divide the random variable into seven levels when using the TSP model to optimize water allocation under uncertainty.

## DISCUSSION

To investigate the reason for the above results, expected benefit and optimized targets obtained by the TSP model under each scenario are compared. The optimized water allocation target for each crop () can be written as by introducing optimized allocation coefficients (). Table 4 shows the optimized allocation coefficients of three crops obtained by the TSP model under the three scenarios. The optimized allocation coefficients under three levels are larger than that under five levels and seven levels. It indicates that in a wet year in which the available water is enough for all water users, larger benefit could be obtained under three levels than that under the other two scenarios, while in a dry year in which the available water is not enough for all water users, higher recourse cost may be achieved under three levels than that under the other two scenarios. Therefore, actual benefit at the first peak under three levels is lower than that under the other two scenarios (see Figure 4).

Inflow level . | Probability . | Inflow range (10^{4} m^{3})
. | Expected inflow (10^{4} m^{3})
. |
---|---|---|---|

Three-level scenario | |||

T 1 | 0.25 | Less than 56,199.4 | 42,934.5 |

T 2 | 0.5 | [56,199.4, 99,294.8] | 76,344.0 |

T 3 | 0.25 | More than 99,294.8 | 124,065.9 |

Five-level scenario | |||

F 1 | 0.125 | Less than 44,726.9 | 35,146.3 |

F 2 | 0.25 | [44,726.9, 65,998.3] | 55,937.5 |

F 3 | 0.25 | [65,998.3, 86,320.9] | 75,852.9 |

F 4 | 0.25 | [86,320.9, 118,549.8] | 100,283.0 |

F 5 | 0.125 | More than 118,549.8 | 140,083.6 |

Seven-level scenario | |||

S 1 | 0.125 | Less than 44,726.9 | 35,146.3 |

S 2 | 0.125 | [44,726.9, 56,199.4] | 50,722.7 |

S 3 | 0.125 | [56,199.4, 65,998.3] | 61,152.9 |

S 4 | 0.25 | [65,998.3, 86,320.9] | 75,852.9 |

S 5 | 0.125 | [86,320.9, 99,294.8] | 92,517.9 |

S 6 | 0.125 | [99,294.8, 118,549.8] | 108,048.2 |

S 7 | 0.125 | More than 118,549.8 | 140,083.6 |

Inflow level . | Probability . | Inflow range (10^{4} m^{3})
. | Expected inflow (10^{4} m^{3})
. |
---|---|---|---|

Three-level scenario | |||

T 1 | 0.25 | Less than 56,199.4 | 42,934.5 |

T 2 | 0.5 | [56,199.4, 99,294.8] | 76,344.0 |

T 3 | 0.25 | More than 99,294.8 | 124,065.9 |

Five-level scenario | |||

F 1 | 0.125 | Less than 44,726.9 | 35,146.3 |

F 2 | 0.25 | [44,726.9, 65,998.3] | 55,937.5 |

F 3 | 0.25 | [65,998.3, 86,320.9] | 75,852.9 |

F 4 | 0.25 | [86,320.9, 118,549.8] | 100,283.0 |

F 5 | 0.125 | More than 118,549.8 | 140,083.6 |

Seven-level scenario | |||

S 1 | 0.125 | Less than 44,726.9 | 35,146.3 |

S 2 | 0.125 | [44,726.9, 56,199.4] | 50,722.7 |

S 3 | 0.125 | [56,199.4, 65,998.3] | 61,152.9 |

S 4 | 0.25 | [65,998.3, 86,320.9] | 75,852.9 |

S 5 | 0.125 | [86,320.9, 99,294.8] | 92,517.9 |

S 6 | 0.125 | [99,294.8, 118,549.8] | 108,048.2 |

S 7 | 0.125 | More than 118,549.8 | 140,083.6 |

Water user . | Maximum water demand (10^{4} m^{3})
. | Benefit (yuan/m^{3})
. | Penalty (yuan/m^{3})
. | |
---|---|---|---|---|

Dongbao | Semi-late rice | 2,044.96 | 1.58 | 2.77 |

Winter rape | 213.25 | 5.49 | 9.62 | |

Cotton | 16.20 | 6.48 | 11.35 | |

Duodao | Semi-late rice | 4,698.67 | 1.50 | 2.77 |

Winter rape | 490.15 | 5.21 | 9.62 | |

Cotton | 37.40 | 6.15 | 11.35 | |

Dangyang | Semi-late rice | 3,152.37 | 1.61 | 2.77 |

Winter rape | 336.80 | 5.58 | 9.62 | |

Cotton | 50.37 | 6.59 | 11.35 | |

Jingzhou | Semi-late rice | 3,154.15 | 1.39 | 2.77 |

Winter rape | 349.31 | 4.84 | 9.62 | |

Cotton | 26.66 | 5.71 | 11.35 | |

Shayang | Semi-late rice | 19,881.18 | 1.34 | 2.77 |

Winter rape | 2,088.47 | 4.65 | 9.62 | |

Cotton | 159.05 | 5.49 | 11.35 | |

Zhanghe | Semi-late rice | 1,353.98 | 1.74 | 2.77 |

Winter rape | 141.20 | 6.05 | 9.62 | |

Cotton | 10.75 | 7.14 | 11.35 | |

Zhongxiang | Semi-late rice | 4,586.73 | 1.42 | 2.77 |

Winter rape | 478.00 | 4.93 | 9.62 | |

Cotton | 36.52 | 5.82 | 11.35 |

Water user . | Maximum water demand (10^{4} m^{3})
. | Benefit (yuan/m^{3})
. | Penalty (yuan/m^{3})
. | |
---|---|---|---|---|

Dongbao | Semi-late rice | 2,044.96 | 1.58 | 2.77 |

Winter rape | 213.25 | 5.49 | 9.62 | |

Cotton | 16.20 | 6.48 | 11.35 | |

Duodao | Semi-late rice | 4,698.67 | 1.50 | 2.77 |

Winter rape | 490.15 | 5.21 | 9.62 | |

Cotton | 37.40 | 6.15 | 11.35 | |

Dangyang | Semi-late rice | 3,152.37 | 1.61 | 2.77 |

Winter rape | 336.80 | 5.58 | 9.62 | |

Cotton | 50.37 | 6.59 | 11.35 | |

Jingzhou | Semi-late rice | 3,154.15 | 1.39 | 2.77 |

Winter rape | 349.31 | 4.84 | 9.62 | |

Cotton | 26.66 | 5.71 | 11.35 | |

Shayang | Semi-late rice | 19,881.18 | 1.34 | 2.77 |

Winter rape | 2,088.47 | 4.65 | 9.62 | |

Cotton | 159.05 | 5.49 | 11.35 | |

Zhanghe | Semi-late rice | 1,353.98 | 1.74 | 2.77 |

Winter rape | 141.20 | 6.05 | 9.62 | |

Cotton | 10.75 | 7.14 | 11.35 | |

Zhongxiang | Semi-late rice | 4,586.73 | 1.42 | 2.77 |

Winter rape | 478.00 | 4.93 | 9.62 | |

Cotton | 36.52 | 5.82 | 11.35 |

Subregion . | Internal available water (10^{4} m^{3})
. | Water use efficiency (η)
. _{dr} | Conveyance efficiency (η)
. _{cr} |
---|---|---|---|

Dongbao | 1,681.3 | 0.65 | 0.92 |

Duodao | 1,425.5 | 0.61 | 0.92 |

Dangyang | 526.1 | 0.63 | 0.96 |

Jingzhou | 519.9 | 0.64 | 0.82 |

Shayang | 1,544.3 | 0.6 | 0.83 |

Zhanghe | 757.7 | 0.66 | 0.98 |

Zhongxiang | 2,440.7 | 0.62 | 0.85 |

Subregion . | Internal available water (10^{4} m^{3})
. | Water use efficiency (η)
. _{dr} | Conveyance efficiency (η)
. _{cr} |
---|---|---|---|

Dongbao | 1,681.3 | 0.65 | 0.92 |

Duodao | 1,425.5 | 0.61 | 0.92 |

Dangyang | 526.1 | 0.63 | 0.96 |

Jingzhou | 519.9 | 0.64 | 0.82 |

Shayang | 1,544.3 | 0.6 | 0.83 |

Zhanghe | 757.7 | 0.66 | 0.98 |

Zhongxiang | 2,440.7 | 0.62 | 0.85 |

Scenario . | Sub-district . | Semi-late rice . | Winter rape . | Cotton . |
---|---|---|---|---|

Three-level | Dongbao | 1 | 1 | 1 |

Duodao | 1 | 1 | 1 | |

Dangyang | 1 | 1 | 1 | |

Jingzhou | 0.77 | 1 | 1 | |

Shayang | 0.77 | 1 | 1 | |

Zhanghe | 1 | 1 | 1 | |

Zhongxiang | 1 | 1 | 1 | |

Five-level | Dongbao | 1 | 1 | 1 |

Duodao | 1 | 1 | 1 | |

Dangyang | 1 | 1 | 1 | |

Jingzhou | 0.77 | 1 | 1 | |

Shayang | 0.77 | 1 | 1 | |

Zhanghe | 1 | 1 | 1 | |

Zhongxiang | 0.87 | 1 | 1 | |

Seven-level | Dongbao | 1 | 1 | 1 |

Duodao | 1 | 1 | 1 | |

Dangyang | 1 | 1 | 1 | |

Jingzhou | 0.77 | 1 | 1 | |

Shayang | 0.77 | 1 | 1 | |

Zhanghe | 1 | 1 | 1 | |

Zhongxiang | 0.81 | 1 | 1 |

Scenario . | Sub-district . | Semi-late rice . | Winter rape . | Cotton . |
---|---|---|---|---|

Three-level | Dongbao | 1 | 1 | 1 |

Duodao | 1 | 1 | 1 | |

Dangyang | 1 | 1 | 1 | |

Jingzhou | 0.77 | 1 | 1 | |

Shayang | 0.77 | 1 | 1 | |

Zhanghe | 1 | 1 | 1 | |

Zhongxiang | 1 | 1 | 1 | |

Five-level | Dongbao | 1 | 1 | 1 |

Duodao | 1 | 1 | 1 | |

Dangyang | 1 | 1 | 1 | |

Jingzhou | 0.77 | 1 | 1 | |

Shayang | 0.77 | 1 | 1 | |

Zhanghe | 1 | 1 | 1 | |

Zhongxiang | 0.87 | 1 | 1 | |

Seven-level | Dongbao | 1 | 1 | 1 |

Duodao | 1 | 1 | 1 | |

Dangyang | 1 | 1 | 1 | |

Jingzhou | 0.77 | 1 | 1 | |

Shayang | 0.77 | 1 | 1 | |

Zhanghe | 1 | 1 | 1 | |

Zhongxiang | 0.81 | 1 | 1 |

Figure 5(b) shows the expected benefits obtained by the TSP model under the three scenarios. The expected benefit is the largest under three levels, while it is the smallest under seven levels. The above results are similar to those of the risks (see Figure 5(a)). Because the expected benefit is larger and recourse cost is higher in a dry year, the expected benefit is harder to achieve under three levels. Therefore, the risk is larger under three levels than that under the other two scenarios.

## CONCLUSIONS

This study develops a methodology to carry out risk analysis of the water allocation scheme obtained by the TSP model. The method mainly includes four parts: uncertain optimization, stochastic simulation, deterministic optimization, and risk assessment. After probability distribution analysis of the hydrological stochastic variable, a suitable TSP model is constructed to calculate the expected benefit and optimized water allocation targets. Then, the Monte Carlo method is used to generate the optimized stochastic sample. After that, a deterministic optimization model called a pre-allocated water optimization model is proposed to obtain the actual benefit under each simulated value in the optimized stochastic sample. Finally, the risk of the TSP model is calculated through comparing all the actual benefits and the expected benefit.

The methodology has been applied to the ZID to carry out risk analysis of a water allocation scheme obtained under the randomness of AIZR. The results show that the risk of the water allocation scheme obtained by the TSP model is 0.372–0.411 and decreases with the increase of number of hydrological levels in the ZID. The obtained water allocation plan and its corresponding risk can provide a good foundation for water resources management in the ZID. Moreover, it is recommended to divide the random variable into seven levels when using the TSP model to obtain a water allocation scheme under uncertainty.

There are various uncertainties in the process of water allocation, such as the randomness of hydrologic features, imprecision of water demand, and ambiguity of social–economic parameters. This paper only considers the impact of stochastic uncertainty on the risk of water allocation. Furthermore, this paper only discusses the allocation risk from the perspective of water quantity, and does not include the risk of joint allocation of water quality and quantity. The risk of the joint allocation of water quality and quantity needs further research.

## ACKNOWLEDGEMENTS

The study was supported by the National Natural Science Foundation of China (No. U2040212 and 51779013), the State Key Research and Development Plan of China (No. 2019YFC0408905), the Research Service Plan of Yangtze River to Huaihe River Diversion Project (HNYJJH/JS/FWKY-2021004), and the Water Conservancy Science and Technology Project of Jiangxi Province (202124ZDKT20).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.