# A Bilevel Optimal Water Allocation Model Considering Water Users’ Satisfaction Degree and Water Rights Transaction: A Case Study in Qingzhang River Basin, China

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Definition and Principles of Water Users’ Satisfaction Degree

#### 2.1.1. Definition of Water Users’ Satisfaction Degree

_{k}is water users’ satisfaction degree in administrative region A

_{k}in the basin, R

_{k}is the water amount allocated to administrative region A

_{k}, and D

_{mink}and D

_{maxk}are the minimum and maximum amount of water demand in administrative region A

_{k}, respectively.

#### 2.1.2. Principles of Water Users’ Satisfaction Degree

_{0}is the minimum satisfaction degree that must be met by each user within the basin, as specified by the watershed management agency.

_{j}is the satisfaction degree in administrative region A

_{j}(j ≠ k) and δ is an error coefficient, which is a minimum positive number close to 0. ‘S

_{k}− S

_{0}′ and ‘S

_{j}− S

_{0}′, respectively, represent the difference between the satisfaction degree in regions A

_{k}and A

_{j}and the minimum satisfaction degree set by the watershed management agency. W

_{k}and w

_{j}are the decision weights of A

_{k}and A

_{j}, respectively, in water resource allocation negotiations. $\frac{{S}_{k}-{S}_{0}}{{w}_{k}}$ and $\frac{{S}_{j}-{S}_{0}}{{w}_{j}}$ denote the satisfaction coefficients of water users A

_{k}and A

_{j}respectively, indicating how well the satisfaction degree matches the decision weight.

_{k}and w

_{j}are highly affected by the adopted principle of water resource allocation. The principle of respecting the historical and current situations places greater emphasis on objective facts and is more suitable to serve as the basis for negotiation among water users. Hence, the decision weights of administrative regions are set mainly according to this principle, including the principles of water source priority, occupation priority, and population priority. The specific methods for applying these principles are as follows.

_{ck}is the decision weight of administrative region A

_{k}on the priority of water sources, C

_{k}represents the water yield in administrative region A

_{k}, and K is the number of administrative regions in the basin.

_{ok}is the decision weight of administrative region A

_{k}on occupation priority and O

_{k}is the current water consumption in administrative region A

_{k}.

_{pk}is the decision weight of administrative region A

_{k}based on population priority and P

_{k}is the total population in administrative region A

_{k}.

_{k}of region A

_{k}can be expressed as the weighted average of the above three types of decision weights, and its mathematical form is

#### 2.2. A Bilevel Optimization Model for Basin Water Resources Allocation

#### 2.2.1. Model Framework

#### 2.2.2. Model Assumptions

_{k}of administrative region A

_{k}is greater than its actual water intake, that is, D

_{k}> Q

_{k}, it is possible to resolve the difference D

_{k}− Q

_{k}in water demand for area A

_{k}by saving water and improving water efficiency. When the actual water intake Q

_{k}of area A

_{k}is greater than its initial water allocation, that is, Q

_{k}> R

_{k}, area A

_{k}can obtain surplus water quantity Q

_{k}− R

_{k}from other regions by water rights trading on the market. In contrast, when the actual water intake Q

_{k}of area A

_{k}is less than the initial water allocation R

_{k}, that is, Q

_{k}< R

_{k}, area A

_{k}can transfer the surplus water R

_{k}− Q

_{k}through a water market to obtain income.

_{d}= V

_{g}− bx(V

_{g}> 0, b > 0), where V

_{d}is the price of water rights transactions, x is the total amount of water rights trading among regions, b is the influence coefficient of water rights supply and demand on water rights transaction price, and V

_{g}is the benchmark price for water rights transactions, which reflects the guiding price for a water rights transaction defined by the government.

#### 2.2.3. Model Construction

_{k}given the maximum net income of the region, under the condition that initial water allocation is known.

_{k}(represented by GDP output value) can be expressed as follows:

_{ka}and b

_{k}

_{0}are the output value of GDP per cubic meter of water when water-saving measures are taken and not taken in area A

_{k}, respectively, and b’ represents the growth rate of GDP per cubic meter of water after water-saving measures are taken, which can be expressed with the percentage of the economic benefit increment per cubic meter before and after water saving.

_{kp}of area A

_{k}can be expressed as follows:

_{r}is the price of water resources (which is an optimization variable to be solved), Q

_{k}· V

_{r}is the cost of water resources in area A

_{k}, ${\phi}_{k}\left({D}_{k}-{Q}_{k}\right)$ is the water-saving cost function of area A

_{k}, and $\left({R}_{k}-{Q}_{k}\right){V}_{d}$ is the water rights trading income of region A

_{k}.

_{k}chooses a strategy (here, it refers to the determination of water intake) that maximizes its net income. Based on this, the optimization model of water rights transactions among water users can be established as follows:

_{T}is the total water resources of the whole basin, R

_{L}is the basic domestic water consumption of the whole basin, R

_{E}is the basic ecological water consumption and basic grain water consumption, and R

_{C}is the basic water use, which is not involved in initial water rights allocation and water rights trading;

_{w}includes an initial water resource allocation sub-objective B

_{w1}and water rights trading regulation sub-objective B

_{w2}. B

_{w1}can be expressed as the weighted sum of water shortage rate and economic benefits after initial distribution, and its mathematical expression is

_{w}

_{2}can be expressed as the sum of water users’ economic benefits in a basin after water rights trading. Considering the unity of the target order of magnitude, it is standardized as follows:

_{w}can be expressed mathematically as

_{k}and S

_{j}can be determined from initial water distribution and water demand according to the formula for water users’ satisfaction.

_{dmax}indicates the maximum value of water rights transaction price, which is also determined by river basin management agencies with the aim of preventing “negative externalities” in water rights transactions and ensuring fairness;

_{k}is initial allocation, V

_{r}is water resource price, and V

_{g}is the water rights trading benchmark price. V

_{r}and V

_{g}are used to guide the water intake behavior and regulate water rights trading;

#### 2.3. Solution to the Bilevel Optimization Model for Water Basin Resource Allocation

#### 2.3.1. Solution Ideas

#### 2.3.2. Algorithm Design Based on Response Surface Methodology

_{k}, V

_{r}, and V

_{g}of the upper model, and its output is the optimization variable Q

_{k}of the lower model. The task of the RSM is to fit a clearly expressed mathematical model Q

_{k}= f(R

_{k}, V

_{r}, V

_{g}) instead of the lower optimization model shown in Formula (10).

_{k}included in the upper model can be calculated directly through the lower response surface model Q

_{k}= f(R

_{k}, V

_{r}, V

_{g}), which simplifies the process of solving the whole bilevel optimization problem and is more conducive to obtaining the global optimal solution. The solution process is shown in Figure 2.

_{k}, V

_{r}, and V

_{g}are the independent variables of the response surface model, and Q

_{k}is a dependent variable. In the feasible region of independent variables, a certain number of independent variable sample points (R

_{k}, V

_{r}, V

_{g}) are selected by an experimental design method. Formula (10) is solved by the optimization algorithm with sample points as basic parameters, and the corresponding dependent variable sample points (Q

_{k})are obtained;

_{k}, V

_{r}, V

_{g}, Q

_{k}) as data to generate a response surface model Q

_{k}= f(R

_{k}, V

_{r}, V

_{g});

_{k}= f(R

_{k}, V

_{r}, V

_{g}) to calculate Q

_{k}, then use the optimization algorithm to solve the upper optimization model to obtain an upper optimal solution (R

_{k}, V

_{r}, V

_{g}) and its corresponding lower optimal solution Q

_{k}.

## 3. Case Analysis

^{3}. The water demand of Shanxi Province and Hebei Province are 36.60 and 106.33 million m

^{3}, respectively. In the case study, they are regarded as the maximum amounts of Shanxi’s and Hebei’s water demands, and half of the two values are assumed as their respective minimum amounts of water demands. Based on field investigation, the output values of GDP per cubic meter of water of Shanxi and Hebei are CNY 114 and CNY 233, respectively, and the priorities of water source, occupation, and population are assumed as 0.3, 0.4, and 0.3, respectively. According to Formula (7), the decision weights of Shanxi and Hebei are 0.35 and 0.65, respectively.

^{3}, ${\widehat{R}}_{2}=$ 1.03 hundred million m

^{3}; the maximum of water shortage rate of the whole basin was 0.06, and the economic benefit was CNY 280.16 hundred million. The other results of Shanxi and Hebei provinces are shown in Table 3.

_{1}, water intake was Q

_{1}, and water-saving cost function was 24(D

_{1}− Q

_{1})

^{2}; Hebei’s initial water allocation was R

_{2}, water intake was Q

_{2}, and water-saving cost function was 26(D

_{2}− Q

_{2})

^{2}. Before and after water saving, the growth rate of water use efficiency in Shanxi was 0.3 and that in Hebei was 0.2. The water resource price was V

_{r}where $0.4\le {V}_{r}\le 2.0$, the benchmark price of water rights transactions was V

_{g}and the price function of water rights transactions was ${V}_{g}-0.13\left({R}_{1}+{R}_{2}-{Q}_{1}-{Q}_{2}\right)$, in which ${V}_{r}\le {V}_{g}\le 4.0$. The minimum satisfaction of Shanxi and Hebei provinces was S

_{0}= 0.8, and the balance error coefficient was δ = 0.1.

_{1}= 34.58 million m

^{3}, R

_{2}= 103.21 million m

^{3}, Q

_{1}= 34.03 million m

^{3}, Q

_{2}= Q

_{1}= 103.76 million m

^{3}, water resource price V

_{r}= CNY 0.54, and water rights transaction benchmark price V

_{g}= CNY 0.75, as shown in Table 4.

^{3}less than its initial distribution, while Hebei’s water intake was 0.55 million m

^{3}more than its initial distribution. This shows that Shanxi saved 0.55 million m

^{3}in water resources, and the saved water was sold to Hebei Province. Hebei Province alleviated its own water shortage problem by purchasing 0.55 million m

^{3}of water from Shanxi. This demonstrates that water rights trading encourages water rights holders to save water and obtain economic benefits by selling water, which promotes water transfer from areas with low efficiency to areas with high efficiency and further realizes an efficient allocation of water resources;

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Table 1.**Water supply and agricultural and industrial water consumption in the Qingzhang River Basin in a given year.

Subregion | Total Water Supply (10^{8} m^{3}) | Current Water Consumption (10 ^{8} m^{3}) | Irrigated Area (10 ^{4} mu) | Agricultural Water Quota (m ^{3}/mu) | Industrial Water Quota (m ^{3}/10^{4} CNY) | Total Water Demand (10 ^{8} m^{3}) |
---|---|---|---|---|---|---|

Qingzhang in Shanxi | 0.50 | 0.50 | 14.69 | 300 | 73.8 | 0.52 |

Qingzhang in Hebei | 1.18 | 1.17 | 18.5 | 548 | 43 | 1.20 |

Subregion | Urban Population (Hundred Million) | Urban Water Use Quota (L/Person-Day) | Rural Population (Hundred Million) | Rural Water Use Quota (L/Person-Day) | Large Livestock (Hundred Million) | Small livestock (Hundred Million) | Water for Large Livestock (L/Person-Day) | Water for Small Livestock (L/Person-Day) |
---|---|---|---|---|---|---|---|---|

Qingzhang in Shanxi | 9.12 | 121.27 | 29.78 | 56.58 | 8.41 | 71.99 | 35 | 15 |

Qingzhang in Hebei | 10.12 | 142 | 35.46 | 45 | 5.26 | 41.23 | 35 | 15 |

**Table 3.**The results for initial water rights allocation, only considering satisfaction degree in the Qingzhang River Basin, in a given year.

Provinces | Water Demand (Million m ^{3}) | Initial Water Distribution (Million m ^{3}) | Water Shortage Rate | Satisfaction Degrees of Water Users | Decision Weight | Economic Benefit (Hundred Million CNY) |
---|---|---|---|---|---|---|

Shanxi | 36.60 | 34.36 | 0.06 | 0.88 | 0.35 | 39.17 |

Hebei | 106.33 | 103.43 | 0.02 | 0.95 | 0.65 | 240.99 |

**Table 4.**The results for the bilevel optimal water allocation of the Qingzhang River Basin in a given year.

Provinces | Water Demand (Million m ^{3}) | Initial Water Distribution (Million m ^{3}) | Water Intake (Million m ^{3}) | Water Resource Fee (CNY) | Trading Benchmark Price(CNY) | Satisfaction of Water Users | Decision Weight | Pre-transaction Benefit (Hundred Million CNY) | Post-transaction Benefit (Hundred Million CNY) |
---|---|---|---|---|---|---|---|---|---|

Shanxi | 36.60 | 34.58 | 34.03 | 0.54 | 0.75 | 0.89 | 0.35 | 39.23 | 50.24 |

Hebei | 106.33 | 103.21 | 103.76 | 0.54 | 0.75 | 0.94 | 0.65 | 239.92 | 289.53 |

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**MDPI and ACS Style**

Chu, Y.; Xiao, Y.; Zhu, J.
A Bilevel Optimal Water Allocation Model Considering Water Users’ Satisfaction Degree and Water Rights Transaction: A Case Study in Qingzhang River Basin, China. *Water* **2023**, *15*, 2650.
https://doi.org/10.3390/w15142650

**AMA Style**

Chu Y, Xiao Y, Zhu J.
A Bilevel Optimal Water Allocation Model Considering Water Users’ Satisfaction Degree and Water Rights Transaction: A Case Study in Qingzhang River Basin, China. *Water*. 2023; 15(14):2650.
https://doi.org/10.3390/w15142650

**Chicago/Turabian Style**

Chu, Yu, Yi Xiao, and Jiulong Zhu.
2023. "A Bilevel Optimal Water Allocation Model Considering Water Users’ Satisfaction Degree and Water Rights Transaction: A Case Study in Qingzhang River Basin, China" *Water* 15, no. 14: 2650.
https://doi.org/10.3390/w15142650