*3.2. Results of the Water Rights Distribution among Farmer Households*

Since the patterns of water rights allocation among farmer households for 411 canals are the same, the canal of Grazing team (4) in Bayangaole Town is taken as an example for calculation and analysis. The irrigation area and the agricultural population of farmer households are selected as the water rights allocation indexes under asymmetric information. On the basis of the calculation formula of the Gini coefficient, the population of farmer households and the corresponding irrigation area data are arranged according to the per capita irrigation area from small to large. The calculation process is shown in Table 3.

**Table 3.** The relevant calculation results for the Gini coefficient under the current condition.


According to the above calculation results, the Gini coefficient for the current distribution of water rights is 0.1968. The above data are substituted into the water rights allocation model among farmer households, and then the balanced per capita irrigation area for eight farmer households of Grazing team (4) are determined through objective function Equations (15)–(17) and constraint Equations (18)–(24).

The current per capita irrigation area of farmer households which exceeds (falls short of) the average per capita irrigation area of the canal system, =0.101 hm2/person, needs to be reduced (compensate). The fairness constraint is:

> *x*<sup>1</sup> ≥ 0.056 *x*<sup>2</sup> ≥ 0.067 *x*<sup>3</sup> ≥ 0.080 *x*<sup>4</sup> ≥ 0.083 *x*<sup>5</sup> ≤ 0.122 *x*<sup>6</sup> ≤ 0.133 *x*<sup>7</sup> ≤ 0.150 *x*<sup>8</sup> ≤ 0.156

A substantial reduction in the per capita irrigation area of farmer households will lead to a reduction in their allocated water rights. In order to ensure a certain amount of basic irrigation water for farmers, we consulted the local water resources management department. This paper sets the reduction ratio to 0.3, and the basic water security constraint is:

$$\begin{cases} \ x\_5 \ge 0.085\\ \ x\_6 \ge 0.093\\ \ x\_7 \ge 0.105\\ \ x\_8 \ge 0.109 \end{cases}$$

The more per capita irrigation area is above or below the average of canal system, *x* = 0.101 hm2/person, the greater the degree of reduction or compensation is; that is, the degree of reduction and compensation is restricted as follows:


After the equilibrium, the per capita irrigation area of each farmer household still satisfies the ranking before the equilibrium, ensuring the fairness of the distribution of water rights among farmer households; that is, the ranking constraint is:

$$6\mathbf{x}\_1 + 8\mathbf{x}\_2 + 5\mathbf{x}\_3 + 4\mathbf{x}\_4 + 6\mathbf{x}\_5 + 7\mathbf{x}\_6 + 4\mathbf{x}\_7 + 3\mathbf{x}\_8 = 4.333$$

After the equilibrium, the Gini coefficient of the farmer households' agricultural population–irrigation area should be smaller than that before the equilibrium, to ensure that the distribution plan is fairer than the current distribution; that is:

$$G\_{
{
{

in i
}}} \le 0.1968$$

After equilibrium, the per capita irrigation area of each farmer household is greater than 0; that is, the non-negative constraint is:

$$x\_i \ge 0 \qquad (i = 1, 2, \dots, 8)$$

The genetic algorithm in MATLAB is used to solve the model, and the per capita irrigation area of the eight farmer households of the canal of Grazing team (4) is balanced. The balanced per capita irrigation area of the farmer households is shown in Table 4.

**Farmer Household Number**

households (m<sup>3</sup>

).


After the equilibrium, the Gini coefficient of the farmer households' agricultural population–irrigation area should be smaller than that before the equilibrium, to ensure that

*G ini* 0.1 9 6 8 After equilibrium, the per capita irrigation area of each farmer household is greater

*x i <sup>i</sup>* 0 ( 1, 2, ,8)

The genetic algorithm in MATLAB is used to solve the model, and the per capita irrigation area of the eight farmer households of the canal of Grazing team (4) is balanced. The balanced per capita irrigation area of the farmer households is shown in Table 4.


*Water* **2021**, *13*, x FOR PEER REVIEW 11 of 15

than 0; that is, the non-negative constraint is:

the distribution plan is fairer than the current distribution; that is:

According to the canal-level water rights allocation method, the allocated water rights of the canal of Grazing team (4) is 13,000 m<sup>3</sup> . According to the per capita irrigation area of each farmer household after the equilibrium, combined with Formula (25), the water rights distributed for each farmer household by the model is calculated. The current per capita irrigated area and the per capita irrigated area after equilibrium are shown in Figure 5 and the current water rights of farmer households and the water rights distributed by the model are shown in Table 5. According to the canal-level water rights allocation method, the allocated water rights of the canal of Grazing team (4) is 13,000 m<sup>3</sup> . According to the per capita irrigation area of each farmer household after the equilibrium, combined with Formula (25), the water rights distributed for each farmer household by the model is calculated. The current per capita irrigated area and the per capita irrigated area after equilibrium are shown in Figure 5 and the current water rights of farmer households and the water rights distributed by the model are shown in Table 5.

**Figure 5.** Per capita irrigation area of farmer households before and after equilibrium. **Figure 5.** Per capita irrigation area of farmer households before and after equilibrium.

**Table 5.** The amount of water rights allocated by the model and the amount of current allocated water rights for farmer **Table 5.** The amount of water rights allocated by the model and the amount of current allocated water rights for farmer households (m<sup>3</sup> ).


#### **4. Discussion**

*4.1. Analysis of Water Rights Distribution for the Canals Diverted Directly from the National Canal System*

The current actual water consumption compared with the permitted water volume shows that the total water consumption volume of the canal system in the Wulanbuhe Irrigation Area is 347.9529 million m<sup>3</sup> , which is greater than the 330 million m<sup>3</sup> permitted. The current water rights allocation needs to be adjusted.

After the completion of the water-saving project, the total amount of water rights distribution for the canal system can be reduced. The Wulanbuhe Irrigation Area mainly saves water through three water-saving projects of canal lining, border field reconstruction, and drip irrigation. The future water-saving amount calculated of three water-saving projects is 77.641 million m<sup>3</sup> , 68.10 million m<sup>3</sup> , and 22.40 million m<sup>3</sup> , respectively. The total water saving in the Wulanbuhe Irrigation Area is 168.0817 million m<sup>3</sup> . After the completion of the water-saving project, the total amount of water rights distribution for the canal system can be reduced. The Wulanbuhe Irrigation Area mainly saves water through three water-saving projects of canal lining, border field reconstruction, and drip irrigation. The future water-saving amount calculated of three water-saving projects is 77.641 million m<sup>3</sup> , 68.10 million m<sup>3</sup> , and 22.40 million m<sup>3</sup> , respectively. The total water saving in the Wulanbuhe Irrigation Area is 168.0817 million m<sup>3</sup> .

, which is greater than the 330 million m<sup>3</sup> permitted.

*4.1. Analysis of Water Rights Distribution for the Canals Diverted Directly from the National* 

The current actual water consumption compared with the permitted water volume shows that the total water consumption volume of the canal system in the Wulanbuhe

According to the water rights allocation model of the national canal system, from the actual current water volume minus the water-saving amount, the total amount of water rights allocated to the 411 canals is 179.8712 million m<sup>3</sup> , which is less than the permitted water volume, and there is a remaining water volume of 150.1288 million m<sup>3</sup> . The remaining water can be traded for water rights to increase the efficiency of water resources utilization. The relevant water volume is shown in Figure 6. According to the water rights allocation model of the national canal system, from the actual current water volume minus the water-saving amount, the total amount of water rights allocated to the 411 canals is 179.8712 million m<sup>3</sup> , which is less than the permitted water volume, and there is a remaining water volume of 150.1288 million m<sup>3</sup> . The remaining water can be traded for water rights to increase the efficiency of water resources utilization. The relevant water volume is shown in Figure 6.

*Water* **2021**, *13*, x FOR PEER REVIEW 12 of 15

Water rights allocated by the model 1284 1888 1310 1072 2064 2562 1600 1220 Current allocated water rights 1000 1600 1200 1000 2200 2800 1800 1400

The current water rights allocation needs to be adjusted.

**4. Discussion**

*Canal System*

hm<sup>2</sup>

Irrigation Area is 347.9529 million m<sup>3</sup>

**Farmer Household 1 2 3 4 5 6 7 8**

**Figure 6.** Water rights allocation of canal system in Wulanbuhe Irrigation Area. **Figure 6.** Water rights allocation of canal system in Wulanbuhe Irrigation Area.

## *4.2. Performance Test of Water Rights Allocation Model among Farmer Households*

*4.2. Performance Test of Water Rights Allocation Model among Farmer Households* After the optimization of the model is solved, the Gini coefficient of the farmer household's population–the balanced irrigation area of the Grazing team (4) is 0.1289, which has been significantly improved compared with the Gini coefficient of 0.1968 of the farmer household's population–the current irrigation area, and the distribution of water rights among farmer households through the model is more equitable. Comparing the per capita irrigation area of farmer households after equilibrium by the model with that of before, the compensation for farmers 1, 2, 3, and 4 is 0.0158 hm<sup>2</sup> /person, 0.0120 hm<sup>2</sup> /person, 0.0073 /person, and 0.006 hm<sup>2</sup> /person, respectively, and the reduction for farmers 5, 6, 7, and 8 is 0.0075 hm<sup>2</sup> /person, 0.0113 hm<sup>2</sup> /person, 0.0167 hm<sup>2</sup> /person, and 0.0189 hm<sup>2</sup> /person, respectively. The water rights distributed by the model for each farmer household have also been compensated or reduced accordingly, compared to before. The amount of compen-After the optimization of the model is solved, the Gini coefficient of the farmer household's population–the balanced irrigation area of the Grazing team (4) is 0.1289, which has been significantly improved compared with the Gini coefficient of 0.1968 of the farmer household's population–the current irrigation area, and the distribution of water rights among farmer households through the model is more equitable. Comparing the per capita irrigation area of farmer households after equilibrium by the model with that of before, the compensation for farmers 1, 2, 3, and 4 is 0.0158 hm2/person, 0.0120 hm2/person, 0.0073 hm2/person, and 0.006 hm2/person, respectively, and the reduction for farmers 5, 6, 7, and 8 is 0.0075 hm2/person, 0.0113 hm2/person, 0.0167 hm2/person, and 0.0189 hm2/person, respectively. The water rights distributed by the model for each farmer household have also been compensated or reduced accordingly, compared to before. The amount of compensation (reduction) is shown in Figure 7.

sation (reduction) is shown in Figure 7. According to the distribution results of the model, for farmer households with a small population and large irrigation area, such as farmer households 5, 6, 7, and 8, the water rights allocated by the model are less than the current allocation. As their irrigation needs cannot be met, they can adjust planting structures or obtain additional water rights through water rights transactions. For farmer households with a large population and a small irrigation area, such as farmer households 1, 2, 3, and 4, the water rights allocated by the model are 754 m<sup>3</sup> more than the current allocation when only the irrigation area is considered. The allocation results by the model take into account the asymmetric factors of farm household population and irrigation area, and is more equitable. For example, the current water rights distributed for farmer household 1 and farmer household 4 are both 1000 m<sup>3</sup> , but the water rights allocated by the model are 1284 m<sup>3</sup> and 1072 m<sup>3</sup> , respectively. This is precisely considering the factor of farmer household population, where relatively more water rights are allocated for farmer households with larger populations.

**Figure 7.** Comparison chart of current water rights and water rights allocated by the model. **Figure 7.** Comparison chart of current water rights and water rights allocated by the model.

#### According to the distribution results of the model, for farmer households with a small *4.3. Overall Analysis of Water Rights Distribution in the Irrigation District*

population and large irrigation area, such as farmer households 5, 6, 7, and 8, the water rights allocated by the model are less than the current allocation. As their irrigation needs cannot be met, they can adjust planting structures or obtain additional water rights through water rights transactions. For farmer households with a large population and a small irrigation area, such as farmer households 1, 2, 3, and 4, the water rights allocated by the model are 754 m<sup>3</sup> more than the current allocation when only the irrigation area is considered. The allocation results by the model take into account the asymmetric factors of farm household population and irrigation area, and is more equitable. For example, the current water rights distributed for farmer household 1 and farmer household 4 are both 1000 m<sup>3</sup> , but the water rights allocated by the model are 1284 m<sup>3</sup> and 1072 m<sup>3</sup> , respectively. This is precisely considering the factor of farmer household population, where relatively more water rights are allocated for farmer households with larger populations. *4.3. Overall Analysis of Water Rights Distribution in the Irrigation District* The results of the water rights distribution at the national canal system level and among farmer households calculated by the double-level water rights allocation model show that the total amount of water rights allocated for each canal in the irrigation district The results of the water rights distribution at the national canal system level and among farmer households calculated by the double-level water rights allocation model show that the total amount of water rights allocated for each canal in the irrigation district has been greatly reduced, which will inevitably lead to a relative decrease in the water rights distributed for farmers in the irrigation district. The distribution of water rights in irrigation areas needs to comprehensively consider fairness and efficiency, but most of the existing studies only consider the area of agricultural land and the actual irrigation area of agricultural land [22], with a lack of consideration for the asymmetry between farmers' population and irrigation area [23,24]. In order to ensure the fairness of agricultural water rights distribution, it is necessary to comprehensively consider the agricultural population and irrigation area in the irrigation water user water rights distribution system. The water rights distribution model among farmers established in this paper is more fair in the process of water rights distribution, and alleviates the contradiction between farmers and water distribution managers to a certain extent. After the establishment of a farmers' water rights market, farmers with more water rights voluntarily sell water rights, while farmers with less water rights actively purchase water rights, which provides an opportunity for water rights trading among farmers in irrigation areas.

#### has been greatly reduced, which will inevitably lead to a relative decrease in the water **5. Conclusions**

rights distributed for farmers in the irrigation district. The distribution of water rights in irrigation areas needs to comprehensively consider fairness and efficiency, but most of the existing studies only consider the area of agricultural land and the actual irrigation area of agricultural land [22], with a lack of consideration for the asymmetry between farmers' population and irrigation area [23,24]. In order to ensure the fairness of agricultural water rights distribution, it is necessary to comprehensively consider the agricultural population The rational distribution of agricultural water rights in irrigation areas is an important basis for improving the agricultural water rights system and establishing a water rights market. This paper establishes a double-level water rights allocation model of canals– farmers in an irrigation district, which is applied to the water rights distribution of the Wulanbuhe Irrigation Area in the Yellow River Basin. The main conclusions are as follows:


nals–farmers in an irrigation district, which is applied to the water rights distribution of

able development of water resources can provide a guarantee for the high-quality development of the Yellow River Basin.

**Author Contributions:** Conceptualization, X.G.; methodology, X.G.; software, Q.D.; validation, Q.D. and B.W.; formal analysis, B.W.; investigation, W.Z.; resources, W.Z.; data curation, W.Z.; writing original draft preparation, Q.D.; writing—review and editing, Q.D., B.W. and W.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Natural Science Foundation of China (No. NSCF-51979119) and by the Basic R & D Special Fund of Central Government for Non-profit Research Institutes (No. HKY-JBYW-2020-17).

**Institutional Review Board Statement:** It is not applicable for this study not involving humans or animals.

**Informed Consent Statement:** It is not applicable for this study not involving humans.

**Data Availability Statement:** The data that support the findings of this study are available in this article.

**Acknowledgments:** The authors would like to thank the research team for their help.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

