Irrigation area water right distribution method based on cooperative Stackelberg game

Abstract

The invention provides an irrigation area water right distribution method based on a cooperative Stackelberg game, relates to the technical field of water resource management and distribution, and permits all irrigation areas to carry out water right transaction in a dispersed manner by constructing an irrigation area water market water right distribution and transaction analysis framework based on the cooperative Stackelberg game so as to realize cooperation and win-win. In the analysis framework, the irrigation district management bureau can set related condition value-at-risk parameters according to the risk preference degree of the irrigation district management bureau so as to balance the risk and the expected income and determine the water supply price, and the analysis framework defines the distribution rule of the income obtained by sharing the water right by using the GNB theory so as to excite the water right transaction among the irrigation districts, so that the water supply cost is reduced. And the fairness of benefit distribution is ensured. According to the method, the risk coping capability of the irrigation area water resource management system can be further improved, and the expected earnings of various participating roles under different risk avoidance levels are improved.

Description

Technical Field

[0001] The invention belongs to the technical field of water resource management and allocation, and in particular relates to an irrigation area water rights allocation method based on cooperative Stackelberg game. Background Art

[0002] With the continuous development of the economy and society, the contradiction between water supply and demand has become increasingly severe, especially in the agricultural sector. As an important pillar for ensuring national food security and a major water-consuming sector, water resource management is particularly important in irrigation areas. In order to effectively alleviate the pressure on agricultural water use and improve the level of water conservation in irrigation areas, it is of great significance to establish and improve the water rights trading system.

[0003] Water rights transactions among irrigation water users in the irrigation area are conducted between water users or water use organizations within the irrigation area. After the water rights and interests are clearly assigned to irrigation water users or water use organizations through water resource use rights confirmation and registration, transactions can be carried out. After the transaction is completed, if the water rights transaction among irrigation water users in the irrigation area is a multi-year transaction, the transferor and the transferee shall report the relevant situation to the irrigation area management bureau.

[0004] Some researchers have proposed adopting an irrigation water market trading model based on cooperative game theory. On the basis of a fixed water price given by the irrigation district administration, each irrigation area independently allocates idle water rights and distributes the benefits based on contribution or contract. However, the limitation of this method is that each irrigation area will passively accept the pricing strategy of the irrigation district administration, and it is impossible to obtain a systematic optimal solution.

[0005] In addition, the actual amount of irrigation water in the irrigation area will be affected by the weather. Due to the uncertainty of precipitation conditions, the number of water rights in the irrigation area will be insufficient or excessive, so the uncertainty of precipitation should be included in the factors considered in water rights management to deal with the randomness in the decision-making process. Existing studies often ignore the cooperative effect between irrigation areas, resulting in the impact of uncertainty not being fully fed back.

[0006] Based on the above problems, this application proposes a cooperation-based Stackelberg game model to solve the price coordination problem between the irrigation district administration and each irrigation area, and combines the incentive mechanism of GNB therapy to promote direct water rights transactions between irrigation areas. Summary of the invention

[0007] In view of the defects and problems existing in the prior art, the present invention provides an irrigation district water rights allocation method based on cooperative Stackelberg game to coordinate the interactive cooperation among various participants in the irrigation district water market.

[0008] The solution adopted by the present invention to solve the technical problem is: a method for allocating water rights in irrigation areas based on cooperative Stackelberg game, comprising the following steps:

[0009] S1. Establish an analytical framework; establish an analytical framework for water rights allocation and trading in the irrigation district water market based on the cooperative Stackelberg game model, and define the roles of the irrigation district administration and each irrigation area; the irrigation district administration, as a leader, is used to formulate strategies for water rights allocation and trading to maximize the comprehensive benefits of the irrigation district; each irrigation area is used to respond to the strategies of the irrigation district administration and maximize its own benefits through water rights trading;

[0010] S2, define model variables and parameters; based on the above step 1, determine the roles and goals, and define the parameters and variables in the Stackelberg game model;

[0011] S3. Constructing a leadership strategy; the irrigation district administration takes the maximum comprehensive benefit of the irrigation district as the management and control goal, while considering the risk control brought by uncertainty, and uses the CvaR stochastic programming method to characterize the expected loss risk of the irrigation district administration;

[0012] S4. Construct follower feedback; each irrigation area provides feedback to the leader’s decision in a cooperative manner, shares idle water rights and distributes benefits according to contribution, while using stochastic programming methods to handle uncertainty;

[0013] S5. Solve the two-level optimization problem; transform the two-level optimization problem into a first-level equivalent single-level mixed integer linear programming problem, including decomposing the GNB problem into two sub-problems: minimizing the cost problem and the bargaining problem; by solving the minimizing the cost problem, the optimal cooperative benefit and the bargaining power of each irrigation area are obtained.

[0014] Furthermore, in step S1, the irrigation district administration uses the price discrimination method to enable each irrigation area to participate in the water rights allocation process to maximize the economic benefits of the irrigation area; and uses CVaR to control the overall risk level from a global perspective; each irrigation area conducts water rights transactions in a decentralized manner to maximize the benefits of water resource utilization.

[0015] Furthermore, the decision model of the irrigation district administration in step S3 is expressed as follows:

[0016]

[0017] The above formula is the objective function, which consists of two parts: 1) the expected net benefit of the irrigation district administration; 2) the trade-off between the risk caused by uncertainty and the expected benefit, where β represents the weight.

[0018] Furthermore, in step S4, GNB is used to describe the water rights transfer process between irrigation areas. Under the premise that the irrigation area manager gives a price, the response of each irrigation area decision maker to the given price is expressed by quantifying the water rights transaction between the irrigation area and the irrigation area management bureau. The follower's reaction model is:

[0019]

[0020] The above formula is the objective function of the irrigation area decision maker's response to price. The objective function consists of two parts: 1) represents the water use cost function of irrigation area i in the absence of water rights trading between irrigation areas: 2) represents the water use cost function of irrigation area i when there is water rights trading between irrigation areas and the net cost paid by the irrigation area for water rights transactions with irrigation area i The sum of

[0021]

[0022] In the above formula, the parameters It reflects the bargaining power of irrigation area i in the bargaining process. and represents the decision variable vector of irrigation basin i with or without water rights trading between irrigation basins; the above formulas represent the internal costs with or without water rights trading between irrigation basins, including the operating costs of reservoirs within the irrigation basin.

[0023] Furthermore, the formula for minimizing the operating cost problem in step S5 is:

[0024]

[0025] The formula for the bargaining problem is:

[0026]

[0027] Furthermore, in step S5, the GNB problem is converted into an equivalent single-layer mixed integer linear programming problem, and the conversion process includes:

[0028] a. By utilizing the KKT condition, the two-level problem is equivalently reformulated as a nonlinear single-level problem by replacing the low-level problem. The nonlinear terms include complementary relaxation conditions and bilinear products.

[0029] b. Use the big-M method to replace the complementary relaxation conditions introduced by the Lagrange multipliers and constraint products;

[0030] c. Based on the strong duality theory, the bilinear product is rewritten into an equivalent linear expression.

[0031] Furthermore, in the bargaining problem, by solving the problem of minimizing the cost, the optimal benefit of cooperation η can be obtained. i，w，* , and obtain the bargaining power of each irrigation area By logarithmically changing the bargaining problem formula, we can get the following formula:

[0032]

[0033] This transforms it into a convex optimization problem, which is solved using MATLAB and the market clearing is obtained.

[0034] Compared with the prior art, the present invention has the following beneficial effects:

[0035] The method for allocating water rights in irrigation areas based on the cooperative Stackelberg game provided by the present invention can allocate water resources in irrigation areas more scientifically, so as to maximize the comprehensive benefits of the entire irrigation area; this method not only takes into account the overall economic benefits of the irrigation area, but also takes into account the specific needs of each irrigation area, thereby realizing the optimal configuration of the irrigation area water resource system.

[0036] This method introduces the CVaR (conditional value at risk) stochastic programming method to characterize the expected losses of the irrigation district administration, so as to effectively control the risks brought by water resource uncertainty. This helps to reduce the potential risk level while ensuring the economic benefits of the irrigation district; in the response model of followers (each irrigation area), a cooperative form is adopted. Each irrigation area shares idle water rights and distributes benefits according to contribution, which helps to promote cooperation and win-win between irrigation areas.

[0037] This cooperation model not only improves the utilization efficiency of water resources, but also enhances the economic connection and interactivity between irrigation areas; this method takes into account the dynamic interactive relationship between the irrigation area management bureau and each irrigation area, making the model highly flexible and adaptable. When faced with different water resource conditions and market environments, the model can quickly adjust its strategy to adapt to new situations. This water rights allocation and trading mechanism helps promote the sustainable development of irrigation areas; it can not only improve the utilization efficiency of water resources, but also enhance the economic vitality of irrigation areas, laying a solid foundation for the long-term development of irrigation areas. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 It is a schematic diagram of the three-dimensional structure of the present invention;

[0039] Figure 2 It is a schematic diagram of the three-dimensional structure of the present invention;

[0040] Figure 3 This is a schematic diagram of the geographical location of the Hetao region;

[0041] Figure 4 This is a curve chart of the planting structure in the Hetao region (2018);

[0042] Figure 5 This is the effective precipitation curve of Hetao area year by year;

[0043] Figure 6 This is a schematic diagram of the P-III curve distribution of effective precipitation in each irrigation area;

[0044] Figure 7 It is the curve diagram of effective water yield and maximum irrigation water requirement of crops under different scenarios in each irrigation area;

[0045] Figure 8 It is a line graph showing the relationship between CVaR and expected returns under different β values;

[0046] Fig. 9 It is a curve diagram of water price in each irrigation area at different β values ​​in each period (without water rights trading between irrigation areas);

[0047] Fig.10 It is a curve diagram of water price in each irrigation area at different β values ​​in each period (when there is water rights trading between irrigation areas);

[0048] Fig.11 This is a curve chart of water rights transaction volume between the irrigation district administration and each irrigation area. The negative sign in the figure represents sale. DETAILED DESCRIPTION

[0049] The present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0050] Embodiment 1:

[0051] This embodiment proposes a method for allocating irrigation district water rights based on cooperative Stackelberg game, which effectively solves the interactive cooperation problem between the irrigation district management bureau and each irrigation area, that is, it allows each irrigation area to trade water rights in a decentralized manner to achieve win-win cooperation. Figure 1-2 , specifically including the following steps:

[0052] 1. Establish an analytical framework

[0053] The framework of water rights allocation and transaction analysis in irrigation water market based on cooperative Stackelberg game is established, that is, the water rights transfer problem involving irrigation district administration and several irrigation areas is expressed as a Stackelberg game model, and two types of participant roles are identified, including leaders (irrigation district administration) and followers (each irrigation area). Figure 1As shown in the figure, the irrigation district administration, as the leader of the irrigation district, is responsible for formulating strategies for water rights allocation and trading. Its goal is to maximize the comprehensive benefits of the irrigation district while considering risk control; each irrigation area, as a participant in the water market, is used to respond to the strategy of the irrigation district administration and maximize its own benefits through water rights trading. At the same time, each irrigation area needs to consider cooperation with other irrigation areas while meeting its own water demand. In this embodiment, the irrigation district administration uses the price discrimination method to enable each irrigation area to participate in the water rights allocation process to maximize the economic benefits of the irrigation district. In order to further consider the risks brought by uncertainty, the irrigation district administration uses CVaR to control the overall risk level from a global perspective. After the irrigation district manager formulates a pricing plan, the irrigation areas negotiate with each other, respond to the pricing plan, and conduct water rights trading in a decentralized manner to maximize the benefits of water resource utilization. This step provides a basic game structure and goals for subsequent steps by defining the roles of the irrigation district administration and each irrigation area.

[0054] 2. Define model variables and parameters

[0055] This embodiment establishes a Stackelberg game based on mutual cooperation among irrigation areas and taking into account price discrimination and risk control. In this model, the irrigation district administration, as the leader, has the right to provide price discrimination and determine risk preferences; the irrigation district decision makers have the right to determine the price specified by the irrigation district administration. Make rational responses, share idle water rights and distribute benefits according to contribution. Based on the roles and goals determined in step 1 above, this step defines the variables and parameters in the model, including the following:

[0056] i Domain Index

[0057] tTime Index

[0058] wScene Index

[0059] Π irrigation domain collection

[0060] Ω Scenario Collection

[0061] Γ time set

[0062] π i，w Probability of occurrence of precipitation scenario

[0063] The price at which the irrigation district administration purchases water rights from the next-level main canal (CNY / m 3 )

[0064] The price at which the irrigation district administration sells water rights to the next-level main canal (CNY / m 3 )

[0065] β risk aversion weight coefficient

[0066] Operation cost of water storage tanks in irrigation area i (CNY / m 3 )

[0067] The effective rainfall available in irrigation area i during period t (m 3 )

[0068] Crop water requirement of irrigation area i under insufficient irrigation in period t (m 3 )

[0069] The maximum limit of water rights purchased by irrigation area i from the irrigation district administration (m 3 )

[0070] The maximum limit of water rights sold by irrigation area i to the irrigation district administration (m 3 )

[0071] The maximum amount of water that can be stored in the reservoir within irrigation area i per unit time period (m 3 )

[0072] The maximum amount of water that can be released from the internal reservoir of irrigation area i within a unit period of time (m 3 )

[0073] Minimum water storage ratio of internal reservoirs in irrigation area i (%)

[0074] Maximum water storage ratio of internal reservoirs in irrigation area i (%)

[0075] Cap i The water storage capacity of the internal reservoir of irrigation area i (m 3 )

[0076] Initial water storage ratio of the internal reservoir of irrigation area i (%)

[0077] Net water storage ratio of the internal reservoir of irrigation area i, that is, the ratio of the original water volume after deducting losses such as evaporation (%)

[0078] Net water release ratio of the internal water storage tank of irrigation area i, that is, the ratio of the original water volume after deducting losses such as evaporation (%)

[0079] The price of water rights purchased by irrigation area i from the irrigation area administration (CNY / m 3 )

[0080] The price of water rights sold by irrigation area i to the irrigation district administration (CNY / m 3 )

[0081] Water rights purchased by irrigation area i from the irrigation district administration (m 3 )

[0082] Water rights sold by irrigation area i to the irrigation district administration (m 3 )

[0083] ζ i Auxiliary variables related to irrigation area i are used to calculate CVaR(CNY), and the optimal value is ζ i*

[0084] γConfidence Level

[0085] η i，w Auxiliary variables related to irrigation area i and scenario w, used to calculate CVaR (CNY)

[0086] Operation cost of irrigation area i without considering water rights trading (CNY)

[0087] Irrigation area i Operating cost when considering water rights trading (CNY)

[0088] Net cost paid for water rights exchange between irrigation areas (CNY)

[0089] The gross water storage volume of the internal reservoir of irrigation area i in period t (m 3 )

[0090] The gross water release volume of the internal reservoir of irrigation area i in period t (m 3 )

[0091] The water rights transaction volume of irrigation area i in period t (m 3 )

[0092] SOC i，t，w Water storage ratio of internal reservoirs in irrigation area i (%)

[0093] Total operating cost of each irrigation area under scenario w (CNY)

[0094] Cooperation benefits of each irrigation area under scenario w (CNY)

[0095] 3. Build a leadership strategy

[0096] As the responsible entity for irrigation district management, the irrigation district administration has the obligation to coordinate the management of each irrigation area to further improve the overall water use efficiency of the irrigation district. The irrigation district administration takes the maximum comprehensive benefit of the irrigation district as the management and control goal, while considering the risk control caused by uncertainty. Therefore, the decision model of the irrigation district administration is expressed as follows:

[0097]

[0098] St

[0099]

[0100]

[0101] The above formula (1) is the objective function, which consists of two parts: 1) the expected net income of the irrigation district administration (the income from selling water rights to each irrigation area minus the cost of purchasing water rights from the upper-level water diversion main canal); 2) the trade-off between the risk caused by uncertainty and the expected income (β represents the weight, and its value depends on the risk preference of the irrigation district administration. The larger the β value, the more averse it is to the risk caused by uncertainty).

[0102] In the constraint condition part, equations (2) and (3) represent the price limit for the irrigation district administration to sell water rights to the irrigation area; equations (4) and (5) represent the average price limit for the irrigation district administration to sell water rights to the irrigation area over a period of time, with the aim of reducing the market power of the irrigation district administration; and in order to make the weight coefficient β affect the comprehensive income of the irrigation district administration, constraint condition (6) is introduced to calculate CVaR.

[0103] 4. Build follower feedback

[0104] The response of followers (each irrigation area) is based on the above-mentioned leader strategy. Followers need to optimize their own strategies through cooperative game theory (GNB) under the conditions given by the leader. In this embodiment, the amount of water rights purchased by each irrigation area from the irrigation district management bureau within a unit calculation period is affected by factors such as weather. If there are more rainy days in a certain irrigation area and the effective rainfall is relatively abundant, the irrigation area can trade water rights with neighboring irrigation areas and irrigation district management bureaus to transfer excess idle water rights. This embodiment uses GNB to study the water rights transfer process between irrigation areas. In the GNB game, each irrigation area cooperates with each other, makes independent decisions, shares idle water rights, maximizes overall benefits, and reasonably distributes benefits according to contributions. This can effectively mobilize the enthusiasm of irrigation area decision makers to participate in water rights management and promote the improvement of agricultural irrigation water conservation levels within the irrigation area.

[0105] Under the premise that the irrigation district manager gives a price, the response of each irrigation district decision maker to the given price is expressed by quantifying the water rights transaction between the irrigation district and the irrigation district management bureau. In various scenarios, in order to characterize the water rights transaction between each irrigation district and the irrigation district management bureau, the modeling of this embodiment is as follows:

[0106]

[0107] St

[0108]

[0109] In the above formula, the parameters It reflects the bargaining power of irrigation area i in the bargaining process. and It represents the decision variable vector of irrigation area i with or without water rights trading between irrigation areas, and is expressed as follows:

[0110]

[0111] The above formula (7) is the objective function of the irrigation area decision maker's response to price. The objective function consists of two parts: 1) represents the water use cost function of irrigation area i in the absence of water rights trading between irrigation areas: 2) represents the water use cost function of irrigation area i when there is water rights trading between irrigation areas and the net cost paid by the irrigation area for water rights transactions with irrigation area i The above formulas (8) and (9) represent the internal costs with and without water rights trading between irrigation areas, including the operating costs of the reservoirs within the irrigation area.

[0112] In terms of constraints: Formula (10) ensures that each irrigation area has a water balance in each scenario and each time period; Formula (11) ensures that the number of water rights sold by the irrigation area with excess water rights is equal to the number of water rights purchased by the irrigation area lacking water rights; the water rights transaction between the irrigation area management bureau and the irrigation area is constrained by Formula (12) and Formula (13); the water storage and release per unit time period of the reservoir within the irrigation area are constrained by Formula (14)-Formula (15), and Formula (16) restricts the water storage state of the reservoir; Formula (17) describes the change of water volume in the reservoir at t = 1, and Formula (18) describes the change of water volume in the reservoir during operation; the expected water storage ratio at t = 24 should satisfy Formula (19); the parameter It reflects the bargaining power of irrigation area i in the bargaining process and can be calculated by the following formula:

[0113]

[0114] Formula (21) ensures that the benefits of cooperation among irrigation areas are greater than the benefits of individual operations; Formula (22) indicates that the sum of all payments should be equal to the sum of all benefits.

[0115] 5. Solve the value of the objective function

[0116] The strategies of the leader and the follower need to be solved through optimization problems. In the present invention, the Karush-Kuhn-Tucker condition is used to transform the above two-layer problem into a first-level equivalent single-layer mixed integer linear programming problem. The solution process is solved by software MATLAB, which specifically includes:

[0117] 1) Decomposition of the GNB problem

[0118] At the level of irrigation area decision makers' response to prices (lower-level planning), the GNB problem is decomposed into two sub-problems that need to be solved in sequence: the cost minimization problem (Q1) and the bargaining problem (Q2).

[0119] Q1: Minimize operating costs

[0120]

[0121] The constraints are (10)-(19). Q1 uses the optimization model to calculate the optimal water rights trading volume of each irrigation area under different scenarios to determine the optimal water rights purchase and sales strategy of each irrigation area under a given water price, so as to minimize its operating costs.

[0122] Q2: Bargaining

[0123]

[0124] The constraints are (20)-(22), where Represents the operating cost saved by irrigation area i through water rights trading. Based on the solution of Q1 above, the bargaining power of each irrigation area in water rights trading and how to distribute benefits according to contribution are determined, so as to further analyze the strategy and benefit distribution of each irrigation area in water rights trading.

[0125] 2) Equivalent single-level mixed integer linear programming problem

[0126] Due to the existence of bilinear products in the above equations (1) and (6), the two-layer problem (1)-(6) and the problem Q1 (10)-(19), (26) is a nonlinear problem. Therefore, when solving the problem using MATLAB, this embodiment converts it into an equivalent single-level mixed integer linear programming (MILP) problem. The conversion process includes:

[0127] a. By using the KKT condition, the two-layer problem is equivalently reformulated as a nonlinear single-layer problem by replacing the low-level problems (10)-(19), (24). The nonlinear terms include complementary relaxation conditions and bilinear products;

[0128] b. The big-M method is used to replace the complementary relaxation conditions introduced by the Lagrange multipliers and constraint products; c. Based on the strong duality theory, the bilinear products in (1) and (6) are rewritten as equivalent linear expressions. The equivalent single-layer MILP formula is as follows:

[0129]

[0130] St(2)-(6),(10)-(19)

[0131]

[0132] Constraints (29)-(35) are dual constraints of constraints (10)-(19). The complementary slack conditions are shown in constraints (36)-(45).

[0133] 3) Bargaining issues

[0134] In the bargaining problem, the GNB theoretical model defines the distribution rules of the benefits obtained from sharing water rights. By solving the cost minimization problem (Q1), the optimal benefit of cooperation η can be obtained. i，w，* , and obtain the bargaining power of each irrigation area The comprehensive benefits obtained from cooperation among irrigation areas can be calculated by formula (27).

[0135] To facilitate MATLAB calculation, equation (27) is logarithmically transformed to obtain equation (46):

[0136]

[0137] st(20)-(22)

[0138]

[0139] After the above transformation, the problem can be transformed into a convex optimization problem, which can be solved using MATLAB and the market clearing can be obtained.

[0140] The method for allocating irrigation district water rights based on cooperative Stackelberg game provided by the present invention establishes a water rights allocation and transaction analysis framework for irrigation district water market based on cooperative Stackelberg game, allowing water rights transactions between irrigation areas in a decentralized manner to achieve win-win cooperation. In this analysis framework, the irrigation district management bureau can set relevant conditional risk value parameters according to its own risk preference level to weigh risks and expected returns and determine the water supply price. At the same time, if water rights transactions between irrigation areas are allowed, the responsiveness of each irrigation area to the water supply price can be further enhanced, and the control ability of the irrigation district management bureau over each irrigation area can be weakened. In addition, the analysis framework uses GNB theory to define the distribution rules for the benefits obtained from shared water rights to encourage water rights transactions and ensure the fairness of benefit distribution. The method of the present invention can further enhance the ability of the irrigation district water resources management system to cope with risks and improve the expected benefits of various participating roles at different risk aversion levels.

[0141] Embodiment 2:

[0142] On the basis of Example 1, in order to verify the feasibility and effectiveness of the present invention, this example uses the actual data of the Hetao Irrigation District, runs the model, compares the model prediction results with the actual observation data, and verifies the effectiveness and actual effect of the model of the present invention through the results of the example experiment, which specifically includes the following contents:

[0143] 1. Study area

[0144] This example uses the Hetao Irrigation District Administration and its five subordinate irrigation areas as research objects to test the model established in Example 1. The Hetao Irrigation District (40°19'-41°18'N, 106°20'-109°19'E) is located in Bayannur City, Inner Mongolia, and includes five irrigation areas: Yongji, Yichang, Ulanbuhe, Urad, and Jiefangzha ( Figure 3 ), located in the arid northwest of China, with an average annual rainfall of 168.5 mm and an annual evaporation of more than 2,000 mm. The irrigation area relies heavily on irrigation from the Yellow River, with an annual water diversion of about 4.84 billion m 3 , which is an area without irrigation and agriculture. The main crops grown in the irrigation area are wheat, corn and sunflower. In recent years, due to the high income of cash crop sunflower, the irrigation area has seen an increase in sunflower planting area and a sharp decline in wheat planting area. The planting structure of the irrigation area is shown in Figure 4 This example is based on the 2018 planting structure.

[0145] 2. Data Collection

[0146] The data used were collected from local yearbooks and research papers, including the attributes of each irrigation area (Table 1) and water prices (Table 2). Meteorological data were obtained from local meteorological stations and downloaded from the website of the China Meteorological Data Service Center. Since precipitation less than 5 mm is distributed on the soil surface, it is easy to evaporate and lose and cannot reach the plant root layer. Therefore, in this example, precipitation greater than 5 mm is regarded as effective precipitation. The effective precipitation in the Hetao region year by year is shown in Figure 2. Figure 5 .

[0147] Table 1 Irrigation domain attribute settings

[0148]

[0149] Table 2 Water price setting

[0150]

[0151]

[0152] The irrigation area is both a water consumption unit and a water production unit. The main sources of its output water are effective precipitation and groundwater. The groundwater exploitation in the Hetao Irrigation Area is relatively stable, but the effective precipitation has a significant fluctuation. This embodiment uses the hydrological frequency analysis method to perform frequency analysis on the effective precipitation in each irrigation area, and divides the inflow into four levels, namely 25%, 50%, 75%, and 90%. The P-III curve distribution of effective precipitation in each irrigation area is shown in Figure 1. Figure 6 .

[0153] In addition, based on the hydrological frequency analysis, this embodiment selects four representative scenarios: wet year (25%), normal year (50%), dry year (75%) and extremely dry year (90%) for analysis. The maximum water requirement of crops is calculated using the Penman formula. The effective water yield and maximum irrigation water requirement of crops under different scenarios in each irrigation area are shown in Figure 7 .

[0154] 3. Impact of risk aversion on irrigation district management decisions

[0155] By analyzing the variables related to β, such as the expected income of the irrigation district administration, CVaR and water price, we can reflect the attitude of decision makers in the face of risks and the impact of β on the decision results. β is defined to vary between 0.001 and 10 and the confidence level γ is set to 0.95.

[0156] 1) Expected Profit and CVaR

[0157] In order to further analyze the impact of the risk aversion weight coefficient β on the expected return of the irrigation district administration, the relationship between CVaR and expected return under different β values ​​is plotted ( Figure 8). As can be seen from the figure, higher expected returns come from higher risk levels. When β = 10, it means that the decision makers of the irrigation district administration are averse to economic losses caused by risks and take a more conservative attitude to avoid risks; on the contrary, when β = 0.001, it means that the decision makers of the irrigation district administration hope to have greater expected returns without considering risks. Therefore, in the actual operation process, the decision makers of the irrigation district can set the β value in the optimization model according to their own risk preference level to weigh the relationship between CVaR and expected returns.

[0158] The expected return of the irrigation district administration is not only affected by β, but also related to whether there is water rights trading between irrigation areas. If there is water rights trading between irrigation areas, as the β value increases, the expected return of the irrigation district administration varies between 1,032,454,052.77 CNY (β = 0.001) and 521,459,388.34 CNY (β = 10), and the corresponding CVaR decreases. In contrast, if there is no water rights trading between irrigation areas, the expected return and CVaR will vary much smaller, which means that in the case of water rights trading, the risk range that can be tolerated is larger.

[0159] 2) Impact of β changes on water prices

[0160] To further reflect the impact of β value on the optimal pricing strategy in the absence of water rights trading between irrigation areas, the water prices of each irrigation area at different β values ​​in different time periods are plotted ( Fig. 9 ) to illustrate the attitude of irrigation district decision makers towards risk. In the figure, the upper curve represents the selling price, which is defined as the price at which each irrigation area sells a unit of water right to the irrigation district administration; the lower curve represents the purchasing price, which is defined as the price at which each irrigation area purchases a unit of water right from the irrigation district administration. Considering the interests of the irrigation district administration, the purchasing price is generally greater than the selling price.

[0161] from Fig. 9 It can be seen that in the planning year, when β increases, the selling price shows an overall downward trend, while the purchasing price shows an upward trend. During the peak water use period in summer, the irrigation district administration provides a relatively high water supply price, forcing each irrigation district to improve water use efficiency and water conservation level, while reducing the operating pressure of the water supply system. Correspondingly, in winter, due to the reduction in irrigation demand, the price per unit water volume is also relatively low. Such a water price setting strategy can further improve the expected benefits of the irrigation district administration and the water conservation awareness of each irrigation district, but the water price is subject to constraints (4) and (5) and needs to be optimized within a reasonable range.

[0162] If water rights trading is allowed between irrigation areas, decision makers in each irrigation area may negotiate with each other to respond to the price, so the irrigation district administration needs to consider the overall demand of each irrigation area to determine the water supply price. In contrast, if water rights trading is allowed between irrigation areas, the difference in water supply prices between irrigation areas can be reduced ( Fig.10 ).from Fig.10 It can be seen that the irrigation district administration raises the water price when the agricultural water demand is high and lowers the water price when the agricultural water demand is low. This strategy can increase its expected benefits. By using this price stimulus method to encourage water rights trading between irrigation areas, the water use efficiency and water conservation level of each irrigation area can be further improved, while maximizing the benefits of surplus water rights.

[0163] 3) Price discrimination

[0164] As the leader of the irrigation district, the irrigation district administration can use price discrimination to set water supply prices to improve the overall water conservation level and economic benefits of the irrigation district. There are many factors affecting water prices, such as the expected natural effective precipitation in each irrigation area and whether water rights transactions can be carried out between irrigation areas.

[0165] In the absence of water rights trading between irrigation areas, the fluctuation of water prices is consistent with the fluctuation of water demand in each irrigation area. Around October, since the net demand of each irrigation area is almost the same, the price of buying and selling water rights is disorderly. However, if water rights trading can be carried out between irrigation areas, the fluctuation of water prices between irrigation areas will be reduced. The reason is that when the irrigation district administration provides price discrimination to different irrigation areas, the decision makers of each irrigation area will actively trade water rights to meet the supply and demand balance of actual irrigation water in the irrigation area. This reduces the dependence of the irrigation area on the irrigation district administration to a certain extent, and also broadens the way for each irrigation area to obtain agricultural water rights. Therefore, compared with the situation where there is no water rights trading between irrigation areas, the irrigation district administration can provide price discrimination in fewer time periods.

[0166] 4) Response from producers and marketers

[0167] The number of water rights transactions between irrigation district administrations and irrigation areas is shown in Fig.11 .from Fig.11It can be seen from the data that if water rights trading is allowed between irrigation areas, it will not have a significant impact on the amount of water rights purchased by each irrigation area from the irrigation district administration, but it will reduce the transaction volume with the irrigation district administration at the selling end. That is, if water rights trading is allowed between irrigation areas, each irrigation area can directly purchase water rights from neighboring irrigation areas without having to trade with the irrigation district administration. Compared with water rights trading between irrigation areas and irrigation district administrations, water rights trading between irrigation areas is more attractive, especially during the period of high agricultural irrigation water demand (May-September). Compared with purchasing water rights during the peak period of agricultural water use, for risk-averse decision makers (β=10), each irrigation area reduces the peak demand for water rights by taking water from the reservoir, and increases demand to replenish the water in the reservoir when the water price is low.

[0168] The above-mentioned model simulation results show that the model can further enhance the ability of irrigation district water resources management system to cope with risks and improve the expected benefits of various participating roles under different risk aversion levels.

[0169] The above description is only a preferred embodiment of the present invention and does not limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for allocating water rights in irrigation areas based on cooperative Stackelberg game, characterized by: The following steps are involved: S1. Establish an analytical framework; establish an analytical framework for water rights allocation and trading in the irrigation district water market based on the cooperative Stackelberg game model, and define the roles of the irrigation district administration and each irrigation area; the irrigation district administration, as a leader, is used to formulate strategies for water rights allocation and trading to maximize the comprehensive benefits of the irrigation district; each irrigation area is used to respond to the strategies of the irrigation district administration and maximize its own benefits through water rights trading; S2, define model variables and parameters; based on the above step 1, determine the roles and goals, and define the parameters and variables in the Stackelberg game model; S3. Constructing a leadership strategy; the irrigation district management bureau takes the maximum comprehensive benefit of the irrigation district as the management and control goal, while considering the risk control brought by uncertainty, and uses the CvaR stochastic programming method to characterize the expected risk of the irrigation district management bureau; S4. Construct follower feedback; each irrigation area provides feedback to the leader’s decision in a cooperative manner, shares idle water rights and distributes benefits according to contribution, while using stochastic programming methods to handle uncertainty; S5. Solve the two-level optimization problem; transform the two-level optimization problem into a first-level equivalent single-level mixed integer linear programming problem, including decomposing the GNB problem into two sub-problems: minimizing the cost problem and the bargaining problem; by solving the minimizing the cost problem, the optimal cooperative benefit and the bargaining power of each irrigation area are obtained.

2. The method for allocating irrigation district water rights based on cooperative Stackelberg game according to claim 1 is characterized by: In step S1, the irrigation district administration uses the price discrimination method to make each irrigation area participate in the water rights allocation process to maximize the economic benefits of the irrigation area; and uses CVaR to control the overall risk level from a global perspective; each irrigation area conducts water rights transactions in a decentralized manner to maximize the benefits of water resource utilization.

3. The method for allocating irrigation district water rights based on cooperative Stackelberg game according to claim 1 is characterized by: The decision model of the irrigation district administration in step S3 is expressed as follows: The above formula is the objective function, which consists of two parts: 1) the expected net benefit of the irrigation district administration; 2) the trade-off between the risk caused by uncertainty and the expected benefit, where β represents the weight.

4. The method for allocating irrigation district water rights based on cooperative Stackelberg game according to claim 1 is characterized by: In step S4, GNB is used to describe the water rights transfer process between irrigation areas. Under the premise that the irrigation area manager gives a price, the response of each irrigation area decision maker to the given price is expressed by quantifying the water rights transaction between the irrigation area and the irrigation area management bureau. The follower's reaction model is: The above formula is the objective function of the irrigation area decision maker's response to price. The objective function consists of two parts: 1) represents the water use cost function of irrigation area i in the absence of water rights trading between irrigation areas: 2) represents the water use cost function of irrigation area i when there is water rights trading between irrigation areas and the net cost paid by the irrigation area for water rights transactions with irrigation area i The sum of In the above formula, the parameters It reflects the bargaining power of irrigation area i in the bargaining process. and represents the decision variable vector of irrigation basin i with or without water rights trading between irrigation basins; the above formulas represent the internal costs with or without water rights trading between irrigation basins, including the operating costs of reservoirs within the irrigation basin.

5. The method for allocating irrigation district water rights based on cooperative Stackelberg game according to claim 1 is characterized by: The formula for minimizing the operating cost problem in step S5 is: The formula for the bargaining problem is:

6. The cooperative Stackelberg game irrigation area water rights allocation method according to claim 1 is characterized by: In step S5, the GNB problem is converted into an equivalent single-layer mixed integer linear programming problem, and the conversion process includes: a. By utilizing the KKT condition, the two-level problem is equivalently reformulated as a nonlinear single-level problem by replacing the low-level problem. The nonlinear terms include complementary relaxation conditions and bilinear products. b. Use the big-M method to replace the complementary relaxation conditions introduced by the Lagrange multipliers and constraint products; c. Based on the strong duality theory, the bilinear product is rewritten into an equivalent linear expression.

7. The cooperative Stackelberg game irrigation area water rights allocation method according to claim 5 is characterized by: In the bargaining problem, by solving the cost minimization problem, the optimal benefit of cooperation η can be obtained i，w，* , and obtain the bargaining power of each irrigation area By logarithmically changing the bargaining problem formula, we can get the following formula: This transforms it into a convex optimization problem, which is solved using MATLAB and the market clearing is obtained.

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