NSGA-II algorithm-based ecological flow process multi-objective optimization method

Abstract

The invention discloses an ecological flow process multi-objective optimization method based on an NSGA-II algorithm, and relates to the technical field of water conservancy projects. Comprising the steps of defining discrete decision variables, setting an objective function, constructing constraint conditions and a mixed penalty function, and configuring algorithm parameters; generating an initial solution meeting the ecological flow constraint, and repairing the solution which does not meet the constraint after genetic manipulation to ensure the feasibility of the solution; performing non-dominated sorting on the population to divide Pareto frontier layers, calculating a congestion distance of a solution in each frontier layer, realizing genetic manipulation through tournament selection, binary crossover simulation and polynomial variation, and updating the population in combination with environment selection; and extracting non-dominated solutions, screening optimization schemes from the non-dominated solutions, and outputting monthly scheduling schemes and water transfer and ecological flow shortage statistics of each reservoir under each scheme. According to the invention, automatic distribution of the ecological base flow, the basic ecological flow and the target ecological flow in the interannual period is realized.

Description

Technical Field

[0001] This invention relates to the field of water conservancy engineering technology, and more specifically to a multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm. Background Technology

[0002] Multi-objective optimization of water supply and ecology based on the NSGA-II algorithm is a key means to achieve water resource allocation and river ecological health. Traditional ecological flow calculation methods are usually based on historical hydrological data or simple percentages, without considering the dynamic scheduling of reservoir groups and the differences in ecological demand in different months. The calculation results are relatively static and lack flexibility. Traditional multi-objective optimization algorithms are prone to getting trapped in local optima and have difficulty handling the complex constraints of joint scheduling of multiple projects. Currently, there is little research on the rational allocation of the interannual composition of ecological flow for multiple water supply and ecology objectives under the influence of water diversion from water conservancy projects. Against this background, there is an urgent need to develop an ecological flow process algorithm that comprehensively considers water supply and ecology objectives and adapts to the influence of water diversion from large-scale water conservancy projects.

[0003] Numerous scholars both domestically and internationally have conducted extensive research on ecological flow and multi-objective optimization algorithms for water supply and ecology. Regarding the determination of ecological flow, Wang Yulian constructed a comprehensive method system for determining the ecological water demand process in water source areas through three key steps: method comparison, objective optimization, and feasibility verification. Based on 10 key dam site sections, she determined the ecological baseflow standard, sensitive ecological water demand threshold, and annual ecological water volume control target for both dry and non-dry seasons. In the field of multi-objective optimization of reservoir groups, traditional research on reservoir optimization scheduling usually focuses on the optimization of a single objective, or simplifies the multi-objective problem into a single objective problem by means of weighted summation, constraint transformation, etc. The research focus is mainly on the following aspects: power generation benefits, water supply security, ecological flow, flood control risks, etc. The joint optimization scheduling of reservoir groups has evolved from a single objective problem to a multi-objective problem that considers power generation, water supply, ecology, etc. The construction of scheduling models has become increasingly complex, exhibiting significant characteristics such as high dimension, dynamic time-varying, and strong nonlinearity. Common calculation methods include: NSGA-II algorithm, multi-objective particle swarm optimization algorithm, hybrid differential particle swarm optimization algorithm, etc. Among them, genetic algorithm and its improved algorithm play an important role in multi-objective optimization of reservoir groups. For example, Wang Bei constructed a multi-objective optimization algorithm for water supply-ecology-power generation of reservoir groups based on NSGA-II algorithm, and Peng Hui used NSGA-II multi-objective optimization algorithm to solve the joint scheduling scheme of reservoir groups. Taking the Longxi River Basin in Chongqing as an example, he compared and analyzed the multi-objective scheduling benefits of reservoir groups under different ecological water demand scenarios. Jin Yawen used a multi-objective particle swarm optimization algorithm to find the optimal solution for maximizing comprehensive benefits in reservoir ecological scheduling. Zhou Xiaoqing introduced an improved cuckoo search algorithm into the multi-objective scheduling optimization of the cascade reservoir group in the Xiangjiang River Basin. By integrating the minimum ecological flow constraint conditions of multiple control sections, a multi-objective collaborative scheduling model that takes into account both ecological protection and efficient water resource utilization was constructed. This algorithm innovatively adopts a dynamic adaptive step size mechanism and an elite retention strategy, demonstrating stronger global search capabilities and convergence accuracy when solving the Pareto front of the scheduling graph, and realizing the multi-objective equilibrium optimization of the reservoir group system at the basin scale, encompassing ecology, water supply, and power generation. Existing multi-objective optimization research on reservoir groups considering ecology has many methods but also has shortcomings. For example, some studies only use a single hydrological method to calculate ecological flow and treat it as a fixed constraint. This approach has the following drawbacks: the ecological flow obtained by different hydrological methods varies greatly, and relying on only one method may lead to overestimation or underestimation of ecological demand, affecting the rationality of the scheduling scheme. At the same time, many studies only treat ecological flow as a hard constraint, rather than incorporating it into the multi-objective optimization framework.To address the aforementioned problems, this invention proposes an innovative multi-objective optimization algorithm for water supply and ecology. Its core contributions are as follows: First, a three-level dynamic ecological flow system: breaking through the constraints of traditional fixed ecological flow, it constructs a three-level discretized coding scheme of base flow, basic ecology, and target ecology. Through the ecological flow matrix, it achieves three major functions: monthly differentiated configuration, independent setting of reservoirs at different cross-sections, and flexible interval constraints (automatic scheme degradation and protection for critical months). Second, compared with the standard NSGA-II algorithm, this invention can handle discrete schemes. For constraint handling, it adopts layered repair + dynamic penalty, providing a new solution for determining ecological flow and for water resource utilization and ecological protection of reservoir groups.

[0004] Therefore, proposing a multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm to address the difficulties in existing technologies is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0005] In view of this, this invention provides a multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm. By constructing a discrete decision variable encoding mechanism, a dynamic constraint repair strategy, and a hybrid penalty function system, it achieves efficient solving of the Pareto optimal solution set for high-dimensional nonlinear systems. This method employs non-dominated sorting and crowding distance calculation to realize population evolution, and maintains population diversity through simulated binary crossover and polynomial mutation. It is suitable for complex system decision problems balancing water supply and ecological objectives, and excels at handling optimization scenarios with integer variable constraints, dynamic state equations, and multi-level constraints.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm includes the following steps:

[0008] S1. Algorithm Initialization and Parameter Setting: Define discrete decision variables to characterize the monthly scheduling scheme of each reservoir in the reservoir group, set the objective function of minimizing water supply shortage and minimizing ecological flow shortage of the reservoir group, construct the constraint conditions and mixed penalty function, and configure the algorithm parameters.

[0009] S2. Feasible solution generation and repair: Generate initial solutions that satisfy ecological flow constraints, repair solutions that do not satisfy constraints after genetic operations, and ensure the feasibility of solutions.

[0010] S3, NSGA-II process: Perform non-dominated sorting on the population to divide the Pareto front layer, calculate the crowding distance of solutions within each front layer, implement genetic operations through tournament selection, simulated binary crossover, and polynomial mutation, and update the population in combination with environmental selection;

[0011] S4. Results Analysis and Output: Extract the non-dominated solutions of the Pareto front, select the optimal scheme from the non-dominated solutions, and output the monthly scheduling scheme of each reservoir and the statistics of water transfer and ecological flow shortage under each scheme.

[0012] Optionally, the discrete decision variables in S1 are defined as follows: the decision variables correspond to the scheduling schemes of each reservoir in the reservoir group for 12 months, and the values ​​include 0, 1, and 2, where 0 represents the ecological base flow, 1 represents the basic ecological flow, and 2 represents the target ecological flow.

[0013] Optionally, the specific content of the objective function in S1, which sets the minimum water supply shortage and the minimum ecological flow shortage of the reservoir group, is as follows:

[0014] Ecological goals:

[0015] In the formula: i is the reservoir number; t is the month; s is the ecological flow scheme; R(i,t,s) is the minimum ecological flow of reservoir i in month t according to scheme s, in m³. 3 / s;Q out (i,t) represents the discharge flow of reservoir i in month t, in m³. 3 / s; T is the number of seconds in a month;

[0016] Water supply target:

[0017] In the formula: D(t) is the water demand in month t, in meters. 3 / s; T(t) is the actual water diversion volume in month t, in m³. 3 / s.

[0018] Optionally, the specific content of constructing the constraints and mixed penalty function in S1 is as follows:

[0019] Hard constraints and dynamic constraints include minimum discharge flow constraints, water balance constraints, reservoir capacity constraints, and outflow constraints.

[0020] Minimum discharge flow constraint

[0021]

[0022] Q out (i,t)≥R bas e(i,t)

[0023] scheme(i,t)≥1,

[0024]

[0025] In the formula: scheme(i,t) is the ecological flow scheme for reservoir i in month t; Q out(i,t) is the discharge flow of reservoir i in month t; R base (i,t) is the ecological base flow of the i-th reservoir in month t; I(C) is the indicator function;

[0026] Water balance constraints:

[0027] V(i,t+1)=V(i,t)+I(i,t)×TQ out (i,t)×T

[0028] In the formula: V(i,t) is the reservoir capacity at the beginning of month t of reservoir i, in meters. 3 I(i,t) is the inflow rate of reservoir i in month t, in m³. 3 / s;Q out (i,t) represents the discharge flow of reservoir i in month t, in m³. 3 / s, T is the number of seconds in a month, and V(i,t+1) is the reservoir capacity at the beginning of the (t+1)th month of reservoir i;

[0029] Storage capacity constraints:

[0030] min_vol(i)≤V(i,t+1)≤max_vol(i)

[0031] In the formula: min_vol(i) is the dead capacity of reservoir i, in meters. 3 max_vol(i) is the reservoir capacity below the normal water level of reservoir i, in meters. 3 ;

[0032] Outflow constraints:

[0033]

[0034] In the formula: excess(i,t) represents the excess water volume of reservoir i in month t, in m³. 3 R(i,t,s) is the minimum ecological flow of reservoir i in month t, calculated using scheme s, in m³. 3 / s;

[0035] Hybrid penalty functions include: static penalty and dynamic penalty;

[0036] Static penalty:

[0037] The formula for the mandatory month constraint penalty is as follows:

[0038]

[0039] In the formula: X (i,t) Let I(·) be the scheduling scheme for reservoir i in month t, and let P be the indicator function. f For a fixed penalty coefficient, P s To enforce monthly restrictions and penalties;

[0040] The plan uses a lower bound penalty:

[0041] P t =max(0,6-N) t )×P l

[0042] P b =max(0,9-N) b )×P l

[0043] Where: N t N b These represent the number of months the target plan will be used, the number of months the target and basic plans will be used, and P. l P is the linear penalty coefficient. t The penalty for the number of months constrained by the target solution, P b Penalties are imposed based on the number of months constrained by the basic and target plans;

[0044] Dynamic penalties:

[0045] Penalty for exceeding storage capacity limits:

[0046]

[0047] In the formula: V (i,t) For the reservoir's capacity in month t, Due to storage capacity limitations, P represents the maximum and minimum storage capacity of reservoir i, respectively. r P is the relative penalty coefficient. vol Penalty for storage capacity constraints;

[0048] Ecological flow shortage penalty:

[0049] In the formula, P eco Punishment for ecological flow shortage;

[0050] Total penalty function: P tot =P s +P t +P b +P vol +P eco

[0051] The objective function is modified based on the total penalty function:

[0052] f'1=f1+w×P tot

[0053] f'2=f2+w×P tot

[0054] In the formula: f1 and f2 are the original water shortage and ecological shortage, respectively; w is the penalty weight coefficient; f'1 and f'2 are the modified water supply objective function and modified ecological objective function, respectively; P tot This is the total penalty item.

[0055] Optionally, the specific content of generating the initial solution that satisfies the ecological flow constraints in S2 is as follows:

[0056] Initialization: Set all months to ecological baseflow scheme 0;

[0057] Forced month processing: Randomly select either basic ecological flow scheme 1 or target ecological flow scheme 2 to fill the forced month;

[0058] Lower limit repair of the plan: If the number of months of the basic and target plans is less than 9 months, randomly upgrade some ecological base flow plans 0 to basic ecological flow plans 1 or target ecological flow plans 2; if the number of months of the target plan is less than 6 months, randomly upgrade some basic ecological flow plans 1 to target ecological flow plans 2.

[0059] Optionally, the specific details of repairing solutions that do not meet the constraints after genetic operations in S2 are as follows:

[0060] Forced monthly repair: If the forced month is ecological base flow scheme 0, it will be randomly set to basic ecological flow scheme 1 or target ecological flow scheme 2.

[0061] Lower limit repair of the scheme: If the number of months of the basic and target schemes is less than 9, randomly upgrade some ecological base flow schemes 0 to basic ecological flow scheme 1 or target ecological flow scheme 2; if the number of months of the target scheme is less than 6, randomly upgrade some basic ecological flow schemes 1 to target ecological flow scheme 2.

[0062] Optionally, in S3, a non-dominated sort is performed on the population to divide it into Pareto front layers. The specific details of calculating the crowding distance of solutions within each front layer are as follows:

[0063] Non-dominated sorting: Calculate the dominating set and dominated count for each solution, and assign solutions with a dominated count of 0 to the first frontier layer, iteratively updating subsequent frontier layers;

[0064] Crowding distance calculation: Sort the solutions of each leading edge layer according to the objective function, and calculate the crowding distance using the following formula:

[0065]

[0066] In the formula, f m (i+1) is the value of solution i+1 at the m-th objective, f m (i-1) is the value of solution i-1 at the m-th objective. These are the maximum and minimum values ​​of the m-th target in the current frontier, respectively.

[0067] Optionally, in S3, genetic operations are implemented through tournament selection, simulated binary crossover, and polynomial mutation. The specific details of updating the population in conjunction with environmental selection are as follows:

[0068] When selecting a tournament, choose the solution with the highest Pareto rating and the longest congestion distance.

[0069] Simulate binary crossover to generate offspring, then round and restrict the values ​​to {0, 1, 2};

[0070] Polynomial variation is expressed by the formula:

[0071] y'=y+δ(y u -y l )

[0072] y' d =round(min(max(y',y l ),y u ))

[0073]

[0074] In the formula, y is the current solution value, y' is the mutated value, and y' d Here, δ represents the discretized value, r is the uniform random number, and η is the variable. m To control the shape of the variation distribution, y l As the lower bound of the variable, y u δ1 represents the upper bound of the variable y, and δ1 represents the distance of the current solution value y from the lower bound y. l The relative distance, δ2 is the distance of the current solution value y from the upper bound y. u The relative distance,

[0075] Environmental selection: The parent and offspring generations are merged to form a 2N-sized population, where N is the population size. New populations are formed by selection based on the priority of the frontier layer and the crowding distance.

[0076] Optionally, the selection method for the optimization scheme in S4 is as follows: the best water transfer scheme is the scheme with the smallest water shortage, the best ecological scheme is the scheme with the smallest ecological water shortage, and the balanced scheme is the scheme with the closest Euclidean distance to the ideal point.

[0077] As can be seen from the above technical solution, compared with the prior art, the present invention discloses a multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm, the beneficial effects of which are:

[0078] 1) Through discrete variable encoding, dynamic constraint repair and hybrid penalty function mechanism, the automatic allocation of ecological base flow, basic ecological flow and target ecological flow in the interannual period is realized, so as to achieve the purpose of ecological protection and efficient use of water resources. It can efficiently output Pareto optimal solution set and is suitable for reservoir group water supply-ecology multi-objective optimization and ecological flow determination.

[0079] 2) Population evolution is achieved by using non-dominated sorting and crowding distance calculation. Population diversity is maintained by simulating binary crossover and polynomial mutation. It is suitable for complex system decision problems that balance water supply and ecological multi-objectives and is good at handling optimization scenarios with integer variable constraints, dynamic state equations and multi-level constraints. Attached Figure Description

[0080] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0081] Figure 1 A flowchart illustrating a multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm provided by this invention;

[0082] Figure 2 The present invention provides an overall flowchart of a multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm. Detailed Implementation

[0083] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0084] See Figure 1 and Figure 2 As shown, this invention discloses a multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm, comprising the following steps:

[0085] S1. Algorithm Initialization and Parameter Setting: Define discrete decision variables to characterize the monthly scheduling scheme of each reservoir in the reservoir group, set the objective function of minimizing water supply shortage and minimizing ecological flow shortage of the reservoir group, construct the constraint conditions and mixed penalty function, and configure the algorithm parameters.

[0086] S2. Feasible solution generation and repair: Generate initial solutions that satisfy ecological flow constraints, repair solutions that do not satisfy constraints after genetic operations, and ensure the feasibility of solutions.

[0087] S3, NSGA-II process: Perform non-dominated sorting on the population to divide the Pareto front layer, calculate the crowding distance of solutions within each front layer, implement genetic operations through tournament selection, simulated binary crossover, and polynomial mutation, and update the population in combination with environmental selection;

[0088] S4. Results Analysis and Output: Extract the non-dominated solutions of the Pareto front, select the optimal scheme from the non-dominated solutions, and output the monthly scheduling scheme of each reservoir and the statistics of water transfer and ecological flow shortage under each scheme.

[0089] Furthermore, the discrete decision variables in S1 are defined as follows: the decision variables correspond to the scheduling plans of each reservoir in the reservoir group for 12 months, and the values ​​include 0, 1, and 2, where 0 represents the ecological base flow, 1 represents the basic ecological flow, and 2 represents the target ecological flow.

[0090] Furthermore, the specific content of the objective function set in S1 to minimize the water supply shortage and the ecological flow shortage of the reservoir group is as follows:

[0091] Ecological goals:

[0092] In the formula: i is the reservoir number; t is the month; s is the ecological flow scheme; R(i,t,s) is the minimum ecological flow of reservoir i in month t according to scheme s, in m³. 3 / s;Q out (i,t) represents the discharge flow of reservoir i in month t, in m³. 3 / s; T is the number of seconds in a month;

[0093] Specifically, s represents the ecological flow scheme, s=0: ecological base flow, s=1: basic ecological flow, s=2: target ecological flow.

[0094] Water supply target:

[0095] In the formula: D(t) is the water demand in month t, in meters. 3 / s; T(t) is the actual water diversion volume in month t, in m³. 3 / s.

[0096] Furthermore, the specific details of constructing the constraints and mixed penalty function in S1 are as follows:

[0097] Hard constraints and dynamic constraints include minimum discharge flow constraints, water balance constraints, reservoir capacity constraints, and outflow constraints.

[0098] Minimum discharge flow constraint

[0099]

[0100] Q out (i,t)≥R base (i,t)

[0101] scheme(i,t)≥1,

[0102]

[0103] In the formula: scheme(i,t) is the ecological flow scheme for reservoir i in month t; Q out (i,t) is the discharge flow of reservoir i in month t; R base (i,t) is the ecological base flow of the i-th reservoir in month t; I(C) is the indicator function;

[0104] Specifically, I(C) is a single entity, and C is scheme(i,t) = 2 or scheme(i,t) ≥ 1.

[0105] Water balance constraints:

[0106] V(i,t+1)=V(i,t)+I(i,t)×TQ out (i,t)×T

[0107] In the formula: V(i,t) is the reservoir capacity at the beginning of month t of reservoir i, in meters. 3 I(i,t) is the inflow rate of reservoir i in month t, in m³. 3 / s;Q out (i,t) represents the discharge flow of reservoir i in month t, in m³. 3 / s, T is the number of seconds in a month, and V(i,t+1) is the reservoir capacity at the beginning of the (t+1)th month of reservoir i;

[0108] Storage capacity constraints:

[0109] min_vol(i)≤V(i,t+1)≤max_vol(i)

[0110] In the formula: min_vol(i) is the dead capacity of reservoir i, in meters. 3 max_vol(i) is the reservoir capacity below the normal water level of reservoir i, in meters. 3 ;

[0111] Outflow constraints:

[0112]

[0113] In the formula: excess(i,t) represents the excess water volume of reservoir i in month t, in m³. 3 R(i,t,s) is the minimum ecological flow of reservoir i in month t, calculated using scheme s, in m³. 3 / s;

[0114] Hybrid penalty functions include: static penalty and dynamic penalty;

[0115] Static penalty:

[0116] The formula for the mandatory month constraint penalty is as follows:

[0117]

[0118] In the formula: x (i,t) Let I(·) be the scheduling scheme for reservoir i in month t, and let P be the indicator function. f For a fixed penalty coefficient, P s To enforce monthly restrictions and penalties;

[0119] The plan uses a lower bound penalty:

[0120] P t =max(0,6-N) t )×P l

[0121] P b =max(0,9-N) b )×P l

[0122] Where: N t N b These represent the number of months the target plan will be used, the number of months the target and basic plans will be used, and P. l P is the linear penalty coefficient. t The penalty for the number of months constrained by the target solution, P b Penalties are imposed based on the number of months constrained by the basic and target plans;

[0123] Dynamic penalties:

[0124] Penalty for exceeding storage capacity limits:

[0125]

[0126] In the formula: V (i,t) For the reservoir's capacity in month t, Due to storage capacity limitations, P represents the maximum and minimum storage capacity of reservoir i, respectively. r P is the relative penalty coefficient. vol Penalty for storage capacity constraints;

[0127] Ecological flow shortage penalty:

[0128] In the formula, P eco Punishment for ecological flow shortage;

[0129] Total penalty function: P tot =P s +P t +P b +P vol +P eco

[0130] The objective function is modified based on the total penalty function:

[0131] f'1=f1+w×P tot

[0132] f'2=f2+w×P tot

[0133] In the formula: f1 and f2 are the original water shortage and ecological shortage, respectively; w is the penalty weight coefficient; f'1 and f'2 are the modified water supply objective function and modified ecological objective function, respectively; P tot This is the total penalty item.

[0134] Specifically, hard constraints include reservoir capacity limits and the selection of plans for sensitive months, while dynamic constraints include minimum discharge flow and water transfer requirements.

[0135] The algorithm parameters are set as follows: population size: 200, maximum number of iterations: 900, crossover probability: 0.9, mutation probability: 0.1.

[0136] Furthermore, the specific content of generating the initial solution that satisfies the ecological flow constraint in S2 is as follows:

[0137] Initialization: Set all months to ecological baseflow scheme 0;

[0138] Forced month processing: Randomly select either basic ecological flow scheme 1 or target ecological flow scheme 2 to fill the forced month;

[0139] Lower limit repair of the plan: If the number of months of the basic and target plans is less than 9 months, randomly upgrade some ecological base flow plans 0 to basic ecological flow plans 1 or target ecological flow plans 2; if the number of months of the target plan is less than 6 months, randomly upgrade some basic ecological flow plans 1 to target ecological flow plans 2.

[0140] Specifically, the initial solution is generated and satisfies the ecological flow constraints, namely: the ecological base flow scheme 0 must be used for at least one month, the target scheme must be used for at least 6 months of the year, and the basic or target scheme must be used for at least 9 months of the year.

[0141] Furthermore, the specific details of repairing solutions that do not meet the constraints after genetic operations in S2 are as follows:

[0142] Forced monthly repair: If the forced month is ecological base flow scheme 0, it will be randomly set to basic ecological flow scheme 1 or target ecological flow scheme 2.

[0143] Lower limit repair of the scheme: If the number of months of the basic and target schemes is less than 9, randomly upgrade some ecological base flow schemes 0 to basic ecological flow scheme 1 or target ecological flow scheme 2; if the number of months of the target scheme is less than 6, randomly upgrade some basic ecological flow schemes 1 to target ecological flow scheme 2.

[0144] Specifically, infeasible solutions are repaired by constraining solutions after mutation or crossover to ensure that all hard constraints are met, and the months are forced to be March, April, August, and September.

[0145] Furthermore, in S3, a non-dominated sort is performed on the population to divide it into Pareto front layers. The specific details of calculating the crowding distance of solutions within each front layer are as follows:

[0146] Non-dominated sorting: Calculate the dominating set and dominated count for each solution, and assign solutions with a dominated count of 0 to the first frontier layer, iteratively updating subsequent frontier layers;

[0147] Specifically, for each solution A, initialize and compute the dominant solution set S. A and the number of dominated individuals n A .

[0148] First-layer frontier, all n A The solution with 0 is assigned to the first-level Pareto front.

[0149] Iterative layering: For each solution A in the current frontier, iterate through the solutions B ∈ S that it governs. A , will n B Decrease by 1; if n B =0, then B is assigned to the next frontier.

[0150] Crowding distance calculation: Sort the solutions of each leading edge layer according to the objective function, and calculate the crowding distance using the following formula:

[0151]

[0152] In the formula, f m (i+1) is the value of solution i+1 at the m-th objective, f m (i-1) is the value of solution i-1 at the m-th objective. These are the maximum and minimum values ​​of the m-th target in the current frontier, respectively.

[0153] Specifically, by sorting the solutions in each front layer according to each objective function value, the normalized distance difference between adjacent solutions in the objective space is calculated to quantify the distribution density of solutions in the objective space, thus avoiding excessive clustering of front-end solutions. For each objective m, after sorting by value...

[0154] Furthermore, in S3, genetic operations are implemented through tournament selection, simulated binary crossover, and polynomial mutation, combined with environmental selection to update the population. The specific details are as follows:

[0155] When selecting a tournament, choose the solution with the highest Pareto rating and the longest congestion distance.

[0156] Simulate binary crossover to generate offspring, then round and restrict the values ​​to {0, 1, 2};

[0157] Specifically, for parent generations p1 and p2, the generated child generations c1 and c2 are:

[0158]

[0159] Where β is determined by the distribution parameter η c control:

[0160]

[0161] In the formula: u is a uniform random number in [0,1]. It is a discrete variable that is rounded after crossover and restricted to {0,1,2}.

[0162] Polynomial variation is expressed by the formula:

[0163] y'=y+δ(y u -y l )

[0164] y' d =round(min(max(y',y l ),y u ))

[0165]

[0166] In the formula, y is the current solution value, y' is the mutated value, and y' d Here, δ represents the discretized value, r is the uniform random number, and η is the variable. m To control the shape of the variation distribution, y l As the lower bound of the variable, y u δ1 represents the upper bound of the variable y, and δ1 represents the distance of the current solution value y from the lower bound y. l The relative distance, δ2 is the distance of the current solution value y from the upper bound y. u The relative distance,

[0167] Specifically, by introducing random perturbations into the solution space, the algorithm avoids getting trapped in local optima. Its core features are: strong controllability: adjusting the mutation amplitude through distribution parameters; boundary protection: ensuring that the mutated value does not exceed the variable's domain; discrete adaptation: designing a rounding mechanism for discrete variables.

[0168] Environmental selection: The parent and offspring generations are merged to form a 2N-sized population, where N is the population size. New populations are formed by selection based on the priority of the frontier layer and the crowding distance.

[0169] Specifically, first, the populations are merged to combine the parent and offspring generations into 2N solutions (N being the population size). Then, a non-dominated sort is performed, that is, the merged solutions are sorted according to the frontier layer. Finally, a new population is filled, that is, solutions at higher frontier levels are selected first, until the population size N is exceeded. For the last frontier layer, the remaining positions are filled according to the crowding distance from largest to smallest. This achieves the goal of merging the parent and offspring populations while preserving the optimal solution.

[0170] Furthermore, the selection method for the optimization scheme in S4 is as follows: the best water transfer scheme is the scheme with the smallest water shortage, the best ecological scheme is the scheme with the smallest ecological water shortage, and the balanced scheme is the scheme that is closest to the ideal point in Euclidean geometry.

[0171] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on the differences from other embodiments. The same or similar parts between the various embodiments can be referred to each other.

[0172] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm, characterized in that, Includes the following steps: S1. Algorithm Initialization and Parameter Setting: Define discrete decision variables to characterize the monthly scheduling scheme of each reservoir in the reservoir group, set the objective function of minimizing water supply shortage and minimizing ecological flow shortage of the reservoir group, construct the constraint conditions and mixed penalty function, and configure the algorithm parameters. S2. Feasible solution generation and repair: Generate initial solutions that satisfy ecological flow constraints, repair solutions that do not satisfy constraints after genetic operations, and ensure the feasibility of solutions. S3, NSGA-II process: Perform non-dominated sorting on the population to divide the Pareto front layer, calculate the crowding distance of solutions within each front layer, implement genetic operations through tournament selection, simulated binary crossover, and polynomial mutation, and update the population in combination with environmental selection; S4. Results Analysis and Output: Extract the non-dominated solutions of the Pareto front, select the optimal scheme from the non-dominated solutions, and output the monthly scheduling scheme of each reservoir and the statistics of water transfer and ecological flow shortage under each scheme.

2. The multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm according to claim 1, characterized in that, The discrete decision variables in S1 are defined as follows: the decision variables correspond to the scheduling plans of each reservoir in the reservoir group for 12 months, and the values ​​include 0, 1, and 2, where 0 represents the ecological base flow, 1 represents the basic ecological flow, and 2 represents the target ecological flow.

3. The multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm according to claim 1, characterized in that, The specific content of the objective function set in S1 to minimize the water supply shortage and the ecological flow shortage of the reservoir group is as follows: Ecological goals: In the formula: i is the reservoir number; t is the month; s is the ecological flow scheme; R(i,t,s) is the minimum ecological flow of scheme s for reservoir i in month t, in m³. 3 / s;Q out (i,t) represents the discharge flow of reservoir i in month t, in m³. 3 / s; T represents the number of seconds in a month; Water supply target: In the formula: D(t) is the water demand in month t, in meters. 3 / s; T(t) is the actual water diversion volume in month t, in m³. 3 / s.

4. The multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm according to claim 1, characterized in that, The specific details of constructing the constraints and mixed penalty function in S1 are as follows: Hard constraints and dynamic constraints include minimum discharge flow constraints, water balance constraints, reservoir capacity constraints, and outflow constraints. Minimum discharge flow constraint In the formula: scheme(i,t) is the ecological flow scheme for reservoir i in month t; Q out (i,t) is the discharge flow of reservoir i in month t; R base (i,t) is the ecological base flow of the i-th reservoir in month t; I(C) is the indicator function; Water balance constraints: V(i,t+1)=V(i,t)+I(i,t)×T-Q out (i,t)×T In the formula: V(i,t) is the reservoir capacity at the beginning of month t of reservoir i, in meters. 3 ; I(i,t) is the inflow rate of reservoir i in month t, in m³. 3 / s;Q out (i,t) represents the discharge flow of reservoir i in month t, in m³. 3 / s, T is the number of seconds in a month, and V(i,t+1) is the reservoir capacity at the beginning of the (t+1)th month of reservoir i; Storage capacity constraints: min_vol(i)≤V(i,t+1)≤max_vol(i) In the formula: min_vol(i) is the dead capacity of reservoir i, in meters. 3 max_vol(i) is the reservoir capacity below the normal water level of reservoir i, in meters. 3 ; Outflow constraints: In the formula: excess(i,t) represents the excess water volume of reservoir i in month t, in m³. 3 R(i,t,s) is the minimum ecological flow of reservoir i in month t, calculated using scheme s, in m³. 3 / s; Hybrid penalty functions include: static penalty and dynamic penalty; Static penalty: The formula for the mandatory month constraint penalty is as follows: In the formula: X (i,t) Let I(·) be the scheduling scheme for reservoir i in month t, and let P be the indicator function. f For a fixed penalty coefficient, P s To enforce monthly restrictions and penalties; The plan uses a lower bound penalty: P t =max(0,6-N t )×P l P b =max(0,9-N b )×P l Where: N t N b These represent the number of months the target plan will be used, the number of months the target and basic plans will be used, and P. l P is the linear penalty coefficient. t The penalty for the number of months constrained by the target solution, P b Penalties are imposed based on the number of months constrained by the basic and target plans; Dynamic penalties: Penalty for exceeding storage capacity limits: In the formula: V (i,t) For the reservoir's capacity in month t, Due to storage capacity limitations, P represents the maximum and minimum storage capacity of reservoir i, respectively. r P is the relative penalty coefficient. vol Penalty for storage capacity constraints; Ecological flow shortage penalty: In the formula, P eco Punishment for ecological flow shortage; Total penalty function: P tot =P s +P t +P b +P vol +P eco The objective function is modified based on the total penalty function: f′1=f1+w×P tot f′2=f2+w×P tot In the formula: f1 and f2 are the original water shortage and ecological shortage, respectively; w is the penalty weight coefficient; f'1 and f'2 are the corrected water supply objective function and the corrected ecological objective function, respectively; P tot This is the total penalty item.

5. A multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm according to claim 2, characterized in that, The specific content of generating the initial solution that satisfies the ecological flow constraint in S2 is as follows: Initialization: Set all months to ecological baseflow scheme 0; Forced month processing: Randomly select either basic ecological flow scheme 1 or target ecological flow scheme 2 to fill the forced month; Lower limit repair of the plan: If the number of months of the basic and target plans is less than 9 months, randomly upgrade some ecological base flow plans 0 to basic ecological flow plans 1 or target ecological flow plans 2; if the number of months of the target plan is less than 6 months, randomly upgrade some basic ecological flow plans 1 to target ecological flow plans 2.

6. The multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm according to claim 5, characterized in that, The specific steps in S2 to repair solutions that do not meet the constraints after genetic operations are as follows: Forced monthly repair: If the forced month is ecological base flow scheme 0, it will be randomly set to basic ecological flow scheme 1 or target ecological flow scheme 2. Lower limit repair of the scheme: If the number of months of the basic and target schemes is less than 9, randomly upgrade some ecological base flow schemes 0 to basic ecological flow scheme 1 or target ecological flow scheme 2; if the number of months of the target scheme is less than 6, randomly upgrade some basic ecological flow schemes 1 to target ecological flow scheme 2.

7. The multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm according to claim 1, characterized in that, In S3, a non-dominated sort is performed on the population to divide it into Pareto front layers. The specific details of calculating the crowding distance of solutions within each front layer are as follows: Non-dominated sorting: Calculate the dominating set and dominated count for each solution, and assign solutions with a dominated count of 0 to the first frontier layer, iteratively updating subsequent frontier layers; Crowding distance calculation: Sort the solutions of each leading edge layer according to the objective function, and calculate the crowding distance using the following formula: In the formula, f m (i+1) is the value of solution i+1 at the m-th objective, f m (i-1) is the value of solution i-1 at the m-th objective. These are the maximum and minimum values ​​of the m-th target in the current frontier, respectively.

8. The multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm according to claim 1, characterized in that, In S3, genetic operations are implemented through tournament selection, simulated binary crossover, and polynomial mutation. The specific details of updating the population in conjunction with environmental selection are as follows: When selecting a tournament, choose the solution with the highest Pareto rating and the longest congestion distance. Simulate binary crossover to generate offspring, then round and restrict the values ​​to {0, 1, 2}; Polynomial variation is expressed by the formula: y′=y+δ(y u -and l ) and' d =round(min(max(y′,y l ),and u )) In the formula, y is the current solution value, y' is the mutated value, and y' d Here, δ represents the discretized value, r is the uniform random number, and η is the variable. m To control the shape of the variation distribution, y l As the lower bound of the variable, y u δ1 represents the upper bound of the variable y, and δ1 represents the distance of the current solution value y from the lower bound y. l The relative distance, δ2 is the distance of the current solution value y from the upper bound y. u The relative distance, Environmental selection: The parent and offspring generations are merged to form a 2N-sized population, where N is the population size. New populations are formed by selection based on the priority of the frontier layer and the crowding distance.

9. A multi-objective optimization method for ecological flow processes based on the NSGA-II algorithm according to claim 1, characterized in that, The selection method for the optimal scheme in S4 is as follows: the best water transfer scheme is the scheme with the smallest water shortage, the best ecological scheme is the scheme with the smallest ecological water shortage, and the balanced scheme is the scheme that is closest to the ideal point in Euclidean algorithm.

Images