Pump station multi-objective optimization scheduling method based on multi-strategy enhanced African vulture algorithm

Abstract

The invention discloses a pump station multi-objective optimization scheduling method based on a multi-strategy enhanced African vulture algorithm, and relates to the technical field of water conservancy projects. The method comprises the following steps: acquiring basic data of pump station operation, and establishing a multi-target optimization scheduling model which takes minimization of electric charge and minimization of the quadratic sum of water shortage of a water receiving area as target functions during the pump station operation period and takes water pump power constraint, water pump pumping capacity constraint and starting-up water pump number constraint as constraint conditions; according to the multi-strategy enhanced multi-target African vulture algorithm, a crowding degree distance sorting mechanism and a hypercube mechanism are adopted to select double leaders to guide a population, an archive-based crossover and mutation strategy is adopted to reduce the possibility that the multi-target algorithm falls into local optimum, and a progressive repair strategy is adopted to repair an out-of-border solution after a main loop is finished. And the multi-strategy enhanced multi-target African vulture algorithm is adopted to carry out pump station multi-target optimization scheduling simulation, so that the operation efficiency of the pump station can be improved.

Description

Technical Field

[0001] This invention relates to the field of water conservancy engineering technology, and in particular to a multi-objective optimization scheduling method for pumping stations based on the multi-strategy enhanced African vulture algorithm. Background Technology

[0002] Pumping stations, as crucial infrastructure in water resource allocation and management, are widely used in various fields such as agricultural irrigation, urban water supply, flood control and drainage, and ecological water replenishment. With rapid socio-economic development and the increasing prominence of water scarcity, pumping station allocation optimization has gradually shifted from traditional single-objective allocation to multi-objective comprehensive optimization allocation that considers economic, ecological, and social benefits. This change means that pumping station operation and management no longer merely pursues the optimization of a specific goal, but rather needs to seek a balance among multiple conflicting or contradictory objectives, such as minimizing pumping station energy consumption, maximizing water supply guarantee rate, maximizing rational utilization of water resources, and maximizing ecological and environmental benefits.

[0003] Traditional methods for optimizing pump station scheduling, such as empirical methods, linear programming, and dynamic programming, typically suffer from high computational complexity, difficulty in effectively handling complex nonlinear problems, and a tendency to get trapped in local optima. Especially when there are significant conflicts between objectives, traditional methods struggle to obtain balanced solutions among multiple objectives, failing to meet the complex and ever-changing demands of practical scheduling. Therefore, achieving efficient solutions to multi-objective problems in pump station scheduling has become one of the critical technical challenges that urgently needs to be overcome in the field.

[0004] In recent years, with the rapid development of intelligent optimization algorithms, swarm intelligence optimization algorithms such as Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Grey Wolf Optimization (GWO), and African Vulture Algorithm (AVOA) have received increasing attention and application due to their strong robustness, adaptability, and flexibility. In particular, the African Vulture Algorithm, as a novel swarm intelligence algorithm proposed in recent years, possesses advantages such as population cooperation, outstanding global optimization capabilities, and fast convergence speed, and has gradually become one of the important tools for solving complex engineering optimization problems. However, existing standard African Vulture algorithms still have shortcomings when dealing with multi-objective optimization problems in pumping station systems, such as insufficient population diversity, susceptibility to local optima in the later stages, and poor optimization accuracy and stability, making it difficult to achieve ideal results when applied to multi-objective scheduling problems in pumping station systems.

[0005] To further improve the solution accuracy and stability of the pump station optimization scheduling model, it is urgent to propose an enhanced multi-objective intelligent optimization method with stronger exploration capabilities and higher accuracy to overcome the bottleneck problems existing in the current pump station optimization scheduling. The multi-strategy enhanced African Vulture algorithm, based on the standard AVOA algorithm, introduces multiple efficient strategies to effectively improve the algorithm's global search capability and local exploitation capability for multi-objective problems, thereby obtaining a more diverse, stable, and higher-quality Pareto non-dominated solution set, providing solid technical support for multi-objective optimization scheduling of pump stations.

[0006] In conclusion, the development of a multi-strategy enhanced African vulture algorithm-based multi-objective optimization scheduling method for pumping stations is of significant theoretical and practical value for improving pumping station operating efficiency, achieving comprehensive and efficient utilization of water resources, balancing economic and ecological benefits, and promoting the intelligent development of the water conservancy industry. Summary of the Invention

[0007] Purpose of the invention:

[0008] This invention overcomes the shortcomings of existing technologies and aims to solve the following technical problem: to provide an enhanced multi-objective African vulture algorithm with strong global optimization capability and good optimization stability to solve the multi-objective scheduling problem of pump station systems.

[0009] Technical solution:

[0010] The multi-objective optimization scheduling method for pumping stations based on the multi-strategy enhanced African vulture algorithm described in this invention includes the following steps:

[0011] Step 1: Obtain basic data on the operation of the pumping station;

[0012] Step 2: Establish a multi-objective optimization scheduling model with the objective functions of minimizing electricity costs and minimizing the sum of squares of water shortage in the water receiving area during pump station operation, and with constraints on pump power, pumping capacity, and the number of pumps in operation.

[0013] Step 3: Construct a multi-strategy enhanced multi-objective African vulture algorithm (EMOAVOA), in which...

[0014] The multi-strategy enhanced multi-objective African vulture algorithm uses a crowding distance sorting mechanism and a hypercube mechanism to select dual leaders to guide the population. It adopts an archive-based crossover and mutation strategy to reduce the possibility of the multi-objective algorithm getting stuck in local optima. It also adopts a progressive repair strategy to repair out-of-bounds solutions after the main loop ends, so as to enhance the competitiveness of the population and ensure its diversity.

[0015] Step four: Use the multi-strategy enhanced multi-objective African vulture algorithm to simulate the multi-objective optimal scheduling of the pumping station.

[0016] Furthermore, the basic data includes: the basic model of the water pump, the rated speed of the water pump, the rated flow rate of the water pump, the motor efficiency of the water pump, the transmission efficiency of the water pump, the efficiency of the frequency converter of the water pump, the characteristic curve of the water pump, and the electricity cost during the operation of the water pump.

[0017] Furthermore, the objective function expression of the multi-objective optimization scheduling model is:

[0018]

[0019] In the formula, F1 is the daily electricity cost of the pumping station; JN is the number of pump units in the pumping station; ρ is the water density; g is the acceleration due to gravity; Q ij Let Hr be the flow rate of the j-th pump in the i-th time period; ij Let η be the head of the j-th pump in the i-th time period; ij Let η be the pump efficiency of the j-th pump in the i-th time period; mot For motor efficiency; η int F2 is the transmission efficiency; F2 is the sum of the squares of the daily water shortage supplied by the pumping station to the water plant; Q i T represents the flow rate of the pumping station during the i-th time period; i Let W be the duration of the i-th time period; W is the daily water demand of the water plant.

[0020] Furthermore, the constraints of the multi-objective optimization scheduling model are expressed as follows:

[0021]

[0022] Where: N ij Represents the actual operating power of the j-th pump in the i-th time period; Ne represents the rated power of a single unit; Q min Q max The minimum and maximum operating flow rates of the pumping station in the i-th time period, respectively; G i G represents the number of pumps started in the i-th time period; imin G imin Let represent the minimum and maximum number of pumps in operation during the i-th time period, respectively.

[0023] Furthermore, in the step of selecting dual leaders to guide the population using the crowding distance ranking mechanism and the hypercube mechanism, the formulas for the crowding distance ranking mechanism and the hypercube mechanism are as follows:

[0024]

[0025] In the formula: d i (p) indicates that in the target f i Crowded distance; f i,max and f i,min They represent the target f respectively iThe maximum and minimum values ​​of f; i (p i+1 ) and f i (p i-1 ) represent solutions p in the objective f i The target values ​​of the left and right neighbors; C(p) is the total congestion distance for solving p; n obj Since the objective number is the number of problems, the solution with the largest C(p) should be selected as the leader.

[0026]

[0027] In the formula: P i Let H represent the probability of selecting each hypercube; H is a constant greater than one; N i It represents the number of Pareto optimal solutions obtained in the i-th segment.

[0028] Furthermore, the formula for the archive-based crossover mutation strategy is as follows:

[0029] mu=Amu+0.1·tan(π·(rand-0.5))

[0030]

[0031] P mu (t)=P(t)+mu·(BestVulture1(t)-P(t)+P r1 (t)-P r2 (t))

[0032] In the formula: mu is the mutation probability; Amu is the successful mutation probability value stored in the archive, rand is a random number; cr i It is the crossover probability; Acr is the successful crossover probability value stored in the archive; P mu P(t) represents the position after mutation at iteration number t; P(t) represents the position at iteration number t; P r1 (t) and P r2 (t) represents the positions of two vultures randomly selected from the population; BestVulture1(t) is the position of the best vulture in the population.

[0033] Furthermore, the formula for the asymptotic repair strategy for out-of-bounds solutions is as follows:

[0034]

[0035] DS = |X d -U d |or DS=|X d -L d |

[0036] progressive_factor = tanh(DS)

[0037]

[0038] In the formula: DS represents the distance between the out-of-bounds solution and the boundary; X d U represents the current position of the individual in dimension d; d L represents the upper bound of an individual in dimension d; d is the lower bound of individual d in dimension d; progressive_factor is the asymptotic factor used to control the intensity or step size of the repair process, which determines the adjustment range of the solution in each iteration, thus affecting the convergence speed and stability of the algorithm.

[0039] The African Vulture Optimization Algorithm (AVOA) simulates the foraging and food competition behavior of African vultures. The algorithm operates based on the following principles:

[0040] (1) Define the hunger rate of vultures:

[0041] Vultures excel at soaring through the air in search of food, especially when locating scarce resources. They can fly for extended periods in search of prey. However, if a vulture becomes extremely hungry, its energy reserves will be insufficient for prolonged flight. In this situation, it will seek food near stronger vultures, and a hungry vulture becomes highly aggressive. This behavior can be differentiated by the hunger rate F. When F is greater than 1, the vulture is more energetic, and the algorithm is in the exploration phase. When F is less than 1, the vulture enters the hunger phase, and the algorithm switches to the development phase. The vulture's hunger rate is defined as follows:

[0042]

[0043] In the formula, F is the hunger rate; rand is a random number generated in the range (0,1); z is a random number generated in the range (-1,1); t is the current iteration number; max_t is the maximum iteration number; G is a variable that changes with the iteration number, and its expression is shown in formula (4).

[0044]

[0045] In the formula, h is a random number generated in the range (-2,2); w is a constant, which is set to 2.5 in this study.

[0046] (2) Exploration phase:

[0047] When a vulture's hunger rate is greater than 1, it has sufficient energy to search for food, exhibiting excellent exploration and flight abilities. AVOA uses two different strategies to locate food. The choice between these strategies is determined by random numbers and a constant P1 that controls the exploration phase.

[0048]

[0049] In the formula, P(t+1) is the position vector of the vulture in the next iteration, and P1 is a constant controlling the exploration phase.

[0050] P(t+1)=R(t)-D(t)×F (4)

[0051] In the formula, R(t) is one of the best vultures in the current iteration number, which is randomly selected by roulette wheel; P(t) is the position vector of the vulture in the iteration; D(t) is determined by formula (7), and F is determined by formula (3).

[0052] D(t)=|2×rand×R(t)-P(t)| (5)

[0053] P(t+1)=R(t)-F+rand×((ub-lb)×rand+lb) (6)

[0054] In the formula, ub and lb represent the upper and lower limits of the variable, respectively.

[0055] (3) First phase of development

[0056] When the hunger rate is between 0.5 and 1, the vultures no longer have enough energy to sustain prolonged high-altitude flight. At this stage, some stronger vultures have already found food sources, and these vultures may clash with other vultures attempting to compete for the same food sources. Meanwhile, other vultures continue low-altitude circling flight to further pinpoint food sources. In the first phase of development, AVOA simulated both behaviors—food source competition and low-altitude circling flight. A random selection was made between these two behaviors for position updates. The formula is as follows:

[0057]

[0058] In the formula: P(t+1) is the position vector of the vulture in the next iteration, and P2 is a constant that controls the first stage of development.

[0059] (i) competing for food

[0060] P(t+1)=D(t)×(F+rand)-d(t) (8)

[0061] In the formula: D(t) is determined by formula (7); F is determined by formula (3); d(t) is determined by formula (11).

[0062] d(t)=R(t)-P(t) (9)

[0063] (ii) Rotational flight

[0064] P(t+1)=R(t)-(S1+S2) (10)

[0065]

[0066] (4) Second phase of development

[0067] When the hunger rate falls below 0.5, vultures become extremely hungry, lacking the energy to compete for food sources. Therefore, some vultures will share a single food source. Meanwhile, some vultures will still strive to move towards the food source. In the second phase of development, AVOA simulated the behaviors of sharing food sources and moving towards them. A random choice was made between these two behaviors for position updates. The formula is as follows:

[0068]

[0069] In the formula: P3 is a constant that controls the second stage of development; Bestvulture1(t) and Bestvulture2(t) are the two best vultures in the current iteration.

[0070]

[0071] P(t+1)=R(t)-|d(t)|×F×Levy(d) (14)

[0072] In the formula: d represents the problem dimension, and the calculation method of Levy(d) is shown in formula (17).

[0073]

[0074] In the formula, u and v are random numbers between 0 and 1; β is a parameter, set to 1.5.

[0075] The basic principles and optimization mechanism of the African Vulture algorithm have been explained in the preceding content of this section. However, it is not appropriate to mechanically embed the multi-objective algorithm framework into the African Vulture algorithm, because the optimization of the African Vulture is led by two leaders. If the same leader selection method is used to select the two leaders, the leader positions will overlap excessively. Therefore, a leader selection framework is designed, which uses the crowding distance sorting mechanism and the hypercube mechanism to select the two leaders respectively. The crowding distance sorting mechanism is shown in Equation (18), and the hypercube mechanism is shown in Equation (19).

[0076]

[0077] In the formula d i (p) indicates that in the target f i Crowded distance; f i,maxand f i,min They represent the target f respectively i The maximum and minimum values ​​of f; i (p i+1 ) and f i (p i-1 ) represent solutions p in the objective f i The target values ​​of the left and right neighbors; C(p) is the total congestion distance for solving p; n obj Since the objective number is the number of the problem, the solution with the largest C(p) should be selected as the leader.

[0078]

[0079] In the formula P i Let H represent the probability of selecting each hypercube; H is a constant greater than one; N i It represents the number of Pareto optimal solutions obtained in the i-th segment.

[0080] The background section of this invention mentions that MOAVOA is prone to getting trapped in local optima in the later stages of iteration. Therefore, an adaptive crossover and mutation strategy based on archives is designed to help the algorithm escape local optima in the later stages. The process of this strategy is as follows:

[0081] First, initialize two files to store the successful crossover and mutation operators (mu, cr), with both files initialized to 0.5.

[0082] Next, a random mu value is selected from the archive and assigned to the Amu value in formula (20). The average cr value in the archive is used to determine the Acr value in formula (21).

[0083] Then use these values ​​to calculate the mu and cr values ​​for crossover and mutation operations, ensuring that both values ​​are in the range of 0 to 1.

[0084] mu=Amu+0.1·tan(π·(rand-0.5)) (18)

[0085]

[0086] Subsequently, the mutated vulture position is calculated using formula (22). For each question dimension of the vulture position, a random value between 0 and 1 is generated. If this random number is less than the cr value, the mutated position will replace the original position for that specific dimension.

[0087] P mu (t)=P(t)+mu·(BestVulture1(t)-P(t)+P r1 (t)-P r2 (t)) (20)

[0088] In the formula: P r1 (t) and P r2 (t) represents the positions of two vultures randomly selected from the population.

[0089] Finally, evaluate the success of the crossover and mutation operations. For multi-objective optimization problems, dominance relations are used to verify success. If the operation is successful, the vulture's position is updated with the new mutation's position, and the mu and cr values ​​in the archive are adjusted accordingly. If the operation fails, neither the position nor the archive is updated.

[0090] In multi-objective optimization algorithms, individuals in the population sometimes exceed the feasible boundary of the problem. Traditional boundary handling methods force individuals that have exceeded the feasible boundary back to the boundary, which can cause the algorithm to miss potential Pareto solutions near the boundary. To address this problem, a progressive boundary repair strategy is designed, which uses the nonlinear characteristics of the tanh function to control the strength of the repair. When the input value is small, the output of the tanh function is also small, which means that the repair adjustment is small when the individual is close to the boundary. Conversely, the repair becomes more important when the individual is far from the boundary. This strategy allows individuals that have exceeded the boundary to gradually move towards the feasible boundary, rather than being suddenly forced to the feasible boundary. The progressive nature of the progressive boundary repair allows the algorithm to retain its exploration and utilization capabilities while correcting out-of-bounds solutions, thereby reducing the possibility of the algorithm getting trapped in local optima. The expression of the tanh function is shown in Equation (25), and the formulas for the progressive boundary repair strategy are shown in Equations (23)-(25).

[0091]

[0092]

[0093]

[0094] In the formula: DS represents the distance between the out-of-bounds solution and the boundary; X d U represents the current position of the individual in dimension d; d L represents the upper bound of an individual in dimension d; d is the lower bound of individual d in dimension d; progressive_factor is the progressive factor.

[0095] Since multi-objective pump station scheduling is usually a constrained multi-objective optimization problem, we first use four constrained multi-objective optimization problems in the CEC2009 multi-objective function test set to verify the performance of EMOAVOA. We select the multi-objective African vulture optimization algorithm (MOAVOA), multi-objective stochastic paint optimization algorithm (MOSPO), and chaotic game multi-objective optimization algorithm (MOCGO) as comparison algorithms. All four algorithms are run independently 30 times with a population size of 100, a solution set archive size of 100, and an iteration count of 1000. We then compare their IGD values ​​(the results are shown in Table 1). The formula for calculating the IGD value is shown in Formula (26). The detailed information of the test function is shown in Formulas (27) to (30). As shown in Table 1, EMOAVOA's best value and mean value are better than the comparison algorithms in the four problems.

[0096]

[0097] In the formula: TPF represents the number of individuals in the true Pareto solution set; d i IGD represents the minimum Euclidean distance between an individual in the true Pareto solution set and the Pareto front derived by the algorithm. A lower IGD value indicates better performance.

[0098] Table 1. IGD values ​​of different algorithms on the test function.

[0099]

[0100]

[0101]

[0102] Equation (27) is the detailed expression of the test function F1. The objective is to minimize f1 and f2. The number of variables n is 10. J1 is odd and J2 is even. The range of values ​​for J1 and J2 is 2 to n.

[0103]

[0104] Equation (28) is the detailed expression of the test function F2. The objective is to minimize f1 and f2. The number of variables n is 10. J1 is odd and J2 is even. The values ​​of J1 and J2 are both from 2 to n.

[0105]

[0106] Equation (29) is the detailed expression of the test function F3. The objective is to minimize f1 and f2. The number of variables n is 10. J1 is odd and J2 is even. The values ​​of J1 and J2 are both from 2 to n.

[0107]

[0108] Equation (30) is the detailed expression of the test function F4. The objective is to minimize f1 and f2. The number of variables n is 10. J1 is odd and J2 is even. The values ​​of J1 and J2 are both from 2 to n.

[0109] The present invention has the following beneficial effects:

[0110] Compared with existing technologies, this scheme uses a crowding distance sorting mechanism and a hypercube mechanism to select two leaders. At the same time, applying two different leader selection methods can significantly reduce the possibility of selecting leaders with similar positions, which will improve the global optimization capability of the multi-objective algorithm.

[0111] Compared with existing technologies, this scheme adopts an archive-based crossover and mutation strategy to reduce the possibility of multi-objective algorithms getting trapped in local optima. This strategy not only provides update formulas for crossover and mutation operators, but also establishes an archive for each operator to evaluate whether the crossover and mutation strategy is successful.

[0112] Compared with existing technologies, this scheme adopts an incremental repair strategy to repair out-of-bounds solutions. Traditional boundary handling methods force individuals that exceed the feasible boundary back to the boundary, which causes the algorithm to miss potential solutions near the boundary. The incremental repair strategy uses the nonlinear characteristics of the tanh function to control the repair intensity. When an individual is close to the boundary, the repair adjustment is smaller, and when an individual is far from the boundary, the repair distance is larger. This strategy allows individuals to gradually move towards the feasible boundary instead of being forcibly forced back to the feasible boundary. This strategy allows the algorithm to retain a certain degree of optimization ability when performing boundary solution repair, increases the competitiveness of the population, and ensures the diversity of the population.

[0113] Compared with existing technologies, this scheme solves the shortcomings of the African Vulture algorithm in solving multi-objective problems, such as insufficient population diversity and easy getting trapped in local optima in the later stages. The multi-strategy enhanced multi-objective African Vulture algorithm (EMOAVOA) is significantly better than the multi-objective African Vulture algorithm (MOAVOA) and multi-objective algorithms proposed in the past three years: Multi-objective Random Drawing Optimization Algorithm (MOSPO) and Multi-objective Chaotic Game Optimization Algorithm (MOCGO) on the multi-objective test function, and significantly improves the ability of MOAVOA in pump station optimization scheduling simulation. Attached Figure Description

[0114] Figure 1 This is a flowchart of the technical solution of the present invention;

[0115] Figure 2 This is a flowchart of the crossover and mutation strategy based on archives in the technical solution of this invention;

[0116] Figure 3This is a comparison diagram of the Pareto solution sets generated by the multi-strategy enhanced multi-objective African vulture algorithm and the multi-objective African vulture algorithm in the pump station optimization scheduling simulation proposed in this invention. Detailed Implementation

[0117] The present invention will be further described below with reference to specific embodiments. The accompanying drawings are for illustrative purposes only and are schematic diagrams, not actual pictures. They should not be construed as limiting the present invention. In order to better illustrate the specific embodiments of the present invention, some parts in the drawings may be omitted, enlarged or reduced, and do not represent the actual product size.

[0118] Taking the daily optimization scheduling of a pumping station system in Chongqing as an example, the specific implementation process of the multi-objective optimization scheduling method for pumping stations based on the multi-strategy enhanced African vulture algorithm proposed in this invention includes the following steps:

[0119] Step 1: Obtain basic data on pump station operation, including: basic pump model, pump rated speed, pump rated flow rate, pump motor efficiency, pump transmission efficiency, pump frequency converter efficiency, pump characteristic curve, and electricity cost during pump operation.

[0120] Step 2: Establish a multi-objective optimization scheduling model with the objective functions of minimizing electricity costs and minimizing the sum of squares of water shortage in the water-receiving area during pump station operation, and with constraints on pump power, pumping capacity, and the number of pumps in operation; where,

[0121] The objective function expression of the multi-objective optimization scheduling model is:

[0122]

[0123] In the formula, F1 is the daily electricity cost of the pumping station; JN is the number of pump units in the pumping station; ρ is the water density; g is the acceleration due to gravity; Q ij Let Hr be the flow rate of the j-th pump in the i-th time period; ij Let η be the head of the j-th pump in the i-th time period; ij Let η be the pump efficiency of the j-th pump in the i-th time period; mot For motor efficiency; η int F2 is the transmission efficiency; F2 is the sum of the squares of the daily water shortage supplied by the pumping station to the water plant; Q i T represents the flow rate of the pumping station during the i-th time period; i Let W be the duration of the i-th time period; W is the daily water demand of the water plant.

[0124] The constraints of the multi-objective optimization scheduling model are expressed as follows:

[0125]

[0126] Where: N ij Represents the actual operating power of the j-th pump in the i-th time period; Ne represents the rated power of a single unit; Q min Q max The minimum and maximum operating flow rates of the pumping station in the i-th time period, respectively; G i G represents the number of pumps started in the i-th time period; imin G imin Let represent the minimum and maximum number of pumps in operation during the i-th time period, respectively.

[0127] Step 3: Construct a multi-strategy enhanced multi-target African vulture algorithm, wherein...

[0128] The multi-strategy enhanced multi-objective African vulture algorithm uses a crowding distance sorting mechanism and a hypercube mechanism to select dual leaders to guide the population. It adopts an archive-based crossover and mutation strategy to reduce the possibility of the multi-objective algorithm getting stuck in local optima. It also adopts a progressive repair strategy to repair out-of-bounds solutions after the main loop ends, so as to enhance the competitiveness of the population and ensure its diversity.

[0129] In the step of selecting dual leaders to guide the population using the crowding distance ranking mechanism and the hypercube mechanism, the formulas for the crowding distance ranking mechanism and the hypercube mechanism are as follows:

[0130]

[0131] In the formula: d i (p) indicates that in the target f i Crowded distance; f i,max and f i,min They represent the target f respectively i The maximum and minimum values ​​of f; i (p i+1 ) and f i (p i-1 ) represent solutions p in the objective f i The target values ​​of the left and right neighbors; C(p) is the total congestion distance for solving p; n obj Since the objective number is the number of problems, the solution with the largest C(p) should be selected as the leader.

[0132]

[0133] In the formula: P i Let H represent the probability of selecting each hypercube; H is a constant greater than one; N i It represents the number of Pareto optimal solutions obtained in the i-th segment.

[0134] The formula for the archive-based crossover mutation strategy is as follows:

[0135] mu=Amu+0.1·tan(π·(rand-0.5))

[0136]

[0137] P mu (t)=P(t)+mu·(BestVulture1(t)-P(t)+P r1 (t)-P r2 (t))

[0138] In the formula: mu is the mutation probability; Amu is the successful mutation probability value stored in the archive, rand is a random number; cr i It is the crossover probability; Acr is the successful crossover probability value stored in the archive; P mu P(t) represents the position after mutation at iteration number t; P(t) represents the position at iteration number t; P r1 (t) and P r2 (t) represents the positions of two vultures randomly selected from the population; BestVulture1(t) is the position of the best vulture in the population.

[0139] The formula for the asymptotic repair strategy for out-of-bounds solutions is as follows:

[0140]

[0141] DS = |X d -U d |or DS=|X d -L d |

[0142] progressive_factor = tanh(DS)

[0143]

[0144] In the formula: DS represents the distance between the out-of-bounds solution and the boundary; X d U represents the current position of the individual in dimension d; d L represents the upper bound of an individual in dimension d; d is the lower bound of individual d in dimension d; progressive_factor is the progressive factor.

[0145] Step 4: Use the multi-strategy enhanced multi-objective African vulture algorithm to simulate the multi-objective optimization scheduling of the pumping station.

[0146] like Figure 3 The comparison diagram of the Pareto solution sets of EMOAVOA and MOAVOA shown is from... Figure 3It can be seen that the vast majority of the solution sets of MOAVOA are above EMOAVOA, meaning that the improved strategy enhances the convergence of MOAVOA. Furthermore, it can be seen that the distribution of EMOAVOA in the upper left and lower right corners of the solution set graph is better than that of MOAVOA, indicating that the improved strategy enhances the uniformity of MOAVOA's distribution. Overall, this invention improves both the uniformity and convergence of MOAVOA in the multi-objective optimization model for pumping stations. In simpler terms, under the same electricity cost conditions, the solutions provided by EMOAVOA can reduce the water shortage in the receiving area. Moreover, when decision-makers in the pumping station system want to reduce the water shortage in the receiving area or reduce the electricity cost of the pumping station units, EMOAVOA can provide decision-makers with more and more economical solutions.

[0147] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.

Claims

1. A multi-objective optimization scheduling method for pumping stations based on the multi-strategy enhanced African vulture algorithm, characterized in that, include: Step 1: Obtain basic data on the operation of the pumping station; Step 2: Establish a multi-objective optimization scheduling model with the objective functions of minimizing electricity costs and minimizing the sum of squares of water shortage in the water receiving area during pump station operation, and with constraints on pump power, pumping capacity, and the number of pumps in operation. Step 3: Construct a multi-strategy enhanced multi-target African vulture algorithm, in which... The multi-strategy enhanced multi-objective African vulture algorithm uses a crowding distance sorting mechanism and a hypercube mechanism to select dual leaders to guide the population. It adopts an archive-based crossover and mutation strategy to reduce the possibility of the multi-objective algorithm getting stuck in local optima. It also adopts a progressive repair strategy to repair out-of-bounds solutions after the main loop ends, so as to enhance the competitiveness of the population and ensure its diversity. Step four: Use the multi-strategy enhanced multi-objective African vulture algorithm to simulate the multi-objective optimal scheduling of the pumping station.

2. The method for multi-objective optimization scheduling of pumping stations based on the multi-strategy enhanced African vulture algorithm according to claim 1, characterized in that, The basic data includes: basic pump model, rated speed of the pump, rated flow rate of the pump, motor efficiency of the pump, transmission efficiency of the pump, frequency converter efficiency of the pump, characteristic curve of the pump, and electricity cost during pump operation.

3. The method for multi-objective optimization scheduling of pumping stations based on the multi-strategy enhanced African vulture algorithm according to claim 1, characterized in that, The objective function expression of the multi-objective optimization scheduling model is: In the formula, F1 is the daily electricity cost of the pumping station; JN is the number of pump units in the pumping station; ρ is the water density; g is the acceleration due to gravity; Q ij Let Hr be the flow rate of the j-th pump in the i-th time period; ij Let J be the head of the j-th pump in the i-th time period; η ij Let be the pump efficiency of the j-th pump in the i-th time period; η mot For motor efficiency; η int F2 is the transmission efficiency; F2 is the sum of the squares of the daily water shortage supplied by the pumping station to the water plant; Q i Let be the flow rate of the pump station in the i-th time period; T i Let be the duration of the i-th time period; W represents the water plant's daily water demand.

4. The method for multi-objective optimization scheduling of pumping stations based on the multi-strategy enhanced African vulture algorithm according to claim 1, characterized in that, The constraints of the multi-objective optimization scheduling model are expressed as follows: Where: N ij Represents the actual operating power of the j-th pump in the i-th time period; Ne represents the rated power of a single unit; Q min Q max The minimum and maximum operating flow rates of the pumping station in the i-th time period, respectively; G i G represents the number of pumps started in the i-th time period; imin G imin Let represent the minimum and maximum number of pumps in operation during the i-th time period, respectively.

5. A multi-objective optimization scheduling method for pumping stations based on the multi-strategy enhanced African vulture algorithm according to claim 1, characterized in that, In the step of selecting dual leaders to guide the population using the crowding distance ranking mechanism and the hypercube mechanism, the formulas for the crowding distance ranking mechanism and the hypercube mechanism are as follows: In the formula: d i (p) indicates that in the target f i Crowded distance; f i,max and f i,min They represent the target f respectively i The maximum and minimum values; f i (p i+1 ) and f i (p i-1 ) represent solutions p in the objective f i The target values ​​of the left and right neighbors; C(p) is the total congestion distance for solving p; n obj Since the objective number is the number of problems, the solution with the largest C(p) should be selected as the leader. In the formula: P i Let H represent the probability of selecting each hypercube; H is a constant greater than one; N i It represents the number of Pareto optimal solutions obtained in the i-th segment.

6. A multi-objective optimization scheduling method for pumping stations based on the multi-strategy enhanced African vulture algorithm according to claim 1, characterized in that, The formula for the archive-based crossover mutation strategy is as follows: mu=Amu+0.1·tan(π·(rand-0.5)) P mu (t)=P(t)+mu·(BestVulture1(t)-P(t)+P r1 (t)-P r2 (t)) In the formula: mu is the mutation probability; Amu is the successful mutation probability value stored in the archive, rand is a random number; cr i It is the crossover probability; Acr is the successful crossover probability value stored in the archive; P mu P(t) represents the position after mutation at iteration number t; P(t) represents the position at iteration number t; P r1 (t) and P r2 (t) represents the positions of two vultures randomly selected from the population; BestVulture1(t) is the position of the best vulture in the population.

7. A multi-objective optimization scheduling method for pumping stations based on the multi-strategy enhanced African vulture algorithm according to claim 1, characterized in that, The formula for the asymptotic repair strategy for out-of-bounds solutions is as follows: DS=|X d -U d |or DS=|X d -L d | progressive_factor = tanh(DS) In the formula: DS represents the distance between the out-of-bounds solution and the boundary; X d U represents the current position of the individual in dimension d; d L represents the upper bound of an individual in dimension d; d is the lower bound of individual d in dimension d; progressive_factor is the progressive factor.

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