Economic emission dispatch method for power system based on improved grey wolf algorithm

Abstract

The application discloses an improved grey wolf algorithm-based power system economic emission scheduling method, which comprises the following steps: firstly, a multi-objective optimization model of power system economic emission scheduling is established; then, an improved multi-objective grey wolf optimization algorithm is run, the generation cost and the emission are taken as the fitness function to calculate the fitness value, the external archive is updated through non-dominated sorting, the grey wolf individual position is updated by using a multi-strategy hybrid mechanism, until the preset maximum iteration number is reached, after the iteration is terminated, the optimal compromise solution is selected from the final non-dominated solution set saved in the external archive, and the corresponding scheduling scheme is output. The application is used for multi-objective optimization solution of the generation cost and the pollutant emission under the condition of meeting the operation constraint, so as to improve the problems of insufficient convergence precision, easy falling into local extreme value and poor distribution uniformity of the non-dominated solution set obtained in the solving process of the prior art, and thus the cost-emission compromise scheduling scheme is obtained.

Description

Technical Field

[0001] This invention belongs to the field of power system operation and control technology, specifically relating to an economic emission dispatch method for power systems based on an improved gray wolf algorithm. Background Technology

[0002] With increasing energy demand and stricter requirements for low-carbon emissions, power system dispatch needs to comprehensively consider generation costs and pollutant emissions while meeting operational constraints. Traditional economic dispatch (ED) typically aims to minimize generation costs. However, when pollutant emissions are incorporated into the objective, forming the economic emission dispatch (EED) problem, a trade-off optimization between generation costs and pollutant emissions is necessary. Due to the turbine valve point effect, the objective function exhibits non-smooth and non-convex characteristics. Furthermore, constraints such as power balance constraints, unit output upper and lower limits, and network losses are nonlinear and strongly coupled, making the EED problem typically a strongly constrained multi-objective nonlinear optimization problem with high solution difficulty.

[0003] Existing solutions to the EED problem mainly fall into two categories: classical mathematical optimization methods and metaheuristic algorithms. Classical mathematical optimization methods typically rely heavily on assumptions regarding the differentiability or convexity of the objective function and constraints, and are sensitive to initial values. When dealing with non-convex, multimodal, and strongly constrained problems, they are prone to obtaining local solutions or struggling to stably obtain a uniformly distributed non-dominated solution set. Metaheuristic algorithms, due to their independence from strict model assumptions and flexible search mechanisms, have been used to solve the EED problem. However, in complex EED problems, they may still suffer from insufficient convergence speed, premature convergence, and uneven distribution of non-dominated solution sets. Taking the Multi-Objective Grey Wolf Optimization Algorithm (MOGWO) as an example, when dealing with EED problems under multi-objective coupling and complex constraints, random initialization may lead to insufficient uniformity and diversity in the initial population distribution; a linearly decreasing convergence factor may cause insufficient global exploration in the early stages of iteration or insufficient local exploitation in the later stages during the nonlinear search process; under the multi-peak objective function condition caused by the threshold effect, the algorithm may get trapped in local extrema, resulting in an uneven distribution of the obtained non-dominated solution set and affecting the acquisition of cost-emission tradeoff solutions. Therefore, there is a need for an economic emission scheduling solution method that can improve the quality of the initial population, enhance the balance between global search and local exploitation, and improve the distribution of non-dominated solution sets under the above constraints. Summary of the Invention

[0004] The purpose of this invention is to provide an economic emission scheduling method for power systems based on an improved gray wolf algorithm. This method is used to perform multi-objective optimization of power generation costs and pollutant emissions under the condition of satisfying operational constraints. It aims to improve the problems that existing technologies may have in the solution process, such as insufficient convergence accuracy, easy getting trapped in local extrema, and poor uniformity of the distribution of the obtained non-dominated solution set, so as to obtain a cost-emission trade-off scheduling scheme.

[0005] The technical solution adopted in this invention is an economic emission dispatch method for power systems based on an improved gray wolf algorithm, which is implemented according to the following steps:

[0006] Step 1: Establish a multi-objective optimization model for economic emission dispatch of the power system, and set the parameter values ​​for the improved multi-objective gray wolf optimization algorithm; Step 2: Run the improved multi-objective gray wolf optimization algorithm, calculate the fitness value using power generation cost and emissions as fitness functions, update the external archives through non-dominated sorting, update the individual gray wolf positions using a multi-strategy mixing mechanism, iterate until the maximum number of iterations is reached, and then terminate the iteration. Step 3: Select compromise solutions and output scheduling schemes. After the iteration terminates, the final non-dominated solution set is saved in the external archive.

[0007] The invention is further characterized in that, Step 1 is implemented in the following steps: Input the basic data of the power system, construct a multi-objective optimization model with total power generation cost and total pollutant emissions as optimization objectives, and set power balance constraints, upper and lower limits of unit output constraints and network loss constraints according to the physical operating characteristics of generator units; Let the decision variables be the active power output vectors of each generator unit. The multi-objective optimization model uses the total power generation cost objective function and the total pollutant emission objective function as fitness functions. The model forms are shown in equations (1) and (2): (1) (2) in Represents the objective function vector of a multi-objective optimization problem; This represents a vector of decision variables, composed of the active power output of each generator unit; This represents the objective function for total power generation cost; The objective function represents the total pollutant emissions; Indicates the first Inequality constraint functions; Indicates the first One equality constraint function; and These represent the numbers of the inequality constraints and the equality constraints, respectively. This represents the total number of inequality constraints. Indicates the total number of equality constraints; Regarding the power generation cost target, a quadratic polynomial model is used to describe the relationship between the power generation cost of each generator unit and the active power output of the unit. The total power generation cost target function is expressed as the sum of the power generation cost functions of each generator unit, as shown in equation (3): (3) in, Indicates the number of generator sets; Indicates the generator set number; , and They represent the first The power generation cost coefficient of the Taiwanese generator unit Indicates the first The active power output of the generator sets; Considering the turbine valve point effect, the fuel cost fluctuates due to throttling losses during valve opening. A valve point effect term is introduced into the quadratic cost function, and the total power generation cost model is shown in equation (4): (4) in and They represent the first The coefficient of the valve point effect term of the generator set; Indicates the first Minimum active power output of generator sets; total emissions Taking SO2 and NO into account x The model is as follows: (5) in , , , and Representing the Emission coefficient of the unit; Represented by natural constant An exponential function with base 0; In the power system optimization and dispatch problem, constraints are an important guarantee for ensuring the safe and stable operation of the system. The power balance constraints are as follows: (6) in, This represents the total active power load requirement of the system. The total active power loss generated by the transmission line is expressed as: (7) in, Number the generator set; These are elements of the transmission loss coefficient matrix, reflecting the impact of power flow between nodes on losses; For the first The active power output of the generator sets; Indicates the relationship with the first Linear network loss coefficient related to the output of each generator unit; This represents the constant term network loss coefficient; in simplified scenarios, network loss is ignored. (8) The upper and lower limits of the unit's output are constrained as follows: (9) in, Indicates the first The maximum active power output allowed for each generator set; In the fitness calculation and candidate solution comparison process, a constraint handling mechanism combining feasibility priority criterion and constraint violation coefficient (CV) is adopted. Each candidate solution contains two key attributes: one is the objective function vector. Second, inequality constraint vectors ,in Indicates the first If a constraint is violated, the constraint violation degree of the solution is defined as the sum of all positive constraint values: (10) in, Indicates candidate solutions The degree of constraint violation; This operation takes the larger of the two values; if and only if When a solution is considered feasible, the dominance rule of this mechanism prioritizes feasibility: feasible solutions are preferred over infeasible solutions; Pareto dominance comparison is used between feasible solutions; among infeasible solutions, the solution with the smaller constraint violation is selected. This strategy ensures that the search converges effectively to the feasible region and avoids premature convergence by preserving infeasible solutions on the boundary to traverse complex constraints, thereby better approximating the Pareto optimal frontier.

[0008] In step 1, the algorithm parameters are set as follows: population size Maximum number of iterations External file capacity .

[0009] Step 2 is implemented in the following steps: Step 2.1: Hybrid initialization strategy of circular chaotic mapping and Latin hypercube sampling (LHS); Step 2.2: Nonlinear convergence factor adjustment strategy; Step 2.3: Differentiated perturbation strategy based on grouping dimension; Step 2.4: Hybrid update strategy based on dynamic Lévy flight.

[0010] Step 2.1 is implemented according to the following steps: Enhancing exploration capabilities by leveraging the ergodicity of circular chaotic mappings, for the previous... For each individual, a chaotic sequence is first generated using equation (11), and then the chaotic sequence is mapped to the solution space to construct the position vector of the individual: (11) in and Represent the circular chaotic mappings respectively. Second and third The chaotic sequence values ​​generated in the next iteration; and The control parameters represent the circular chaotic mapping; Indicates the number of iterations; This represents the modulo operation with respect to 1; Let represent a sine function; after mapping the generated chaotic sequence to the solution space, the position vector of the corresponding individual can be obtained. Then, stratified sampling using LHS is used to ensure spatial uniformity. For the remaining individuals, LHS generation is employed. Uniform sample points in 3D space: (12) in This represents the sample matrix generated using Latin hypercube sampling; Indicates the sample point number; Indicates the first The location vectors corresponding to each sample point; D represents the question dimension; Indicates the first The sample point at the th th Values ​​on the dimension; Indicates the dimension number; Indicates the first The sample point at the th th The value that can be taken in the dimension.

[0011] Step 2.2 is implemented according to the following steps: The equation is as follows: (13) in, Indicates the first The nonlinear convergence factor at the next iteration; This represents the current iteration number; and These represent the minimum and maximum values ​​of the convergence factor, respectively. This represents the maximum number of iterations.

[0012] Step 2.3 is implemented according to the following steps: During the algorithm's exploration phase, some individuals were selected as those to be disturbed and divided into groups using a clustering strategy. Each subgroup employs a differentiated perturbation strategy to gradually activate the potential search capabilities of its members. The first 90% of iterations are defined as the exploration phase. Each subgroup contains... Individuals, among which Population size; For the Group of individuals, Only some dimensions are perturbed and updated; specifically, random selection is used. Update each dimension: (14) in Indicates the first The number of dimensions that need to be updated for each individual in the group; Indicates the subgroup number; For random selection Each dimension, for a randomly selected set The perturbation step size is controlled by a cosine decay factor: (15) in Indicates the first Perturbation step size control factor in the next iteration; Represents the cosine function; Represents the smallest positive constant to prevent the step size from degenerating to 0; step size control factor Initially, the value approaches 1, achieving a large-scale jump; as iterations proceed... Gradually approaching 0 to ensure smooth convergence, the individual position update formula is: (16) in Indicates the first The individual in the first The value after the dimension update; Indicates the individual ID; Indicates the dimension number; Indicates the first The individual in the first The value before the update; and These represent the upper and lower bounds of the dimension, respectively. Indicates that it conforms to the interval Uniformly distributed random numbers; This represents a uniform distribution between 0 and 1; If an updated individual exceeds the boundary, an adaptive repair strategy is adopted: For high-dimensional problems, a dimension cross-replacement mechanism is used, randomly selecting the corresponding dimension value of an individual from the current population for replacement. This mechanism utilizes existing information within the population for knowledge transfer, avoiding invalid searches. For low-dimensional problems, a random re-initialization mechanism is used to maintain the randomness and diversity of the population. The update formula is as follows: (17) in Indicates the first The individual in the first The value after repair; This indicates the first random selection from the current population. The individual in the first Values ​​on the dimension; This represents a random individual index; through the synergistic effect of the above-mentioned grouping differential perturbation, nonlinear step size control and adaptive boundary repair, the algorithm's escape ability and optimization accuracy in complex high-dimensional search spaces are significantly improved.

[0013] Step 2.4 is implemented according to the following steps: To enhance the ability to escape local optima, an adaptive hybrid update strategy is constructed, which generates random numbers during position updates. ,like Then execute Lévy flight: (18) in Indicates the first The position vector of the individual at the next iteration; This represents the reference position vector used to guide position updates; This represents the Lévy flight stride coefficient; Indicates the first The position vector of the individual is updated in the next iteration; otherwise, the position is updated using the standard GWO triple leadership mechanism. This strategy effectively utilizes long step jumps to expand the search range and avoid the algorithm getting stuck in local optima.

[0014] Step 3 is implemented in the following steps: Using the normalized distance criterion based on the ideal point, the objective function values ​​of each candidate solution in the non-dominated solution set are normalized using the minimum-maximum normalization method. The Euclidean distance from each candidate solution to the ideal point is then calculated based on the ideal point distance criterion, according to equation (19): (19) in Indicates the first Euclidean distances from each candidate solution to the ideal point; Indicates the candidate solution number; and These represent the normalized target values ​​for power generation cost and pollutant emissions, respectively. The ideal point is taken as the origin in the normalized target space. The candidate solution with the smallest Euclidean distance is selected as the compromise solution, and the unit scheduling scheme is output accordingly. The calculation is explained using the min-maximum standardized and weighted comprehensive scoring method. The standardized formula is shown in equation (20): (20) in Indicators Normalized values; This represents the original index value to be normalized. and These represent the minimum and maximum values ​​of all comparison algorithms for a certain evaluation index, respectively. After standardization, each index value is linearly mapped to the [0,1] interval, and it is unified that the smaller the value, the better the performance under that evaluation index. The final comprehensive score is calculated using the following formula: (twenty one) Weight , The smaller the overall score, the closer the compromise result under the evaluation criteria is to the expectation.

[0015] The beneficial effects of this invention are as follows: The power system economic emission dispatch method based on the improved Grey Wolf algorithm, by employing a hybrid initialization strategy combining circular chaotic mapping and Latin hypercube sampling, can improve the coverage uniformity and diversity of the initial population in the solution space, thereby improving the quality of the initial solution; through a nonlinear convergence factor and a differentiated perturbation mechanism based on grouping dimensions, it can dynamically balance global search and local exploitation capabilities during the iteration process, and with the dynamic Lévy flight update strategy, it enhances the ability to escape local extrema under non-convex, multi-peak objective function conditions caused by the threshold effect; through external archive maintenance and non-dominated solution set screening mechanisms, it can obtain a relatively uniformly distributed non-dominated solution set, and further select a compromise solution based on a trade-off criterion. Therefore, this invention can provide a cost-emission trade-off dispatch scheme for power system economic emission dispatch under the conditions of satisfying operational constraints such as power balance, unit output upper and lower limits, and network losses. Attached Figure Description

[0016] Figure 1 This is a flowchart of the improved multi-objective gray wolf optimization algorithm; Figure 2(a) is a schematic diagram of the initial population distribution after random initialization; Figure 2(b) is a schematic diagram of the initial population distribution during mixed initialization; Figure 3(a) is a schematic diagram of the variation curve of the linear convergence factor; Figure 3(b) is a schematic diagram of the variation curve of the improved nonlinear convergence factor; Figures 4(a1), 4(a2), 4(a3), and 4(a4) are comparison charts of the distribution of non-dominated solution sets of the IMOGWO algorithm under the test functions MW3, MW6, MW7, and MW12, respectively. Figures 4(b1), 4(b2), 4(b3), and 4(b4) are comparison diagrams of the distribution of non-dominated solution sets of the MOGWO algorithm under the test functions MW3, MW6, MW7, and MW12, respectively. Figures 4(c1), 4(c2), 4(c3), and 4(c4) are comparison diagrams of the distribution of non-dominated solution sets of the IMOPSO algorithm under the test functions MW3, MW6, MW7, and MW12, respectively. Figures 4(d1), 4(d2), 4(d3), and 4(d4) are comparison charts of the distribution of non-dominated solution sets of the NSGAII algorithm under the test functions MW3, MW6, MW7, and MW12, respectively. Figures 4(e1), 4(e2), 4(e3), and 4(e4) are comparison charts of the distribution of non-dominated solution sets of the NSGAIII algorithm under the test functions MW3, MW6, MW7, and MW12, respectively. Figure 5(a) is a comparison of the non-dominated solution set distribution of IMOGWO under 6 units; Figure 5(b) is a comparison of the non-dominated solution set distribution of MOGWO under 6 units; Figure 6(a) is a comparison of the non-dominated solution set distribution of IMOGWO under Unit 10; Figure 6(b) is a comparison of the non-dominated solution set distribution of MOGWO under Unit 10; Figure 7(a) is a comparison of the non-dominated solution set distribution of IMOGWO under Unit 40; Figure 7(b) is a comparison of the non-dominated solution set distribution of MOGWO under Unit 40. Detailed Implementation

[0017] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0018] This invention addresses the optimization problem of economic emission scheduling in power systems under multi-objective coupling, strong non-convexity, and multiple constraints. It proposes a power system economic emission scheduling method based on a multi-strategy improved multi-objective gray wolf optimization algorithm. In one embodiment, the method includes: generating an initial population using a hybrid initialization strategy combining circular chaotic mapping and Latin hypercube sampling to improve population diversity and distribution uniformity; employing a nonlinear convergence factor adjustment strategy based on exponential variation to dynamically balance global search and local exploitation capabilities during iteration; introducing a differentiated perturbation mechanism based on grouping dimensions to enhance exploration capabilities and ensure the feasibility of candidate solutions with boundary repair; introducing a dynamically triggered Lévy flight strategy during position updates to enhance the ability to escape local extrema; maintaining an external archive to store the non-dominated solution set, and after normalizing the objectives, selecting a compromise solution from the non-dominated solution set using a criterion based on ideal point distance, and outputting the unit scheduling scheme. Through the above technical solutions, the method of the present invention can improve the quality of the initial population distribution, enhance the solution stability under strong non-convex, multimodal and strong constraint conditions, and help obtain a more uniformly distributed non-dominated solution set, thereby providing a usable compromise decision scheme for economic emission scheduling.

[0019] Example 1 This invention presents an economic emission dispatch method for power systems based on an improved gray wolf algorithm, the flowchart of which is shown below. Figure 1 As shown, please follow these steps: Step 1: Establish a multi-objective optimization model for economic emission dispatch of the power system, and set the parameter values ​​for the improved multi-objective gray wolf optimization algorithm; Step 1 is implemented in the following steps: Input the basic data of the power system, construct a multi-objective optimization model with total power generation cost and total pollutant emissions as optimization objectives, and set power balance constraints, upper and lower limits of unit output constraints and network loss constraints according to the physical operating characteristics of generator units; Let the decision variables be the active power output vectors of each generator unit. The multi-objective optimization model uses the total power generation cost objective function and the total pollutant emission objective function as fitness functions. The model forms are shown in equations (1) and (2): (1) (2) in Represents the objective function vector of a multi-objective optimization problem; This represents a vector of decision variables, composed of the active power output of each generator unit; This represents the objective function for total power generation cost; The objective function represents the total pollutant emissions; Indicates the first Inequality constraint functions; Indicates the first One equality constraint function; and These represent the numbers of the inequality constraints and the equality constraints, respectively. This represents the total number of inequality constraints. Indicates the total number of equality constraints; Regarding the power generation cost target, a quadratic polynomial model is used to describe the relationship between the power generation cost of each generator unit and the active power output of the unit. The total power generation cost target function is expressed as the sum of the power generation cost functions of each generator unit, as shown in equation (3): (3) in, Indicates the number of generator sets; Indicates the generator set number; , and They represent the first The power generation cost coefficient of the Taiwanese generator unit Indicates the first The active power output of the generator sets; Considering the turbine valve point effect, the fuel cost fluctuates due to throttling losses during valve opening. A valve point effect term is introduced into the quadratic cost function, and the total power generation cost model is shown in equation (4): (4) in and They represent the first The coefficient of the valve point effect term of the generator set; Indicates the first Minimum active power output of generator sets; total emissions Taking SO2 and NO into account x The model is as follows: (5) in , , , and Representing the Emission coefficient of the unit; Represented by natural constant An exponential function with base 0; In the power system optimization and dispatch problem, constraints are an important guarantee for ensuring the safe and stable operation of the system. The power balance constraints are as follows: (6) in, This represents the total active power load requirement of the system. The total active power loss generated by the transmission line is expressed as: (7) in, Number the generator set; These are elements of the transmission loss coefficient matrix, reflecting the impact of power flow between nodes on losses; For the first The active power output of the generator sets; Indicates the relationship with the first Linear network loss coefficient related to the output of each generator unit; This represents the constant term network loss coefficient; in simplified scenarios, network loss is ignored. (8) The upper and lower limits of the unit's output are constrained as follows: (9) in, Indicates the first The maximum active power output allowed for each generator set; In the fitness calculation and candidate solution comparison process, a constraint handling mechanism combining feasibility priority criterion and constraint violation coefficient (CV) is adopted. Each candidate solution contains two key attributes: one is the objective function vector. Second, inequality constraint vectors ,in Indicates the first If a constraint is violated, the constraint violation degree of the solution is defined as the sum of all positive constraint values: (10) in, Indicates candidate solutions The degree of constraint violation; This operation takes the larger of the two values; if and only if When a solution is considered feasible, the dominance rule of this mechanism prioritizes feasibility: feasible solutions are preferred over infeasible solutions; Pareto dominance comparison is used between feasible solutions; among infeasible solutions, the solution with the smaller constraint violation is selected. This strategy ensures that the search converges effectively to the feasible region and avoids premature convergence by preserving infeasible solutions on the boundary to traverse complex constraints, thereby better approximating the Pareto optimal frontier.

[0020] In step 1, the algorithm parameters are set as follows: population size Maximum number of iterations External file capacity The above parameters can also be adjusted according to the system size and constraint complexity.

[0021] Step 2: Run the improved multi-objective gray wolf optimization algorithm. Calculate the fitness value using power generation cost and emissions as fitness functions. Update the external archives through non-dominated sorting. Update the individual gray wolf positions using a multi-strategy hybrid mechanism. Iterate until the maximum number of iterations is reached, then terminate the iteration. To facilitate explanation of the overall execution process of the improved multi-objective gray wolf optimization algorithm of this invention, Figure 1 A flowchart illustrating the method of the present invention is shown.

[0022] Step 2 is implemented in the following steps: The detailed process involved is as follows: To address the problems of insufficient initial population diversity, susceptibility to local optima, and low convergence accuracy in the standard multi-objective gray wolf optimization algorithm, this invention proposes an improved multi-objective gray wolf optimization algorithm. The specific improvement strategy is as follows: Step 2.1: Hybrid initialization strategy of circular chaotic mapping and Latin hypercube sampling (LHS); Step 2.1 is implemented according to the following steps: In swarm intelligence algorithms, a uniformly distributed initial population is beneficial for improving algorithm performance. Traditional MOGWO algorithms initialize the population randomly, resulting in a relatively uneven initial population distribution with clusters and gaps, which significantly impacts the algorithm's convergence speed and optimization accuracy. This invention employs a hybrid strategy combining circular chaotic mapping and LHS (Low Harmonic Hierarchy Mapping) for population initialization. The ergodicity of circular chaotic mapping enhances exploration capabilities for the initial population. For each individual, a chaotic sequence is first generated using equation (11), and then the chaotic sequence is mapped to the solution space to construct the position vector of the individual: (11) in and Represent the circular chaotic mappings respectively. Second and third The chaotic sequence values ​​generated in the next iteration; and The control parameters represent the circular chaotic mapping; Indicates the number of iterations; This represents the modulo operation with respect to 1; Let represent a sine function; after mapping the generated chaotic sequence to the solution space, the position vector of the corresponding individual can be obtained. Then, stratified sampling using LHS is used to ensure spatial uniformity. For the remaining individuals, LHS generation is employed. Uniform sample points in 3D space: (12) in This represents the sample matrix generated using Latin hypercube sampling; Indicates the sample point number; Indicates the first The location vectors corresponding to each sample point; D represents the question dimension; Indicates the first The sample point at the th th Values ​​on the dimension; Indicates the dimension number; Indicates the first The sample point at the th th Values ​​on the dimension; This invention divides the population into two parts: one part is generated using circular chaotic mapping, and the other part is generated using LHS. To illustrate the impact of the hybrid initialization strategy on the initial population distribution, Figures 2(a) and 2(b) show the initial individual distributions obtained by random initialization and the hybrid initialization strategy of this invention. As can be seen from Figures 2(a) and 2(b), compared with random initialization, the hybrid initialization of this invention provides more uniform coverage in the solution space, which can be used to improve the diversity of the initial population, thereby providing a more sufficient basis for the distribution of candidate solutions for subsequent iterative searches.

[0023] Step 2.2: Nonlinear convergence factor adjustment strategy; Step 2.2 is implemented according to the following steps: Global search capability and local exploitation capability are two common attributes describing the performance of metaheuristic algorithms, and effectively balancing these two aspects is a primary consideration in metaheuristic algorithms. In GWO, the convergence factor... The update method used in the iteration employs a linear decreasing strategy. Linear strategies often struggle to adapt to complex nonlinear search processes. Therefore, this invention proposes a nonlinear convergence factor update method based on a natural exponential change, the equation of which is as follows: (13) in, Indicates the first The nonlinear convergence factor at the next iteration; This represents the current iteration number; and These represent the minimum and maximum values ​​of the convergence factor, respectively. This represents the maximum number of iterations. To illustrate the difference in the variation between the nonlinear convergence factor and the linear convergence factor, Figures 3(a) and 3(b) show the curves of the linear convergence factor and the nonlinear convergence factor of this invention as a function of the number of iterations. Compared with the linear decreasing method, the nonlinear convergence factor of this invention maintains a larger value in the early stage of iteration to enhance the global search capability, and gradually decreases in the middle and later stages of iteration to enhance the local exploitation capability, thereby achieving a dynamic balance between exploration and exploitation.

[0024] Step 2.3: Differentiated perturbation strategy based on grouping dimension; Step 2.3 is implemented according to the following steps: To enhance the global search capability of the Grey Wolf optimizer in early iterations and address the problem of the algorithm easily getting trapped in local optima, this invention introduces a differentiated perturbation mechanism based on grouping dimensions. This strategy effectively improves population diversity and balances global exploration with local exploitation by dividing the population into subgroups at different levels and applying differentiated dimension updates and boundary treatments to different subgroups.

[0025] The specific implementation steps are as follows: During the algorithm's exploration phase, some individuals were selected as those to be disturbed and divided into groups using a clustering strategy. Each subgroup employs a differentiated perturbation strategy to gradually activate the potential search capabilities of its members. The first 90% of iterations are defined as the exploration phase. Each subgroup contains... Individuals, among which Population size; For the Group of individuals, Only certain dimensions are perturbed and updated to achieve the effect of local forgetting and global exploration. Specifically, random selection... Update each dimension: (14) in Indicates the first The number of dimensions that need to be updated for each individual in the group; This indicates the subgroup number; this design allows the earlier subgroups to be perturbed in more dimensions and undertake stronger exploration tasks; while the subsequent subgroups are perturbed in fewer dimensions and focus on local adjustments. For random selection Each dimension, for a randomly selected set The perturbation step size is controlled by a cosine decay factor: (15) in Indicates the first Perturbation step size control factor during the next iteration; Represents the cosine function; Represents the smallest positive constant to prevent the step size from degenerating to 0; step size control factor Initially, the value approaches 1, achieving a large-scale jump; as iterations proceed... Gradually approaching 0 to ensure smooth convergence, the individual position update formula is: (16) in Indicates the first The individual in the first The updated value; Indicates the individual ID; Indicates the dimension number; Indicates the first The individual in the first The value before the update; and These represent the upper and lower bounds of the dimension, respectively. Indicates that it conforms to the interval Uniformly distributed random numbers; This represents a uniform distribution between 0 and 1; If an individual exceeds the boundary after the update, an adaptive repair strategy is adopted: For high-dimensional problems (D>15), a dimension cross-replacement mechanism is used, randomly selecting the corresponding dimension value of an individual from the current population for replacement. This mechanism utilizes existing information within the population for knowledge transfer, avoiding invalid searches. For low-dimensional problems, a random re-initialization mechanism is used to maintain the randomness and diversity of the population. The update formula is as follows: (17) in Indicates the first The individual in the first The value after repair; This indicates the first random selection from the current population. The individual in the first Values ​​on the dimension; This represents a random individual index; through the synergistic effect of the above-mentioned grouping differential perturbation, nonlinear step size control and adaptive boundary repair, the algorithm's escape ability and optimization accuracy in complex high-dimensional search spaces are significantly improved.

[0026] Step 2.4: Hybrid update strategy based on dynamic Lévy flight.

[0027] Step 2.4 is implemented according to the following steps: To enhance the ability to escape local optima, an adaptive hybrid update strategy is constructed, which generates random numbers during position updates. ,like (Dynamic trigger probability), then Lévy flight will be executed: (18) in Indicates the first The position vector of the individual at the next iteration; This represents the reference position vector used to guide position updates; This represents the Lévy flight stride coefficient; Indicates the first The position vector of the individual in the next iteration; otherwise, the position is updated using the standard GWO triple leadership mechanism. This strategy effectively utilizes long step jumps to expand the search range and avoid the algorithm getting stuck in local optima. Step 3: Select compromise solutions and output scheduling schemes. After the iteration terminates, the final non-dominated solution set is saved in the external archive.

[0028] Step 3 is implemented in the following steps: Since the dimensions and numerical scales of power generation costs and pollutant emissions are different, in order to select a compromise solution for scheduling output from the non-dominated solution set, this invention adopts a normalized distance criterion based on the ideal point. The objective function values ​​of each candidate solution in the non-dominated solution set are normalized using the minimum-maximum normalization method, and the Euclidean distance from each candidate solution to the ideal point is calculated based on the ideal point distance criterion, according to equation (19): (19) in Indicates the first Euclidean distances from each candidate solution to the ideal point; Indicates the candidate solution number; and These represent the normalized target values ​​for power generation cost and pollutant emissions, respectively. The ideal point is taken as the origin in the normalized target space. The candidate solution with the smallest Euclidean distance is selected as the compromise solution, and the unit scheduling scheme is output accordingly.

[0029] To facilitate the quantitative expression of the cost and emission indicators corresponding to the compromise solution, the minimum-maximum standardization and weighted comprehensive scoring method can be used for calculation and explanation. The standardization formula is shown in equation (20): (20) in Indicators Normalized values; This represents the original index value to be normalized. and These represent the minimum and maximum values ​​of all comparison algorithms for a certain evaluation index, respectively. After standardization, each index value is linearly mapped to the [0,1] interval, and it is unified that the smaller the value, the better the performance under that evaluation index. The final comprehensive score is calculated using the following formula: (twenty one) Weight , This reflects the engineering decision-making preference of "cost priority, while also considering emission reduction." The smaller the overall score, the closer the compromise result under this evaluation criterion is to the expectation.

[0030] Example 2 This invention presents an economic emission dispatch method for power systems based on an improved gray wolf algorithm, the flowchart of which is shown below. Figure 1 As shown, please follow these steps: Step 1: Establish a multi-objective optimization model for economic emission dispatch of the power system, and set the parameter values ​​for the improved multi-objective gray wolf optimization algorithm; Step 2: Run the improved multi-objective gray wolf optimization algorithm. Calculate the fitness value using power generation cost and emissions as fitness functions. Update the external archives through non-dominated sorting. Update the individual gray wolf positions using a multi-strategy hybrid mechanism. Iterate until the maximum number of iterations is reached, then terminate the iteration. To facilitate explanation of the overall execution process of the improved multi-objective gray wolf optimization algorithm of this invention, Figure 1 A flowchart illustrating the method of the present invention is shown.

[0031] Step 3: Select compromise solutions and output scheduling schemes. After the iteration terminates, the final non-dominated solution set is saved in the external archive.

[0032] Example 3 This invention presents an economic emission dispatch method for power systems based on an improved gray wolf algorithm, the flowchart of which is shown below. Figure 1 As shown, please follow these steps: Step 1: Establish a multi-objective optimization model for economic emission dispatch of the power system, and set the parameter values ​​for the improved multi-objective gray wolf optimization algorithm; Step 1 is implemented in the following steps: Input the basic data of the power system, construct a multi-objective optimization model with total power generation cost and total pollutant emissions as optimization objectives, and set power balance constraints, upper and lower limits of unit output constraints and network loss constraints according to the physical operating characteristics of generator units; Let the decision variables be the active power output vectors of each generator unit. The multi-objective optimization model uses the total power generation cost objective function and the total pollutant emission objective function as fitness functions. The model forms are shown in equations (1) and (2): (1) (2) in Represents the objective function vector of a multi-objective optimization problem; This represents a vector of decision variables, composed of the active power output of each generator unit; This represents the objective function for total power generation cost; The objective function represents the total pollutant emissions; Indicates the first Inequality constraint functions; Indicates the first One equality constraint function; and These represent the numbers of the inequality constraints and the equality constraints, respectively. This represents the total number of inequality constraints. Indicates the total number of equality constraints; Regarding the power generation cost target, a quadratic polynomial model is used to describe the relationship between the power generation cost of each generator unit and the active power output of the unit. The total power generation cost target function is expressed as the sum of the power generation cost functions of each generator unit, as shown in equation (3): (3) in, Indicates the number of generator sets; Indicates the generator set number; , and They represent the first The power generation cost coefficient of the Taiwanese generator unit Indicates the first The active power output of the generator sets; Considering the turbine valve point effect, the fuel cost fluctuates due to throttling losses during valve opening. A valve point effect term is introduced into the quadratic cost function, and the total power generation cost model is shown in equation (4): (4) in and They represent the first The coefficient of the valve point effect term of the generator set; Indicates the first Minimum active power output of generator sets; total emissions Taking SO2 and NO into account x The model is as follows: (5) in , , , and Representing the Emission coefficient of the unit; Represented by natural constant An exponential function with base 0; In the power system optimization and dispatch problem, constraints are an important guarantee for ensuring the safe and stable operation of the system. The power balance constraints are as follows: (6) in, This represents the total active power load requirement of the system. The total active power loss generated by the transmission line is expressed as: (7) in, Number the generator set; These are elements of the transmission loss coefficient matrix, reflecting the impact of power flow between nodes on losses; For the first The active power output of the generator sets; Indicates the relationship with the first Linear network loss coefficient related to the output of each generator unit; This represents the constant term network loss coefficient; in simplified scenarios, network loss is ignored. (8) The upper and lower limits of the unit's output are constrained as follows: (9) in, Indicates the first The maximum active power output allowed for each generator set; In the fitness calculation and candidate solution comparison process, a constraint handling mechanism combining feasibility priority criterion and constraint violation coefficient (CV) is adopted. Each candidate solution contains two key attributes: one is the objective function vector. Second, inequality constraint vectors ,in Indicates the first If a constraint is violated, the constraint violation degree of the solution is defined as the sum of all positive constraint values: (10) in, Indicates candidate solutions The degree of constraint violation; This operation takes the larger of the two values; if and only if When a solution is considered feasible, the dominance rule of this mechanism prioritizes feasibility: feasible solutions are preferred over infeasible solutions; Pareto dominance comparison is used between feasible solutions; among infeasible solutions, the solution with the smaller constraint violation is selected. This strategy ensures that the search converges effectively to the feasible region and avoids premature convergence by preserving infeasible solutions on the boundary to traverse complex constraints, thereby better approximating the Pareto optimal frontier.

[0033] In step 1, the algorithm parameters are set as follows: population size Maximum number of iterations External file capacity The above parameters can also be adjusted according to the system size and constraint complexity.

[0034] Step 2: Run the improved multi-objective gray wolf optimization algorithm. Calculate the fitness value using power generation cost and emissions as fitness functions. Update the external archives through non-dominated sorting. Update the individual gray wolf positions using a multi-strategy hybrid mechanism. Iterate until the maximum number of iterations is reached, then terminate the iteration. To facilitate explanation of the overall execution process of the improved multi-objective gray wolf optimization algorithm of this invention, Figure 1 A flowchart illustrating the method of the present invention is shown.

[0035] Step 3: Select compromise solutions and output scheduling schemes. After the iteration terminates, the final non-dominated solution set is saved in the external archive.

[0036] Example 4 This invention presents an economic emission dispatch method for power systems based on an improved gray wolf algorithm, the flowchart of which is shown below. Figure 1 As shown, please follow these steps: Step 1: Establish a multi-objective optimization model for economic emission dispatch of the power system, and set the parameter values ​​for the improved multi-objective gray wolf optimization algorithm; Step 1 is implemented in the following steps: Input the basic data of the power system, construct a multi-objective optimization model with total power generation cost and total pollutant emissions as optimization objectives, and set power balance constraints, upper and lower limits of unit output constraints and network loss constraints according to the physical operating characteristics of generator units; Let the decision variables be the active power output vectors of each generator unit. The multi-objective optimization model uses the total power generation cost objective function and the total pollutant emission objective function as fitness functions. The model forms are shown in equations (1) and (2): (1) (2) in Represents the objective function vector of a multi-objective optimization problem; This represents a vector of decision variables, composed of the active power output of each generator unit; This represents the objective function for total power generation cost; The objective function represents the total pollutant emissions; Indicates the first Inequality constraint functions; Indicates the first One equality constraint function; and These represent the numbers of the inequality constraints and the equality constraints, respectively. This represents the total number of inequality constraints. Indicates the total number of equality constraints; Regarding the power generation cost target, a quadratic polynomial model is used to describe the relationship between the power generation cost of each generator unit and the active power output of the unit. The total power generation cost target function is expressed as the sum of the power generation cost functions of each generator unit, as shown in equation (3): (3) in, Indicates the number of generator sets; Indicates the generator set number; , and They represent the first The power generation cost coefficient of the Taiwanese generator unit Indicates the first The active power output of the generator sets; Considering the turbine valve point effect, the fuel cost fluctuates due to throttling losses during valve opening. A valve point effect term is introduced into the quadratic cost function, and the total power generation cost model is shown in equation (4): (4) in and They represent the first The coefficient of the valve point effect term of the generator set; Indicates the first Minimum active power output of generator sets; total emissions Taking SO2 and NO into account x The model is as follows: (5) in , , , and Representing the Emission coefficient of the unit; Represented by natural constant An exponential function with base 0; In the power system optimization and dispatch problem, constraints are an important guarantee for ensuring the safe and stable operation of the system. The power balance constraints are as follows: (6) in, This represents the total active power load requirement of the system. The total active power loss generated by the transmission line is expressed as: (7) in, Number the generator set; These are elements of the transmission loss coefficient matrix, reflecting the impact of power flow between nodes on losses; For the first The active power output of the generator sets; Indicates the relationship with the first Linear network loss coefficient related to the output of each generator unit; This represents the constant term network loss coefficient; in simplified scenarios, network loss is ignored. (8) The upper and lower limits of the unit's output are constrained as follows: (9) in, Indicates the first The maximum active power output allowed for each generator set; In the fitness calculation and candidate solution comparison process, a constraint handling mechanism combining feasibility priority criterion and constraint violation coefficient (CV) is adopted. Each candidate solution contains two key attributes: one is the objective function vector. Second, inequality constraint vectors ,in Indicates the first If a constraint is violated, the constraint violation degree of the solution is defined as the sum of all positive constraint values: (10) in, Indicates candidate solutions The degree of constraint violation; This operation takes the larger of the two values; if and only if When a solution is considered feasible, the dominance rule of this mechanism prioritizes feasibility: feasible solutions are preferred over infeasible solutions; Pareto dominance comparison is used between feasible solutions; among infeasible solutions, the solution with the smaller constraint violation is selected. This strategy ensures that the search converges effectively to the feasible region and avoids premature convergence by preserving infeasible solutions on the boundary to traverse complex constraints, thereby better approximating the Pareto optimal frontier.

[0037] In step 1, the algorithm parameters are set as follows: population size Maximum number of iterations External file capacity The above parameters can also be adjusted according to the system size and constraint complexity.

[0038] Step 2: Run the improved multi-objective gray wolf optimization algorithm. Calculate the fitness value using power generation cost and emissions as fitness functions. Update the external archives through non-dominated sorting. Update the individual gray wolf positions using a multi-strategy hybrid mechanism. Iterate until the maximum number of iterations is reached, then terminate the iteration. To facilitate explanation of the overall execution process of the improved multi-objective gray wolf optimization algorithm of this invention, Figure 1 A flowchart illustrating the method of the present invention is shown.

[0039] Step 2 is implemented in the following steps: The detailed process involved is as follows: To address the problems of insufficient initial population diversity, susceptibility to local optima, and low convergence accuracy in the standard multi-objective gray wolf optimization algorithm, this invention proposes an improved multi-objective gray wolf optimization algorithm. The specific improvement strategy is as follows: Step 2.1: Hybrid initialization strategy of circular chaotic mapping and Latin hypercube sampling (LHS); Step 2.1 is implemented according to the following steps: In swarm intelligence algorithms, a uniformly distributed initial population is beneficial for improving algorithm performance. Traditional MOGWO algorithms initialize the population randomly, resulting in a relatively uneven initial population distribution with clusters and gaps, which significantly impacts the algorithm's convergence speed and optimization accuracy. This invention employs a hybrid strategy combining circular chaotic mapping and LHS (Low Harmonic Hierarchy Mapping) for population initialization. The ergodicity of circular chaotic mapping enhances exploration capabilities for the initial population. For each individual, a chaotic sequence is first generated using equation (11), and then the chaotic sequence is mapped to the solution space to construct the position vector of the individual: (11) in and Represent the circular chaotic mappings respectively. Second and third The chaotic sequence values ​​generated in the next iteration; and The control parameters represent the circular chaotic mapping; Indicates the number of iterations; This represents the modulo operation with respect to 1; Let represent a sine function; after mapping the generated chaotic sequence to the solution space, the position vector of the corresponding individual can be obtained. Then, stratified sampling using LHS is used to ensure spatial uniformity. For the remaining individuals, LHS generation is employed. Uniform sample points in 3D space: (12) in This represents the sample matrix generated using Latin hypercube sampling; Indicates the sample point number; Indicates the first The location vectors corresponding to each sample point; D represents the question dimension; Indicates the first The sample point at the th th Values ​​on the dimension; Indicates the dimension number; Indicates the first The sample point at the th th Values ​​on the dimension; This invention divides the population into two parts: one part is generated using circular chaotic mapping, and the other part is generated using LHS. To illustrate the impact of the hybrid initialization strategy on the initial population distribution, Figures 2(a) and 2(b) show the initial individual distributions obtained by random initialization and the hybrid initialization strategy of this invention. As can be seen from Figures 2(a) and 2(b), compared with random initialization, the hybrid initialization of this invention provides more uniform coverage in the solution space, which can be used to improve the diversity of the initial population, thereby providing a more sufficient basis for the distribution of candidate solutions for subsequent iterative searches.

[0040] Step 2.2: Nonlinear convergence factor adjustment strategy; Step 2.3: Differentiated perturbation strategy based on grouping dimension; Step 2.4: Hybrid update strategy based on dynamic Lévy flight.

[0041] Step 3: Select compromise solutions and output scheduling schemes. After the iteration terminates, the final non-dominated solution set is saved in the external archive.

[0042] Example 5 This invention presents an economic emission dispatch method for power systems based on an improved gray wolf algorithm, the flowchart of which is shown below. Figure 1 As shown, please follow these steps: Step 1: Establish a multi-objective optimization model for economic emission dispatch of the power system, and set the parameter values ​​for the improved multi-objective gray wolf optimization algorithm; Step 1 is implemented in the following steps: Input the basic data of the power system, construct a multi-objective optimization model with total power generation cost and total pollutant emissions as optimization objectives, and set power balance constraints, upper and lower limits of unit output constraints and network loss constraints according to the physical operating characteristics of generator units; Let the decision variables be the active power output vectors of each generator unit. The multi-objective optimization model uses the total power generation cost objective function and the total pollutant emission objective function as fitness functions. The model forms are shown in equations (1) and (2): (1) (2) in Represents the objective function vector of a multi-objective optimization problem; This represents a vector of decision variables, composed of the active power output of each generator unit; This represents the objective function for total power generation cost; The objective function represents the total pollutant emissions; Indicates the first Inequality constraint functions; Indicates the first One equality constraint function; and These represent the numbers of the inequality constraints and the equality constraints, respectively. This represents the total number of inequality constraints. Indicates the total number of equality constraints; Regarding the power generation cost target, a quadratic polynomial model is used to describe the relationship between the power generation cost of each generator unit and the active power output of the unit. The total power generation cost target function is expressed as the sum of the power generation cost functions of each generator unit, as shown in equation (3): (3) in, Indicates the number of generator sets; Indicates the generator set number; , and They represent the first The power generation cost coefficient of the Taiwanese generator unit Indicates the first The active power output of the generator sets; Considering the turbine valve point effect, the fuel cost fluctuates due to throttling losses during valve opening. A valve point effect term is introduced into the quadratic cost function, and the total power generation cost model is shown in equation (4): (4) in and They represent the first The coefficient of the valve point effect term of the generator set; Indicates the first Minimum active power output of generator sets; total emissions Taking SO2 and NO into account x The model is as follows: (5) in , , , and Representing the Emission coefficient of the unit; Represented by natural constant An exponential function with base 0; In the power system optimization and dispatch problem, constraints are an important guarantee for ensuring the safe and stable operation of the system. The power balance constraints are as follows: (6) in, This represents the total active power load requirement of the system. The total active power loss generated by the transmission line is expressed as: (7) in, Number the generator set; These are elements of the transmission loss coefficient matrix, reflecting the impact of power flow between nodes on losses; For the first The active power output of the generator sets; Indicates the relationship with the first Linear network loss coefficient related to the output of each generator unit; This represents the constant term network loss coefficient; in simplified scenarios, network loss is ignored. (8) The upper and lower limits of the unit's output are constrained as follows: (9) in, Indicates the first The maximum active power output allowed for each generator set; In the fitness calculation and candidate solution comparison process, a constraint handling mechanism combining feasibility priority criterion and constraint violation coefficient (CV) is adopted. Each candidate solution contains two key attributes: one is the objective function vector. Second, inequality constraint vectors ,in Indicates the first If a constraint is violated, the constraint violation degree of the solution is defined as the sum of all positive constraint values: (10) in, Indicates candidate solutions The degree of constraint violation; This operation takes the larger of the two values; if and only if When a solution is considered feasible, the dominance rule of this mechanism prioritizes feasibility: feasible solutions are preferred over infeasible solutions; Pareto dominance comparison is used between feasible solutions; among infeasible solutions, the solution with the smaller constraint violation is selected. This strategy ensures that the search converges effectively to the feasible region and avoids premature convergence by preserving infeasible solutions on the boundary to traverse complex constraints, thereby better approximating the Pareto optimal frontier.

[0043] In step 1, the algorithm parameters are set as follows: population size Maximum number of iterations External file capacity The above parameters can also be adjusted according to the system size and constraint complexity.

[0044] Step 2: Run the improved multi-objective gray wolf optimization algorithm. Calculate the fitness value using power generation cost and emissions as fitness functions. Update the external archives through non-dominated sorting. Update the individual gray wolf positions using a multi-strategy hybrid mechanism. Iterate until the maximum number of iterations is reached, then terminate the iteration. To facilitate explanation of the overall execution process of the improved multi-objective gray wolf optimization algorithm of this invention, Figure 1 A flowchart illustrating the method of the present invention is shown.

[0045] Step 2 is implemented in the following steps: The detailed process involved is as follows: To address the problems of insufficient initial population diversity, susceptibility to local optima, and low convergence accuracy in the standard multi-objective gray wolf optimization algorithm, this invention proposes an improved multi-objective gray wolf optimization algorithm. The specific improvement strategy is as follows: Step 2.1: Hybrid initialization strategy of circular chaotic mapping and Latin hypercube sampling (LHS); Step 2.1 is implemented according to the following steps: In swarm intelligence algorithms, a uniformly distributed initial population is beneficial for improving algorithm performance. Traditional MOGWO algorithms initialize the population randomly, resulting in a relatively uneven initial population distribution with clusters and gaps, which significantly impacts the algorithm's convergence speed and optimization accuracy. This invention employs a hybrid strategy combining circular chaotic mapping and LHS (Low Harmonic Hierarchy Mapping) for population initialization. The ergodicity of circular chaotic mapping enhances exploration capabilities for the initial population. For each individual, a chaotic sequence is first generated using equation (11), and then the chaotic sequence is mapped to the solution space to construct the position vector of the individual: (11) in and Represent the circular chaotic mappings respectively. Second and third The chaotic sequence values ​​generated in the next iteration; and The control parameters represent the circular chaotic mapping; Indicates the number of iterations; This represents the modulo operation with respect to 1; Let represent a sine function; after mapping the generated chaotic sequence to the solution space, the position vector of the corresponding individual can be obtained. Then, stratified sampling using LHS is used to ensure spatial uniformity. For the remaining individuals, LHS generation is employed. Uniform sample points in 3D space: (12) in This represents the sample matrix generated using Latin hypercube sampling; Indicates the sample point number; Indicates the first The location vectors corresponding to each sample point; D represents the question dimension; Indicates the first The sample point at the th th Values ​​on the dimension; Indicates the dimension number; Indicates the first The sample point at the th th Values ​​on the dimension; This invention divides the population into two parts: one part is generated using circular chaotic mapping, and the other part is generated using LHS. To illustrate the impact of the hybrid initialization strategy on the initial population distribution, Figures 2(a) and 2(b) show the initial individual distributions obtained by random initialization and the hybrid initialization strategy of this invention. As can be seen from Figures 2(a) and 2(b), compared with random initialization, the hybrid initialization of this invention provides more uniform coverage in the solution space, which can be used to improve the diversity of the initial population, thereby providing a more sufficient basis for the distribution of candidate solutions for subsequent iterative searches.

[0046] Step 2.2: Nonlinear convergence factor adjustment strategy; Step 2.2 is implemented according to the following steps: Global search capability and local exploitation capability are two common attributes describing the performance of metaheuristic algorithms, and effectively balancing these two aspects is a primary consideration in metaheuristic algorithms. In GWO, the convergence factor... The update method used in the iteration employs a linear decreasing strategy. Linear strategies often struggle to adapt to complex nonlinear search processes. Therefore, this invention proposes a nonlinear convergence factor update method based on a natural exponential change, the equation of which is as follows: (13) in, Indicates the first The nonlinear convergence factor at the next iteration; This represents the current iteration number; and These represent the minimum and maximum values ​​of the convergence factor, respectively. This represents the maximum number of iterations. To illustrate the difference in the variation between the nonlinear convergence factor and the linear convergence factor, Figures 3(a) and 3(b) show the curves of the linear convergence factor and the nonlinear convergence factor of this invention as a function of the number of iterations. Compared with the linear decreasing method, the nonlinear convergence factor of this invention maintains a larger value in the early stage of iteration to enhance the global search capability, and gradually decreases in the middle and later stages of iteration to enhance the local exploitation capability, thereby achieving a dynamic balance between exploration and exploitation.

[0047] Step 2.3: Differentiated perturbation strategy based on grouping dimension; Step 2.3 is implemented according to the following steps: To enhance the global search capability of the Grey Wolf optimizer in early iterations and address the problem of the algorithm easily getting trapped in local optima, this invention introduces a differentiated perturbation mechanism based on grouping dimensions. This strategy effectively improves population diversity and balances global exploration with local exploitation by dividing the population into subgroups at different levels and applying differentiated dimension updates and boundary treatments to different subgroups.

[0048] The specific implementation steps are as follows: During the algorithm's exploration phase, some individuals were selected as those to be disturbed and divided into groups using a clustering strategy. Each subgroup employs a differentiated perturbation strategy to gradually activate the potential search capabilities of its members. The first 90% of iterations are defined as the exploration phase. Each subgroup contains... Individuals, among which Population size; For the Group of individuals, Only certain dimensions are perturbed and updated to achieve the effect of local forgetting and global exploration. Specifically, random selection... Update each dimension: (14) in Indicates the first The number of dimensions that need to be updated for each individual in the group; This indicates the subgroup number; this design allows the earlier subgroups to be perturbed in more dimensions and undertake stronger exploration tasks; while the subsequent subgroups are perturbed in fewer dimensions and focus on local adjustments. For random selection Each dimension, for a randomly selected set The perturbation step size is controlled by a cosine decay factor: (15) in Indicates the first Perturbation step size control factor in the next iteration; Represents the cosine function; Represents the smallest positive constant to prevent the step size from degenerating to 0; step size control factor Initially, the value approaches 1, achieving a large-scale jump; as iterations proceed... Gradually approaching 0 to ensure smooth convergence, the individual position update formula is: (16) in Indicates the first The individual in the first The value after the dimension update; Indicates the individual ID; Indicates the dimension number; Indicates the first The individual in the first The value before the update; and These represent the upper and lower bounds of the dimension, respectively. Indicates that it conforms to the interval Uniformly distributed random numbers; This represents a uniform distribution between 0 and 1; If an individual exceeds the boundary after the update, an adaptive repair strategy is adopted: For high-dimensional problems (D>15), a dimension cross-replacement mechanism is used, randomly selecting the corresponding dimension value of an individual from the current population for replacement. This mechanism utilizes existing information within the population for knowledge transfer, avoiding invalid searches. For low-dimensional problems, a random re-initialization mechanism is used to maintain the randomness and diversity of the population. The update formula is as follows: (17) in Indicates the first The individual in the first The value after repair; This indicates the first random selection from the current population. The individual in the first Values ​​on the dimension; This represents a random individual index; through the synergistic effect of the above-mentioned grouping differential perturbation, nonlinear step size control and adaptive boundary repair, the algorithm's escape ability and optimization accuracy in complex high-dimensional search spaces are significantly improved.

[0049] Step 2.4: Hybrid update strategy based on dynamic Lévy flight.

[0050] Step 2.4 is implemented according to the following steps: To enhance the ability to escape local optima, an adaptive hybrid update strategy is constructed, which generates random numbers during position updates. ,like (Dynamic trigger probability), then Lévy flight will be executed: (18) in Indicates the first The position vector of the individual at the next iteration; This represents the reference position vector used to guide position updates; This represents the Lévy flight stride coefficient; Indicates the first The position vector of the individual in the next iteration; otherwise, the position is updated using the standard GWO triple leadership mechanism. This strategy effectively utilizes long step jumps to expand the search range and avoid the algorithm getting stuck in local optima. The MW benchmark functions can be used for comparative evaluation of constrained multi-objective optimization algorithms. To illustrate the solution performance of the IMOGWO method of this invention on constrained multi-objective problems, 11 two-dimensional test functions from the MW benchmark suite are selected for simulation illustration, including MW1 to MW3, MW5 to MW7, and MW9 to MW13. These test functions contain coupled forms of objective functions and constraints, and can be used to construct nonlinear, non-convex, and strongly constrained test scenarios. Simulations can be performed on general-purpose computing devices.

[0051] In this embodiment, the maximum number of iterations was set to 1000, and each algorithm was run independently 30 times for each problem. IMOGWO was compared with MOGWO, AGEMOEA-II, PPS, ARMOEA, MOPSO, NSGA-III, and NSGA-II, and the average of the inverse generational distance (IGD) and the hypervolume (HV) was used as evaluation metrics. A smaller IGD value indicates a higher degree of proximity between the non-dominated solution set and the reference front, while a larger HV value indicates a higher degree of coverage of the non-dominated solution set in the target space. The test results are shown in Tables 1 and 2, respectively.

[0052] Table 1. Comparison of IGD mean values ​​between IMOGWO and the contrasting algorithms on the MW test set.

[0053] Table 1 presents the mean IGD results of each algorithm on MW1, MW2, MW3, MW5, MW6, MW7, MW9, MW10, MW11, MW12, and MW13. The values ​​in bold are the optimal values ​​for the same test problem. As shown in Table 1, the IMOGWO method of this invention achieves the minimum mean IGD in 9 out of 11 test problems, and its mean IGD is smaller than that of the standard MOGWO across all test problems. For example, on MW1, the mean IGD of IMOGWO is 0.0048, while that of MOGWO is 0.3724.

[0054] Table 2 Comparison of HV mean values ​​between IMOGWO and the contrasting algorithms on the MW test set.

[0055] Table 2 presents the mean HV values ​​of each algorithm on the aforementioned test problems. The values ​​in bold are the optimal values ​​for the same test problem. As shown in Table 2, the IMOGWO method of this invention achieves the maximum mean HV value in 9 out of 11 test problems, and its mean HV value is larger than that of the standard MOGWO across all test problems. For example, on MW1, the mean HV value of IMOGWO is 0.5525, while that of MOGWO is 0.2125.

[0056] Figures 4(a1) to 4(e4) show the comparison results of the non-dominated solution set distributions obtained by the proposed method IMOGWO with the comparative algorithms MOGWO, MOPSO, NSGA-II, and NSGA-III under the test functions MW3, MW6, MW7, and MW12. The solid lines in the figures represent the reference front, and the scattered points represent the non-dominated solution sets obtained by each algorithm under the same iteration settings. As can be seen from Figures 4(a1), 4(a2), 4(a3), and 4(a4), the non-dominated solution set obtained by the proposed method closely approximates the reference front distribution and maintains good continuity and uniformity in the non-convex region, discontinuous region, and both ends of the target space. In comparison, the MOGWO solutions in Figures 4(b1), 4(b2), 4(b3), and 4(b4) exhibit significant deviations from the reference front or sparse distribution of non-dominated solutions under some test functions; the MOPSO solutions in Figures 4(c1), 4(c2), 4(c3), and 4(c4) show solution set gaps or insufficient coverage in some intervals; the NSGA-II solutions in Figures 4(d1), 4(d2), 4(d3), and 4(d4), and the NSGA-III solutions in Figures 4(e1), 4(e2), 4(e3), and 4(e4) also show varying degrees of distribution gaps or insufficient local coverage under some test functions. The distribution characteristics shown in Figures 4(a1) to 4(e4) correspond to the changing trends of the IGD and HV evaluation indices in Tables 1 and 2.

[0057] Step 3: Select compromise solutions and output scheduling schemes. After the iteration terminates, the final non-dominated solution set is saved in the external archive.

[0058] Example 6 To illustrate the solution results of the method of this invention under different system scales and modeling conditions, the method of this invention was applied to the economic emission scheduling models of IEEE 30-node 6-unit, 10-unit, and 40-unit systems for simulation. To compare the solution results under different conditions, three test scenarios were constructed: Regarding network losses, the 6-unit and 10-unit system examples included network losses, while the 40-unit system example ignored network losses for comparison; Regarding objective function modeling, the 6-unit system example used the standard generation cost function, while the 10-unit and 40-unit system examples introduced a threshold effect term into the generation cost function.

[0059] (1) IEEE 30-node 6-unit system In one embodiment, the IEEE 30-node 6-unit system is selected as Case 1 to solve the economic emission dispatch model that takes network losses into account and uses the standard cost function. The total system load demand is set at 2.834 pu, and the baseline capacity is 100 MVA. In this case, while calculating the total power loss of transmission lines, the generation cost function adopts a quadratic polynomial form and does not include a threshold effect term; the emission objective function is calculated using the following nonlinear model: (twenty two) In this embodiment, the compromise solution obtained using the IMOGWO method of the present invention corresponds to the following indices: power generation cost 616.8518, emissions 0.2000, and overall score 0.1014. The distribution of the non-dominated solution set after 2000 iterations of the method of the present invention and the standard MOGWO is compared, as shown in Figures 5(a) and 5(b). The non-dominated solution obtained by the method of the present invention (Figure 5(b)) is continuously distributed within the cost-emission trade-off region, with a relatively complete coverage of the solution points; it also exhibits a relatively stable point density in the middle trade-off region and both ends of the region. In contrast, the solution points obtained by the standard MOGWO (Figure 5(a)) are relatively sparsely distributed in some intervals, and the point spacing increases in some local areas.

[0060] Under the same parameter settings, the method of this invention is compared with MOGWO, ABC-NSGA-II, HBB-BC, MOPSO, and MOEA / D. Each algorithm is run independently 30 times for this example, and the results are averaged. The comparison results are shown in Table 3, where the bold text represents the optimal value under the same evaluation index. As can be seen from Table 3, the generation cost corresponding to the method of this invention is relatively small, and the comprehensive score is also at a relatively low level, which can be used for output cost-emission trade-off dispatch schemes.

[0061] Table 3 Comparison of trade-off indices between IMOGWO and the comparative algorithm for 6 units

[0062] In summary, the results shown in Figures 5(a), 5(b) and Table 3 are used to describe the compromise solution output of the method of the present invention under the condition of taking network loss into account and using the standard cost function.

[0063] (2) Unit 10 system In one embodiment, a 10-unit system is selected as Case 2, with a total system load demand of 2000MW, taking into account the total power loss of transmission lines. The method of this invention is used to solve the case, and the resulting compromise solution corresponds to the following indices: generation cost 112795.96, emissions 4188.73, and a comprehensive score of 0.4000.

[0064] The distribution of non-dominated solution sets obtained by the IMOGWO method of this invention and the standard MOGWO method are compared, as shown in Figures 6(a) and 6(b). The non-dominated solution generated by the method of this invention (Figure 6(b)) is more continuously distributed within the trade-off region and maintains a certain coverage in both the low-cost and low-emission regions. The non-dominated solution corresponding to the standard MOGWO method (Figure 6(a)) has insufficient coverage in some intervals, the solution points are more concentrated in local areas, and the point spacing is increased.

[0065] Under the same parameter settings, the method of this invention is compared with MOGWO, GSA, MODE, MOAOA, and NSGA-II. The comparison results are shown in Table 4, where the bold text represents the optimal value under the same evaluation index. As can be seen from Table 4, the power generation cost and comprehensive score corresponding to the method of this invention are relatively small, and it can be used for output cost-emission trade-off dispatch schemes.

[0066] Table 4. Comparison of trade-off indices between IMOGWO and the comparative algorithm for 10 units.

[0067] In summary, Figures 6(a) and 6(b) and the results shown in Table 4 are used to describe the compromise solution output of the method of the present invention under medium-scale system conditions.

[0068] (3) Large-scale system of 40 units In one embodiment, a 40-unit system is selected as Case 3, with a total system load demand of 10500MW, comprising 40 generating units, and a threshold effect term is introduced into the generation cost function. Network losses are ignored in the calculation of this case. The method of the present invention is used to solve the case, and the compromise solution obtained corresponds to the following indices: generation cost 125468.98, emissions 203415.33, and a comprehensive score of 0.2680.

[0069] The distribution of non-dominated solution sets obtained by the IMOGWO method of this invention and the standard MOGWO method are compared, as shown in Figures 7(a) and 7(b). Under the conditions of a 40-unit scale and the introduction of a valve point effect term, the non-dominated solution obtained by the method of this invention (Figure 7(b)) can cover a relatively complete tradeoff region, and the solution points are distributed more continuously along the tradeoff band; the point density remains relatively stable in some intervals. In contrast, the standard MOGWO (Figure 7(a)) has a relatively sparse distribution of solution points in some intervals, and there is insufficient coverage in some local areas.

[0070] Under the same parameter settings, the method of this invention was compared with MOGWO, RCCRO, MABC, BSA, and MOMVO. The comparison results are shown in Table 5, where the bold text represents the optimal value under the same evaluation index. As can be seen from Table 5, the emissions and comprehensive score values ​​corresponding to the method of this invention are smaller; for example, compared with MOMVO, the difference in emissions value of the method of this invention is 6502.81; compared with MOGWO, the comprehensive score value of the method of this invention is smaller.

[0071] Table 5. Comparison of trade-off indices between IMOGWO and the comparative algorithm for 40 units.

[0072] In summary, Figures 7(a) and 7(b) and the results shown in Table 5 are used to describe the compromise solution output of the method of the present invention under large-scale system conditions.

Claims

1. A power system economic emission dispatch method based on an improved gray wolf algorithm, characterized in that, The specific steps are as follows: Step 1: Establish a multi-objective optimization model for economic emission dispatch of the power system, and set the parameter values ​​for the improved multi-objective gray wolf optimization algorithm; Step 2: Run the improved multi-objective gray wolf optimization algorithm, calculate the fitness value using power generation cost and emissions as fitness functions, update the external archives through non-dominated sorting, update the individual gray wolf positions using a multi-strategy mixing mechanism, iterate until the maximum number of iterations is reached, and then terminate the iteration. Step 3: Select compromise solutions and output scheduling schemes. After the iteration terminates, the final non-dominated solution set is saved in the external archive.

2. The power system economic emission dispatch method based on the improved gray wolf algorithm according to claim 1, characterized in that, Step 1 is implemented in the following steps: Input the basic data of the power system, construct a multi-objective optimization model with total power generation cost and total pollutant emissions as optimization objectives, and set power balance constraints, upper and lower limits of unit output constraints and network loss constraints according to the physical operating characteristics of generator units; Let the decision variables be the active power output vectors of each generator unit. The multi-objective optimization model uses the total power generation cost objective function and the total pollutant emission objective function as fitness functions. The model forms are shown in equations (1) and (2): （1） （2） in Represents the objective function vector of a multi-objective optimization problem; This represents a vector of decision variables, composed of the active power output of each generator unit; This represents the objective function for total power generation cost; The objective function represents the total pollutant emissions; Indicates the first Inequality constraint functions; Indicates the first One equality constraint function; and These represent the numbers of the inequality constraints and the equality constraints, respectively. This represents the total number of inequality constraints. Indicates the total number of equality constraints; Regarding the power generation cost target, a quadratic polynomial model is used to describe the relationship between the power generation cost of each generator unit and the active power output of the unit. The total power generation cost target function is expressed as the sum of the power generation cost functions of each generator unit, as shown in equation (3): （3） in, Indicates the number of generator sets; Indicates the generator set number; , and They represent the first The power generation cost coefficient of the Taiwanese generator unit Indicates the first The active power output of the generator sets; Considering the turbine valve point effect, the fuel cost fluctuates due to throttling losses during valve opening. A valve point effect term is introduced into the quadratic cost function, and the total power generation cost model is shown in equation (4): （4） in and They represent the first The coefficient of the valve point effect term of the generator set; Indicates the first Minimum active power output of generator sets; total emissions Taking SO2 and NO into account x The model is as follows: （5） in , , , and Representing the Emission coefficient of the unit; Represented by natural constant An exponential function with base 0; In the power system optimization and dispatch problem, constraints are an important guarantee for ensuring the safe and stable operation of the system. The power balance constraints are as follows: （6） in, This represents the total active power load requirement of the system. The total active power loss generated by the transmission line is expressed as: （7） in, Number the generator set; These are elements of the transmission loss coefficient matrix, reflecting the impact of power flow between nodes on losses; For the first The active power output of the generator sets; Indicates the relationship with the first Linear network loss coefficient related to the output of each generator unit; This represents the constant term network loss coefficient; in simplified scenarios, network loss is ignored. （8） The upper and lower limits of the unit's output are constrained as follows: （9） in, Indicates the first The maximum active power output allowed for each generator set; In the fitness calculation and candidate solution comparison process, a constraint handling mechanism combining feasibility priority criterion and constraint violation coefficient (CV) is adopted. Each candidate solution contains two key attributes: one is the objective function vector. Second, inequality constraint vectors ,in Indicates the first If a constraint is violated, the constraint violation degree of the solution is defined as the sum of all positive constraint values: （10） in, Indicates candidate solutions The degree of constraint violation; This operation takes the larger of the two values; if and only if When a solution is considered feasible, the dominance rule of this mechanism prioritizes feasibility: feasible solutions are preferred over infeasible solutions; Pareto dominance comparison is used between feasible solutions; among infeasible solutions, the solution with the smaller constraint violation is selected. This strategy ensures that the search converges effectively to the feasible region and avoids premature convergence by preserving infeasible solutions on the boundary to traverse complex constraints, thereby better approximating the Pareto optimal frontier.

3. The power system economic emission dispatch method based on the improved gray wolf algorithm according to claim 2, characterized in that, In step 1, the algorithm parameters are set as follows: population size. Maximum number of iterations External file capacity .

4. The power system economic emission dispatch method based on the improved gray wolf algorithm according to claim 2, characterized in that, Step 2 is implemented in the following steps: Step 2.1: Hybrid initialization strategy of circular chaotic mapping and Latin hypercube sampling (LHS); Step 2.2: Nonlinear convergence factor adjustment strategy; Step 2.3: Differentiated perturbation strategy based on grouping dimension; Step 2.4: Hybrid update strategy based on dynamic Lévy flight.

5. The power system economic emission dispatch method based on the improved gray wolf algorithm according to claim 4, characterized in that, Step 2.1 is implemented in the following steps: Enhancing exploration capabilities by leveraging the ergodicity of circular chaotic mappings, for the previous... For each individual, a chaotic sequence is first generated using equation (11), and then the chaotic sequence is mapped to the solution space to construct the position vector of the individual: （11） in and Represent the circular chaotic mappings respectively. Second and third The chaotic sequence values ​​generated in the next iteration; and The control parameters represent the circular chaotic mapping; Indicates the number of iterations; This represents the modulo operation with respect to 1; Let represent a sine function; after mapping the generated chaotic sequence to the solution space, the position vector of the corresponding individual can be obtained. Then, stratified sampling using LHS is used to ensure spatial uniformity. For the remaining individuals, LHS generation is employed. Uniform sample points in 3D space: （12） in This represents the sample matrix generated using Latin hypercube sampling; Indicates the sample point number; Indicates the first The location vectors corresponding to each sample point; D represents the question dimension; Indicates the first The sample point at the th th Values ​​on the dimension; Indicates the dimension number; Indicates the first The sample point at the th th The value that can be taken in the dimension.

6. The power system economic emission dispatch method based on the improved gray wolf algorithm according to claim 5, characterized in that, Step 2.2 is implemented in the following steps: The equation is as follows: （13） in, Indicates the first The nonlinear convergence factor at the next iteration; This represents the current iteration number; and These represent the minimum and maximum values ​​of the convergence factor, respectively. This represents the maximum number of iterations.

7. The power system economic emission dispatch method based on the improved gray wolf algorithm according to claim 6, characterized in that, Step 2.3 is implemented in the following steps: During the algorithm's exploration phase, some individuals were selected as those to be disturbed and divided into groups using a clustering strategy. Each subgroup employs a differentiated perturbation strategy to gradually activate the potential search capabilities of its members. The first 90% of iterations are defined as the exploration phase. Each subgroup contains... Individuals, among which Population size; For the Group of individuals, Only some dimensions are perturbed and updated; specifically, random selection is used. Update each dimension: （14） in Indicates the first The number of dimensions that need to be updated for each individual in the group; Indicates the subgroup number; For random selection Each dimension, for a randomly selected set The perturbation step size is controlled by a cosine decay factor: （15） in Indicates the first Perturbation step size control factor in the next iteration; Represents the cosine function; Represents the smallest positive constant to prevent the step size from degenerating to 0; step size control factor Initially, the value approaches 1, achieving a large-scale jump; as iterations proceed... Gradually approaching 0 to ensure smooth convergence, the individual position update formula is: （16） in Indicates the first The individual in the first The value after the dimension update; Indicates the individual ID; Indicates the dimension number; Indicates the first The individual in the first The value before the update; and These represent the upper and lower bounds of the dimension, respectively. Indicates that it conforms to the interval Uniformly distributed random numbers; This represents a uniform distribution between 0 and 1; If an updated individual exceeds the boundary, an adaptive repair strategy is adopted: For high-dimensional problems, a dimension cross-replacement mechanism is used, randomly selecting the corresponding dimension value of an individual from the current population for replacement. This mechanism utilizes existing information within the population for knowledge transfer, avoiding invalid searches. For low-dimensional problems, a random re-initialization mechanism is used to maintain the randomness and diversity of the population. The update formula is as follows: （17） in Indicates the first The individual in the first The value after repair; This indicates the first random selection from the current population. The individual in the first Values ​​on the dimension; This represents a random individual index; through the synergistic effect of the above-mentioned grouping differential perturbation, nonlinear step size control and adaptive boundary repair, the algorithm's escape ability and optimization accuracy in complex high-dimensional search spaces are significantly improved.

8. The power system economic emission dispatch method based on the improved gray wolf algorithm according to claim 7, characterized in that, Step 2.4 is implemented in the following steps: To enhance the ability to escape local optima, an adaptive hybrid update strategy is constructed, which generates random numbers during position updates. ,like Then execute Lévy flight: （18） in Indicates the first The position vector of the individual at the next iteration; This represents the reference position vector used to guide position updates; This represents the Lévy flight stride coefficient; Indicates the first The position vector of the individual is updated in the next iteration; otherwise, the position is updated using the standard GWO triple leadership mechanism. This strategy effectively utilizes long step jumps to expand the search range and avoid the algorithm getting stuck in local optima.

9. The power system economic emission dispatch method based on the improved gray wolf algorithm according to claim 8, characterized in that, Step 3 is implemented in the following steps: Using the normalized distance criterion based on the ideal point, the objective function values ​​of each candidate solution in the non-dominated solution set are normalized using the minimum-maximum normalization method. The Euclidean distance from each candidate solution to the ideal point is then calculated based on the ideal point distance criterion, according to equation (19): （19） in Indicates the first Euclidean distances from each candidate solution to the ideal point; Indicates the candidate solution number; and These represent the normalized target values ​​for power generation cost and pollutant emissions, respectively. The ideal point is taken as the origin in the normalized target space. The candidate solution with the smallest Euclidean distance is selected as the compromise solution, and the unit scheduling scheme is output accordingly. The calculation is explained using the min-maximum standardized and weighted comprehensive scoring method. The standardized formula is shown in equation (20): （20） in Indicators Normalized values; This represents the original index value to be normalized. and These represent the minimum and maximum values ​​of all comparison algorithms for a certain evaluation index, respectively. After standardization, each index value is linearly mapped to the [0,1] interval, and it is unified that the smaller the value, the better the performance under that evaluation index. The final comprehensive score is calculated using the following formula: （21） Weight , The smaller the overall score, the closer the compromise result under the evaluation criteria is to the expectation.

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