Super-multi-objective optimization method and system based on dual-file coevolution and dual-distance index

Abstract

The invention discloses a super-multi-objective optimization method and system based on dual-file coevolution and dual-distance indexes, and the method comprises the steps: generating an initial population based on the parameters of a power system, initializing the parameters and a reference vector set, and storing the population into a convergence file and a diversity file; calculating an ideal point and a worst point based on the convergence file, and updating the convergence file in combination with a double-distance index; maintaining population global distribution through a reference vector association operation; evaluating a population distribution state in combination with information entropy to determine a local neighborhood size; calculating individual local neighborhood density based on the parallel distance and updating a diversity file; constructing a mating pool based on the double archives and generating a filial generation population; and iteratively executing until the non-dominated solution set in the diversity file is output. According to the method, the convergence and diversity retention capability of the algorithm to the complex leading edge is enhanced, and a power system scheduling scheme with multi-target balance of the power generation cost, the transmission loss, the line utilization rate and the power generation adjustment amount can be efficiently generated.

Description

Technical Field

[0001] This invention belongs to the field of power system optimization and scheduling technology, and particularly relates to a multi-objective optimization method and system based on dual-archive co-evolution and dual-distance index. Background Technology

[0002] In the field of power system optimization and dispatching, grid operators need to coordinate multiple physical devices such as generators, transmission lines, and transformers simultaneously. Their optimization objectives typically include minimizing generation costs, minimizing transmission losses, minimizing line utilization, and minimizing generation adjustments. These objectives are inherently conflicting; for example, pursuing the lowest generation cost may lead to excessive line load, while ensuring safe line operation may require increasing generation costs and generator adjustment frequency. This complex trade-off between multiple objectives makes power system optimization and dispatching essentially a complex multi-objective optimization problem. Traditional single-objective optimization methods, such as weighted summation, often rely on dispatchers' prior experience to transform the multi-objective problem into a single-objective problem. This method struggles to obtain a diverse range of trade-off solutions, resulting in a final dispatching scheme that may not adapt to the real-time changing operating conditions of the grid, affecting the system's economic operation and safety margin. To address this challenge, researchers have proposed several multi-objective evolutionary algorithms, mainly including three categories: (1) Algorithms based on improved dominance relations. Such methods enhance selection pressure by relaxing dominance conditions. However, when dealing with the high-dimensional target space of power systems, excessive selection pressure can easily lead to premature convergence of the search process, making it impossible to explore a wide range of scheduling strategies. The resulting solution set lacks diversity, which is reflected in the actual operation of the power grid as a single scheduling scheme that is difficult to cope with diverse load demands and operating conditions.

[0003] (2) Decomposition algorithm based on reference vector. This type of method guides the search direction through a preset reference vector. However, when dealing with the complex and irregular Pareto front common in power systems, its fixed reference vector system is difficult to flexibly adapt to the actual operating constraints of the power grid. This results in insufficient convergence pressure of the algorithm, which cannot accurately approximate the true high-efficiency operating range of the system. The generated scheduling scheme may perform poorly in terms of either economy or security.

[0004] (3) Performance index-based algorithms. These algorithms use indices to guide the search, but a single fixed index can easily cause the optimization process to get stuck in local optima. In engineering terms, this means that the scheduling scheme gets stuck in a certain fixed operating mode and cannot discover innovative scheduling strategies that can significantly improve the overall operating efficiency of the system. On the other hand, introducing multiple indices will significantly increase the computational overhead and make it difficult to meet the real-time requirements of large-scale power systems for optimization calculations.

[0005] While existing dual-archive algorithm frameworks provide a foundation for balancing convergence and diversity, their main shortcomings in power system applications lie in: insufficient adaptability of the convergence archive update strategy to complex power grid front shapes, and a lack of targeted design for the diversity archive maintenance mechanism based on system operating characteristics. These limitations often lead to optimization results biased towards local optima, making it difficult to generate a set of high-quality dispatch schemes that guarantee both economy and security constraints. Therefore, there is an urgent need to develop a novel optimization method that can deeply integrate power system operating characteristics and achieve better performance in both convergence speed and distribution quality, in order to meet the practical needs of modern power grids for efficient and reliable dispatch decisions. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention proposes a multi-objective optimization method and system based on dual-archive co-evolution and dual-distance indexes, thereby resolving the issues present in the prior art.

[0007] To achieve the above objectives, this invention provides a multi-objective optimization method based on dual-archive co-evolution and dual-distance metrics, comprising: S1. Generate an initial population based on the parameters of the power system to be optimized, and initialize the algorithm parameters and reference vector set. Store the initial population in the convergence file and the diversity file respectively. S2. Calculate the ideal point and the worst point based on the convergence archive, and update the convergence archive by combining the dual distance index; S3. Associate the individuals in the diversity archive with the reference vector set to maintain the global distribution of the population; S4. Combine information entropy to evaluate the population distribution status of the diversity archive, and determine the size of the local neighborhood based on the evaluation results; S5. Based on the local neighborhood size, calculate the local neighborhood density of individuals in the diversity profile using parallel distance, and update the diversity profile based on the local neighborhood density; S6. Construct a mating pool based on the updated convergence file and diversity file, and generate a progeny population through the mating pool; S7. Iterate through steps S2-S6 until the termination condition is met, and output the solution set in the diversity archive as the optimal scheduling scheme for the power system.

[0008] Optionally, in step S2, the ideal point and worst point are calculated based on the convergence profile, and the convergence profile is updated using a dual-distance metric, including: Calculate the ideal point and worst point based on the target value of individuals in the current population; calculate the dual distance index value of each individual in the convergence archive based on the ideal point and worst point; filter individuals in the convergence archive based on the dual distance index value and update the convergence archive.

[0009] Optionally, the calculation formula for the dual-distance index in S2 is as follows: ; in, Individual The Euclidean distance to the ideal point is calculated using the following formula: ; Individual The Euclidean distance to the worst-case scenario is calculated using the following formula: ; In the current population The minimum value, In the current population The maximum value, It is the first The ideal value for a target It is the first The worst value of each target It is the target quantity. Individual In the The objective function value on each target.

[0010] Optionally, in step S4, the population distribution status of the diversity archive is evaluated using information entropy, and the size of the local neighborhood is determined based on the evaluation results, including: Calculate the current population distribution entropy and the optimal distribution entropy; determine the size of the local neighborhood based on the ratio of the current population distribution entropy to the optimal distribution entropy.

[0011] Optionally, calculate the current population distribution entropy. and optimal distribution entropy The formula is: ; ; Where L is the number of reference vectors, For the first Reference vectors, For diversity archives and the first The proportion of individuals associated with each reference vector, i.e. , To the reference vector The number of associated individuals, For the size of the diversity archive, This represents the probability value under an ideal uniform distribution.

[0012] Optionally, the size of the local neighborhood is determined based on the ratio of the current population distribution entropy to the optimal distribution entropy, as shown in the following formula: ; in, For the floor function, This is the function for finding the maximum value.

[0013] Optionally, in step S5, the calculation of the local neighborhood density of individuals in the diversity profile using parallel distance includes: calculating the parallel distance between individuals based on the normalized fitness value of the individuals; and calculating the local neighborhood density of each individual based on the parallel distance and the local neighborhood size.

[0014] Optionally, the parameters of the power system to be optimized include the upper and lower limits of generator output and cost coefficient, the impedance and capacity limits of transmission lines, and the load demand of each node. The solution set is a non-dominated solution set for multiple objectives, including power generation cost, transmission loss, line utilization, and power generation adjustment.

[0015] A multi-objective optimization system based on dual-archive co-evolution and dual-distance metrics is provided to implement the above method, comprising: An initialization module is used to generate an initial population based on the parameters of the power system to be optimized, and to initialize the algorithm parameters and reference vector set, and to store the initial population in the convergence file and the diversity file respectively. The convergence archive update module is used to calculate the ideal point and the worst point based on the convergence archive, and update the convergence archive by combining the dual distance index. A global distributed maintenance module is used to associate individuals in the diversity archive with the reference vector set; The state assessment module is used to assess the population distribution state of the diversity archive by combining information entropy, and to determine the size of the local neighborhood based on the assessment results. The local diversity maintenance module is used to calculate the local neighborhood density of individuals in the diversity profile based on the local neighborhood size using parallel distance, and to update the diversity profile based on the local neighborhood density. The mating pool generation module is used to construct a mating pool based on the updated convergence profile and diversity profile, and to generate a progeny population through the mating pool. The iterative control module is used to control the iterative execution of the convergence file update module, global distribution maintenance module, state evaluation module, local diversity maintenance module and mating pool generation module until the termination condition is met, and outputs the solution set in the diversity file as the optimal scheduling scheme of the power system.

[0016] Compared with the prior art, the present invention has the following advantages and technical effects: This invention introduces a dual distance index (DDI) to simultaneously consider the distances of an individual to both the ideal point (e.g., the lowest generation cost point) and the worst-case point (e.g., the highest transmission loss point), effectively enhancing convergence pressure and preventing the algorithm from biasing towards the population center during the search process. This improves the adaptability to complex Pareto fronts of power systems (e.g., nonlinear, high-dimensional objective spaces) and accelerates convergence towards a high-quality scheduling scheme that balances economy and safety. Regarding diversity maintenance, a reference vector and information entropy evaluation mechanism are combined to dynamically adjust the size of the local neighborhood and use parallel distance to calculate local density, balancing global distribution and local diversity. This ensures the algorithm can comprehensively explore the solution space, generating a set of scheduling schemes covering various trade-offs from low cost to high reliability, providing grid operators with diverse decision-making options. Through a dual-archive collaboration mechanism, convergence and diversity are maintained separately, avoiding the mutual interference problem in traditional methods, thus significantly improving the overall performance and robustness of the algorithm in multi-objective optimization. The output of this method directly corresponds to a series of executable and selectable power system dispatching instructions (such as generator output planning, line load control, etc.), which can be directly applied to the power grid control center to solve the practical technical problem of balancing economy, transmission efficiency, line safety and power generation adjustment stability, and significantly improve the intelligence level of power grid operation, resource utilization efficiency and system stability. Attached Figure Description

[0017] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a flowchart of a multi-objective optimization method based on dual-archive co-evolution and dual-distance index according to an embodiment of the present invention; Figure 2 This is the first part of the parallel coordinate diagram of the 15-dimensional WFG8 problem in this embodiment of the invention; where (a), (b), (c), and (d) respectively show the final population distribution of SSCEA, RVEAIGNG, Two_Arch2, and MaOEAIGD; Figure 3 The second part of the parallel coordinate diagram of the 15-dimensional WFG8 problem in this embodiment of the invention; wherein (a), (b), and (c) show the final population distribution of hpaEA, RVEA-DWC, and MDCS, respectively; Figure 4 This is a convergence curve of the 15-dimensional problem IGDplus index as a function of the number of iterations in an embodiment of the present invention; wherein (a) to (d) correspond to the DTLZ6, MaF6, WFG5 and WFG7 test problems, respectively; Figure 5This is a convergence curve of the HV index of the 15-dimensional problem in this embodiment of the invention as a function of the number of iterations; where (a) to (d) correspond to the DTLZ6, MaF6, WFG5 and WFG7 test problems, respectively. Figure 6 This is a schematic diagram illustrating the change in total power generation cost as a function of the number of iterations in an embodiment of the present invention. Figure 7 This is a schematic diagram illustrating the change in transmission loss as a function of iteration number in an embodiment of the present invention. Figure 8 This is a schematic diagram illustrating the variation of maximum line utilization with the number of iterations in an embodiment of the present invention; Figure 9 This is a schematic diagram illustrating the change of the total power generation adjustment amount with the number of iterations in an embodiment of the present invention; Figure 10 This is a parallel coordinate diagram of the final population decision variables in an embodiment of the present invention. Detailed Implementation

[0018] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0019] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0020] Example 1 This invention deeply couples the algorithm search process with the physical operating characteristics of the power system. It collaboratively manages convergence and diversity through a dual-archive mechanism (convergence archive and diversity archive), and enhances convergence pressure by combining dual-distance indicators (considering the distance between an individual and both the ideal and worst-case points). It dynamically maintains global and local diversity through information entropy assessment and parallel distance calculation, thereby achieving precise optimization and robust control of the power grid's operating state. Figure 1 As shown, this embodiment provides a multi-objective optimization method based on dual-archive co-evolution and dual-distance metrics, including: Step S1: First, an optimization model is constructed and initialized based on the specific parameters of the power system. These parameters include the upper and lower limits of generator output and cost coefficients, the impedance and capacity limits of transmission lines, and the load demand of each node. Then, under the premise of satisfying system power balance and equipment safety operation constraints, a set of initial scheduling schemes is randomly generated to form an initial population, with each scheme representing a complete generator output plan. Simultaneously, parameters such as the population size and number of iterations required by the algorithm, as well as the reference vector set used to guide the search direction, are initialized. The initial populations are stored in a convergence archive focused on convergence and a diversity archive responsible for maintaining distribution, laying the foundation for subsequent multi-objective collaborative optimization. The technical effect of this step is that by generating a set of diverse initial scheduling schemes feasible under safety constraints, a solid foundation is laid for subsequent deep optimization. These schemes cover different generator output combinations, representing various potential operating strategies from cost-first to safety-first, ensuring that the optimization search can start from the actual feasible operating space of the power grid, comprehensively exploring various possible scheduling trade-offs, thus providing a key guarantee for ultimately obtaining a set of Pareto optimal scheduling schemes that can be directly applied to the power grid control center.

[0021] The system uses real-time and historical data of the power system as core inputs, including the cost characteristics and output limits of generator units, the topology parameters and branch capacity constraints of the transmission network, and load forecast data of system nodes. By constructing multi-objective optimization models for generation cost, transmission loss, line utilization, and generation adjustment, the above data is mapped into standardized inputs that the algorithm can process. After the system is running, the output is a set of Pareto optimal scheduling schemes covering different operational trade-off points. This scheme set consists of a series of specific generator active power output plans, which can be directly used for issuance and execution by the power grid control center, thereby achieving comprehensive optimization and closed-loop control of the power system's economy, security, and stability.

[0022] Power system details: The total generation cost, total transmission loss, maximum branch utilization, and total generation adjustment of the power system are optimized, and the output power of the system generators is adjusted reasonably to minimize the optimization target.

[0023] The objective function for the total power generation cost of the system is approximately a quadratic function: in, For the number of generators, For the first The output power of the generator , , For the first Cost coefficient of a generator.

[0024] The total transmission loss of the system is defined as the total power generation minus the total load: in, For the number of generators, For the first The output power of the generator Number of busbars For the first The load power on each busbar.

[0025] The maximum branch utilization rate of the system is defined as the maximum power utilization rate of all branches: in For the number of branch roads, For from the busbar to bus The branch power flow, For the first Capacity limit for each branch.

[0026] The total power generation adjustment of the system is defined as the total absolute deviation of all generator outputs relative to the base / initial power generation: in, For the number of generators, For the first The output power of the generator For the first The initial output power of the generator.

[0027] Step S2: A dual-distance index is used to screen scheduling schemes in the population. This index calculates the positive distance between each scheduling scheme and the ideal operating point of the power system (the target point consisting of the theoretical lowest generation cost and minimum transmission loss), and the negative distance between it and the unacceptable operating point (the worst point corresponding to the highest generation cost and line overload risk), and considers both comprehensively. This mechanism ensures that the algorithm prioritizes scheduling schemes that simultaneously approach the ideal economic operating state and are far from the safety hazard boundary in the target space, thereby effectively guiding the optimization process towards a practical scheduling strategy that combines economy and safety. This significantly enhances the algorithm's search motivation towards the high-performance solution region and effectively avoids the optimization process from prematurely converging to homogeneous schemes with a single tendency. This mechanism enables the algorithm to accurately handle the complex nonlinear trade-offs between multiple objectives in the power system, thereby quickly guiding a series of high-quality scheduling schemes with low generation cost, low line loss, high safety margin, and smooth adjustment characteristics within the solution space that includes economy, safety, and stability.

[0028] The specific formula for calculating the dual-distance index is as follows: in, Individual The Euclidean distance to the ideal point is calculated using the following formula: Individual The Euclidean distance to the worst-case scenario is calculated using the following formula: In the current population The minimum value, In the current population The maximum value, It is the first The ideal value for a target It is the first The worst value of each target It is the target quantity. Individual In the The objective function value on each target.

[0029] In the power system optimal dispatch implementation, the technical significance of the dual-distance index lies in its ability to establish a mapping relationship between mathematical indicators and the physical operating state of the power grid by quantifying the relative position of each dispatch scheme with the ideal operating point (representing theoretical optimal economy) and the worst operating point (representing the safety risk threshold). This index can dynamically sense the distribution of the current population relative to the system's economic and safety boundaries. When it detects that a scheme is biased towards a safety-conservative region, it automatically increases the search pressure towards economic efficiency; when a scheme excessively pursues economy and approaches the safety limit, it strengthens the consideration of safety. This dynamic balancing mechanism enables the search process to adaptively respond to the complex game relationships among multiple objectives in the power system, ensuring that the optimization process always progresses along a practical direction that balances economy and safety, ultimately generating a set of dispatch schemes that meet the actual operating needs of the power grid.

[0030] Step S3: Maintain the global distribution of the population through reference vector association operations, mapping scheduling schemes representing different operating strategies to uniformly distributed reference vector directions. This mechanism ensures that the optimization process can systematically explore the complete trade-offs between multiple objectives such as generation cost, transmission loss, line utilization, and generation adjustment. Its technical effect is to generate a set of scheduling schemes that broadly cover different operating preferences, providing dispatchers with comprehensive decision-making support for diverse grid operating conditions such as ensuring safety during peak periods and reducing costs during off-peak periods, effectively avoiding the concentration of schemes in localized areas and the omission of key operating modes.

[0031] Step S4: Quantitatively assess the distribution characteristics of the current set of scheduling schemes in the diversity archive using information entropy, calculate the degree of difference between it and the ideal uniform distribution, and adaptively adjust the neighborhood range of the local search based on this entropy ratio. The technical effect is that when the entropy value is low, indicating that the scheduling schemes are too concentrated in certain specific operating modes, the search range is automatically expanded to explore neglected operating areas, avoiding overlooking key scheduling strategies for dealing with extreme conditions; when the entropy value is close to the ideal distribution, the focus is on fine-tuning the local search, improving the ability to fine-tune generator output parameters. This dynamic adjustment mechanism enables the algorithm to intelligently adapt to complex operating conditions such as grid load fluctuations and faults, significantly enhancing the adaptability and robustness of the scheduling schemes in the face of actual operational uncertainties.

[0032] Information entropy assessment specifically includes calculating the current population distribution entropy and the optimal distribution entropy, as follows: ; ; Where L is the number of reference vectors, For the first Reference vectors, For diversity archives and the first The proportion of individuals associated with each reference vector, i.e. , To the reference vector The number of associated individuals, For the size of the diversity archive, This represents the probability value under an ideal uniform distribution.

[0033] Based on the current population distribution entropy and the optimal distribution entropy, the neighborhood size... Calculated using the following formula: in, For the floor function, This is the function for finding the maximum value.

[0034] In the power system optimal dispatch implementation, the neighborhood size The technical significance lies in the fact that this parameter dynamically reflects the degree of matching between the distribution breadth of the current scheduling scheme set and the system's operating conditions through information entropy. When the entropy value is low, it indicates that the existing schemes are concentrated in a few operating modes; in this case, increasing the entropy value is appropriate. The entropy value expands the search scope, guiding the algorithm to explore alternative scheduling strategies under special conditions such as extreme weather and equipment failure. When the entropy value is high, it indicates that the solution has fully covered common operating modes, and at this time, it should be reduced. The value then shifts to fine-tuning, corresponding to precise verification of generator output under normal operating conditions such as slight load changes and frequency fluctuations. This adaptive mechanism enables the algorithm to intelligently identify the diverse needs of the power grid's operating state, autonomously balancing global exploration and local optimization, and ultimately forming a comprehensive and accurate set of scheduling decisions.

[0035] Step S5: The distribution density of each dispatch scheme within its local neighborhood is evaluated using the parallel distance calculation method. This distance metric effectively eliminates the interference of complex coupling relationships between multiple optimization objectives such as generation cost and transmission loss. Local density is quantified by calculating the average parallel distance between each dispatch scheme and its neighboring schemes, and the diversity profile is updated based on this density index, prioritizing the retention of dispatch schemes located in sparse regions. This effectively overcomes the local optimum dilemma that traditional optimization algorithms easily fall into when dealing with grid nonlinear constraints, accurately identifying those dispersed high-quality solutions with unique advantages under specific operating conditions. This ensures that the final dispatch scheme set not only includes conventional operating modes but also covers special dispatch strategies for extreme conditions, significantly enhancing the comprehensiveness and robustness of grid dispatch schemes.

[0036] The calculation of local neighborhood density is as follows: First, calculate the distance between individuals using the parallel distance formula: in, Indicates the first Individual and the first Parallel distance between individuals For the target quantity, For the first The individual in the first Normalized fitness values ​​for each objective For the first The individual in the first Normalized fitness values ​​for each objective.

[0037] Based on the parallel distance, calculate the local neighborhood density for each individual: in, Indicates the first Local neighborhood density of an individual The neighborhood size is as determined in claim 2. For the first The individual and its first Parallel distance between neighboring individuals.

[0038] In the power system optimal dispatch implementation, the technical significance of local neighborhood density lies in the fact that this parameter quantifies the scarcity of dispatch strategies within a specific operating area through parallel distance. Low-density areas correspond to special operating modes that are difficult to achieve under multiple objective constraints such as generation cost and line load, such as fine-tuning strategies that balance transmission bottleneck avoidance and rapid frequency regulation requirements. The parallel distance metric effectively isolates the coupling effects between various objectives, accurately identifying dispatch schemes with truly unique operational value. This density-aware mechanism ensures that the algorithm can discover key backup strategies for dealing with extreme operating conditions, avoiding the optimization process from getting trapped in local optima of common operating modes, thereby significantly enhancing the coverage and emergency response capabilities of the dispatch scheme set for complex power grid operating environments.

[0039] Step S6: Based on the elite scheduling schemes in the convergence archive and the distinctive scheduling schemes in the diversity archive, a mating pool is constructed. A hybrid selection strategy of elite preservation and diversity priority is adopted. Through simulating genetic operations such as binary crossover and polynomial mutation, a new generation of scheduling scheme population is generated. The technical effect is the intelligent integration and innovative exploration of excellent scheduling strategies. It inherits the efficiency and reliability of existing schemes while introducing new combinations of operating modes. It effectively simulates the comprehensive decision-making process of scheduling experts based on various operating experiences, thereby continuously promoting the synergistic evolution of scheduling schemes across multiple objective dimensions such as power generation economy, system security, and operational stability.

[0040] Step S7: Iteratively execute steps S2 to S6. When the evolutionary process meets the preset termination condition, the algorithm terminates the iterative optimization process. The final output is a set of Pareto optimal scheduling schemes stored in the diversity archive. This set contains a series of scheduling schemes that achieve the best trade-off among multiple objectives such as generation cost, transmission loss, line utilization, and generation adjustment. Grid dispatchers can select the most suitable scheduling plan from this set of schemes based on real-time operational needs and issue specific generator output commands to each power plant for execution. Thus, this method completes the entire technical closed loop from multi-objective optimization calculation to actual power system scheduling, realizing comprehensive optimization and direct control of grid economy, security, and stability.

[0041] Experiments show that the proposed dual distance index (DDI) and dual diversity maintenance mechanism significantly improve algorithm performance. First, DDI, by simultaneously considering the distances of individuals to both the ideal and worst-case points, effectively enhances the convergence pressure on the two ends of the Pareto front, avoiding the center-clustering problem caused by traditional single distance indices, making it particularly suitable for multi-objective optimization problems with concave or irregular fronts. Second, the dual diversity maintenance mechanism combines the global distribution of the reference vector with the locally neighborhood density adaptively adjusted based on information entropy, balancing the overall population distribution and local sparsity, effectively improving the algorithm's exploration capability in high-dimensional target spaces. Furthermore, using parallel distance to calculate local density reduces sensitivity to front shape, enhancing the algorithm's adaptability to complex front structures. Through the dual-archive co-evolutionary framework, this invention achieves decoupled management of convergence and diversity, avoiding mutual interference between the two in environment selection, and significantly improving the algorithm's robustness and solution quality in multi-objective problems.

[0042] As shown in the algorithm flow, this invention first initializes the population and reference vectors, then enters the main loop. In each generation, the algorithm updates the convergence profile (CA) using the dual distance index (DDI), prioritizing individuals close to the ideal point and far from the worst point, thereby increasing convergence pressure. Simultaneously, it maintains the global distribution of the diversity profile (DA) using reference vector association operations and assesses the population distribution state using information entropy, adaptively determining the size of the local neighborhood. Next, it calculates the local density of individuals based on parallel distances, further selecting individuals with good local diversity to enter the DA. Subsequently, the algorithm constructs a mating pool based on CA and DA, generating offspring populations using elite selection and diversity-first strategies, respectively. If the termination condition is not met, the above process continues iteratively. This process achieves parallel optimization of convergence and diversity through a dual-profile collaborative mechanism, effectively addressing the problems of convergence difficulties and diversity loss in multi-objective optimization.

[0043] This embodiment also discloses a multi-objective optimization system based on dual-archive co-evolution and dual-distance index, used to implement the above method, including: An initialization module is used to generate an initial population based on the parameters of the power system to be optimized, and to initialize the algorithm parameters and reference vector set, and to store the initial population in the convergence file and the diversity file respectively. The convergence archive update module is used to calculate the ideal point and the worst point based on the convergence archive, and update the convergence archive by combining the dual distance index. A global distributed maintenance module is used to associate individuals in the diversity archive with the reference vector set; The state assessment module is used to assess the population distribution state of the diversity archive by combining information entropy, and to determine the size of the local neighborhood based on the assessment results. The local diversity maintenance module is used to calculate the local neighborhood density of individuals in the diversity profile based on the local neighborhood size using parallel distance, and to update the diversity profile based on the local neighborhood density. The mating pool generation module is used to construct a mating pool based on the updated convergence profile and diversity profile, and to generate a progeny population through the mating pool. The iterative control module is used to control the iterative execution of the convergence file update module, global distribution maintenance module, state evaluation module, local diversity maintenance module and mating pool generation module until the termination condition is met, and outputs the solution set in the diversity file as the optimal scheduling scheme of the power system.

[0044] This study systematically evaluates the performance of the proposed MDCS algorithm through three sets of experiments. The first set of experiments compares the Multi-Distance Cooperative Selection Super Multi-Objective Evolutionary Algorithm (MDCS) with six current mainstream super multi-objective evolutionary algorithms, including SSCEA, RVEAIGNG, MaOEAIGD, hpaEA, Two_Arch2, and RVEAI-DWC. The second set of experiments analyzes the convergence and stability of MDCS on typical test functions, focusing on its performance under different objective dimensions and frontier structures. The third set of experiments verifies the contribution of the dual distance index (DDI) and dual diversity maintenance strategy to the overall performance of the algorithm through ablation experiments.

[0045] To verify the effectiveness of MDCS, this paper compares it with six advanced multi-objective evolutionary algorithms on three major benchmark datasets: DTLZ, WFG, and MaF. The benchmark datasets contain 19 test problems, covering various Pareto front features such as linear, concave, degenerate, discontinuous, multimodal, and inverted problems, with objective dimensions set to 5, 10, 15, and 20, respectively. All algorithms use the same population size and maximum function evaluation count, where the population size is set to 200–240 depending on the number of objectives, and the maximum number of evaluations is uniformly set to M × 10000, where M is the number of objectives. The parameters of the comparison algorithms all follow the recommended settings in their original literature. The experimental environment consists of a Windows 10 64-bit operating system, an Intel i7 processor, 32GB of RAM, and the simulation platform is PlatEMO (Matlab version).

[0046] To reduce the impact of random errors on the experimental results, each algorithm was run independently 30 times on each test problem, and the statistical results of the HV performance index, including the mean and standard deviation, were recorded. All experiments were conducted in the same hardware and software environment to ensure the fairness and reproducibility of the comparison results.

[0047] Tables 1 to 4 present the HV statistics of seven algorithms on the WFG, DTLZ, and MaF test sets for 5, 10, 15, and 20 objectives, respectively. The symbols "+", "−", and "=" indicate that MDCS is significantly inferior to, significantly superior to, or equivalent to the compared algorithms, respectively. By counting the number of times MDCS dominates on each test problem, its overall solution accuracy and robustness on the DTLZ, WFG, and MaF test sets can be quantified.

[0048] Table 1 Table 2 Table 3 Table 4 Statistical results show that MDCS demonstrates strong competitiveness in low-objective-dimensional (5-objective) scenarios, but its advantage is not yet prominent. As the objective-dimensionality increases to 15 and 20, its HV (High Value) index significantly outperforms the other six comparative algorithms on most DTLZ, WFG, and MaF test problems, with Friedman ranking consistently first. This indicates that MDCS is more suitable for high-dimensional, multi-objective scenarios. Further observation of function types reveals that MDCS shows the most significant improvement on concave, degenerate, and discontinuous frontier problems. While it leads overall in multimodal and mixed functions (such as MaF3 and MaF4), it slightly lags behind RVEAIGNG and RVEA-DWC, which are specifically designed for irregular frontiers, on a few 20-objective instances, indicating that the algorithm still has room for further development in these types of functions.

[0049] To more intuitively demonstrate the performance of the method of the present invention, Figure 2 and Figure 3 The final population parallel coordinate plots of different algorithms on the 15-dimensional WFG8 test problem are shown. Figure 2 and Figure 3 As shown, on the WFG8 problem, the solution set obtained by the MDCS algorithm is evenly distributed and has a wide coverage on the parallel coordinate axes, indicating that it can effectively maintain the diversity of high-dimensional space; in contrast, the solution sets of other comparative algorithms are sparsely distributed or concentrated locally.

[0050] To gain a deeper understanding of the convergence behavior of MDCS, Figure 4 and Figure 5The average IGDplus and HV convergence curves of seven algorithms on 15-objective DTLZ6, MaF6, WFG5, and WFG7 are presented. It can be seen that as the number of evaluations increases, the MDCS index value enters the rapid decline phase earliest and is the first to stabilize, verifying its convergence efficiency in high-dimensional objective spaces. MDCS exhibits the fastest early convergence speed on concave, degenerate, and discontinuous frontier problems, mainly due to the dual distance index (DDI) applying bidirectional pressure from both the ideal point and the worst point, prompting the population to rapidly approach the true frontier; while the dual diversity maintenance mechanism adaptively adjusts the neighborhood size through information entropy, avoiding getting trapped in local Pareto sub-regions. In summary... Figure 4 and Figure 5 Statistical results show that MDCS leads in stability on degenerate frontiers such as DTLZ6 and MaF6, as well as on deceptive problems such as WFG5 and WFG7, significantly outperforming MaOEAIGD, hpaEA, Two_Arch2, and SSCEA. Its standard deviation fluctuation range is also smaller than that of RVEAIGNG and RVEA-DWC, indicating that the proposed dual-archive synergy and dual-distance index strategy effectively reduces random fluctuations in high-dimensional environments while maintaining convergence speed, resulting in the best overall robustness.

[0051] MDCS introduces three key designs around dual-archive update: (1) Dual Distance Index (DDI) is used to screen convergent archive CAs; (2) Global diversity maintenance guided by reference vectors; and (3) Local diversity maintenance based on information entropy and parallel distance. To verify the contribution of each component, the algorithm is divided into the following six sub-versions for ablation comparison: MDCS1: CA updates use only ideal points; MDCS2: CA update only ; MDCS3: CA updates use only double distance; MDCS4: DA update removes global diversity maintenance; MDCS5: DA update removes local diversity maintenance; MDCS6: DA update removes adaptive neighborhood density calculation.

[0052] The six sub-versions were compared with the full MDCS on the DTLZ, WFG, and MaF test sets for objectives 5, 10, 15, and 20. The results are listed in Table 5. Each version was run independently 30 times for each test question, and statistical significance was determined using the mean of the HV index and the Wilcoxon rank-sum test.

[0053] Table 5 Table 5 shows that all three major strategy modules relied upon by MDCS are indispensable, and the conclusions can be summarized as follows: (1) Both global and local diversity maintenance significantly affect algorithm performance. In all 5, 10, 15 and 20 objectives, MDCS4, which removes the global maintenance of the reference vector, drops sharply in ranking and has the largest HV loss; MDCS5, which removes the local maintenance, is second, and its disadvantage further expands on the high-dimensional degradation front of the 15 / 20 objectives, indicating that the two must work together to maintain sufficient distribution breadth and local sparsity.

[0054] (2) The global diversity strategy is more sensitive to dimensional expansion. As the number of targets increases from 5 to 20, the relative performance of MDCS4 degrades the most, indicating that the reference vector alone is no longer sufficient to cover high-dimensional sparse spaces, and the global module still needs to be further strengthened.

[0055] (3) Adaptive neighborhood density calculation is robust to complex frontiers. MDCS6 tied for first place with the full MDCS in most tests, but occasionally outperformed the degenerate / irregular PF, showing that the strategy is not sensitive to problem complexity and can steadily improve local exploration capabilities.

[0056] In summary, the dual diversity maintenance mechanism is indispensable in high-dimensional multi-objective optimization. Global maintenance is most sensitive to dimensionality, while local adaptation is robust to the shape of the frontier. Only by combining the two can the algorithm maintain diversity and stability throughout the entire objective space.

[0057] Example 2 This embodiment applies the method described in this invention to power system optimization scheduling to simultaneously optimize four objectives: generation cost, transmission loss, line utilization, and generation adjustment.

[0058] First, an initial scheduling scheme population is generated based on real-time power grid data, and parameters and reference vector sets are initialized and stored in the convergence file (CA) and diversity file (DA), respectively.

[0059] In the main loop, the Dual Distance Index (DDI) is used to update the CA (Calendar Analysis), which increases the convergence pressure on high-quality scheduling schemes by calculating the distances between individuals and the ideal and worst points. Global distribution is maintained through reference vector association, ensuring that the solution set covers multiple operational trade-offs. The neighborhood size is dynamically adjusted using information entropy: the search range is expanded when entropy is low, and local fine-tuning is focused when entropy is high, improving adaptability to load fluctuations. Local density is calculated using parallel distance to update the DA (Calendar Analysis) to maintain diversity and avoid local optima.

[0060] A mating pool is constructed based on CA and DA, and offspring are generated through genetic operations. After iteration to the termination condition, the non-dominated solution set in DA is output. This series of scheduling schemes balances economy and security and can be directly used by the power grid control center for execution, realizing optimized control of multi-objective operation of the power system.

[0061] In practical power system optimization scheduling applications, the convergence process of the method of this invention is as follows: Figures 6 to 9 As shown. Figure 6 The curve showing the change of total power generation cost with the number of iterations demonstrates that as the iterations proceed, the cost target decreases rapidly and tends to stabilize, proving the effective convergence pressure of the dual-distance index. Figure 7 , Figure 8 and Figure 9 The changes in transmission loss, maximum line utilization, and total power generation adjustment with the number of iterations were shown respectively. These indicators all showed a good convergence trend during the algorithm optimization process, and finally converged to the Pareto optimal frontier, realizing the synergistic optimization of multiple objectives.

[0062] at last, Figure 10 The diagram presents a parallel coordinate graph of the decision variables for the final generated set of optimal power system dispatch schemes. This graph visually illustrates the distribution of generator output under different dispatch schemes, providing dispatchers with visualized decision support and validating the practicality and versatility of the system's output.

[0063] This invention introduces a dual distance index (DDI) to simultaneously consider the distances of an individual to both the ideal point (e.g., the lowest generation cost point) and the worst-case point (e.g., the highest transmission loss point), effectively enhancing convergence pressure and preventing the algorithm from biasing towards the population center during the search process. This improves the adaptability to complex Pareto fronts of power systems (e.g., nonlinear, high-dimensional objective spaces) and accelerates convergence towards a high-quality scheduling scheme that balances economy and safety. Regarding diversity maintenance, a reference vector and information entropy evaluation mechanism are combined to dynamically adjust the size of the local neighborhood and use parallel distance to calculate local density, balancing global distribution and local diversity. This ensures the algorithm can comprehensively explore the solution space, generating a set of scheduling schemes covering various trade-offs from low cost to high reliability, providing grid operators with diverse decision-making options. Through a dual-archive collaboration mechanism, convergence and diversity are maintained separately, avoiding the mutual interference problem in traditional methods, thus significantly improving the overall performance and robustness of the algorithm in multi-objective optimization. The output of this method directly corresponds to a series of executable and selectable power system dispatching instructions (such as generator output planning, line load control, etc.), which can be directly applied to the power grid control center to solve the practical technical problem of balancing economy, transmission efficiency, line safety and power generation adjustment stability, and significantly improve the intelligence level of power grid operation, resource utilization efficiency and system stability.

[0064] This invention deeply couples a multi-objective optimization method and system based on dual-archive co-evolution and dual-distance index with the power system scheduling problem. Through algorithmic innovation, it directly addresses the practical challenges of multi-objective conflicts such as generation cost, transmission loss, line utilization, and generation adjustment.

[0065] The above are merely preferred embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A multi-objective optimization method based on dual-archive co-evolution and dual-distance index, characterized in that, Includes the following steps: S1. Generate an initial population based on the parameters of the power system to be optimized, and initialize the algorithm parameters and reference vector set. Store the initial population in the convergence file and the diversity file respectively. S2. Calculate the ideal point and the worst point based on the convergence archive, and update the convergence archive by combining the dual distance index; S3. Associate the individuals in the diversity archive with the reference vector set to maintain the global distribution of the population; S4. Combine information entropy to evaluate the population distribution status of the diversity archive, and determine the size of the local neighborhood based on the evaluation results; S5. Based on the local neighborhood size, calculate the local neighborhood density of individuals in the diversity profile using parallel distance, and update the diversity profile based on the local neighborhood density; S6. Construct a mating pool based on the updated convergence file and diversity file, and generate a progeny population through the mating pool; S7. Iterate through steps S2-S6 until the termination condition is met, and output the solution set in the diversity archive as the optimal scheduling scheme for the power system.

2. The multi-objective optimization method based on dual-archive co-evolution and dual-distance index as described in claim 1, characterized in that, In step S2, the ideal point and worst point are calculated based on the convergence profile, and the convergence profile is updated using a dual-distance metric, including: Calculate the ideal point and worst point based on the target value of individuals in the current population; calculate the dual distance index value of each individual in the convergence archive based on the ideal point and worst point; filter individuals in the convergence archive based on the dual distance index value and update the convergence archive.

3. The multi-objective optimization method based on dual-archive co-evolution and dual-distance index as described in claim 2, characterized in that, The formula for calculating the dual-distance index in S2 is as follows: ； in, Individual The Euclidean distance to the ideal point is calculated using the following formula: ； Individual The Euclidean distance to the worst-case scenario is calculated using the following formula: ； In the current population The minimum value, In the current population The maximum value, It is the first The ideal value for a target It is the first The worst value of each target It is the target quantity. Individual In the The objective function value on each target.

4. The multi-objective optimization method based on dual-archive co-evolution and dual-distance index as described in claim 1, characterized in that, In step S4, the population distribution status of the diversity archive is evaluated using information entropy, and the size of the local neighborhood is determined based on the evaluation results, including: Calculate the current population distribution entropy and the optimal distribution entropy; determine the size of the local neighborhood based on the ratio of the current population distribution entropy to the optimal distribution entropy.

5. The multi-objective optimization method based on dual-archive co-evolution and dual-distance index as described in claim 4, characterized in that, Calculate the current population distribution entropy and optimal distribution entropy The formula is: ； ； Where L is the number of reference vectors, For the first Reference vectors, For diversity archives and the first The proportion of individuals associated with each reference vector, i.e. , To the reference vector The number of associated individuals, For the size of the diversity archive, This represents the probability value under an ideal uniform distribution.

6. The multi-objective optimization method based on dual-archive co-evolution and dual-distance index as described in claim 4, characterized in that, The size of the local neighborhood is determined based on the ratio of the current population distribution entropy to the optimal distribution entropy, as shown in the following formula: ； in, For the floor function, This is the function for finding the maximum value.

7. The multi-objective optimization method based on dual-archive co-evolution and dual-distance index as described in claim 1, characterized in that, The S5 step of calculating the local neighborhood density of individuals in the diversity profile using parallel distance includes: calculating the parallel distance between individuals based on their normalized fitness values; and calculating the local neighborhood density of each individual based on the parallel distance and the local neighborhood size.

8. The multi-objective optimization method based on dual-archive co-evolution and dual-distance index as described in claim 1, characterized in that, The parameters of the power system to be optimized include the upper and lower limits of generator output and cost coefficient, the impedance and capacity limits of transmission lines, and the load demand of each node. The solution set is a non-dominated solution set for multiple objectives, including power generation cost, transmission loss, line utilization, and power generation adjustment.

9. A multi-objective optimization system based on dual-archive co-evolution and dual-distance index, used to implement the method described in any one of claims 1-8, characterized in that, include: An initialization module is used to generate an initial population based on the parameters of the power system to be optimized, and to initialize the algorithm parameters and reference vector set, and to store the initial population in the convergence file and the diversity file respectively. The convergence archive update module is used to calculate the ideal point and the worst point based on the convergence archive, and update the convergence archive by combining the dual distance index. A global distributed maintenance module is used to associate individuals in the diversity archive with the reference vector set; The state assessment module is used to assess the population distribution state of the diversity archive by combining information entropy, and to determine the size of the local neighborhood based on the assessment results. The local diversity maintenance module is used to calculate the local neighborhood density of individuals in the diversity profile based on the local neighborhood size using parallel distance, and to update the diversity profile based on the local neighborhood density. The mating pool generation module is used to construct a mating pool based on the updated convergence profile and diversity profile, and to generate a progeny population through the mating pool. The iterative control module is used to control the iterative execution of the convergence file update module, global distribution maintenance module, state evaluation module, local diversity maintenance module and mating pool generation module until the termination condition is met, and outputs the solution set in the diversity file as the optimal scheduling scheme of the power system.

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