High cable city power grid optimization method and device for balancing target and constraint satisfaction
By establishing a multi-objective reactive power optimization model and a two-stage constrained multi-objective evolution algorithm, the reactive power optimization problem of high-cable rate urban power grids is solved, the economic and environmental benefits of the power grid are improved, and the safe and stable operation of the power grid is ensured.
Patent Information
- Application Number
- CN202510689066.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-05-27
- Publication Date
- 2025-08-26
AI Technical Summary
The existing technology is difficult to effectively solve the reactive power optimization problem of high-cable urban power grids, especially in complex power grid scenarios, which makes it difficult to balance the optimization and constraint satisfaction, resulting in inaccurate optimization results and high computational complexity.
The high-cable urban grid optimization method with balanced goals and constraint satisfaction is adopted. By establishing a multi-objective reactive power optimization model and a two-stage constraint multi-objective evolution algorithm (CMOEA-MS), dynamic balanced target optimization and constraint satisfaction are used. Mathematical models of SVC, load, transmission lines, transformers and thermal power units are used to combine cost, active loss and pollution emission targets to achieve efficient reactive power optimization.
It significantly improves the economic and environmental benefits of high-cable urban power grids, quickly finds high-quality Pareto optimal solution, ensures safe and stable operation of the power grid, and reduces operating costs and pollution emissions.
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Figure CN120546045A_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field of power grid optimization, and specifically relates to a method and device for optimizing a high-cable urban power grid with balanced objectives and constraint satisfaction, and is particularly suitable for multi-objective reactive power optimization of power grids. Background Art
[0002] In the field of power system planning and operation, reactive power optimization is a key step in ensuring stable, economical, and efficient grid operation. Its importance is particularly prominent in urban power grids with high cable ratios. However, current reactive power optimization technologies for urban power grids with high cable ratios still have many shortcomings and cannot meet the optimization needs in complex grid scenarios.
[0003] High-cable urban power grids, characterized by a high proportion of cable lines, widely varying line parameters, and complex and variable operating modes, present numerous challenges for reactive power optimization. Firstly, the significant capacitance effect of cable lines, their high charging power, and their significant influence on operating voltages complicate the distribution and flow characteristics of reactive power, increasing the difficulty of developing reactive power optimization models. Secondly, the highly variable operating conditions of urban power grids, with significant variations in load characteristics and power supply access across different regions, necessitates considering reactive power optimization strategies under various operating conditions to ensure stable grid operation.
[0004] Currently, common reactive power optimization methods include conventional sensitivity analysis, linear programming, and quadratic programming. Sensitivity analysis analyzes the sensitivity relationship between reactive power and control variables to guide the configuration and adjustment of reactive compensation equipment. However, this method typically only yields locally optimal solutions and suffers from low computational accuracy in complex power grids. Linear programming simplifies the reactive power optimization problem into a linearly constrained optimization problem. While computationally fast, it ignores the nonlinear characteristics of the power grid, leading to significant deviations between the optimization results and actual operation. Quadratic programming, while accounting for some nonlinear factors, is computationally complex and difficult to guarantee convergence when dealing with reactive power optimization problems in large, complex power grids. Furthermore, these traditional methods often exhibit significant limitations when addressing the complex constraints of high-cable urban power grids. For example, they struggle to comprehensively consider and effectively coordinate the installation location and capacity limitations of reactive compensation equipment, as well as various operational constraints (such as voltage and power constraints). Summary of the Invention
[0005] In view of the above deficiencies in the prior art, the purpose of the present invention is to provide a high-cable urban power grid optimization method and device that balances objectives and constraint satisfaction. Through modeling and a two-stage constrained multi-objective evolutionary algorithm, it can dynamically balance objective optimization and constraint satisfaction, efficiently solve the problem of reactive power optimization in high-cable urban power grids, and significantly improve economic and environmental benefits.
[0006] To achieve the above objectives, the present invention provides a high-cable urban power grid optimization method that balances objectives and constraint satisfaction, comprising the following steps: S1. For the basic structure of a high-cable urban power grid including static VAR compensator (SVC), loads, transmission lines, transformers, and thermal power units, a mathematical model of SVC is established. S2. Taking cost, active power loss and pollution emission as targets, a multi-objective reactive power optimization model of high-cable urban power grid is established and its constraints are set; S3. Based on the mathematical definition of multi-objective problems, a constrained multi-objective evolutionary algorithm based on equilibrium objectives and constraint satisfaction is established to solve the multi-objective reactive power optimization model and obtain the optimization plan for the high-cable urban power grid.
[0007] As a preferred embodiment of the present invention, in S1, the mathematical model of SVC is: (1); Where, is the equivalent reactance of SVC; V is the connection node voltage; Reactive power injected into the node.
[0008] As a preferred solution of the present invention, in S2, the multi-objective reactive power optimization model includes three objective functions, namely: Objective function 1: The active power generated by the thermal power unit and the fuel cost are expressed in the form of a quadratic function: (2); Where, is the total fuel cost of the thermal power unit; g is the index of the thermal power unit, and NTG is the number of thermal power units; 、 、 、 、 is the cost coefficient of the g-th thermal power unit; is the active power output by the g-th thermal power unit; is the lower limit of the active power output of the g-th thermal power unit; Objective function 2, active power loss The expression is: (3); Where i and j are the indexes of the nodes, and Nl is the total number of nodes; 、 are the voltage amplitudes at nodes i and j respectively; 、 They are 、 The corresponding phase angle; is the conductance between node i and node j; Objective function 3: total amount of atmospheric pollutant emissions from thermal power units for: (4); Where, 、 、 、 、 is the emission coefficient of the g-th thermal power unit.
[0009] As a preferred embodiment of the present invention, the control variables in the multi-objective reactive power optimization model include the active output power of the thermal power unit, the voltage amplitude of the thermal power unit bus, the tap setting value of the adjustable transformer, and the position and rated value of the SVC, which can be expressed as: (5); Where X represents the set of control variables; Indicates the active power output from thermal power unit 2 to thermal power unit NTG; Indicates the bus voltage amplitude from thermal power unit 1 to thermal power unit NTG; Indicates the tap setting value of transformer 1 to transformer NT, where NT is the total number of transformers; Indicates the node location where the SVC is installed, and NL is the total number of SVCs; Indicates the reactive power input to the SVC line at the corresponding SVC installation node.
[0010] As a preferred solution of the present invention, the constraints of the multi-objective reactive power optimization model include: Equality constraints, the active and reactive power balance of the power grid must satisfy: (6); (7); Where, is the active power of node i; is the active power demand of the load at node i; is the reactive power of node i; is the reactive power demand of the load at node i; is the susceptance between node i and node j; Inequality constraints define the prohibited operation zone (POZ) of the thermal power unit. POZ refers to the operation zone that needs to be restricted or prohibited due to abnormal operating conditions during the operation of the thermal power unit. The inequality constraint containing POZ is expressed as: (8); Where z is the index of POZ, is the total number of POZs; is the lower limit of the first POZ; is the lower limit of the zth POZ; 、 are the upper limits of the z-1th and zth POZs, respectively; is the upper limit of the active power output of the g-th thermal power unit.
[0011] As a preferred solution of the present invention, in S3, the mathematical definition of the multi-objective problem is: (9); Where, is a solution consisting of d decision variables, are the decision variables corresponding to the elements in the set X of control variables, which together constitute the solution x. is the decision space; subject to means subject to the constraints; objective function Contains multiple targets, Indicates the values of the 1st to mth targets; represents the inequality constraint, the number of inequality constraints is p; Represents an equality constraint, and the number of equality constraints is q.
[0012] As a preferred embodiment of the present invention, in S3, the constrained multi-objective evolutionary algorithm based on equilibrium objectives and constraint satisfaction is a two-stage constrained multi-objective evolutionary algorithm, which starts with random initialization of a population of size n; during the reproduction process, n parents are selected from the current population P based on the fitness of the solution, and then n offspring solutions are generated based on the parents and combined with the current population; thereafter, the algorithm enters stage A or stage B, where stage A is used to determine the dominance relationship between the set of all solutions controlled by x and all solutions that dominate x, and stage B gives the objective a lower priority than the constraint; In the environmental selection process, n solutions with the best fitness values are selected and survive to the next generation: if there are more than n solutions in P with fitness values less than 1, then n solutions are selected according to the Euclidean distance truncation method between the solutions, that is, for solutions with fitness values greater than 1, they are compared according to their fitness values, and for solutions with fitness values less than 1, they are compared according to the Euclidean distance between them.
[0013] As a preferred solution of the present invention, the solution process of the constrained multi-objective evolutionary algorithm based on the equilibrium goal and constraint satisfaction is as follows: Step 1: Algorithm input, including population size n and parameters of the current stage ; Step 2: Randomly initialize a population P of size n; Step 3: Calculate the fitness of each solution in the population P using formulas (2) and (3) to obtain the fitness set F; Step 4: Determine whether the termination condition is met. If not, proceed to step 5; if so, jump to step 10. Step 5: Select n parents from the current population p according to fitness to form a parent set ; Step 6: Collect the parents Merge into the current population P to form a new population P; Step 7. Calculate the proportion of feasible solutions in the current population P: If the proportion of feasible solutions is less than , enter phase A, recalculate the fitness of each solution in population P through formulas (2) and (3), and update F; If the proportion of feasible solutions is greater than or equal to , enter phase B, recalculate the fitness of each solution in population P through formulas (2) and (6), and update F; Step 8: Determine whether there are more than n solutions in the population P whose fitness is less than 1: If so, use the truncation method based on Euclidean distance to select n solutions from the population P whose fitness is less than 1 to form a new population P; Otherwise, directly select n solutions with smaller fitness values from the population P to form a new population P; Step 9: Return to step 4 and continue the iterative loop; Step 10: When the termination condition is met, output the final population P.
[0014] As a preferred solution of the present invention, in the constrained multi-objective evolutionary algorithm based on the equilibrium objective and constraint satisfaction, the fitness calculation formula of the solution x is as follows: (10); Where, is the fitness value of solution x; All solutions that dominate x are stored, and y is one of the solutions; is the set of all solutions controlled by y; For the The nearest value of x; Represents x and The Euclidean distance between Phase A determines the set of all solutions controlled by x by considering the following two new objectives: and Advantages of relationship: (11); Where, represents the objective function of stage A, is the minimum distance between x and other solutions in the population P based on the displacement-based density estimate; is the total constraint violation of x; Expressed as: (12); Where, represents other solutions in the population P except x; represents the value of the rth target corresponding to y; represents the value of the rth target corresponding to x; Expressed as: (13); Phase B aims to give lower priority to the objective than to the constraints. A solution x is better than another solution y if the following conditions are met: (14); Where, is the total constraint violation of y; h is the index of the target value, j = 1,…,m; Phase B always prefers the solution with the lower violation constraint; if two solutions are both feasible, they are compared based on Pareto dominance.
[0015] A high-cable urban power grid optimization device that balances objectives and constraint satisfaction includes a memory, a processor, and a computer program stored in the memory and capable of running on the processor. The above method is implemented by executing the computer program through the processor.
[0016] The beneficial effects of the present invention are: This paper proposes a two-stage constrained multi-objective evolutionary algorithm (CMOEA-MS), which adaptively balances objective optimization and constraint satisfaction during the evolutionary process. In the early stages of evolution, when the proportion of feasible solutions in the population is low, the algorithm enters Phase A, where the objective and constraints are given equal priority, allowing some infeasible solutions to remain, helping the population to cross the infeasible region. When the proportion of feasible solutions in the population reaches a certain threshold, the algorithm switches to Phase B, which prioritizes constraint satisfaction, giving the objective lower priority than the constraints, and promoting the population to diffuse along the feasible boundary. This adaptive adjustment mechanism effectively addresses the problems of traditional constrained multi-objective evolutionary algorithms, such as their tendency to fall into local optimality, low solution efficiency, and difficulty balancing the objective and constraints when dealing with complex feasible regions. This significantly improves the algorithm's adaptability and robustness for reactive power optimization problems in urban power grids with high cable rates.
[0017] The multi-objective reactive power optimization model of the high-cable urban power grid constructed by the present invention comprehensively and accurately covers multiple key objective functions and complex constraints. In the selection of objective functions, the three interrelated and mutually restrictive key factors of cost, active power loss and pollution emission are comprehensively considered to maximize the comprehensive benefits of power grid operation. In the setting of constraint conditions, it not only includes conventional upper and lower limit constraints, but also fully considers special constraints such as the active and reactive power balance constraints of the power grid and the prohibited operation areas of thermal power units, ensuring the feasibility and reliability of the optimization results. Combined with the powerful global search capability and efficient processing capability of the CMOEA-MS algorithm for complex constraints, the present invention can quickly and accurately find a set of high-quality Pareto optimal solutions, providing a scientific and effective decision-making basis for the reactive power optimization of high-cable urban power grids, effectively ensuring the safe and stable operation of the power grid, while reducing the operating costs and environmental pollution of the power grid, and having significant economic and social benefits. BRIEF DESCRIPTION OF THE DRAWINGS
[0018] Figure 1 It is a schematic diagram of the process of the present invention; Figure 2 It is a solution flow chart of the present invention. DETAILED DESCRIPTION
[0019] The embodiments of the present invention are further described below with reference to the accompanying drawings: Example 1: Figure 1 As shown in the figure, the high-cable urban power grid optimization method with balanced objectives and constraint satisfaction includes the following steps: S1. For the basic structure of a high-cable urban power grid including static VAR compensator (SVC), loads, transmission lines, transformers, and thermal power units, a mathematical model of SVC is established. S2. Taking cost, active power loss and pollution emission as targets, a multi-objective reactive power optimization model of high-cable urban power grid is established and its constraints are set; S3. Based on the mathematical definition of multi-objective problems, a constrained multi-objective evolutionary algorithm based on equilibrium objectives and constraint satisfaction is established to solve the multi-objective reactive power optimization model and obtain the optimization plan for the high-cable urban power grid.
[0020] In S1, the SVC can be used for both inductive and capacitive compensation. In power flow research, the SVC is usually modeled as a reactive injection device connected to the node. The mathematical model of the SVC is: (1); Where, is the equivalent reactance of SVC; V is the connection node voltage; Reactive power injected into the node.
[0021] In S2, the multi-objective reactive power optimization model includes three objective functions, namely: Objective function 1: The active power generated by the thermal power unit and the fuel cost are expressed in the form of a quadratic function: (2); Where, is the total fuel cost of the thermal power unit; g is the index of the thermal power unit, and NTG is the number of thermal power units; 、 、 、 、 is the cost coefficient of the g-th thermal power unit (obtained through publicly known technologies, manufacturer data, experimental measurements, literature references, historical operating data, etc.; the same applies to the emission coefficient); is the active power output by the g-th thermal power unit; is the lower limit of the active power output of the g-th thermal power unit; Objective function 2: During the transmission of electricity in the power grid, active power loss is generated due to the resistance and conductivity in the transmission line, also known as network loss. The expression is: (3); Where i and j are the indexes of the nodes, and Nl is the total number of nodes; 、 are the voltage amplitudes at nodes i and j respectively; 、 They are 、 The corresponding phase angle; is the conductance between node i and node j; Objective function 3: total amount of atmospheric pollutant emissions from thermal power units for: (4); Where, 、 、 、 、 is the emission coefficient of the g-th thermal power unit.
[0022] The control variables in the multi-objective reactive power optimization model include the active output power of the thermal power unit, the voltage amplitude of the thermal power unit bus, the tap setting value of the adjustable transformer, and the position and rated value of the SVC, which can be expressed as: (5); Where X represents the set of control variables; Indicates the active power output from thermal power unit 2 to thermal power unit NTG; Indicates the bus voltage amplitude from thermal power unit 1 to thermal power unit NTG; Indicates the tap setting value of transformer 1 to transformer NT, where NT is the total number of transformers; Indicates the node location where the SVC is installed, and NL is the total number of SVCs; Indicates the reactive power input to the SVC line at the corresponding SVC installation node.
[0023] The constraints of the multi-objective reactive power optimization model include: Equality constraints, the active and reactive power balance of the power grid must satisfy: (6); (7); Where, is the active power of node i (e.g. thermal power plant node); is the active power demand of the load at node i; is the reactive power of node i; is the reactive power demand of the load at node i (e.g. thermal power plant node); is the susceptance between nodes i and j; Equation (6) is the active power equality constraint, and Equation (7) is the reactive power equality constraint. The termination condition of the Newton-Raphson method is that Equations (6) and (7) are satisfied.
[0024] Inequality constraints, conventional constraints are upper and lower limit constraints, including upper and lower limit constraints on the active power and reactive power output by the generator, upper and lower limit constraints on the transformer tap value, and upper and lower limit constraints on the reactive power output by the SVC device. This application additionally defines a prohibited operating zone (POZ) for thermal power units. POZ refers to the operating area that needs to be restricted or prohibited due to abnormal operating conditions during the operation of the thermal power unit. The inequality constraint containing POZ is expressed as: (8); Where z is the index of POZ, is the total number of POZs; is the lower limit of the first POZ; is the lower limit of the zth POZ; 、 are the upper limits of the z-1th and zth POZs, respectively; is the upper limit of the active power output of the g-th thermal power unit.
[0025] In S3, the mathematical definition of the multi-objective problem is: (9); Where, is a solution consisting of d decision variables, are the decision variables corresponding to the elements in the set X of control variables, which together constitute the solution x. is the decision space; subject to means subject to the constraints; objective function Contains multiple targets, Indicates the values of the 1st to mth targets; represents an inequality constraint, the number of inequality constraints is p, specifically in this embodiment, p=3; represents an equality constraint, the number of equality constraints is q, and specifically in this embodiment, q=2.
[0026] The constrained multi-objective evolutionary algorithm based on equilibrium objectives and constraint satisfaction is a two-stage constrained multi-objective evolutionary algorithm. It starts with a randomly initialized population of size n. During the reproduction process, n parents are selected from the current population P based on the fitness of the solution. Then, n offspring solutions are generated based on the parents and combined with the current population. After that, it enters phase A or phase B. Phase A is used to determine the dominance relationship between the set of all solutions controlled by x and all solutions that dominate x. Phase B gives priority to the objectives over the constraints. In the environmental selection process, n solutions with the best fitness values are selected and survive to the next generation: if there are more than n solutions in P with fitness values less than 1 (that is, these solutions are non-dominated), then n solutions are selected according to the Euclidean distance truncation method between the solutions, that is, for solutions with fitness values greater than 1, they are compared according to their fitness values, and for solutions with fitness values less than 1, they are compared according to the Euclidean distance between them.
[0027] By comparing the Euclidean distances between solutions, we select those that are farther away from the others. This ensures that the population contains solutions with different characteristics, thus maintaining diversity. Specifically, we calculate the minimum Euclidean distance from each solution to all other solutions; we sort the solutions based on these minimum distances; and we select solutions with larger minimum Euclidean distances, as these solutions are more different from the others and help maintain diversity in the population.
[0028] like Figure 2 As shown in the figure, the solution process of the constrained multi-objective evolutionary algorithm based on equilibrium objectives and constraint satisfaction is as follows: Step 1: Algorithm input, including population size n and parameters of the current stage ; Step 2: Randomly initialize a population P of size n; Step 3: Calculate the fitness of each solution in the population P using formulas (2) and (3) to obtain the fitness set F; Step 4: Determine whether the termination condition is met. If not, proceed to step 5; if so, jump to step 10. Step 5: Select n parents from the current population p according to fitness to form a parent set ; Step 6: Collect the parents Merge into the current population P to form a new population P; Step 7. Calculate the proportion of feasible solutions in the current population P: If the proportion of feasible solutions is less than , enter phase A, recalculate the fitness of each solution in population P through formulas (2) and (3), and update F; If the proportion of feasible solutions is greater than or equal to , enter phase B, recalculate the fitness of each solution in population P through formulas (2) and (6), and update F; Step 8: Determine whether there are more than n solutions in the population P whose fitness is less than 1: If so, use the truncation method based on Euclidean distance to select n solutions from the population P whose fitness is less than 1 to form a new population P; Otherwise, directly select n solutions with smaller fitness values from the population P to form a new population P; Step 9: Return to step 4 and continue the iterative loop; Step 10: When the termination condition is met, output the final population P.
[0029] It consists of m targets.
[0030] The fitness of the solution plays a crucial role, and the two-stage fitness evaluation strategy is also the core contribution of the algorithm. The proposed fitness evaluation strategy is elaborated in detail below. In the constrained multi-objective evolutionary algorithm based on equilibrium objectives and constraint satisfaction, the fitness calculation formula of the solution x is as follows: (10); Where, is the fitness value of solution x; All solutions that dominate x are stored, and y is one of the solutions; is the set of all solutions controlled by y; For the The nearest value of x; Represents x and The Euclidean distance between The smaller the fitness value, the better the solution quality. Indicates that a solution is not dominated by other solutions.
[0031] Phase A determines the set of all solutions controlled by x by considering the following two new objectives: and Advantages of relationship: (11); Where, represents the objective function of stage A, It is the minimum distance between x and other solutions in the population P based on the displacement-based density estimation, which can be used to evaluate the quality of the solution in terms of convergence and diversity; is the total constraint violation of x, and the quality of the solution is evaluated based on the constraint satisfaction; Expressed as: (12); Where, represents other solutions in the population P except x; represents the value of the rth target corresponding to y; represents the value of the rth target corresponding to x; Expressed as: (13); Phase B aims to give lower priority to the objective than to the constraints. A solution x is better than another solution y if the following conditions are met: (14); Where, is the total constraint violation of y; h is the index of the target value, j = 1,…,m; It means "to all" or "to every one", and is used for universal quantification; It means "there is at least one" or "there is one", and is used for existential quantification; Phase B always prefers the solution with the lower violation constraint; if two solutions are both feasible, they are compared based on Pareto dominance.
[0032] Based on the method of this embodiment, the IEEE 30 bus can be improved by adding two SVCs, whose positions are not fixed. During the optimization process, the SVC positions are continuously adjusted as control variables. The proposed optimization method has been verified to effectively reduce the active power loss of the IEEE 30-bus system. By rationally adjusting the SVC positions and parameters, the optimized system significantly reduces active power loss compared to the original system, improving system transmission efficiency and reducing energy loss.
[0033] After optimization, the voltage amplitude at each node in the system has become more stable, and voltage deviations have been effectively controlled. The SVC's dynamic reactive power compensation function can quickly respond to voltage fluctuations in the system, maintaining node voltages within a set range, thereby enhancing the system's voltage stability and power supply reliability.
[0034] An optimization method with cost minimization as one of the objective functions effectively reduces system operating costs. By optimizing the active power distribution of generators and the reactive power compensation of SVCs, fuel consumption and equipment operation and maintenance costs are reduced, improving the economic operation of the power system.
[0035] By incorporating pollution emissions into the objective function, the optimization method significantly reduced the system's total atmospheric pollutant emissions. By optimizing unit operation and reactive power compensation strategies, pollutant emissions during the power generation process were reduced, demonstrating positive environmental benefits.
[0036] Example 2: A high-cable urban power grid optimization device that balances objectives and constraint satisfaction includes a memory, a processor, and a computer program stored in the memory and capable of running on the processor. The method in Example 1 is implemented by executing the computer program through the processor.
Claims
1. A high-cable urban power grid optimization method that balances objectives and constraint satisfaction, characterized by The following steps are involved: S1. For the basic structure of a high-cable urban power grid including static VAR compensator (SVC), loads, transmission lines, transformers, and thermal power units, a mathematical model of SVC is established. S2. Taking cost, active power loss and pollution emission as targets, a multi-objective reactive power optimization model of high-cable urban power grid is established and its constraints are set; S3. Based on the mathematical definition of multi-objective problems, a constrained multi-objective evolutionary algorithm based on equilibrium objectives and constraint satisfaction is established to solve the multi-objective reactive power optimization model and obtain the optimization plan for the high-cable urban power grid.
2. The high-cable urban power grid optimization method with balanced objectives and constraint satisfaction according to claim 1 is characterized in that: In the above S1, the mathematical model of SVC is: （1）； Where, is the equivalent reactance of SVC; V is the connection node voltage; Reactive power injected into the node.
3. The high-cable urban power grid optimization method with balanced objectives and constraint satisfaction according to claim 1 is characterized in that: In S2, the multi-objective reactive power optimization model includes three objective functions, namely: Objective function 1: The active power generated by the thermal power unit and the fuel cost are expressed in the form of a quadratic function: （2）； Where, is the total fuel cost of the thermal power unit; g is the index of the thermal power unit, and NTG is the number of thermal power units; 、 、 、 、 is the cost coefficient of the g-th thermal power unit; is the active power output of the g-th thermal power unit; is the lower limit of the active power output of the g-th thermal power unit; Objective function 2, active power loss The expression is: （3）； Where i and j are the indexes of the nodes, and Nl is the total number of nodes; 、 are the voltage amplitudes at nodes i and j respectively; 、 They are 、 The corresponding phase angle; is the conductance between node i and node j; Objective function 3: total amount of atmospheric pollutant emissions from thermal power units for: （4）； Where, 、 、 、 、 is the emission coefficient of the g-th thermal power unit.
4. The high-cable urban power grid optimization method with balanced objectives and constraint satisfaction according to claim 3 is characterized in that: The control variables in the multi-objective reactive power optimization model include the active output power of the thermal power unit, the voltage amplitude of the thermal power unit bus, the tap setting value of the adjustable transformer, and the position and rated value of the SVC, which can be expressed as: （5）； Where X represents the set of control variables; Indicates the active power output from thermal power unit 2 to thermal power unit NTG; Indicates the bus voltage amplitude from thermal power unit 1 to thermal power unit NTG; Indicates the tap setting value of transformer 1 to transformer NT, where NT is the total number of transformers; Indicates the node location where the SVC is installed, and NL is the total number of SVCs; Indicates the reactive power input to the SVC line at the corresponding SVC installation node.
5. The high-cable urban power grid optimization method with balanced objectives and constraint satisfaction according to claim 4 is characterized in that: The constraints of the multi-objective reactive power optimization model include: Equality constraints, the active and reactive power balance of the power grid must satisfy: （6）； （7）； Where, is the active power of node i; is the active power demand of the load at node i; is the reactive power of node i; is the reactive power demand of the load at node i; is the susceptance between node i and node j; Inequality constraints define the prohibited operation zone (POZ) of the thermal power unit. POZ refers to the operation zone that needs to be restricted or prohibited due to abnormal operating conditions during the operation of the thermal power unit. The inequality constraint containing POZ is expressed as: （8）； Where z is the index of POZ, is the total number of POZs; is the lower limit of the first POZ; is the lower limit of the zth POZ; 、 are the upper limits of the z-1th and zth POZs, respectively; is the upper limit of the active power output of the g-th thermal power unit.
6. The high-cable urban power grid optimization method of balancing objectives and constraint satisfaction according to claim 5 is characterized in that: In S3, the mathematical definition of the multi-objective problem is: （9）； Where, is a solution consisting of d decision variables, are the decision variables corresponding to the elements in the set X of control variables, which together constitute the solution x. is the decision space; subject to means subject to the constraints; objective function Contains multiple targets, Indicates the values of the 1st to mth targets; represents the inequality constraint, the number of inequality constraints is p; Represents an equality constraint, and the number of equality constraints is q.
7. The high-cable urban power grid optimization method for balancing objectives and constraint satisfaction according to claim 6 is characterized in that: In S3, the constrained multi-objective evolutionary algorithm based on equilibrium objectives and constraint satisfaction is a two-stage constrained multi-objective evolutionary algorithm that starts with a random initialization of a population of size n. During the reproduction process, n parents are selected from the current population P based on the fitness of the solution, and then n offspring solutions are generated based on the parents and combined with the current population. After that, the algorithm enters stage A or stage B. Stage A is used to determine the dominance relationship between the set of all solutions controlled by x and all solutions that dominate x. Stage B gives priority to the objectives over the constraints. In the environmental selection process, n solutions with the best fitness values are selected and survive to the next generation: if there are more than n solutions in P with fitness values less than 1, then n solutions are selected according to the Euclidean distance truncation method between the solutions, that is, for solutions with fitness values greater than 1, they are compared according to their fitness values, and for solutions with fitness values less than 1, they are compared according to the Euclidean distance between them.
8. The high-cable urban power grid optimization method with balanced objectives and constraint satisfaction according to claim 7 is characterized in that: The solution process of the constrained multi-objective evolutionary algorithm based on equilibrium objectives and constraint satisfaction is as follows: Step 1: Algorithm input, including population size n and parameters of the current stage ; Step 2: Randomly initialize a population P of size n; Step 3: Calculate the fitness of each solution in the population P using formulas (2) and (3) to obtain the fitness set F; Step 4: Determine whether the termination condition is met. If not, proceed to step 5; if so, jump to step 10. Step 5: Select n parents from the current population p according to fitness to form a parent set ; Step 6: Collect the parents Merge into the current population P to form a new population P; Step 7. Calculate the proportion of feasible solutions in the current population P: If the proportion of feasible solutions is less than , enter phase A, recalculate the fitness of each solution in population P through formulas (2) and (3), and update F; If the proportion of feasible solutions is greater than or equal to , enter phase B, recalculate the fitness of each solution in population P through formulas (2) and (6), and update F; Step 8: Determine whether there are more than n solutions in the population P whose fitness is less than 1: If so, use the truncation method based on Euclidean distance to select n solutions from the population P whose fitness is less than 1 to form a new population P; Otherwise, directly select n solutions with smaller fitness values from the population P to form a new population P; Step 9: Return to step 4 and continue the iterative loop; Step 10: When the termination condition is met, output the final population P.
9. The high-cable urban power grid optimization method with balanced objectives and constraint satisfaction according to claim 8 is characterized in that: In the constrained multi-objective evolutionary algorithm based on equilibrium objectives and constraint satisfaction, the fitness calculation formula of solution x is as follows: （10）； Where, is the fitness value of solution x; All solutions that dominate x are stored, and y is one of the solutions; is the set of all solutions controlled by y; For the The nearest value of x; Represents x and The Euclidean distance between Phase A determines the set of all solutions controlled by x by considering the following two new objectives: and Advantages of relationship: （11）； Where, represents the objective function of stage A, is the minimum distance between x and other solutions in the population P based on the displacement-based density estimate; is the total constraint violation of x; Expressed as: （12）； Where, represents other solutions in the population P except x; represents the value of the rth target corresponding to y; represents the value of the rth target corresponding to x; Expressed as: （13）； Phase B aims to give lower priority to the objective than to the constraints. A solution x is better than another solution y if the following conditions are met: （14）； Where, is the total constraint violation of y; h is the index of the target value, j = 1,…,m; Phase B always prefers the solution with the lower violation constraint; if two solutions are both feasible, they are compared based on Pareto dominance.
10. A high-cable urban power grid optimization device that balances objectives and constraint satisfaction, characterized by: The method comprises a memory, a processor, and a computer program stored in the memory and capable of running on the processor, wherein the method according to any one of claims 1 to 9 is implemented by executing the computer program by the processor.