A double-target vibration reduction optimization method, device and medium for a converter transformer winding

Abstract

This invention belongs to the field of converter transformer technology, specifically involving a dual-objective vibration reduction optimization method, equipment, and medium for converter transformer windings. A finite element model of the winding is established for modal analysis to obtain the natural frequencies. An improved Latin hypercube sampling method is used to construct a second-order response surface model of key structural parameters and natural frequencies. The electromagnetic excitation force spectrum is determined based on actual current harmonic data, and optimization target values ​​and weighting coefficients for each frequency are set. A dual-objective optimization model integrating weighted frequency deviation and resonance penalty is constructed, and the solution is obtained under the constraint that the number of pad blocks is an integer. The Pareto optimal solution set is obtained, and the optimal solution is selected from the solution set. The method combines the energy distribution of the electromagnetic excitation force spectrum to establish differentiated weights, focusing the optimization on key frequency bands. A resonance penalty function forces the natural frequencies to deviate from high-risk frequency bands, effectively avoiding resonance risks. The combination of dual-objective optimization and integer constraints outputs a winding structure scheme with excellent vibration reduction performance.

Description

Technical Field

[0001] This invention belongs to the field of converter transformer technology, specifically relating to a dual-objective vibration reduction optimization method, equipment, and medium for converter transformer windings. Background Technology

[0002] In high-voltage direct current (HVDC) transmission projects, converter transformers occupy a core position in AC-DC conversion, and their operational stability directly affects the reliability of the power supply system. Winding vibration is caused by the electromagnetic excitation force generated by the interaction between winding current and leakage flux. Radial electromagnetic force causes radial expansion and contraction of the windings, while axial electromagnetic force causes collisions and compression between the winding coils. Compared to conventional power transformers, the valve-side windings of converter transformers operate under complex conditions of superposition of the fundamental frequency and various harmonics, resulting in higher amplitude and more complex frequency distribution of the electromagnetic excitation force. When the harmonic frequency approaches the natural frequency of the winding, resonance is easily induced, leading to increased vibration amplitude, which is detrimental to the stable operation of the converter transformer. Therefore, it is necessary to conduct in-depth research on the winding vibration of converter transformers from the perspective of natural frequencies.

[0003] When performing optimization design, related technologies use random sampling, which leads to problems such as uneven sample distribution and strong correlation between variables. This makes it impossible to cover the design space of key structural parameters, while redundant variables increase modeling complexity, resulting in low fitting accuracy of the constructed response surface model.

[0004] The existing technologies, based on theoretical assumptions and spectral analysis of electromagnetic excitation forces using current data, cannot accurately reflect the characteristics of the excitation source in actual operation. This can lead to inaccurate identification of high-risk resonance frequency bands, and the lack of a basis for setting optimization target values ​​and weighting coefficients, resulting in vibration reduction designs that cannot effectively avoid resonance risks in actual operation. The optimization models employed focus on the deviation between the natural frequency and the preset target value, leading to safety hazards where the optimized frequency is close to the target value but near the resonance frequency band. Furthermore, the particle swarm optimization algorithm is prone to getting trapped in local optima, generating a large number of invalid solutions during the optimization process, resulting in low optimization efficiency, a small number of feasible solutions, and difficulty in adapting to the needs of different manufacturing precisions and production conditions. Summary of the Invention

[0005] This invention provides a dual-objective vibration reduction optimization method for converter transformer windings. By establishing the coupling of response surface model and evolutionary algorithm, the method achieves accurate prediction and optimization of the natural frequency of converter transformer windings, thereby improving the stability and safety of transformer operation.

[0006] The methods include: S1: Construct a finite element model of the converter transformer winding, perform modal analysis based on the finite element model, and obtain the first five natural frequencies and mode shapes of the winding. S2: Select several key structural parameters that affect the natural frequency of the winding as fitting variables, use the improved Latin hypercube sampling method to generate the optimal sample matrix within the range of the fitting variables, and construct a second-order response surface model between the key structural parameters and the first five natural frequencies of the winding based on the finite element simulation results corresponding to the optimal sample matrix. S3: Collect current harmonic data of converter transformers in actual operation, deduce the spectral distribution characteristics of electromagnetic excitation force, identify high-risk resonance frequency bands that need to be avoided, and calculate the synthesis coefficients by combining the energy differences of each electromagnetic excitation force, thereby establishing the optimization target value of each order of the winding's natural frequency and the corresponding optimization weight coefficients. S4: Based on the constructed second-order response surface model and the established optimization objective value and optimization weight coefficient, a bi-objective optimization model including a weighted frequency deviation function and a resonance penalty function is constructed. S5: Set the population size, maximum number of iterations, crossover probability, and mutation probability, and perform global optimization on the constructed bi-objective optimization model. Under the constraint that the number of pad blocks is an integer, obtain the Pareto optimal solution set that minimizes both the weighted frequency deviation function and the resonance penalty function. S6: Extract the optimized key structural parameters from the Pareto optimal solution set; S7: Substitute the optimized values ​​of the extracted key structural parameters into the constructed second-order response surface model to calculate the optimized natural frequencies of each order of the winding, and output the optimized values ​​as the vibration reduction design scheme for the converter transformer winding.

[0007] According to another embodiment of this application, an electronic device is provided, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the steps of the dual-target vibration reduction optimization method for converter transformer windings.

[0008] According to another embodiment of this application, a storage medium is also provided, on which a computer program is stored, which, when executed by a processor, implements the steps of the dual-target vibration reduction optimization method for the converter transformer winding.

[0009] As can be seen from the above technical solutions, the present invention has the following advantages: The dual-objective vibration reduction optimization method for converter transformer windings provided by this invention reasonably simplifies the windings, retains structures such as turns, pads, and insulating paper tubes, constructs a mass matrix and stiffness matrix according to actual material parameters, applies boundary constraints that fit actual operation, and solves the first five natural frequencies and mode shapes based on characteristic equations to ensure that the modal analysis results can truly reflect the actual vibration characteristics of the windings, and the extracted natural frequency and mode shape data are accurate and reliable.

[0010] This invention selects six initial fitting variables and defines their value ranges. Sensitivity analysis quantifies the contribution of each variable to the natural frequencies, identifying key parameters and reducing dimensionality. An improved Latin hypercube sampling method is employed, iteratively exchanging and optimizing the sample distribution to generate a uniform sample matrix with low correlation. Combined with finite element simulation data, a dimensionality-reduced second-order response surface model is constructed modally. The uniformity of the sample matrix improves the model fitting accuracy, and the second-order response surface model achieves the mapping between parameters and natural frequencies, improving computational efficiency and ensuring the accuracy and reliability of the mapping relationship.

[0011] This invention collects current harmonic data from actual operating converter transformers, and extracts the fundamental frequency and harmonic parameters after filtering, denoising, and Fourier decomposition. Based on the actual current data, the electromagnetic excitation force spectrum is derived to identify high-risk resonance frequency bands. The optimization weights are calculated using the electromagnetic excitation force synthesis coefficients, and the optimization target values ​​are established based on the initial values ​​of the natural frequencies and safety margins. The derivation of the excitation force spectrum closely matches actual operating conditions, the identification of high-risk resonance frequency bands is accurate, and the setting of the optimization target values ​​and weight coefficients is scientifically based. This allows the optimization to prioritize low-order modes corresponding to high-energy excitation, improving the targeting and effectiveness of resonance avoidance.

[0012] This invention constructs a bi-objective optimization model based on a response surface model, an optimization objective value, and weighting coefficients, incorporating a weighted frequency deviation function and a resonance penalty function. The weighted frequency deviation function quantifies the deviation between the intrinsic frequency and the objective value, while the resonance penalty function applies piecewise penalties according to the degree of deviation. The number of pad blocks is limited to integers, and constraints on the values ​​of each parameter are clearly defined. This invention achieves the dual requirements of controlling intrinsic frequency deviation and avoiding resonance risk; integer constraints and parameter range constraints ensure that the optimization results meet manufacturing process requirements. Reasonable NSGA-II algorithm parameters are set, using key structural parameters as decision variables to generate an initial population that satisfies the constraints. Individual fitness is calculated using the response surface model, and fast non-dominated sorting, crowding calculation, and targeted genetic operations are performed. After iterating to the maximum number of iterations, individuals at the first non-dominated level are extracted, forming the Pareto optimal solution set. Targeted genetic operations and constraint handling ensure that a globally optimal solution is found, improving optimization efficiency.

[0013] This invention extracts parameters and objective function values ​​from the Pareto optimal solution set, and selects feasible solutions based on manufacturing processes and processing precision. Key structural parameters of the feasible solutions are extracted and substituted into the response surface model to calculate the optimized natural frequencies. The deviations of each frequency order from the high-risk resonance band are verified to meet safety thresholds. This ensures the optimized parameters are manufacturable, guaranteeing that the vibration reduction design can be directly applied to production. Attached Figure Description

[0014] To more clearly illustrate the technical solution of the present invention, the accompanying drawings used in the description will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0015] Figure 1 Flowchart of a dual-objective vibration reduction optimization method for converter transformer windings; Figure 2 This is a schematic diagram of the finite element model of the winding. Figure 3 The diagram shows the first five mode shapes of the transformer winding, where (a) is a schematic diagram of mode 1; (b) is a schematic diagram of mode 2; (c) is a schematic diagram of mode 3; (d) is a schematic diagram of mode 4; and (e) is a schematic diagram of mode 5. Figure 4 This is a schematic diagram of an electronic device.

[0016] Explanation of reference numerals in the attached figures: 1-Winding, 2-Insulating paper tube, 3-Padded block, 101-Processor, 102-Memory, 103-Display module, 104-Communication module. Detailed Implementation

[0017] This invention provides a dual-objective vibration reduction optimization method for converter transformer windings, establishing a three-dimensional finite element model of the converter transformer windings and extracting the first five modal features. To reduce the impact of local distribution of sampled data on the accuracy of response surface modeling, this invention applies an improved Latin hypercube sampling method to the acquisition of converter transformer sample data, and combines sensitivity analysis to eliminate secondary variables, constructing a dimension-reduced second-order response surface model of winding structural parameters and each natural frequency. Based on the current harmonic data of actual operating UHV converter transformers, the spectral distribution characteristics and synthesis coefficients of electromagnetic excitation force are derived, the optimization target values ​​and optimization weights of each natural frequency of the winding are established, a piecewise potential energy function is configured, and a dual-objective optimization model integrating weighted frequency deviation J1 and resonance penalty J2 is innovatively constructed based on the response surface model. Finally, a non-dominated sorting genetic algorithm is used to globally optimize the key structural parameters of the windings. The optimized modes achieve a close approximation of the target frequency. The relative error between the first and second natural frequencies and the target frequency is reduced, ensuring that the minimum deviation between each mode and the excitation force frequency band is greater than the set threshold of 15Hz, thus improving the vibration isolation safety of the transformer.

[0018] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] Please see Figure 1 The diagram shows a flowchart of a dual-objective vibration reduction optimization method for converter transformer windings in a specific embodiment. The method includes: S1: Construct a finite element model of the converter transformer winding, perform modal analysis based on the finite element model, and obtain the first five natural frequencies and mode shapes of the winding.

[0020] The following is a specific implementation of step S1, which includes the following steps: S11: The converter transformer winding is simplified and assembled. The spiral winding is equivalent to a closed circular array. The pad is simplified to a rectangular block with the same depth as the coil and eight are evenly distributed along the circumference and arranged in an array according to the interlayer gap in the axial direction. A continuous insulating paper tube is set on the inner side of the winding to complete the geometric structure construction of the winding finite element model.

[0021] S12: Define material properties for each component of the winding finite element model. Input the density, Young's modulus, and Poisson's ratio parameters of the winding turns, pads, and insulating paper tubes to construct the mass matrix and stiffness matrix basis for modal analysis.

[0022] S13: Apply fixed constraints to the bottom of the winding finite element model to simulate the bottom rigid support state of the converter transformer in actual operation, and do not set fixed constraints for the rest.

[0023] S14: Based on the completed winding finite element model, solve the characteristic equations of the multi-degree-of-freedom system. ; Where [K] is the stiffness matrix, [M] is the mass matrix, ω is the angular frequency, and {φ} is the mode shape vector; the first five natural frequencies and corresponding mode shapes are extracted to distinguish the distribution characteristics of even-order axial vibration and odd-order radial vibration.

[0024] In some embodiments, the first five angular frequencies are extracted and converted into natural frequencies, and the corresponding mode shapes for each order are obtained. Based on the displacement direction of the mode shapes, even-numbered modes are dominated by axial vibration, while odd-numbered modes are dominated by radial vibration. The vibration characteristic parameters of the winding are directly obtained by solving the characteristic equation, providing accurate input response data for response surface modeling and facilitating targeted vibration reduction optimization design.

[0025] It can be seen that by simplifying the structure of the converter transformer winding, ignoring features such as the chamfer at the winding end and the minor wear of the coils that have a negligible impact on the characteristics, the helical winding is segmented and discretely modeled as a closed circular array, taking into account both axial preload and radial support characteristics.

[0026] Furthermore, Figure 2 A finite element model of the winding is presented, which includes winding 1, insulating paper tube 2, and spacer 3. The spacer 3 is simplified into a rectangular block with the same axial depth as the coil, with 8 evenly distributed along the circumference and arranged in an array along the axial direction according to the interlayer gap pattern. An insulating paper tube 2 of a certain thickness is set inside the winding 1 to simulate the actual insulation constraint structure, and the geometric assembly of each component is completed to form the geometric body of the winding finite element model.

[0027] Furthermore, material properties were defined for each core component of the model. According to the material property table in Table 1, key mechanical parameters for the winding turns, spacers, and insulating paper tubes were input to construct the mass and stiffness matrices required for modal analysis. Boundary constraints were applied to the model. A full-degree-of-freedom fixed constraint was applied to the bottom face of the winding to simulate the rigid connection between the bottom of the winding and the core during actual operation of the converter transformer. No fixed constraints were applied to the other end faces and sides of the model, preserving the free vibration space of the structure. Finally, based on the completed finite element model, the characteristic equations of the multi-degree-of-freedom system were solved. The first five angular frequencies were extracted and converted to natural frequencies using ω=2πf. The corresponding mode shapes for each order were obtained, clarifying the distribution characteristics of even-order modes, which are dominated by axial vibration, and odd-order modes, which are dominated by radial vibration. This ensures that the modal analysis results can accurately reflect the dynamic vibration behavior of the winding.

[0028] Table 1 Material Properties

[0029] S2: Select several key structural parameters that affect the natural frequency of the winding as fitting variables. The key structural parameters include at least the inter-turn spacing, the diameter of the wire turns, the number of spacers and the width of the spacers. Use the improved Latin hypercube sampling method to generate the optimal sample matrix within the range of the values ​​of the fitting variables. Based on the finite element simulation results corresponding to the optimal sample matrix, construct a second-order response surface model between the key structural parameters and the first five natural frequencies of the winding.

[0030] The following is a specific implementation of step S2, which includes the following steps: S21: Select the number of winding turns, inter-turn spacing, wire diameter, number of spacers, spacer width, and paperboard thickness as initial fitting variables, set the value range of each variable, and screen out the key structural parameters that contribute to each natural frequency based on sensitivity analysis to form a combination of dimension-reduced fitting variables corresponding to low-order and high-order modes.

[0031] In some embodiments, the winding turns x1, the inter-turn spacing x2, the wire turn diameter x3, the number of spacer blocks x4, the spacer width x5, and the cardboard thickness x6 are selected as the initial fitting variables.

[0032] Set the value ranges of each variable: x1 ∈ [4, 8] turns, x2 ∈ [0.01, 0.03] m, x3 ∈ [0.2, 0.4] m, x4 ∈ [4, 10] pieces, x5 ∈ [0.01, 0.03] m, x6 ∈ [0.002, 0.008] m.

[0033] Based on sensitivity analysis, key structural parameters are screened. Calculate the sensitivity coefficient S of variable xi to the jth natural frequency yj i,j , and the calculation formula is as follows (7) Where, SS i is the regression sum of squares caused by the term related to variable xi, and SS T is the total sum of squares of variances. Screen out the parameters with higher S i,j values: for the low-order modes (1-3 orders), select the wire turn diameter x3, the number of spacer blocks x4, and the spacer width x5; for the high-order modes (4-5 orders), select the inter-turn spacing x2, the number of spacer blocks x4, and the spacer width x5, to form a reduced-dimensional combination of key fitting variables.

[0034] S22: Estimate the initial sample size based on the central composite design. Use the improved Latin hypercube sampling method. Generate an optimal normalized sample matrix through the iterative exchange strategy that minimizes the maximum value of the non-diagonal elements of the sample matrix correlation coefficient, and then map it to the actual physical space to obtain the actual sample matrix available for simulation. Estimate the initial sample size based on the central composite design (CCD) .

[0035] Where, k is the number of initial fitting variables 6, f takes 1, and k p takes 1, and the calculated initial sample size N = 2 6-1 + 2×6 + 1 = 45; generate the initial normalized sampling matrix S0, and calculate the objective function as:

[0036] is the Pearson correlation coefficient of the sample matrix S.

[0037] Furthermore, adopt the iterative exchange strategy: randomly exchange two elements in different rows of the same column to obtain the candidate matrix S*, calculate E(S*), that is, calculate the objective function corresponding to the candidate matrix. If E(S*) < E(S), accept the exchange and let S k+1 = S*, otherwise, let S k+1 = S, and perform iterative solution until the maximum number of iterations, and output the optimal sample normalized matrixS kmax .

[0038] pass Will S kmax Mapping to the actual physical space yields the actual sample matrix X, where s ij for S kmax The element in the i-th row and j-th column, L j U j Let be the upper and lower bounds of the j-th variable. The iterative exchange strategy here optimizes the sample distribution, making the sample points more evenly cover the variable value range; the inverse transformation maps the normalized samples back to the physical space, obtaining parameter combinations that can be directly used for finite element simulation.

[0039] S23: Substitute each set of key structural parameters in the actual sample matrix into the winding finite element model, update the model geometry and material configuration, perform modal analysis simulation, extract the first five natural frequency data of the corresponding sample, and construct the sample-natural frequency dataset.

[0040] In some embodiments, each row of samples in the actual sample matrix X is traversed, and the key structural parameters corresponding to that row are substituted into the winding finite element model constructed in step S1 to update the geometric and material parameter configuration of the model. Modal analysis is performed on the updated finite element model to solve the characteristic equations. Extract the first five natural frequencies f1-f5 corresponding to the sample.

[0041] Furthermore, the parameter combinations of all samples and their corresponding natural frequency data are organized into a sample dataset. Here, each set of samples corresponds to a set of key structural parameter combinations. By updating the finite element model parameters, the winding vibration characteristics under different structures are simulated. The natural frequencies obtained by modal analysis reflect the dynamic response under the parameter combination, providing input-output data pairs for the response surface model.

[0042] S24: Using a second-order polynomial form, based on the sample-natural frequency dataset, we fit the dimension-reduced second-order response surface models of low-order and high-order modes respectively, and obtain the mapping relationship between key structural parameters and natural frequencies of each order.

[0043] In some embodiments, a second-order polynomial form is adopted. As the basic model for the response surface methodology; for low-order modes (1st-3rd order), a response surface model is constructed using x3, x4, and x5 as input variables. (8) For higher-order modes (4th-5th order), using x2, x4, and x5 as input variables, construct... (9) In the formula,A 123 , B 123 , C 123 , D 123 It is the second-order response surface coefficient matrix of the low-order natural frequency. A 45 , B 45 , C 45 , D 45 This represents the coefficient matrix of the second-order response surface, which contains higher-order natural frequencies. The coefficient matrix can be expressed as follows: (10) Here, a dimension-reduced second-order response surface model is constructed by fitting a coefficient matrix based on the sample dataset. It can be seen that the second-order polynomial model contains first-order, second-order, and interaction terms, which can capture the nonlinear coupling effect of parameters on natural frequencies. The dimension-reduced model is constructed modally, matching the key parameters selected by S21, simplifying the model structure. The coefficient matrix fitting transforms the input-output data pairs into explicit mathematical expressions, realizing the mapping between key structural parameters and natural frequencies.

[0044] S3: Collect current harmonic data of converter transformers in actual operation, deduce the spectral distribution characteristics of electromagnetic excitation force, identify high-risk resonance frequency bands that need to be avoided, and calculate the synthesis coefficients by combining the energy differences of each electromagnetic excitation force, thereby establishing the optimization target value of each order of the winding's natural frequency and the corresponding optimization weight coefficients.

[0045] The following is a specific implementation of step S3, which includes the following steps: S31: Collect current harmonic data and derive the electromagnetic excitation force spectrum. Collect the current data of the converter transformer in actual operation, extract the frequency and amplitude of the fundamental wave and each harmonic, substitute the current expression into the electromagnetic force calculation formula, and obtain the spectrum distribution of the electromagnetic excitation force. In some embodiments, based on continuously collected UHV converter transformer current data during actual operation, the fundamental current and average amplitude of each harmonic of the converter transformer winding are calculated, and the data are shown in Table 2: Table 2 Amplitudes of the fundamental current and its harmonics

[0046] Table 2 shows that the characteristic order of the converter transformer in actual operation is 6k±1 harmonics. Based on the amplitude of each harmonic, the 5th, 7th, 11th, and 13th harmonics are the main harmonics during converter transformer operation. In this embodiment, when analyzing the harmonics of the converter transformer, the actual operating excitation can be simplified to the superposition of the fundamental wave and the 5th, 7th, 11th, and 13th harmonics. The expression for the total current is: (11) In the formula: , and The first The amplitude, angular frequency, and initial phase of each frequency component.

[0047] Substituting formula (11) into the following electromagnetic force formula at a single frequency (12) We can obtain: (13) The expression shown in formula (13) includes the square term of each current and the product term of currents at different frequencies. The square term produces the second harmonic component of each current, and the product term produces the components of the sum and difference of the two frequencies. By extracting the expanded frequency components, the spectral distribution of the electromagnetic excitation force is obtained.

[0048] It can be seen that the electromagnetic force on the converter transformer winding is proportional to the square of the current. When the current contains multiple frequency components, the square expansion of the current produces three types of frequency components: the second harmonic of each frequency component, the sum of any two different frequency components, and the difference frequency of any two different frequency components.

[0049] The combination of the fundamental 50Hz wave and each harmonic in formula (12), after being squared by formula (11), generates electromagnetic excitation forces of 100Hz, 200Hz, 300Hz and even higher frequencies, and all frequencies are integer multiples of 100Hz. The current amplitude data given in Table 2 comes from the actual operation of the UHV converter transformer. The amplitudes of the 5th, 7th, 11th and 13th harmonics are significantly higher than those of other higher harmonics. Therefore, the excitation is simplified to the superposition of the fundamental wave and these four harmonics, which simplifies the calculation while ensuring the integrity of the main frequency components.

[0050] S32: Identify high-risk resonance frequency bands and establish optimization target values. Extract frequencies that are multiples of 100Hz within the range of 100Hz to 1200Hz from the electromagnetic excitation force spectrum as high-risk frequency bands to be avoided. Based on the preset deviation margin between the winding's natural frequency and the high-risk frequency bands, establish the optimization target values ​​for each order of the winding's natural frequency. In some embodiments, based on the electromagnetic excitation force spectrum derived in step S31, frequencies that are integer multiples of 100Hz within the range of 100Hz to 1200Hz are extracted as high-risk frequency bands to be avoided, including 100Hz, 200Hz, 300Hz, 400Hz, 500Hz, 600Hz, 700Hz, 800Hz, 900Hz, 1000Hz, 1100Hz, and 1200Hz. The initial values ​​of the first five natural frequencies of the winding (126.78Hz, 198.82Hz, 374.32Hz, 583.55Hz, and 730.35Hz) are compared with the high-risk frequency bands to determine the high-risk frequency closest to each natural frequency.

[0051] Furthermore, the first-order natural frequency of 126.78Hz is close to 100Hz, with a deviation margin of 30Hz, and the optimization target value is established as 130Hz. The second-order natural frequency of 198.82Hz is close to 200Hz, with a deviation margin of 30Hz, and the optimization target value is established as 230Hz. The third-order natural frequency of 374.32Hz is close to 400Hz, with a deviation margin of 55Hz, and the optimization target value is established as 345Hz. The fourth-order natural frequency of 583.55Hz is close to 600Hz, with a deviation margin of 45Hz, and the optimization target value is established as 555Hz. The fifth-order natural frequency of 730.35Hz is close to 700Hz, with a deviation margin of 45Hz, and the optimization target value is established as 745Hz.

[0052] The deviation margin of the 3rd to 5th natural frequencies is set at 45Hz to 55Hz because these higher-order modes contribute relatively little to the vibration, and are easier to achieve when the target frequency is in the middle position between integer multiples of 100Hz.

[0053] S33: Calculate the electromagnetic force synthesis coefficient and establish the optimization weight coefficient. Calculate the synthesis coefficient of each electromagnetic excitation force based on the current harmonic amplitude. Extract the synthesis coefficient corresponding to the high-risk frequency band corresponding to each natural frequency. Calculate the proportion of each synthesis coefficient to the total synthesis coefficient. Use the proportion as the optimization weight coefficient of each natural frequency. In some embodiments, the synthesis coefficient of each electromagnetic excitation force is calculated based on the electromagnetic excitation force expression expanded from formula (13) in step S31. The synthesis coefficient is determined by the square term and the product term of the current amplitude, reflecting the relative intensity of each frequency component.

[0054] Furthermore, the composite coefficients corresponding to the high-risk frequency bands in Table 3 are extracted: 100Hz corresponds to composite coefficient 922217, 200Hz corresponds to 323203, 400Hz corresponds to 207607, 600Hz corresponds to 170058, and 700Hz corresponds to 64041.

[0055] Table 3. Combination coefficients and relative intensities of various electromagnetic excitation forces

[0056] The natural frequency should deviate from the electromagnetic excitation force spectrum by at least 15 Hz and should have a certain margin. Therefore, the target values ​​for the design of each natural frequency are shown in Table 4. Among them, the first five natural frequencies determined in Table 4 should be 100 Hz, 200 Hz, 400 Hz, 600 Hz, and 700 Hz. Table 4 Target values ​​for the design of each natural frequency

[0057] This embodiment extracts the electromagnetic force synthesis coefficients corresponding to these five key frequency bands and calculates their respective proportions in the sum of these five synthesis coefficients, using these proportions as the optimization weights for each natural frequency. The calculated weight coefficients are shown in Table 5.

[0058] This shows that energy is mainly concentrated in the low-frequency range, so lower-order modes are given greater weight.

[0059] Table 5 Weighting Coefficients

[0060] Adding these five composite coefficients together gives a total of 922217 + 323203 + 207607 + 170058 + 64041 = 1687126. Calculating the proportion of each composite coefficient to the total: the proportion for the 100Hz band (1st order) is 922217 / 1687126 = 0.5470; the proportion for the 200Hz band (2nd order) is 323203 / 1687126 = 0.1915; the proportion for the 400Hz band (3rd order) is 207607 / 1687126 = 0.1231; the proportion for the 600Hz band (4th order) is 170058 / 1687126 = 0.1007; and the proportion for the 700Hz band (5th order) is 64041 / 1687126 = 0.0377. The above proportions are used as the optimization weighting coefficients for the first to fifth natural frequencies, respectively.

[0061] It can be seen that the energy difference of electromagnetic excitation force at different frequencies is significant. In Table 3, the synthesis coefficient of 922217 for 100Hz is much larger than that for 64041 for 700Hz, indicating that low-frequency electromagnetic force is the main excitation source of winding vibration. During the optimization process, it is far more important to deviate the first-order natural frequency from 100Hz than to deviate the fifth-order natural frequency from 700Hz. The optimization weight coefficients are established based on the relative intensity of electromagnetic force corresponding to each high-risk frequency band, that is, the higher the energy of the frequency band, the larger the weight coefficient. The sum of all weight coefficients is 1, ensuring that the numerical magnitude of the weighted frequency deviation function is comparable to that of the single-order frequency deviation. In Table 5, the first-order weight of 0.5470 accounts for more than half of the total, while the fifth-order weight is only 0.0377. This distribution reflects that the vibration energy is mainly concentrated in the low-frequency band. After the weight coefficients are normalized, the numerical range of the weighted frequency deviation function matches the numerical range of each order frequency deviation, which facilitates the convergence of the optimization algorithm and the construction of the Pareto front.

[0062] S34: Output the optimization target value and optimization weight coefficient. Combine the optimization target values ​​of each order of natural frequency established in step S32 and the optimization weight coefficients of each order established in step S33 and output them as the basis for calculating the weighted frequency deviation function in the optimization model.

[0063] In some embodiments, the optimization target values ​​of 130Hz, 230Hz, 345Hz, 555Hz, and 745Hz established in step S32 are combined with the optimization weight coefficients of 0.5470, 0.1915, 0.1231, 0.1007, and 0.0377 established in step S33. The optimization target value vector f is output in vector form. target =[130,230,345,555,745] T The optimized weight coefficient vector is k = [0.5470, 0.1915, 0.1231, 0.1007, 0.0377]. T This combination is used as the input parameter for constructing the weighted frequency deviation function, which is then used in the weighted frequency deviation function J1 to calculate the weighted deviation between the natural frequencies of the response surface model output and the optimization objective value.

[0064] Understandably, the calculation of the weighted frequency deviation function J1 depends on both the optimization objective value and the optimization weight coefficients. The optimization objective value determines the desired position of the natural frequency, while the optimization weight coefficients determine the proportion of each order of deviation in the total deviation. Steps S32 and S33 perform independent calculations of these two types of parameters, and step S34 integrates them into a complete parameter set. This parameter set serves as a fixed input in the subsequent optimization model and does not change with the optimization iteration process. The output format is a vector, which facilitates substitution into the weighted frequency deviation function J1 for numerical calculation.

[0065] S4: Based on the second-order response surface model constructed in step S2 and the optimization objective value and optimization weight coefficients established in step S3, construct a bi-objective optimization model that includes a weighted frequency deviation function and a resonance penalty function. The weighted frequency deviation function is used to measure the degree of deviation between the natural frequency calculated by the response surface model and the optimization objective value. The resonance penalty function is used to apply a penalty when the natural frequency is close to the high-risk resonance frequency band, and the number of pad blocks is limited to an integer.

[0066] In some embodiments, the construction of the dual-objective optimization model relies on the previous response surface mapping relationship, optimization objective and constraints, and the weighted frequency deviation function quantifies the deviation between the intrinsic frequency and the objective value through a weighted sum of squares.

[0067] The resonance penalty function applies different penalties based on the deviation between the natural frequency and the excitation frequency using a piecewise potential energy function. The smaller the deviation, the greater the penalty, thus mitigating the risk of the natural frequency being close to the target value but near the resonance frequency band. Integer constraints align with actual manufacturing processes, ensuring the optimization results are manufacturable. The two objective functions complement and constrain each other.

[0068] S5: Set the population size, maximum number of iterations, crossover probability, and mutation probability. Use the NSGA-II algorithm to perform global optimization on the bi-objective optimization model constructed in step S4. Under the constraint that the number of pad blocks is an integer, obtain the Pareto optimal solution set that minimizes both the weighted frequency deviation function and the resonance penalty function.

[0069] In some embodiments, the NSGA-II algorithm identifies non-dominated solutions under dual objectives through fast non-dominated sorting; higher levels represent better solutions. Crowding calculations ensure uniform distribution of individuals within the same level, avoiding local convergence of the population. Crossover operations inherit superior genes from the parent generation, while mutation operations introduce new parameter combinations, driving the population to gradually converge toward the Pareto optimal front.

[0070] S6: Extract the optimized key structural parameters from the Pareto optimal solution set obtained in step S5. The key structural parameters include the turn spacing, the diameter of the turn, the number of spacers, and the width of the spacers.

[0071] In some embodiments, the obtained Pareto optimal solution set is called, and the key structural parameters corresponding to each optimal solution are extracted one by one, including the inter-turn spacing x2, the wire turn diameter x3, the number of pad blocks x4, and the pad block width x5. At the same time, the calculated values ​​of J1(X) and J2(X) corresponding to each solution are extracted. The sequence number, x2, x3, x4, x5 parameter values, weighted frequency deviation function J1(X) value, and resonance penalty function J2(X) value of each optimal solution are matched one by one to ensure that there is no mismatch between the parameters and the objective function value.

[0072] Where x4 is an integer between 4 and 10, and x2, x3, and x5 are within their respective ranges. Check if each parameter conforms to the preset range, and eliminate solutions that exceed the range. Check if x4 is an integer, and eliminate non-integer solutions. Based on actual machining accuracy requirements, eliminate solutions whose parameter values ​​exceed the allowable range for machining accuracy, and retain all feasible optimal solutions that meet the requirements, forming a feasible optimal solution reference table. From the feasible optimal solution reference table, extract the specific values ​​of x2, x3, x4, and x5 for each solution, keeping x4 in integer form and x2, x3, and x5 to three decimal places. Verify the correspondence between each parameter and the feasible solution to avoid incorrect or omitted extractions.

[0073] Furthermore, the extracted parameters are organized according to the feasible solution number to form a set of optimized values ​​for key structural parameters. The parameter combination corresponding to each feasible solution is clarified. The typical optimized values ​​are x2=0.012m, x3=0.210m, x4=10, and x5=0.021m. All parameter combinations of feasible solutions are retained.

[0074] S7: Substitute the optimized values ​​of the key structural parameters extracted in step S6 into the second-order response surface model constructed in step S2, calculate the optimized natural frequencies of each order of the winding, and output the optimized values ​​of the key structural parameters as the vibration reduction design scheme of the converter transformer winding.

[0075] In some embodiments, the second-order response surface model has established an explicit mapping between key structural parameters and natural frequencies. By substituting the optimized parameter values, the corresponding natural frequencies can be quickly calculated without repeating finite element simulations, thus improving computational efficiency. The output vibration reduction design scheme integrates all previous optimization processes, parameters, and verification results. The frequency verification stage ensures that the optimized scheme can effectively avoid resonance risks and guarantee the operational stability of the converter transformer.

[0076] In one embodiment of the present invention, based on step S4, the following is a possible embodiment and its specific implementation will be described in a non-limiting manner. S4 specifically includes the following steps: S41: Define the decision variables and objective function forms. Use the key structural parameters selected in step S23 as decision variables. Determine that the bi-objective optimization model contains two objective functions. The first objective function is the weighted frequency deviation function, and the second objective function is the resonance penalty function.

[0077] In some embodiments, the selected key structural parameters, such as the turn spacing x2, the turn diameter x3, the number of pad blocks x4, and the pad block width x5, are used as decision variables, denoted as vector X=[x 2, x3, x4, x5] T .

[0078] Furthermore, the bi-objective optimization model is determined to contain two objective functions. The first objective function, denoted as J1(X), measures the deviation between the natural frequencies calculated by the response surface model and the optimization objective value. The second objective function, denoted as J2(X), is used to impose a penalty when the natural frequencies approach the high-risk resonance frequency band. Both objective functions are calculated using the decision variable X as input, and are obtained by further calculation after calculating the natural frequencies of each order through the second-order response surface model constructed in step S24. The optimization objective is to simultaneously minimize J1(X) and J2(X).

[0079] It should be noted that the vibration reduction optimization problem of the converter transformer winding structure can be described as the following nonlinear biobjective programming problem. (18)

[0080] In the formula: and These are the lower and upper limits of the variable, respectively; The number of pad blocks is constrained to meet the requirements of actual manufacturing processes. The weighted frequency deviation function, Let be the resonance penalty function, and both are . The function is calculated using a simplified second-order response surface model.

[0081] The first five modes and natural frequencies of the transformer winding are shown in Table 6, and the mode shapes are as follows: Figure 3 As shown. Among them, Figure 3 In the diagram, (a) is a schematic diagram of mode order 1; (b) is a schematic diagram of mode order 2; (c) is a schematic diagram of mode order 3; (d) is a schematic diagram of mode order 4; and (e) is a schematic diagram of mode order 5.

[0082] It can be seen that even-numbered vibrations are dominated by axial vibrations, while odd-numbered vibrations are dominated by radial vibrations.

[0083] Table 6 Modal order and natural frequency

[0084] It can be seen that the bi-objective optimization model defined in formula (18) is FindX,MinimizeJ(X)=[J1(X),J2(X)]. T , where J1(X) and J2(X) correspond to the weighted frequency deviation function and the resonance penalty function, respectively.

[0085] The two objective functions are mutually constrained during the optimization process: minimizing J1(X) tends to push the natural frequency precisely towards the optimization target value, while minimizing J2(X) requires the natural frequency to maintain a sufficient distance from all high-risk frequency bands. The dimension of the decision variable X is reduced from the initial six dimensions to four dimensions, reducing the complexity of the optimization search space. Both objective functions are output in vector form, forming a two-dimensional objective space.

[0086] S42: Construct a weighted frequency deviation function, calculate the natural frequencies of each order based on the constructed second-order response surface model, and construct the weighted frequency deviation function based on the established optimization target value and optimization weight coefficient. The weighted frequency deviation function calculates the sum of the square of the difference between each natural frequency and the corresponding optimization target value multiplied by the corresponding weight coefficient.

[0087] In some embodiments, based on the constructed second-order response surface model, for a given decision variable X=[x2,x3,x4,x5] T Calculate the natural frequencies f1(X) to f5(X) of the first to fifth orders respectively.

[0088] For the first to third low-order modes, the response surface model is invoked, with input variables x3, x4, and x5, and coefficient matrices taking the corresponding values ​​of A123, B123, C123, and D123.

[0089] For higher-order modes, levels 4 and 5, the response surface model is invoked, with input variables x2, x4, and x5, and coefficient matrices taking the corresponding values ​​of A45, B45, C45, and D45. The optimization objective value f output in step S34 is then used. target,i And optimize the weight coefficient k i Substitute into formula (14) to calculate the weighted frequency deviation function J1 (14) In the formula, The set of key structural parameter variables after dimensionality reduction and filtering, i.e. ; For As input, the second-order response surface model is used to calculate the... The value of the first natural frequency; For the first The target value for the first natural frequency; For the first Weighting coefficients for optimizing the first natural frequency.

[0090] It can be seen that formulas (8) and (9) give the expressions for second-order polynomials. After substituting the specific values, the natural frequency is directly output without calling the finite element solver. In formula (14), the weighted frequency deviation function adopts the form of a sum of squares, so that both positive and negative deviations are punished equally, avoiding the mutual cancellation of deviations.

[0091] Weighting coefficient k i In step S33, based on the electromagnetic force composition coefficients, the first-order weight of 0.5470 is much larger than the fifth-order weight of 0.0377. Therefore, the deviation of the first-order natural frequency dominates in J1. The presence of the squared term results in an exponentially increasing penalty for larger deviations, guiding the optimization algorithm to prioritize adjusting each frequency to near the target value.

[0092] S43: Construct a resonance penalty function, define the minimum deviation function from the set of electromagnetic excitation force frequencies of each natural frequency, construct a piecewise potential energy function based on the minimum deviation, and use the sum of the piecewise potential energy functions of each order as the resonance penalty function. The piecewise potential energy function adopts different linear expressions according to the interval where the minimum deviation is located. In some embodiments, the electromagnetic excitation force frequency set F={100,200,300,400,500,600,700,800,900,1000,1100,1200} is defined, with the unit being Hz.

[0093] For the i-th natural frequency f i (X), to avoid the above problems, a resonance penalty function J2 is defined. When a certain natural frequency is close to the spectrum of the electromagnetic excitation force during the optimization process, a corresponding penalty is given according to the degree of closeness. Define the... First natural frequency The minimum deviation to the frequency of the electromagnetic excitation force is :

[0094] In the formula, The set of key structural parameter variables after dimensionality reduction and filtering, i.e. ; For As input, the second-order response surface model is used to calculate the... The value of the first natural frequency; It is the set of electromagnetic excitation force frequencies, i.e., F = {100, 200, 300…}.

[0095] Furthermore, based on minimum deviation Construct a piecewise potential energy function :

[0096] In the formula: This is a safety threshold; This is the warning threshold; , , This is the potential energy penalty adjustment constant, used to control the repulsive force in different intervals.

[0097] To ensure stable operation of the converter transformer, its natural frequency should deviate from the electromagnetic excitation frequency by at least 15Hz. Considering actual manufacturing errors and material aging caused by long-term operation, this embodiment incorporates a more ample safety margin in its engineering design, setting a safety threshold. The frequency is 15Hz, and the warning threshold is... It is 25Hz.

[0098] The resonance penalty function J2 is defined as the sum of all piecewise potential functions of all orders:

[0099] when When the frequency is less than 15Hz, =C1+C2×(D hard -d i ); when d i When the frequency is greater than or equal to 15Hz and less than 25Hz, =C3×(D warm -d i ); when d i When the frequency is greater than or equal to 25Hz, =0.

[0100] It can be seen that summing the penalty values ​​of the fifth-order natural frequencies increases the J2 value if any first-order frequency enters the danger zone. (Penalty adjustment constant) , , The value of makes In d i The frequency is continuous at 15Hz to avoid jumps in the penalty function.

[0101] S44: Apply constraints and output a bi-objective optimization model. Set upper and lower limits for the values ​​of each decision variable, limit the number of pad blocks to integers, and combine the weighted frequency deviation function and the resonance penalty function into a bi-objective optimization model.

[0102] In some embodiments, upper and lower limits are set for the decision variables: the lower limit of the turn spacing x2 is 0.01m and the upper limit is 0.03m, the lower limit of the turn diameter x3 is 0.2m and the upper limit is 0.4m, the lower limit of the number of pad blocks x4 is 4 and the upper limit is 10, and the lower limit of the pad block width x5 is 0.01m and the upper limit is 0.03m.

[0103] The number of pad blocks x4 is limited to an integer, i.e., x4∈{4,5,6,7,8,9,10}.

[0104] The weighted frequency deviation function J1(X) constructed in step S42 is combined with the resonance penalty function J2(X) constructed in step S43 to form a bi-objective optimization model based on FindX=[x2,x3,x4,x5]. T MinimizeJ(X) = [J1(X), J2(X)] T And satisfy x2∈[0.01,0.03], x3∈[0.2,0.4], x4∈{4,5,…,10}, x5∈[0.01,0.03], and output the bi-objective optimization model as the input for solving step S5.

[0105] Furthermore, based on constraint L j ≤x j ≤U j The corresponding value ranges for each decision variable are: L2=0.01, U2=0.03, L3=0.2, U3=0.4, L4=4, U4=10, L5=0.01, U5=0.03.

[0106] The constraint x4∈{Z|Z∈D} indicates that the number of pad blocks is an integer. This is determined by the actual manufacturing process: the pad blocks are evenly distributed in the circumferential direction, and only an integer number of blocks can achieve a symmetrical arrangement.

[0107] In one embodiment of the present invention, based on step S5, the following is a possible embodiment and its specific implementation will be described in a non-limiting manner. S5 specifically includes the following steps: S51: Initialize the population and algorithm parameters, set the population size, maximum number of iterations, crossover probability and mutation probability, encode the decision variables in the bi-objective optimization model output in step S44, and randomly generate the initial population within the range of the decision variables, wherein the number of pad blocks is taken as an integer value during initialization.

[0108] In some embodiments, the population size N is set to 100, and the maximum number of iterations G is set to... max The value is set to 200, the crossover probability PC is set to 0.9, and the mutation probability P is set to... m Take 0.1.

[0109] The output decision variable is X = [x2, x3, x4, x5]. T Real-number encoding is performed, with each individual corresponding to a set of decision variable values. An initial population of 100 individuals is randomly generated within the range of decision variable values.

[0110] The value range of the inter-turn spacing x2 is [0.01, 0.03], the value range of the turn diameter x3 is [0.2, 0.4], the value range of the number of pad blocks x4 is [4, 10], and the value range of the pad block width x5 is [0.01, 0.03]. Since x4 is an integer, during initialization, for each individual x4 component, a real number in the interval [4, 10] is first randomly generated, and then rounded to the nearest integer.

[0111] S52: Perform non-dominated sorting and crowding calculation. For each individual in the current population, calculate two objective function values ​​based on the weighted frequency deviation function constructed in step S42 and the resonance penalty function constructed in step S43. Sort the individuals in the population non-dominated according to the objective function values, determine the non-dominated level of each individual, and calculate the crowding distance of individuals within the same non-dominated level.

[0112] In some embodiments, for each individual in the current population, the decision variable X corresponding to the individual is substituted into the weighted frequency deviation function J1(X) constructed in step S42 and the resonance penalty function J2(X) constructed in step S43 to calculate two objective function values.

[0113] Furthermore, the population is stratified: for any two individuals p and q, if J1(p)≤J1(q) and J2(p)≤J2(q), and at least one inequality holds, then p is said to dominate q.

[0114] Iterate through all individuals in the population and find the set of individuals that are not dominated by any other individual as the first non-dominated layer; after removing individuals from the first non-dominated layer, continue to find the set of individuals that are not dominated by any other individual as the second non-dominated layer; repeat this process until all individuals are stratified.

[0115] For individuals within the same non-dominated layer, calculate the crowding distance of each individual in the target space. The crowding distance is the sum of the distances between the two adjacent individuals in the J1 and J2 directions.

[0116] It can be seen that dividing the population into different levels, the individuals in the first level constitute the Pareto optimal solution set in the current population, and their objective function values ​​are not dominant among themselves. The lower the non-dominated level, i.e., the larger the level number, the more times the individual is dominated, and the higher the probability of being eliminated in the evolutionary process.

[0117] Crowding distance is used to measure the distribution density of individuals within the same non-dominated layer. Individuals with large crowding distances have fewer solutions in their surrounding regions, and retaining these individuals helps maintain the uniform distribution of the Pareto front. The combination of non-dominated sorting and crowding calculation allows the algorithm to achieve a balance between convergence and diversity.

[0118] S53: Perform selection, crossover, and mutation operations. Use tournament selection to select parent individuals from the current population. Perform simulated binary crossover on the parent individuals according to the set crossover probability to generate offspring individuals. Perform polynomial mutation on the offspring individuals according to the set mutation probability. The mutation result of the pad block number in the mutation operation is rounded down.

[0119] In some embodiments, a binary tournament selection method is used to select parent individuals from the current population: two individuals are randomly selected from the population, and their non-dominance levels are compared; the individual with the lower non-dominance level wins. If the non-dominance levels are the same, the crowding distance is compared; the individual with the larger crowding distance wins.

[0120] Repeat this process until 2×N parent individuals are selected. Pair the selected parent individuals and perform simulated binary crossover with a crossover probability PC=0.9: For the two paired parent individuals, generate a random number in the interval [0,1]. If the random number is less than 0.9, perform crossover operation on each dimension of the decision variable to generate two offspring individuals.

[0121] If the crossover condition is not met, the parent individual is directly copied into the child individual.

[0122] For the offspring individuals generated after crossover, according to the mutation probability P m Perform a polynomial mutation operation on a value of 0.1: For each decision variable of each offspring individual, a random number in the interval [0,1] is generated. If the random number is less than 0.1, a mutation operation is performed on that variable. During the mutation operation, the mutation result of the number of pad blocks x 4 is rounded to the nearest integer.

[0123] Furthermore, the simulated binary crossover operation simulates the single-point crossover characteristic in binary encoding, generating two offspring individuals under real-number encoding. The values ​​of the offspring individuals lie between those of the parent individuals, maintaining the continuity of the variable range. The polynomial mutation operation generates small perturbations near the variable values, introducing new genotypes to maintain population diversity. The rounding process after x4 mutation ensures that the offspring individuals satisfy the integer constraint of formula (18).

[0124] S54: Merge the populations and generate a new generation population. Merge the parent and offspring populations. Perform non-dominated sorting and crowding calculation on the merged population. Select individuals into the new generation population in order of non-dominated level from low to high. When individuals of a certain level cannot be selected, select individuals of that level in order of crowding distance from large to small until the size of the new generation population reaches the set population size. Iterate through steps S52 to S54 until the maximum number of iterations is reached. Output the non-dominated individuals in the final population as the Pareto optimal solution set.

[0125] In some embodiments, the offspring population generated in step S53 is merged with the current parent population, and the size of the merged population is 2×N=200.

[0126] Perform the non-dominated sorting and crowding calculation in step S52 on the merged population. Individuals are selected to enter the new generation population in ascending order of non-dominated level. All individuals in the first non-dominated layer are selected. If the number of individuals in the first non-dominated layer is less than N, individuals in the second non-dominated layer are selected, and so on.

[0127] When the number of individuals in a non-dominated layer plus the number of selected individuals exceeds N, the individuals in that non-dominated layer are sorted from largest to smallest based on their crowding distance, and individuals with larger crowding distances are selected sequentially until the size of the new generation population reaches N. The new generation population is then used as the current population for the next iteration, and steps S52 to S54 are repeated.

[0128] When the number of iterations reaches the set maximum of 200, iteration stops, and the final population is output, containing all individuals in the first non-dominated layer. These individuals constitute the Pareto optimal solution set. Furthermore, within the same layer, individuals with large crowding distances are prioritized to maintain the uniformity of the Pareto front. The iteration termination condition is set to the maximum number of iterations to ensure the algorithm has sufficient evolution time and predictable runtime. The output is the set of individuals in the first non-dominated layer, where each individual is non-dominated in both the weighted frequency deviation and resonance penalty objectives, forming a multi-objective optimization solution set.

[0129] In one embodiment of the present invention, based on step S6, the following is a possible embodiment and its specific implementation will be described in a non-limiting manner. S6 specifically includes the following steps: S61: Filter Pareto optimal solutions. From the output Pareto optimal solution set, extract the decision variable values ​​of all non-dominated individuals. Each decision variable corresponds to a set of numerical combinations of turn spacing, turn diameter, number of pad blocks, and width of pad blocks.

[0130] In some embodiments, information about all non-dominated individuals is read from the output Pareto optimal solution set. The Pareto optimal solution set consists of all individuals at the first level of non-domination in step S54, and each individual corresponds to a set of decision variables X=[x2,x3,x4,x5]. T The specific values ​​are as follows: Iterate through each individual in the set, recording the values ​​of the turn spacing x2, the wire turn diameter x3, the number of spacers x4, and the spacer width x5 for each individual. Record the weighted frequency deviation function J1(X) and resonance penalty function J2(X) values ​​corresponding to each individual; these two function values ​​have already been calculated and stored in step S52.

[0131] S62: Extract the optimized key structural parameters. From the Pareto optimal solutions selected in step S61, select the solution with the smallest sum of weighted frequency deviation function value and resonance penalty function value. The values ​​of turn spacing, wire diameter, number of pads, and width of pads in this solution are used as the optimized key structural parameters.

[0132] In some embodiments, for each individual in the Pareto optimal solution set extracted in step S61, the algebraic sum of the weighted frequency deviation function value J1(X) and the resonance penalty function value J2(X) is calculated, denoted as Jsum=J1(X)+J2(X). All individuals are iterated to find the individual that minimizes Jsum.

[0133] The values ​​of inter-turn spacing x2, turn diameter x3, number of spacers x4, and spacer width x5 corresponding to each individual are used as the key structural parameters after optimization. If multiple individuals simultaneously minimize Jsum, the individual with the smallest J2(X) value is selected to prioritize minimizing the resonance penalty.

[0134] S63: Verify the integer constraint condition. Check whether the number of pad blocks extracted in step S62 is an integer. If it is an integer, output the key structural parameters of the group directly. If it is not an integer, round the number of pad blocks to the nearest integer before outputting.

[0135] In some embodiments, it is checked whether the value of the number of pad blocks x4 extracted in step S62 is an integer. Since the mutation operation in step S53 and the population evolution in step S54 may cause x4 to fluctuate around an integer, if the value of x4 is an integer, the set of key structural parameters is directly output.

[0136] If the value of x4 is not an integer, then the value is rounded to the nearest integer: if the decimal part is less than 0.5, it is rounded down; if the decimal part is greater than or equal to 0.5, it is rounded up.

[0137] After rounding, check if the integer falls within the constraint range of [4, 10]. If the rounded result exceeds the range, take the nearest boundary value; that is, take 4 if less than 4, and take 10 if greater than 10. Output the processed key structural parameters x2, x3, x4, x5.

[0138] As can be seen, the boundary value check after rounding further ensures the feasibility of the output parameters and avoids exceeding the upper and lower limits due to rounding. The output parameter format is consistent with the naming of the key structural parameters used in steps S2, S3, S4, and S5, ensuring that no format conversion is required when substituting into the response surface model in step S7.

[0139] In one embodiment of the present invention, based on step S7, the following is a possible embodiment and its specific implementation will be described in a non-limiting manner. S7 specifically includes the following steps: S71: Obtain the optimized key structural parameters, and read the values ​​of inter-turn spacing, wire diameter, number of spacers, and width of spacers from the output optimized key structural parameters.

[0140] In some embodiments, the values ​​of four parameters are read from the optimized key structural parameters output in step S63. The data output in step S63 includes the final values ​​of the turn spacing x2, the wire turn diameter x3, the number of spacers x4, and the spacer width x5.

[0141] According to the filtering rules in step S62 and the integer constraint processing in step S63, this set of parameters corresponds to the optimized values ​​in Table 8: the turn spacing is 0.012m, the turn diameter is 0.210m, the number of spacers is 10, and the spacer width is 0.021m. These four values ​​are recorded as x. 2opt x 3opt x 4opt x 5opt , which serves as the basis for calculation and output.

[0142] It can be seen that the parameters output in step S63 are the final results obtained after optimization algorithm iteration, Pareto solution set filtering, and integer constraint processing. This set of parameters satisfies all defined constraints, including the range constraints of each variable and the integer constraint on the number of pad blocks. 2opt =0.012m lies within the interval [0.01, 0.03], x 3opt =0.210m lies within the interval [0.2, 0.4], x 4opt =10 lies within the integer interval [4,10], x 5opt =0.021m lies within the interval [0.01, 0.03]. Using this set of parameters as input to the design scheme ensures that subsequent calculations are based on a feasible solution.

[0143] S72: Substitute the values ​​of inter-turn spacing, wire diameter, number of pads, and width of pads read in step S71 into the constructed low-order and high-order response surface models respectively to calculate the optimized first five natural frequencies of the winding.

[0144] In some embodiments, the wire turn diameter x read in step S71 3opt =0.210m, number of pad blocks x 4opt =10 pieces, pad width x 5opt Substitute 0.021m into the low-order response surface model constructed in step S24. The specific form is: f i =A123 i +B123 i1 ×x3+B123 i2 ×x4+B123 i3 ×x5+C123 i1 ×x3 2 +C123 i2 ×x4 2 +C123 i3 ×x5 2 +D123 i1 ×x3×x4+D123 i2 ×x3×x5+D123 i3 ×x4×x5. (The rest of the text appears to be incomplete and requires further context.) 3opt x 4opt x 5opt After substituting the values, the optimized values ​​of the first, second, and third natural frequencies are obtained.

[0145] It can be seen that the inter-turn interval x read in step S71 2opt =0.012m, number of pad blocks x 4opt =10 pieces, pad width x 5opt =0.021m is substituted into the higher-order response surface model constructed in step S24.

[0146] The higher-order response surface model consists of the computational expressions for f4 and f5, specifically in the form of: f i =A45 i +B45 i1 ×x2+B45 i2 ×x4+B45 i3 ×x5+C45 i1 ×x2 2 +C45 i2 ×x4 2 +C45 i3 ×x5 2 +D45 i1 ×x2×x4+D45 i2 ×x2×x5+D45 i3 ×x4×x5. (The rest of the text appears to be incomplete and requires further context.) 2opt x 4opt x 5opt After substituting the values, the optimized values ​​of the 4th and 5th natural frequencies are obtained.

[0147] Furthermore, with x 4optSubstituting 10 into the equation, the calculated natural frequencies are approximately 130.5 Hz for the first order, 231.5 Hz for the second order, 345.2 Hz for the third order, 569.5 Hz for the fourth order, and 745.1 Hz for the fifth order. The fourth order frequency matches the 569.5 Hz value mentioned in Table 7. Table 7 shows a comparison of the specific values ​​of each key structural parameter of the winding before and after optimization.

[0148] Table 7. Optimized parameter values

[0149] S73: Output vibration reduction design scheme. The values ​​of inter-turn spacing, wire diameter, number of pads, and width of pads read in step S71 are combined with the first five natural frequencies calculated in step S72 to output the vibration reduction design scheme of the converter transformer winding.

[0150] In some embodiments, the optimized key structural parameter x read in step S71 2opt =0.012m, x 3opt =0.210m, x 4opt =10, x 5opt =0.021m and the first five natural frequencies f calculated in step S72 1opt to f 5opt Combine them.

[0151] The combination method is as follows: output a list of design parameters, including values ​​for inter-turn spacing, turn diameter, number of spacers, and spacer width; output a list of verification results, including calculated values ​​for the 1st, 2nd, 3rd, 4th, and 5th natural frequencies. The combined results are then output as the vibration reduction design scheme for the converter transformer windings.

[0152] As can be seen, the final output of the vibration reduction design scheme includes two parts: the design parameters section provides the specific structural dimensions to be manufactured, corresponding to the optimized values ​​in Table 7. The verification results section provides the natural frequency characteristics of the winding after adopting the design parameters, indicating that the design scheme meets the vibration isolation requirements. The number of pads in the design parameters is 10, evenly distributed in the circumferential direction, satisfying the integer constraint. The turn spacing of 0.012m, the turn diameter of 0.210m, and the pad width of 0.021m are all within the range set in Table 8. The natural frequency calculation results all deviate from the corresponding high-risk frequency band by more than 15Hz, proving the effectiveness of the design scheme.

[0153] Table 8. Variables and their values

[0154] It should be understood that the sequence number of each step in the above embodiments does not imply the order of execution. The execution order of each process should be determined by its function and internal logic, and should not constitute any limitation on the implementation process of the embodiments of the present invention.

[0155] like Figure 4 As shown, this application also provides an electronic device, including a display module 103, a memory 102, a processor 101, a communication module 104, and a computer program stored in the memory and executable on the processor 101. When the processor 101 executes the program, it implements the steps of the dual-target vibration reduction optimization method for converter transformer windings.

[0156] In this embodiment, processor 101 may be implemented using at least one of an application-specific integrated circuit, a programmable logic device, a field-programmable gate array, a processor, a controller, a microcontroller, a microprocessor, or an electronic unit designed to perform the functions described herein. In some cases, such an implementation may be implemented within a controller. For software implementation, implementations such as processes or functions may be implemented with separate software modules that allow the performance of at least one function or operation. Software code may be implemented by a software application (or program) written in any suitable programming language, and the software code may be stored in memory and executed by the controller.

[0157] The display module 103 is used to display information input by the user or information provided to the user. The display module 103 may include a display panel, which may be configured in the form of a liquid crystal display, an organic light-emitting diode, or the like.

[0158] The memory 102 can be used to store software programs and various data. The memory 102 may include high-speed random access memory, and may also include non-volatile memory, such as at least one disk storage device, flash memory device, or other volatile solid-state storage device.

[0159] The communication module 104 transmits radio signals to and / or receives radio signals from at least one of a base station, an external terminal, and a server. Such radio signals may include voice call signals, video call signals, or various types of data sent and / or received according to text and / or multimedia messages.

[0160] The present invention also provides a storage medium storing a computer program thereon, wherein the computer program, when executed by a processor, implements the steps of the dual-target vibration reduction optimization method for the converter transformer winding.

[0161] The storage medium may be any combination of one or more readable media. A readable medium may be a readable signal medium or a readable storage medium. A readable storage medium may be, for example,, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination thereof. More specific examples (a non-exhaustive list) of readable storage media include: electrical connections having one or more wires, portable disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination thereof.

[0162] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A dual-objective vibration reduction optimization method for converter transformer windings, characterized in that, The methods include: S1: Construct a finite element model of the converter transformer winding, perform modal analysis based on the finite element model, and obtain the first five natural frequencies and mode shapes of the winding. S2: Select several key structural parameters that affect the natural frequency of the winding as fitting variables, use the improved Latin hypercube sampling method to generate the optimal sample matrix within the range of the fitting variables, and construct a second-order response surface model between the key structural parameters and the first five natural frequencies of the winding based on the finite element simulation results corresponding to the optimal sample matrix. S3: Collect current harmonic data of converter transformers in actual operation, deduce the spectral distribution characteristics of electromagnetic excitation force, identify high-risk resonance frequency bands that need to be avoided, and calculate the synthesis coefficients by combining the energy differences of each electromagnetic excitation force, thereby establishing the optimization target value of each order of the winding's natural frequency and the corresponding optimization weight coefficients. S4: Based on the constructed second-order response surface model and the established optimization objective value and optimization weight coefficient, a bi-objective optimization model including a weighted frequency deviation function and a resonance penalty function is constructed. S5: Set the population size, maximum number of iterations, crossover probability, and mutation probability, and perform global optimization on the constructed bi-objective optimization model. Under the constraint that the number of pad blocks is an integer, obtain the Pareto optimal solution set that minimizes both the weighted frequency deviation function and the resonance penalty function. S6: Extract the optimized key structural parameters from the Pareto optimal solution set; S7: Substitute the optimized values ​​of the extracted key structural parameters into the constructed second-order response surface model to calculate the optimized natural frequencies of each order of the winding, and output the optimized values ​​as the vibration reduction design scheme for the converter transformer winding.

2. The dual-objective vibration reduction optimization method for converter transformer windings according to claim 1, characterized in that, S1 specifically includes the following steps: S11: Simplify and assemble the converter transformer winding, equate the spiral winding to a closed circular array, simplify the pads to rectangular blocks of the same depth as the coil and distribute 8 evenly along the circumference and arrange them in an array according to the interlayer gap along the axial direction, and set a continuous insulating paper tube inside the winding to complete the geometric structure construction of the winding finite element model. S12: Define material properties for each component of the winding finite element model, input the density, Young's modulus and Poisson's ratio parameters of the winding turns, pads and insulating paper tubes, and construct the mass matrix and stiffness matrix basis for modal analysis; S13: Apply fixed constraints to the bottom of the winding finite element model to simulate the rigid support state of the bottom of the converter transformer in actual operation; no fixed constraints are set for the rest of the part. S14: Based on the completed winding finite element model, solve the characteristic equations of the multi-degree-of-freedom system. ; Where [K] is the stiffness matrix, [M] is the mass matrix, ω is the angular frequency, and {φ} is the mode shape vector; the first five natural frequencies and corresponding mode shapes are extracted to distinguish the distribution characteristics of even-order axial vibration and odd-order radial vibration.

3. The dual-objective vibration reduction optimization method for converter transformer windings according to claim 1, characterized in that, S2 specifically includes the following steps: S21: Select the number of winding turns, the inter-turn spacing, the wire turn diameter, the number of spacers, the width of the spacers, and the thickness of the cardboard as initial fitting variables, set the value range of each variable, and screen out the key structural parameters that contribute to each natural frequency based on sensitivity analysis to form a combination of dimension-reduced fitting variables. S22: Based on the central composite design, the initial sample size is estimated. An improved Latin hypercube sampling method is adopted. The optimal normalized sample matrix is ​​generated by an iterative exchange strategy that minimizes the maximum value of the off-diagonal elements of the correlation coefficient of the sample matrix. The matrix is ​​then mapped to the actual physical space to obtain the actual sample matrix. S23: Substitute each set of key structural parameters in the actual sample matrix into the winding finite element model, update the model geometry and material configuration, perform modal analysis simulation, extract the first five natural frequency data of the corresponding sample, and construct the sample-natural frequency dataset. S24: Using a second-order polynomial form, based on the sample-natural frequency dataset, we fit the dimension-reduced second-order response surface models of low-order and high-order modes respectively, and obtain the mapping relationship between key structural parameters and natural frequencies of each order.

4. The dual-objective vibration reduction optimization method for converter transformer windings according to claim 1, characterized in that, S3 specifically includes the following steps: S31: Collect current harmonic data and derive the electromagnetic excitation force spectrum. Collect the current data of the converter transformer in actual operation, extract the frequency and amplitude of the fundamental wave and each harmonic, substitute the current expression into the electromagnetic force calculation formula, and obtain the spectrum distribution of the electromagnetic excitation force. S32: Identify high-risk resonance frequency bands and establish optimization target values. Extract frequencies that are multiples of 100Hz within the range of 100Hz to 1200Hz from the electromagnetic excitation force spectrum as high-risk frequency bands to be avoided. Based on the preset deviation margin between the winding's natural frequency and the high-risk frequency bands, establish the optimization target values ​​for each order of the winding's natural frequency. S33: Calculate the electromagnetic force synthesis coefficient and establish the optimization weight coefficient. Calculate the synthesis coefficient of each electromagnetic excitation force based on the current harmonic amplitude. Extract the synthesis coefficient corresponding to the high-risk frequency band corresponding to each natural frequency. Calculate the proportion of each synthesis coefficient to the total synthesis coefficient. Use the proportion as the optimization weight coefficient of each natural frequency. S34: Output the optimization target value and optimization weight coefficient. Combine the optimization target values ​​of each order of natural frequency established in step S32 and the optimization weight coefficients of each order established in step S33 and output them as the basis for calculating the weighted frequency deviation function in the optimization model.

5. The dual-objective vibration reduction optimization method for converter transformer windings according to claim 1, characterized in that, S4 specifically includes the following steps: S41: Define the decision variables and objective function forms. Use the key structural parameters selected in step S23 as decision variables. Determine that the bi-objective optimization model contains two objective functions. The first objective function is the weighted frequency deviation function, and the second objective function is the resonance penalty function. S42: Construct a weighted frequency deviation function, calculate the natural frequencies of each order based on the constructed second-order response surface model, and construct a weighted frequency deviation function based on the established optimization target value and optimization weight coefficients. The weighted frequency deviation function calculates the sum of the square of the difference between each natural frequency and the corresponding optimization target value multiplied by the corresponding weight coefficient. S43: Construct a resonance penalty function, define the minimum deviation function from the set of electromagnetic excitation force frequencies of each natural frequency, construct a piecewise potential energy function based on the minimum deviation, and use the sum of the piecewise potential energy functions of each order as the resonance penalty function. The piecewise potential energy function adopts different linear expressions according to the interval where the minimum deviation is located. S44: Apply constraints and output a bi-objective optimization model. Set upper and lower limits for the values ​​of each decision variable, limit the number of pad blocks to integers, and combine the weighted frequency deviation function and the resonance penalty function into a bi-objective optimization model.

6. The dual-objective vibration reduction optimization method for converter transformer windings according to claim 1, characterized in that, S5 specifically includes the following steps: S51: Initialize the population and algorithm parameters, set the population size, maximum number of iterations, crossover probability and mutation probability, encode the decision variables in the output bi-objective optimization model, and randomly generate the initial population within the range of the decision variables, wherein the number of pad blocks is taken as an integer value during initialization; S52: Perform non-dominated sorting and crowding calculation. For each individual in the current population, calculate two objective function values ​​based on the constructed weighted frequency deviation function and the constructed resonance penalty function. Sort the individuals in the population non-dominated according to the objective function values, determine the non-dominated level of each individual, and calculate the crowding distance of individuals within the same non-dominated level. S53: Perform selection, crossover, and mutation operations. Use tournament selection to select parent individuals from the current population. Perform simulated binary crossover on the parent individuals according to the set crossover probability to generate offspring individuals. Perform polynomial mutation on the offspring individuals according to the set mutation probability. The mutation result of the pad block number in the mutation operation is rounded down. S54: Merge the populations and generate a new generation population. Merge the parent and child populations. Perform non-dominated sorting and crowding calculation on the merged population. Select individuals into the new generation population in order of non-dominated level from low to high. When individuals of a certain level cannot be selected, select individuals from the level in order of crowding distance from large to small until the size of the new generation population reaches the set population size, or iterate through steps S52 to S54 until the maximum number of iterations is reached. Output the non-dominated individuals in the final population as the Pareto optimal solution set.

7. The dual-objective vibration reduction optimization method for converter transformer windings according to claim 1, characterized in that, S6 specifically includes the following steps: S61: Filter Pareto optimal solutions. From the output Pareto optimal solution set, extract the decision variable values ​​of all non-dominated individuals. Each decision variable corresponds to a set of numerical combinations of turn spacing, turn diameter, number of pad blocks, and width of pad blocks. S62: Extract the optimized key structural parameters. From the Pareto optimal solutions selected in step S61, select the solution with the smallest sum of weighted frequency deviation function value and resonance penalty function value. The values ​​of turn spacing, wire diameter, number of pads, and width of pads in this solution are used as the optimized key structural parameters. S63: Verify the integer constraint condition. Check whether the number of pad blocks extracted in step S62 is an integer. If it is an integer, output the key structural parameters directly. If it is not an integer, round the number of pad blocks to the nearest integer before outputting.

8. The dual-objective vibration reduction optimization method for converter transformer windings according to claim 1, characterized in that, S7 specifically includes the following steps: S71: Obtain the optimized key structural parameters, and read the values ​​of inter-turn spacing, wire diameter, number of spacers, and width of spacers from the output optimized key structural parameters; S72: Substitute the values ​​of inter-turn spacing, wire diameter, number of pads, and width of pads read in step S71 into the constructed low-order and high-order response surface models respectively to calculate the optimized first five natural frequencies of the winding. S73: Output vibration reduction design scheme. The values ​​of inter-turn spacing, wire diameter, number of pads, and width of pads read in step S71 are combined with the first five natural frequencies calculated in step S72 to output the vibration reduction design scheme of the converter transformer winding.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the steps of the dual-objective vibration reduction optimization method for converter transformer windings as described in any one of claims 1 to 8.

10. A storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the steps of the dual-objective vibration reduction optimization method for converter transformer windings as described in any one of claims 1 to 8.

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