Method for simulating and optimizing mechanical properties of photovoltaic support structure

Abstract

The application discloses a kind of mechanical property simulation and optimization design method of photovoltaic support structure, belong to the structural design field of photovoltaic power generation system.The application is in view of the problem that existing design wind vibration response simulation is not accurate and optimization efficiency is low, first constructs the three-dimensional and microscale dynamic wind field model of photovoltaic support and carries out fluid-solid coupling simulation, exports the displacement-time curve of key node;Modal decomposition is carried out to displacement curve and the time history of generalized coordinate of dominant mode is extracted, establishes its time sequence mapping relationship with wind pressure load, constructs wind vibration response reduced model characterized by reduced modal coordinate;With the minimum displacement peak value predicted by reduced model and the minimum total weight of support as double target, with component stress ratio as constraint, automatically optimized to generate Pareto solution set using multi-objective evolutionary algorithm;Simulation verification is carried out to selected solution set scheme, and the reduced model is corrected according to error;Repeat iteration until convergence and output final optimization design parameters.

Description

Technical Field

[0001] This invention relates to the field of photovoltaic power generation technology. More specifically, this invention relates to a method for simulating and optimizing the mechanical properties of photovoltaic support structures. Background Technology

[0002] As a crucial component of photovoltaic (PV) power generation systems, the structural safety and economic efficiency of PV support structures directly impact the long-term stable operation and return on investment of PV power plants. PV support structures typically consist of components such as columns, beams, and purlins, and are exposed to the outdoor environment for extended periods, bearing loads including their own weight, wind loads, and snow loads, with wind loads often being the controlling factor. Because PV modules are usually installed at significant tilt angles, the support structure is highly sensitive to wind loads, especially in areas prone to strong winds or typhoons, where wind-induced vibrations can lead to fatigue damage, loosening of connections, or even complete failure of the support structure. Therefore, accurately simulating and evaluating the mechanical performance of PV support structures under wind loads and conducting reasonable structural optimization design are of significant engineering importance.

[0003] However, in current engineering practice, the structural design and analysis of photovoltaic (PV) support structures still face several technical challenges. First, the accuracy of wind load simulation needs improvement. Traditional design methods often rely on building structural load codes, using average wind pressure or quasi-static wind loads for calculations, which struggle to reflect the spatiotemporal variations of actual wind fields, especially the microscale effects of the wind field around the support structure. PV supports are typically located in open areas or on rooftops, with relatively low heights but wide distribution areas. The wind field is significantly affected by ground roughness, topography, and module tilt angles, resulting in significant local turbulence and eddy current shedding, leading to uneven wind pressure distribution on the support surface that changes rapidly over time. Simplified wind load models cannot accurately capture these complex wind-induced response characteristics, thus affecting the reliability of structural safety assessments.

[0004] Secondly, while fluid-structure interaction (FSI) simulations can accurately simulate the interaction between structures and wind fields, their high computational cost makes them difficult to directly apply to optimization designs. Photovoltaic support structures often contain numerous members and nodes. Establishing a detailed three-dimensional finite element model and performing time-domain FSI analysis requires significant computational resources and time, especially when considering multiple load conditions and parameter combinations, where the computational load increases exponentially. This difficulty forces designers to select only a limited number of typical load conditions for verification during the design phase, making it difficult to comprehensively examine the dynamic response of the structure under wind loads and to fully optimize parameters such as cross-sectional dimensions and support arrangements.

[0005] In modal analysis and response reconstruction, traditional methods often employ the modal superposition method for the entire structure. However, for structures like photovoltaic supports, which exhibit dense modes and local vibration characteristics, low-order modes may not fully reflect the main contribution of wind-induced vibration response, and neglecting high-order modes can lead to distorted response predictions. Furthermore, the mapping relationship between wind load and structural response is complex, making it difficult to establish a simple and effective reduced-order model, thus lacking a reliable theoretical foundation for rapid prediction and reconstruction of wind-induced vibration response.

[0006] In the structural optimization design phase, traditional photovoltaic (PV) support designs often rely on experience or simple parameter verification, lacking a systematic multi-objective optimization method. Designers typically select component cross-sections based on experience from similar projects, then verify them through finite element analysis. If the strength or stiffness is insufficient, the cross-sectional dimensions are increased. This trial-and-error method is inefficient and struggles to obtain a globally optimal solution. Because there is a contradictory relationship between wind-induced vibration response and structural weight, simply pursuing lightweighting may lead to excessive wind-induced displacement, affecting module safety and power generation efficiency; conversely, excessively pursuing stiffness increases material usage and cost. How to achieve a comprehensive optimization of peak displacement and structural weight while meeting stress ratio limits is a pressing problem in engineering design.

[0007] Furthermore, verifying the reliability of optimization results is also a weak link in the design process. Existing optimization designs are mostly based on simplified models or empirical formulas, and the optimization results often lack verification through high-precision simulations. Once there is a deviation between the simplified model and the actual physical process, the actual performance of the optimized solution may differ significantly from expectations, posing risks of safety hazards or material waste. Especially for complex dynamic loads such as wind loads, the errors in the simplified model may be amplified, leading to design failure.

[0008] In summary, existing photovoltaic support design methods have shortcomings in terms of wind load simulation accuracy, computational efficiency, modal contribution identification, multi-objective optimization, and result verification. They are difficult to balance the requirements of structural safety and economy, and there is an urgent need for a technical solution that can perform mechanical performance simulation and optimization design efficiently and accurately. Summary of the Invention

[0009] One object of the present invention is to solve at least the above-mentioned problems and to provide at least the advantages that will be described later.

[0010] Another objective of this invention is to provide a method for simulating and optimizing the mechanical properties of photovoltaic support structures. This method addresses the technical problems in existing technologies, such as insufficient accuracy in wind load simulation, difficulty in reflecting the time-varying characteristics of actual microscale wind fields, high computational cost of fluid-structure interaction simulation making it difficult to directly use for optimization iteration, inability of constant-scale mode analysis to accurately identify dominant modes and establish a simple mapping relationship between wind load and response, difficulty of achieving multi-objective comprehensive optimization of peak displacement and structural weight under stress ratio constraints using traditional trial-and-error design methods, and insufficient reliability of design schemes due to lack of high-precision simulation verification of optimization results and inability to correct model deviations in a timely manner.

[0011] To achieve these objectives and other advantages of the present invention, a method for simulating and optimizing the mechanical properties of a photovoltaic support structure is provided, comprising the following steps: S1. Construct a three-dimensional geometric model and a microscale dynamic wind field model of the photovoltaic support structure, and perform fluid-structure interaction simulation calculations to output the dynamic response of the support structure in the time domain. The dynamic response should include at least the displacement-time curves of the key nodes. S2. Perform modal decomposition on the displacement-time curve output in step S1, identify the top N dominant modes that contribute the most to the wind vibration response, and extract the generalized coordinate time history of each dominant mode. S3. Establish the temporal mapping relationship between the generalized coordinates of each dominant mode and the wind pressure load, and construct a reduced-order model of the wind vibration response of the support structure characterized by the reduced-order dominant mode coordinates. S4. With minimizing the peak displacement response predicted by the reduced-order model and minimizing the total weight of the support as optimization objectives, and with the stress ratio of each component not exceeding the allowable value as a constraint, a multi-objective evolutionary algorithm is used to automatically optimize the cross-sectional dimensions of the components and generate a Pareto optimal solution set. S5. Select candidate solutions from the Pareto optimal solution set to update the geometric model. Repeat step S1 to perform fluid-structure interaction simulation verification and determine whether the error between the simulation results and the predicted values ​​of the reduced-order model exceeds the preset threshold. If so, add the current simulation data to the training set to correct the reduced-order model. S6. Repeat steps S2 to S5 until the convergence condition is met, and output the final optimized photovoltaic support structure design parameters.

[0012] Preferably, the mechanical performance simulation and optimization design method for the photovoltaic support structure includes, in step S1, constructing a microscale dynamic wind field model, specifically comprising: S101. Obtain the elevation data and surface roughness length of the terrain where the photovoltaic support is located, and generate the computational domain grid. S102. Using the large eddy simulation method, the fluctuating wind speed time history is generated at the entrance of the computational domain (based on the Davenport wind speed spectrum); S103. Measure the current height H of the photovoltaic support above the ground and the spacing d between it and the adjacent support, and calculate the ratio of spacing to height d / H; S104. When d / H < 5, the additional turbulence intensity correction is calculated using the following formula: ; S105. The additional turbulence intensity calculated in step S104. I add It is superimposed on the pulsating wind speed time history generated in step S102 as the corrected inlet boundary condition.

[0013] Preferably, in the mechanical performance simulation and optimization design method for the photovoltaic support structure, step S2 involves modal decomposition of the displacement-time curve to identify the dominant mode, specifically including: S201. Set the penalty factor α=2000 for the variational mode decomposition algorithm and the preset number of mode decompositions K=10; S202. Input the displacement-time curve x(t) of the key node into the variational mode decomposition algorithm, and output 10 eigenmode function components. u k (t) , k =1,2,...,10; S203. Calculate the energy contribution rate of each intrinsic mode function component using the following formula: ; in, E k For the first k Energy contribution rate of the first mode, expressed as a percentage; u k ( t ) is the first k The intrinsic mode function components, in meters; t is time, in seconds; ∫ represents integration over time. S204. For each intrinsic modal function component, calculate its coherence function γ with the wind pressure load time history p(t). k(f) The maximum value of the coherence function is extracted and denoted as the peak coherence coefficient γ. k,max ; S205. Determine whether each mode simultaneously satisfies E. k >5% and γ k,max >0.6; S206. Sort the modes that satisfy the conditions in step S205 from high to low according to their energy contribution rate, select the top N as the dominant modes, and output the corresponding generalized coordinate time history. q i (t) ,ini =1,2,...,N.

[0014] Preferably, in the mechanical performance simulation and optimization design method for the photovoltaic support structure, step S3 involves constructing a reduced-order model of the support structure's wind vibration response characterized by reduced-order dominant mode coordinates, specifically including: S301. A complete candidate function library Φ has been constructed, containing the following five types of elements: ; in, q i For the first i Generalized coordinates of order, in meters; express q i The first derivative with respect to time, in m / s; q i 3 express q i cubed, unit is m 3 ; q i 2 q j express q i The square of the first j generalized coordinates q j The unit is m 3 ; p ( t () represents the wind pressure load time history, in Ps; S302, the first i The second derivative of the generalized coordinate with respect to time is expressed as: ; in, express q i The second derivative with respect to time, in m / s 2 ; x The vector of sparse coefficients to be determined is given, and the dimensions of each component are determined according to the corresponding terms. S303, Solved using the sequence threshold Ridge regression algorithm. x Set a threshold for regression residuals e =10 -3 The regression residuals will be less than e The coefficients of are retained, and the remaining coefficients are set to zero; S304. Based on the retained coefficients, output the following differential governing equations: ; in, gi For the first i The damping ratio of the first mode is dimensionless; oh i For the first i The natural frequency of the first mode, expressed in rad / s; β ij The nonlinear coupling coefficient is expressed in units of 1 / (m). 2 ⋅s 2 ); c i The wind pressure participation factor is expressed in units of 1 / (kg⋅m). S305. Assign the coefficients retained in step S303 to the corresponding physical parameters: and q ˙ i The coefficients corresponding to the terms are used to calculate the damping ratio. g i ,and q i The coefficients corresponding to the terms are used to calculate the natural frequencies. oh i ,and q i 3 The coefficients corresponding to the terms are nonlinear coupling coefficients. β ij ,and p ( t The coefficient corresponding to the term is the wind pressure participation factor. c i .

[0015] Preferably, in the mechanical performance simulation and optimization design method for the photovoltaic support structure, step S4 employs a multi-objective evolutionary algorithm to automatically optimize the cross-sectional dimensions of the components, specifically including: S401. Initialize the population P, where each individual in the population corresponds to a set of cross-sectional dimension parameter vectors for the photovoltaic support. X =[x1,x2, ...,x m ] T Where m is the total number of design variables, which is determined according to the structural form of the photovoltaic support; when the photovoltaic support adopts a common structure consisting of columns, beams and diagonal braces, the design variables include at least the column wall thickness, beam flange width and diagonal brace section height, all in millimeters; when the photovoltaic support includes other components, the design variables are increased accordingly by the corresponding cross-sectional dimension parameters. S402. The peak displacement response f1(X) predicted by the reduced-order model is approximated by the Kriging surrogate model, and the total weight f2(X) of the support is directly calculated based on the cross-sectional dimension parameters. S403. Construct the Pareto front F based on the non-dominated solution set of the current generation, and calculate the value of each individual X on the front. iThe minimum Euclidean distance to all other individuals is taken as the discreteness d of that individual. min (X i ); S404. Mark the three individuals with the largest dispersion as filling sampling points, and call the fluid-structure interaction simulation in step S1 to perform real finite element analysis on these individuals to obtain the real displacement response peak value. S405. Add the real analysis results obtained in step S404 to the training sample set, and retrain and update the Kriging proxy model. S406. Repeat steps S403 to S405 until any of the following stopping conditions are met: RMSE < 5% or iter ≥ iter max ; Where RMSE is the root mean square error of the Kriging surrogate model, iter is the current iteration number, and iter max The maximum number of iterations is preset. S407. Output the final Pareto optimal solution set F*, each solution containing a set of cross-sectional dimension parameters X and their corresponding displacement response peak f1(X) and total support weight f2(X).

[0016] Preferably, in the mechanical performance simulation and optimization design method for the photovoltaic support structure, step S5, determining whether the error between the simulation result and the predicted value of the reduced-order model exceeds a preset threshold, specifically includes: S501. Extract the peak displacement response of key nodes in the time domain and record the peak values ​​from the fluid-structure interaction simulation. u sim Peak value predicted by the reduced-order model u pred ; S502. Calculate the relative error using the following formula: ; in, d peak This represents relative error, expressed as a percentage; |⋅| indicates taking the absolute value. S503. Extract the displacement response sequence over the entire time period, and denote them as simulation sequences. u sim and predicted sequence u pred Calculate the dynamic time-normalized distance between two sequences. D dtw ; S504, Set the first threshold T 1 = 15%, second threshold T 2 = 0.05 × max( usim ); S505, if d peak > T 1 and D dtw > T 2. If the error exceeds the limit, proceed to step S506; otherwise, the model is deemed valid, and the current reduced-order model continues to be used. S506, triggers the downgraded model correction procedure.

[0017] Preferably, in the mechanical performance simulation and optimization design method for the photovoltaic support structure, step S5 involves modifying the reduced-order model, specifically including: S507. Compare the wind pressure load time history obtained from the current fluid-structure interaction simulation with the generalized coordinate time history data. Add to the training dataset, where the superscript l Indicates the first l One sample, l =1,2,..., L , L This represents the current total number of samples; S508. Extract the vector of parameters to be corrected in the differential control equation: ; Where, ζ i Dimensionless, ω i The unit is radians per second. β ij The unit is 1 / (meter) 2 ⋅ seconds 2 ), c i The unit is 1 / (kg·m); S509. Constructing the objective function: ; in q i (pred) (t;θ) For the current parameter i The generalized coordinate time history predicted by the model, q i (sim,l) (t) is the th l The first simulation sample i Generalized coordinate time history, unit is meters T represents the total duration of the time series, in seconds; ∫ denotes the integral over time. S510, Solve for J(θ) Minimize parameters θ* The iterative update formula is: ; in, i k For the first k The parameter vector for the next iteration; J Let be the Jacobian matrix, and each element be the partial derivative of the residual with respect to the parameter; r Let be the residual vector, with elements of . ; l The damping factor is dimensionless. I It is the identity matrix; S511, Apply the new parameters obtained from the optimization solution. θ* Replace the corresponding parameters in the original reduced-order model to complete the model correction.

[0018] Preferably, in the mechanical performance simulation and optimization design method for the photovoltaic support structure, step S510, in which the Levenberg-Marquardt algorithm is used to solve for the parameter θ*, further includes: S510a, Update parameter θ in each iteration k+1 Then, the updated parameters are physically validated, specifically including: Determine the updated damping ratio g i ( k+1) Does it satisfy 0 < g i (k+1) <1; if not satisfied, then g i (k+1) Corrected to the preset reasonable range. g min , g max [Inside, among which] g min =0.001, g max =0.1; Determine the updated natural angular frequency oh i (k+1) Compared with theoretical values ​​calculated based on the finite element model oh i FEM Does the relative deviation satisfy: ; in d ω =20%; if not satisfied, cancel the current iteration update, increase the damping factor λ, and resolve the parameter update amount; S510b, Calculate the residual vector r Sensitivity matrix for each parameter to be corrected S Its elements are: ; According to the sensitivity matrix S Identify the factors that have the greatest impact on model prediction error. M These parameters are updated first during subsequent iterations and corrections. M One parameter is used, and the remaining parameters retain their current values.

[0019] The present invention has at least the following beneficial effects: 1. This invention, by constructing a three-dimensional geometric model and a microscale dynamic wind field model and performing fluid-structure interaction simulation, can more accurately simulate the dynamic response of photovoltaic supports under real wind field environments, overcoming the shortcomings of insufficient accuracy in traditional simplified wind load models. By performing modal decomposition on the displacement-time curve and identifying the dominant modes, minor components that contribute little to the wind-induced vibration response are effectively filtered out, reducing model complexity. Based on this, a reduced-order model of the wind-induced vibration response, characterized by reduced-order modal coordinates, is constructed, enabling rapid prediction of wind-induced response and significantly improving computational efficiency. A multi-objective evolutionary algorithm is used for automatic optimization with the dual objectives of minimizing peak displacement and minimizing the total weight of the support, achieving a Pareto optimal solution set that balances safety and economy while satisfying stress ratio constraints. By introducing high-precision fluid-structure interaction simulation to verify the optimization results and correcting the reduced-order model based on errors, a closed-loop iterative optimization process is formed, effectively improving the reliability and engineering applicability of the final design scheme.

[0020] 2. This invention constructs a more targeted microscale dynamic wind field model by incorporating topographic elevation data, surface roughness length, and the ratio of spacing to height between adjacent supports. A large eddy simulation method is used to generate fluctuating wind speed time histories, and an additional turbulence intensity correction is applied to adjust the inlet boundary conditions. This allows for a more accurate reflection of the local turbulence enhancement effect caused by wake interference when photovoltaic arrays are densely arranged. This correction mechanism makes the wind field simulation closer to the actual service environment of photovoltaic supports, especially for arrays with close spacing. It effectively improves the accuracy of wind load input and provides more reliable boundary conditions for subsequent fluid-structure interaction simulations, thereby enhancing the realism and credibility of the overall simulation results.

[0021] 3. This invention employs a variational mode decomposition algorithm to adaptively decompose the displacement-time curve, avoiding the mode aliasing or endpoint effects problems present in traditional Fourier transform or empirical mode decomposition. By calculating the energy contribution rate of each intrinsic mode function and its peak coherence coefficient with the wind pressure load time history, and setting dual screening conditions, the true dominant modes that contribute significantly to the wind-induced vibration response and are strongly correlated with the wind load can be accurately identified, while spurious modes with weak energy or unclear physical meaning are eliminated. This method ensures that only key dynamic features are retained when constructing the subsequent order-reduction model, reducing the model order while preserving core response information, laying a solid foundation for establishing a concise and effective wind-induced vibration response order-reduction model.

[0022] 4. This invention achieves the goal of automatically discovering the system's governing equations from data by constructing an overcomplete candidate function library containing generalized coordinates and their derivatives, nonlinear terms, and wind pressure load terms, and by employing a sequential threshold Ridge regression algorithm to solve for sparse coefficients. The obtained differential governing equations have clear physical meaning, and the coefficients of each term can directly correspond to physical parameters such as damping ratio, natural frequency, nonlinear coupling coefficient, and wind pressure participation factor, making the reduced-order model not only predictive but also interpretable. Compared to traditional black-box surrogate models, this physics-based reduced-order model can more accurately reflect the intrinsic laws of structural dynamic response, has better generalization ability when extrapolating samples, and provides a reliable and rapid evaluation tool for subsequent multi-objective optimization.

[0023] 5. This invention combines a multi-objective evolutionary algorithm with a Kriging surrogate model, employing a discrete-degree-of-dispersion-based filling sampling strategy. During the optimization process, it adaptively selects the most informative individuals for high-precision fluid-structure interaction simulation, effectively balancing computational accuracy and efficiency. By continuously updating the surrogate model and monitoring its root mean square error, the approximate accuracy of the displacement response peak is ensured to gradually improve until the convergence condition is met. This method overcomes the dilemma in traditional optimization where either relying entirely on high-precision simulation leads to excessive computation, or relying entirely on simplified models results in unreliable optimization results. It can obtain a Pareto optimal solution set that truly meets multi-objective requirements within an acceptable computational cost.

[0024] Other advantages, objectives and features of the present invention will become apparent in part from the following description, and in part from those skilled in the art through study and practice of the invention. Detailed Implementation

[0025] The present invention will be further described in detail below with reference to embodiments, so that those skilled in the art can implement it based on the description.

[0026] It should be understood that terms such as “having,” “comprising,” and “including” as used herein do not exclude the presence or addition of one or more other elements or combinations thereof.

[0027] It should be noted that, unless otherwise specified, the experimental methods described in the following implementation plan are all conventional methods, and the reagents and materials described are all commercially available unless otherwise specified.

[0028] This invention provides a method for simulating and optimizing the mechanical properties of photovoltaic support structures, which includes the following steps: S1. Construct a three-dimensional geometric model and a microscale dynamic wind field model of the photovoltaic support structure, and perform fluid-structure interaction simulation calculations to output the dynamic response of the support structure in the time domain. The dynamic response should at least include the displacement-time curves of key nodes. Key nodes refer to representative locations in the photovoltaic support structure that can reflect the overall structural wind vibration response, typically including: the connection point between the column and the foundation, the mid-span point of the beam, the connection point between the brace and the column, and the support point of the photovoltaic module. The selection of key nodes should be based on the following principles: displacement-sensitive points, selecting nodes with large displacement responses in the preliminary static analysis or modal analysis; stress concentration points, selecting structural connections or cross-sectional changes to reflect weak parts of the structure; engineering experience points, combining common failure modes of photovoltaic supports to select locations of key concern in the project; uniform distribution principle, selecting several nodes in different areas of the support (such as the edge, middle, and top) to ensure the representativeness of the response; in practical applications, it is recommended to select no less than 5 key nodes for time history response output to comprehensively evaluate the wind vibration performance of the structure. S2. Perform modal decomposition on the displacement-time curve output in step S1, identify the top N dominant modes that contribute the most to the wind vibration response, and extract the generalized coordinate time history of each dominant mode. S3. Establish the temporal mapping relationship between the generalized coordinates of each dominant mode and the wind pressure load, and construct a reduced-order model of the wind vibration response of the support structure characterized by the reduced-order dominant mode coordinates. S4. With minimizing the peak displacement response predicted by the reduced-order model and minimizing the total weight of the support as optimization objectives, and with the stress ratio of each component not exceeding the allowable value as a constraint, a multi-objective evolutionary algorithm is used to automatically optimize the cross-sectional dimensions of the components and generate a Pareto optimal solution set. S5. Select candidate solutions from the Pareto optimal solution set to update the geometric model. Repeat step S1 to perform fluid-structure interaction simulation verification and determine whether the error between the simulation results and the predicted values ​​of the reduced-order model exceeds the preset threshold. If so, add the current simulation data to the training set to correct the reduced-order model. S6. Repeat steps S2 to S5 until the convergence condition is met, and output the final optimized photovoltaic support structure design parameters. It should be noted that during the iteration process in step S6, the "construction of a reduced-order model" described in step S3 is only performed on the first execution. In subsequent iterations, the model structure is not reconstructed (i.e., the candidate function library and equation form remain unchanged). Instead, the physical parameters in the model are corrected based on the simulation data added in step S5. That is: First iteration: Execute step S3 to identify the model structure and initialize parameters from the initial simulation data; Subsequent iterations: Skip step S3 and directly proceed to steps S4 and S5, using the correction algorithm in step S5 to optimize and update the existing model parameters. This approach ensures the consistency of the model structure while improving model accuracy using continuously accumulated simulation data, avoiding computational waste and model drift problems caused by repeated modeling.

[0029] The convergence condition is satisfied when any of the following conditions are met: Condition 1: In two consecutive iterations, the relative error between the peak displacement response predicted by the reduced-order model and the fluid-structure interaction simulation result is less than 5%; Condition 2: In two consecutive iterations, the change in all design variables in the Pareto optimal solution set is less than 1%; Condition 3: The number of iterations reaches the preset maximum number of iterations (iter) max =30; Condition 4: The change in parameters after the reduction model is less than a preset threshold (e.g., ||θ||). new −θ old ||<10 −3 When any condition is met, the iteration terminates and the current optimal design parameters are output.

[0030] The above technical solution discloses a method for simulating and optimizing the mechanical performance of photovoltaic support structures. Through a series of operations such as constructing a high-precision simulation model, identifying the dominant mode and establishing a reduced-order model, carrying out multi-objective optimization and closed-loop verification correction, the method achieves efficient and accurate analysis and optimization design of photovoltaic support structures under wind load.

[0031] In step S1 of this technical solution, a three-dimensional geometric model and a microscale dynamic wind field model of the photovoltaic support are first constructed, and fluid-structure interaction simulation calculations are performed. The microscale dynamic wind field model refers to a computational model that can accurately describe the wind speed and turbulence characteristics of the local area of ​​the photovoltaic support. Its construction process needs to fully consider the terrain and the mutual interference effects between adjacent supports. Specifically, the elevation data and surface roughness length of the terrain where the photovoltaic support is located are first obtained, and a computational domain mesh containing the support structure is generated. The large eddy simulation method is used to generate a fluctuating wind speed time history at the entrance of the computational domain. To consider the wake interference between adjacent supports, the ground clearance H of the current photovoltaic support and the arrangement spacing d with adjacent supports are measured, and the ratio of spacing to height d / H is calculated. When d / H is less than 5, it indicates that the spacing between adjacent supports is too close, and the wake interference is significant. In this case, an additional turbulence intensity correction Iadd needs to be calculated according to the formula, and this correction is superimposed on the generated fluctuating wind speed time history as the corrected entrance boundary condition. After completing the above settings, perform fluid-structure interaction simulation calculations and output the dynamic response of the support structure in the time domain, including at least the displacement-time curves of key nodes.

[0032] After completing step S1 and obtaining the displacement-time curve, step S2 is performed to decompose the displacement-time curve into modes, identifying the top N dominant modes that contribute the most to the wind-induced vibration response, and extracting the generalized coordinate time histories of each dominant mode. Variational mode decomposition (VMD) is a non-recursive signal decomposition method that can decompose complex signals into several eigenmode functions with specific sparsity characteristics. In practice, the penalty factor α of the VMD algorithm is set to 2,000, and the preset number of mode decompositions K is 10. The displacement-time curves of key nodes are input into the algorithm, which outputs ten eigenmode function components. To select the modes that contribute the most to the wind-induced vibration response, the energy contribution rate of each eigenmode function component needs to be calculated, i.e., the percentage of the total energy of that component. Simultaneously, the coherence function between each component and the wind pressure load time histories is calculated, and its maximum value is extracted and denoted as the peak coherence coefficient. Determine whether each mode simultaneously satisfies the conditions that the energy contribution rate is greater than 5% and the peak coherence coefficient is greater than 0.6. Sort the modes that meet the conditions from high to low according to their energy contribution rate, take the top N as the dominant modes, and output the corresponding generalized coordinate time history.

[0033] After identifying the dominant modes and their generalized coordinate time histories, step S3 establishes the time-series mapping relationship between the generalized coordinates of each dominant mode and the wind pressure load, constructing a reduced-order model of the support structure's wind-induced vibration response characterized by the reduced-order dominant mode coordinates. First, an overcomplete candidate function library is constructed, containing five types of elements: the generalized coordinate itself, the first derivative of the generalized coordinate, the cube of the generalized coordinate, the product of the squares of different-order generalized coordinates with another-order generalized coordinate, and the wind pressure load time history. The second derivative of the i-th-order generalized coordinate with respect to time is expressed as a linear combination of the terms in the aforementioned candidate function library, with the combination coefficients being the sparse coefficient vector to be determined. The sequence threshold Ridge regression algorithm is used to solve for this sparse coefficient vector, setting the regression residual threshold to 10^-9. Coefficients with regression residuals less than this threshold are retained, while the remaining coefficients are set to zero. Based on the retained coefficients, a nonlinear differential control equation is output, which clearly expresses the motion law of the dominant mode. Specifically, the coefficients corresponding to the first derivative term of the generalized coordinates are used to calculate the damping ratio of the modes, the coefficients corresponding to the generalized coordinate terms are used to calculate the natural frequencies, the coefficients corresponding to the cubic term of the generalized coordinates are the nonlinear coupling coefficients, and the coefficients corresponding to the wind pressure load term are the wind pressure participation factors. Thus, a reduced-order model of the wind-induced vibration response of the support structure, characterized by the reduced dominant mode coordinates, is obtained. This model can quickly predict the dynamic response of the structure under different wind loads.

[0034] After constructing the reduced-order model, step S4 proceeds to minimize the peak displacement response predicted by the reduced-order model and the total weight of the support structure as dual optimization objectives, with the stress ratio of each component not exceeding the allowable value as a constraint. A multi-objective evolutionary algorithm is used to automatically optimize the cross-sectional dimensions of the components, generating a Pareto optimal solution set. First, a population is initialized, with each individual in the population corresponding to a set of cross-sectional dimension parameter vectors for the photovoltaic support structure. Design variables include at least the column wall thickness, beam flange width, and brace cross-sectional height. Since directly calling fluid-structure interaction (FSI) simulations for optimization is too costly, this step introduces a Kriging surrogate model to approximate the peak displacement response predicted by the reduced-order model, while simultaneously calculating the total weight of the support structure directly based on the cross-sectional dimension parameters. A Pareto front is constructed based on the current generation's non-dominated solution set. The dispersion of each individual on the front is calculated, and the three individuals with the largest dispersion are marked as filling sampling points. The FSI simulation from step S1 is called to perform real finite element analysis on these individuals to obtain the true peak displacement response. These real analysis results are added to the training sample set, and the Kriging surrogate model is retrained and updated. Repeat the above process of discrete-time sampling and surrogate model update until the root mean square error of the surrogate model is less than 5% or the number of iterations reaches a preset maximum value. The final Pareto optimal solution set output contains a set of cross-sectional dimension parameters and their corresponding peak displacement response and total support weight for each solution.

[0035] After obtaining the Pareto optimal solution set, step S5 selects candidate solutions from the Pareto optimal solution set to update the geometric model. Step S1 is repeated for fluid-structure interaction (FSI) simulation verification, and it is determined whether the error between the simulation results and the predicted values ​​of the reduced-order model exceeds a preset threshold. If so, the current simulation data is added to the training set to correct the reduced-order model. Specifically, candidate solutions are selected from the solution set, the geometric model is updated, and step S1 is repeated for FSI simulation verification. The peak displacement response of key nodes in the time domain is extracted, and the peak values ​​of the FSI simulation and the predicted peak values ​​of the reduced-order model are recorded respectively, and the relative error between the two is calculated. At the same time, the displacement response sequence over the entire time history is extracted, and the dynamic time warping distance between the simulation sequence and the predicted sequence is calculated. The first threshold is set to 15%, and the second threshold is 5% of the simulated displacement peak value. If the relative error of the peak value is greater than the first threshold and the dynamic time warping distance is greater than the second threshold, the error of the reduced-order model is determined to be out of limit, and the correction procedure needs to be triggered. The correction procedure of the reduced-order model is a data-driven parameter optimization process. The wind pressure load time history and generalized coordinate time history data obtained from the current FSI simulation are added to the training dataset. Extract the parameter vector to be corrected from the differential control equations, including damping ratio, natural frequency, nonlinear coupling coefficient, and wind pressure participation factor. Construct an objective function, whose physical meaning is the sum of squared cumulative errors between the generalized coordinate time history predicted by the model under the current parameters and the generalized coordinate time history of all simulation samples. Solve for the parameter vector that minimizes this objective function, using the Levenberg-Marquardt algorithm for iterative updates. After each iteration of parameter updates, the physical validity of the updated parameters needs to be verified. Verification includes determining whether the updated damping ratio is within a reasonable range of zero to one, and whether the relative deviation between the updated natural frequency and the theoretical value calculated based on the finite element model exceeds 20%. If the verification fails, cancel the iteration update and resolve the algorithm with adjusted parameters. In addition, to improve the correction efficiency, calculate the sensitivity matrix of the residual vector to each parameter to be corrected, identify the top M parameters that have the greatest impact on the model prediction error, and prioritize updating these parameters in subsequent iterations, while keeping the remaining parameters unchanged. Replace the corresponding parameters in the original reduced-order model with the new parameters obtained from the optimized solution, thus completing one model correction.

[0036] Finally, step S6 repeats steps S2 to S5 until the convergence condition is met, outputting the final optimized photovoltaic support structure design parameters. This closed-loop iterative process ensures the accuracy of the reduced-order model during the optimization process, enabling the final output design parameters to possess both high confidence and high performance.

[0037] This technical solution organically combines high-precision simulation, dominant modal identification, reduced-order model construction, multi-objective optimization, and closed-loop verification and correction to form a complete technical solution. It solves a series of technical problems in existing technologies, such as insufficient accuracy of wind load simulation, high computational cost, inaccurate modal analysis, difficulty in multi-objective optimization, and low reliability of results.

[0038] In another technical solution, the method for simulating and optimizing the mechanical performance of the photovoltaic support structure, step S1, involves constructing a microscale dynamic wind field model, specifically including: S101. Obtain the elevation data and surface roughness length of the terrain where the photovoltaic support is located, and generate the computational domain grid. S102. Using the large eddy simulation method, the fluctuating wind speed time history is generated at the entrance of the computational domain; S103. Measure the current height H of the photovoltaic support above the ground and the spacing d between it and the adjacent support, and calculate the ratio of spacing to height d / H; S104. When d / H < 5, the additional turbulence intensity correction is calculated using the following formula: ; S105. The additional turbulence intensity calculated in step S104. I add It is superimposed on the pulsating wind speed time history generated in step S102 as the corrected inlet boundary condition.

[0039] In step S1 of the present invention, constructing a microscale dynamic wind field model is the basis for realizing high-precision fluid-structure interaction simulation. The above technical solution further defines the specific implementation method of constructing the microscale dynamic wind field model. Its purpose is to more accurately simulate the actual wind field characteristics around the photovoltaic support, especially considering the topography and the mutual interference effect between adjacent supports.

[0040] In specific implementation, step S101 first acquires the elevation data and surface roughness length of the terrain where the photovoltaic support is located. The elevation data can be obtained from a geographic information system, UAV aerial surveying, or field measurements, and its spatial resolution should be sufficient to reflect the main undulations of the terrain. The surface roughness length is a parameter describing the obstruction effect of the ground on airflow, and its value can be determined according to the surface type (e.g., grassland, farmland, sparse towns, etc.) with reference to relevant standards or literature. Based on the acquired elevation data and roughness length, a computational domain mesh containing the photovoltaic support structure is generated. When meshing, local refinement should be performed in the area near the support to accurately capture boundary layer flow and vortex structures around the support; a sparser mesh can be used in areas far from the support to save computational resources. The size of the computational domain should be large enough to avoid adverse effects of boundary conditions on the flow field near the support. It is generally recommended that the distance from the inlet to the front end of the support, the distance from the outlet to the rear end of the support, and the distances from the sides and top to the support should all be no less than 5 to 10 times the height of the support.

[0041] Step S102 employs the Large Eddy Simulation (LES) method to generate the fluctuating wind speed time history at the inlet of the computational domain. LES is a numerical simulation method for turbulence, its basic idea being to directly solve the equations of motion for large-scale eddies, while using a subgrid model to model small-scale eddies. Compared to the Reynolds-averaged Navier-Stokes method, LES can more accurately reflect the fluctuating characteristics of turbulence, showing significant advantages for simulating dynamic loads such as wind loads. The generation of the inlet fluctuating wind speed time history can be achieved as follows: first, determine the average wind speed profile, typically described using exponential or logarithmic laws; then, superimpose fluctuating components onto the average wind speed. The generation of these fluctuating components can be achieved using synthesis methods based on turbulence spectra, such as the Davenport spectrum, Kaimal spectrum, or von Kármán spectrum. For the LES inlet boundary, a fluctuating velocity field satisfying spatial correlation also needs to be generated, which can be achieved using broadband random noise synthesis or eddy scale superposition methods.

[0042] Step S103 measures the current photovoltaic support's height H above the ground and its spacing d with adjacent supports, and calculates the ratio of spacing to height, d / H. The height H above the ground refers to the vertical distance from the lowest point of the support to the ground, typically taken as the height from the bottom of the support column to the crossbeam or the lower edge of the module. The spacing d between adjacent supports refers to the center-to-center distance or net distance between adjacent supports in the same row, and should be determined based on the actual arrangement of the supports. The d / H ratio reflects the relative distance between adjacent supports in the height direction and is an important parameter for judging whether the wake interference effect is significant.

[0043] Step S104 determines whether additional turbulence intensity correction needs to be considered based on the d / H ratio. When d / H < 5, it indicates that the distance between adjacent supports is relatively short, and the wake generated by the upstream support will significantly affect the wind field at the downstream support. In this case, additional turbulence intensity correction needs to be considered. Additional turbulence intensity I add It can be calculated using the following formula: ; Where α is an empirical coefficient, determined based on the arrangement and tilt angle of the photovoltaic array, typically ranging from 0.1 to 0.3; d is the spacing between adjacent supports; and H is the height of the supports above the ground. The physical meaning of this formula is: the smaller d / H is, the closer the supports are, the stronger the wake interference, and the greater the additional turbulence intensity; when d / H approaches 0, the additional turbulence intensity approaches α; and when d / H = 5, the additional turbulence intensity drops to 0. It should be noted that the above formula is only an exemplary implementation; in practical applications, other modified formulas can be obtained by fitting wind tunnel test data or field measurement data.

[0044] Step S105 calculates the additional turbulence intensity I. addThis is superimposed onto the fluctuating wind speed time history generated in step S102 as the corrected inlet boundary condition. The specific implementation of the superposition is as follows: keeping the average wind speed constant, multiply the root mean square value of the fluctuating wind speed component by (1+1...). add / I0), where I0 is the background turbulence intensity at the inlet without correction; or the target turbulence intensity can be set directly to I0+I in the synthesized fluctuating wind speed time history. add This correction enhances the turbulence intensity at the inlet, thus reflecting the disturbance effect of the upstream support wake on the downstream support wind field.

[0045] Through steps S101 to S105 above, this invention achieves a refined construction of a microscale dynamic wind field model for photovoltaic supports. The beneficial effects of this construction method are as follows: First, by introducing terrain elevation and surface roughness data, the wind field model can reflect the influence of actual terrain on airflow; second, the use of large eddy simulation (LES) to generate fluctuating wind speed time histories can accurately simulate the spatiotemporal evolution characteristics of turbulence; third, by introducing an additional turbulence intensity correction based on the d / H ratio, the wake interference effect between adjacent supports is considered, making the wind field model closer to the actual arrangement of the photovoltaic array; finally, the corrected turbulence intensity is superimposed on the inlet boundary conditions, providing a more physically realistic wind load input for subsequent fluid-structure interaction simulations. Compared with existing methods using average wind pressure or simple turbulence models, the microscale dynamic wind field model of this invention can more accurately simulate the wind loads experienced by photovoltaic supports in the actual environment, laying a solid foundation for structural mechanical performance analysis and optimization design.

[0046] In another technical solution, the mechanical performance simulation and optimization design method for the photovoltaic support structure, step S2, involves modal decomposition of the displacement-time curve to identify the dominant mode, specifically including: S201. Set the penalty factor α=2000 for the variational mode decomposition algorithm and the preset number of mode decompositions K=10; S202. Input the displacement-time curve x(t) of the key node into the variational mode decomposition algorithm, and output 10 eigenmode function components. u k (t) , k =1,2,...,10; S203. Calculate the energy contribution rate of each intrinsic mode function component using the following formula: ; in, E k For the first k Energy contribution rate of the first mode, expressed as a percentage; u k ( t) is the first k The intrinsic mode function components, in meters; t is time, in seconds; ∫ represents integration over time. S204. For each intrinsic modal function component, calculate its coherence function γ with the wind pressure load time history p(t). k(f) The maximum value of the coherence function is extracted and denoted as the peak coherence coefficient γ. k,max ; S205. Determine whether each mode simultaneously satisfies E. k >5% and γ k,max >0.6; S206. Sort the modes that satisfy the conditions in step S205 from high to low according to their energy contribution rate, select the top N as the dominant modes, and output the corresponding generalized coordinate time history. q i (t) ,in i =1,2,...,N.

[0047] In step S2 of this invention, modal decomposition of the displacement-time curve and identification of the dominant mode are key prerequisites for building a reduced-order model. The above technical solution further defines the specific implementation of the modal decomposition and dominant mode identification. Its purpose is to accurately extract the vibration component that contributes the most to wind vibration and is most correlated with wind load from the complex structural vibration response, so as to provide effective generalized coordinate input for the subsequent reduced-order model.

[0048] In specific implementation, step S201 first sets the parameters of the variational mode decomposition algorithm. Variational mode decomposition is a non-recursive signal decomposition method. Its core idea is to decompose the original signal into several eigenmode functions with specific center frequencies and finite bandwidths by solving a variational problem. To achieve accurate decomposition, two key parameters need to be set: a penalty factor α and a preset number of mode decompositions K. In this embodiment, the penalty factor α is set to 2000. This parameter controls the bandwidth of each mode. The larger α is, the narrower the bandwidth of each mode and the higher the frequency resolution. The preset number of mode decompositions K is set to 10. This parameter represents the total number of modes expected to be decomposed. It should be noted that the above parameter values ​​can be adjusted according to the complexity of the actual signal. For structures with relatively simple vibration patterns, the value of K can be appropriately reduced to improve computational efficiency; for structures with complex vibration patterns, the value of K can be appropriately increased to avoid mode aliasing.

[0049] Step S202 inputs the displacement-time curve x(t) of the key node into the variational mode decomposition algorithm, which outputs 10 intrinsic mode function components u. k(t), k=1,2,…,10. The intrinsic mode functions (EMFs) are functions that satisfy the following two conditions: throughout the entire data sequence, the number of extrema is equal to or at most differs by one from the number of zero-crossings; at any point, the mean of the upper envelope defined by local maxima and the lower envelope defined by local minima is zero. Each EMF component represents a vibrational component of a specific frequency band in the original signal, reflecting the vibrational characteristics of the structure within that frequency band.

[0050] Step S203: Calculate the energy contribution rate E of each intrinsic mode function component. k The energy contribution rate refers to the percentage of energy of a particular modal component relative to the total energy of all modal components, reflecting the importance of that mode in the overall vibration response. The specific calculation method is as follows: ; Among them, ∫u k 2 (t) dt represents the energy of the k-th eigenmode function component in the time domain, in meters. 2 · second (m) 2 ·s); the denominator is the sum of the energies of all modal components; E k The unit is percentage (%). The higher the energy contribution rate, the more significant the contribution of the mode to the structural vibration.

[0051] Step S204: Calculate the coherence function γ between each intrinsic mode function component and the wind pressure load time history p(t). k (f), and extract its maximum value as the peak coherence coefficient γ. k,max The coherence function is an index describing the degree of correlation between two signals in the frequency domain. It is defined as the square of the magnitude of the cross-power spectrum of the two signals divided by the product of their individual power spectra, and its value ranges from 0 to 1. The closer the coherence function value is to 1, the stronger the correlation between the two signals at that frequency; the closer it is to 0, the weaker the correlation. For each intrinsic mode function component u... k γ(t) is used to calculate its coherence function with the wind pressure load time history p(t). k (f) yields a curve that varies with frequency, and the maximum value of this curve is extracted as γ. k,max The peak coherence coefficient reflects the maximum correlation between this mode and wind load across the entire frequency domain.

[0052] Step S205 determines whether each mode satisfies the dual screening condition. Specifically, it determines whether each mode simultaneously satisfies the energy contribution rate E. k >5% and peak coherence coefficient γ k,maxModes with an energy contribution rate greater than 5% have a non-negligible contribution to structural vibration; modes with a peak coherence coefficient greater than 0.6 are strongly correlated with wind load. Modes that simultaneously meet both conditions not only make a significant contribution to the structural response but are also closely related to wind load, and are the main components constituting the wind-induced vibration response.

[0053] Step S206 sorts the modes that satisfy the conditions of step S205 from high to low according to their energy contribution rate, selects the top N as the dominant modes, and outputs the corresponding generalized coordinate time history q. i (t), where i = 1, 2, ..., N. The generalized coordinate time history is the intrinsic mode function component itself corresponding to the dominant mode, describing the law of change of the amplitude of the dominant mode with time. The value of N can be determined according to actual needs, and usually the first 2 to 5 modes can cover most of the response energy. For example, if there are 5 modes that meet the conditions, the first 3 with the largest energy contribution rate can be selected as the dominant modes; if the number of modes that meet the conditions is less than the preset value of N, then the actual number of modes that meet the conditions shall prevail.

[0054] Through steps S201 to S206 above, this invention achieves accurate modal decomposition and dominant mode identification of displacement-time curves. The beneficial effects of this implementation are as follows: First, by employing a variational modal decomposition algorithm, the modal aliasing problem commonly found in empirical modal decomposition can be effectively avoided, resulting in eigenmode functions with clearer spectral separation. Second, by introducing dual screening conditions of energy contribution rate and peak coherence coefficient, the dominance of modes is comprehensively evaluated from two dimensions: energy importance and load correlation, avoiding the one-sidedness of relying solely on energy or correlation. Third, by sorting by energy contribution rate and selecting the top N modes, it ensures that the reduced-order model can capture as much response information as possible with as few modes as possible, laying a solid foundation for the subsequent construction of a concise and effective reduced-order model. Compared with existing methods that directly use the first few intrinsic modes or empirically select modes, the modal identification method of this invention is more scientific and targeted, and can more accurately characterize the vibration characteristics of photovoltaic supports under wind loads.

[0055] In another technical solution, the mechanical performance simulation and optimization design method for the photovoltaic support structure, step S3, involves constructing a reduced-order model of the support structure's wind vibration response characterized by reduced-order dominant mode coordinates, specifically including: S301. A complete candidate function library Φ has been constructed, containing the following five types of elements: ; in, q i For the first i Generalized coordinates of order, in meters; express q iThe first derivative with respect to time, in m / s; q i 3 express q i cubed, unit is m 3 ; q i 2 q j express q i The square of the first j generalized coordinates q j The unit is m 3 ; p ( t () represents the wind pressure load time history, in Ps; S302, the first i The second derivative of the generalized coordinate with respect to time is expressed as: ; in, express q i The second derivative with respect to time, in m / s 2 ; x The vector of sparse coefficients to be determined is given, and the dimensions of each component are determined according to the corresponding terms. S303, Solved using the sequence threshold Ridge regression algorithm. x Set a threshold for regression residuals e =10 -3 The regression residuals will be less than e The coefficients of are retained, and the remaining coefficients are set to zero; S304. Based on the retained coefficients, output the following differential governing equations: ; in, g i For the first i The damping ratio of the first mode is dimensionless; oh i For the first i The natural frequency of the first mode, expressed in rad / s; β ij The nonlinear coupling coefficient is expressed in units of 1 / (m). 2 ⋅s 2 ); c i The wind pressure participation factor is expressed in units of 1 / (kg⋅m). S305. Assign the coefficients retained in step S303 to the corresponding physical parameters: and q ˙ iThe coefficients corresponding to the terms are used to calculate the damping ratio. g i ,and q i The coefficients corresponding to the terms are used to calculate the natural frequencies. oh i ,and q i 3 The coefficients corresponding to the terms are nonlinear coupling coefficients. β ij ,and p ( t The coefficient corresponding to the term is the wind pressure participation factor. c i .

[0056] In step S3 of this invention, establishing the temporal mapping relationship between the generalized coordinates of the dominant mode and the wind pressure load, and constructing a wind vibration response reduction model characterized by the reduced dominant mode coordinates, is the core link to achieve efficient optimization iteration. The above technical solution further defines the specific construction method of the reduction model. Its purpose is to automatically discover the control equations of the system from the simulation data, obtain a reduction model that is both concise and retains key physical characteristics, and provide a fast-response calculation tool for subsequent multi-objective optimization.

[0057] In its implementation, step S301 first constructs a complete candidate function library Φ. This library contains all candidate function terms that may describe the dynamic behavior of the system. Its design philosophy is to cover linear terms, nonlinear terms, and external excitation terms to fully express the various mapping relationships that may exist between generalized coordinates and wind pressure loads. Specifically, the candidate function library Φ contains the following five types of elements: The generalized coordinate itself q i : The displacement-related force of the corresponding system; First derivative of generalized coordinates with respect to time : The velocity-dependent force of the corresponding system, i.e., the damping force; cubed q of generalized coordinates i 3 : Corresponds to the nonlinear restoring force of the system, used to describe the nonlinear characteristics of stiffness; The product q of the squares of different orders of generalized coordinates and another order of generalized coordinates. i 2 q j (When i≠j or i=j, it degenerates into) q i 3 ): This corresponds to the nonlinear coupling effect between different modes; Wind pressure load time history p(t): corresponds to the effect of external excitation on the system.

[0058] In the above items, q iand The dimensions of q are meters (m) and meters per second (m / s), respectively; i 3 Both and have dimensions of m. 3 The dimension of p(t) is Pascal (Pa). By constructing such a comprehensive function library, a wealth of candidate basis functions are provided for subsequent identification of system control equations from data.

[0059] Step S302 calculates the second derivative of the i-th generalized coordinate with respect to time. This is represented as a linear combination of all terms in the candidate function library. The mathematical expression is: in, This represents the second derivative of the i-th generalized coordinate with respect to time, i.e., the generalized acceleration, with dimensions in meters per second. 2 (m / s) 2 ); Φ is the candidate function library row vector constructed in step S301, containing the values ​​of all candidate function terms at the current time; ξ is the sparse coefficient vector to be determined, the dimensions of each component are determined according to the corresponding candidate function terms, such that the dimensions of the combined coefficients are consistent with those of the candidate function terms. Consistent. The physical meaning of this expression is that it assumes the dynamic behavior of the system can be described by a linear combination of several terms in the candidate function library, while the coefficient vector ξ represents the contribution weight of each term to the system acceleration.

[0060] Step S303 uses the Sequence Thresholding Ridge Regression algorithm to solve for the sparse coefficient vector ξ. Sequence Thresholding Ridge Regression is a sparse identification algorithm that combines the regularization properties of Ridge Regression with a threshold operation. Its specific implementation process is as follows: First, Ridge Regression is used to solve for the initial coefficient estimation. Ridge Regression, by adding an L2 regularization term to the loss function, can handle the multicollinearity problem among candidate function terms; then, the regression residual threshold ε is set to 10. −3 The algorithm retains coefficients whose regression residuals are less than a threshold, and sets the remaining coefficients to zero; it repeats this process until the coefficient vector converges. The resulting coefficient vector ξ is sparse, meaning most coefficients are zero, with only a few terms contributing significantly to the system dynamics retained. This sparsity ensures the simplicity of the reduced-order model and avoids overfitting.

[0061] Step S304, based on the retained coefficients, outputs the following differential governing equations: This equation is the standard form for a single-degree-of-freedom nonlinear oscillator, where each term has a definite physical meaning: The term represents the linear damping force ζ of the system. iω is the damping ratio of the i-th mode, dimensionless; i Let be the natural frequency of the i-th mode, expressed in radians per second (rad / s). ω i 2 q i The term represents the linear restoring force of the system; ∑ j β ij q i 2 q j The term represents the nonlinear restoring force of the system, β ij The nonlinear coupling coefficient is expressed in units of 1 / (m). 2 ⋅s 2 This reflects the nonlinear interaction between modes; γ i The term p(t) represents the excitation of the system by the wind load, γ i The wind pressure participation factor, expressed in units of 1 / (kg⋅m), characterizes the efficiency of converting wind pressure load into modal generalized force.

[0062] Step S305 assigns the coefficients retained in step S303 to the corresponding physical parameters. The specific correspondence is as follows: and The coefficients corresponding to the terms are used to calculate the damping ratio ζ. i The calculation method is to divide the coefficient of this term by 2ω. i ; With q i The coefficients corresponding to the terms are used to calculate the natural frequency ω. i The calculation method is to take the square root of the coefficient of the term; With q i 3 Item or q i 2 q j The coefficients corresponding to the terms are directly used as the nonlinear coupling coefficients β. ij ; The coefficient corresponding to the p(t) term is directly used as the wind pressure participation factor γ. i .

[0063] Through this assignment process, the regression coefficients of pure mathematics are transformed into system parameters with clear physical meaning, enabling the reduced-order model not only to accurately predict the response, but also to explain the physical characteristics of the system, such as the magnitude of damping, natural frequency, strength of nonlinearity, and excitation efficiency of wind load.

[0064] Through steps S301 to S305 above, this invention achieves automated construction of a reduced-order model from simulation data. The beneficial effects of this approach are as follows: First, by constructing a comprehensive candidate function library, it provides rich candidate basis functions for system identification, capable of capturing various dynamic behaviors such as linear, nonlinear, and coupled models. Second, by employing a sequential threshold Ridge regression algorithm, it achieves coefficient sparsity while ensuring fitting accuracy, avoiding overly complex models and overfitting. Third, the output differential control equations have clear physical meaning, and each coefficient can be interpreted as a specific physical parameter, facilitating engineer understanding and subsequent correction. Finally, this reduced-order model can quickly predict the structural response under different wind loads, providing an efficient computational tool for subsequent multi-objective optimization and significantly shortening the optimization iteration cycle. Compared with existing methods that rely on empirical formulas or simplified assumptions, the reduced-order model construction method of this invention is more data-driven and universal, adaptable to the wind vibration response characteristics of different types of photovoltaic supports.

[0065] In another technical solution, the mechanical performance simulation and optimization design method for the photovoltaic support structure, step S4 employs a multi-objective evolutionary algorithm to automatically optimize the cross-sectional dimensions of the components, specifically including: S401. Initialize the population P, where each individual in the population corresponds to a set of cross-sectional dimension parameter vectors for the photovoltaic support. X =[x1,x2, ...,x m ] T Where m is the total number of design variables, which is determined according to the structural form of the photovoltaic support; when the photovoltaic support adopts a common structure consisting of columns, beams and diagonal braces, the design variables include at least the column wall thickness, beam flange width and diagonal brace section height, all in millimeters; when the photovoltaic support includes other components, the design variables are increased accordingly by the corresponding cross-sectional dimension parameters. S402. Approximately estimate the peak displacement response f1(X) predicted by the reduced-order model, and directly calculate the total weight f2(X) of the support based on the cross-sectional dimension parameters. S403. Construct the Pareto front F based on the non-dominated solution set of the current generation, and calculate the value of each individual X on the front. i The minimum Euclidean distance to all other individuals is taken as the discreteness d of that individual. min (X i ); S404. Mark the three individuals with the largest dispersion as filling sampling points, and call the fluid-structure interaction simulation in step S1 to perform real finite element analysis on these individuals to obtain the real displacement response peak value. S405. Add the real analysis results obtained in step S404 to the training sample set, and retrain and update the Kriging proxy model. S406. Repeat steps S403 to S405 until any of the following stopping conditions are met: RMSE < 5% or iter ≥ iter max ; Where RMSE is the root mean square error of the Kriging surrogate model, iter is the current iteration number, and iter max The maximum number of iterations is preset. S407. Output the final Pareto optimal solution set F*, each solution containing a set of cross-sectional dimension parameters X and their corresponding displacement response peak f1(X) and total support weight f2(X).

[0066] In step S4 of this invention, minimizing the peak displacement response predicted by the reduced-order model and minimizing the total weight of the support are the dual optimization objectives. With the stress ratio of each component not exceeding the allowable value as a constraint, a multi-objective evolutionary algorithm is used to automatically optimize the cross-sectional dimensions of the components and generate a Pareto optimal solution set. The above technical solution further defines the specific implementation method of the multi-objective optimization process. Its purpose is to efficiently and accurately search the design space and obtain the optimal design scheme that takes into account both structural safety and economy.

[0067] In practical implementation, step S401 first initializes the population P. The population is the foundation of the multi-objective evolutionary algorithm, with each individual representing a candidate design scheme. Specifically, each individual in the population corresponds to a set of cross-sectional dimension parameter vectors X=[x1,x2,…,x...] for the photovoltaic support structure. m ] T Where m represents the total number of design variables, determined based on the actual structural form of the photovoltaic support system. When the photovoltaic support system adopts a common structure consisting of columns, beams, and braces, the design variables should at least include the column wall thickness, beam flange width, and brace section height, all in millimeters (mm). The column wall thickness directly affects the bending stiffness and stability of the column; the beam flange width affects the beam's section modulus and bending capacity; and the brace section height affects the brace's compression member stability and overall stiffness. When the photovoltaic support system includes other components, such as purlins, foundation connectors, and stiffening ribs, the corresponding cross-sectional dimension parameters should be added to the design variables. The initial population can be generated using random sampling or Latin hypercube design methods to ensure that individuals are evenly distributed within the design space, covering as many design schemes as possible.

[0068] Step S402 estimates the optimization objective. While directly using a reduced-order model for optimization is faster, the model itself contains approximation errors; while directly using fluid-structure interaction (FSI) simulation offers high accuracy but is computationally extremely expensive. To resolve this contradiction, this step introduces a Kriging surrogate model to approximate the peak displacement response f1(X) predicted by the reduced-order model. The Kriging surrogate model is an interpolation method based on statistical theory, which not only provides the predicted value but also estimates the uncertainty of the prediction, making it suitable for approximating complex nonlinear functions. Constructing the Kriging model requires a certain number of initial sample points, which can be selected using experimental design methods (such as orthogonal design, uniform design, etc.) and the corresponding true peak displacement response obtained by calling the FSI simulation in step S1. Simultaneously, based on the cross-sectional dimension parameter XX and the material density, the total weight f2(X) of the support is directly calculated. This calculation is deterministic and does not require surrogate model approximation.

[0069] Step S403: Construct the Pareto front based on the non-dominated solution set of the current generation. F In multi-objective optimization, a non-dominated solution is a solution for which no other solution is inferior in all objectives and superior in at least one objective. The set of all non-dominated solutions is the Pareto optimal solution set, and its projection onto the objective space is called the Pareto front. After constructing the Pareto front, the values ​​of each individual X on the front are calculated. i The minimum value of the Euclidean distance to all other individuals is taken as the dispersion of that individual. d min (X i The calculation formula is as follows: ; in, This represents the Euclidean norm. Dispersion reflects the sparsity of the individual on the Pareto front; the greater the dispersion, the fewer design options are available around the individual, and the less design information is available in that region.

[0070] Step S404 marks the three individuals with the highest dispersion as fill sampling points. These individuals are located in the sparse region of the Pareto front. Performing high-precision simulations on these regions can maximize the supplementation of information in the design space and improve the global accuracy of the surrogate model. The fluid-structure interaction simulation from step S1 is then used to perform a true finite element analysis on these individuals to obtain the true peak displacement response. It should be noted that the number of fill sampling points can be adjusted according to the actual situation, such as taking the two or five with the highest dispersion, but the first three are the preferred option to balance computational cost and information gain.

[0071] Step S405 adds the true analysis results obtained in step S404 to the training sample set, retraining and updating the Kriging surrogate model. With the addition of new sample points, the prediction accuracy of the Kriging model in key regions of the design space is improved, particularly its fitting ability to sparse regions of the Pareto front is enhanced. The updated Kriging model will be used for objective function estimation in the next round of optimization.

[0072] Step S406 repeats steps S403 to S405 until either of the following stopping conditions is met: RMSE < 5% or iter ≥ iter max Wherein, RMSE is the root mean square error of the Kriging surrogate model, and its calculation formula is: ; In the formula, n is the number of samples in the validation set, and y i This represents the true response value of the i-th verification sample (obtained from fluid-structure interaction simulation). These are the predicted values ​​from the Kriging model. This represents the mean of the actual response values. `iter` represents the current iteration number. max This is a preset maximum number of iterations. When the RMSE is less than 5%, it indicates that the surrogate model has sufficient accuracy and iteration can stop; when the number of iterations reaches the preset maximum value, it indicates that computing resources have been exhausted and iteration should also stop. The preset maximum number of iterations is `iter`. max The value can be set according to project requirements and computing resources, with typical values ​​ranging from 20 to 50.

[0073] Step S407 outputs the final Pareto optimal solution set F*. Each solution in this set contains a set of cross-sectional dimension parameters X and their corresponding peak displacement response f1(X) and total support weight f2(X). The peak displacement response f1(X) can be predicted by the updated Kriging surrogate model, or it can be output after verifying the key individuals in the final solution set through fluid-structure interaction simulation; the total support weight f2(X) is calculated directly based on the cross-sectional dimensions. The output Pareto optimal solution set provides designers with several trade-off options: the option located at the left end of the Pareto front has the smallest peak displacement response but a larger weight, suitable for scenarios with strict deformation requirements; the option located at the bottom has the lightest weight but a larger peak displacement response, suitable for cost-sensitive projects; the option in the middle of the front achieves a balance between the two. Designers can select the most suitable option from these options based on specific engineering requirements.

[0074] Through steps S401 to S407 above, this invention achieves multi-objective optimization based on a surrogate model. The beneficial effects of this implementation are as follows: First, by introducing a Kriging surrogate model to approximate the predictions of the reduced-order model, it utilizes the computational efficiency of the reduced-order model while continuously refining the surrogate model using real simulation data, ensuring optimization accuracy. Second, by employing a discrete-degree-of-dispersion-based filling sampling strategy, it prioritizes high-precision simulation of the sparse region of the Pareto front, obtaining the maximum information gain with minimal computational cost. Third, by using dual stopping conditions of RMSE and maximum iteration count, it controls computational cost while ensuring optimization quality. Finally, it outputs a complete Pareto optimal solution set, providing designers with a rich set of preferred solutions rather than a single solution, fully demonstrating the advantages of multi-objective optimization. Compared with existing methods that rely on empirical trial and error or single-objective optimization, the optimization method of this invention can systematically explore the design space and truly achieve the comprehensive optimization of peak displacement and structural weight while satisfying stress ratio constraints.

[0075] In another technical solution, the mechanical performance simulation and optimization design method for the photovoltaic support structure, step S5, which determines whether the error between the simulation result and the predicted value of the reduced-order model exceeds a preset threshold, specifically includes: S501. Extract the peak displacement response of key nodes in the time domain and record the peak values ​​from the fluid-structure interaction simulation. u sim Peak value predicted by the reduced-order model u pred ; S502. Calculate the relative error using the following formula: ; in, d peak This represents relative error, expressed as a percentage; |⋅| indicates taking the absolute value. S503. Extract the displacement response sequence over the entire time period, and denote them as simulation sequences. u sim and predicted sequence u pred Calculate the dynamic time-normalized distance between two sequences. D dtw ; S504, Set the first threshold T 1 = 15%, second threshold T 2 = 0.05 × max( u sim ); S505, if d peak > T 1 and D dtw> T 2. If the error exceeds the limit, proceed to step S506; otherwise, the model is deemed valid, and the current reduced-order model continues to be used. S506, triggers the downgraded model correction procedure.

[0076] In step S5 of this invention, selecting candidate solutions from the Pareto optimal solution set for fluid-structure interaction simulation verification and determining whether the error between the simulation result and the predicted value of the reduced-order model exceeds a preset threshold is a key step to ensure the reliability of the reduced-order model. The above technical solution further defines the specific implementation method of the error judgment. Its purpose is to comprehensively evaluate the prediction accuracy of the reduced-order model from two dimensions: peak error and waveform similarity, and to provide a scientific basis for whether to trigger model correction.

[0077] In practical implementation, step S501 first extracts the peak displacement response of key nodes in the time domain. The peak displacement response refers to the maximum absolute value of the displacement of the key nodes throughout the entire time history analysis period. It reflects the maximum deformation that the structure can achieve under wind load and is an important indicator for evaluating structural stiffness and safety. For the same candidate scheme, the peak displacement response u from the fluid-structure interaction simulation is recorded. sim The peak displacement response u predicted by the reduced-order model pred Peak value u in fluid-structure interaction simulation sim The value obtained through high-precision simulation in step S1 is considered a reference value for the true response; the peak value u predicted by the reduced-order model... pred These are the predicted values ​​obtained through rapid calculation using the reduced-order model constructed in step S3, and are to be verified.

[0078] Step S502: Calculate the peak relative error δ peak The specific calculation formula is as follows: ; Among them, | u sim− u pred | represents the absolute value of the difference between the simulated peak value and the predicted peak value, in meters (m); | u sim | represents the absolute value of the simulated peak value, in meters (m); d The peak value represents the relative error, expressed as a percentage (%). This metric reflects the accuracy of the reduced-order model in predicting the maximum response amplitude. d The smaller the peak value, the more accurate the reduced-order model's prediction of the peak response. d The larger the peak, the greater the deviation between the predicted value and the actual value.

[0079] Step S503: Extract the displacement response sequence over the entire time history, and denot it as the simulation sequence. u sim (t ) and predicted sequence u pred ( t ), calculate the dynamic time-normalized distance between two sequences. D dtw Dynamic time warping (VTW) is an algorithm for measuring the similarity between two time series. Its core idea is to find the optimal alignment path between the two series by allowing nonlinear scaling along the time axis, thereby calculating the cumulative distance. Compared to traditional metrics such as Euclidean distance, VTW effectively handles situations where the two series have phase shifts or local rate differences along the time axis, making it more suitable for comparing non-stationary time series signals such as vibration responses. (Dynamic time warping distance) D dtw The smaller the value, the more similar the waveforms of the two sequences are; D dtw The larger the value, the more significant the waveform difference. The calculation can be performed using a standard dynamic programming algorithm, with appropriate step size and boundary conditions set.

[0080] Step S504 sets the threshold for error judgment. This invention uses a dual threshold for comprehensive judgment: the first threshold... T 1 = 15%, used to measure the acceptable range of relative peak error; second threshold T 2 = 0.05 × max( u sim The threshold, 5% of the simulated displacement peak value, is used to measure the acceptable range of dynamic time-warped distance. These two thresholds are set based on engineering experience: a peak error within 15% is generally considered an acceptable engineering approximation; controlling waveform differences within 5% of the peak value ensures that the main characteristics of the dynamic response are accurately captured. It should be noted that these thresholds can be adjusted according to the accuracy requirements and structural importance of the specific project. For photovoltaic power plants with high safety levels, the thresholds can be appropriately reduced; for the preliminary design stage, the thresholds can be appropriately relaxed.

[0081] Step S505 determines the subsequent operation based on the threshold judgment result. If the peak relative error d peak> T 1 and dynamic time-normalized distance D dtw > T2. If the prediction error of the reduced-order model exceeds the limit, it is determined that the error exceeds the limit. The AND logic here means that two conditions must be met simultaneously to trigger correction. The design idea is as follows: if only the peak error is large but the waveform similarity is high, it may be due to amplitude scaling, which can be handled with simple correction; if only the waveform difference is large but the peak error is small, it may be caused by phase shift, which has limited impact on the maximum deformation of engineering concern; only when both exceed the limit does it indicate that the reduced-order model has significant deviations in both amplitude and dynamic characteristics, and correction is necessary. If the error exceeds the limit, step S506 is executed to trigger the reduced-order model correction procedure; otherwise, the model is considered valid, and the current reduced-order model continues to be used for subsequent optimization iterations.

[0082] Step S506 triggers the reduced-order model correction procedure. Its core idea is to utilize newly acquired fluid-structure interaction simulation data to optimize and adjust the physical parameters of the reduced-order model, thereby improving the accuracy of model predictions. After triggering the correction, the current simulation data is added to the training set to correct the reduced-order model. The corrected model will then be used in the next round of optimization iterations.

[0083] Through steps S501 to S506 above, this invention achieves closed-loop verification and error determination of the prediction accuracy of the reduced-order model. The beneficial effects of this implementation are as follows: First, by employing two complementary indicators—peak relative error and dynamic time warping distance—the model performance is comprehensively evaluated from two dimensions: extreme value accuracy and waveform similarity, avoiding the one-sidedness of a single indicator. Second, by setting a reasonable threshold range, a quantified error tolerance standard is provided for engineering applications. Third, by using AND logic for joint determination, the rigor of triggering corrections is ensured, while avoiding frequent invalid corrections due to accidental exceedances of a single indicator. Finally, timely triggering of the correction procedure and inclusion of new data into the training set enable the reduced-order model to continuously learn and improve during the optimization process, forming a closed-loop iterative mechanism of simulation-prediction-verification-correction. Compared with existing methods that only compare peak values ​​or simple correlation coefficients, the error determination method of this invention is more comprehensive and scientific, effectively ensuring the reliability of the reduced-order model throughout the entire optimization process.

[0084] In another technical solution, the mechanical performance simulation and optimization design method for the photovoltaic support structure, step S5 involves modifying the reduced-order model, specifically including: S507. Compare the wind pressure load time history obtained from the current fluid-structure interaction simulation with the generalized coordinate time history data. Add to the training dataset, where the superscript l Indicates the first l One sample, l =1,2,..., L , L This represents the current total number of samples; S508. Extract the vector of parameters to be corrected in the differential control equation: ; Where, ζ i Dimensionless, ω i The unit is radians per second. β ij The unit is 1 / (meter) 2 ⋅ seconds 2 ), c i The unit is 1 / (kg·m); S509. Constructing the objective function: ; in q i (pred) (t;θ) For the current parameter i The generalized coordinate time history predicted by the model, q i (sim,l) (t) is the th l The first simulation sample i Generalized coordinate time history, unit is meters T represents the total duration of the time series, in seconds; ∫ denotes the integral over time. S510, Solve for J(θ) Minimize parameters θ* The iterative update formula is: ; in, i k For the first k The parameter vector for the next iteration; J Let be the Jacobian matrix, and each element be the partial derivative of the residual with respect to the parameter; r Let be the residual vector, with elements of . ; l The damping factor is dimensionless. I It is the identity matrix; S511, Apply the new parameters obtained from the optimization solution. θ* Replace the corresponding parameters in the original reduced-order model to complete the model correction.

[0085] In step S5 of the present invention, when it is determined that the prediction error of the reduced-order model exceeds the limit, a correction procedure needs to be triggered to update the model. The above technical solution further defines the specific implementation method of the reduced-order model correction. Its purpose is to use the newly obtained fluid-structure interaction simulation data to optimize and adjust the physical parameters of the reduced-order model so that the corrected model can more accurately predict the wind vibration response of the photovoltaic support.

[0086] In specific implementation, step S507 first adds the wind pressure load time history and generalized coordinate time history data pairs obtained from the current fluid-structure interaction simulation to the training dataset. Specifically, for the l-th simulation verification, a set of data pairs {p} can be obtained. (l) (t),q i (l) (t)}, where p (l) (t) represents the time history of wind pressure load, in Pascals (Pa); q i (l) (t) represents the generalized coordinate time history of the i-th dominant mode, in meters; superscript l Indicates the first l One sample, l =1,2,...,L, where L is the current total number of samples. These data pairs form the basis for the order reduction model correction, directly reflecting the mapping relationship between wind load input and structural modal response output. As the optimization iteration proceeds, new data pairs are added each time a correction is triggered, continuously enriching the training dataset and providing more information for the accurate identification of model parameters.

[0087] Step S508 extracts the parameter vector θ to be corrected in the differential control equation. Based on the reduced-order model constructed in step S3, its differential control equation takes the form of: ; The parameters to be corrected in this equation include: the damping ratio ζ of the i-th mode. i , dimensionless; the natural angular frequency ω of the i-th mode. i The unit is radians per second; the nonlinear coupling coefficient β ij The unit is 1 / (m) 2 ·s 2 Wind pressure participation factor γ i The unit is 1 / (kg·m). These parameters are combined into a parameter vector to be corrected: θ=[ζ1,ω1,β] 11 ,β 12 ,...,γ1,ζ2,ω2,β 21 ,β 22 ,...,γ2,...] T The specific dimension of the vector depends on the number of dominant modes N and the number of nonlinear coupling terms. For an Nth-order dominant mode, considering all possible q... i 2 q j The coupling terms then consist of N damping ratios, N natural frequencies, and N... 2 There are one nonlinear coupling coefficient and N wind pressure participation factors, totaling 2N+N. 2 +N parameters to be corrected.

[0088] Step S509 constructs the objective function J(θ). The physical meaning of the objective function is to measure the overall deviation between the generalized coordinate time history predicted by the reduced-order model and the generalized coordinate time history of all simulation samples under the current parameters θ. The specific expression is: ; where q i (pred) (t;θ) represents the result obtained by solving the differential governing equation under the current parameter θ. i The time history of the generalized coordinate prediction is expressed in meters; q i (sim,l) (t) represents the time history of the i-th generalized coordinate of the l-th simulation sample, in meters; T is the total duration of the time history, in seconds; ∫ represents the integral over time from 0 to T; Σ represents the summation over all L simulation samples. The smaller the value of this objective function, the closer the prediction result of the reduced-order model under the current parameters is to the high-precision simulation result. It should be noted that for multiple dominant modes, an objective function can be constructed for each order separately, or the errors of all orders can be weighted and summed to form a comprehensive objective function.

[0089] Step S510 solves for the parameter vector θ* that minimizes the objective function J(θ). This is a nonlinear least squares optimization problem, which is solved using the Levenberg-Marquardt algorithm. The Levenberg-Marquardt algorithm combines the advantages of the Gauss-Newton method and gradient descent, enabling efficient solutions to nonlinear least squares problems. Its iterative update formula is: ; where θ k Let be the parameter vector for the k-th iteration; J is the Jacobian matrix, whose elements are the partial derivatives of the residuals with respect to each parameter ∂r / ∂θ, reflecting the degree of influence of parameter changes on the prediction error; r is the residual vector, whose elements are r {l,i} (t)=q i (pred) (t;θ k )-q i (sim,l) (t) represents the difference between the predicted time history and the simulated time history of the i-th mode of the l-th sample under the current parameters; λ is the damping factor, dimensionless, used to adjust the convergence characteristics of the algorithm. When λ is large, the algorithm is close to the gradient descent method; when λ is small, the algorithm is close to the Gauss-Newton method; I is the identity matrix. During the iteration process, the damping factor λ can be adaptively adjusted according to the decrease of the objective function after each iteration: if the objective function decreases significantly, λ is decreased to accelerate convergence; if the objective function does not decrease significantly or increases, λ is increased to ensure convergence stability.

[0090] Step S511 replaces the corresponding parameters in the original reduced-order model with the new parameters θ* obtained from the optimization solution, completing the model correction. The corrected reduced-order model is as follows: ; Where, ζ i * ω i * β ij * γ i * To optimize the obtained optimal parameters, this corrected model will be used in subsequent optimization iterations until the next correction is triggered. Through this parameter update, the reduced-order model can continuously absorb newly acquired simulation data, and its prediction accuracy gradually improves as the optimization iterations proceed.

[0091] Through steps S507 to S511 above, this invention achieves data-driven correction of the reduced-order model. The beneficial effects of this implementation are as follows: First, newly acquired fluid-structure interaction simulation data is continuously added to the training set, allowing the correction process to fully utilize the ever-increasing amount of information and achieve continuous model improvement. Second, an optimization problem is constructed with the sum of squared errors between the predicted and simulated time histories as the objective function, directly optimizing the model's dynamic prediction capability from the perspective of time-domain waveform fitting. Third, the Levenberg-Marquardt algorithm is used to solve the nonlinear least squares problem, possessing both fast convergence speed and good stability, enabling efficient finding of the optimal parameters. Finally, the optimized parameters replace the original model, completing the correction loop and ensuring the reduced-order model maintains high prediction accuracy throughout the optimization process. Compared to existing methods that fix model parameters or only perform simple scaling corrections, the correction method of this invention can systematically optimize multiple physical parameters of the model, more comprehensively improving the model's prediction capability and providing a reliable and rapid analysis tool for subsequent optimization iterations.

[0092] In another technical solution, the mechanical performance simulation and optimization design method for the photovoltaic support structure, in step S510, further includes the following steps in solving the parameter θ* using the Levenberg-Marquardt algorithm: S510a, Update parameter θ in each iteration k+1 Then, the updated parameters are physically validated, specifically including: Determine the updated damping ratio g i ( k+1) Does it satisfy 0 < g i (k+1) <1; if not satisfied, then g i (k+1) Corrected to the preset reasonable range. g min , gmax [Inside, among which] g min =0.001, g max =0.1; Determine the updated natural angular frequency oh i (k+1) Compared with theoretical values ​​calculated based on the finite element model oh i FEM Does the relative deviation satisfy: ; in d ω =20%; if not satisfied, cancel the current iteration update, increase the damping factor λ, and resolve the parameter update amount; S510b, Calculate the residual vector r Sensitivity matrix for each parameter to be corrected S Its elements are: According to the sensitivity matrix S Identify the factors that have the greatest impact on model prediction error. M These parameters are updated first during subsequent iterations and corrections. M One parameter is used, and the remaining parameters retain their current values.

[0093] In step S510 of the present invention, the Levenberg-Marquardt algorithm is used to solve for the parameter θ* that minimizes the objective function. This is the core algorithm for achieving accurate correction of the reduced-order model. The above technical solution further defines two key links in the solution process: physical validity verification and sensitivity analysis. The purpose is to ensure that the corrected parameters can not only accurately fit the simulation data, but also conform to physical laws, while improving the computational efficiency of the correction process.

[0094] In practical implementation, step S510a first updates the parameter θ in each iteration. {k+1} Next, the updated parameters undergo physical validity verification. This verification step is introduced based on the following considerations: The Levenberg-Marquardt algorithm is essentially a mathematical optimization process, aiming to minimize prediction error, without guaranteeing that the optimized parameters have physical meaning. Without constraints, physical errors may occur, such as negative damping ratios or excessive deviations from theoretical values ​​in natural frequencies. This could lead to absurd results in extrapolation predictions, even if the corrected model can fit the current data. Therefore, it is necessary to verify the rationality of the parameters from a physics perspective.

[0095] Physical validity verification specifically includes two aspects: Firstly, determine the updated damping ratio ζ. i (k+1) Does 0 < ζ? i (k+1) <1. The damping ratio is a parameter describing a system's ability to dissipate energy. According to vibration theory, for general engineering structures, the damping ratio should be positive and usually less than 1. Negative damping means that the system continuously absorbs energy during vibration, which is impossible in passively energy-dissipating structures. A damping ratio greater than or equal to 1 corresponds to an overdamped or critically damped state; lightweight steel structures such as photovoltaic supports typically exhibit low damping characteristics. Therefore, the reasonable range for the damping ratio is preset to [ζ]. min ,ζ max ], where ζ min =0.001, ζ max =0.1. If the updated damping ratio ζ i (k+1) If the value falls within this interval, the check passes; otherwise, ζ is set to... i (k+1) The correction is made to fit within this interval. The specific correction method can be based on the interval boundary value, i.e., if ζ... i (k+1) ≤ζ min Then take ζ min If ζ i (k+1) ≥ζ max Then take ζ max .

[0096] Secondly, determine the updated natural angular frequency ω. i (k+1) Compared with the theoretical value ω calculated based on the finite element model i FEM Whether the relative deviation meets the preset requirements. The natural frequency is an inherent characteristic of the structure, mainly determined by the structure's mass and stiffness distribution. During the optimization design process, although the cross-sectional dimensions may change, the topological form of the structure remains essentially unchanged; therefore, the natural frequency should vary within a certain range near the theoretical value calculated by the finite element method. The specific judgment condition is: |(ω i (k+1) -ω i FEM ) / ω i FEM |≤δ ω ; where δ ω To define the upper limit of the permissible relative deviation, δ is set in this invention. ω =20%. This threshold allows for some adjustment of the parameters to adapt to the actual dynamic characteristics under wind load, while preventing the natural frequency from deviating too far from the theoretical value, which would lead to model distortion. If this condition is not met, it indicates that the parameters updated in this iteration are physically unreasonable, and the iteration should be canceled, i.e., θ is maintained.{k+1} =θ k Then, increase the damping factor λ (e.g., multiply λ by 10) and recalculate the parameter update. Increasing the damping factor can reduce the algorithm's step size, improve convergence stability, and prevent the algorithm from jumping out of the physically reasonable region due to an excessively large step size.

[0097] Step S510b calculates the sensitivity matrix S of the residual vector r with respect to each parameter to be corrected, and identifies key parameters based on the sensitivity. The purpose of sensitivity analysis is to quantify the influence of each parameter to be corrected on the model prediction error, so that in subsequent iterations, parameters with greater influence are updated first, while parameters with less influence are ignored, thereby improving correction efficiency.

[0098] The elements of the sensitivity matrix S are defined as follows: ; where r l (t) represents the nth element in the residual vector. l The element corresponding to the element l The prediction error r of a sample at time t l (t)=q i (pred) (t;θ)−q i (sim,l) (t), θ j For the j-th parameter to be corrected (e.g., ζ) i ,ω i ,β ij ,γ i ); The partial derivative reflects the parameter i j Tiny changes at time t Residual r l ( t The greater the absolute value of the parameter's influence, the more sensitive it is to the prediction error at that moment.

[0099] Based on the sensitivity matrix S, for each parameter i j Calculate the overall sensitivity index, for example, by summing the absolute values ​​of sensitivity for all times and all samples: ; Alternatively, the sum of squares form can be used: Based on the comprehensive sensitivity index S j Sort the parameters from largest to smallest to identify the top M parameters that have the greatest impact on the model's prediction error. The value of M can be determined according to the actual situation, usually taking 20% ​​to 30% of the total number of parameters, or directly taking a fixed value (such as M=5).

[0100] In subsequent iterative corrections, these M key parameters are updated first, while the remaining parameters remain unchanged. This strategy concentrates computational resources on optimizing the parameters that contribute most to the prediction error, achieving maximum accuracy improvement with minimal computational cost, while reducing the risk of overfitting.

[0101] Through the aforementioned steps S510a and S510b, this invention enhances and optimizes the Levenberg-Marquardt algorithm. The beneficial effects of this implementation are as follows: First, by introducing physical validity verification, it ensures that the corrected parameters conform to basic physical laws, avoiding parameter distortion problems that may arise from purely mathematical optimization, and improving the generalization ability and reliability of the corrected model. Second, the mechanism for canceling updates when the damping ratio is corrected to a reasonable range and when the natural frequency exceeds the limit ensures that the optimization process always takes place within a physically reasonable search space, improving the algorithm's robustness. Third, by identifying key parameters through sensitivity analysis and prioritizing their updates, the number of parameters requiring optimization is significantly reduced, computational complexity is lowered, and correction efficiency is improved. Finally, sensitivity analysis also provides quantitative evidence for understanding the influence of each physical parameter on the structural wind-induced vibration response, contributing to a deeper understanding of the structure's dynamic characteristics. Compared with existing methods that blindly optimize all parameters or select correction parameters solely based on experience, the parameter correction method of this invention is more scientific and targeted, significantly improving computational efficiency while ensuring the correction effect.

[0102] The number of devices and processing scale described herein are for the purpose of simplifying the description of the invention. Applications, modifications, and variations of the invention will be readily apparent to those skilled in the art.

[0103] Although the embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. They can be applied to various fields suitable for the present invention. For those skilled in the art, other modifications can be easily made. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details.

Claims

1. A method for simulating and optimizing the mechanical properties of a photovoltaic support structure, characterized in that, Includes the following steps: S1. Construct a three-dimensional geometric model and a microscale dynamic wind field model of the photovoltaic support structure, and perform fluid-structure interaction simulation calculations to output the dynamic response of the support structure in the time domain. The dynamic response should include at least the displacement-time curves of the key nodes. S2. Perform modal decomposition on the displacement-time curve output in step S1, identify the top N dominant modes that contribute the most to the wind vibration response, and extract the generalized coordinate time history of each dominant mode. S3. Establish the temporal mapping relationship between the generalized coordinates of each dominant mode and the wind pressure load, and construct a reduced-order model of the wind vibration response of the support structure characterized by the reduced-order dominant mode coordinates. S4. With minimizing the peak displacement response predicted by the reduced-order model and minimizing the total weight of the support as optimization objectives, and with the stress ratio of each component not exceeding the allowable value as a constraint, a multi-objective evolutionary algorithm is used to automatically optimize the cross-sectional dimensions of the components and generate a Pareto optimal solution set. S5. Select candidate solutions from the Pareto optimal solution set to update the geometric model. Repeat step S1 to perform fluid-structure interaction simulation verification and determine whether the error between the simulation results and the predicted values ​​of the reduced-order model exceeds the preset threshold. If so, add the current simulation data to the training set to correct the reduced-order model. S6. Repeat steps S2 to S5 until the convergence condition is met, and output the final optimized photovoltaic support structure design parameters.

2. The method for simulating and optimizing the mechanical properties of photovoltaic support structures according to claim 1, characterized in that, Step S1 involves constructing a microscale dynamic wind field model, specifically including: S101. Obtain the elevation data and surface roughness length of the terrain where the photovoltaic support is located, and generate the computational domain grid. S102. Using the large eddy simulation method, the fluctuating wind speed time history is generated at the entrance of the computational domain; S103. Measure the current height H of the photovoltaic support above the ground and the spacing d between it and the adjacent support, and calculate the ratio of spacing to height d / H; S104. When d / H < 5, the additional turbulence intensity correction is calculated using the following formula: ； S105. The additional turbulence intensity calculated in step S104. I add It is superimposed on the pulsating wind speed time history generated in step S102 as the corrected inlet boundary condition.

3. The method for simulation and optimization design of the mechanical properties of photovoltaic support structures as described in claim 1, characterized in that, Step S2 involves modal decomposition of the displacement-time curve to identify the dominant modes, specifically including: S201. Set the penalty factor α=2000 for the variational mode decomposition algorithm and the preset number of mode decompositions K=10; S202. Input the displacement-time curve x(t) of the key node into the variational mode decomposition algorithm, and output 10 eigenmode function components. u k (t) , k =1,2,...,10; S203. Calculate the energy contribution rate of each intrinsic mode function component using the following formula: ； in, E k For the first k Energy contribution rate of the first mode, expressed as a percentage; u k ( t ) is the first k The intrinsic mode function components, in meters; t is time, in seconds; ∫ represents integration over time. S204. For each intrinsic modal function component, calculate its coherence function γ with the wind pressure load time history p(t). k(f) The maximum value of the coherence function is extracted and denoted as the peak coherence coefficient γ. k,max ; S205. Determine whether each mode simultaneously satisfies E. k >5% and γ k,max >0.6; S206. Sort the modes that satisfy the conditions in step S205 from high to low according to their energy contribution rate, select the top N as the dominant modes, and output the corresponding generalized coordinate time history. q i (t) ,in i =1,2,...,N.

4. The method for simulation and optimization design of the mechanical properties of photovoltaic support structures as described in claim 1, characterized in that, Step S3 involves constructing a reduced-order model of the wind-induced vibration response of the support structure, characterized by the reduced-order dominant mode coordinates. This specifically includes: S301. A complete candidate function library Φ has been constructed, containing the following five types of elements: ； in, q i For the first i Generalized coordinates of order, in meters; express q i The first derivative with respect to time, in m / s; q i 3 express q i cubed, unit is m 3 ; q i 2 q j express q i The square of the first j generalized coordinates q j The unit is m 3 ; p ( t () represents the wind pressure load time history, in Ps; S302, the first i The second derivative of the generalized coordinate with respect to time is expressed as: ; in, express q i The second derivative with respect to time, in m / s 2 ; ξ The vector of sparse coefficients to be determined is given, and the dimensions of each component are determined according to the corresponding terms. S303, Solved using the sequence threshold Ridge regression algorithm. ξ Set a threshold for regression residuals ε =10 -3 The regression residuals will be less than ε The coefficients of are retained, and the remaining coefficients are set to zero; S304. Based on the retained coefficients, output the following differential governing equations: ； in, ζ i For the first i The damping ratio of the first mode is dimensionless; ω i For the first i The natural frequency of the first mode, expressed in rad / s; β ij The nonlinear coupling coefficient is expressed in units of 1 / (m). 2 ⋅s 2 ); γ i The wind pressure participation factor is expressed in units of 1 / (kg⋅m). S305. Assign the coefficients retained in step S303 to the corresponding physical parameters: and q ˙ i The coefficients corresponding to the terms are used to calculate the damping ratio. ζ i ,and q i The coefficients corresponding to the terms are used to calculate the natural frequencies. ω i ,and q i 3 The coefficients corresponding to the terms are nonlinear coupling coefficients. β ij ,and p ( t The coefficient corresponding to the term is the wind pressure participation factor. γ i .

5. The method for mechanical performance simulation and optimization design of photovoltaic support structures as described in claim 1, characterized in that, Step S4 employs a multi-objective evolutionary algorithm to automatically optimize the cross-sectional dimensions of the component, specifically including: S401. Initialize the population P, where each individual in the population corresponds to a set of cross-sectional dimension parameter vectors for the photovoltaic support. X =[x1,x2,...,x m ] T Where m is the total number of design variables, which is determined according to the structural form of the photovoltaic support; when the photovoltaic support adopts a common structure consisting of columns, beams and diagonal braces, the design variables include at least the column wall thickness, beam flange width and diagonal brace section height, all in millimeters; when the photovoltaic support includes other components, the design variables are increased accordingly by the corresponding cross-sectional dimension parameters. S402. Approximately estimate the peak displacement response f1(X) predicted by the reduced-order model, and directly calculate the total weight f2(X) of the support based on the cross-sectional dimension parameters. S403. Construct the Pareto front F based on the non-dominated solution set of the current generation, and calculate the value of each individual X on the front. i The minimum Euclidean distance to all other individuals is taken as the discreteness d of that individual. min (X i ); S404. Mark the three individuals with the largest dispersion as filling sampling points, and call the fluid-structure interaction simulation in step S1 to perform real finite element analysis on these individuals to obtain the real displacement response peak value. S405. Add the real analysis results obtained in step S404 to the training sample set, and retrain and update the Kriging proxy model. S406. Repeat steps S403 to S405 until any of the following stopping conditions are met: RMSE < 5% or iter ≥ iter max ; Where RMSE is the root mean square error of the Kriging surrogate model, iter is the current iteration number, and iter max The maximum number of iterations is preset. S407. Output the final Pareto optimal solution set F*, each solution containing a set of cross-sectional dimension parameters X and their corresponding displacement response peak f1(X) and total support weight f2(X).

6. The method for simulation and optimization design of the mechanical properties of photovoltaic support structures as described in claim 1, characterized in that, Step S5 involves determining whether the error between the simulation result and the predicted value of the reduced-order model exceeds a preset threshold, specifically including: S501. Extract the peak displacement response of key nodes in the time domain and record the peak values ​​from the fluid-structure interaction simulation. u sim Peak value predicted by the reduced-order model u pred ; S502. Calculate the relative error using the following formula: ； in, δ peak This represents relative error, expressed as a percentage; |⋅| indicates taking the absolute value. S503. Extract the displacement response sequence over the entire time period, and denote them as simulation sequences. u sim and predicted sequence u pred Calculate the dynamic time-normalized distance between two sequences. D dtw ; S504, Set the first threshold T 1 = 15%, second threshold T 2 = 0.05 × max( u sim ); S505, if δ peak > T 1 and D dtw > T 2. If the error exceeds the limit, proceed to step S506; otherwise, the model is deemed valid, and the current reduced-order model continues to be used. S506, triggers the downgraded model correction procedure.

7. The method for simulation and optimization design of the mechanical properties of photovoltaic support structures as described in claim 4, characterized in that, Step S5 involves revising the reduced-order model, specifically including: S507. Compare the wind pressure load time history obtained from the current fluid-structure interaction simulation with the generalized coordinate time history data. Add to the training dataset, where the superscript l Indicates the first l One sample, l =1,2,..., L , L This represents the current total number of samples; S508. Extract the vector of parameters to be corrected in the differential control equation: ； Where, ζ i Dimensionless, ω i The unit is radians per second. β ij The unit is 1 / (meter) 2 ⋅ seconds 2 ), γ i The unit is 1 / (kg·m); S509. Constructing the objective function: ； in q i (pred) (t;θ) For the current parameter θ The generalized coordinate time history predicted by the model, q i (sim,l) (t) is the th l The first simulation sample i Generalized coordinate time history, unit is meters T represents the total duration of the time series, in seconds; ∫ denotes the integral over time. S510, Solve for J(θ) Minimize parameters θ* The iterative update formula is: ； in, θ k For the first k The parameter vector for the next iteration; J Let be the Jacobian matrix, and each element be the partial derivative of the residual with respect to the parameter; r Let be the residual vector, with elements of . ; λ The damping factor is dimensionless. I It is the identity matrix; S511, Apply the new parameters obtained from the optimization solution. θ* Replace the corresponding parameters in the original reduced-order model to complete the model correction.

8. The method for simulation and optimization design of the mechanical properties of photovoltaic support structures according to claim 7, characterized in that, In step S510, the process of solving for the parameter θ* using the Levenberg-Marquardt algorithm further includes: S510a, Update parameter θ in each iteration k+1 Then, the updated parameters are physically validated, specifically including: Determine the updated damping ratio ζ i ( k+1) Does it satisfy 0 < ζ i (k+1) <1; if not satisfied, then ζ i (k+1) Corrected to the preset reasonable range. ζ min , ζ max [Inside, among which] ζ min =0.001, ζ max =0.1; Determine the updated natural angular frequency ω i (k+1) Compared with theoretical values ​​calculated based on the finite element model ω i FEM Does the relative deviation satisfy: ； in δ ω =20%; if not satisfied, cancel the current iteration update, increase the damping factor λ, and resolve the parameter update amount; S510b, Calculate the residual vector r Sensitivity matrix for each parameter to be corrected S Its elements are: According to the sensitivity matrix S Identify the factors that have the greatest impact on model prediction error. M These parameters are updated first during subsequent iterations and corrections. M One parameter is used, and the remaining parameters retain their current values.

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