Ice-coated power transmission line galloping simulation method and simulation system based on composite time integral

Abstract

The invention relates to an icing power transmission line galloping simulation method and simulation system based on a composite time integral. The method comprises the following steps: S1, establishing a nonlinear finite element model of an icing power transmission line; s2, carrying out time domain solution by adopting a composite time integration algorithm; calculating the solution of the moment in two sub-steps according to the solution of the moment, namely from the first sub-step to the second sub-step; the first sub-step adopts a trapezoidal rule for integration, and the second sub-step adopts a three-point backward difference format based on information of three moments; s3, coupling and solving a nonlinear aerodynamic load related to the speed; s4, obtaining the galloping response of the power transmission line; the problems that in a transmission line galloping simulation technology based on a traditional Newmark method, contradiction between calculation efficiency and numerical precision exists in the strong nonlinear coupling problem, accurate tracking of low-frequency galloping and efficient suppression of high-frequency noise are difficult to achieve at the same time under the large time step length, simulation calculation efficiency is low, and convergence is poor are solved.

Description

Technical Field

[0001] This invention belongs to the field of overhead transmission line icing technology, and relates to a method and system for simulating the galloping of iced transmission lines based on composite time integration. Background Technology

[0002] Currently, a massive amount of electrical energy needs to be transmitted from the west to the east via ultra-high voltage (UHV) transmission lines. These major transmission arteries inevitably traverse mountainous areas with high altitudes, complex terrain, and harsh climates. These regions experience long winters, low temperatures, and high humidity, providing natural conditions for widespread and severe icing of transmission lines. Simultaneously, frequent special wind fields such as corridor winds and valley winds create typical high-risk environments for icing-induced galloping of transmission lines. Therefore, accurate prediction and effective prevention of galloping disasters on these critical transmission channels have become a major technical challenge for ensuring the smooth implementation of the national energy strategy and the safety of the power grid's "lifeline."

[0003] Under the aforementioned harsh meteorological conditions, irregular icing on the surface of transmission line conductors alters their aerodynamic shape, easily inducing a low-frequency, high-amplitude self-excited vibration—galloping. Severe galloping can cause devastating damage to transmission lines, including: fatigue damage to conductors, hardware, and insulators; insufficient phase-to-phase spacing leading to flashovers or short-circuit trips; and in extreme cases, excessive dynamic tension may even cause tower instability or collapse, resulting in widespread power outages, enormous economic losses, and severe social impacts.

[0004] To explore the complex mechanism of galloping, the academic community has proposed a variety of theories. In terms of numerical simulation, the nonlinear finite element method has become the mainstream tool for analyzing transmission line galloping problems because it can effectively handle the geometric nonlinear characteristics such as large displacement and large rotation angle during the galloping process.

[0005] However, accurate simulation of galloping on iced power transmission lines faces severe technical bottlenecks, fundamentally due to the system's inherent strong nonlinear characteristics. This strong nonlinearity stems primarily from two aspects:

[0006] (1) Geometric nonlinearity: As a typical flexible cable structure, the power transmission line will generate large displacement and large rotation that are much larger than its cross-sectional size during the galloping process, which causes the stiffness characteristics of the structure to change in real time with its geometric configuration.

[0007] (2) Aerodynamic nonlinearity: The aerodynamic forces (lift, drag and torque) acting on the icing conductor are not constant, but are highly related to the instantaneous motion state of the conductor relative to the incoming wind (i.e. translational velocity and torsional angular velocity). This complex aerodynamic-structural coupling effect is the key to the self-excitation and sustainability of the galloping.

[0008] These strong nonlinear characteristics make the dynamic equations describing the dancing behavior highly complex and rigid, placing extremely stringent requirements on the time integration algorithm for solving these equations. In existing technologies, unconditionally stable implicit time integration methods have become the mainstream choice, among which the Newmark average constant acceleration method is widely used due to its second-order accuracy and good energy preservation characteristics. However, this method has a recognized fatal flaw, particularly prominent when dealing with strongly nonlinear problems: the lack of an intrinsic numerical damping mechanism. This means that it cannot effectively dissipate and suppress the high-frequency numerical noise introduced during the computation process by spatial discretization or nonlinear iterative solutions, which is inconsistent with physical reality.

[0009] These spurious numerical oscillations severely contaminate the true low-frequency physical response, not only reducing the accuracy of simulation results but also potentially hindering the convergence of nonlinear iterative processes. To suppress these numerical oscillations, researchers are often forced to use time steps much smaller than those required for stability, which largely negates the large step size advantage that implicit algorithms should possess, resulting in low simulation efficiency and poor convergence. This problem is particularly severe when simulating the entire galloping process of long-distance, high-risk railway lines, as well as complex dynamic phenomena such as beat vibrations that may occur.

[0010] In summary, existing transmission line galloping simulation techniques based on the traditional Newmark method generally suffer from a sharp contradiction between computational efficiency and numerical accuracy when dealing with strongly nonlinear coupled problems. Therefore, there is an urgent need in this field for a novel numerical simulation method that can both accurately track the low-frequency dominant physical phenomenon of galloping and efficiently suppress high-frequency numerical noise, thereby maintaining good convergence and stability over large time steps. Ultimately, this will provide an efficient and high-fidelity simulation tool for the safety of transmission lines in major national engineering projects. Summary of the Invention

[0011] In view of this, in order to solve the problem that the above-mentioned transmission line galloping simulation technology based on the traditional Newmark method faces the contradiction between computational efficiency and numerical accuracy in strongly nonlinear coupling problems, it is difficult to simultaneously achieve accurate tracking of low-frequency galloping and efficient suppression of high-frequency noise at large time steps, and the simulation computation efficiency is low and the convergence is poor, the present invention provides a method and simulation system for simulating galloping of icing transmission lines based on composite time integration.

[0012] To achieve the above objectives, the present invention provides the following technical solution:

[0013] A method for simulating galloping of icing transmission lines based on composite time integration, characterized by comprising the following steps:

[0014] S1. Establish a nonlinear finite element model of the ice-covered transmission line;

[0015] The transmission line is discretized into multiple two-node cable elements, each node containing translational degrees of freedom and torsional degrees of freedom about the element axis, thus obtaining a semi-discrete nonlinear control equation describing the dynamic behavior of the entire transmission line system, which includes the mass matrix, damping matrix, and internal force vectors:

[0016] (1)

[0017] Where M is the mass matrix and C is the damping matrix of the conductor. For the nodal forces inside the conductor, The external aerodynamic load on the conductor. The acceleration at the conductor node, The velocity at the conductor node, For conductor node displacement;

[0018] S2. Solve the time domain problem using a composite time integral algorithm;

[0019] according to The solution at time step is calculated in two sub-steps. The solution at any given moment, the first sub-step from arrive The second sub-step from arrive First step Integrating using the trapezoidal rule, the second sub-step Adopting based on , and Three-point backward difference format for information at three time points;

[0020] S21. In the first sub-step calculation... The time-discretized equilibrium equation is:

[0021] (2)

[0022] Velocity and acceleration are related to displacement by the assumption of the trapezoidal law as follows:

[0023] (3)

[0024] (4)

[0025] Substituting equations (3) and (4) into equation (2), we obtain the following about The nonlinear equation system is solved iteratively using the Newton-Raphson method; let... This is the result of the previous iteration. For the current iteration step, the displacement increment is Then the first The linearized equation for the next iteration is:

[0026] (5)

[0027] in Is The tangent stiffness matrix calculated at the point, By incorporating the aerodynamic stiffness matrix into the equations and iteratively converging, we obtain... displacement at time The updated velocity and acceleration are obtained by calculating using equations (6) and (7);

[0028] The acceleration update is as follows:

[0029] ; ;

[0030] ; ; (6)

[0031] The speed update is as follows:

[0032] ; ;

[0033] ; ; (7);

[0034] S22. In the second sub-step calculation... The time-discretized equilibrium equation is:

[0035] (8)

[0036] in:

[0037] (9)

[0038] but The update formulas for velocity and acceleration at any given time are:

[0039] (10)

[0040] (11)

[0041] Substituting equations (9), (10), and (11) into equation (8), and using the Newton-Raphson method iteratively to solve, the th... The linearized equation for the next iteration is:

[0042] (12)

[0043] in Is The calculated tangent stiffness matrix can be obtained after iterative convergence. displacement at time And the corresponding velocity and acceleration can be calculated using equations (10) and (11);

[0044] S3. Coupled solution of velocity-dependent nonlinear aerodynamic loads;

[0045] Nonlinear aerodynamic loads include dynamic lift. Aerodynamic drag and aerodynamic torque They are determined by: using data fitted from wind tunnel tests, and the instantaneous effective angle of attack. The relevant lift coefficient, drag coefficient, and torque coefficient; combined with air density, incoming wind speed, and conductor characteristic dimensions, the Newton-Raphson nonlinear iterative algorithm is used in the solution process of each sub-step of step S2. In each iteration, the following operations are performed: based on the instantaneous translational velocity and instantaneous torsional angular velocity of the conductor nodes in the current iteration step, the instantaneous effective angle of attack of the conductor cross-section relative to the incoming wind is calculated in real time. Thus, the aerodynamic force and aerodynamic torque acting per unit length of the conductor can be calculated;

[0046] S4. Obtain the transmission line galloping response:

[0047] Repeat steps S2 and S3 to calculate the displacement, velocity, and acceleration response of the transmission line within a set time range, thereby simulating the galloping process of the icy transmission line.

[0048] Furthermore, this method can stably and accurately simulate the modal coupling and beat vibration phenomena of transmission lines under specific Irvine parameters. By effectively suppressing numerical noise, it clearly reveals the envelope, frequency composition, and complex attractor structure in phase space of the beat vibration.

[0049] The composite time integral-based icing transmission line galloping simulation system, based on the aforementioned icing transmission line galloping simulation method, includes:

[0050] The model building module is used to perform step S1 above and establish a nonlinear finite element model of the ice-covered transmission line;

[0051] The time-domain solution module is configured to execute steps S2 and S3 as described above. It integrates a composite time integration algorithm and a coupled iterative solution logic for nonlinear aerodynamic loads.

[0052] A processor for executing instructions from the model building module and the time-domain solution module;

[0053] The memory is used to store the finite element model data, intermediate variables during the calculation process, and the final dancing response results.

[0054] The beneficial effects of this invention are as follows:

[0055] This invention discloses a composite time-integration-based method for simulating galloping of iced transmission lines. Addressing the common challenges of strong geometric and aerodynamic nonlinearity in galloping analysis of iced transmission lines, this invention innovatively introduces a composite time-integration method. By decomposing the time step into two sub-steps for integration, it significantly improves the convergence performance and computational stability of the nonlinear system solution. Simultaneously, this invention directly integrates the velocity-dependent aerodynamic forces unique to iced conductors into the iterative solution framework, achieving real-time and accurate updates of aerodynamic loads, and significantly improving the accuracy and speed of simulating the initiation and evolution of galloping. This method successfully overcomes the bottlenecks of low efficiency and poor convergence in traditional simulations, providing advanced technical support for galloping prediction and prevention in power grid safety operation and maintenance.

[0056] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0057] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0058] Figure 1 This is a logic block diagram of the method for simulating galloping of iced transmission lines based on composite time integration according to the present invention.

[0059] Figure 2 The displacement-time history curve of the midpoint obtained using the composite time integration method of this invention is shown below. Figure 2 (a) is the vertical displacement diagram of the midpoint. Figure 2 (b) is a diagram showing the horizontal displacement of the midpoint;

[0060] Figure 3 The displacement-time history curve of the midpoint obtained using the traditional Newmark implicit time integration method is shown below. Figure 3 (a) is the vertical displacement diagram of the midpoint. Figure 3 (b) is a diagram showing the horizontal displacement of the midpoint;

[0061] Figure 4A diagram of a nonlinear motion model of a rod considering galloping at both ends, for a specific example;

[0062] Figure 5 A comparison chart of vertical displacement versus time displacement obtained by using the composite time integration method of this invention and Matlab calculation;

[0063] Figure 6 This is a mechanical model diagram of the transmission line in the icing-covered transmission line galloping simulation method of the present invention. Detailed Implementation

[0064] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention.

[0065] like Figure 1 The method for simulating galloping of icing transmission lines based on composite time integral, as shown, is applied to the dynamic analysis of transmission lines and includes the following steps:

[0066] S1. Establish a nonlinear finite element model of the ice-covered transmission line;

[0067] This model is constructed by discretizing the transmission line into multiple two-node cable elements, where each node contains translational degrees of freedom and torsional degrees of freedom about the element axis. This yields a semi-discrete nonlinear control equation that describes the dynamic behavior of the entire transmission line system, including the mass matrix, damping matrix, and internal force vectors. The specific formula is as follows:

[0068] (1)

[0069] Where M is the mass matrix and C is the damping matrix of the conductor. For the nodal forces inside the conductor, This refers to the external aerodynamic load on the conductor. The acceleration at the conductor node, The velocity at the conductor node, This represents the displacement of the conductor node.

[0070] S2. Solve the time domain problem using a composite time integral algorithm;

[0071] Bathe's algorithm is a composite time integration scheme. Assume time... The solution at time is fully known; the goal is to compute the time. The solution. This algorithm introduces a parameter. Time step It is divided into two sub-steps. The first sub-step starts from... arrive The second step from arrive .

[0072] The core feature of this algorithm lies in the first sub-step. The trapezoidal rule is used for integration. The trapezoidal rule is second-order accurate and unconditionally stable, and it can well preserve the accuracy of the low-frequency response. In the second sub-step... Adopting based on , and A three-point backward difference scheme for three time-phase information. This scheme has L-stability and can effectively dissipate high-frequency pseudo-oscillations introduced by model discretization or nonlinear iteration, thereby improving the overall stability and robustness of the algorithm.

[0073] S21. In the first sub-step calculation, a solution is required. The unknown quantity at time t. The discretized equilibrium equation at that time is:

[0074] (2)

[0075] Velocity and acceleration are related to displacement through the assumption of the trapezoidal law:

[0076] (3)

[0077] (4)

[0078] Substituting equations (3) and (4) into equation (2), we obtain the following about The nonlinear equation system is solved iteratively using the Newton-Raphson method. Let... This is the result of the previous iteration. For the current iteration step, the displacement increment is Then the first The linearized equation for the next iteration is:

[0079] (5)

[0080] in Is The tangent stiffness matrix calculated at the point, This part is one of the innovative aspects of this patent: introducing the aerodynamic stiffness matrix into the equations, which results in better stability and convergence. After iterative convergence, the following can be obtained: displacement at time The corresponding velocity and acceleration can be calculated using equations (3) and (4).

[0081] The acceleration update is as follows:

[0082] ; ;

[0083] ; ; (6)

[0084] The speed update is as follows:

[0085] ; ;

[0086] ; ; (7)

[0087] S22. In the second sub-step, the solution is required. The unknown quantity at time t. The discretized equilibrium equation at that time is:

[0088] (8)

[0089] make

[0090] (9)

[0091] The update formulas for velocity and acceleration are then:

[0092] (10)

[0093] (11)

[0094] Substituting equations (9), (10), and (11) into equation (8), and using the Newton-Raphson method iteratively to solve, the th... The linearized equation for the next iteration is:

[0095] (12)

[0096] in Is The tangent stiffness matrix is ​​calculated at the given location. After iterative convergence, the result is obtained. displacement at time The corresponding velocity and acceleration can be calculated using equations (10) and (11).

[0097] S3. Coupled solution of velocity-dependent nonlinear aerodynamic loads;

[0098] Nonlinear aerodynamic loads include dynamic lift. Aerodynamic drag and aerodynamic torque They are determined by: using data fitted from wind tunnel tests, and the instantaneous effective angle of attack. The relevant lift coefficient, drag coefficient, and torque coefficient; combined with air density, incoming wind speed, and conductor characteristic dimensions, the Newton-Raphson nonlinear iterative algorithm is used in the solution process of each sub-step of step S2. In each iteration, the following operations are performed: based on the instantaneous translational velocity and instantaneous torsional angular velocity of the conductor nodes in the current iteration step, the instantaneous effective angle of attack of the conductor cross-section relative to the incoming wind is calculated in real time. Thus, the aerodynamic force and aerodynamic torque acting per unit length of the conductor can be calculated;

[0099] Instantaneous effective angle of attack The specific calculation process is as follows: When the icing conductor is stationary, the aerodynamic load on the icing conductor can be measured through wind tunnel experiments. The lift force per unit length of the icing conductor is measured under different angles of attack by fixing the conductor's rotation axis. ,resistance and torque The specific calculation formula is as follows:

[0100] (13a)

[0101] (13b)

[0102] (13c)

[0103] (14)

[0104] The actual wind speed U, along the positive horizontal direction, causes vertical velocity, horizontal, and torsional motion of the conductor. θ0 is the initial wind angle of the icing conductor. t Let θ be the rotation angle of the conductor under the action of external torque, Δθ1 be the change in the direction of relative wind speed caused by the angular velocity of the conductor's rotation under the quasi-static assumption and the small rotation assumption, Δθ2 be the change in the direction of relative wind speed caused by the vertical velocity of the conductor under the quasi-static assumption and the small angle assumption, and Ur be the relative wind speed caused by the combination of various factors.

[0105] R is the characteristic radius, which can be determined based on experimental parameters; generally, the radius of the bare conductor is chosen. In the formula... The wind speed is relative to the cross-section of the conductor.

[0106] Among them, the instantaneous effective angle of attack is calculated by combining (13a), (13b) and (13c). Its instantaneous effective angle of attack The calculation comprehensively considers the following factors: the horizontal incoming wind speed, the instantaneous translational velocity of the conductor cross-section in the horizontal and vertical directions, the tangential linear velocity of the conductor at its characteristic radius due to torsional motion, and the instantaneous torsion angle of the conductor.

[0107] The instantaneous torsion angle at any point on the conductor axis is obtained by linear interpolation of the instantaneous torsion angle vector at the end node of the conductor. The calculation principle is as follows:

[0108] like Figure 6 As shown, assuming the span of the power transmission line suspended at both ends is L, the sag under its own weight is f, and the conductor will form a non-circular cross-section after being covered with ice, and will be subjected to lift force F under wind load. L Resistance F D And torque M. The spatial coordinate of any point on the centerline of the conductor under wind load is x. To describe the shape of the conductor, it is divided into n-1 elements by n nodes. The current spatial coordinate vector of any point on the axis of the conductor element is... It can be obtained through the coordinate vectors of the two endpoints. and Linear interpolation yields:

[0109] (15)

[0110] In the above formula, the node coordinate vector is The linear interpolation function is: and ,in s is an arc coordinate. This represents the current length of the element. The torsion angle at any point on the element. The torsion angle at both ends can be used. and Perform linear interpolation: ,here The superscript indicates a node The angle of twist.

[0111] resistance The relative wind speed Ur is in the same direction, while the lift... Perpendicular to the relative wind speed Ur, when obtaining the dynamic equations for the conductor's motion, it is necessary to project the changing drag and lift onto the coordinate axes to obtain the lift and drag projected onto the coordinate axes:

[0112] (16a)

[0113] (16b)

[0114] After determining the quasi-static aerodynamic force components and torques acting per unit length in the global coordinate system, the quasi-static nodal load vector of the element is... This can be obtained by weighted averaging using shape functions. Assuming axial aerodynamic forces are negligible and integration is performed using natural coordinates, then:

[0115] (17)

[0116] Points obtained It is the nodal load vector in the global coordinate system, which can be directly used to assemble the global finite element equations.

[0117] S4. Obtain the transmission line galloping response:

[0118] By repeatedly executing steps S2 and S3, the displacement, velocity, and acceleration response of the transmission line within a set time range are calculated, thereby simulating the galloping process of the icy transmission line.

[0119] This method can stably and accurately simulate the modal coupling and beat vibration phenomena of transmission lines under specific Irvine parameters. By effectively suppressing numerical noise, it clearly reveals the envelope, frequency composition, and complex attractor structure in phase space of the beat vibration.

[0120] This simulation system for galloping of icing transmission lines based on composite time integral includes:

[0121] The model building module is used to execute step S1 as described in claim 1 to establish a nonlinear finite element model of the ice-covered transmission line;

[0122] The time-domain solution module is configured to execute steps S2 and S3 as described in claim 1, and it integrates a composite time integration algorithm and a coupled iterative solution logic for nonlinear aerodynamic loads.

[0123] A processor for executing instructions from the model building module and the time-domain solution module;

[0124] The memory is used to store the finite element model data, intermediate variables during the calculation process, and the final dancing response results.

[0125] Specific examples

[0126] A conductor with a span of 100m is selected, the sag at the lowest point is 0.8802m, and it is divided into 51 nodes and 50 units. The cross-sectional area of ​​the conductor is 0.000452mm². 2 The conductor tension is 24 kN. Below... Figure 2 The method provided in this patent calculates the dancing results, and it can be seen that the data is smooth and has good stability. Figure 3The results of the dancing motion were obtained by the traditional Newmark implicit time integration method. As can be seen from the figure, the data is highly discrete and unstable.

[0127] Calculation example:

[0128] Figure 4 Considering a nonlinear motion model of a rod with constraints at both ends and considering galloping, where the mass per unit length of the rod is m and the density is ρ, the equivalent mass at node 2 is: Then the dynamic equations for node 2 can be given:

[0129]

[0130] Performing a Taylor expansion on the above polynomial yields the ordinary differential equation of a cubic polynomial:

[0131]

[0132] The parameters for working condition 1 are: EA0=1, A0=0.001, E=1000, l0=1, θ0=45°, and the mass per unit length m=1. Based on this assumption, the equivalent mass of element node 2 is 0.5.

[0133] Based on operating condition 1, some parameters related to wind speed, density, and cross-sectional diameter were added, and an aerodynamic force Fy was modified.

[0134]

[0135] Substituting the cubic polynomial into the ordinary differential equation, we get:

[0136]

[0137] in: The calculated aerodynamic forces; air density; The incoming flow velocity; The diameter of the ice-covered conductor; The characteristic length; This is the drag coefficient.

[0138] Finally, displacement maps were obtained over a time interval of 0 to 40 seconds. These maps were then plotted against the results obtained using Matlab code, yielding the following results. Figure 5 It can be seen that the simulation results are basically consistent with the results calculated by the Matlab code, proving that the simulation method can not only ensure accurate tracking of the low-frequency dominant physical phenomenon of dancing, but also effectively suppress high-frequency numerical noise, thus maintaining good convergence and stability even with a large time step.

[0139] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for simulating galloping of icing transmission lines based on composite time integral, characterized in that, Includes the following steps: S1. Establish a nonlinear finite element model of the ice-covered transmission line; The transmission line is discretized into multiple two-node cable elements, each node containing translational degrees of freedom and torsional degrees of freedom about the element axis, thus obtaining a semi-discrete nonlinear control equation describing the dynamic behavior of the entire transmission line system, which includes the mass matrix, damping matrix, and internal force vectors: （1） Where M is the mass matrix and C is the damping matrix of the conductor. For the nodal forces inside the conductor, The external aerodynamic load on the conductor. The acceleration at the conductor node, The velocity at the conductor node, For conductor node displacement; S2. Solve the time domain problem using a composite time integral algorithm; according to The solution at time step is calculated in two sub-steps. The solution at any given moment, the first sub-step from arrive The second sub-step from arrive First step Integrating using the trapezoidal rule, the second sub-step Adopting based on , and Three-point backward difference format for information at three time points; S21. In the first sub-step calculation... The time-discretized equilibrium equation is: (2) Velocity and acceleration are related to displacement by the assumption of the trapezoidal law as follows: (3) (4) Substituting equations (3) and (4) into equation (2), we obtain the following about The nonlinear equation system is solved iteratively using the Newton-Raphson method; let... This is the result of the previous iteration. For the current iteration step, the displacement increment is Then the first The linearized equation for the next iteration is: (5) in Is The tangent stiffness matrix calculated at the point, By incorporating the aerodynamic stiffness matrix into the equations and iteratively converging, we obtain... displacement at time The updated velocity and acceleration are obtained by calculating using equations (6) and (7); The acceleration update is as follows: ； ； ; ; (6) The speed update is as follows: ； ； ; ; (7)； S22. In the second sub-step calculation... The time-discretized equilibrium equation is: (8) in: (9) but The update formulas for velocity and acceleration at any given time are: (10) (11) Substituting equations (9), (10), and (11) into equation (8), and using the Newton-Raphson method iteratively to solve, the th... The linearized equation for the next iteration is: (12) in Is The calculated tangent stiffness matrix can be obtained after iterative convergence. displacement at time And calculate the corresponding velocity and acceleration using equations (10) and (11); S3. Coupled solution of velocity-dependent nonlinear aerodynamic loads; Nonlinear aerodynamic loads include dynamic lift. Aerodynamic drag and aerodynamic torque They are determined by: using data fitted from wind tunnel tests, and the instantaneous effective angle of attack. The relevant lift coefficient, drag coefficient, and torque coefficient; combined with air density, incoming wind speed, and conductor characteristic dimensions, the Newton-Raphson nonlinear iterative algorithm is used in the solution process of each sub-step of step S2. In each iteration, the following operations are performed: based on the instantaneous translational velocity and instantaneous torsional angular velocity of the conductor nodes in the current iteration step, the instantaneous effective angle of attack of the conductor cross-section relative to the incoming wind is calculated in real time. Thus, the aerodynamic force and aerodynamic torque acting per unit length of the conductor can be calculated; S4. Obtain the transmission line galloping response: Repeat steps S2 and S3 to calculate the displacement, velocity, and acceleration response of the transmission line within a set time range, thereby simulating the galloping process of the icy transmission line.

2. The method for simulating galloping of iced transmission lines based on composite time integration as described in claim 1, characterized in that, In step S1, the damping matrix of the cable element is constructed using the Rayleigh damping model. This damping matrix is ​​represented as a linear combination of the element mass matrix and the initial elastic stiffness matrix, and its damping coefficient is determined based on the first two modal frequencies of the structure and the corresponding modal damping ratios.

3. The method for simulating galloping of iced transmission lines based on composite time integration as described in claim 1, characterized in that, In the solution process of each sub-step of step S2, the Newton-Raphson nonlinear iterative algorithm is adopted, and the following operations are performed in each iteration: based on the instantaneous translational velocity and instantaneous torsional angular velocity of the conductor node in the current iteration step, the instantaneous effective angle of attack of the conductor cross section relative to the incoming wind is calculated in real time. .

4. The method for simulating galloping of iced transmission lines based on composite time integration as described in claim 1, characterized in that, In step S21, the tangent stiffness matrix is ​​defined as the sum of the material stiffness matrix and the geometric stiffness matrix. The geometric stiffness matrix is ​​a function of the current axial internal force of the element, used to accurately capture the strong geometric nonlinear effects of the conductor caused by large displacement and large rotation.

5. The method for simulating galloping of iced transmission lines based on composite time integration as described in claim 1, characterized in that, Instantaneous effective angle of attack in step S3 The specific calculation process is as follows: When the icing conductor is stationary, the aerodynamic load on the icing conductor is measured through wind tunnel experiments, with different angles of attack measured while the conductor's rotation axis is fixed. Under these conditions, the icing conductor experiences aerodynamic lift per unit length. Aerodynamic drag and aerodynamic torque The specific calculation formula is as follows: (13a) (13b) (13c) (14) Among them, the actual wind speed U is along the positive horizontal direction, causing the conductor to move in the vertical, horizontal and torsional directions; The initial wind angle of the icing conductor. Let be the angle of rotation of the conductor under the action of an external torque. This refers to the change in relative wind speed direction caused by the angular velocity of the conductor's rotation under the quasi-static and small rotation assumptions. The relative wind speed direction is determined by the vertical velocity of the conductor under the quasi-static and small-angle assumptions. Ur represents the relative wind speed caused by a combination of various factors; R is the radius of the bare conductor. The wind speed is relative to the cross-section of the conductor.

6. The method for simulating galloping of iced transmission lines based on composite time integration as described in claim 5, characterized in that, In step S3, the instantaneous torsion angle at any point on the conductor axis is obtained by linear interpolation of the instantaneous torsion angle vector at the end node of the conductor, and the resistance... The relative wind speed Ur is in the same direction, while the lift... Perpendicular to the relative wind speed Ur, when obtaining the dynamic equations for the conductor's motion, it is necessary to project the changing drag and lift onto the coordinate axes to obtain the lift and drag projected onto the coordinate axes: (16a) (16b) After determining the quasi-static aerodynamic force components and torques acting per unit length in the global coordinate system, the quasi-static nodal load vector of the element is... The weighted average is obtained through shape functions; assuming axial aerodynamic forces are negligible and integration is performed using natural coordinates, then: (17) Points obtained It is the nodal load vector in the global coordinate system, which can be directly used to assemble the global finite element equations.

7. The method for simulating galloping of iced transmission lines based on composite time integration as described in claim 6, characterized in that, This method can stably and accurately simulate the modal coupling and beat vibration phenomena of transmission lines under specific Irvine parameters. By effectively suppressing numerical noise, it clearly reveals the envelope, frequency composition, and complex attractor structure in phase space of the beat vibration.

8. A system for simulating galloping of iced transmission lines based on composite time integration, wherein the method for simulating galloping of iced transmission lines as described in any one of claims 1 to 7 is characterized in that, include: The model building module is used to execute step S1 as described in claim 1 to establish a nonlinear finite element model of the ice-covered transmission line; The time-domain solution module is configured to execute steps S2 and S3 as described in claim 1, and it integrates a composite time integration algorithm and a coupled iterative solution logic for nonlinear aerodynamic loads. A processor for executing instructions from the model building module and the time-domain solution module; The memory is used to store the finite element model data, intermediate variables during the calculation process, and the final dancing response results.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the method for simulating the galloping of icing power transmission lines as described in any one of claims 1 to 7.

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