Method for solving conductor galloping trajectory equation of high-voltage transmission line

Abstract

The present invention relates to the technical field of power transmission conductor galloping, and in particular to a method for solving a conductor galloping trajectory equation of a high-voltage transmission line. The method comprises the following steps: establishing a coordinate system; analyzing a nonlinear motion of any point on a conductor; solving a conductor galloping trajectory equation; solving a torsion angle and an acceleration under an arc length constraint; and carrying out convex optimization on an analytic expression of the torsion angle. According to the present invention, an inter-station RTK differential mode is used, and Beidou observation data analyzed by a monitoring station and Beidou observation data analyzed by a reference station are combined, thereby improving the accuracy of a monitoring result; a finite element analysis method is used to carry out discretization processing on the problem of conductor motion integral, so that conductor displacement can be effectively restored to obtain a good curve fitting effect; and a calculation method for the torsion angle under the limitation of the arc length of the conductor is provided, overcoming the error of calculating a trajectory integral by using the acceleration, and an optimized calculation method is provided.

Description

A method for solving the galloping trajectory equation of high-voltage transmission line conductors Technical Field

[0001] The present invention relates to the technical field of transmission line galloping, and in particular to a method for solving a high-voltage transmission line conductor galloping trajectory equation. Background Art

[0002] Galloping of high-voltage transmission lines is a common problem plaguing the power engineering community both domestically and internationally. With the development of my country's national economy, and particularly the uneven geographical distribution of modern industry, extensive, long-distance transmission networks are becoming increasingly necessary. The safe and stable operation of transmission lines is fundamental to the development of the power industry. When external forces (such as wind and icing pressure) act on conductors, they induce a low-frequency, high-amplitude, self-excited vibration that produces an up-and-down oscillation. Transmission line galloping is a particularly destructive threat to power systems, often causing damage to conductors, hardware, and tower components. In severe cases, it can even lead to tower collapse.

[0003] Conductor galloping is the oscillatory motion of a transmission line under the influence of external forces. Using Beidou positioning to obtain conductor galloping posture information can restore the conductor galloping trajectory for safety monitoring. The conductor galloping trajectory can be described by a smooth spatial curve. Unlike general curve fitting and interpolation problems, the conductor galloping trajectory curve has a fixed length. To obtain a relatively accurate conductor shape, these geometric constraints must be considered. Therefore, restoring the conductor galloping trajectory is a curve fitting problem under arc length constraints. Traditional discrete models using broken line segment chains replace nonlinear arc length constraints with linear constraints, and can only solve small deformation problems. Furthermore, the divergence of motion acceleration caused by conductor torsion causes singularities in the conductor galloping trajectory.

[0004] In view of this, we propose a method to solve the galloping trajectory equation of high-voltage transmission line conductors.

[0005] Summary of the Invention

[0006] The object of the present invention is to provide a method for solving the galloping trajectory equation of a high-voltage transmission line conductor, so as to solve the problems raised in the above background technology.

[0007] In order to solve the above-mentioned technical problems, one of the purposes of the present invention is to provide a transmission line galloping monitoring system, that is, a pre-built transmission line galloping monitoring system based on Beidou RTK differential technology, which is used to solve the problems of acceleration divergence and curvature limitation in conductor galloping; by numerically simulating the galloping of the line, a polynomial is used as an approximate function in the displacement mode of the unit in the finite element method, so that the shape of the conductor galloping trajectory under the action of gravity is approximately an ellipse; and an equation of the conductor displacement with respect to coordinates and torsion angle is established to maximize the restoration of the conductor galloping trajectory.

[0008] As a further improvement to this technical solution, a complete transmission line gallop monitoring system consists of a base station receiver, one or more gallop sensors (mobile station receivers), a Beidou antenna, and a backend computer. The system's receiver and antenna are connected via cables, while receivers themselves and the computer are connected via wireless devices to transmit and receive data. The base station receiver and its Beidou antenna are installed in an open area near the control room, while the gallop sensors (mobile station receivers) and their Beidou antennas are mounted on high-voltage conductors.

[0009] A second object of the present invention is to provide a method for solving the galloping trajectory equation of a high-voltage transmission line conductor. The specific solution method includes the following steps:

[0010] S1. Establish a coordinate system;

[0011] S2. Analyze the nonlinear motion of any point on the wire;

[0012] S3, solve the conductor dancing trajectory equation;

[0013] S4. Solve the torsion angle and acceleration under the arc length constraint;

[0014] S5. Perform convex optimization on the analytical expression of the torsion angle.

[0015] As a further improvement of this technical solution, in step S1, the process of establishing the coordinate system is as follows:

[0016] Establish a global Cartesian coordinate system X, Y, Z and establish local coordinates along the wire axis. Assume that the displacement components of any point on the wire in the X, Y, and Z directions are U, D, and W respectively, and the torsion angle is θ. The displacement vector of the node is q. Where: q = (U, D, W, θ)

[0017] Assume that the horizontal span between two towers is K and the length of the inextensible transmission line is L. The corresponding length on the conductor is L. q , multiple BeiDou positioning units are installed at points q = 1, 2, ..., N. After the torsion information obtained at time t is converted, the tangent direction of the wire is obtained as T lq, where the length of the curve is L q The unit tangent vector at t is used to fit the spatial trajectory of the entire conductor at time t. Taking the mathematical model of a two-dimensional plane curve as an example, it is described as follows. With the left tower as the coordinate origin P0 = (0, 0), a spatial coordinate system is established. The direction toward the right tower is the positive direction of the X-axis, and the coordinates of the right endpoint of the conductor are P1 = (K, 0).

[0018] As a further improvement of the present technical solution, in step S2, the motion analysis of any point on the wire is as follows:

[0019] The speed of the wire is V, and the speeds in the X and Y directions are V x ,V y , the acceleration is a; then the balance equation of the wire can be derived according to the relevant method as follows:

[0020] Among them, M is the mass matrix of the unit, C is the damping matrix, K is the stiffness matrix, and F is the external force matrix;

[0021] Taking the X direction as an example, the velocity can be expressed as: V x (t) = V x0 +∫0 t a(t)dt

[0022] Among them, V x0 is the initial velocity; then the horizontal displacement U can be expressed as: U(t)=U0+∫0 t V x (t)dt=U0+∫0 t (V x0 +∫0 t a(t)dt)dt

[0023] Among them, U0 is the initial displacement component of the point in the X direction;

[0024] Furthermore, at a certain time interval [0,t s ] is divided into several equal moments: t0, t1, ..., t s , each small time period is: Δt1=t1-t0, Δt2=t2-t1,...,Δt s =t s -t s-1 , the speeds of each time period are: V1, V2, ..., V s , then we can get: ΔV i =a(t i )Δt i ,i=1,2,...,s

[0025] Where, ΔV iis the speed change value in each small period of time;

[0026] Discretizing the velocity and displacement in the X direction, we can obtain:

[0027] Similarly, the calculation formula for the displacement D in the Y direction can be obtained as follows:

[0028] Among them, D y (t i ) is the discretized value of the displacement in the Y direction, and D0 is the initial displacement component of the point in the Y direction.

[0029] As a further improvement of the present technical solution, in step S3, the method for solving the conductor dancing trajectory is a method combining Newmark-β and Newton-Raphson, and the specific process is as follows:

[0030] First, the following assumptions are made about the node displacement vector q using the Newmark-β method:

[0031] Among them, ξ and λ are parameters required for integration accuracy and stability, and Δt is the time increment. Combining the equilibrium equation, the displacement without iteration can be obtained as:

[0032] Where R represents the modal matrix;

[0033] Let n be the number of iteration steps, then:

[0034] After updating the stiffness matrix K and iterating, the displacement after iteration can be obtained as:

[0035] Considering the influence of torsion angle on acceleration under the limitation of wire arc length, it is necessary to discuss the relationship between acceleration and angle to reduce the error in the acceleration calculation process.

[0036] As a further improvement of the present technical solution, in step S4, the analysis of the wire torsion angle is as follows:

[0037] Find an arc length parametric curve that satisfies the following conditions: r(s),0≤s≤L r(0)=P0,r(L)=P1,r′(L q )=T lq

[0038] Obviously, there are countless such curves, and we usually hope to find the smoothest one, that is, the minimum energy curve. In the process of finding the minimum energy curve, the angle between the tangent and the positive direction of the Y axis is taken as θ(s), and the curve energy is: E=∫0 L K 2 (s)ds=∫0 L (θ′(s)) 2 ds

[0039] Where K(s) is the curvature; the above problem can be described by the optimization model as follows: min E Subjectto:r(0)=P0,r(L)=P1,r′(L q )=T lq

[0040] Among them, r(0)=P0 represents the starting point of the curve. At the origin of the coordinate system, r′(L q )=T lq , the form is relatively simple. The more complicated part is the constraint r(L)=P1 on the right end of the curve, because in most cases the integral on the left end has no analytical form. Next, we approximate r(L) by discretizing θ(s). Discretize the arc length parameter curve with N broken line segments, each of which is of equal length and L / N. The angle between the i-th broken line segment and the positive direction of the Y axis is θ i (θ i ∈[0,π]), then the above optimization problem is discretized into the following form:

[0041] The starting point is the origin of the coordinate system, so r(0)=P0 naturally satisfies, where k=KN / L; θ lq is the angle corresponding to the broken line segment; because T lq is a unit vector, so r′(L q )=T lq It can be further simplified to cosθ lq =T lq,y ,θ lq is the tangent direction angle of the monitoring point with a curve length of lq.

[0042] As a further improvement of the present technical solution, in step S5, the torsion angle is optimized, and the specific method is as follows:

[0043] Because the constraints contain sine and cosine functions, in actual calculations, in order to transform the model into an optimization model that is easier to handle, we can continue to make variable substitutions and let: cosθ i =x i , and obtain the following optimization model:

[0044] The expression of E in the objective function is also re-expressed;

[0045] According to actual needs, through a series of transformations, a curve fitting optimization model under arc length constraints is derived; the nonlinear constraint is approximated by a first-order Taylor approximation at the optimal solution obtained at the previous sampling point, and then a convex optimization model with linear constraints is obtained. The objective function is a convex function, and the constraints are all linear constraints, which are also convex. The existing optimization algorithm can easily find the global minimum, that is:

[0046] Where f(x) is a linear constraint equation.

[0047] The third object of the present invention is to provide a solution method running platform device, including a processor, a memory, and a computer program stored in the memory and running on the processor, and the processor is used to implement the steps of the solution method of the high-voltage transmission line conductor dancing trajectory equation when executing the computer program.

[0048] A fourth object of the present invention is to provide a computer-readable storage medium, which stores a computer program, and when the computer program is executed by a processor, it implements the steps of the method for solving the above-mentioned high-voltage transmission line conductor dancing trajectory equation.

[0049] Compared with the prior art, the present invention has the following beneficial effects:

[0050] 1. This method for solving the high-voltage transmission line conductor galloping trajectory equation uses inter-station RTK differential mode, combining BeiDou observation data analyzed by the monitoring station and the base station to improve the accuracy of the monitoring results.

[0051] 2. The method for solving the galloping trajectory equation of the high-voltage transmission line conductor adopts the finite element analysis method to discretize the conductor motion integral problem, which can effectively restore the conductor displacement and obtain good curve fitting results;

[0052] 3. In the solution method of the high-voltage transmission line conductor dancing trajectory equation, a calculation method of the torsion angle under the conductor arc length limitation is proposed, which overcomes the error caused by using acceleration to calculate the trajectory integral, and proposes an optimized calculation method. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] FIG1 is an exemplary architecture diagram of a transmission line galloping monitoring system according to the present invention;

[0054] FIG2 is a flow chart of an exemplary method for solving the galloping trajectory equation of a high-voltage transmission line conductor in the present invention;

[0055] FIG3 is a structural diagram of an exemplary electronic computer platform device in the present invention. DETAILED DESCRIPTION

[0056] The following will provide a clear and complete description of the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0057] Example 1

[0058] As shown in Figure 1, this embodiment provides a transmission line gallop monitoring system based on Beidou RTK differential technology. This system can address the issues of acceleration divergence and arc limitation in conductor galloping. By numerically simulating the line galloping and using polynomials as approximate functions in the displacement mode of the unit in the finite element method, the conductor galloping trajectory, under the influence of gravity, is approximately elliptical. This system offers high reliability, strong resistance to electromagnetic interference, and accurate gallop measurement, making it suitable for long-term online monitoring of conductor galloping.

[0059] In this embodiment, the above-mentioned transmission line galloping monitoring system can realize real-time monitoring of the galloping trajectory of a certain point on the conductor. A complete conductor galloping monitoring system consists of a base station receiver, one or more galloping sensors (mobile station receivers), a Beidou antenna, and a background computer. The receiver and antenna of the system are connected by cables, and the receivers and the computer are connected by wireless devices to send and receive data. Among them, the base station receiver and its Beidou antenna are installed in a relatively open area near the control room, and the galloping sensor (mobile station receiver) and its Beidou antenna are installed on the high-voltage conductor. When the conductor gallops, the mobile station antenna located on the conductor moves with the conductor. After a series of complex calculations, the position information of the mobile station antenna is sent to the background computer in real time. The monitoring software performs real-time analysis and processing on the received data, extracts the coordinate value of the mobile station antenna in the geocentric rectangular coordinate system, and draws its motion trajectory in the geocentric rectangular coordinate system based on the changes in the coordinate value.

[0060] Specifically, when a transmission line gallops, the line will produce coupled vibrations in the vertical, lateral, and torsional directions, especially the vertical and lateral motion amplitudes are large. During the galloping process, the motion state is nonlinear. The cross-section of the wire is generally asymmetric. In addition to drag and lift, the aerodynamic force it is subjected to in the external force field also includes torque, which causes torsional motion. Considering the acceleration dispersion caused by the change in the torsion angle, in order to restore the wire galloping trajectory to the greatest extent possible, it is necessary to establish an equation for the wire displacement with respect to the coordinates and torsion angle.

[0061] As shown in FIG2 , this embodiment further provides a method for solving the galloping trajectory equation of a high-voltage transmission line conductor. First, a nonlinear motion equation is established. The specific method is as follows: a displacement function with three translational coordinates X, Y, and Z and one torsional degree of freedom θ is used, and the information of discrete points is curve-fitted using a finite element analysis method to obtain the galloping equation of the conductor. Then, a corresponding nonlinear iterative solution method is provided, a method for calculating the torsion angle under the arc length constraint of the conductor is provided, and convex optimization processing is performed, and a corresponding calculation program is compiled. The specific solution method includes the following steps:

[0062] S1. Establish a coordinate system: Establish three-dimensional coordinates at a certain point, and establish a coordinate system in the entire wire dancing device model, with one wire rod as the origin of the coordinate system and the other wire rod as the end point of the coordinate system. Convert the longitude, latitude, and altitude information obtained by Beidou positioning into standard values ​​in this coordinate system;

[0063] In this step, the process of establishing the coordinate system is as follows:

[0064] Establish a global Cartesian coordinate system X, Y, Z and establish local coordinates along the wire axis. Assume that the displacement components of any point on the wire in the X, Y, and Z directions are U, D, and W respectively, and the torsion angle is θ. The displacement vector of the node is q. Where: q = (U, D, W, θ)

[0065] Assume that the horizontal span between two towers is K and the length of the inextensible transmission line is L. The corresponding length on the conductor is L. q , multiple BeiDou positioning units are installed at points q = 1, 2, ..., N. After the torsion information obtained at time t is converted, the tangent direction of the wire is obtained as T lq , where the length of the curve is L q The unit tangent vector at t is used to fit the spatial trajectory of the entire conductor at time t. Taking the mathematical model of a two-dimensional plane curve as an example, it is described as follows. With the left tower as the coordinate origin P0 = (0, 0), a spatial coordinate system is established. The direction toward the right tower is the positive direction of the X-axis, and the coordinates of the right endpoint of the conductor are P1 = (K, 0).

[0066] S2. By analyzing the nonlinear motion of any point on the conductor, its displacement change characteristics and equilibrium equations can be derived, and the four-dimensional information of the horizontal, vertical, height and torsion of a certain point on the conductor can be obtained;

[0067] In this step, the motion analysis of any point on the wire is as follows:

[0068] The speed of the wire is V, and the speeds in the X and Y directions are V x ,V y , the acceleration is a; then the balance equation of the wire can be derived according to the relevant method as follows:

[0069] Among them, M is the mass matrix of the unit, C is the damping matrix, K is the stiffness matrix, and F is the external force matrix;

[0070] Taking the X direction as an example, the velocity can be expressed as: V x (t) = V x0 +∫0 t a(t)dt

[0071] Among them, V x0 is the initial velocity; then the horizontal displacement U can be expressed as: U(t)=U0+∫0 t V x (t)dt=U0+∫0 t (V x0 +∫0 t a(t)dt)dt

[0072] Among them, U0 is the initial displacement component of the point in the X direction;

[0073] Furthermore, at a certain time interval [0,t s ] is divided into several equal moments: t0, t1, ..., t s , each small time period is: Δt1=t1-t0, Δt2=t2-t1,...,Δt s =t s -t s-1 , the speeds of each time period are: V1, V2, ..., V s , then we can get: ΔV i =a(t i )Δt i ,i=1,2,...,s

[0074] Where, ΔV i is the speed change value in each small period of time;

[0075] Discretizing the velocity and displacement in the X direction, we can obtain:

[0076] Similarly, the calculation formula for the displacement D in the Y direction can be obtained as follows:

[0077] Among them, D y (t i ) is the discretized value of the displacement in the Y direction, and D0 is the initial displacement component of the point in the Y direction.

[0078] S3. Solve the conductor's trajectory equation: Place multiple Beidou positioning terminals at equal distances at multiple locations on the conductor and use a specific broken line or curve fitting method to fit the entire conductor's trajectory.

[0079] In this step, the method for solving the conductor dancing trajectory is a combination of the Newmark-β and Newton-Raphson methods. The specific process is as follows:

[0080] First, the following assumptions are made about the node displacement vector q using the Newmark-β method:

[0081] Among them, ξ and λ are parameters required for integration accuracy and stability, and Δt is the time increment. Combining the equilibrium equation, the displacement without iteration can be obtained as:

[0082] Where R represents the modal matrix;

[0083] Let n be the number of iteration steps, then:

[0084] After updating the stiffness matrix K and iterating, the displacement after iteration can be obtained as:

[0085] Considering the influence of torsion angle on acceleration under the limitation of wire arc length, it is necessary to discuss the relationship between acceleration and angle to reduce the error in the acceleration calculation process.

[0086] S4. Torsion angle analysis, solving the torsion angle and acceleration under arc length constraints: Analyze the torsion angle under the wire arc length constraint, obtain the optimal angle solution under the constraint conditions, and substitute the torsion angle into the specific acceleration calculation formula;

[0087] In this step, the analysis of the wire torsion angle is as follows:

[0088] Find an arc length parametric curve that satisfies the following conditions: r(s),0≤s≤L r(0)=P0,r(L)=P1,r′(L q )=T lq

[0089] Obviously, there are countless such curves, and we usually hope to find the smoothest one, that is, the minimum energy curve. In the process of finding the minimum energy curve, the angle between the tangent and the positive direction of the Y axis is taken as θ(s), and the curve energy is: E=∫0 L K 2 (s)ds=∫0 L (θ′(s)) 2 ds

[0090] Where K(s) is the curvature; the above problem can be described by the optimization model as follows: min E Subjectto:r(0)=P0,r(L)=P1,r′(L q )=T lq

[0091] Among them, r(0)=P0 represents the starting point of the curve. At the origin of the coordinate system, r′(L q )=T lq , the form is relatively simple. The more complicated part is the constraint r(L)=P1 on the right end of the curve, because in most cases the integral on the left end has no analytical form. Next, we approximate r(L) by discretizing θ(s). Discretize the arc length parameter curve with N broken line segments, each of which is of equal length and L / N. The angle between the i-th broken line segment and the positive direction of the Y axis is θ i (θ i ∈[0,π]), then the above optimization problem is discretized into the following form:

[0092] The starting point is the origin of the coordinate system, so r(0)=P0 naturally satisfies, where k=KN / L; θ lq is the angle corresponding to the broken line segment; because T lq is a unit vector, so r′(L q )=T lq It can be further simplified to cosθ lq =T lq,y ,θ lq is the tangent direction angle of the monitoring point with a curve length of lq.

[0093] S5. Perform convex optimization on the analytical expression of the torsion angle: Perform a first-order Taylor approximation on the optimal solution of the nonlinear constraint to obtain a convex optimization model with linear constraints, which makes it easy to find the global minimum.

[0094] In this step, the torsion angle is optimized, and the specific method is as follows:

[0095] Because the constraints contain sine and cosine functions, in actual calculations, in order to transform the model into an optimization model that is easier to handle, we can continue to make variable substitutions and let: cosθi =x i , and obtain the following optimization model:

[0096] The expression of E in the objective function has also been re-expressed. On the one hand, it makes the problem form simpler, and on the other hand, it does not deviate from the meaning of making the curve as smooth as possible (because the cosine values ​​of the corresponding direction angles at two adjacent broken line segments are also approximately equal, and conversely, the corresponding tangent angles with similar cosine values ​​are also similar, because the cosine function is monotonic in the interval [0,π], which is why sine is not used for variable substitution).

[0097] According to actual needs, through a series of transformations, a curve fitting optimization model under arc length constraints is derived. Because it contains nonlinear constraints, solving the above problem belongs to the category of nonlinear programming. The main optimization tools are sequential quadratic programming and interior point method. Because the actual dancing frequency is much lower than the sampling frequency, the shape of the curve corresponding to the two adjacent sampling points will not change much, and there is no need to run the above optimization model every time data is obtained. The following processing is performed to make a first-order Taylor approximation of the nonlinear constraint at the optimal solution obtained at the previous sampling point, and then a convex optimization model with linear constraints is obtained. The objective function is a convex function, and the constraints are all linear constraints, which are also convex. The existing optimization algorithm can easily find the global minimum, that is:

[0098] Where f(x) is a linear constraint equation.

[0099] As shown in FIG3 , this embodiment further provides a solution method running platform device, which includes a processor, a memory, and a computer program stored in the memory and running on the processor.

[0100] The processor includes one or more processing cores, and the processor is connected to the memory via a bus. The memory is used to store program instructions. When the processor executes the program instructions in the memory, the steps of the method for solving the above-mentioned high-voltage transmission line conductor dancing trajectory equation are implemented.

[0101] Alternatively, the memory may be implemented by any type of volatile or non-volatile memory device or a combination thereof, such as static random access memory (SRAM), electrically erasable programmable read-only memory (EEPROM), erasable programmable read-only memory (EPROM), programmable read-only memory (PROM), read-only memory (ROM), magnetic memory, flash memory, magnetic disk or optical disk.

[0102] In addition, the present invention also provides a computer-readable storage medium, which stores a computer program. When the computer program is executed by a processor, it implements the steps of the method for solving the above-mentioned high-voltage transmission line conductor dancing trajectory equation.

[0103] Optionally, the present invention further provides a computer program product comprising instructions, which, when executed on a computer, enables the computer to execute the steps of the method for solving the above-mentioned high-voltage transmission line conductor galloping trajectory equation.

[0104] Those skilled in the art will appreciate that the process of implementing all or part of the steps of the above embodiments may be accomplished by hardware, or may be accomplished by instructing the relevant hardware through a program, and the program may be stored in a computer-readable storage medium, and the above-mentioned storage medium may be a read-only memory, a disk, or an optical disk, etc.

[0105] The above shows and describes the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions are merely preferred examples of the present invention and are not intended to limit the present invention. Various changes and improvements may be made to the present invention without departing from the spirit and scope of the present invention. Such changes and improvements fall within the scope of the present invention. The scope of protection claimed in the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for solving the dancing trajectory equation of a high-voltage transmission line conductor, characterized in that: Firstly, a transmission line galloping monitoring system based on Beidou RTK differential technology is constructed to solve the problems of acceleration divergence and curvature limitation in conductor galloping. By numerically simulating the galloping of the line, a polynomial is used as an approximate function in the displacement mode of the unit in the finite element method, so that the shape of the conductor galloping trajectory under the action of gravity is approximately an ellipse. And an equation of the conductor displacement with respect to coordinates and torsion angle is established to restore the conductor galloping trajectory to the maximum extent. The specific solution method includes the following steps: S1. Establish a coordinate system; S2. Analyze the nonlinear motion of any point on the wire; S3, solving the dancing trajectory equation of the wire; S4, solving the torsion angle and acceleration under the arc length constraint; S5. Perform convex optimization on the analytical expression of the torsion angle.

2. The method for solving the high-voltage transmission line conductor dancing trajectory equation according to claim 1 is characterized in that: In step S1, the process of establishing the coordinate system is as follows: Establish a global Cartesian coordinate system X, Y, Z, and establish local coordinates along the wire axis; assume that the displacement components of any point on the wire in the X, Y, and Z directions are U, D, and W respectively, the torsion angle is θ, and the displacement vector of the node is q; where: q=(U,D,W,θ) Assume that there is an inextensible transmission line on two towers with a horizontal span of K, with a length of L, and a corresponding length on the conductor of L q , multiple Beidou positioning units are installed at points q = 1, 2, ..., N. After the torsion information obtained at time t is converted, the tangent direction of the wire is obtained as T lq , where the length of the curve is L q The unit tangent vector at ; these data are used to fit the spatial trajectory of the entire wire at time t.

3. The method for solving the high-voltage transmission line conductor dancing trajectory equation according to claim 2 is characterized in that: In step S2, the motion analysis of any point on the wire is as follows: The speed of the wire is V, and the speeds in the X and Y directions are V x ,V y , the acceleration is a; then the balance equation of the wire can be derived according to the relevant method as follows: Among them, M is the mass matrix of the unit, C is the damping matrix, K is the stiffness matrix, and F is the external force matrix; Taking the X direction as an example, the speed can be expressed as: Among them, V x0 is the initial velocity; then the horizontal displacement U can be expressed as: Among them, U0 is the initial displacement component of the point in the X direction; Furthermore, at a certain time interval [0,t s ] is divided into several equal moments: t0, t1, ..., t s , each small period of time is: Δt1 = t1-t0, Δt2 = t2-t1, ..., Δt s =t s -t s-1 , the speeds in each time period are: V1, V2, ..., V s , then we can get: ΔV i =a(t i )Δt i ,i=1,2,...,s Where, ΔV i is the speed change value in each small period of time; Discretizing the velocity and displacement in the X direction, we can obtain: Similarly, the calculation formula for the displacement D in the Y direction can be obtained as follows: Among them, D y (t i ) is the discretized value of the displacement in the Y direction, and D0 is the initial displacement component of the point in the Y direction.

4. The method for solving the high-voltage transmission line conductor dancing trajectory equation according to claim 3 is characterized in that: In step S3, the method for solving the wire dancing trajectory is a method combining Newmark-β and Newton-Raphson, and the specific process is as follows: First, the following assumptions are made about the node displacement vector q using the Newmark-β method: Among them, ξ,λ are the parameters required for integration accuracy and stability, Δt is the time increment, and the displacement without iteration can be obtained by combining the equilibrium equation: Where R represents the modal matrix; Let n be the number of iterations, then: After updating the stiffness matrix K and iterating, the displacement after iteration can be obtained as: Next, we need to discuss the relationship between acceleration and angle to reduce the error in the acceleration calculation process.

5. The method for solving the high-voltage transmission line conductor dancing trajectory equation according to claim 4 is characterized in that: In step S4, the analysis of the wire twist angle is as follows: Find an arc length parametric curve that satisfies the following conditions: r(s),0≤s≤L r(0)=P0,r(L)=P1,r′(L q )=T lq In the process of finding the minimum energy curve, the angle between the tangent line and the positive direction of the Y axis is taken as θ(s), and the curve energy is: Among them, K(s) is the curvature; the above problem is described by the optimization model as follows: min E Subjectto:r(0)=P0,r(L)=P1,r′(L q )=T lq Among them, r(0) = P0 represents the starting point of the curve. At the origin of the coordinate system, r′(L q )=T lq ; Approximate r(L) by discretizing θ(s); discretize the arc length parameter curve with N broken line segments, each broken line segment is of equal length and is L / N, and the angle between the i-th broken line segment and the positive direction of the Y axis is θ i (θ i ∈[0,π]), then the above optimization problem is discretized into the following form: The starting point is the origin of the coordinate system, so r(0)=P0 naturally satisfies, where k=KN / L; θ lq is the angle corresponding to the polyline segment; because T lq is a unit vector, so r′(L q )=T lq It can be further simplified to cosθ lq =T lq,y ,θ lq is the tangent direction angle of the monitoring point with a curve length of lq.

6. The method for solving the high-voltage transmission line conductor dancing trajectory equation according to claim 5 is characterized in that: In step S5, the torsion angle is optimized, and the specific method is as follows: Because the constraints contain sine and cosine functions, in actual calculations, in order to transform the model into an optimization model that is easier to handle, we can continue to make variable substitutions, and let: cosθ i =x i , and get the following optimization model: The expression of E in the objective function is also re-expressed; According to actual needs, through a series of transformations, the curve fitting optimization model under arc length constraints is derived; the nonlinear constraint is approximated by a first-order Taylor approximation at the optimal solution obtained at the previous sampling point, and then the convex optimization model of linear constraints is obtained. The objective function is a convex function, and the constraints are all linear constraints, which are also convex. The existing optimization algorithm can easily find the global minimum, that is: Where f(x) is a linear constraint equation.

Images