Sag simulation method and device for power transmission conductor
By constructing a multi-field coupling model and finite element analysis, the problem of low accuracy of transmission conductor sag simulation is solved, high-precision three-dimensional sag simulation is achieved, and the accuracy and optimization capabilities of transmission line design are improved.
Patent Information
- Application Number
- CN202411666259.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-11-20
- Publication Date
- 2025-04-04
AI Technical Summary
In the prior art, the accuracy of sag simulation of transmission conductors is low, especially in extreme climatic conditions and complex terrain, it is difficult to achieve high-precision prediction and optimized design.
By determining the target area to be deployed in the transmission conductor, obtain terrain and meteorological information, and construct a three-dimensional spatial model. Based on this model, a multi-field coupling model is established, including mechanical and meteorological coupling, thermal effect and mechanical performance coupling, multi-conductor interaction coupling, and interaction coupling between terrain and wire coupling model. These models are converted into finite element models, and the equilibrium state is solved through a nonlinear finite element algorithm to obtain the node values of each sub-region, thereby constructing the sag simulation results.
High-precision three-dimensional sag simulation of transmission conductors under complex operating conditions is realized, which improves the accuracy of the simulation and can more accurately predict and optimize the design of transmission lines.
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Figure CN119761097A_ABST
Abstract
Description
Technical Field
[0001] The present application relates to the field of power data information processing, and in particular to a method and device for simulating sag of a transmission line. Background Art
[0002] With the continuous growth of global energy demand and the rapid expansion of power transmission networks, the design and maintenance of high-voltage transmission lines face unprecedented challenges. Combined conductors have become the first choice for long-distance, large-capacity transmission lines due to their superior current-carrying capacity, low loss and strong mechanical properties. However, the three-dimensional sag problem of combined conductors in complex natural environments (such as mountainous areas, windy areas, and ice-covered areas) directly affects the safe operation of the line, the efficiency of power transmission, and the maintenance cost. The two-dimensional analysis methods and empirical estimates in related technologies have been unable to meet the needs of high-precision design, especially in extreme climatic conditions and complex terrain. The simulation accuracy needs to be improved urgently. The transmission line design software in related technologies is difficult to achieve high-precision prediction and optimization design of three-dimensional sag.
[0003] With regard to the problem of low accuracy in sag simulation of power transmission lines in related technologies, no effective solution has been proposed so far. Summary of the invention
[0004] The main purpose of the present application is to provide a method and device for simulating the sag of a transmission line, so as to solve the problem of low accuracy of sag simulation of a transmission line in the related art.
[0005] In order to achieve the above-mentioned purpose, according to one aspect of the present application, a method for simulating the sag of a transmission line is provided. The method includes: determining a target area where the transmission line is to be deployed, obtaining terrain information and meteorological information of the target area, and obtaining attribute information of the transmission line; receiving the end point location information of the transmission line deployed in the target area input by the user, and constructing a three-dimensional space model of the transmission line based on the end point location information and terrain information; constructing a multi-field coupling model of the transmission line based on the three-dimensional space model, meteorological information and attribute information, wherein the multi-field coupling model includes at least one of the following: a mechanical and meteorological coupling model, a thermal effect and mechanical property coupling model, a multi-conductor interaction coupling model and a terrain and conductor interaction coupling model; converting the multi-field coupling model into a finite element model to obtain a plurality of interconnected sub-areas, solving the equilibrium state of the finite element model by a preset nonlinear finite element algorithm, and obtaining the node value of each sub-area, wherein the node value is used to characterize the physical quantity in the sub-area, and the physical quantity includes at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line; constructing the sag simulation result of the transmission line by the node value of each sub-area.
[0006] Optionally, constructing a multi-field coupling model of a transmission line based on a three-dimensional space model, meteorological information and attribute information includes: determining the physical fields to be coupled, and determining the initial conditions and boundary conditions of each physical field based on the three-dimensional space model and the meteorological information, wherein the physical field includes at least one of the following: a mechanical field, a thermal field, an electromagnetic field and a fluid field, and the initial conditions are used to characterize the initial values of physical quantities in the physical field, and the boundary conditions are used to characterize the initial values of physical quantities at the edge of the physical field; constructing a mathematical model of each physical field through partial differential equations to obtain multiple mathematical models, wherein the mathematical model is used to describe the behavior of physical quantities in the physical field, and the mathematical model includes at least one of the following: a heat conduction model, a fluid dynamics model and a structural mechanics model; determining the coupling conditions between the physical fields through the three-dimensional space model, meteorological information and attribute information, and constructing a multi-field coupling model based on the coupling conditions and multiple mathematical models.
[0007] Optionally, the multi-field coupling model includes a mechanical and meteorological coupling model, and determining the coupling conditions between physical fields through a three-dimensional space model, meteorological information and attribute information includes: extracting wind speed and wind direction of the target area from the meteorological information, and extracting the geometry and material of the transmission line from the attribute information, and determining the wind load of the transmission line based on the wind speed, wind direction, geometry and material, wherein the wind load is used to characterize the force exerted by the wind on the transmission line; extracting icing information of the target area from the meteorological information, and extracting the surface area of the transmission line from the attribute information, and determining the icing load of the transmission line based on the icing information and the surface area, wherein the icing load is used to characterize the force exerted by icing on the transmission line; extracting the weight of the transmission line from the attribute information, and determining the gravity exerted on the transmission line based on the weight; and determining the wind load, icing load and gravity as coupling conditions.
[0008] Optionally, the multi-field coupling model includes a thermal effect and mechanical property coupling model, and determining the coupling conditions between physical fields through a three-dimensional space model, meteorological information and attribute information includes: extracting the temperature of the target area from the meteorological information, and determining the material of the transmission line from the attribute information, determining the stress, strain and tension of the transmission line in the target area based on the temperature and material; and determining the stress, strain and tension as coupling conditions.
[0009] Optionally, the multi-field coupling model includes a multi-conductor interaction coupling model, and determining the coupling conditions between physical fields through a three-dimensional space model, meteorological information and attribute information includes: determining the distance between transmission lines from the three-dimensional space model, extracting the cross-sectional area of the transmission lines from the attribute information, and calculating the attraction or repulsion force on the transmission lines based on the distance and the cross-sectional area; and determining the attraction or repulsion force as the coupling condition.
[0010] Optionally, the multi-field coupling model includes an interaction coupling model between terrain and conductors, and determining the coupling conditions between physical fields through the three-dimensional space model, meteorological information and attribute information includes: determining obstacles and terrain information that the transmission lines contact from the three-dimensional space model; determining the local wind field in the target area through the terrain information, and determining the wind force of the local wind field on the transmission lines, and determining the contact force of the transmission lines based on the obstacles; and determining the wind force and contact force as coupling conditions.
[0011] Optionally, constructing a multi-field coupling model based on coupling conditions and multiple mathematical models includes: determining coupling variables from the coupling conditions, wherein the coupling variables include at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line; combining multiple mathematical models to obtain a coupling equation group, and determining the coupling variables and the coupling equation group as a multi-field coupling model.
[0012] Optionally, after converting the multi-field coupling model into a finite element model to obtain a plurality of interconnected sub-regions, the method further includes: determining nodes at the boundaries of each sub-region, wherein the nodes are used to characterize connection nodes of transmission lines between different sub-regions; for each sub-region, constructing a local set of equations based on the coupling equations of the multi-field coupling model, wherein the local set of equations is used to characterize changes in physical quantities of nodes within the sub-region; and constructing a global set of equations through the coupling conditions of the multi-field coupling model, boundary conditions and the local set of equations of each sub-region.
[0013] Optionally, solving the equilibrium state of the finite element model by a preset nonlinear finite element algorithm to obtain the node value of each sub-area includes: determining an initial solution of the global equation group, wherein the initial solution includes the initial value of each node of all sub-areas, and the initial value is used to characterize the physical quantity of the transmission line when it is not affected by environmental changes; decomposing the coupling conditions into a set of increments, and iteratively solving the global equation group based on the set of increments, wherein each iteration adds one increment in the set of increments to the global equation group; after all increments in a set of increments are added to the global equation group, solving the global equation group to obtain a target solution, and determining the node value of each sub-area in the target solution.
[0014] In order to achieve the above objective, according to another aspect of the present application, a sag simulation device for a transmission line is provided. The device comprises: a first determining unit, used for determining a target area where a transmission line is to be deployed, obtaining terrain information and meteorological information of the target area, and obtaining attribute information of the transmission line; a receiving unit, used for receiving the end point position information of the transmission line deployed in the target area input by a user, and constructing a three-dimensional space model of the transmission line based on the end point position information and terrain information; a first constructing unit, used for constructing a multi-field coupling model of the transmission line based on the three-dimensional space model, meteorological information and attribute information, wherein the multi-field coupling model comprises at least one of the following: a mechanical and meteorological coupling model, a thermal effect and mechanical property coupling model, a multi-conductor interaction coupling model and a terrain and conductor interaction coupling model; a converting unit, used for converting the multi-field coupling model into a finite element model, obtaining a plurality of interconnected sub-areas, solving the equilibrium state of the finite element model by a preset nonlinear finite element algorithm, and obtaining a node value of each sub-area, wherein the node value is used to characterize the physical quantity in the sub-area, and the physical quantity comprises at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line; and a second constructing unit, used for constructing a sag simulation result of the transmission line by the node value of each sub-area.
[0015] In order to achieve the above-mentioned purpose, according to another aspect of the present application, a computer program product is provided, including a computer program, which, when executed by a processor, implements the steps of the sag simulation method for a transmission line described in each embodiment of the present application.
[0016] Through the present application, the following steps are adopted: determining the target area where the transmission line is to be deployed, obtaining the terrain information and meteorological information of the target area, and obtaining the attribute information of the transmission line; receiving the endpoint position information of the transmission line deployed in the target area input by the user, and constructing a three-dimensional space model of the transmission line based on the endpoint position information and the terrain information; constructing a multi-field coupling model of the transmission line based on the three-dimensional space model, the meteorological information and the attribute information, wherein the multi-field coupling model includes at least one of the following: a mechanical and meteorological coupling model, a thermal effect and mechanical property coupling model, a multi-conductor interaction coupling model and a terrain and conductor interaction coupling model; converting the multi-field coupling model into a finite element model to obtain a plurality of interconnected sub-areas, solving the equilibrium state of the finite element model by a preset nonlinear finite element algorithm, and obtaining the node value of each sub-area, wherein the node value is used to characterize the physical quantity in the sub-area, and the physical quantity includes at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line; constructing the sag simulation result of the transmission line by the node value of each sub-area, solving the problem of low accuracy of the sag simulation of the transmission line in the related art. By introducing multi-physics field coupling models and nonlinear finite element analysis, high-precision three-dimensional sag simulation of transmission lines under complex working conditions is achieved, thereby achieving the effect of improving the accuracy of sag simulation of transmission lines. BRIEF DESCRIPTION OF THE DRAWINGS
[0017] The drawings constituting a part of the present application are used to provide a further understanding of the present application. The illustrative embodiments and descriptions of the present application are used to explain the present application and do not constitute an improper limitation on the present application. In the drawings:
[0018] Figure 1 is a flow chart of a method for simulating sag of a transmission line provided in an embodiment of the present application;
[0019] Figure 2 is a schematic diagram of a combined conductor three-dimensional sag simulation system provided according to an embodiment of the present application;
[0020] Figure 3 is a schematic diagram of a sag simulation device for a power transmission line provided according to an embodiment of the present application;
[0021] Figure 4 is a schematic diagram of an electronic device provided according to an embodiment of the present application. DETAILED DESCRIPTION
[0022] It should be noted that, in the absence of conflict, the embodiments and features in the embodiments of the present application can be combined with each other. The present application will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.
[0023] In order to enable those skilled in the art to better understand the solution of the present application, the technical solution in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by ordinary technicians in this field without creative work should fall within the scope of protection of the present application.
[0024] It should be noted that the terms "first", "second", etc. in the specification and claims of the present application and the above-mentioned drawings are used to distinguish similar objects, and are not necessarily used to describe a specific order or sequence. It should be understood that the data used in this way can be interchanged where appropriate, so that the embodiments of the present application described here. In addition, the terms "including" and "having" and any of their variations are intended to cover non-exclusive inclusions, for example, a process, method, system, product or device comprising a series of steps or units is not necessarily limited to those steps or units clearly listed, but may include other steps or units that are not clearly listed or inherent to these processes, methods, products or devices.
[0025] It should be noted that the user information (including but not limited to user device information, user personal information, etc.) and data (including but not limited to data for display, data for analysis, etc.) involved in this disclosure are all information and data authorized by the user or fully authorized by all parties.
[0026] It should be noted that the information collected is information and data authorized by the user or fully authorized by all parties, and the collection, storage, use, processing, transmission, provision, disclosure and application of relevant data comply with the relevant laws, regulations and standards of the relevant regions, take necessary confidentiality measures, do not violate public order and good customs, and provide corresponding operation entrances for users to choose to authorize or refuse.
[0027] The present invention is described below in conjunction with preferred implementation steps. Figure 1 is a flow chart of a method for simulating sag of a transmission line according to an embodiment of the present application, such as Figure 1 As shown, the method comprises the following steps:
[0028] Step S101, determining a target area where the transmission line is to be deployed, obtaining terrain information and meteorological information of the target area, and obtaining attribute information of the transmission line.
[0029] In step S101, the terrain information may be the elevation information, slope, terrain type (such as plain, mountain, hill), etc. of the surface of the target area, and the starting point and end point of the transmission line in the target area and the main path along the line are determined through a geographic information system or a satellite map. For example, a GIS system (Geographic Information System), drone aerial photography, or satellite remote sensing technology is used to obtain high-precision terrain data of the target area.
[0030] Meteorological information can be wind speed, wind direction, temperature, humidity, precipitation type, ice thickness, etc. Meteorological information of the target area can be collected from meteorological monitoring networks such as meteorological stations, meteorological satellites, and radars. It can also be connected to the meteorological forecast system to obtain short-term or long-term meteorological forecast data in real time or regularly, including changing trends in wind speed and temperature. This helps to dynamically adjust simulation parameters and predict the behavior of the conductor under different climatic conditions.
[0031] The attribute information of the transmission line may include the type of conductor material, cross-sectional area, elastic modulus, density, thermal expansion coefficient, thermal conductivity, resistivity, conductivity, capacitance characteristics, combination of conductors (such as single conductor, split conductor), diameter, mass per unit length, tension, etc.
[0032] It should be noted that after obtaining the terrain information and meteorological information of the target area, as well as the attribute information of the transmission line, the collected terrain, meteorological and wire attribute information will be integrated to ensure the consistency and integrity of the data. The original data will be cleaned, verified and formatted to meet the needs of subsequent modeling and analysis. For example, for terrain information, terrain simplification or terrain feature extraction will be performed; for meteorological information, trend analysis or outlier filtering will be performed.
[0033] Step S102, receiving the terminal position information of the power transmission line deployed in the target area input by the user, and constructing a three-dimensional space model of the power transmission line based on the terminal position information and terrain information.
[0034] In step S102, by receiving the geographic coordinates of the starting point and the end point of the transmission line deployed in the target area input by the user, if the line path of the transmission line passes through complex terrain, the user is also required to provide the location information of key points along the way to ensure that the three-dimensional space model can accurately reflect the actual direction of the transmission line. The endpoint location information and the terrain information of the target area are imported into the three-dimensional modeling software. The three-dimensional modeling software automatically constructs a three-dimensional space model of the target area.
[0035] Step S103, constructing a multi-field coupling model of the transmission line based on the three-dimensional space model, meteorological information and attribute information, wherein the multi-field coupling model includes at least one of the following: a mechanical and meteorological coupling model, a thermal effect and mechanical property coupling model, a multi-conductor interaction coupling model and a terrain and conductor interaction coupling model.
[0036] In step S103, the construction of the mechanical and meteorological coupling model can be based on the three-dimensional space model to establish a mechanical model of the transmission line, and the elastic properties, tension, gravity and connection conditions of the line with the tower are considered in the mechanical model. A meteorological model is established based on the collected meteorological information, and the meteorological model includes the influence of wind load and ice load on the mechanical behavior of the line. The output of the meteorological model (such as wind load and ice load) is used as the input load of the mechanical model, and the conditions in the mechanical model are updated in real time or periodically to reflect the influence of meteorological changes on the mechanical state of the line, thereby obtaining a mechanical and meteorological coupling model.
[0037] The construction of the coupled model of thermal effect and mechanical properties can establish a thermodynamic model that takes into account the temperature change of the wire based on the thermodynamic parameters such as thermal expansion coefficient and thermal conductivity in the wire property information. Based on the temperature change calculated by the thermodynamic model, the material properties in the mechanical model, such as elastic modulus and yield strength, are updated to reflect the impact of temperature change on the mechanical properties of the wire. The interaction between the thermodynamic model and the mechanical model is based on the energy conservation and heat conduction equations to calculate the temperature distribution of the wire, which in turn affects its mechanical behavior, thereby constructing a coupled model of thermal effect and mechanical properties.
[0038] The construction of the multi-conductor interaction coupling model is to establish a model of the interaction between conductors in the case of multiple transmission conductors, including electrodynamic force, magnetic force and contact friction. Considering the dynamic swing of the conductors under wind load and ice load, as well as the collision and friction between the conductors, a dynamic response model is established. The interaction force of the multi-conductor system under complex loads is calculated, and the mechanical model of each conductor is updated to reflect the influence of the interaction between the conductors, so as to construct a multi-conductor interaction coupling model.
[0039] The construction of the terrain-conductor interaction coupling model can be based on the terrain information in the three-dimensional space model to establish a terrain model, taking into account the support and influence of terrain undulations on the conductor. In areas with complex terrain, a fluid mechanics model is established to analyze the influence of terrain on wind field distribution and the influence of wind field changes on conductor stress. The coupling between the terrain model and the conductor mechanics model ensures the correct mechanical behavior of the conductor model under specific terrain conditions through wind field calculation and terrain support effect, thereby constructing a terrain-conductor interaction coupling model.
[0040] When constructing a multi-field coupling model, the boundary conditions of the model need to be set to ensure that the model can accurately reflect the actual working conditions. For example, in the mechanical and meteorological coupling model, the initial tension and boundary support conditions of the conductor should match the actual conditions of the target area; in the thermal effect and mechanical properties coupling model, the initial temperature and temperature boundary conditions of the conductor should consider the actual environment of the target area; in the multi-conductor interaction coupling model, the initial distance and relative position between the conductors should be consistent with the plan; in the terrain and conductor interaction coupling model, the boundary conditions of the terrain should reflect the boundary characteristics of the actual terrain.
[0041] The construction of the multi-field coupling model needs to follow the physical laws of energy conservation, momentum conservation, and charge conservation to ensure the physical rationality of the simulation results. In addition, the parameters of the model are calibrated according to actual measurement data or engineering experience to improve the prediction accuracy of the model.
[0042] Step S104, converting the multi-field coupling model into a finite element model to obtain a plurality of interconnected sub-regions, solving the equilibrium state of the finite element model by a preset nonlinear finite element algorithm, and obtaining the node value of each sub-region, wherein the node value is used to characterize the physical quantity in the sub-region, and the physical quantity includes at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line.
[0043] In step S104, the transmission line and its surrounding environment (including terrain, pole towers, etc.) in the multi-field coupling model are divided into multiple small, interconnected sub-areas (finite element units). The grid division can be set according to the physical size of the wire, the complexity of the load change, and the limitation of computing resources. In each finite element unit, a series of nodes are defined, which will be used to approximate the physical quantities in the continuous physical field. The lines between the nodes are called edges, and the changes in physical quantities on the edges are described by interpolation functions. Corresponding physical properties are assigned to each finite element, such as the elastic modulus, Poisson's ratio, density, etc. of the wire material, as well as the rigidity, friction coefficient, etc. of the terrain. The attribute settings of the nodes match the physical fields in the multi-field coupling model.
[0044] Based on the divided finite element units and nodes, a set of control equations reflecting the coupling relationship of each physical field is established. These equations include but are not limited to mechanical equilibrium equations, thermal equilibrium equations, fluid dynamics equations, etc. Boundary conditions are set to describe the behavior of the model at the boundary, such as the fixed or free end of the wire, and the supporting effect of the terrain. At the same time, according to the multi-field coupling model, loads such as wind loads, ice loads, and temperature gradients are applied.
[0045] Due to the material nonlinearity, geometric nonlinearity and contact nonlinearity of the transmission line in the target area, a preset nonlinear finite element algorithm is used to solve the system of equations. For example, the incremental iteration method, the Newton-Raphson method and its modified versions, such as the linearized incremental iteration method. The solution of the system of equations is gradually approached through the iterative process until the convergence criterion is met. The goal of the preset nonlinear finite element algorithm is to find the distribution of physical quantities such as displacement, velocity, stress, strain and pressure of each node when the conductor and its surrounding environment are in equilibrium under a given load, that is, the node value of each sub-area. In the algorithm solution process, the principles of conservation of energy and momentum are applied to verify the physical rationality of the solution, and the stability of the solution process and results is optimized by adjusting the algorithm parameters or meshing.
[0046] Step S105, constructing the sag simulation result of the transmission line through the node value of each sub-area.
[0047] By analyzing the node values of each sub-area, detailed information on the behavior of the conductor under different conditions is determined, such as sag changes, strain distribution, temperature gradient, etc. The node values are converted into three-dimensional graphical outputs to intuitively display the deformation, stress distribution and temperature changes of the conductor, and the sag simulation results of the transmission line are obtained.
[0048] For example, the node values of each sub-area are sorted out, and these node values correspond to the displacement, stress, strain and temperature of the transmission line at the node. Based on the displacement data of the node, the node displacement is mapped to the entire transmission line model using the built-in or customized interpolation algorithm of the finite element analysis software, so as to construct the three-dimensional deformation morphology of the transmission line under given load conditions. This can intuitively display the sag change, bending and torsion state of the wire. By analyzing the displacement data of the node, the sag curve of the transmission line between different support points (i.e., nodes) is generated. These curves reflect the natural droop state of the wire under the action of gravity, wind load, ice load, etc., and are used to evaluate the performance and safety of the wire. Using the stress and strain data of the node, the mechanical state of the transmission line under different environmental conditions is analyzed, including the maximum stress, strain concentration area and the elastic and plastic deformation state of the wire, which is used to evaluate the carrying capacity and service life of the transmission line.
[0049] The node values at different time steps are converted into dynamic three-dimensional simulation results, showing the sag changes and swing states of the transmission lines under dynamic environmental conditions such as wind speed and temperature changes, which are used to evaluate the line safety under extreme conditions. The constructed sag simulation results are verified with the principles of conservation of energy and momentum to ensure the physical rationality and accuracy of the results.
[0050] The sag simulation results are displayed in the form of three-dimensional graphics, curve charts, etc., which are easy for decision makers with non-technical backgrounds to understand. At the same time, a detailed analysis report is generated, including model parameters, solution process, result evaluation and recommended measures, etc., to provide a scientific basis for line design, construction and operation and maintenance.
[0051] The sag simulation method of the transmission line provided in the embodiment of the present application determines the target area where the transmission line is to be deployed, obtains the terrain information and meteorological information of the target area, and obtains the attribute information of the transmission line; receives the endpoint position information of the transmission line deployed in the target area input by the user, and constructs a three-dimensional space model of the transmission line based on the endpoint position information and the terrain information; constructs a multi-field coupling model of the transmission line based on the three-dimensional space model, the meteorological information and the attribute information, wherein the multi-field coupling model includes at least one of the following: a mechanical and meteorological coupling model, a thermal effect and mechanical property coupling model, a multi-conductor interaction coupling model and a terrain and conductor interaction coupling model; converts the multi-field coupling model into a finite element model to obtain a plurality of interconnected sub-areas, solves the equilibrium state of the finite element model by a preset nonlinear finite element algorithm, and obtains the node value of each sub-area, wherein the node value is used to characterize the physical quantity in the sub-area, and the physical quantity includes at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line; constructs the sag simulation result of the transmission line by the node value of each sub-area, and solves the problem of low accuracy of the sag simulation of the transmission line in the related art. By introducing multi-physics field coupling models and nonlinear finite element analysis, high-precision three-dimensional sag simulation of transmission lines under complex working conditions is achieved, thereby achieving the effect of improving the accuracy of sag simulation of transmission lines.
[0052] In order to simulate the performance of transmission lines under complex environmental conditions, it is necessary to construct a multi-field coupling model of the transmission line. Optionally, in the sag simulation method of the transmission line provided in the embodiment of the present application, constructing the multi-field coupling model of the transmission line based on the three-dimensional space model, meteorological information and attribute information includes: determining the physical field to be coupled, and determining the initial conditions and boundary conditions of each physical field based on the three-dimensional space model and the meteorological information, wherein the physical field includes at least one of the following: mechanical field, thermal field, electromagnetic field and fluid field, the initial conditions are used to characterize the initial values of physical quantities in the physical field, and the boundary conditions are used to characterize the initial values of physical quantities at the edge of the physical field; constructing a mathematical model of each physical field through partial differential equations to obtain multiple mathematical models, wherein the mathematical model is used to describe the behavior of physical quantities in the physical field, and the mathematical model includes at least one of the following: heat conduction model, fluid dynamics model and structural mechanics model; determining the coupling conditions between the physical fields through the three-dimensional space model, meteorological information and attribute information, and constructing a multi-field coupling model based on the coupling conditions and multiple mathematical models.
[0053] In some examples, the physical fields to be coupled may include mechanical fields, thermal fields, electromagnetic fields, and fluid fields. For example, when simulating conductor sag, the mechanical field is used to describe the elastic behavior and stress conditions of the transmission line, the thermal field is used to analyze the impact of temperature changes on the length and mechanical properties of the transmission line, and the fluid field is used to calculate the impact of wind loads on the swing of the line. For each physical field, its initial conditions and boundary conditions are determined based on the three-dimensional space model and meteorological information. The initial condition refers to the initial value of the physical quantity in the physical field at a specific moment (such as the beginning of the simulation), such as the initial temperature and initial tension of the conductor. The boundary condition can be the physical value of the physical field at the boundary, such as the fixed constraint of the conductor on the tower, the distribution of wind speed on the terrain, etc.
[0054] Select appropriate partial differential equations for each physical field to build a mathematical model. For example, for the thermal field, the heat conduction equation is used to describe the change of temperature over time and space; for the fluid field, the Navier-Stokes equations are used to describe the flow of the wind field; for the mechanical field, the partial differential equations of elasticity are used to describe the deformation and stress distribution of the conductor under load. Coupling conditions are used to describe how different physical fields interact and influence each other. For example, changes in wind speed (fluid field) will affect the force on the transmission line (mechanical field), thereby causing changes in the temperature of the transmission line (thermal field); similarly, changes in the temperature of the transmission line will affect its material properties, thereby affecting the mechanical response. Coupling conditions can include coupling expressions between physical quantities, such as the relationship between temperature and elastic modulus, the relationship between wind speed and conductor swing, etc.
[0055] By integrating all mathematical models and coupling conditions, a comprehensive multi-field coupling model is constructed. The multi-field coupling model reflects the comprehensive behavior of all physical fields and their mutual influence under specific environmental conditions. When constructing the model, the time and space scales of different physical fields must be considered to ensure the coordination and consistency of the model between different fields.
[0056] This embodiment can simulate the performance of transmission lines under complex environmental conditions more comprehensively and accurately by constructing a multi-field coupling model, providing a powerful tool for the design, construction and operation and maintenance of power projects. The use of multi-field coupling models can not only improve the reliability of engineering design, but also provide accurate decision support during the maintenance phase, reduce unnecessary on-site inspections and maintenance work, and significantly reduce operation and maintenance costs.
[0057] When constructing a multi-field coupling model, a mechanical and meteorological coupling model can be constructed. Optionally, in the sag simulation method for a transmission line provided in an embodiment of the present application, the multi-field coupling model includes a mechanical and meteorological coupling model. Determining the coupling conditions between physical fields through a three-dimensional space model, meteorological information, and attribute information includes: extracting the wind speed and wind direction of the target area from the meteorological information, and extracting the geometry and material of the transmission line from the attribute information, and determining the wind load of the transmission line based on the wind speed, wind direction, geometry, and material, wherein the wind load is used to characterize the force exerted by the wind on the transmission line; extracting icing information of the target area from the meteorological information, and extracting the surface area of the transmission line from the attribute information, and determining the icing load of the transmission line based on the icing information and the surface area, wherein the icing load is used to characterize the force exerted by icing on the transmission line; extracting the weight of the transmission line from the attribute information, and determining the gravity exerted on the transmission line based on the weight; and determining the wind load, icing load, and gravity as coupling conditions.
[0058] In some examples, wind speed and direction data of the target area are extracted from meteorological information, and the geometry (such as diameter, length, wire type) and material properties (such as density, elastic modulus, surface roughness) of the transmission line are extracted from the attribute information. The wind load of the transmission line is calculated based on the wind speed, wind direction, wire geometry and material properties. Ice information of the target area is extracted from the meteorological information, such as ice thickness, density, etc. The surface area of the transmission line is extracted from the attribute information. The ice load is calculated based on the ice thickness, density and surface area of the wire. The weight of the transmission line is extracted from the attribute information, which can be calculated based on the material density and volume of the wire. Based on the weight of the wire and the gravitational acceleration of the target area, the impact of the gravity load on the wire is calculated.
[0059] The calculated wind load, ice load and gravity load are used as coupling conditions of the mechanical and meteorological coupling model and input into the nonlinear finite element solver. These loads will act as external excitations and interact with the mechanical behavior of the transmission line, affecting the sag, displacement, stress and strain of the line.
[0060] This embodiment determines the coupling conditions for constructing a mechanical and meteorological coupling model to accurately predict the three-dimensional sag state of the transmission line under complex environmental conditions, provide a scientific basis for line design, and improve its adaptability and safety to complex environments.
[0061] When constructing a multi-field coupling model, a thermal effect and mechanical property coupling model can be constructed. Optionally, in the sag simulation method for a transmission line provided in an embodiment of the present application, the multi-field coupling model includes a thermal effect and mechanical property coupling model. The coupling conditions between physical fields are determined through a three-dimensional space model, meteorological information, and attribute information, including: extracting the temperature of a target area from the meteorological information, determining the material of the transmission line from the attribute information, determining the stress, strain, and tension of the transmission line in the target area based on the temperature and material; and determining the stress, strain, and tension as coupling conditions.
[0062] In some examples, temperature data of the target area is extracted from meteorological information, which may include ambient temperature changes, daytime temperature fluctuations, and seasonal temperature changes. Temperature data can be measured values or predicted values. For complex terrain, it can also include the local impact of local wind fields and solar radiation on temperature. Based on the three-dimensional spatial model, the heat conduction model and convection model are applied to calculate the temperature distribution of each part of the transmission line and surrounding structures such as towers and terrain, especially the temperature gradient and thermal deformation degree of the transmission line.
[0063] The material properties of the transmission line are determined from the attribute information, including the thermal expansion coefficient, elastic modulus, yield strength, etc. Based on the temperature change and material properties, the temperature-stress relationship curve in material mechanics is used to determine the stress, strain and tension of the transmission line at different temperatures. When the temperature rises, the transmission line will increase in length and reduce tension due to thermal expansion; when the temperature drops, the transmission line will shrink and increase tension. At the same time, temperature changes will affect the elastic modulus and yield strength of the material, thereby affecting the mechanical response of the wire. The calculated stress, strain and tension are used as coupling conditions for the thermal effect and mechanical property coupling model. These conditions reflect the direct impact of temperature changes on the mechanical properties of the wire.
[0064] This embodiment determines the coupling conditions for constructing the thermal effect and mechanical performance coupling model to construct the thermal effect and mechanical performance coupling model, accurately reflects the mechanical response of the transmission line under temperature changes, including sag changes, stress distribution and tension state, and provides a scientific basis for line design, construction and operation and maintenance, especially in areas with high and low temperatures and large temperature differences between day and night. It can help evaluate the thermodynamic stability of the conductor, prevent line failures caused by temperature changes, and ensure the safe operation of the power system.
[0065] When constructing a multi-field coupling model, a multi-conductor interaction coupling model can be constructed. Optionally, in the sag simulation method for a transmission line provided in an embodiment of the present application, the multi-field coupling model includes a multi-conductor interaction coupling model. Determining the coupling conditions between physical fields through a three-dimensional space model, meteorological information, and attribute information includes: determining the distance between the transmission lines from the three-dimensional space model, extracting the cross-sectional area of the transmission lines from the attribute information, and calculating the attraction or repulsion on the transmission lines based on the distance and the cross-sectional area; and determining the attraction or repulsion as a coupling condition.
[0066] In some examples, the layout information of the transmission lines is extracted from the three-dimensional spatial model, including the relative position and distance between the transmission lines. Based on the three-dimensional model of the transmission lines, the shortest spatial distance between each pair of transmission lines is calculated. The cross-sectional area of each transmission line is extracted from the attribute information to calculate the electrostatic interaction force (repulsion force) or electromagnetic induction force (attraction force) between the transmission lines. Based on the distance and cross-sectional area between the wires, the repulsion force between the wires is calculated using Coulomb's law. When current flows through the transmission lines, the attraction between the transmission lines is calculated based on Ampere's law and Biot-Savart's law. The calculated attraction or repulsion force is used as an external load of the mechanical model to solve the deformation and displacement of the wires under the action of these forces.
[0067] The interaction force (attractive or repulsive force) between the wires is input as a coupling condition into the multi-wire interaction coupling model.
[0068] This embodiment determines the coupling conditions for constructing the multi-conductor interaction coupling model as described above to construct a multi-conductor interaction coupling model, and accurately simulates the mechanical and electrical behaviors of the transmission line under the multi-conductor configuration, including sag changes, interaction forces between conductors, and deformation and displacement of the conductors under electromagnetic interaction, which provides a more scientific basis for line design, construction, and operation and maintenance, especially in applications under high-voltage transmission, compact lines, or special environments (such as lightning, ice coating), which can help evaluate the stability and safety of the line and prevent line failures caused by contact or excessive proximity of conductors.
[0069] When constructing a multi-field coupling model, a terrain-conductor interaction coupling model can be constructed. Optionally, in the transmission line sag simulation method provided in an embodiment of the present application, the multi-field coupling model includes a terrain-conductor interaction coupling model. Determining the coupling conditions between physical fields through a three-dimensional space model, meteorological information, and attribute information includes: determining obstacles and terrain information contacted by the transmission line from the three-dimensional space model; determining the local wind field in the target area through the terrain information, and determining the wind force of the local wind field on the transmission line, and determining the contact force of the transmission line based on the obstacles; and determining the wind force and the contact force as coupling conditions.
[0070] In some examples, terrain features on the transmission line path are determined from a three-dimensional spatial model, including the shape, size, and location of obstacles such as mountains, valleys, forests, and buildings. Specific information about obstacles, such as height, width, and material properties, is extracted to assess the potential risk of contact with the conductors. Based on terrain information and meteorological information (such as wind speed and wind direction), a fluid mechanics model is used to determine the local wind field in the target area, such as calculating the wind speed distribution, eddy current effect, and wind pressure distribution around obstacles.
[0071] Based on the simulated local wind field, calculate the wind forces acting on the transmission lines. For example, the lateral and vertical forces on the lines due to wind speed, and the impact of wind direction changes on the swing and sag of the lines. Evaluate the contact risk between the transmission lines and obstacles or terrain, and calculate the contact force between the transmission lines and obstacles based on the geometric information and material properties of the contact points. The contact forces can include friction, collision, etc.
[0072] The calculated wind force and contact force are used as coupling conditions and input into the coupling model of terrain and conductor interaction.
[0073] This embodiment determines the coupling conditions for constructing a terrain-conductor interaction coupling model to construct a terrain-conductor interaction coupling model, and more comprehensively simulates the mechanical behavior of the transmission line under complex terrain conditions, including the response of the line to terrain changes, the swing of the line under wind action, and the risk of contact with obstacles, so as to provide a scientific basis for line design, construction, and operation and maintenance, especially in applications under strong winds, ice cover, and complex terrain, which can help evaluate the stability and safety of the line and reduce line failures and maintenance costs caused by terrain and wind action.
[0074] After determining the coupling conditions, a multi-field coupling model is constructed based on the coupling conditions and multiple mathematical models. Optionally, in the sag simulation method for a transmission line provided in an embodiment of the present application, constructing the multi-field coupling model based on the coupling conditions and multiple mathematical models includes: determining coupling variables from the coupling conditions, wherein the coupling variables include at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line; combining multiple mathematical models to obtain a coupling equation group, and determining the coupling variables and the coupling equation group as a multi-field coupling model.
[0075] In some examples, coupling variables may include the displacement, velocity, stress, strain, and pressure of the transmission line, wherein the displacement of the transmission line is the spatial displacement of the line under the action of wind load, gravity, ice load, etc., and is an important variable connecting the mechanical model with the terrain model. Velocity is the speed at which the line changes with dynamic loads such as wind and ice cover. Stress is the internal stress distribution of the line under various loads, reflecting the stress state of the line material, and is a key indicator of the coupling of material nonlinearity and thermal effects. Strain is related to stress, indicating the degree of deformation of the line material under load, and is an important basis for evaluating the thermal expansion and contraction and mechanical deformation of the line. Pressure is the pressure when the line contacts an obstacle, as well as the air pressure inside the line or in the surrounding environment.
[0076] The mechanical model is based on the principles of material mechanics and structural mechanics to describe the mechanical behavior of the conductor, such as elastic deformation, tension changes, etc., as well as the contact mechanics with terrain obstacles. The thermal effect model can describe the impact of temperature changes on the mechanical properties of the conductor for the thermal conduction, thermal expansion and thermodynamic-mechanical coupling effects of the conductor. The fluid dynamics model is used to simulate the interaction between the wind field, airflow and conductors, and calculate the impact of wind loads and ice loads on the conductors. By combining multiple mathematical models, each model describes the behavior of a specific physical field, and the coupled equations merge these models to reflect the interaction between them.
[0077] The above models are combined to form a coupled equation group, which takes the coupled variables (displacement, velocity, stress, strain, pressure, etc.) as unknowns, and solves the coupling problems of multiple physical fields at the same time by solving the coupled equation group. For example, the coupled equation group may include the following equations: equilibrium equations, which describe the equilibrium state of the conductor under the action of mechanical loads, including static equilibrium equations and dynamic equilibrium equations. Thermodynamic equations, which describe the heat conduction and temperature changes inside the conductor, and the influence of temperature on mechanical properties. Fluid dynamics equations, which describe the influence of wind fields and icing loads on the mechanical behavior of the conductor, including wind speed, wind direction and airflow distribution. Contact mechanics equations, which describe the contact forces such as friction and collision force when the conductor contacts obstacles, and the influence of contact forces on the displacement and stress of the conductor.
[0078] This embodiment constructs a multi-field coupling model to more comprehensively reflect the real behavior of transmission lines under complex environmental conditions, and provide a more scientific basis for line design, construction, and operation and maintenance. In particular, when it is necessary to comprehensively consider mechanical, thermal, and fluid mechanics effects, the multi-field coupling model can help predict the three-dimensional sag changes, thermal stability, and dynamic response of the conductor under the action of wind fields and obstacles, thereby optimizing the line design and improving the operational safety and economy of the power system.
[0079] After determining the finite element model, it is necessary to determine the global equation group of the finite element model. Optionally, in the sag simulation method for the transmission line provided in the embodiment of the present application, after converting the multi-field coupling model into a finite element model to obtain multiple interconnected sub-areas, the method further includes: determining the nodes at the boundary of each sub-area, wherein the nodes are used to characterize the connection nodes of the transmission line between different sub-areas; for each sub-area, constructing a local equation group based on the coupling equation group of the multi-field coupling model, wherein the local equation group is used to characterize the changes in physical quantities of the nodes within the sub-area; and constructing a global equation group through the coupling conditions, boundary conditions and local equation group of each sub-area of the multi-field coupling model.
[0080] In some examples, after the multi-field coupling model is divided into multiple sub-regions according to the grid, the nodes on the boundary of each sub-region are determined. These nodes are used to connect adjacent sub-regions and are also the key points of multi-physical field coupling. The physical quantity changes (such as displacement, stress, temperature) of the nodes are affected by multiple physical fields around them. For each sub-region, a local set of equations is constructed based on the coupled equations of the multi-field coupling model. The local set of equations describes the physical quantity changes of the nodes in the sub-region, including but not limited to: Mechanical equations: describe the mechanical response of the transmission line in the sub-region under various loads (such as wind load, gravity load, ice load), involving the displacement, velocity, acceleration and stress distribution of the nodes. Thermodynamic equations: consider the influence of temperature changes in the sub-region on the mechanical properties of the transmission line, describe the temperature distribution and heat conduction behavior inside the transmission line, and the geometric changes of the transmission line caused by temperature changes (such as thermal expansion and contraction). Fluid dynamics equations: simulate the wind field and airflow behavior around the transmission line in the sub-region, and the influence of airflow changes on the mechanical behavior of the transmission line. Contact mechanics equation: describes the friction and collision force when the transmission line contacts the obstacle in the sub-area, and the influence of these forces on the displacement and stress of the transmission line.
[0081] The coupling conditions of the multi-field coupling model are applied to the boundary nodes to reflect the interaction between different physical fields. Boundary conditions are set for the overall system of the entire transmission line, such as fixed endpoints (fixed points of the transmission line at the tower), free endpoints (free ends of the transmission line in the air) and physical boundaries (terrain boundaries). These conditions limit the scope and mode of system behavior. The local equations of each sub-area, as well as the coupling conditions and boundary conditions, are combined to form a global equation group that describes the overall system behavior of the entire transmission line. The global equation group can include multiple unknowns, such as displacements, stresses, temperatures, etc. of all nodes, as well as multiple equations that describe the interdependence between these unknowns.
[0082] This embodiment constructs a global set of equations to fully reflect the multi-physical field coupling behavior of transmission lines under complex working conditions, providing more accurate and reliable analysis results for line design, construction, and operation and maintenance. In particular, for applications in special geographical environments and extreme climatic conditions, the multi-field coupling model can help evaluate the stability and safety of the line, optimize line design, reduce potential fault risks, and improve the operating efficiency and economy of the power system.
[0083] After determining the global equation group, the global equation group is solved by a preset nonlinear finite element algorithm. Optionally, in the sag simulation method for the transmission line provided in the embodiment of the present application, the equilibrium state of the finite element model is solved by a preset nonlinear finite element algorithm to obtain the node value of each sub-area, including: determining an initial solution of the global equation group, wherein the initial solution includes the initial value of each node of all sub-areas, and the initial value is used to characterize the physical quantity of the transmission line when it is not affected by environmental changes; decomposing the coupling conditions into a set of increments, and iteratively solving the global equation group based on the set of increments, wherein each iteration adds one increment in a set of increments to the global equation group; after all increments in a set of increments are added to the global equation group, solving the global equation group to obtain a target solution, and determining the node value of each sub-area in the target solution.
[0084] In some examples, the initial solution may include physical quantities such as initial displacement, velocity, stress, strain, and temperature of each node in all sub-areas. The initial value may reflect the natural state of the transmission line without external loads or environmental changes, for example, the state of the transmission line without wind load, ice load, or temperature change. The initial solution is the distribution of physical quantities of each part of the overall system of the transmission line under static or equilibrium conditions. For the transmission line, the initial solution is that the transmission line is in its geometric equilibrium position without external loads and the internal stress distribution is uniform.
[0085] The preset nonlinear finite element algorithm can be an incremental iteration method, which decomposes the coupling conditions (such as various loads, environmental changes, and changes in material properties) into a series of increments, each increment represents a small change in the load or condition, and uses the increments to gradually approach the equilibrium state of the overall system of the transmission line. Based on a set of increments, the global equations are iteratively solved, and each iteration only increases one increment in a set of increments. Physical quantities such as displacement, stress, and strain are gradually updated to gradually approach the final state of the transmission line under given load conditions. During the iteration process, the nonlinear finite element algorithm recalculates the loads, constraints, and material properties based on the updated physical quantities until an equilibrium state is reached, that is, the balance of external and internal forces, and the satisfaction of conservation of energy and conservation of momentum.
[0086] When all increments in a set of increments have been added to the global set of equations and the iterative process meets the convergence criteria (set according to expert experience), the algorithm will obtain the final solution of the overall system of the transmission line, that is, the target solution. The target solution contains the node values of each sub-area, which is used to describe the distribution of physical quantities such as displacement, stress, strain and temperature of the transmission line under specific load and environmental conditions. Finally, the physical rationality of the simulation results is checked by the principles of conservation of energy and momentum to ensure the accuracy and stability of the final target solution.
[0087] This embodiment gradually modifies the model state of the transmission line through the incremental iteration method until all coupling conditions and boundary conditions are met, thereby obtaining an accurate transmission line response. It is suitable for simulating the mechanical and electrical behavior of transmission lines under complex terrain and extreme weather conditions, as well as the interaction between the lines and the surrounding environment, providing a scientific basis for line design and operation and maintenance, helping to optimize the line structure, and improving the safety and economy of the power system.
[0088] For example, the incremental iteration method gradually approaches the solution of the problem in the process of continuous iterations. Each iteration can be expressed by the following formula:
[0089]
[0090] in, It is The point at the iteration (that is, the value of a physical quantity in the global equations), It is The step size of the iteration (that is, the size of the increment), which is a positive scalar used to control the size of each step. Function ∫ at point x K The gradient vector at , represents the steepest ascent direction of the function at that point; the above iterative formula uses a variant of the gradient descent method, which is an iterative method for finding the local minimum of an unconstrained optimization problem. The basic idea of the incremental iteration method is to gradually adjust the variable along the negative direction of the function gradient (that is, the opposite direction of the fastest increasing direction of the function). The above formula shows how to get from the current point x K Update to next point By subtracting the gradient and multiplying it by the step size Gradually approach the minimum point of the function. Step size It can be fixed or dynamically adjusted according to different strategies, such as line search method to ensure faster convergence speed and better global convergence performance.
[0091] It should be noted that the above formula is only one of many forms of the incremental iteration method, and the specific form will vary depending on the nature of the problem (such as whether it is a constrained optimization problem, the specific form of the function, etc.).
[0092] The Newton-Raphson method is an efficient iterative method for solving nonlinear equations. The root of is a The Newton-Raphson method is a vector function that maps a 2-dimensional Euclidean space to itself. Each iteration is based on a linear approximation of the current point and uses the Jacobian matrix to improve the estimate of the solution. The iterative formula of the Newton-Raphson method is as follows:
[0093]
[0094] in, It is The estimated value of the solution at the iteration, J F (x K ) is the function F at the point The Jacobian matrix at The matrix composed of partial derivatives of F (x K ) -1 represents the inverse matrix of the Jacobian matrix, F(x K ) is at point The value of the function F at , that is, a column vector, is obtained through the Raphson method and the modified iteration. The Raphson method uses the Taylor series expansion to linearize the nonlinear equation (that is, the global equation group) near a certain point (that is, the node). Solving the linearized equation to obtain the solution for the next iteration requires calculating the Jacobian matrix and its inverse as well as the residual vector. After each iteration, the modified iteration adjusts the step size or updates the approximation of the Jacobian matrix according to the size of the residual to improve convergence.
[0095] The above formula shows that in each iteration, the function value F(x K ) and the Jacobian matrix J F (x K ), and then use the inverse Jacobian matrix of the current point to "correct" the solution of each iteration so that it is closer to the root of the nonlinear equation;
[0096] It should be noted that directly finding the inverse of the Jacobian matrix may be impractical or computationally expensive in actual operations, especially for large-scale problems. Therefore, in practical applications, iterative methods or decomposition methods (such as Gaussian elimination) are used to indirectly solve J F (x K ) -1 F(x K) problem, or use specialized algorithms in numerical linear algebra packages to solve such linear systems. In addition, to ensure convergence and stability, damping factors are introduced or more complex line search techniques are used to determine the optimal step size for each step.
[0097] According to another embodiment of the present application, a combined conductor three-dimensional sag simulation system is also provided. Figure 2 Schematic diagram of a three-dimensional sag simulation system for a combined conductor provided according to an embodiment of the present application. Figure 2 As shown, the system includes: an information module, a three-dimensional modeling and meshing module, a physical field coupling module, a nonlinear solution module, a dynamic simulation module and a report module.
[0098] Among them, the information module includes a data input module and a preprocessing module. The data input module is used to receive the basic line information input by the user. The three-dimensional modeling and meshing module constructs a three-dimensional model of the transmission line, tower and terrain according to the input data, and reasonably meshes the model to prepare for finite element analysis. In this embodiment, the data input module is used to receive the basic line information (conductor type, combination method, span length, etc.), environmental parameters (wind speed, temperature, ice thickness, etc.), terrain data, etc. input by the user. The preprocessing module automatically verifies the rationality of the data, supports the import of multiple formats, and preliminarily processes the data for subsequent analysis.
[0099] The 3D modeling and meshing module constructs 3D models of transmission lines, towers, and terrain based on input data, and performs reasonable meshing of the models in preparation for finite element analysis. It supports accurate simulation of complex terrain, automatically optimizes mesh density, and ensures a balance between computational efficiency and accuracy.
[0100] The physical field coupling module cooperates with the nonlinear solution module to realize the establishment and solution of multi-field coupling models such as mechanics (elasticity, plasticity), thermal, fluid dynamics, etc. based on the finite element method. Taking nonlinear effects into consideration, an efficient iterative algorithm (Newton-Raphson method) is used to solve nonlinear equations. It supports parallel computing acceleration to ensure the rapid convergence of large-scale models.
[0101] The dynamic simulation module dynamically simulates the impact of environmental changes on conductor sag at different time steps, supports real-time data access, dynamically adjusts model parameters according to monitoring data, and can provide a dynamic visualization interface to intuitively display the three-dimensional shape of the conductor over time, facilitating real-time evaluation and adjustment.
[0102] The report results generated by the reporting module facilitate team collaboration and communication.
[0103] The combined conductor three-dimensional sag simulation system provided in this embodiment realizes high-precision three-dimensional sag simulation of combined conductors under complex working conditions by introducing multi-physical field coupling models and nonlinear finite element analysis, effectively solving the problem of insufficient accuracy of sag simulation under complex environments. The system-integrated design process is fully automated from data input to result output, reducing manual intervention and repeated calculations, significantly improving the efficiency of line design and optimization, and shortening the project cycle. The dynamic simulation and real-time feedback mechanism can timely evaluate the safety margin of the line under various extreme conditions, prevent line failures caused by improper sag, and enhance the stability and safety of the power system. The system fully considers the influence of factors such as terrain and meteorology on sag, provides a scientific basis for the design of transmission lines in special geographical environments, and improves the adaptability of lines to complex environments. By accurately predicting the operating status of the conductor, maintenance strategies can be planned in advance, reducing unnecessary on-site inspections and repairs, thereby significantly reducing operating and maintenance costs, promoting scientific and technological progress in the field of power engineering design, providing advanced research tools for relevant scientific researchers, and promoting the development and application of new technologies and theories.
[0104] It should be noted that the steps shown in the flowcharts of the accompanying drawings can be executed in a computer system such as a set of computer executable instructions, and that, although a logical order is shown in the flowcharts, in some cases, the steps shown or described can be executed in an order different from that shown here.
[0105] The embodiment of the present application also provides a sag simulation device for a transmission line. It should be noted that the sag simulation device for a transmission line in the embodiment of the present application can be used to execute the sag simulation method for a transmission line provided in the embodiment of the present application. The sag simulation device for a transmission line provided in the embodiment of the present application is introduced below.
[0106] Figure 3 Schematic diagram of a sag simulation device for a power transmission line provided in accordance with an embodiment of the present application. Figure 3 As shown, the device comprises:
[0107] The first determining unit 301 is used to determine the target area where the transmission line is to be deployed, obtain terrain information and meteorological information of the target area, and obtain attribute information of the transmission line;
[0108] The receiving unit 302 is used to receive the terminal position information of the power transmission line deployed in the target area input by the user, and construct a three-dimensional space model of the power transmission line based on the terminal position information and terrain information;
[0109] The first construction unit 303 is used to construct a multi-field coupling model of the transmission line based on the three-dimensional space model, meteorological information and attribute information, wherein the multi-field coupling model includes at least one of the following: a mechanical and meteorological coupling model, a thermal effect and mechanical property coupling model, a multi-conductor interaction coupling model and a terrain and conductor interaction coupling model;
[0110] The conversion unit 304 is used to convert the multi-field coupling model into a finite element model to obtain a plurality of interconnected sub-regions, and solve the equilibrium state of the finite element model by a preset nonlinear finite element algorithm to obtain a node value of each sub-region, wherein the node value is used to characterize a physical quantity in the sub-region, and the physical quantity includes at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line;
[0111] The second constructing unit 305 is used to construct the sag simulation result of the transmission line according to the node value of each sub-area.
[0112] The sag simulation device for a transmission line provided in an embodiment of the present application determines the target area where the transmission line is to be deployed, obtains the terrain information and meteorological information of the target area, and obtains the attribute information of the transmission line through a first determination unit 301; a receiving unit 302 receives the endpoint position information of the transmission line deployed in the target area input by the user, and constructs a three-dimensional space model of the transmission line based on the endpoint position information and the terrain information; a first construction unit 303 constructs a multi-field coupling model of the transmission line based on the three-dimensional space model, the meteorological information and the attribute information, wherein the multi-field coupling model includes at least one of the following: a mechanical and meteorological coupling model, a thermal effect and mechanical property coupling model, a multi-conductor interaction coupling model and a terrain and conductor interaction coupling model; a conversion unit 304 is used to construct a multi-field coupling model of the transmission line based on the three-dimensional space model, the meteorological information and the attribute information. 04, converting the multi-field coupling model into a finite element model to obtain multiple interconnected sub-regions, solving the equilibrium state of the finite element model through a preset nonlinear finite element algorithm, and obtaining the node value of each sub-region, wherein the node value is used to characterize the physical quantity in the sub-region, and the physical quantity includes at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line; the second construction unit 305, constructing the sag simulation result of the transmission line through the node value of each sub-region, solves the problem of low accuracy of the sag simulation of the transmission line in the related technology, and realizes high-precision three-dimensional sag simulation of the transmission line under complex working conditions by introducing the multi-physical field coupling model and nonlinear finite element analysis, thereby achieving the effect of improving the accuracy of the sag simulation of the transmission line.
[0113] Optionally, in the sag simulation device for a transmission line provided in an embodiment of the present application, the first construction unit 303 includes: a first determination module, used to determine the physical field to be coupled, and determine the initial conditions and boundary conditions of each physical field based on the three-dimensional space model and meteorological information, wherein the physical field includes at least one of the following: a mechanical field, a thermal field, an electromagnetic field, and a fluid field, and the initial conditions are used to characterize the initial values of physical quantities in the physical field, and the boundary conditions are used to characterize the initial values of physical quantities at the edge of the physical field; a first construction module, used to construct a mathematical model of each physical field through partial differential equations to obtain multiple mathematical models, wherein the mathematical model is used to describe the behavior of physical quantities in the physical field, and the mathematical model includes at least one of the following: a heat conduction model, a fluid dynamics model, and a structural mechanics model; a second determination module, used to determine the coupling conditions between physical fields through the three-dimensional space model, meteorological information, and attribute information, and construct a multi-field coupling model based on the coupling conditions and multiple mathematical models.
[0114] Optionally, in the sag simulation device for a transmission line provided in an embodiment of the present application, the second determination module includes: a first extraction submodule, used to extract the wind speed and wind direction of the target area from the meteorological information, and extract the geometry and material of the transmission line from the attribute information, and determine the wind load of the transmission line based on the wind speed, wind direction, geometry and material, wherein the wind load is used to characterize the force exerted by the wind on the transmission line; a second extraction submodule, used to extract icing information of the target area from the meteorological information, and extract the surface area of the transmission line from the attribute information, and determine the icing load of the transmission line based on the icing information and the surface area, wherein the icing load is used to characterize the force exerted by the icing on the transmission line; a third extraction submodule, used to extract the weight of the transmission line from the attribute information, and determine the gravity exerted on the transmission line based on the weight; and a first determination submodule, used to determine the wind load, icing load and gravity as coupling conditions.
[0115] Optionally, in the sag simulation device for a transmission line provided in an embodiment of the present application, the second determination module includes: a fourth extraction submodule, used to extract the temperature of a target area from meteorological information, and determine the material of the transmission line from the attribute information, and determine the stress, strain and tension of the transmission line in the target area based on the temperature and material; and a second determination submodule, used to determine the stress, strain and tension as coupling conditions.
[0116] Optionally, in the sag simulation device for a transmission line provided in an embodiment of the present application, the second determination module includes: a third determination submodule, used to determine the distance between the transmission lines from a three-dimensional space model, extract the cross-sectional area of the transmission lines from the attribute information, and calculate the attraction or repulsion force on the transmission lines based on the distance and the cross-sectional area; and a fourth determination submodule, used to determine the attraction or repulsion force as a coupling condition.
[0117] Optionally, in the sag simulation device for a transmission line provided in an embodiment of the present application, the second determination module includes: a fifth determination submodule, used to determine obstacles and terrain information contacted by the transmission line from a three-dimensional space model; a sixth determination submodule, used to determine the local wind field in the target area through terrain information, and determine the wind force of the local wind field on the transmission line, and determine the contact force of the transmission line based on the obstacle; and a seventh determination submodule, used to determine the wind force and the contact force as coupling conditions.
[0118] Optionally, in the sag simulation device for a transmission line provided in an embodiment of the present application, the second determination module includes: an eighth determination submodule, used to determine coupling variables from coupling conditions, wherein the coupling variables include at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line; a combined submodule, used to combine multiple mathematical models to obtain a coupling equation group, and determine the coupling variables and the coupling equation group as a multi-field coupling model.
[0119] Optionally, in the sag simulation device for a transmission line provided in an embodiment of the present application, the device further includes: a second determination unit, used to determine the nodes at the boundary of each sub-region, wherein the nodes are used to characterize the connection nodes of the transmission line between different sub-regions; a third construction unit, used to construct a local set of equations for each sub-region based on the coupling equations of the multi-field coupling model, wherein the local set of equations is used to characterize the changes in physical quantities of the nodes within the sub-region; and a fourth construction unit, used to construct a global set of equations through the coupling conditions of the multi-field coupling model, boundary conditions and the local set of equations for each sub-region.
[0120] Optionally, in the sag simulation device for a transmission line provided in an embodiment of the present application, the conversion unit 304 includes: a third determination module, used to determine an initial solution of the global set of equations, wherein the initial solution includes an initial value of each node of all sub-areas, and the initial value is used to characterize the physical quantity of the transmission line when it is not affected by environmental changes; a decomposition module, used to decompose the coupling conditions into a set of increments, and iteratively solve the global set of equations based on the set of increments, wherein each iteration adds one increment in a set of increments to the global set of equations; an addition module, used to solve the global set of equations after all increments in a set of increments are added to the global set of equations, obtain a target solution, and determine the node value of each sub-area in the target solution.
[0121] The sag simulation device for a transmission line includes a processor and a memory. The above-mentioned first determination unit 301, receiving unit 302, first construction unit 303, conversion unit 304 and second construction unit 305 are all stored in the memory as program units, and the processor executes the above-mentioned program units stored in the memory to realize corresponding functions.
[0122] The processor includes a kernel, and the kernel calls the corresponding program unit from the memory. One or more kernels can be set, and the accuracy of the sag simulation of the transmission line can be improved by adjusting the kernel parameters.
[0123] The memory may include non-permanent memory in a computer-readable medium, random access memory (RAM) and / or non-volatile memory, such as read-only memory (ROM) or flash RAM, and the memory includes at least one memory chip.
[0124] An embodiment of the present invention provides a computer-readable storage medium on which a program is stored. When the program is executed by a processor, a method for simulating the sag of a transmission line is implemented.
[0125] An embodiment of the present invention provides a processor, which is used to run a program, wherein the program executes a sag simulation method for a power transmission line when the program is running.
[0126] Figure 4 is a schematic diagram of an electronic device provided according to an embodiment of the present application. Figure 4 As shown, the electronic device 401 includes a processor, a memory, and a program stored in the memory and executable on the processor. When the processor executes the program, the following steps are implemented: determining a target area for the deployment of the transmission line, obtaining terrain information and meteorological information of the target area, and obtaining attribute information of the transmission line; receiving the endpoint position information of the transmission line deployed in the target area input by the user, and constructing a three-dimensional space model of the transmission line based on the endpoint position information and the terrain information; constructing a multi-field coupling model of the transmission line based on the three-dimensional space model, the meteorological information, and the attribute information, wherein the multi-field coupling model includes at least one of the following: a mechanical and meteorological coupling model, a thermal effect and mechanical property coupling model, a multi-conductor interaction coupling model, and a terrain and conductor interaction coupling model; converting the multi-field coupling model into a finite element model to obtain a plurality of interconnected sub-areas, solving the equilibrium state of the finite element model by a preset nonlinear finite element algorithm, and obtaining a node value of each sub-area, wherein the node value is used to characterize the physical quantity in the sub-area, and the physical quantity includes at least one of the following: displacement, velocity, stress, strain, and pressure of the transmission line; and constructing the sag simulation result of the transmission line by the node value of each sub-area. The devices in this article can be servers, PCs, PADs, mobile phones, etc.
[0127] The present application also provides a computer program product, which, when executed on a data processing device, is suitable for executing a program that initializes the following method steps: determining a target area for the deployment of transmission lines, obtaining terrain information and meteorological information of the target area, and obtaining attribute information of the transmission lines; receiving the endpoint position information of the transmission lines deployed in the target area input by the user, and constructing a three-dimensional space model of the transmission lines based on the endpoint position information and terrain information; constructing a multi-field coupling model of the transmission lines based on the three-dimensional space model, meteorological information and attribute information, wherein the multi-field coupling model includes at least one of the following: a mechanical and meteorological coupling model, a thermal effect and mechanical property coupling model, a multi-conductor interaction coupling model and a terrain and conductor interaction coupling model; converting the multi-field coupling model into a finite element model to obtain a plurality of interconnected sub-areas, solving the equilibrium state of the finite element model by a preset nonlinear finite element algorithm, and obtaining a node value of each sub-area, wherein the node value is used to characterize the physical quantity in the sub-area, and the physical quantity includes at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line; constructing the sag simulation result of the transmission line by the node value of each sub-area.
[0128] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems, or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment, or an embodiment in combination with software and hardware. Moreover, the present application may adopt the form of a computer program product implemented in one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that include computer-usable program code.
[0129] The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.
[0130] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.
[0131] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.
[0132] In a typical configuration, a computing device includes one or more processors (CPU), input / output interfaces, network interfaces, and memory.
[0133] The memory may include non-permanent memory in a computer-readable medium, random access memory (RAM) and / or non-volatile memory in the form of read-only memory (ROM) or flash RAM. The memory is an example of a computer-readable medium.
[0134] Computer readable media include permanent and non-permanent, removable and non-removable media that can be implemented by any method or technology to store information. Information can be computer readable instructions, data structures, program modules or other data. Examples of computer storage media include, but are not limited to, phase change memory (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), other types of random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technology, compact disk read-only memory (CD-ROM), digital versatile disk (DVD) or other optical storage, magnetic cassettes, magnetic tape disk storage or other magnetic storage devices or any other non-transmission media that can be used to store information that can be accessed by a computing device. As defined herein, computer readable media does not include temporary computer readable media (transitory media), such as modulated data signals and carrier waves.
[0135] It should also be noted that the terms "include", "comprises" or any other variations thereof are intended to cover non-exclusive inclusion, so that a process, method, commodity or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, commodity or device. In the absence of more restrictions, the elements defined by the sentence "comprises a ..." do not exclude the existence of other identical elements in the process, method, commodity or device including the elements.
[0136] Those skilled in the art will appreciate that the embodiments of the present application may be provided as methods, systems or computer program products. Therefore, the present application may adopt the form of a complete hardware embodiment, a complete software embodiment or an embodiment in combination with software and hardware. Moreover, the present application may adopt the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that contain computer-usable program code.
[0137] The above are only embodiments of the present application and are not intended to limit the present application. For those skilled in the art, the present application may have various changes and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application should be included within the scope of the claims of the present application.
Claims
1. A method for simulating the sag of a transmission line, characterized in that: include: Determine a target area where the transmission line is to be deployed, obtain terrain information and meteorological information of the target area, and obtain attribute information of the transmission line; Receiving the endpoint location information of the transmission line deployed in the target area input by the user, and constructing a three-dimensional space model of the transmission line based on the endpoint location information and the terrain information; Constructing a multi-field coupling model of the transmission line based on the three-dimensional space model, the meteorological information and the attribute information, wherein the multi-field coupling model includes at least one of the following: a mechanical and meteorological coupling model, a thermal effect and mechanical property coupling model, a multi-conductor interaction coupling model and a terrain and conductor interaction coupling model; The multi-field coupling model is converted into a finite element model to obtain a plurality of interconnected sub-regions, and the equilibrium state of the finite element model is solved by a preset nonlinear finite element algorithm to obtain a node value of each sub-region, wherein the node value is used to characterize a physical quantity in the sub-region, and the physical quantity includes at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line; The sag simulation result of the transmission line is constructed by using the node value of each sub-area.
2. The method according to claim 1, characterized in that Constructing the multi-field coupling model of the transmission line based on the three-dimensional space model, the meteorological information and the attribute information includes: Determine the physical fields to be coupled, and determine the initial conditions and boundary conditions of each physical field based on the three-dimensional space model and the meteorological information, wherein the physical field includes at least one of the following: a mechanical field, a thermal field, an electromagnetic field, and a fluid field, the initial conditions are used to characterize the initial values of physical quantities in the physical field, and the boundary conditions are used to characterize the initial values of physical quantities at the edges of the physical field; Constructing a mathematical model of each physical field through partial differential equations to obtain a plurality of mathematical models, wherein the mathematical model is used to describe the behavior of physical quantities in the physical field, and the mathematical model includes at least one of the following: a heat conduction model, a fluid dynamics model, and a structural mechanics model; The coupling conditions between the physical fields are determined through the three-dimensional space model, the meteorological information and the attribute information, and the multi-field coupling model is constructed based on the coupling conditions and the multiple mathematical models.
3. The method according to claim 2, characterized in that The multi-field coupling model includes a mechanical and meteorological coupling model, and determining the coupling conditions between the physical fields through the three-dimensional space model, the meteorological information and the attribute information includes: Extracting the wind speed and wind direction of the target area from the meteorological information, and extracting the geometric shape and material of the transmission line from the attribute information, and determining the wind load of the transmission line based on the wind speed, wind direction, the geometric shape and the material, wherein the wind load is used to characterize the force exerted by the wind on the transmission line; Extracting icing information of the target area from the meteorological information, and extracting the surface area of the transmission line from the attribute information, and determining an icing load of the transmission line based on the icing information and the surface area, wherein the icing load is used to characterize the force exerted by icing on the transmission line; Extracting the weight of the power transmission line from the attribute information, and determining the gravity exerted on the power transmission line based on the weight; The wind load, the ice load and the gravity are determined as the coupling conditions.
4. The method according to claim 2, characterized in that: The multi-field coupling model includes a thermal effect and mechanical property coupling model, and determining the coupling conditions between the physical fields through the three-dimensional space model, the meteorological information and the attribute information includes: Extracting the temperature of the target area from the meteorological information, determining the material of the transmission line from the attribute information, and determining the stress, strain and tension of the transmission line in the target area based on the temperature and the material; The stress, strain and tension are determined as the coupling conditions.
5. The method according to claim 2, characterized in that: The multi-field coupling model includes a multi-conductor interaction coupling model, and determining the coupling conditions between the physical fields through the three-dimensional space model, the meteorological information and the attribute information includes: Determine the distance between the transmission wires from the three-dimensional space model, extract the cross-sectional area of the transmission wire from the attribute information, and calculate the attractive force or repulsive force on the transmission wire based on the distance and the cross-sectional area; The attractive force or the repulsive force is determined as the coupling condition.
6. The method according to claim 2, characterized in that The multi-field coupling model includes a terrain and wire interaction coupling model, and determining the coupling conditions between the physical fields through the three-dimensional space model, the meteorological information and the attribute information includes: Determining obstacles and terrain information that the transmission line contacts from the three-dimensional space model; Determine a local wind field in the target area through the terrain information, determine the wind force of the local wind field on the transmission line, and determine the contact force of the transmission line based on the obstacle; The wind force and the contact force are determined as the coupling condition.
7. The method according to claim 2, characterized in that Constructing the multi-field coupling model based on the coupling condition and the multiple mathematical models includes: Determining coupling variables from the coupling conditions, wherein the coupling variables include at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line; The multiple mathematical models are combined to obtain a coupling equation group, and the coupling variables and the coupling equation group are determined as the multi-field coupling model.
8. The method according to claim 1, characterized in that: After converting the multi-field coupling model into a finite element model to obtain a plurality of interconnected sub-regions, the method further includes: Determine nodes at the boundary of each sub-region, wherein the nodes are used to represent connection nodes of the transmission wires between different sub-regions; For each sub-region, constructing a local set of equations based on the coupled equations of the multi-field coupling model, wherein the local set of equations is used to characterize changes in physical quantities of nodes within the sub-region; The global equations are constructed by the coupling conditions, boundary conditions and local equations of each sub-region of the multi-field coupling model.
9. The method according to claim 1, characterized in that: The equilibrium state of the finite element model is solved by a preset nonlinear finite element algorithm, and the node value of each sub-area is obtained, including: Determine an initial solution of the global equations, wherein the initial solution includes an initial value of each node of all sub-regions, and the initial value is used to characterize the physical quantity of the transmission line when it is not affected by environmental changes; Decomposing the coupling condition into a set of increments, and iteratively solving the global set of equations based on the set of increments, wherein each iteration increases the global set of equations by one increment in the set of increments; After all increments in the group of increments are added to the global set of equations, the global set of equations is solved to obtain a target solution, and a node value of each of the sub-regions in the target solution is determined.
10. A sag simulation device for a power transmission line, characterized in that: include: A first determining unit, configured to determine a target area where the transmission line is to be deployed, obtain terrain information and meteorological information of the target area, and obtain attribute information of the transmission line; A receiving unit, configured to receive the endpoint location information of the transmission line deployed in the target area input by a user, and to construct a three-dimensional spatial model of the transmission line based on the endpoint location information and the terrain information; A first construction unit is used to construct a multi-field coupling model of the transmission line based on the three-dimensional space model, the meteorological information and the attribute information, wherein the multi-field coupling model includes at least one of the following: a mechanical and meteorological coupling model, a thermal effect and mechanical property coupling model, a multi-conductor interaction coupling model and a terrain and conductor interaction coupling model; A conversion unit is used to convert the multi-field coupling model into a finite element model to obtain a plurality of interconnected sub-regions, and solve the equilibrium state of the finite element model by a preset nonlinear finite element algorithm to obtain a node value of each sub-region, wherein the node value is used to characterize the physical quantity in the sub-region, and the physical quantity includes at least one of the following: displacement, velocity, stress, strain and pressure of the transmission line; a second construction unit is used to construct a sag simulation result of the transmission line through the node value of each sub-region.
Citation Information
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