A deep learning optimization method for transmission efficiency of super copper low-voltage cable

Abstract

The application discloses a kind of super copper low-voltage cable transmission efficiency deep learning optimization method, belong to intelligent power grid distribution equipment energy efficiency promotion technical field.The application constructs the multi-source data set covering material microstructure, cable geometric configuration and operating condition;Physical information neural network model is trained, physical constraint term based on skin effect theory is embedded in loss function, to ensure that the predicted AC resistance value is not lower than the theoretical minimum value, so as to ensure that the prediction conforms to electromagnetic law;Based on the physically credible proxy model, the gradient descent algorithm with constraints is used to optimize the cable structure parameters in reverse, and the cable optimization design scheme that meets the safety constraints is automatically generated.The application overcomes the defects of traditional methods, such as insufficient modeling of multi-dimensional strong coupling relationship and physical untrustworthiness of pure data-driven models, through deep integration of physical mechanism and data-driven, and realizes high-precision prediction and collaborative optimization of cable transmission efficiency.

Description

Technical Field

[0001] This invention belongs to the field of energy efficiency improvement technology for smart grid power distribution equipment, specifically involving a deep learning optimization method for the transmission efficiency of super copper low-voltage cables. Background Technology

[0002] As modern power distribution systems transform towards higher efficiency, intelligence, and green low-carbon practices, low-voltage cables, as a key medium connecting end-users and the power distribution network, directly impact the overall energy efficiency of the system. With the application of super copper—a copper-based conductor whose conductivity is enhanced through nano-doping and grain boundary modulation—in the field of low-voltage power transmission, fully leveraging the potential of such advanced materials has become a crucial breakthrough for improving the performance of power distribution equipment.

[0003] However, the actual transmission efficiency of a cable is not solely determined by the intrinsic conductivity of the conductor. It is the result of a complex coupling effect of multiple factors, including the material's microstructure, cable geometry such as strand pitch and filament diameter, and operating conditions such as frequency, temperature, and harmonic content. Traditional design methods primarily rely on empirical formulas or simplified electromagnetic models from IEC standards, which struggle to accurately characterize highly nonlinear interactions and effectively assess the loss behavior of super copper at high frequencies. This leads to design results remaining at local optima and limiting the full realization of the material's potential.

[0004] Some studies have attempted to introduce artificial intelligence technology to improve the intelligence level of cable systems, but neither the technical approach nor the core objective has been aimed at improving the transmission efficiency of the cable itself.

[0005] Patent CN120672745A discloses a method for detecting surface defects in cable sheaths. It utilizes YOLOv5 computer vision technology to automatically identify small-sized defects in the cable's outer sheath. While valuable in improving quality inspection efficiency in the manufacturing process, this approach focuses entirely on the visual inspection of the cable's external appearance. It does not involve modeling the electromagnetic behavior inside the conductor, nor does it establish any physical relationship between material properties, structural parameters, and power loss. Its data input is limited to optical images, lacking key electromagnetic parameters such as conductivity and frequency, thus lacking the ability to model or optimize transmission efficiency.

[0006] Patent CN114205981A discloses a method for generating extreme ultraviolet (EUV) light using artificial intelligence and nanosecond pulsed fast ionization waves. Deep learning is used to inversely derive the driving voltage waveform to optimize plasma radiation efficiency. Although copper conductors are used in the high-voltage transient system and artificial intelligence is employed, the physical model and optimization objectives revolve entirely around the plasma excitation process, focusing on spectral selectivity and energy conversion efficiency, rather than the conductor's power transmission characteristics in steady-state or quasi-steady-state conditions. The model does not consider conductor loss mechanisms under low-voltage, power frequency, and harmonic conditions, and its framework is highly heterogeneous to the technical requirements for solving cable energy efficiency problems, making direct transfer or adaptation impossible.

[0007] Existing technologies either limit themselves to surface quality inspection or serve entirely different physical systems, failing to construct an intelligent optimization framework that deeply integrates material properties, geometry, and operating conditions to improve cable transmission efficiency and ensures that model output conforms to fundamental physical laws. The empirical limitations of traditional methods and the deviation from the goals of existing AI applications together constitute the key obstacles preventing the systematic and reliable optimization of the energy efficiency of current super copper low-voltage cables.

[0008] To address this, this invention proposes a novel technical approach: constructing an end-to-end, physically embedded deep learning collaborative optimization framework. This framework is used to establish a high-fidelity, physically reliable mapping model from multi-dimensional inputs of materials, structure, and operating conditions to active power loss output. Based on this model, automated and constrained reverse design of cable structural parameters is achieved, thereby deeply exploring and releasing the power transmission efficiency potential of super copper materials while ensuring engineering safety. Summary of the Invention

[0009] This invention solves the technical problem that existing technologies cannot accurately model the strong coupling nonlinear relationships between multiple variables such as material properties, cable geometry and operating conditions, and the prediction results of pure data-driven models lacking physical constraints are physically unreliable, thus making it impossible to systematically and reliably optimize the transmission efficiency of super copper low-voltage cables.

[0010] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows: This invention provides a deep learning optimization method for the transmission efficiency of super copper low-voltage cables, comprising the following steps: S1: Construct a multi-source heterogeneous training dataset; the multi-source heterogeneous training dataset includes a super copper conductor microstructure parameter dataset, a cable geometric configuration parameter dataset, an operating condition and environmental variable dataset, and a unit length active power loss label dataset corresponding to the combination of the first three types of data. S2: Construct a physical information neural network model; the physical information neural network model takes the feature vector obtained by fusing the microstructure parameter dataset of the super copper conductor, the cable geometric configuration parameter dataset, and the operating condition and environmental variable dataset as input, and outputs the predicted active power loss value per unit length; the loss function of the physical information neural network model includes a data fitting loss term and a physical constraint loss term. The physical constraint loss term is constructed based on the principles of electromagnetic physics and is used to force the AC resistance value predicted by the model to be no less than the theoretical minimum AC resistance value calculated based on the skin effect theory during the model training process; S3: Perform reverse design of cable structure based on the physical information neural network model; specifically: use the physical information neural network model as a differentiable surrogate model, take minimizing the predicted active power loss per unit length as the objective function, adjust the designable structural parameters of the cable, and perform iterative optimization under the condition of meeting the preset safety constraints until the preset energy efficiency target is achieved; S4: Output cable optimization design scheme; the cable optimization design scheme includes the combination of cable structure parameters obtained by S3 optimization.

[0011] Furthermore, the super copper conductor microstructure parameter dataset in S1 includes: average grain diameter, standard deviation of grain size distribution, volume fraction of nano-precipitates, average grain size of nano-precipitates, dislocation density, and principal value of residual stress; the cable geometry parameter dataset includes: total conductor cross-sectional area, single filament diameter, stranding pitch, stranding angle, insulation layer thickness, relative permittivity of insulation material, shielding layer structure type, and shielding layer thickness; the operating condition and environmental variable dataset includes: operating frequency, effective value of fundamental current, content of specific harmonic current, ambient temperature, and thermal resistance coefficient of laying method.

[0012] Furthermore, the feature vector input to the physical information neural network model in S2 also includes multiple dimensionless combination features calculated based on the microstructure parameters, geometric configuration parameters, and operating conditions and environmental variables; the multiple dimensionless combination features include at least: the Struhal number calculated based on the operating frequency and stranding pitch, the Reynolds number calculated based on the conductivity and single filament diameter, the skin ratio calculated based on the single filament diameter and theoretical skin depth, and the total harmonic distortion rate calculated based on the harmonic current content.

[0013] Furthermore, the physical information neural network model described in S2 employs a multi-layer fully connected network structure with a physical constraint loss term in the loss function, which is used to force the model to predict AC resistance values ​​for samples with frequencies higher than a preset threshold (e.g., 50 Hz) that are not lower than the minimum theoretical AC resistance value calculated based on the theoretical skin effect formula.

[0014] Furthermore, the physical constraint loss term in S2 is constructed in the following way: for each training sample, the theoretical minimum AC resistance value is calculated based on the conductor resistivity, geometric dimensions, and operating frequency; the predicted AC resistance value corresponding to the active power loss value per unit length predicted by the model is calculated; when the predicted AC resistance value is lower than the theoretical minimum AC resistance value, the square of the difference between the two is included in the loss function as a penalty term.

[0015] Furthermore, the designable structural parameters of the cable described in S3 include the diameter of a single filament, the stranding pitch, and the thickness of the insulation layer; the preset safety constraints include: the maximum operating temperature constraint of the conductor, the minimum insulation resistance constraint, the minimum diameter of a single filament constraint, and the minimum stranding pitch constraint.

[0016] Furthermore, the iterative optimization described in S3 employs a constrained gradient descent algorithm, specifically including: constructing an augmented Lagrangian function containing the objective function and all safety constraint violation penalty terms; calculating the gradient of the augmented Lagrangian function with respect to the designable structure parameters through automatic differentiation; updating the designable structure parameters along the gradient descent direction, and updating them using the projection operator. Ensure that the updated parameters fall within a physically feasible range of values; projection operator Truncate each variable to the preset upper and lower bounds; repeat the above steps until the preset convergence condition is met.

[0017] Furthermore, S2 also includes a transfer learning fine-tuning step: pre-training the physical information neural network model using the multi-source heterogeneous training dataset; obtaining a small amount of measured power loss data for a specific batch of super copper conductors; and fine-tuning at least some network layer parameters of the pre-trained model using the measured power loss data as new label data.

[0018] Furthermore, in the inference stage of the physical information neural network model in S2, the Monte Carlo Dropout mechanism is enabled. By performing multiple random forward propagations on the same input feature vector, the mean and standard deviation of the predicted active power loss per unit length are obtained, which are used to evaluate the uncertainty of the design scheme.

[0019] Furthermore, it also includes S5: based on the cable optimization design scheme output by S3, a lightweight deep learning model is deployed to an edge computing device for online energy efficiency monitoring and dynamic evaluation during cable operation; the lightweight deep learning model is obtained by knowledge distillation of the physical information neural network model.

[0020] Furthermore, the online energy efficiency monitoring and dynamic evaluation described in S5 includes: real-time acquisition of current waveform and ambient temperature data during cable operation; calculation of equivalent thermal effect current; inputting the equivalent thermal effect current, ambient temperature, and cable fixing structure parameters into the lightweight deep learning model to obtain real-time power loss prediction values; and generating and outputting energy efficiency optimization strategy suggestions when the real-time power loss prediction values ​​continuously exceed a preset threshold.

[0021] Furthermore, in S3, when the optimization objective involves multiple parallel-laid super copper low-voltage cables, the input feature vector of the physical information neural network model also includes the current phase difference and relative position parameters of adjacent loops; the objective function is expanded to simultaneously minimize the power loss of this loop and the mutual inductance coupling effect between adjacent loops; the iterative optimization adopts a multi-objective optimization algorithm, and outputs a set of non-dominated solutions to form a Pareto front.

[0022] Furthermore, the physical information neural network model described in S2 is a field-loss collaborative prediction physical information neural network model; The output layer of the field-loss collaborative prediction physical information neural network model includes two branches: The first branch outputs the predicted active power loss per unit length. ; The second branch outputs the predicted axial component complex amplitude of the time-harmonic magnetic field at N preset sampling points on the cross-section of the conductor. ;in , The coordinates of the sampling point in the local coordinate system of the conductor cross-section; the loss function of the field-loss collaborative prediction physical information neural network model. Including data fitting loss term Physical constraint loss item The residual loss term of partial differential equations and boundary condition loss terms : ; in, , , , These are the weighting coefficients for each loss term; By predicting the complex amplitude of the axial component of the time-harmonic magnetic field Substituting the Helmholtz equation under time-harmonic magnetic field, the residuals of the sampling points in the domain are calculated to obtain the result. The complex amplitude of the axial component of the time-harmonic magnetic field is predicted by calculation. Whether the normal derivative at the conductor-insulator boundary satisfies the preset continuity condition is obtained.

[0023] Among them, the residual loss term of the partial differential equation The specific construction method is as follows: for a uniform, isotropic conductor region, the axial component of the time-harmonic magnetic field... Satisfies the Helmholtz equation: ; in, For the Laplace operator, , The imaginary unit, Angular frequency, Permeability, Effective conductivity; during training, in the conductor cross-sectional area Internal random sampling Points , Calculate the predicted value Laplace operator Then the residual loss term of the partial differential equation for: ; The effective conductivity The following parameters are dynamically calculated from the microstructure of the super copper conductor in the input feature vector: ; in, The basic electrical conductivity of the super copper matrix is ​​expressed in S / m. The conductivity enhancement factor for nanoprecipitated phases is dimensionless. The volume fraction of the nano-precipitated phase is dimensionless. The average grain diameter is expressed in meters (m). Dislocation density, in units of ; The grain boundary and dislocation scattering correction factor is calculated using the following formula: ; in, is the grain boundary scattering coefficient, in meters; This is the dislocation scattering coefficient, in units of .

[0024] In addition, the present invention also discloses a cable design system, the system comprising: The data acquisition and preprocessing module is used to acquire and construct the multi-source heterogeneous training dataset; The model building and training module is used to build, train, and fine-tune the physical information neural network model. The reverse design and optimization module is used to call the trained physical information neural network model to perform the reverse design of the cable structure; The results output and application module is used to output the cable optimization design scheme or deploy a lightweight model to achieve online monitoring.

[0025] Compared with the prior art, the present invention has the following beneficial effects: The intelligent optimization framework constructed by this invention, which deeply integrates physical mechanisms and data-driven approaches, brings a systematic and substantial technological improvement to solving the energy efficiency bottleneck problem of super copper low-voltage cables.

[0026] By constructing a physical information neural network model, this invention overcomes the shortcomings of pure data-driven models in scenarios with strong physical constraints. The skin effect theory constraint embedded in the model training transforms the physical law that the AC resistance is not lower than the lower limit determined by the basic electromagnetic principle into a calculable loss penalty term, forcing the surrogate model to comply with the basic laws of energy conservation and field distribution while fitting data. This makes the model prediction not only fast, but also possesses the physical reliability and extrapolation robustness lacking in traditional black-box models. Even under operating conditions not covered by the training data or under new material parameters, its prediction results will not produce physically impossible low loss values, ensuring the engineering reliability of subsequent optimization design.

[0027] Based on this differentiable and physically reliable surrogate model, this invention achieves efficient and directional reverse design of cable structure parameters. Utilizing automatic differentiation technology, the design optimization problem is transformed into a constrained gradient descent solution for adjustable parameters of a neural network. It abandons the traditional random search mode that relies on trial and error or time-consuming simulation, realizing a shift from discrete iteration of simulation and evaluation to continuous optimization of gradient and update. The optimization process can proceed along a clear direction of loss reduction within seconds, and conductor temperature rise, insulation strength, and safety constraints are rigidly embedded into the iterative process through projection operators, ensuring that each set of output design parameters simultaneously satisfies optimal electrical performance and engineering safety and reliability.

[0028] The aforementioned technical principles work together to form a complete technical closed loop from the perception of microscopic material properties to the optimization of macroscopic system energy efficiency. This not only generates customized high-efficiency cable structure solutions for specific super copper materials and operating scenarios during the design phase, releasing the potential of the materials, but also enables real-time energy efficiency monitoring and dynamic strategy adjustment of cables in operation through lightweight model deployment. This achieves continuous maximization of transmission efficiency throughout the cable's entire life cycle and provides a core tool for building efficient and intelligent new power distribution systems. Attached Figure Description

[0029] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation of the scope. For those skilled in the art, other related drawings can be obtained from these drawings without creative effort.

[0030] Figure 1 This is a schematic diagram of the overall process of the present invention; Figure 2 This is a schematic diagram of the system architecture of the present invention; Figure 3 This is a flowchart of the iterative optimization process of the present invention. Detailed Implementation

[0031] In the following description, only certain exemplary embodiments are briefly described. As those skilled in the art will recognize, the described embodiments can be modified in various ways without departing from the spirit or scope of the embodiments of the invention. Therefore, the drawings and description are considered to be exemplary in nature and not restrictive.

[0032] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0033] Example 1: See Figures 1-3 This embodiment discloses a deep learning optimization method for the transmission efficiency of super copper low-voltage cables. It constructs an end-to-end deep neural network collaborative optimization framework with embedded physical information to achieve high-fidelity modeling and reverse design guidance of the complex nonlinear mapping relationship between multidimensional input variables such as the microstructure of super copper conductor materials, cable geometry, operating conditions and environmental boundary conditions and active power loss per unit length.

[0034] In the data acquisition and preprocessing stage, a multi-source heterogeneous training dataset covering all dimensions of materials, structures, and working conditions was constructed. This dataset consists of four subsets, each requiring a strict standardized acquisition process. The first subset is the microstructure parameter dataset for super copper materials, which relies on high-resolution microscopic characterization techniques. Field emission scanning electron microscopy (FE-SEM) combined with an electron backscatter diffraction (EBSD) system was used to scan the conductor cross-section at an accelerating voltage of 20 kV and a step size of 0.2 μm to obtain grain orientation maps. The average grain diameter (unit: μm) and its standard deviation were calculated using OIM Analysis software. Transmission electron microscopy (TEM) was used to image ultrathin sections (approximately 80 nm thick) at an accelerating voltage of 200 kV. The number density and area ratio of nanoprecipitates were statistically analyzed using the ImageJ plugin, converted to volume fraction (%), and their average particle size (nm) was measured. Dislocation density was determined by X-ray diffraction (XRD) linewidth method using Cu-Kα radiation (λ= The scanning range is 40°–90° (2θ), with a step size of 0.02°. The integral half-width at half-maximum (FWHM) is calculated after instrument broadening correction using the Williamson-Hall method and then substituted into the following formula: ; in: Dislocation density, in units (square meters) - ¹); : Integral half-width at half-maximum (FWHM) of XRD diffraction peaks, in rad (radians). : Burgers vector modulus of copper, a constant, with a value of 0.2556 nm (nanometers); : Coefficients derived from the formula, dimensionless; The wavelength of Cu-Kα radiation; The residual stress tensor is determined by measuring the interplanar spacing changes in the three principal directions using the XRD sin²ψ method, and the principal stress components (MPa) are calculated. The above six parameters together constitute the basis of the material's eigenvector.

[0035] The second category is a dataset of cable geometric configuration parameters, covering macroscopic design variables of the conductor and insulation shielding system; the total conductor cross-sectional area (mm²) is classified and set according to the IEC 60228 standard, with typical values ​​including 10 mm², 16 mm², 25 mm², 35 mm², and 50 mm²; the single wire diameter (mm) is discretely sampled in increments of 0.05 mm within the range of 0.5 mm to 2.0 mm; the stranding pitch (mm) is defined as the distance a single wire travels axially in one revolution around the central axis, with a value range of 10 mm to 50 mm; the stranding angle (°) is calculated from the pitch p and the conductor outer diameter D. ; in: : Twist angle, unit ° (degrees); : Hinge pitch, unit mm (millimeters); : Outer diameter of conductor, in mm (millimeters); Pi (π), a constant, with a value of 3.1416, is dimensionless. The arctangent function outputs values ​​in radians, which are then converted to degrees. The insulation layer thickness (mm) is set according to the voltage level, and is usually 0.8 mm to 2.0 mm for low-voltage cables; the relative permittivity of the insulation material is measured at 1 kHz using an impedance analyzer; whether the shielding layer is continuous is represented by a binary variable (1 for continuous metal foil, 0 for braided shielding); the shielding layer thickness (mm) is obtained by actual measurement; these eight parameters fully describe the mechanical and electromagnetic geometric characteristics of the cable.

[0036] The third category is the operating condition and environmental variable dataset, reflecting the actual service conditions of the cable; the operating frequency range covers 0 Hz (DC) to 10 kHz, with a focus on sampling typical power frequencies and intermediate frequencies of 50 Hz, 400 Hz, 1 kHz, and 5 kHz; the effective value of the load current (A) is set according to the current carrying capacity standard based on the cross-sectional area, such as 100 A for 25 mm²; the harmonic content is based on the fundamental frequency, recording the percentage (%) of the effective value of the 3rd, 5th, and 7th harmonic currents relative to the fundamental frequency; the ambient temperature (°C) is set to five levels: -20°C, 0°C, 20°C, 40°C, and 60°C; the thermal resistance coefficient (K·m / W) of the laying method is determined by referring to a table according to IEC 60287-3-1, such as 1.2 for direct burial, 2.5 for conduit laying, and 0.8 for free laying in the air; these seven variables constitute the input of the dynamic operating scenario.

[0037] The fourth category is tag data, namely active power loss per unit length (W / m), which is generated through high-precision three-dimensional full-wave electromagnetic field finite element simulation. The simulation is executed on the COMSOL Multiphysics 6.0 platform, establishing a complete cross-sectional model including the conductor, insulation, shielding, and surrounding air domain. The axial length is set to a twist pitch to match the periodic boundary conditions. The excitation source is a specified frequency f and amplitude. sinusoidal current density Apply to the conductor region; solve the Maxwell equations in the frequency domain: ; in: : Vector differential operators (NaBLAO operators) are used to describe spatial gradient, divergence, and curl; they are dimensionless operators. Relative permeability, dimensionless, with an isotropic scalar value of 1.00001 for supercopper; Magnetic vector potential, a vector quantity, with units of Wb / m (Weber per meter). Imaginary unit, Dimensionless; Angular frequency, measured in rad / s (radians per second). ( (for operating frequency) Spatial position vector, unit m (meter); Excitation current density, vector, unit A / m² (amperes per square meter); Spatial location The electrical conductivity at a given point, in units of S / m (Siemens per meter); obtained by mapping local microstructure parameters: ; in: Local conductivity, in units of S / m (Siemens per meter); : The basic electrical conductivity of the super copper matrix, in S / m (Siemens per meter); : Conductivity enhancement factor of nanoprecipitated phase, dimensionless, typical value 1.0~1.2; Volume fraction of locally precipitated nanophase, dimensionless (must be converted to a decimal when inputting, e.g., 0.15 represents 15%). The base coefficient in the formula is dimensionless. Active power loss is calculated using volume fractional joule heat density: ; in: Active power loss per unit length, measured in W / m (watts per meter). Integration region (total volume of conductors), unit m³ (cubic meters); Electrical conductivity, in units of S / m (Siemens per meter); Electric field strength, a vector quantity, measured in volts per meter (V / m). : The magnitude of electric field strength, in V / m (volts per meter); Volume element, unit m³ (cubic meter); : The average coefficient of active power under sinusoidal steady state, dimensionless; electric field strength The integration region V covers the entire conductor; in: Electric field strength, a vector quantity, measured in volts per meter (V / m). Imaginary unit, Dimensionless; Angular frequency, measured in rad / s (radians per second). Magnetic vector potential, a vector quantity, with units of Wb / m (Weber per meter). : Negative sign, a symbol used in formula derivation, dimensionless; Mesh generation employs an adaptive refinement strategy, ensuring that the minimum cell size at the conductor surface and interface is less than 1 / 5 of the skin depth δ(f), thus guaranteeing analytical accuracy of the field distribution at high frequencies. Each set of input parameters generates a... A total of 12,800 training samples were constructed using tags.

[0038] Furthermore, a Physical Information Neural Network (PINN) model is constructed based on the aforementioned dataset; the input layer receives a 28-dimensional feature vector, which, in addition to the aforementioned 21 original parameters, also includes 7 dimensionless combined features: the Struhal number. ; in: Take 1 / 3 of the speed of light ( This can be used as an approximation of the speed of current propagation. is the Struhal number (dimensionless); Operating frequency (unit: Hz); The pitch is the twist pitch (unit: m). This is an approximate value for the speed of electric current propagation (unit: m / s); Reynolds electric numbers ; in: Reynolds number, dimensionless; Electrical conductivity, in units of S / m (Siemens per meter); Electric field strength, estimated by Ohm's law, unit V / m (volts per meter); : Diameter of a single filament, in meters (m). Equivalent viscosity of copper electron fluid, in Pa·s (Pascal-second), with a value of 1.2 × 10⁻⁶. -5 Pa·s; Skin ratio d / δ, where δ is derived from the formula calculate; in: Skin depth, measured in meters (m), and related to operating frequency. Related; : Conductor resistivity, unit Ω·m (ohm-meter). ( (conductivity); Angular frequency, measured in rad / s (radians per second). ; Vacuum permeability, a constant, with various values. (Henry per meter); Relative permeability, dimensionless, 1.00001 for super copper; : Coefficients derived from the formula, dimensionless; Square root operator, dimensionless; Total Harmonic Distortion ; in: Total harmonic distortion (THD), dimensionless; : No. The effective value of the second harmonic current, in amperes (A). ; : Fundamental current RMS value, unit A (ampere); : The sum of the squares of the effective values ​​of each harmonic current, in A² (amperes squared). Square root operator, dimensionless; Thermoelectric coupling factor ; in: is the thermoelectric coupling factor (dimensionless). The temperature coefficient of copper (values) ); Ambient temperature (unit: °C); Reference temperature (value) ); In addition, it also includes the ratio of twist pitch to diameter. Insulation thickness to conductor diameter ratio ; in: : Hinge pitch, unit mm (millimeters); : Single filament diameter, unit mm (millimeters); Ratio: dimensionless; Insulation layer thickness, in mm (millimeters); Conductor diameter, in mm (millimeters); These derived features enhance the model's ability to perceive physical mechanisms.

[0039] The neural network consists of a six-layer fully connected structure, with the number of neurons in each layer being 256, 512, 1024, 1024, 512, and 256, respectively; the activation function used is the Swish function. ; in: The output value of the Swish activation function is dimensionless. The input value to the activation function (the output of the neural network layer) is dimensionless. Sigmoid function Dimensionless (as opposed to conductivity) distinguish); : Learnable parameters of the Swish function, dimensionless, initially set to 1.0, and automatically adjusted during training through backpropagation; The input value of the Sigmoid function is dimensionless. The output layer consists of single-neuron linear units that directly output... (W / m); Loss function Loss due to data fitting With physical constraint loss Weighted composition: ; in: Total loss function value, dimensionless; : Weighting coefficients for data fitting loss, dimensionless, typical value 1.0; : Weighting coefficient for physical constraint loss, dimensionless, typical value 0.1; Physical constraint loss term, dimensionless; The data fitting loss term is dimensionless and represents the mean squared error between the predicted and labeled values. ; in: : Data fitting loss term, dimensionless; Total number of training samples, dimensionless; Sample index, dimensionless. ; : No. Predicted active power loss per unit length for each sample, in W / m (watts per meter). : No. The actual active power loss per unit length of each sample (label data), in W / m (watts per meter). : No. The squared prediction error of each sample, in units of (W / m)²; Average coefficient, dimensionless; Used to force the model to satisfy the lower limit constraint of AC resistance under the skin effect; For training samples with operating frequencies higher than a preset threshold (e.g., 50 Hz), their theoretical minimum AC resistance is first calculated. : ; in: Theoretical minimum AC resistance, in Ω / m (ohms per meter); DC resistance of a conductor, measured in Ω / m (ohms per meter). : Single wire diameter (equivalent conductor diameter), unit m (meter); Skin depth, in meters (m). : The fourth power of the ratio of monofilament diameter to skin depth, dimensionless; The enhancement factor of AC resistance relative to DC resistance, dimensionless; The base coefficient in the formula is dimensionless. The AC resistance predicted by the model based on the input is: ; in: Predicted AC resistance, in Ω / m (ohms per meter); : Predicted active power loss per unit length, in W / m (watts per meter). : Fundamental current RMS value, unit A (ampere); : The square of the effective value of the fundamental current, in A² (amperes squared); Physical constraint loss Then it is defined as a violation The penalty for this relationship: ; in: Physical constraint loss term, dimensionless; The number of training samples with a frequency higher than a preset threshold, dimensionless; Sample index, dimensionless. ; : No. The theoretical minimum AC resistance value of a sample, in Ω / m (ohms per meter). : No. Predicted AC resistance values ​​for each sample, in Ω / m (ohms per meter). Constraint violation quantity ( (Take 0 at this time), unit Ω / m (ohms per meter); : The square of the constraint violation, in units of (Ω / m)²; Average coefficient, dimensionless; The model was trained on an NVIDIA A100 GPU with a batch size of 256, using the Adam optimizer and an initial learning rate of 1×10⁻⁶. - ³, decaying by 0.5 every 50 rounds, training until the validation loss converges (usually within 300 rounds); to improve generalization ability with small samples, transfer learning is introduced: first, the base model is pre-trained on 12,800 sets of simulation data; when the measured data of a specific batch of materials is obtained (e.g., 50 sets), the previous data is frozen. During the construction phase, the reverse ladder (e.g., 0.8 W / m), material parameters (e.g., SC-02 batch, 105% IACS), operating conditions ( (And security constraints, treating the neural network as a differentiable surrogate model, optimizing the designable variables) Define an augmented Lagrangian function that includes the objective function and constraint penalty terms. : ; in: : The augmented Lagrange function value is dimensionless; A variable vector can be designed, with components representing the diameter of a single filament. (mm), hinge pitch (mm), insulation layer thickness (mm); The model predicts the active power loss per unit length (about (a function of W / m, unit W / m); : Preset target loss value, in W / m (watts per meter). Objective function term, unit (W / m)²; Total number of safety constraints, dimensionless; Constrained index, dimensionless. ; : No. The penalty coefficient for each safety constraint is dimensionless and is set according to the importance of the constraint. : No. The violation amount of each safety constraint, with the unit matching the constraint type (e.g., temperature constraint in °C, dimensional constraint in mm). This indicates a violation of the constraints; Nonnegation of constraint violation quantities ( (Take 0 at time), unit and Consistent; : The square term of the constraint violation, with units equal to the square of the constraint violation unit.

[0040] The function is calculated using automatic differentiation for the designable variables. gradient The update rule for variables is: ; in: : No. The designable variable vector after each iteration has components in mm (millimeters). Projection operator: projects variables onto the physically feasible region. , dimensionless operator; The physical feasible domain of the designable variables, i.e. the preset upper and lower bounds of each variable (e.g., monofilament diameter 0.5~2.0 mm). : No. The designable variable vector for each iteration has components in mm (millimeters). Number of iterations, dimensionless. ; : The learning rate for the optimization iteration is dimensionless and adjusted according to the convergence speed; : No. In the next iteration, the augmented Lagrange function is... The gradient, in units of (W / m)² / mm (gradient of the objective function term), is the superposition of the gradient of the penalty term (dimensionless); : Gradient step size term, in mm (millimeters); The optimization process is iterative until the preset convergence conditions are met, such as the difference between the predicted loss and the target loss being less than a certain tolerance and all safety constraints being met.

[0041] To quantify the model performance, comparative experiments were conducted; three types of super copper cables were prepared: SC-01 (102% IACS, Al2O3 doped 0.15 vol%), SC-02 (105% IACS, Al2O3 doped 0.15 vol%), and SC-02 (105% IACS, Al2O3 doped 0.15 vol%). Doped 0.20 vol%), SC-03 (108% IACS, Doping 0.25 vol%; five sets of samples with different stranding pitches (15, 20, 25, 30, 35 mm) were prepared for each type, with a conductor cross-sectional area of ​​25 mm², a single wire diameter of 0.8 mm, and an insulation thickness of 1.0 mm; AC resistance was measured at 50 Hz, 400 Hz, 1 kHz, and 5 kHz using the double bridge method: DC bias was provided by a Keysight 6681A (100 A), and AC components were superimposed by an Agilent 33500B (amplitude 10). Voltage sampling was performed using a Keithley 2182A nanovoltmeter, with synchronous triggering to ensure phase alignment; simultaneously, predictions were made using the IEC60287 standard formula and the PINN model of this invention. .

[0042] The following table shows a comparison between the measured and predicted AC resistance of the SC-02 batch of super copper cables at a frequency of 1 kHz: Table 1:

[0043] Mean absolute percentage error (MAPE) across the entire frequency band: 18.7% according to IEC 60287, 2.3% for the PINN of this invention; in reverse engineering tasks, for SC-02 at 1 kHz, 100 A... =0.8 W / m, the model outputs the optimal parameters: =0.85 mm, p=22.3 mm, =1.2 mm. Actual measurement. =0.792 W / m, a 14.6% reduction compared to the conventional design (p=30 mm).

[0044] Furthermore, to support online monitoring, a lightweight edge deployment model was developed. Through knowledge distillation, the six-layer teacher model was compressed into a three-layer student network (128-64-32 neurons), and the input features were simplified to 15 key variables. The distillation loss includes output matching and intermediate layer feature alignment terms. The compressed model has 48 KB of parameters and an inference latency of 4.2 ms (on an NVIDIA Jetson AGX Xavier), meeting real-time requirements. During online operation, the system collects current waveforms (after FFT harmonic decomposition), ambient temperature, and historical load data to calculate the equivalent thermal current. ; in: Equivalent thermal current, unit A (ampere); : Fundamental current RMS value, unit A (ampere); : The square of the effective value of the fundamental current, in A² (amperes squared); Harmonic order, dimensionless. ; : No. The thermal effect weighting coefficient of subharmonics is dimensionless and is set according to the characteristics of harmonic thermal effects. : No. The effective value of the subharmonic current, in amperes (A). : No. The square of the effective value of the second harmonic current, in A² (amperes squared). : No. The weighted thermal effect current square of the subharmonic, in A² (ampere square). :Weighted sum of squares of thermal effect currents from the 2nd to the 7th harmonics, in A² (amperes squared). Square root operator, dimensionless; If prediction If the load exceeds the threshold by 10% for 10 consecutive minutes, a suggestion will be issued to adjust the load phase balance or consider replacing the cable with a low strand pitch cable.

[0045] Furthermore, to quantify design uncertainty, Monte Carlo Dropout is enabled during the inference phase: 100 forward propagations are performed on the same input (Dropout rate 0.2), and the output... mean with standard deviation ;like + > If a design is deemed high-risk, it is marked as high-risk and triggers a re-optimization; for example, a candidate solution. =0.78 W / m, =0.02 W / m, then + =0.82 W / m > 0.8 W / m, the judgment is unreliable.

[0046] This invention deeply integrates materials science, electromagnetic theory, and deep learning to construct a complete technical closed loop from microscopic characterization to macroscopic design, and from forward prediction to reverse optimization. All implementation steps have clear engineering parameters, reproducible experimental conditions, and quantifiable performance indicators, ensuring that technical personnel in the relevant field can implement this solution without obstacles, thereby improving the transmission efficiency and energy efficiency of super copper low-voltage cables.

[0047] Example 2: Based on Example 1, this example addresses the technical shortcomings of insufficient physical reliability and limited extrapolation capability of the pure data-driven model in the problem of optimizing the transmission efficiency of super copper low-voltage cables. It proposes a deep physical embedding method based on a field-loss collaborative prediction physical information neural network model.

[0048] This embodiment directly embeds the core governing equations of Maxwell's equations—the Helmholtz equations—and boundary conditions into the neural network loss function, forcing the neural network to not only learn the input-output mapping relationship during training but also learn and obey the inherent physical laws of electromagnetic field distribution, thereby obtaining a physically self-consistent and extrapolation-robust field-loss co-prediction model.

[0049] 2.1 Construction of a neural network model for predicting physical information in conjunction with field-loss: The field-loss collaborative prediction physical information neural network model adopts a dual-branch output architecture. The input layer receives feature vectors. The input feature vector of the physical information neural network model in Example 1 has the same dimension as that in Example 1, which is 28-dimensional. It includes the microstructure parameters of the super copper conductor, the cable geometric configuration parameters, the operating conditions and environmental variables, and the calculated dimensionless combination features.

[0050] The main structure of the hidden layer of the model is consistent with that of Example 1, employing a six-layer fully connected network with the number of neurons in each layer being 256, 512, 1024, 1024, 512, and 256 respectively. The activation function used is the Swish function. After the last hidden layer (256 dimensions), the network splits into two independent branches: Branch 1 (Loss Prediction Branch): This branch has the same structure and function as the physical information neural network model in Example 1. It outputs the predicted active power loss per unit length through a 256-dimensional fully connected layer and a linear output layer. The unit is W / m.

[0051] Branch Two (Magnetic Field Prediction Branch): This branch is used to predict the complex amplitude distribution of the axial components of the time-harmonic magnetic field on the conductor's cross-section. First, the output of the last hidden layer (256 dimensions) is concatenated with the output of a coordinate encoding layer. The coordinate encoding layer receives the spatial coordinates of the conductor's cross-section. (Coordinates have been normalized to) This is then mapped to a higher-dimensional space to enhance the network's ability to fit high-frequency spatial variations. Encoding function Defined as: ; in, Normalized coordinate vector; The encoding matrix is ​​randomly initialized, with elements having a mean of 0 and a variance of 0. Sampling from a Gaussian distribution, For the encoding dimension, this embodiment takes , ; and For element-wise cosine and sine functions. Encoded coordinate features. Dimensions dimension.

[0052] The 256-dimensional hidden layer semantic features are concatenated with the 256-dimensional encoded coordinate features to obtain a 512-dimensional combined feature vector. This vector is then passed sequentially through three fully connected layers (dimensions of 256, 128, and 64 respectively), with the final output layer being a linear layer with an output dimension of 2, corresponding to the coordinate points. The real part of the complex amplitude of the axial component of the predicted time-harmonic magnetic field. and the virtual part The unit is A / m. The complex amplitude predicted by the network. for ,in It is the imaginary unit.

[0053] During training and inference, the magnetic field prediction branch requires a fixed set of coordinate points for each training sample. , The number of sampling points for each sample. In this embodiment, the radius is the conductor radius. Within the circular conductor region, the Hammersley sequence sampling method is used to generate... Each point ensures that the sampling is spatially uniform and has low variability.

[0054] 2.2 Construction of the deep physical embedding loss function: Overall loss function of the field-loss co-prediction physical information neural network model It consists of four weighted components, as shown in claim 10. The construction details of each loss term are as follows: 2.2.1 Data Fitting Loss Data fitting loss The definition is the same as in Example 1, measuring the branch-one prediction. Tag data generated by finite element simulation The mean square error between them. Weighting coefficients. The value is 1.0.

[0055] 2.2.2 Physical constraint loss Physical constraint loss The definition is the same as in Example 1, where the AC resistance value used to force the model prediction is not lower than the theoretical minimum calculated based on the skin effect theory. Weighting coefficient The value is 0.1.

[0056] 2.2.3 Residual Loss of Partial Differential Equations Residual loss of partial differential equations Magnetic field distribution used to predict the output of the forced magnetic field branch It satisfies the Helmholtz equations describing the propagation of time-harmonic electromagnetic fields in a conductor.

[0057] For each training sample, firstly, based on the working frequency in its input feature vector... Basic conductivity Volume fraction of nano-precipitated phase Average grain diameter and dislocation density Calculate the effective conductivity according to the formula in claim 11. The material constants are: (dimensionless) , Vacuum permeability Super copper relative permeability Therefore, the total permeability .

[0058] Calculate angular frequency and the square of complex wavenumber .

[0059] In the conductor region (Circle, radius) Within this range, besides those used for magnetic field prediction... In addition to the fixed sampling points, additional random sampling Points For each sampling point, the predicted magnetic field is calculated using automatic differentiation techniques. Spatial coordinates and The second-order partial derivatives yield the Laplace operator. .

[0060] but Calculated according to the formula in claim 11. Weighting coefficients. The value is 0.05.

[0061] 2.2.4 Boundary Condition Loss Boundary condition loss This is used to force the magnetic field distribution at the boundary between the conductor and the insulation layer to meet specific continuity conditions. For the low-voltage cable studied in this embodiment, the conductivity of the insulation layer (such as XLPE) is approximately 0. Ideally, the conductor surface ( The normal derivative of the axial component of the harmonic magnetic field at point () should approach 0.

[0062] At the conductor-insulator boundary Uniform sampling Points For each boundary point, the predicted magnetic field is calculated. Along the outer normal direction of the boundary directional derivative The directional derivative is also calculated using automatic differentiation. The boundary condition loss is defined as: ; Weighting coefficient The value is 0.01.

[0063] 2.3 Model Training and Evaluation: The training platform, optimizer, learning rate strategy, and batch size of the field-loss co-prediction physical information neural network model are the same as in Example 1. The total number of training rounds is 500, with each round iterating through the entire training dataset (12,800 samples). Due to the increased... and The loss term, the forward propagation of each training sample needs to be in Calculating the magnetic field prediction and its derivative at each spatial point significantly increases the computational load, but by using GPU parallel computing, the training time for a single round is controlled to about 45 minutes (using NVIDIA A100 GPU).

[0064] To verify the effectiveness of deep physical embeddings, an extrapolation generalization capability test was designed. A challenging combination of parameters outside the training data distribution was selected as the test sample: Super copper material: New model SC-04, its microstructure parameters are: average grain diameter Standard deviation of grain size distribution Nano-precipitated phase ( Volume fraction average particle size dislocation density Residual stress Actual measurements show that its DC conductivity is... (Approximately 110% IACS).

[0065] Cable geometry: Total cross-sectional area of ​​conductors monofilament diameter (Exceeding the maximum training set by 2.0 mm), hinge pitch Twist angle Insulation layer thickness .

[0066] Operating conditions: operating frequency (Far exceeding the training set's maximum of 5 kHz), fundamental current RMS value No harmonics, ambient temperature .

[0067] High-precision three-dimensional finite element method (COMSOL Multiphysics) was used, with extremely fine mesh and solver tolerance. Simulations were performed on this configuration to obtain the baseline active power loss per unit length. .

[0068] Using the original physical information neural network model from Example 1 (only) and The test sample is predicted using the field-loss co-prediction physical information neural network model of this embodiment. Simultaneously, to evaluate the physical correctness of the magnetic field prediction, the magnetic field distribution at the same coordinate points on the conductor cross-section is extracted from the finite element simulation results. And calculate the magnetic field predicted by the model. Distribution similarity (using normalized cross-correlation coefficient) value range (The closer to 1, the more similar they are).

[0069] The test results are compared in Table 2 below: Table 2:

[0070] The results are analyzed as follows: Extrapolation prediction accuracy: Under extreme parameters that are completely outside the range of the training data, the prediction in Example 1 showed severe physical distortion, with the predicted loss value (1.147 W / m) being far lower than the finite element benchmark (1.815 W / m), and the error reaching -36.8%. In contrast, the model prediction value (1.776 W / m) in this example is in high agreement with the benchmark, with an error of only -2.15%, demonstrating excellent extrapolation generalization ability.

[0071] The magnetic field distribution predicted by the model in this embodiment has a similarity of 0.937 with the high-precision finite element simulation results, proving that the mapping relationship learned within the model is highly consistent with the basic laws of electromagnetic fields.

[0072] This embodiment addresses the core technical challenge of shallow and simplistic physical constraints in traditional physical information neural networks, which leads to a sharp decline in model reliability outside the training data coverage. By deeply embedding the governing equations (PDEs) and boundary conditions (BCs) into the loss function, the model is forced to obey physical laws in the function space, not just the output scalar. This significantly improves the model's predictive reliability for unverified operating conditions such as novel super copper materials, higher frequencies, and larger sizes; enhances the reliability of optimization design results, avoiding invalid or dangerous design schemes due to physical distortion of the model; and provides additional physical field outputs (magnetic field distribution) besides losses, enabling multi-physics collaborative design for cables, including electromagnetic compatibility and thermal analysis.

[0073] The field-loss co-prediction physical information neural network model constructed in this embodiment can directly replace the physical information neural network model in step S2 of embodiment 1 and be applied to the cable structure reverse design in S3. It is suitable for innovative cable design for future new materials or extreme application scenarios.

[0074] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0075] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. It should be noted that any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A deep learning optimization method for the transmission efficiency of super copper low-voltage cables, characterized in that, Includes the following steps: S1: Construct a multi-source heterogeneous training dataset; the multi-source heterogeneous training dataset includes a super copper conductor microstructure parameter dataset, a cable geometric configuration parameter dataset, an operating condition and environmental variable dataset, and a unit length active power loss label dataset corresponding to the combination of the first three types of data. S2: Construct a physical information neural network model; the physical information neural network model takes the feature vector obtained by fusing the microstructure parameter dataset of the super copper conductor, the cable geometric configuration parameter dataset, and the operating condition and environmental variable dataset as input, and outputs the predicted active power loss value per unit length; the loss function of the physical information neural network model includes a data fitting loss term and a physical constraint loss term. The physical constraint loss term is constructed based on the principles of electromagnetic physics and is used to force the AC resistance value predicted by the model to be no less than the theoretical minimum AC resistance value calculated based on the skin effect theory during the model training process; S3: Perform reverse design of cable structure based on the physical information neural network model; specifically: use the physical information neural network model as a differentiable surrogate model, take minimizing the predicted active power loss per unit length as the objective function, adjust the designable structural parameters of the cable, and perform iterative optimization under the condition of meeting the preset safety constraints until the preset energy efficiency target is achieved; S4: Optimized design scheme for output cable; The cable optimization design scheme includes a combination of cable structure parameters obtained by S3 optimization.

2. The deep learning optimization method for the transmission efficiency of a super copper low-voltage cable according to claim 1, characterized in that, The microstructure parameter dataset of the super copper conductor described in S1 includes: average grain diameter, standard deviation of grain size distribution, volume fraction of nano-precipitates, average grain size of nano-precipitates, dislocation density, and principal value of residual stress; the cable geometric configuration parameter dataset includes: total conductor cross-sectional area, single wire diameter, stranding pitch, stranding angle, insulation layer thickness, relative permittivity of insulation material, shielding layer structure type, and shielding layer thickness; the operating condition and environmental variable dataset includes: operating frequency, effective value of fundamental current, content of specific harmonic current, ambient temperature, and thermal resistance coefficient of laying method.

3. The deep learning optimization method for the transmission efficiency of a super copper low-voltage cable according to claim 2, characterized in that, The feature vector of the physical information neural network model input in S2 also includes multiple dimensionless combination features calculated based on the microstructure parameters, geometric configuration parameters, and operating conditions and environmental variables.

4. The deep learning optimization method for the transmission efficiency of a super copper low-voltage cable according to claim 1, characterized in that, The physical information neural network model described in S2 adopts a multi-layer fully connected network structure; the physical constraint loss term of the loss function is used to force the model to predict AC resistance values ​​for samples with frequencies higher than a preset threshold that are not lower than the minimum theoretical AC resistance value calculated based on the theoretical skin effect formula.

5. The deep learning optimization method for the transmission efficiency of a super copper low-voltage cable according to claim 4, characterized in that, The physical constraint loss term described in S2 is constructed as follows: for each training sample, the theoretical minimum AC resistance value is calculated based on the conductor resistivity, geometric dimensions, and operating frequency; the predicted AC resistance value corresponding to the active power loss per unit length predicted by the model is calculated; when the predicted AC resistance value is lower than the theoretical minimum AC resistance value, the square of the difference between the two is included as a penalty term in the loss function.

6. The deep learning optimization method for the transmission efficiency of a super copper low-voltage cable according to claim 1, characterized in that, The designable structural parameters of the cable described in S3 include single wire diameter, strand pitch, and insulation layer thickness; The preset safety constraints include: maximum operating temperature of the conductor, minimum insulation resistance, minimum diameter of a single filament, and minimum stranding pitch.

7. The deep learning optimization method for the transmission efficiency of a super copper low-voltage cable according to claim 6, characterized in that, The iterative optimization described in S3 employs a constrained gradient descent algorithm, specifically including: constructing an augmented Lagrangian function containing the objective function and all safety constraint violation penalty terms; calculating the gradient of the augmented Lagrangian function with respect to the designable structure parameters through automatic differentiation; updating the designable structure parameters along the gradient descent direction, and ensuring that the updated parameters fall within a physically feasible range of values ​​through a projection operator; repeating the above steps until the preset convergence condition is met.

8. The deep learning optimization method for the transmission efficiency of a super copper low-voltage cable according to claim 1, characterized in that, S2 also includes a transfer learning fine-tuning step: pre-training the physical information neural network model using the multi-source heterogeneous training dataset; obtaining a small amount of measured power loss data for a specific batch of super copper conductors; and fine-tuning at least some network layer parameters of the pre-trained model using the measured power loss data as new label data.

9. The deep learning optimization method for the transmission efficiency of a super copper low-voltage cable according to claim 1, characterized in that, In S2, during the inference phase of the physical information neural network model, the Monte Carlo Dropout mechanism is enabled. By performing multiple random forward propagations on the same input feature vector, the mean and standard deviation of the predicted active power loss per unit length are obtained, which are used to evaluate the uncertainty of the design scheme.

10. A deep learning optimization method for the transmission efficiency of a super copper low-voltage cable according to any one of claims 1-9, characterized in that: The physical information neural network model described in S2 is a field-loss collaborative prediction physical information neural network model; The output layer of the field-loss collaborative prediction physical information neural network model includes two branches: The first branch outputs the predicted active power loss per unit length. ; The second branch outputs the predicted axial component complex amplitude of the time-harmonic magnetic field at N preset sampling points on the cross-section of the conductor. ;in , The coordinates of the sampling point in the local coordinate system of the conductor cross-section are given.

Images