Pulse power coil temperature rise limit evaluation method based on sparse simulation and order reduction extrapolation

Abstract

The present application relates to a pulse power coil temperature rise limit evaluation method based on sparse simulation and order reduction extrapolation, belonging to the technical field of electrical equipment thermal management and performance evaluation. The method includes five steps of establishing a high-fidelity physical model and defining the working condition parameter space, sparse sampling and short-term simulation of the working condition space, constructing and calibrating the equivalent thermal circuit order reduction model, order reduction model verification and precision evaluation, and temperature rise limit extrapolation prediction. The present application solves the technical problems of high cost and long cycle of physical experiments, low calculation efficiency of full-size simulation, and difficulty in supporting design optimization in the existing pulse power coil temperature rise limit evaluation method. The present application converts the original full-size long-time FEM simulation which takes several days or even several weeks into several short-time FEM simulations and order reduction model calculations completed instantaneously, and the total evaluation efficiency is improved by 2-3 orders of magnitude.

Description

Technical Field

[0001] This invention belongs to the field of electrical equipment thermal management and performance evaluation technology, specifically involving a method for evaluating the temperature rise limit of pulse power coils based on sparse simulation and reduced-order extrapolation. Background Technology

[0002] Pulsed power coil systems are indispensable key components in fields such as shipbuilding, precision instrument manufacturing, and medical equipment. They are used to generate large-amplitude pulsed magnetic fields to meet specific requirements, necessitating the input of specific high-current pulses into the coil. These pulse sequences typically feature long periods, variable duty cycles, and large variations in current amplitude. When current flows through the coil, Joule heating is generated due to the resistance effect, causing the coil temperature to rise. If the temperature rise exceeds the tolerance limit of the insulation material, it will lead to aging and damage of the insulation layer, and may even cause serious faults such as short circuits and burnout, directly threatening equipment safety and operational reliability.

[0003] Therefore, accurately assessing the long-term temperature rise limit of the pulse power coil under the most demanding operating conditions is a crucial task during the design and verification phase.

[0004] Currently, there are two main methods for assessing the temperature rise limit: (a) Physical prototype experiment method By manufacturing a 1:1 physical prototype of the actual system and applying a designed pulsed current, the temperature change is directly measured using equipment such as thermocouples and infrared thermal imagers. The advantage of this method is that the results are accurate and reliable. However, its disadvantages are also extremely prominent: (1) High cost: The cost of manufacturing large pulse power coil prototypes, testing sites and dedicated high-power pulse power supplies is enormous.

[0005] (2) Long cycle: From design and manufacturing to testing, the entire process may take several months, which seriously slows down the R&D progress.

[0006] (3) Poor flexibility: Once the design parameters (such as the number of coil turns, wire material, cooling method) change, the prototype needs to be remanufactured, and it is not possible to quickly find the best solution from multiple options.

[0007] (ii) Full-size high-fidelity simulation method A refined simulation model of the electro-thermal-fluid multiphysics coupling is established using finite element method (FEM) software (such as ANSYS, COMSOL, etc.). This method can simulate the detailed temperature distribution inside the coil with high accuracy. However, for long-duration, complex pulse conditions such as pulsed power coils, this method has a fatal flaw: (1) Extremely low computational efficiency: A complete simulation of a long-cycle (e.g., several hours) temperature rise process requires extremely dense transient time step solutions. On a high-performance computing workstation, a single simulation may take several days or even weeks.

[0008] (2) Huge resource consumption: Fine mesh generation and multiphysics coupling require massive amounts of computing memory and storage space.

[0009] (3) Difficult to support design exploration: Due to the excessive time required for a single simulation, it is impossible to quickly iterate and evaluate multiple pulse conditions or combinations of design parameters, which greatly limits the optimization space of the design.

[0010] In summary, existing technologies for evaluating the temperature rise limit of pulse power coils are either hampered by high costs and long cycles, or limited by unacceptable computational efficiency. There is an urgent need for a new evaluation method that can balance high efficiency, high accuracy, and low cost. Overcoming the shortcomings of existing technologies is a problem that urgently needs to be solved in this field. Summary of the Invention

[0011] The purpose of this invention is to address the shortcomings of existing technologies and provide a method for evaluating the temperature rise limit of pulse power coils based on sparse simulation and reduced-order extrapolation.

[0012] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A method for evaluating the temperature rise limit of pulse power coils based on sparse simulation and reduced-order extrapolation includes the following steps: Step 1: Establish a high-fidelity physical model and define the operating parameter space: A three-dimensional electro-thermal coupled finite element model was established using finite element analysis software. This model includes the geometry, material properties, and boundary conditions of the pulsed power coil. At the same time, a working condition parameter space is defined, and each combination of parameter values ​​in the space corresponds to a pulse current working condition. Step 2, Sparse Sampling and Short-Term Simulation of the Working Condition Space: The operating condition parameter space is divided into three subsets: calibration area, verification area, and exploration area. Short-term simulations are performed using a three-dimensional electro-thermal coupled finite element model only for the operating conditions in the calibration area and verification area to obtain the temperature change curves of key temperature measurement points of the coil over time under these operating conditions, thus obtaining the simulation data of the calibration area and the simulation data of the verification area. Step 3: Construct and calibrate the equivalent thermal path order reduction model: To address the macroscopic thermal dynamics of a pulse power coil system, an equivalent thermal path reduction model is constructed; the reduction model is a first-order or second-order equivalent thermal path model. Using the simulation data of the calibration area obtained in step two, the unknown parameters in the reduced-order model are identified and determined through a curve fitting optimization algorithm, and the calibrated reduced-order model is obtained. Step 4: Validation and accuracy evaluation of the reduced-order model: Using the calibrated reduced-order model obtained in step three, predict the temperature response curve under the operating conditions in the verification area; compare it with the simulation data of the verification area obtained in step two, and calculate the relative error δ between the two at the test point: Set an acceptable error threshold δ limit ; If δ≤δ limit If so, the calibrated reduced-order model is considered to have passed the validation. If δ>δ limit If the result is negative, return to step three and consider using a higher-order reduced-order model or re-dividing the calibration and validation areas. Repeat steps two through four until the calibrated reduced-order model passes validation. Step 5, extrapolation prediction of temperature rise limit: For the validated and calibrated reduced-order model, the temperature rise limit is evaluated under the operating conditions of the exploration zone.

[0013] Furthermore, preferably, in step one, the material properties include electrical conductivity, thermal conductivity, density, and specific heat capacity; the boundary conditions include ambient temperature and convective heat transfer coefficient.

[0014] Furthermore, preferably, in step one, the operating parameter space is based on the pulse current amplitude. Single pulse conduction time Pulse period and continuous running time As operating condition parameters, construct the operating condition parameter space.

[0015] Furthermore, preferably, in step two, the three subsets are specifically: Subset 1 is the calibration area, where 1-2 basic pulse conditions are selected; Subset 2 is the verification area, in which 1 to 2 verification pulse conditions are selected. The verification pulse conditions have the same number of pulse sequence segments, current amplitude and duty cycle are on the same order of magnitude, and thermal time constant is on the same order of magnitude as the calibration area conditions. However, the specific current amplitude, duty cycle, and thermal time constant of the verification pulse conditions are different from those of the calibration area conditions. Subset 3 is the exploration zone, which includes all pulse conditions that require extreme evaluation; In step two, the short-term simulation duration is 10 to 20 minutes.

[0016] Furthermore, preferably, in step three, the reduced-order model is a first-order equivalent thermal circuit model, whose temperature response is described by the following equation: (1) In the formula, the thermal time constant τ and the equivalent average thermal power P eff They are determined by the following formulas respectively: (2) (3) In the formula, T(t) is the pulse power coil temperature predicted by the reduced-order model, in Kelvin; T amb Ambient temperature, in Kelvin. P eff The equivalent average thermal power of the pulse current over one cycle is expressed in watts. R th The equivalent thermal resistance of the pulse power coil system is expressed in Kelvin per watt. τ is the thermal time constant of the pulse power coil system, in seconds; C th This is the equivalent heat capacity of the pulse power coil system, measured in joules per Kelvin. R coil This is the DC resistance of the coil, measured in ohms. D is the duty cycle, which is dimensionless; t represents time, in seconds; Identify and determine the unknown parameter R in the reduced-order model. th and C th .

[0017] Furthermore, preferably, in step three, the curve fitting optimization algorithm is a nonlinear least squares method.

[0018] Furthermore, preferably, in step four, the assessment point is the simulation endpoint; (4) In the formula, δ represents the relative error, expressed as a percentage. T ROM The temperature predicted at the test point by the calibrated reduced-order model, in Kelvin; T FEM The temperature simulated at the same test point for the three-dimensional electro-thermal coupled finite element model is expressed in Kelvin. δ limit It is 5%.

[0019] Furthermore, preferably, in step five, for each pulse condition within the exploration zone, the theoretical saturation temperature of the pulse power coil system is calculated according to the following formula based on the equivalent average thermal power of that pulse condition: (7) In the formula, This is the theoretical saturation temperature of the pulse power coil system during long-term operation under the corresponding pulse condition, serving as the temperature rise limit under that pulse condition, and is expressed in Kelvin. This refers to ambient temperature, expressed in Kelvin. The equivalent average thermal power of the pulse current over one cycle is expressed in watts. The equivalent thermal resistance of the calibrated pulse power coil, in Kelvin per watt; The theoretical saturation temperature corresponding to each pulse condition in the exploration area is calculated using Equation (7), and the theoretical saturation temperature is used as the temperature rise limit under the corresponding pulse condition to complete the temperature rise limit assessment.

[0020] The evaluation object of this method is a pulse power coil system, which consists of a pulse power coil body, a cooling structure and its working environment. The "system" in the "system equivalent thermal resistance", "system equivalent thermal capacity" and "system macroscopic thermal dynamics" of this invention refers to the pulse power coil system.

[0021] In this invention, the operating condition parameter space is defined as follows: based on current amplitude I and pulse width... Using the duty cycle D as the coordinate system, a multi-dimensional parameter space is constructed. Each combination of parameter values ​​in this space corresponds to a specific pulse current operating condition. The range of values ​​for each parameter is determined by the actual operating requirements of the pulse power coil.

[0022] In this invention, the operating condition parameter space is divided into three subsets: Subset 1, Calibration Zone: Select 1 to 2 basic pulse conditions. The basic pulse conditions refer to conditions where the parameter values ​​are located in the middle range of the condition parameter space, the temperature curve has deviated from the initial linear rise segment within the short time-term simulation duration, and the conditions can reflect the interaction process of Joule heat generation and convective heat dissipation.

[0023] Subset 2, Validation Zone: Select 1 to 2 validation pulse conditions. The validation pulse conditions have the same number of pulse sequence segments, current amplitude and duty cycle on the same order of magnitude, and thermal time constant as the calibration conditions, but the specific parameter values ​​are different. This is used to test the predictive ability of the reduced-order model within a similar parameter range.

[0024] Subset 3, Exploration Zone: Contains pulsed operating conditions that require temperature rise limit assessment, wherein the pulsed operating conditions include at least one of long-duration operating conditions and high-power operating conditions; The long-duration operating condition refers to continuous operating time. Not less than Double heating time constant The pulse condition, among which, The preset values ​​are 3 to 5; in the first-order reduced-order model, the running time reaches respectively and At that time, the system temperature rise reached approximately 95.0% and 99.3% of the theoretical saturation temperature rise, respectively, therefore the continuous operating time should not be less than The operating condition is considered as a long-term operating condition that has reached a state of near thermal saturation; The high-power operating condition refers to the equivalent average thermal power. Not less than the preset equivalent average heat power threshold Under pulsed operating conditions, the preset equivalent average thermal power threshold The power range is predetermined based on the rated power of the pulse power coil system, the allowable temperature, or the power range of the parameter space to be evaluated.

[0025] For the operating conditions in the calibration and verification areas only, a three-dimensional electro-thermal coupled finite element model (FEM) was used to perform short-term simulations (e.g., 10 to 20 minutes) to obtain the temperature change curves of key temperature measurement points of the pulse power coil under these operating conditions, thus obtaining simulation data for the calibration area and simulation data for the verification area.

[0026] This invention addresses the macroscopic thermal dynamics of pulsed power coil systems by constructing a reduced-order equivalent thermal path model (ROM). This reduced-order model can be a first-order or second-order equivalent thermal path model. Taking the first-order model as an example, its temperature response is described by the following equation: (1) In the formula, the thermal time constant τ and the equivalent average thermal power P eff They are determined by the following formulas respectively: (2) (3) In the formula, T(t) is the pulse power coil temperature predicted by the reduced-order model, and the unit is Kelvin (K). T amb The ambient temperature is expressed in Kelvin (K). P eff The equivalent average thermal power of the pulse current over one cycle is expressed in watts (W). R th The equivalent thermal resistance of the pulse power coil system is expressed in Kelvin per watt (K / W), which characterizes the temperature rise caused by the dissipation of a unit power of heat in the system. τ is the thermal time constant of the pulse power coil system, in seconds (s). C thThe equivalent heat capacity of a pulse power coil system is expressed in joules per Kelvin (J / K), which characterizes the system's ability to store heat. R coil This is the DC resistance of the coil, measured in ohms (Ω). D is the duty cycle, which is dimensionless.

[0027] Using the simulation data of the calibration area obtained in step two, the unknown parameter R in the reduced-order model is identified and determined through curve fitting optimization algorithms (such as nonlinear least squares method). th and C th .

[0028] In step four of this invention, the reduced-order model calibrated in step three (i.e., having a certain R) is used. th and C th (Value reduction model), predict the temperature response curve T under the operating conditions of the verification area. ROM Compare it with the simulation data T of the verification area obtained in step two. FEM By comparison, the relative error δ between the two at the assessment point (such as the simulation endpoint) is calculated: (4) In the formula, δ represents the relative error, expressed as a percentage (%); T ROM The temperature predicted at the test point by the calibrated reduced-order model, in Kelvin (K); T FEM Temperature at the same test point for the simulation of a three-dimensional electro-thermal coupled finite element model (FEM), in Kelvin (K).

[0029] Set error threshold δ limit (For example, 5%). If δ≤δ limit If the calibrated reduced-order model passes verification; if δ>δ limit If the result is negative, return to step three, use a higher-order (e.g., second-order) reduced-order model or re-divide the calibration and verification areas, and repeat the calibration and verification process until δ≤δ limit .

[0030] This invention uses a validated, calibrated, reduced-order model to assess the temperature rise limit under long-term and high-power operating conditions within the exploration region. For the calibrated first-order reduced-order model, with ambient temperature, equivalent average heat power, and heat dissipation boundary conditions remaining constant, its temperature response is: (5) in, The pulse power coil predicted by the reduced-order model at time... Temperature; This is the time calculated from the start of operation under the corresponding working condition; The ambient temperature; As the running time t approaches infinity, the exponential term approaches zero, therefore: (6) The theoretical saturation temperature (i.e., temperature rise limit) T of a pulse power coil system sat Calculated by the following formula: (7) In the formula, This is the theoretical saturation temperature of the pulse power coil system during long-term operation under the corresponding pulse condition, serving as the temperature rise limit under that pulse condition, and is expressed in Kelvin. This refers to ambient temperature, expressed in Kelvin. The equivalent average thermal power of the pulse current over one cycle is expressed in watts. The equivalent thermal resistance of the calibrated pulse power coil, in Kelvin per watt; The theoretical saturation temperature corresponding to each pulse condition in the exploration area is calculated using Equation (7), and the theoretical saturation temperature is used as the temperature rise limit under the corresponding pulse condition to complete the temperature rise limit assessment.

[0031] In this invention, the key temperature measuring point of the coil is a commonly existing temperature measuring point, and this invention does not impose any special restrictions on it, such as a place where cables are densely packed.

[0032] Those skilled in the art should know that all temperature units involved in this invention, Kelvin, can also be converted to degrees Celsius.

[0033] This invention provides a method for efficiently and accurately assessing the temperature rise limit of coil systems operating under complex pulse conditions. It aims to solve the technical problems of high physical experiment costs, long cycles, low efficiency of full-scale simulation calculations, and difficulty in supporting design optimization in existing pulse power coil temperature rise limit assessment methods.

[0034] The flowchart of the pulse power coil temperature rise limit assessment method based on sparse simulation and reduced-order extrapolation of this invention is as follows: Figure 1 As shown. Figure 1 It clearly demonstrates a scientific process of "extracting patterns, verifying patterns, and applying patterns" from complex problems. Its innovation lies not in a single algorithm, but in the organic integration of multiple techniques such as high-fidelity simulation, sparse sampling, reduced-order modeling, and closed-loop verification into a completely new, efficient, and reliable evaluation workflow. For example... Figure 1As shown, firstly, a three-dimensional electro-thermal coupled finite element model (FEM) is used as the "precision anchor point." A "sparse sampling" strategy, which performs short-term simulations on only a few representative operating conditions in the "calibration zone and verification zone," is employed to obtain key data with extremely high efficiency. Next, this data is used to calibrate a "high-speed engine"—an equivalent thermal path reduced-order model (ROM)—that describes the macroscopic thermal characteristics of the system. Finally, this reduced-order model must undergo an "accuracy guarantee" process with independent data verification. Only after confirming that its error is within the allowable range can it be used for instantaneous extrapolation prediction of the temperature rise limit under arbitrary long-term and harsh operating conditions in the "exploration zone." This constitutes the complete technical pillar of "high-precision anchoring, low-cost sampling, high-efficiency modeling, and closed-loop verification." The core idea of ​​this invention can be summarized as follows: using a small amount of short-term "high-precision anchor point data" to calibrate an "ultra-high-speed simplified model," and then using this simplified model to instantaneously solve arbitrarily long-term and complex "limit problems." This is a typical hybrid modeling strategy that achieves significant results with minimal investment and replaces slow processes with fast ones. It cleverly avoids the two extremes of traditional methods: it avoids the cost and time pitfalls of "full-physical prototypes," eliminating the need to manufacture expensive physical prototypes for lengthy experiments; and it avoids the efficiency and resource pitfalls of "full-scale simulations," eliminating the need for brute-force solutions that take days or even weeks to complete for every demanding operating condition. The essence of this approach is a shift from "process reproduction" to "law capture." Traditional methods attempt to completely reproduce the entire heating process that takes several hours, while this invention argues that as long as the inherent thermodynamic laws of the system (such as equivalent thermal resistance and heat capacity) can be accurately captured, it is unnecessary to concern oneself with every intermediate detail, and the final steady state can be directly derived.

[0035] Compared with the prior art, the beneficial effects of this invention are as follows: (1) Shorten the evaluation time. Existing methods use full-scale, long-term FEM simulations to obtain the temperature rise results of pulse power coils, and a single evaluation usually takes several days to several weeks. This method decomposes it into several short-term FEM simulations (totaling several hours) and reduced-order model extrapolation calculations (completed in seconds), which shortens the overall evaluation time by about 2 to 3 orders of magnitude.

[0036] (2) Maintain evaluation accuracy. The R-value of the reduced-order model... th C th The parameters are identified from high-fidelity FEM simulation data and calibrated using the simulation results. Within the operating range of this embodiment, the relative error between the temperature rise obtained by extrapolation from the reduced-order model and the full-scale FEM simulation results does not exceed 5%.

[0037] (3) Reduce computing and R&D costs. This method reduces the long-term occupation of high-performance computing resources; temperature rise assessment and iteration can be completed in the design stage, which can reduce the physical prototype modification caused by thermal problems in the later stage.

[0038] (4) Supports rapid analysis under multiple operating conditions. The reduced-order model has low calculation time and can perform analysis on different combinations of pulse operating conditions and different cooling conditions (corresponding to R). th The values ​​are varied, and batch calculations and comparisons are performed to determine the operating boundary corresponding to the temperature rise limit, providing a basis for parameter selection. Attached Figure Description

[0039] Figure 1 This is a flowchart of the pulse power coil temperature rise limit evaluation method based on sparse simulation and reduced-order extrapolation of the present invention. Figure 2 This is a comparison chart of the fitting curve of the drooping-order model (ROM) for the calibration condition and the simulation data of the three-dimensional electro-thermal coupled finite element model (FEM) in Application Example 1; Figure 3 This is a comparison chart of the ROM model prediction curve and the simulation data of the three-dimensional electro-thermal coupled finite element model (FEM) under the verification working condition in Application Example 1; Figure 4 The fitting effect of the first-order model on the calibration data of the SLXC-B type pulse power coil system in Application Example 2. Figure 5 This is a comparison chart of the calibration results of the second-order ROM model and the FEM data in Application Example 2; Figure 6 This is an evaluation chart of the prediction accuracy of the ROM model for the verification working condition in Application Example 2. Detailed Implementation

[0040] The present invention will now be described in further detail with reference to the embodiments.

[0041] Those skilled in the art will understand that the following embodiments are for illustrative purposes only and should not be construed as limiting the scope of the invention. Where specific techniques or conditions are not specified in the embodiments, they are performed in accordance with the techniques or conditions described in the literature in the field or according to the product instructions. Materials or equipment whose manufacturers are not specified are all conventional products that can be obtained by purchase.

[0042] Example 1

[0043] A method for evaluating the temperature rise limit of pulse power coils based on sparse simulation and reduced-order extrapolation includes the following steps: Step 1: Establish a high-fidelity physical model and define the operating parameter space: A three-dimensional electro-thermal coupled finite element model was established using finite element analysis software. This model includes the geometry, material properties, and boundary conditions of the pulsed power coil. At the same time, a working condition parameter space is defined, and each combination of parameter values ​​in the space corresponds to a pulse current working condition. Step 2, Sparse Sampling and Short-Term Simulation of the Working Condition Space: The operating condition parameter space is divided into three subsets: calibration area, verification area, and exploration area. Short-term simulations are performed using a three-dimensional electro-thermal coupled finite element model only for the operating conditions in the calibration area and verification area to obtain the temperature change curves of key temperature measurement points of the coil over time under these operating conditions, thus obtaining the simulation data of the calibration area and the simulation data of the verification area. Step 3: Construct and calibrate the equivalent thermal path order reduction model: To address the macroscopic thermal dynamics of a pulse power coil system, an equivalent thermal path reduction model is constructed; the reduction model is a first-order or second-order equivalent thermal path model. Using the simulation data of the calibration area obtained in step two, the unknown parameters in the reduced-order model are identified and determined through a curve fitting optimization algorithm, and the calibrated reduced-order model is obtained. Step 4: Validation and accuracy evaluation of the reduced-order model: Using the calibrated reduced-order model obtained in step three, predict the temperature response curve under the operating conditions in the verification area; compare it with the simulation data of the verification area obtained in step two, and calculate the relative error δ between the two at the test point: Set an acceptable error threshold δ limit ; If δ≤δ limit If so, the calibrated reduced-order model is considered to have passed the validation. If δ>δ limit If the result is negative, return to step three and consider using a higher-order reduced-order model or re-dividing the calibration and validation areas. Repeat steps two through four until the calibrated reduced-order model passes validation. Step 5, extrapolation prediction of temperature rise limit: For the validated and calibrated reduced-order model, the temperature rise limit is evaluated under the operating conditions of the exploration zone.

[0044] Example 2

[0045] A method for evaluating the temperature rise limit of pulse power coils based on sparse simulation and reduced-order extrapolation includes the following steps: Step 1: Establish a high-fidelity physical model and define the operating parameter space: A three-dimensional electro-thermal coupled finite element model was established using finite element analysis software. This model includes the geometry, material properties, and boundary conditions of the pulsed power coil. At the same time, a working condition parameter space is defined, and each combination of parameter values ​​in the space corresponds to a pulse current working condition. Step 2, Sparse Sampling and Short-Term Simulation of the Working Condition Space: The operating condition parameter space is divided into three subsets: calibration area, verification area, and exploration area. Short-term simulations are performed using a three-dimensional electro-thermal coupled finite element model only for the operating conditions in the calibration area and verification area to obtain the temperature change curves of key temperature measurement points of the coil over time under these operating conditions, thus obtaining the simulation data of the calibration area and the simulation data of the verification area. Step 3: Construct and calibrate the equivalent thermal path order reduction model: To address the macroscopic thermal dynamics of a pulse power coil system, an equivalent thermal path reduction model is constructed; the reduction model is a first-order or second-order equivalent thermal path model. Using the simulation data of the calibration area obtained in step two, the unknown parameters in the reduced-order model are identified and determined through a curve fitting optimization algorithm, and the calibrated reduced-order model is obtained. Step 4: Validation and accuracy evaluation of the reduced-order model: Using the calibrated reduced-order model obtained in step three, predict the temperature response curve under the operating conditions in the verification area; compare it with the simulation data of the verification area obtained in step two, and calculate the relative error δ between the two at the test point: Set an acceptable error threshold δ limit ; If δ≤δ limit If so, the calibrated reduced-order model is considered to have passed the validation. If δ>δ limit If the result is negative, return to step three and consider using a higher-order reduced-order model or re-dividing the calibration and validation areas. Repeat steps two through four until the calibrated reduced-order model passes validation. Step 5, extrapolation prediction of temperature rise limit: For the validated and calibrated reduced-order model, the temperature rise limit is evaluated under the operating conditions of the exploration zone.

[0046] In step one, material properties include electrical conductivity, thermal conductivity, density, and specific heat capacity; boundary conditions include ambient temperature and convective heat transfer coefficient.

[0047] In step one, the operating parameter space is based on the pulse current amplitude. Single pulse conduction time Pulse period and continuous running time As operating condition parameters, construct the operating condition parameter space.

[0048] In step two, the three subsets are specifically: Subset 1 is the calibration area, where 1-2 basic pulse conditions are selected; Subset 2 is the verification area, in which 1 to 2 verification pulse conditions are selected. The verification pulse conditions have the same number of pulse sequence segments, current amplitude and duty cycle are on the same order of magnitude, and thermal time constant is on the same order of magnitude as the calibration area conditions. However, the specific current amplitude, duty cycle, and thermal time constant of the verification pulse conditions are different from those of the calibration area conditions. Subset 3 is the exploration zone, which includes all pulse conditions that require extreme evaluation; In step two, the short-term simulation duration is 10 to 20 minutes.

[0049] In step three, the reduced-order model is a first-order equivalent thermal circuit model, whose temperature response is described by the following equation: (1) In the formula, the thermal time constant τ and the equivalent average thermal power P eff They are determined by the following formulas respectively: (2) (3) In the formula, T(t) is the pulse power coil temperature predicted by the reduced-order model, in Kelvin; T amb Ambient temperature, in Kelvin. P eff The equivalent average thermal power of the pulse current over one cycle is expressed in watts. R th The equivalent thermal resistance of the pulse power coil system is expressed in Kelvin per watt. τ is the thermal time constant of the pulse power coil system, in seconds; C th This is the equivalent heat capacity of the pulse power coil system, measured in joules per Kelvin. R coil This is the DC resistance of the coil, measured in ohms. D is the duty cycle, which is dimensionless; t represents time, in seconds; Identify and determine the unknown parameter R in the reduced-order model. th and C th .

[0050] In step three, the curve fitting optimization algorithm is a nonlinear least squares method.

[0051] In step four, the assessment point is the simulation endpoint; (4) In the formula, δ represents the relative error, expressed as a percentage. T ROMThe temperature predicted at the test point by the calibrated reduced-order model, in Kelvin; T FEM The temperature simulated at the same test point for the three-dimensional electro-thermal coupled finite element model is expressed in Kelvin. δ limit It is 5%.

[0052] In step five, for each pulsed operating condition within the exploration zone, the theoretical saturation temperature of the pulsed power coil system is calculated according to the equivalent average thermal power of that pulsed operating condition using the following formula: (7) In the formula, This is the theoretical saturation temperature of the pulse power coil system during long-term operation under the corresponding pulse condition, serving as the temperature rise limit under that pulse condition, and is expressed in Kelvin. This refers to ambient temperature, expressed in Kelvin. The equivalent average thermal power of the pulse current over one cycle is expressed in watts. The equivalent thermal resistance of the calibrated pulse power coil, in Kelvin per watt; The theoretical saturation temperature corresponding to each pulse condition in the exploration area is calculated using Equation (7), and the theoretical saturation temperature is used as the temperature rise limit under the corresponding pulse condition to complete the temperature rise limit assessment. Application Example 1: Evaluation of an Air-Cooled Pulse Power Coil System This embodiment aims to illustrate in detail how the method of the present invention can be applied to a specific engineering object—the air-cooled pulsed power coil system FLXC-A—to evaluate its long-term temperature rise limit under a specific operating regime with high efficiency and high accuracy.

[0053] Step 1: Establish a high-fidelity physical model and define the operating parameter space: The purpose of this step is to establish a high-fidelity finite element model that can characterize the electro-thermal physical behavior of the pulsed power coil, and to define a parameter space consisting of pulse current amplitude, pulse on-time, pulse period, and continuous operating time, based on the design current, pulse duty cycle, and continuous operating time of the coil system. This high-fidelity finite element model is used for subsequent sparse simulations and to obtain the temperature response data required for model calibration and verification.

[0054] (1.1) Modeling Platform and Geometric Construction: ANSYS, a widely recognized commercial multiphysics simulation software, was selected as the modeling platform. In the ANSYS geometric modeling environment, a three-dimensional geometric model of the FLXC-A air-cooled pulsed power coil system was created based on its actual design drawings. The coil body is made of tightly wound enameled copper wire. To simplify calculations without losing key physical properties, thousands of independent enameled copper wires were equivalent to a circular ring entity with uniform properties. Key geometric parameters are defined as follows: Total number of turns of the coil: turn Inner radius of the coil loop: ; outer radius of the coil ring: Coil axial length (thickness): (1.2) Assigning Material Properties: Assigning accurate material properties to different parts of the model is crucial to ensuring the realism of the simulation results. The main materials and their properties involved in this example are shown in Table 1.

[0055] Table 1 Main materials and their property parameters

[0056] (1.3) Boundary conditions and load application: To simulate the actual operating environment of the coil, reasonable boundary conditions need to be applied.

[0057] 1) Thermal boundary conditions: The system uses natural air cooling. The ambient temperature (far-field temperature) of the model is set to... .

[0058] Convective heat transfer boundary conditions are applied to all air-exposed external surfaces of the coil. The overall convective heat transfer coefficient is estimated based on empirical formulas and the actual operating environment of the system. This boundary condition describes the heat exchange capacity between the coil surface and the surrounding air.

[0059] 2) Electrical load: The DC resistance of the coil at the reference temperature is obtained by performing DC conductance analysis using the electromagnetic module in ANSYS, or by calculating it according to theoretical formulas. In the subsequent transient thermal analysis, the applied pulsed current will generate Joule heat through this resistor, serving as the volumetric heat source for the thermal analysis.

[0060] (1.4) Definition of operating condition parameter space Based on the design operating regime of the air-cooled pulse power coil system FLXC-A, with pulse current amplitude... Single pulse conduction time Pulse period and continuous running time As operating condition parameters, construct an operating condition parameter vector: (1-1) in, This represents the current amplitude during the pulse conduction period, in amperes. The time during which the current is in the conducting state within each pulse cycle, in seconds; The pulse period is measured in seconds. The duration of operation under the corresponding pulse condition is measured in seconds or hours.

[0061] Pulse duty cycle Determined according to the following formula: (1-2) In the formula, It is a dimensionless parameter and satisfies: (1-3) This example uses a periodic rectangular pulse current. Within any pulse period, the coil current... Represented as: (1-4) Ignoring the change in coil resistance during the pulse period, and using the coil DC resistance at the reference temperature... Under the condition of equivalent average heat power, the corresponding operating condition for: (1-5) In the formula, It represents the equivalent average thermal power of the pulse current over one cycle, measured in watts.

[0062] Based on the design current range and operating regime of the FLXC-A system, this example will use the operating parameter space. Defined as: (1-6) in, ,therefore The parameters are not independently variable; only the combination of parameters that simultaneously satisfies the above constraints on pulse width, pulse period, and duty cycle is considered as the effective operating condition.

[0063] Within the aforementioned operating condition parameter space, calibration conditions, verification conditions, and exploratory conditions for which temperature rise limit assessment will be performed are selected. Representative operating conditions are shown in Table 2.

[0064] Table 2 Representative Pulse Conditions of the FLXC-A System

[0065] The equivalent average heat power in Table 2 is based on... The calculations are as follows: the calibration condition is used to identify the equivalent thermal resistance and equivalent heat capacity of the reduced-order model; the verification condition is used to verify the prediction accuracy of the calibrated reduced-order model under different pulse parameters; and the exploration condition is used to evaluate the theoretical saturation temperature of the coil system under long-term operating conditions or higher equivalent average thermal power conditions.

[0066] In step one, since the reduced-order model has not yet been calibrated and the system's thermal time constant has not yet been determined, candidate operating conditions for the exploration area are selected based on the predetermined continuous operating time and equivalent average thermal power. After the reduced-order model parameters are identified in step three, the thermal time constant is determined based on the identified parameters. Further determine whether the continuous operating time for the corresponding working condition meets the requirements: (1-7) In the formula, The preset value is 3 to 5; when the reduced-order model has multiple thermal time constants, the thermal time constant with the largest value is taken as the thermal time constant. Operating conditions meeting the above conditions are identified as long-term operating conditions. For a first-order reduced-order model, when the running time reaches... and At that time, their temperature rise reached approximately 95.0% and 99.3% of the theoretical saturation temperature rise, respectively.

[0067] High-power operation is based on a preset equivalent average thermal power threshold. A judgment will be made. In this example, [the judgment will be made]. Set as the high-power judgment threshold, which is to meet the following conditions. The operating condition was determined to be a high-power condition. Therefore, exploratory condition 2 in Table 2 belongs to the high-power condition; whether exploratory condition 1 and exploratory condition 2 both belong to long-term conditions is determined based on the relationship between their respective continuous operating time and dominant thermal time constant after the parameter identification of the reduced-order model is completed.

[0068] Thus, the definition of the FLXC-A system operating condition parameter space and the initial division of calibration, verification, and exploration operating conditions were completed.

[0069] At this point, a complete three-dimensional electro-thermal coupled finite element model (FEM) has been constructed.

[0070] Step 2, Sparse Sampling and Short-Term Simulation of the Working Condition Space: This step aims to obtain “sparse” temperature history data that reflects the core thermal dynamics of the system at an acceptable computational cost by performing short-term transient simulations on a three-dimensional electro-thermal coupled finite element model (FEM).

[0071] (2.1) Definition of typical working conditions: Three representative working systems were selected, as shown in Table 3.

[0072] Table 3 Three representative work systems

[0073] (2.2) Perform sparse simulation: In the ANSYS transient thermal analysis module, simulations were performed for the "calibration condition" and the "verification condition". To reflect the concept of "sparse" sampling, the simulation duration was set to only 20 minutes. The simulation tracked and recorded the temperature change over time at the preset highest temperature point (hottest point) inside the coil.

[0074] A 20-minute simulation of the "calibration condition" was performed to obtain the hottest temperature curve. This simulation took approximately 3 hours on a high-performance workstation.

[0075] A 20-minute simulation of the "verification condition" was performed to obtain the temperature curve of the hottest spot. The simulation took approximately 3.2 hours.

[0076] Although these two sets of data only describe the thermal behavior of the system in the early stage of startup, they already contain key information on thermal resistance and thermal capacity, which is the data foundation for building a reduced-order model in the next step.

[0077] Step 3: Construct and calibrate the equivalent thermal path order reduction model: This step uses the sparse simulation data obtained in step two to identify the parameters of a simple and computationally fast mathematical model, namely the reduced-order model (ROM model).

[0078] (3.1) ROM Model Construction: For the air-cooled pulsed power coil system FLXC-A, its overall thermal dynamic characteristics are described using a first-order lumped-parameter equivalent thermal circuit model. In the first-order lumped-parameter equivalent thermal circuit model, the pulsed power coil is equivalent to having an equivalent heat capacity. The temperature node, which is achieved through equivalent thermal resistance Maintain at temperature The environment undergoes heat exchange. The heat balance equation of the first-order lumped parameter equivalent thermal circuit model is: (1-8) In the formula, The pulse power coil predicted by the reduced-order model at time... Temperature, in Kelvin (K); This is the time taken from the start of operation under the corresponding working condition, in seconds (s). The ambient temperature is expressed in Kelvin (K). The equivalent average thermal power in one pulse cycle for the corresponding pulsed operating condition is expressed in watts (W). The equivalent thermal resistance of the pulse power coil system characterizes the ability of the coil system to dissipate heat to the environment, and is measured in Kelvin per watt (K / W). This is the equivalent heat capacity of the pulse power coil system, characterizing the thermal inertia of the coil system, and is measured in joules per Kelvin (J / K).

[0079] The thermal time constant of the first-order reduced-order model is defined as: (1-9) In the formula, is the thermal time constant, with units of seconds (s).

[0080] At ambient temperature Equivalent average heat power Equivalent thermal resistance and equivalent heat capacity It remains constant during the corresponding temperature rise, and the initial conditions are: In this case, the general analytical solution of equation (1-8) is: (1-10) Or equivalently expressed as: (1-11) In the formula, The initial temperature of the coil at the start of operation corresponding to the pulse condition is expressed in Kelvin (K).

[0081] In this embodiment, both the calibration and verification conditions begin applying pulsed current after the coil and environment reach initial thermal equilibrium. Therefore: (1-12) Substituting equation (1-12) into equation (1-10) or equation (1-11), we get: (1-13) Further Substituting into equation (1-13), we get: (1-14) Equation (1-13) or Equation (1-14) is consistent with the temperature response formula of the reduced-order model used in the claims.

[0082] As the running time approaches infinity, the exponential term in equation (1-10) approaches zero, therefore the theoretical saturation temperature under pulsed conditions is: (1-15) Therefore, the initial temperature The transient process affects the temperature response, but does not affect the theoretical saturation temperature if the model parameters and boundary conditions remain unchanged. .

[0083] (3.2) Model parameter identification (calibration): Parameter identification is performed using the data from the “calibration condition”.

[0084] First, calculate the equivalent average power under calibration conditions. : (1-16) Then, the nonlinear least squares fitting function lsqcurvefit is used in the MATLAB software environment. Equation (1-10) is used as the fitting function to fit the FEM simulation data. As the target curve, for unknown parameters and To identify.

[0085] After optimized calculation, the identification results are obtained: Equivalent thermal resistance of pulse power coil system Equivalent heat capacity of pulse power coil system like Figure 2 As shown, the ROM model prediction curve obtained by substituting the identified parameters into equation (1-13) was compared with the original FEM simulation data points. Figure 2 The two curves in the figure highly overlap, proving that the first-order ROM model is sufficient to accurately describe the thermal dynamic characteristics of the system under calibration conditions.

[0086] Figure 2 The discrete data points in the diagram represent precise temperature values ​​obtained through short-time simulations using a three-dimensional electro-thermal coupled finite element model (FEM), while the solid red line is the fitted curve generated by a reduced-order model (ROM), which is several orders of magnitude faster in computation. The high degree of overlap between the curve and the data points strongly demonstrates that, despite its extreme simplification, the ROM model has successfully learned and captured the core thermodynamic laws of the system accurately from a small amount of short-term "anchor data." This successful fitting not only verifies the effectiveness of the reduced-order model but also lays a solid foundation of accuracy for subsequent rapid and reliable extrapolation predictions of temperature rise limits under arbitrary long-term operating conditions.

[0087] Step 4: Validation and accuracy evaluation of the reduced-order model: This step verifies the predictive ability of the reduced-order model obtained in step three for different operating conditions.

[0088] Calculate the equivalent average power P under the verification condition. eff_val : (1-17) P eff_val With the identified Rth =0.048K / W, C th Substituting 42000 J / K into equation (1-10), calculate the predicted temperature T at each time point under the verification condition. ROM (t).

[0089] Accuracy assessment: Compare the temperature values ​​at the end of 20 minutes: FEM model simulation value: T FEM (t=20min)=65.8℃ ROM model prediction: T ROM (t=20min)=66.2℃ The relative error δ is calculated using the following formula: (1-18) Substitute the values: (1-19) like Figure 3 As shown, the prediction curve T of the ROM model for the verification condition is... ROM (t) The data points from the FEM model simulation under the validation condition are plotted in the same graph. The square points in the graph represent the temperature data obtained from the FEM model simulation under the validation condition, and the dashed line represents the predicted curve calculated by the calibrated reduced-order model based solely on the parameters of this condition. The relative error between the two over a 20-minute time history is 0.98%, which is less than the engineering acceptable threshold of 5%. The results show that the predicted temperature of the reduced-order model is consistent with the FEM simulation results under the validation condition, which is different from the calibration condition, and the validation is successful.

[0090] Step 5, extrapolation prediction of temperature rise limit: Using a validated reduced-order model, the long-term temperature rise limit of the exploratory working condition is extrapolated.

[0091] When the system reaches thermal steady state, dT(t) / dt=0, calculate : (1-20) For exploration conditions ( (This should be entered according to the actual operating conditions of FLXC-A, and should not be less than the power of the verification operating conditions). Its temperature rise limit is: (1-21) The calculation of equation (1-21) takes less than 0.1 seconds in MATLAB.

[0092] To verify the extrapolation results, a long-term complete FEM simulation (or actual measurement using a prototype) was performed on the exploratory working condition, and the steady-state temperature was found to be 103.9℃. The relative error compared with the extrapolated value of Equation (1-21) was 1%, which is within the 5% threshold, indicating that the reduced-order model still maintains the prediction accuracy under long-term extrapolation.

[0093] This example demonstrates the application of this method in the air-cooled pulsed power coil system FLXC-A.

[0094] Application Example 2: Forced Water-Cooled Pulse Power Coil System and Second-Order Model Application This example aims to further demonstrate the adaptability and scalability of the method of the present invention. Its application is to an SLXC-B high-power pulsed power coil system with a more complex structure and heat dissipation mode, characterized by forced water cooling. Unlike the natural air-cooled system in Application Example 1, the heat of this system is dissipated through a multi-stage path of "coil → cooling plate → cooling water," and its thermal dynamic behavior cannot be accurately described by a single time constant. Therefore, a higher-order reduced-order model (ROM) is required for characterization.

[0095] Step 1: Establish a high-fidelity physical model and define the operating parameter space: Similar to Application Example 1, a three-dimensional electro-thermal coupled finite element model (FEM) of the forced water-cooled high-power pulsed power coil system SLXC-B is first established in ANSYS software.

[0096] (1.1) Geometry and Materials: The geometric model of the SLXC-B forced water-cooled high-power pulse power coil system was constructed according to its design specifications, and the precise material properties of each component (copper coil, insulation material, cooling plate, etc.) were assigned.

[0097] Table 4. Main materials and properties of the forced water cooling system SLXC-B

[0098] (1.2) Boundary Condition Setting: This step is the key difference between this example and Example 1. The forced water-cooled high-power pulsed power coil system SLXC-B uses forced water cooling, and its cooling circuit is tightly integrated with the coil. To simulate this efficient heat dissipation method, the following boundary conditions are applied in the FEM model: 1) Forced Convection Boundary: A boundary condition representing the forced convection effect of cooling water is applied to the surface of the flow channel within the cooling plate to which the coil is attached. This boundary condition contains two core parameters: Bulk Temperature (Reference Temperature for Coolant): This is the cooling water inlet temperature, which is assumed to have a negligible temperature rise during circulation and is considered a constant heat dissipation endpoint.

[0099] Heat transfer coefficient (HFT): Based on fluid dynamics calculations or empirical data, set a relatively high value (e.g., This reflects the powerful heat exchange capacity of forced water cooling.

[0100] (1.3) Definition of electrical load and operating parameters: The electrical load and operating parameters of the SLXC-B forced water-cooled pulse power coil system at the reference temperature are obtained through ANSYS electrical conduction analysis or calculation based on the coil conductor length, effective cross-sectional area and material resistivity. DC resistance below :

[0101] In this example, the DC resistance is used as a reference resistance for calculating the equivalent average thermal power under different pulse conditions. In the finite element model, the Joule heat generated by the pulse current is used as the volumetric heat source for the coil region.

[0102] This example uses a periodic rectangular pulse current, with the pulse current amplitude... On-time within a single pulse cycle Pulse period and continuous running time As pulse condition parameters, construct a condition parameter vector: (2-1) In the formula, The current amplitude during the pulse conduction period is expressed in amperes (A). The time during which the pulse current is in the conducting state within each pulse cycle, in seconds (s). The pulse period is measured in seconds (s). The duration of operation under the corresponding pulse condition is measured in seconds (s) or hours (h).

[0103] Pulse duty cycle Determined according to the following formula: (2-2) In the formula, Let be a dimensionless parameter, and satisfy: (2-3) Within any pulse period, the rectangular pulse current applied to the coil can be expressed as: (2-4) When using the DC resistance of the coil at the reference temperature And ignoring the change in coil resistance within one pulse cycle, the equivalent average heat power of the corresponding pulse condition within one pulse cycle is: (2-5) In the formula, It is the equivalent average thermal power of the pulse current over one cycle, measured in watts (W).

[0104] Based on the rated current, pulse duty cycle, and predetermined continuous operating time of the SLXC-B system, this example defines the pulse duty cycle parameter space as follows: (2-6) Due to duty cycle Depend on and The ratio is determined, therefore These are not independently varying operating parameters. Only parameter combinations that simultaneously satisfy the constraints shown in equations (2-3) and (2-6) constitute an effective pulse operating condition.

[0105] Within the aforementioned parameter space, calibration conditions are selected for parameter identification of the second-order reduced model, verification conditions for model accuracy checking, and exploratory conditions for extrapolation prediction of the temperature rise limit. Representative pulse conditions selected for this example are shown in Table 5.

[0106] Table 5 Representative pulse conditions of the forced water-cooled SLXC-B system

[0107] The equivalent average heat power in Table 5 is based on equation (2-5) and uses... Calculated.

[0108] The calibration and verification conditions involve performing a 10-minute high-fidelity transient finite element simulation to obtain short-term temperature response data required for the calibration and verification of the second-order reduced model. The exploration condition does not perform a full-term finite element simulation corresponding to the continuous running time. Instead, after the second-order reduced model is verified, the theoretical saturation temperature of the coil is obtained by extrapolation through the second-order reduced model.

[0109] It should be noted that the “Continuous running time” in Table 5 refers to the target duration of the corresponding pulse duty cycle in actual operation or temperature rise limit assessment; the “Finite element simulation duration” for calibration and verification conditions refers to the short-term simulation time set to obtain sparse temperature response data, and the two have different meanings.

[0110] This example ensures that the equivalent average heat power is not less than a preset equivalent average heat power threshold. The operating condition is determined to be a high-power condition. The preset power threshold is determined based on the rated average power, allowable temperature, and design heat dissipation capacity of the cooling system of the SLXC-B system. In this example, it is set as follows: Therefore, when When the pulse condition is applied, it is determined to be a high-power condition. According to this criterion, both exploration condition 1 and exploration condition 2 in Table 5 belong to high-power conditions.

[0111] For long-duration operating conditions, since the parameter identification of the second-order reduced-order model has not yet been completed in step one, the system's thermal time constant is not yet determined. Therefore, in this step, operating conditions with longer continuous operating times are first classified as candidate operating conditions in the exploration zone. After the second-order reduced-order model calibration is completed in step three, the two identified thermal time constants are used as the basis for further analysis. and The thermal time constant with the larger value is defined as the dominant thermal time constant: (2-7) When the duration of operation of the corresponding pulse condition meets the following requirements: (2-8) At that time, the pulse condition is determined to be a long-term condition, wherein, The preset value is 3 to 5. Using the thermal time constant with the largest value as the criterion ensures that even the thermal dynamic component with the slowest decay rate in the second-order reduced model has decayed to the predetermined level.

[0112] In this example, the forced water-cooled boundary conditions are maintained as follows:

[0113] That is, the coolant reference temperature and convective heat transfer coefficient are not considered as varying parameters in the parameter space of this example. Therefore, the temperature response differences between different operating conditions mainly originate from the pulse current amplitude, conduction time, pulse period, and their corresponding equivalent average heat power.

[0114] Through the above settings, the operating parameter space definition of the forced water-cooled pulse power coil system SLXC-B and the initial division of calibration, verification and exploration operating conditions are completed.

[0115] Step 2, Sparse Sampling and Short-Term Simulation of the Working Condition Space: Typical "calibration conditions" and "verification conditions" applicable to the SLXC-B forced water-cooled high-power pulsed power coil system were selected, and short-term simulations were performed using the established FEM model. Given the system's faster thermal response, the simulation duration was shortened to 10 minutes. The simulations yielded the hottest spot temperature curves under the calibration conditions. and verification working conditions A 10-minute short-term FEM simulation was performed on the calibration condition, and the temperature of the hottest spot in the coil was recorded at fixed time intervals (e.g., every 10 seconds) to obtain the temperature-time data sequence of the calibration condition. This data serves as the target data for parameter identification in step three. A 10-minute short-term FEM simulation is performed on the verification condition, and the temperature of the hottest spot in the coil is recorded at equal intervals to obtain the temperature-time data sequence for the verification condition. , serving as independent baseline data for the cross-validation in step four.

[0116] Table 6. Three representative operating systems of the SLXC-B forced water cooling system.

[0117] Step 3: Construct and calibrate the equivalent thermal path order reduction model: (3.1) Model order exploration and determination: First, try to use the first-order thermal circuit model described in Application Example 1 to test the calibration data. Perform fitting. For example... Figure 4 As shown, the fitting curve (dashed line) of the first-order model and the FEM simulation data points (solid points) have a systematic deviation in shape, especially in the initial heating stage, where the calculated fitting error is large and cannot meet the accuracy requirements.

[0118] Analysis suggests that the failure of the first-order model stems from the existence of two significantly different physical processes in the heat transfer path of the forced water-cooled high-power pulsed power coil system SLXC-B, each corresponding to a primary thermal time constant: Process 1: Joule heat accumulates inside the coil, causing the coil itself to heat up (corresponding to a relatively fast time constant).

[0119] Process 2: Heat is transferred from the coil to the cooling plate through the insulation layer and contact surface with limited thermal conductivity, and then from the cooling plate to the cooling water (corresponding to a slower time constant).

[0120] Therefore, a second-order reduced-order model capable of describing both time scales must be adopted.

[0121] Second-order ROM model construction: A second-order equivalent thermal network model is constructed. This network contains two nodes, representing the coil and the cooling plate, respectively. Its physical behavior is described by the following set of differential equations: (2-9) In the formula, The temperature of the pulse power coil, which is the output we are interested in, is expressed in Kelvin (K). The equivalent temperature of the cooling plate is expressed in Kelvin (K). The equivalent average heat power of the pulse current over one cycle, in units of: . Coolant temperature, unit: Kelvin (K). These represent the equivalent thermal resistance from the pulse power coil to the cooling plate and the equivalent thermal capacity of the coil itself, respectively. These represent the equivalent thermal resistance from the cooling plate to the coolant and the equivalent heat capacity of the cooling plate itself, respectively.

[0122] exist Under constant conditions, the temperature of the pulse power coil in the differential equation system (2-9) The analytical solution can be expressed as: (2-10) In the formula, and The two thermal time constants of the second-order reduced-order model are given by , , and To be determined jointly; and The temperature amplitude coefficient of the corresponding characteristic mode is determined by four equivalent thermal parameters, the equivalent average thermal power, and the initial temperatures of the coil and the cooling plate.

[0123] In this example, the calibration, verification, and exploration modes all begin from the state where the coil, cooling plate, and coolant reach initial thermal equilibrium, i.e.: (2-11) Under these initial conditions, the amplitude coefficients satisfy: (2-12) Model parameter identification (calibration): Using equation (2-10) as the fitting function, and the FEM simulation data of the calibration condition... To achieve this goal, optimization algorithms (such as lsqcurvefit) are used in software like MATLAB to identify four unknown parameters. The optimal value.

[0124] like Figure 5 As shown, the prediction curve (solid line) of the second-order ROM model plotted using the identified parameters perfectly matches the FEM simulation data points (solid dots), proving the accurate ability of the second-order model to capture the thermal dynamics of this complex system.

[0125] Step 4: Validation and accuracy evaluation of the reduced-order model: To ensure that the calibrated second-order ROM model has reliable predictive capabilities, it is cross-validated using the "validation test".

[0126] Prediction using ROM: Calculate the equivalent power corresponding to the "verification condition". Compare it with the already identified parameters Substituting both into the system of differential equations (Equation (2-9)) or its analytical solution (Equation (2-10)), the temperature prediction curve under the verification condition is calculated. .

[0127] Accuracy assessment: The parameters for the verification working condition are:

[0128] Its equivalent average heat power is:

[0129] Will Substituting into equation (2-9), under the initial conditions Below, the calculation shows that at the end of 10 minutes, that is... The predicted ROM temperature at that time is:

[0130] The temperature of the hottest spot in the coil obtained from the FEM simulation under the verification condition at the same moment is:

[0131] Using the temperature rise relative to the coolant temperature as the error calculation benchmark, we obtain:

[0132] The calculated relative error is less than the preset error threshold. Therefore, the second-order reduced-order model has been verified and can be used to extrapolate and predict the temperature rise limit under various operating conditions. The temperatures mentioned above can also be uniformly converted to Kelvin for calculation. Since the Celsius temperature difference and the Kelvin temperature difference have the same numerical value, using either Celsius or Kelvin as the error denominator will not change the error calculation result when using temperature rise as the error denominator.

[0133] like Figure 6 As shown, the prediction curve of the ROM model for the verification condition is highly consistent with the FEM simulation data throughout the entire time history. This result proves that by appropriately increasing the order of the reduced-order model, the method of the present invention can effectively handle systems with more complex internal heat conduction paths while maintaining high accuracy.

[0134] Step 5, extrapolation prediction of temperature rise limit: The validated second-order reduced-order model was used to extrapolate and predict the temperature rise limit of pulsed conditions within the exploration zone.

[0135] When the pulsed power coil system reaches its theoretical thermal steady state, both the coil node temperature and the cooling plate node temperature no longer change with time, that is:

[0136] Substituting the above steady-state conditions into equation (2-9), we get: (2-13) In the formula, The temperature of the coil node when the pulse power coil system reaches thermal steady state is the theoretical saturation temperature of the hottest spot in the coil, expressed in Kelvin (K). The equivalent temperature of the cooling plate node when the pulse power coil system reaches thermal steady state is the theoretical saturation temperature of the cooling plate, expressed in Kelvin (K).

[0137] as well as: (2-14) Therefore, the theoretical saturation temperature of the hottest spot in the coil satisfies the following relationship with the coolant temperature: (2-15) Define the total equivalent thermal resistance of the second-order equivalent thermal path. for: (2-16) The theoretical saturation temperature of the hottest spot in the coil is: (2-17) In the formula, To correspond to the theoretical saturation temperature of the hottest spot in the coil under the exploration condition, and to serve as the temperature rise limit under this condition; Reference temperature for coolant; To explore the equivalent average thermal power corresponding to the operating conditions; It is the total equivalent thermal resistance between the hottest point of the coil and the coolant.

[0138] In this example, the pulse parameters for the exploratory working condition are:

[0139]

[0140]

[0141] Therefore, the duty cycle for the exploration condition is:

[0142] The equivalent average heat power for the exploratory operating condition is:

[0143] Based on the parameter identification results in step three, the total equivalent thermal resistance of the system is:

[0144] Therefore, the theoretical saturation temperature rise of the hottest spot in the coil relative to the coolant under exploratory operating conditions is:

[0145] The reference temperature for the coolant is:

[0146] This yields the theoretical saturation temperature of the hottest spot in the coil under the exploratory operating conditions:

[0147] Converted to Kelvin:

[0148] Therefore, while the coolant temperature is maintained at The convective heat transfer coefficient remains at . The reference resistance of the coil is Under the condition that the pulse parameters remain unchanged, the theoretical saturation temperature rise corresponding to the exploratory working condition is: The theoretical saturation temperature of the hottest spot in the coil is ,Right now The theoretical saturation temperature is used as the temperature rise limit assessment result under the aforementioned exploratory conditions.

[0149] In this example, the dominant thermal time constant of the second-order reduced model is:

[0150] The corresponding 3-fold and 5-fold dominant thermal time constants are as follows:

[0151]

[0152] If the target continuous operating time for the exploration condition is 8 hours, then:

[0153] Therefore, this exploratory operating condition belongs to the long-term operating condition. At the end of the 8-hour operation, both exponential decay terms in equation (2-10) are close to zero, and the temperature of the hottest spot in the coil is very close to the theoretical saturation temperature. .

[0154] Application: For any given severe operating condition, it is only necessary to calculate its equivalent power. The final temperature rise limit can be calculated directly in milliseconds using equation (2-17) without any additional time-consuming simulation.

[0155] This embodiment successfully applies the method of the present invention to a complex pulse power coil system employing forced water cooling and featuring multi-stage heat dissipation paths. Practical application has proven that: Adaptability of the method: This invention is not limited to simple systems that can be described by a first-order model. When dealing with more complex physical objects, the order of the reduced-order model (ROM) can be increased (e.g., from first-order to second-order) to effectively address the problem.

[0156] Process completeness: This invention includes a built-in verification step that can determine whether the selected ROM model order is sufficient. If the model calibration or verification error is too large, it guides the user to increase the model order until the accuracy requirements are met.

[0157] The core advantage continues: even when using higher-order models, once the model parameters are identified and verified, the computational cost for extrapolation prediction is almost negligible, maintaining a significant efficiency advantage over traditional full-time FEM methods.

[0158] In summary, Application Example 2 powerfully demonstrates that the method of the present invention has excellent universality, adaptability and scalability, and can flexibly handle thermal systems of different complexities, making it a robust and efficient engineering evaluation tool.

[0159] Table 7 is a comparison table of computational efficiency between traditional full-time FEM simulation and the method of the present invention in Embodiment 1 of the present invention.

[0160] Table 7 Comparison of computational efficiency between traditional full-time FEM simulation and the method of this invention.

[0161] Analysis of Table 7 shows that: (1) Significant time cost savings: Through the core idea of ​​“sparse simulation + reduced-order model extrapolation”, this invention shortens the evaluation cycle that originally required several hours or even longer to the minute level.

[0162] (2) The great release of computing resources avoids the long-term occupation of expensive computing servers, and enables temperature rise assessment to be completed efficiently on ordinary workstations, significantly reducing hardware threshold and enterprise costs.

[0163] (3) Revolutionary improvement in design iteration efficiency: The sharp reduction in evaluation time means that R&D engineers can conduct rapid and multiple "what-if" analyses and solution optimizations during the product design phase, which greatly accelerates the product development process and iteration speed.

[0164] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.

Claims

1. A method for evaluating the temperature rise limit of pulse power coils based on sparse simulation and reduced-order extrapolation, characterized in that, Includes the following steps: Step 1: Establish a high-fidelity physical model and define the operating parameter space: A three-dimensional electro-thermal coupled finite element model was established using finite element analysis software. This model includes the geometry, material properties, and boundary conditions of the pulsed power coil. At the same time, a working condition parameter space is defined, and each combination of parameter values ​​in the space corresponds to a pulse current working condition. Step 2, Sparse Sampling and Short-Term Simulation of the Working Condition Space: The operating condition parameter space is divided into three subsets: calibration area, verification area, and exploration area. Short-term simulations are performed using a three-dimensional electro-thermal coupled finite element model only for the operating conditions in the calibration area and verification area to obtain the temperature change curves of key temperature measurement points of the coil over time under these operating conditions, thus obtaining the simulation data of the calibration area and the simulation data of the verification area. Step 3: Construct and calibrate the equivalent thermal path order reduction model: To address the macroscopic thermal dynamics of a pulse power coil system, an equivalent thermal path reduction model is constructed; the reduction model is a first-order or second-order equivalent thermal path model. Using the simulation data of the calibration area obtained in step two, the unknown parameters in the reduced-order model are identified and determined through a curve fitting optimization algorithm, and the calibrated reduced-order model is obtained. Step 4: Validation and accuracy evaluation of the reduced-order model: Using the calibrated reduced-order model obtained in step three, predict the temperature response curve under the operating conditions in the verification area; compare it with the simulation data of the verification area obtained in step two, and calculate the relative error δ between the two at the test point: Set an acceptable error threshold δ limit ; If δ≤δ limit If so, the calibrated reduced-order model is considered to have passed the validation. If δ>δ limit If the result is negative, return to step three and consider using a higher-order reduced-order model or re-dividing the calibration and validation areas. Repeat steps two through four until the calibrated reduced-order model passes validation. Step 5, extrapolation prediction of temperature rise limit: For the validated and calibrated reduced-order model, the temperature rise limit is evaluated under the operating conditions of the exploration zone.

2. The pulse power coil temperature rise limit assessment method based on sparse simulation and reduced-order extrapolation according to claim 1, characterized in that, In step one, material properties include electrical conductivity, thermal conductivity, density, and specific heat capacity; boundary conditions include ambient temperature and convective heat transfer coefficient.

3. The pulse power coil temperature rise limit assessment method based on sparse simulation and reduced-order extrapolation according to claim 1, characterized in that, In step one, the operating parameter space is based on the pulse current amplitude. Single pulse conduction time Pulse period and continuous running time As operating condition parameters, construct the operating condition parameter space.

4. The pulse power coil temperature rise limit assessment method based on sparse simulation and reduced-order extrapolation according to claim 1, characterized in that, In step two, the three subsets are specifically: Subset 1 is the calibration area, where 1-2 basic pulse conditions are selected; Subset 2 is the verification area, where one or two verification pulse conditions are selected; The verification pulse condition and the calibration area condition have the same number of pulse sequence segments, the same current amplitude and duty cycle, and the same thermal time constant; however, the specific current amplitude, duty cycle, and thermal time constant of the verification pulse condition are different from those of the calibration area condition. Subset 3 is the exploration zone, which includes all pulse conditions that require extreme evaluation; In step two, the short-term simulation duration is 10 to 20 minutes.

5. The method for evaluating the temperature rise limit of pulse power coils based on sparse simulation and reduced-order extrapolation according to claim 1, characterized in that, In step three, the reduced-order model is a first-order equivalent thermal circuit model, whose temperature response is described by the following equation: （1） In the formula, the thermal time constant τ and the equivalent average thermal power P eff They are determined by the following formulas respectively: （2） （3） In the formula, T(t) is the pulse power coil temperature predicted by the reduced-order model, in Kelvin; T amb Ambient temperature, in Kelvin. P eff The equivalent average thermal power of the pulse current over one cycle is expressed in watts. R th The equivalent thermal resistance of the pulse power coil system is expressed in Kelvin per watt. τ is the thermal time constant of the pulse power coil system, in seconds; C th This is the equivalent heat capacity of the pulse power coil system, measured in joules per Kelvin. R coil This is the DC resistance of the coil, measured in ohms. D is the duty cycle, which is dimensionless; t represents time, in seconds; Identify and determine the unknown parameter R in the reduced-order model. th and C th .

6. The method for evaluating the temperature rise limit of pulse power coils based on sparse simulation and reduced-order extrapolation according to claim 1, characterized in that, In step three, the curve fitting optimization algorithm is a nonlinear least squares method.

7. The method for evaluating the temperature rise limit of pulse power coils based on sparse simulation and reduced-order extrapolation according to claim 1, characterized in that, In step four, the assessment point is the simulation endpoint; （4） In the formula, δ represents the relative error, expressed as a percentage. T ROM The temperature predicted at the test point by the calibrated reduced-order model, in Kelvin; T FEM The temperature simulated at the same test point for the three-dimensional electro-thermal coupled finite element model is expressed in Kelvin. δ limit It is 5%.

8. The method for evaluating the temperature rise limit of pulse power coils based on sparse simulation and reduced-order extrapolation according to claim 1, characterized in that, In step five, for each pulsed operating condition within the exploration zone, the theoretical saturation temperature of the pulsed power coil system is calculated according to the equivalent average thermal power of that pulsed operating condition using the following formula: （7） In the formula, This is the theoretical saturation temperature of the pulse power coil system during long-term operation under the corresponding pulse condition, serving as the temperature rise limit under that pulse condition, and is expressed in Kelvin. This refers to ambient temperature, expressed in Kelvin. The equivalent average thermal power of the pulse current over one cycle is expressed in watts. The equivalent thermal resistance of the calibrated pulse power coil, in Kelvin per watt; The theoretical saturation temperature corresponding to each pulse condition in the exploration area is calculated using Equation (7), and the theoretical saturation temperature is used as the temperature rise limit under the corresponding pulse condition to complete the temperature rise limit assessment.

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