A data-driven method for modeling differential equations of a dynamic system

Abstract

The present invention relates to the field of differential modeling, and particularly to a data-driven method for modeling differential equations of a dynamic system. By collecting time series data of the dynamic system, extracting the dynamic characteristics of the time series data, preprocessing the time series data based on the dynamic characteristics to generate corresponding pre-data sequences, calculating the oscillation coefficient of the dynamic system and selecting the form of the differential equation, determining the corresponding differential equation parameters, constructing a differential equation model corresponding to the dynamic system to learn the pre-data sequences, generating simulation data within a preset time, comparing it with the real data corresponding to the preset time, generating the prediction accuracy of the differential equation model and comparing it with the accuracy threshold to determine whether to adjust the differential equation model, it can learn the non-linear relationship from the time series data, objectively evaluate the prediction performance of the model, timely discover the deficiencies of the model, and construct a differential equation model that conforms to the actual situation.

Description

Technical Field

[0001] The present invention relates to the field of differential modeling, and in particular to a data-driven method for modeling differential equations of dynamic systems. Background Art

[0002] Modeling of dynamic systems is an important field in engineering and scientific research, and is widely applied to multiple systems such as mechanical, fluid, biological, and economic systems.

[0003] By extracting the dynamic characteristics in the time series data of a dynamic system through a data-driven algorithm and combining physical principles to construct a differential equation model, both the data information is fully utilized and the interpretability of the physical model is retained.

[0004] Chinese Patent Application Publication No.: CN106227978A discloses a method for modeling the suction surface primitive curve of a compressor blade based on a second-order ordinary differential equation. After preprocessing the compressor blade primitive curve data, a second-order constant coefficient linear non-homogeneous ordinary differential system is selected for fitting to obtain the expression of the primitive curve. According to the requirements of the boundary conditions, the problem of obtaining a second-order curve that coincides with the head and tail points of the given data is transformed into a two-point boundary value problem, and the implicit Euler method is selected to solve the intermediate points of the primitive curve, completing the reconstruction of the primitive curve. The invention solves the problem that the first-order system cannot simultaneously ensure the interpolation conditions at the head and tail points, obtains a larger optimization operation space, and reduces the sensitivity to the parameter matrix. The invention can accurately interpolate the head and tail points of the data scatter points, which has great advantages for connecting the piecewise or segmented fitting data scatter points, and can restore the shape of the compressor blade with high precision.

[0005] Chinese Patent Application Publication No.: CN113378310A discloses a method for modeling fatigue crack growth based on uncertain differential equations. Aiming at the crack growth process considering crack closure and high-cycle retardation effects, first, the cognitive uncertainties in four aspects of physical properties, external factors, time dimension, and threshold are quantified based on the uncertainty theory to construct a fatigue crack growth model based on uncertain differential equations; then, the model parameters are determined through statistical analysis; then, the certainty reliability function is derived to evaluate the certainty reliability; finally, the crack growth process and fatigue life are predicted. The above method for modeling fatigue crack growth based on uncertain differential equations provided by the invention gives an option for reasonably quantifying the cognitive uncertainties in fatigue crack growth experiments, makes the understanding of fatigue laws more accurate, and describes the crack growth process based on uncertain differential equations, which helps to improve the accuracy of crack growth and fatigue life prediction.

[0006] However, the above methods have the following problems: when the system parameters change or the system structure changes, the model needs to be readjusted, and the adaptability is poor. Summary of the Invention

[0007] To this end, the present invention provides a data-driven method for modeling differential equations of a power system to overcome the problem in the prior art that when the system parameters change or the system structure changes, the model needs to be readjusted, resulting in poor adaptability.

[0008] To achieve the above object, the present invention provides a data-driven method for modeling differential equations of a power system, including:

[0009] Collect time series data of the power system, where

[0010] the time series data includes experimental measurement data, sensor data, and numerical simulation results;

[0011] Use a data-driven algorithm to extract the dynamic characteristics of the time series data, and preprocess the time series data based on the dynamic characteristics to generate corresponding sequence pre-data, where

[0012] the dynamic characteristics include the oscillation frequency, growth rate, and fluctuation of the power system;

[0013] Calculate the oscillation coefficient of the power system using the dynamic characteristics, select the differential equation form according to the comparison result between the oscillation coefficient and the oscillation coefficient threshold, determine the corresponding differential equation parameters, and construct the differential equation model corresponding to the power system;

[0014] Use the differential equation model to learn the sequence pre-data, generate simulation data within a preset time, compare it with the real data corresponding to the preset time, generate the prediction accuracy rate of the differential equation model, compare the prediction accuracy rate with the accuracy rate threshold, and determine whether to adjust the differential equation model according to the comparison result, where

[0015] the accuracy rate threshold is the lowest standard for the prediction accuracy rate of the differential equation model and is related to the driving performance of the power system.

[0016] Further, the step of collecting time series data of the power system includes:

[0017] Determine the measurement variables and sampling frequency of the power system, where

[0018] the measurement variables include physical variables and state variables;

[0019] Continuously sample the measurement variables at the sampling frequency to obtain the time series data;

[0020] Store the obtained time series data in a data storage device.

[0021] Furthermore, the steps of extracting kinetic features using a data-driven algorithm include:

[0022] Analyze the time series data using a data-driven algorithm to extract the corresponding feature vectors;

[0023] Map the feature vectors to an infinite-dimensional linear space;

[0024] Analyze the feature vectors using linear algebra tools to extract the corresponding kinetic features.

[0025] Furthermore, the steps of preprocessing time series data based on kinetic features include:

[0026] Perform standardization processing on the time series data based on the kinetic features and generate corresponding preprocessed data sequences;

[0027] Among them, the standardization processing is to divide the time series data according to the standard cutting rate;

[0028] The standard cutting rate is the cutting rate that the differential equation model can identify, and for single learning, the corresponding standard cutting rate is a single cutting rate.

[0029] Furthermore, the steps of calculating the oscillation coefficient of the dynamic system include:

[0030] Perform Fourier transform on the kinetic features and convert the kinetic features into frequency domain data;

[0031] Fit the frequency domain data through a sparse regression algorithm to obtain the dynamic spectrum corresponding to the frequency domain data;

[0032] Select the highest peak value of the dynamic spectrum and output the highest peak value as the oscillation coefficient.

[0033] Furthermore, when the oscillation coefficient of the dynamic system is greater than the oscillation coefficient threshold, select a high-order differential equation, and when the oscillation coefficient of the dynamic system is less than the oscillation coefficient threshold, select a linear differential equation.

[0034] Furthermore, the oscillation coefficient threshold is a critical value used to judge the oscillation characteristics of the dynamic system and is related to the driving performance of the dynamic system.

[0035] Furthermore, the steps of the differential equation model learning the preprocessed data sequences include:

[0036] Determine the corresponding differential equation parameters according to the differential equation form to obtain the corresponding differential equation model:

[0037] Input the preset time and the sequence pre-data into the differential equation model;

[0038] Calculate the output result of the differential equation model at the corresponding preset time through a numerical solution method, and output the output result as simulation data.

[0039] Further, collect the real data when the power system reaches the preset time, compare the real data with the simulation data to generate the prediction accuracy rate of the differential equation model, and compare the prediction accuracy rate with the accuracy threshold. When the prediction accuracy rate is greater than the accuracy threshold, do not adjust the differential equation model.

[0040] Further, when the prediction accuracy rate is less than the accuracy threshold, use the gradient descent method to adjust the differential equation parameters.

[0041] Compared with the prior art, the present invention collects the time series data of the power system, extracts the dynamic characteristics of the time series data, preprocesses the time series data based on the dynamic characteristics to generate the corresponding sequence pre-data, calculates the oscillation coefficient of the power system and selects the differential equation form, determines the corresponding differential equation parameters, constructs the differential equation model corresponding to the power system to learn the sequence pre-data, generates the simulation data within the preset time, and compares it with the real data corresponding to the preset time to generate the prediction accuracy rate of the differential equation model and compare it with the accuracy threshold to determine whether to adjust the differential equation model. It can learn the non-linear relationship from the time series data, objectively evaluate the prediction performance of the model, timely discover the deficiencies of the model, and construct a differential equation model that conforms to the actual situation.

[0042] Further, by determining the measurement variables of the power system, including physical variables and state variables, it can comprehensively reflect the operating state and physical characteristics of the power system, provide a rich and accurate data basis for subsequent modeling and analysis, and avoid model deviation caused by missing important variables. Selecting an appropriate sampling frequency can ensure that the collected data can reflect the dynamic changes of the system, and will not generate unnecessary data redundancy due to too high sampling frequency or lose important information due to too low sampling frequency. This helps to improve the quality and usability of the data.

[0043] Further, by using a data-driven algorithm to analyze the time series data, it can automatically discover hidden patterns and rules from a large amount of data without relying on prior physical model assumptions. This method can better adapt to the non-linear characteristics and dynamic changes of complex power systems, so as to extract more accurate feature vectors. This automatic adaptability enables the model to better handle different types of power systems and has stronger generalization ability.

[0044] Furthermore, by standardizing time series data based on kinetic characteristics and segmenting it according to the standard cutting rate, the quality and consistency of the data can be significantly improved, and the learning efficiency and prediction accuracy of the model can be enhanced. This process not only simplifies the model construction and adjustment process, but also improves the reliability and stability of the system, providing strong support for the modeling and analysis of dynamic systems.

[0045] Furthermore, through this process of calculating the oscillation coefficient of the dynamic system, key information reflecting the dynamic characteristics of the system can be effectively extracted from the kinetic characteristics, providing an important basis for the selection of differential equation models and the determination of parameters. This process not only improves the extraction accuracy of the oscillation coefficient, simplifies the model selection and parameter determination process, but also improves the interpretability and understandability of the system, providing strong support for the modeling and analysis of dynamic systems.

[0046] Furthermore, by selecting the form of the differential equation according to the comparison result between the oscillation coefficient and the oscillation coefficient threshold, the dynamic characteristics of the dynamic system can be more accurately matched, thereby constructing a more accurate differential equation model. This method not only improves the adaptability and accuracy of the model, simplifies the model construction and optimization process, but also improves the interpretability and understandability of the system.

[0047] Furthermore, through this step of learning the sequence pre-data by the differential equation model, the preprocessed data can be effectively used to train and verify the differential equation model, thereby improving the prediction accuracy and reliability of the model. This method not only improves the adaptability and accuracy of the model, simplifies the model construction and optimization process, but also improves the interpretability and understandability of the system, providing strong support for the modeling and analysis of dynamic systems.

[0048] Furthermore, through this method of model evaluation and adjustment based on the comparison between real data and simulation data, the parameters of the differential equation model can be dynamically optimized to ensure that its prediction accuracy meets the actual application requirements. This method not only improves the prediction accuracy and adaptability of the model, but also enhances the operation efficiency and stability of the system, providing strong support for the modeling, analysis and optimization of dynamic systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 It is a flowchart of a data-driven differential equation modeling method for a dynamic system according to an embodiment of the present invention;

[0050] Figure 2 It is a flowchart of collecting time series data of a dynamic system according to an embodiment of the present invention;

[0051] Figure 3 It is a flowchart of extracting kinetic characteristics using a data-driven algorithm according to an embodiment of the present invention;

[0052] Figure 4 Flow chart for calculating the oscillation coefficient of the power system in an embodiment of the present invention. Detailed implementation manners

[0053] In order to make the objectives and advantages of the present invention clearer, the present invention will be further described below in conjunction with embodiments; it should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0054] The preferred embodiments of the present invention will be described below with reference to the accompanying drawings. Those skilled in the art should understand that these embodiments are only used to explain the technical principles of the present invention and do not limit the protection scope of the present invention.

[0055] It should be noted that in the description of the present invention, the terms indicating directions or positional relationships such as "upper", "lower", "left", "right", "inner", "outer", etc. are based on the directions or positional relationships shown in the drawings. This is only for description and does not indicate or imply that the device or element must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the present invention.

[0056] In addition, it should be noted that in the description of the present invention, unless otherwise clearly specified and limited, the terms "installation", "connection", and "connection" should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integral connection; it can be a mechanical connection or an electrical connection; it can be directly connected or indirectly connected through an intermediate medium, and it can be the communication inside two elements. For those skilled in the art, the specific meanings of the above terms in the present invention can be understood according to specific situations.

[0057] Please refer to Figure 1 As shown, it is a flow chart of a data-driven differential equation modeling method for a power system in an embodiment of the present invention, including:

[0058] Step S1, collect the time series data of the power system, where

[0059] The time series data includes experimental measurement data, sensor data, and numerical simulation results;

[0060] Step S2, use a data-driven algorithm to extract the dynamic characteristics of the time series data, and preprocess the time series data based on the dynamic characteristics to generate corresponding pre-data sequences, where

[0061] The dynamic characteristics include the oscillation frequency, growth rate, and fluctuation of the power system;

[0062] Step S3: Calculate the oscillation coefficient of the dynamic system using the kinetic characteristics. Select the form of the differential equation according to the comparison result between the oscillation coefficient and the oscillation coefficient threshold, and determine the corresponding differential equation parameters to construct the differential equation model corresponding to the dynamic system.

[0063] Step S4: Use the differential equation model to learn the sequence pre-data, generate simulation data within a preset time, compare it with the real data corresponding to the preset time, generate the prediction accuracy rate of the differential equation model, and compare the prediction accuracy rate with the accuracy threshold. Determine whether to adjust the differential equation model according to the comparison result. Among them,

[0064] The accuracy threshold is the minimum standard of the prediction accuracy rate of the differential equation model and is related to the driving performance of the dynamic system.

[0065] By collecting the time series data of the dynamic system, extracting the kinetic characteristics of the time series data, preprocessing the time series data based on the kinetic characteristics to generate the corresponding sequence pre-data, calculating the oscillation coefficient of the dynamic system and selecting the form of the differential equation, determining the corresponding differential equation parameters, constructing the differential equation model corresponding to the dynamic system to learn the sequence pre-data, generating simulation data within a preset time, comparing it with the real data corresponding to the preset time, generating the prediction accuracy rate of the differential equation model and comparing it with the accuracy threshold, and determining whether to adjust the differential equation model, it is possible to learn the non-linear relationship from the time series data, objectively evaluate the prediction performance of the model, timely discover the deficiencies of the model, and construct a differential equation model that conforms to the actual situation.

[0066] Please refer to Figure 2 as shown, which is the flowchart of collecting the time series data of the dynamic system in the embodiment of the present invention, including:

[0067] Step S11: Determine the measurement variables and sampling frequency of the dynamic system. Among them,

[0068] The measurement variables include physical variables and state variables.

[0069] Step S12: Continuously sample the measurement variables according to the sampling frequency to obtain the time series data.

[0070] Step S13: Store the obtained time series data in the data storage device.

[0071] In specific implementation, the obtained time series data is stored in a data storage device, facilitating unified management and maintenance of the data. This makes the data storage more secure and reliable, and also facilitates subsequent data query and analysis. The stored data can be traced and verified at any time, which is crucial for model verification and optimization. When the model needs to be adjusted or verified, the original data can be directly obtained from the data storage device to ensure the authenticity and integrity of the data.

[0072] High-quality time series data is the basis for data-driven modeling. Through a standardized data acquisition and storage process, accurate and reliable data support can be provided for subsequent dynamic system differential equation modeling, thereby improving the accuracy and reliability of the model. By analyzing and modeling time series data, potential problems and fault hazards in the system can be detected in a timely manner, and measures can be taken in advance for prevention and repair, thereby enhancing the reliability and stability of the system.

[0073] By determining the measurement variables of the dynamic system, including physical variables and state variables, the operating state and physical characteristics of the dynamic system can be comprehensively reflected, providing a rich and accurate data basis for subsequent modeling and analysis, and avoiding model deviation caused by missing important variables. Selecting an appropriate sampling frequency can ensure that the collected data can reflect the dynamic changes of the system, and at the same time, it will not generate unnecessary data redundancy due to too high sampling frequency or lose important information due to too low sampling frequency. This helps to improve the quality and usability of the data.

[0074] Please refer to Figure 3 as shown, which is a flowchart of using a data-driven algorithm to extract dynamic characteristics in an embodiment of the present invention, including:

[0075] Step S21, using a data-driven algorithm to analyze the time series data and extract the corresponding feature vectors;

[0076] Step S22, mapping the feature vectors to an infinite-dimensional linear space;

[0077] Step S23, using linear algebra tools to analyze the feature vectors and extract the corresponding dynamic characteristics.

[0078] In specific implementation, machine learning or deep learning algorithms (such as principal component analysis PCA, autoencoder, long short-term memory network LSTM, etc.) are used to analyze the time series data. These algorithms can automatically learn the internal structure and patterns of the data from the data. The algorithms will extract the key information in the time series data to form feature vectors. These feature vectors are low-dimensional representations of the data, which can capture the main change trends and dynamic characteristics in the data.

[0079] Through certain mathematical transformations (such as kernel methods, Hilbert space mapping, etc.), the feature vectors are mapped from the original finite-dimensional space to an infinite-dimensional linear space. This mapping can transform non-linear problems into linear problems for subsequent analysis. In the infinite-dimensional linear space, the tools and theories of linear algebra can be used to process data, enabling complex non-linear relationships to be understood and analyzed in a simpler and more intuitive way.

[0080] Use methods in linear algebra (such as Singular Value Decomposition SVD, Eigenvalue Decomposition, etc.) to analyze the mapped feature vectors. These tools can effectively extract the main components and dynamic characteristics in the data. The finally extracted dynamic characteristics include oscillation frequency, growth rate, fluctuation conditions, etc., which can directly reflect the operating state and dynamic behavior of the dynamic system. The data-driven algorithm can handle complex non-linear relationships, making the extracted dynamic characteristics better reflect the actual dynamic characteristics of the dynamic system, thereby improving the adaptability of the model to different types of dynamic systems. The extracted dynamic characteristics can be directly used for the construction and optimization of differential equation models, accelerating the modeling and adjustment speed and improving the modeling efficiency.

[0081] Mapping the feature vectors to an infinite-dimensional linear space can transform complex non-linear problems into linear problems for processing. In the infinite-dimensional linear space, linear algebra tools can be used to more effectively analyze the feature vectors and extract the corresponding dynamic characteristics, such as oscillation frequency, growth rate, and fluctuation conditions. This mapping method can make full use of the theories and tools of linear algebra to improve the accuracy and efficiency of feature extraction. For dynamic systems with complex dynamic characteristics, traditional physics-based model feature extraction methods may be difficult to accurately describe their behavior. The data-driven algorithm can effectively extract the dynamic characteristics reflecting the system's dynamic characteristics by learning the essential laws from the data, thus providing support for constructing more accurate differential equation models.

[0082] By using data-driven algorithms to analyze time series data, hidden patterns and laws can be automatically discovered from a large amount of data without relying on prior physical model assumptions. This method can better adapt to the non-linear characteristics and dynamic changes of complex dynamic systems, thereby extracting more accurate feature vectors. This automatic adaptability enables the model to better handle different types of dynamic systems and has stronger generalization ability. After mapping the feature vectors to an infinite-dimensional linear space and using linear algebra tools for analysis, dynamic characteristics can be extracted more efficiently, improving the accuracy and efficiency of feature extraction.

[0083] Specifically, the steps for preprocessing time series data based on dynamic characteristics include:

[0084] Standardize the time series data based on dynamic characteristics and generate corresponding sequence pre-data;

[0085] Among them, the standardization process is to divide the time series data according to the standard cutting rate;

[0086] The standard cutting rate is the cutting rate that the differential equation model can recognize, and for single learning, the corresponding standard cutting rate is a single cutting rate.

[0087] In specific implementation, the time series data is converted into a unified format or range so that the differential equation model can process these data more effectively. This usually includes operations such as normalizing, denoising, and smoothing the data. The standard cutting rate refers to the frequency or time interval at which the time series data is divided into several segments, and this cutting rate is recognizable by the differential equation model. For example, if the model needs to learn at a fixed time step, then the standard cutting rate is this fixed time step. For a single learning process, the standard cutting rate is a fixed value, that is, a single cutting rate. This means that the length of each data segment processed is the same, which helps the model maintain consistency and stability during the learning process. Through the standardization process, the time series data is converted into a unified format, eliminating the differences in dimension, range, and time step of the data. This enables data from different sources to be directly compared and analyzed, improving the consistency and comparability of the data. The standardized data reduces the number of parameters that the model needs to adjust when processing the data. This makes the model construction and adjustment process simpler and more straightforward, reducing the difficulty of modeling and the complexity and error of the model when processing data in different formats.

[0088] By standardizing the time series data based on dynamic characteristics and dividing it according to the standard cutting rate, the quality and consistency of the data can be significantly improved, and the learning efficiency and prediction accuracy of the model can be enhanced. This process not only simplifies the model construction and adjustment process but also improves the reliability and stability of the system, providing strong support for the modeling and analysis of dynamic systems.

[0089] Please refer to Figure 4 as shown, which is the flowchart for calculating the oscillation coefficient of the dynamic system in the embodiment of the present invention, including:

[0090] Step S31, perform Fourier transform on the dynamic characteristics and convert the dynamic characteristics into frequency domain data;

[0091] Step S32, fit the frequency domain data through a sparse regression algorithm to obtain the dynamic spectrum corresponding to the frequency domain data;

[0092] Step S33, select the highest peak of the dynamic spectrum and output the highest peak as the oscillation coefficient.

[0093] In a specific implementation, the Fourier transform is a mathematical method that converts a time-domain signal into a frequency-domain signal. By performing a Fourier transform on dynamic characteristics (such as the oscillation frequency, growth rate, and fluctuation in time-series data), these characteristics can be transformed from the time domain to the frequency domain, enabling a more intuitive observation of the frequency components of the signal. The transformed frequency-domain data contains frequency information of the original dynamic characteristics, such as the magnitude, distribution, and intensity of the oscillation frequency, etc.

[0094] Sparse regression algorithms (such as LASSO, Elastic Net, etc.) are algorithms that introduce regularization terms in the regression model. They can effectively select the most important features from a large number of features while suppressing the overfitting phenomenon. By fitting the frequency-domain data with a sparse regression algorithm, a dynamic spectrum can be obtained. This spectrum shows the intensity and distribution of different frequency components in the frequency-domain data. The dynamic spectrum can intuitively reflect the oscillation characteristics of the dynamic system. In the dynamic spectrum, the highest peak represents the frequency component with the largest intensity in the frequency-domain data, that is, the main oscillation frequency of the dynamic system. Outputting the highest peak as the oscillation coefficient can provide an important basis for the selection and parameter determination of the differential equation model. The magnitude and position of the oscillation coefficient can reflect the oscillation characteristics of the dynamic system, such as the intensity and frequency of the oscillation, etc.

[0095] By converting dynamic characteristics into frequency-domain data through the Fourier transform, the frequency components of the signal can be observed more intuitively, thereby improving the extraction accuracy of the oscillation coefficient. Sparse regression algorithms can effectively select the most important frequency components from the frequency-domain data while suppressing the overfitting phenomenon, further improving the extraction accuracy of the oscillation coefficient. Outputting the highest peak as the oscillation coefficient can provide a direct basis for the selection and parameter determination of the differential equation model. This simplifies the process of model selection and parameter determination and reduces the difficulty of modeling.

[0096] An accurate oscillation coefficient can provide an important reference for the decision-making and control of the system, helping to predict potential problems of the system in advance and taking corresponding control measures to improve the operating efficiency and reliability of the system.

[0097] Through this process of calculating the oscillation coefficient of the dynamic system, key information reflecting the dynamic characteristics of the system can be effectively extracted from the dynamic characteristics, providing an important basis for the selection and parameter determination of the differential equation model. This process not only improves the extraction accuracy of the oscillation coefficient, simplifies the model selection and parameter determination process, but also improves the interpretability and understandability of the system, providing strong support for the modeling and analysis of the dynamic system.

[0098] Specifically, when the oscillation coefficient of the dynamic system is greater than the oscillation coefficient threshold, a high-order differential equation is selected; when the oscillation coefficient of the dynamic system is less than the oscillation coefficient threshold, a linear differential equation is selected.

[0099] Specifically, the oscillation coefficient threshold is a critical value used to judge the oscillation characteristics of the dynamic system and is related to the driving performance of the dynamic system.

[0100] In a specific implementation, the oscillation coefficient is calculated through the above process (Fourier transform, sparse regression fitting of the dynamic spectrum, and selection of the highest peak value), which reflects the main oscillation characteristics of the dynamic system, such as the intensity and frequency of oscillation. This is a preset critical value used to determine whether the oscillation characteristics of the dynamic system are significant. This threshold is closely related to the driving performance of the dynamic system, and different dynamic systems may have different threshold settings.

[0101] When the oscillation coefficient is greater than the oscillation coefficient threshold, it indicates that the oscillation characteristics of the dynamic system are relatively significant, and there may be complex nonlinear dynamic behaviors. At this time, choosing a higher-order differential equation for modeling can better capture the complex dynamic characteristics of the system.

[0102] When the oscillation coefficient is less than the oscillation coefficient threshold, it indicates that the oscillation characteristics of the dynamic system are relatively weak, and the system may be closer to linear dynamic behavior. At this time, choosing a linear differential equation for modeling can simplify the model structure and improve the calculation efficiency.

[0103] Selecting an appropriate form of the differential equation according to the comparison result between the oscillation coefficient and the threshold can more accurately match the actual dynamic characteristics of the dynamic system. For a system with significant oscillation characteristics, a higher-order differential equation can better capture the nonlinear dynamics; for a system with weak oscillation characteristics, a linear differential equation can simplify the model structure and improve the adaptability and accuracy of the model. This method can automatically adapt to different types of dynamic systems and has strong generalization ability. Regardless of how the oscillation characteristics of the system change, the model can select an appropriate equation form according to the magnitude of the oscillation coefficient, thereby improving the applicability of the model in different scenarios.

[0104] By selecting the form of the differential equation according to the comparison result between the oscillation coefficient and the oscillation coefficient threshold, the dynamic characteristics of the dynamic system can be more accurately matched, thereby constructing a more accurate differential equation model. This method not only improves the adaptability and accuracy of the model, simplifies the model construction and optimization process, but also improves the interpretability and understandability of the system.

[0105] Specifically, the steps for the differential equation model to learn the sequence pre-data include:

[0106] Determine the corresponding differential equation parameters according to the form of the differential equation to obtain the corresponding differential equation model:

[0107] Input the preset time and the sequence pre-data into the differential equation model;

[0108] The output result of the differential equation model at the corresponding preset time is calculated by a numerical solution method, and the output result is output as simulation data.

[0109] In the specific implementation, for the selected differential equation form, its corresponding parameters need to be determined. These parameters can be determined by data fitting, optimization algorithms (such as least squares method, gradient descent method, etc.) or machine learning methods (such as neural networks). The parameter determination process is intended to enable the differential equation model to better fit the sequence data.

[0110] Preset time: The preset time refers to the time range or time point for which prediction is required.

[0111] Pre-sequence data: Pre-sequence data is the time series data that has been preprocessed (standardized, segmented, etc.) and will serve as the input of the differential equation model.

[0112] Input model: Input preset time and sequence data into the differential equation model to prepare for model solving and prediction.

[0113] Numerical solution methods (such as Euler's method, Runge-Kutta method, etc.) are commonly used methods for solving differential equations. These methods can convert continuous differential equations into discrete difference equations, which can then be solved on a computer. Through numerical solution methods, the output results of the differential equation model at a preset time can be calculated. These output results are the predicted values of the model, also known as simulation data.

[0114] By comparing the simulated data with the real data, the prediction performance and reliability of the model can be verified. If the simulated data is highly consistent with the real data, it means that the model has a high reliability. If there is a large difference between the simulated data and the real data, the model can be adjusted and optimized to improve the reliability of the model.

[0115] Through the steps of learning the sequence pre-data by this differential equation model, the pre-processed data can be effectively used to train and verify the differential equation model, thereby improving the prediction accuracy and reliability of the model. This method not only improves the adaptability and accuracy of the model, simplifies the model construction and optimization process, but also improves the interpretability and understanding of the system, providing strong support for the modeling and analysis of dynamic systems.

[0116] Specifically, real data when the power system reaches the preset time is collected, and the real data is compared with the simulation data to generate the prediction accuracy of the differential equation model, and the prediction accuracy is compared with the accuracy threshold. When the prediction accuracy is greater than the accuracy threshold, the differential equation model is not adjusted.

[0117] Specifically, when the prediction accuracy is less than the accuracy threshold, the gradient descent method is used to adjust the parameters of the differential equation.

[0118] In a specific implementation, at a preset time point, the real operation data of the power system is obtained through sensors, experimental measurements, or other data acquisition means. These data reflect the real state of the power system during actual operation. Ensure that the collected real data is synchronized with the simulation data in time for accurate comparison.

[0119] Calculate the error between the real data and the simulation data. Commonly used error metrics include mean square error (MSE), mean absolute error (MAE), etc. Calculate the prediction accuracy of the model according to the error metric. The accuracy can be expressed as the reciprocal of the error or other forms to reflect the accuracy of the model prediction.

[0120] The gradient descent method adjusts the model parameters step by step by calculating the gradient of the loss function (such as the sum of squared errors) to minimize the loss function.

[0121] Parameter adjustment: When the prediction accuracy is less than the accuracy threshold, the gradient descent method is used to adjust the parameters of the differential equation. The specific steps are as follows:

[0122] Define the loss function: Select a suitable loss function, such as mean square error (MSE).

[0123] Calculate the gradient: Calculate the gradient of the loss function with respect to the model parameters.

[0124] Update the parameters: Update the model parameters according to the gradient direction and the learning rate.

[0125] Iterative optimization: Repeat the above steps until the prediction accuracy reaches or exceeds the accuracy threshold.

[0126] Through this model evaluation and adjustment method based on the comparison of real data and simulation data, the parameters of the differential equation model can be dynamically optimized to ensure that its prediction accuracy meets the actual application requirements. This method not only improves the prediction accuracy and adaptability of the model, but also enhances the operation efficiency and stability of the system, providing strong support for the modeling, analysis, and optimization of the power system.

[0127] So far, the technical solution of the present invention has been described in combination with the preferred embodiments shown in the accompanying drawings. However, it is easy for those skilled in the art to understand that the protection scope of the present invention is obviously not limited to these specific embodiments. Without departing from the principle of the present invention, those skilled in the art can make equivalent changes or substitutions to the relevant technical features, and the technical solutions after these changes or substitutions will fall within the protection scope of the present invention.

[0128] The foregoing are only the preferred embodiments of the present invention and are not intended to limit the present invention; for those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A data-driven differential equation modeling method for a dynamic system, characterized in that, Including: Collecting time - series data of the power system, where the time - series data includes experimental measurement data, sensor data, and numerical simulation results; Using a data - driven algorithm to extract the dynamic characteristics of the time - series data, and pre - processing the time - series data based on the dynamic characteristics to generate corresponding pre - sequence data, where the dynamic characteristics include the oscillation frequency, growth rate, and fluctuation of the power system; Calculating the oscillation coefficient of the power system using the dynamic characteristics, selecting the form of the differential equation according to the comparison result between the oscillation coefficient and the oscillation - coefficient threshold, and determining the corresponding differential - equation parameters to construct the differential - equation model corresponding to the power system; Using the differential - equation model to learn the pre - sequence data, generating simulation data within a preset time, comparing it with the real data corresponding to the preset time, generating the prediction accuracy of the differential - equation model, and comparing the prediction accuracy with the accuracy threshold, and determining whether to adjust the differential - equation model according to the comparison result, where The accuracy threshold is related to the lowest standard of the prediction accuracy of the differential - equation model and the driving performance of the power system; The steps of calculating the oscillation coefficient of the power system include: Performing a Fourier transform on the dynamic characteristics and converting the dynamic characteristics into frequency - domain data; Fitting the frequency - domain data through a sparse regression algorithm to obtain the dynamic spectrum corresponding to the frequency - domain data; Selecting the highest peak value of the dynamic spectrum and outputting the highest peak value as the oscillation coefficient; When the oscillation coefficient of the power system is greater than the oscillation - coefficient threshold, a high - order differential equation is selected; when the oscillation coefficient of the power system is less than the oscillation - coefficient threshold, a linear differential equation is selected; The steps of the differential - equation model learning the pre - sequence data include: Determining the corresponding differential - equation parameters according to the form of the differential equation to obtain the corresponding differential - equation model: Inputting the preset time and the pre - sequence data into the differential - equation model; Calculating the output result of the differential - equation model at the corresponding preset time through a numerical solution method and outputting the output result as simulation data.

2. The data-driven differential equation modeling method for a power system according to claim 1, wherein The steps of collecting time - series data of the power system include: Determining the measurement variables and sampling frequency of the power system, where the measurement variables include physical variables and state variables; Continuously sampling the measurement variables at the sampling frequency to obtain the time - series data; Storing the obtained time - series data in a data - storage device.

3. The data-driven dynamic system differential equation modeling method according to claim 2, characterized in that The steps of using a data - driven algorithm to extract dynamic characteristics include: Using a data - driven algorithm to analyze the time - series data and extract the corresponding feature vectors; Mapping the feature vectors to an infinite - dimensional linear space; Using linear - algebra tools to analyze the feature vectors and extract the corresponding dynamic characteristics.

4. The data-driven dynamic system differential equation modeling method according to claim 3, wherein The steps of pre - processing the time - series data based on the dynamic characteristics include: Normalize the time series data based on the kinetic characteristics and generate corresponding pre-sequence data; wherein, the normalization process is to segment the time series data according to a standard cutting rate. The standard cutting rate is the cutting rate that can be recognized by the differential equation model, and for single learning, the corresponding standard cutting rate is a single cutting rate.

5. The data-driven dynamic system differential equation modeling method according to claim 4, characterized in that The oscillation coefficient threshold is a critical value used to judge the oscillation characteristics of the dynamic system and is related to the driving performance of the dynamic system.

6. The data-driven dynamic system differential equation modeling method according to claim 5, wherein Collect the real data when the dynamic system reaches the preset time, compare the real data with the simulation data to generate the prediction accuracy of the differential equation model, and compare the prediction accuracy with the accuracy threshold. When the prediction accuracy is greater than the accuracy threshold, do not adjust the differential equation model.

7. The data-driven differential equation modeling method for a power system according to claim 6, wherein When the prediction accuracy is less than the accuracy threshold, adjust the differential equation parameters using the gradient descent method.

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