A frequency response characteristic decomposition method considering spatial distribution difference

Abstract

The application discloses a frequency response characteristic decomposition method considering spatial distribution difference, and relates to the technical field of power system transient frequency stability analysis. The technical problems that the existing method cannot effectively decouple global components and local relative oscillation components in transient frequency response and cannot accurately identify different oscillation characteristics of local components of nodes are solved. The technical scheme comprises the following steps: S1, a global component screening model based on an improved Pearson similarity is constructed; S2, an identification model of local relative oscillation components based on multivariate variational modal decomposition is constructed, and a unified oscillation center constraint is introduced; S3, the local relative oscillation components of nodes are decomposed into several eigenmode function components with different oscillation centers; S4, the effectiveness of screening of global components based on the improved Pearson similarity is verified; and S5, the multivariate variational modal decomposition is verified. The application realizes accurate decomposition of different node frequency response difference characteristics.

Description

Technical Field

[0001] This invention relates to the field of transient frequency stability analysis technology for power systems, and in particular to a frequency response characteristic decomposition method that considers spatial distribution differences. Background Technology

[0002] With the increasing penetration of new energy sources, the spatial distribution differences in the frequency response of power systems are becoming increasingly significant. Therefore, how to characterize the differences in the spatial distribution of node frequency response has become one of the key issues in the field of transient stability analysis.

[0003] Existing methods for extracting the spatial distribution differences of frequency response mainly include analytical modeling and data-driven methods. In analytical modeling, the frequency divider theory proposed by Milano et al. established for the first time an algebraic relationship between node frequency and generator speed, revealing the propagation mechanism of frequency response in the power grid. This theory simplifies the frequency response of complex networks into an algebraic relationship based on network topology, provides an analytical expression for node frequency, and achieves a quantitative description of the spatial distribution of node frequency, providing an important theoretical tool for understanding frequency spatial distribution. Building on this, some scholars have proposed frequency-dependent models for simulating the spatial distribution of frequency in transient stability analysis. This model, based on frequency divider theory, dynamically corrects frequency-related component parameters during transient processes. However, this type of method focuses on the algebraic expression of frequency and fails to deeply reveal the formation mechanism of frequency spatial distribution. Therefore, some scholars have further analyzed the frequency response from a modal perspective, developing frequency mode decomposition methods. Some literature has proposed defining the concept of modal frequency in renewable energy power systems, decomposing the frequency response into a common-mode component reflecting global consistency and several differential-mode components characterizing relative oscillations between nodes. It points out that the differential-mode components, representing relative frequency oscillations between nodes, are the root cause of spatial frequency distribution differences. Addressing the difficulty in analyzing differential-mode frequencies in renewable energy power systems, a frequency response decomposition method based on unified structure simplified modeling and quadratic eigenvalue analysis has been proposed. This research reveals the formation mechanism of frequency spatial distribution from a modal perspective, providing a theoretical basis for the quantitative analysis of frequency spatial distribution. However, such analytical methods rely on accurate equipment models and small-signal assumptions, making them difficult to apply to the highly nonlinear scenarios caused by high-proportion renewable energy integration.

[0004] To overcome the dependence of analytical methods on the accuracy of equivalent models, scholars both domestically and internationally have begun to explore data-driven frequency spatial distribution feature extraction methods. Data-driven frequency feature extraction methods mainly consist of two steps: feature extraction and frequency feature identification. In frequency signal feature extraction, existing research has utilized time-frequency analysis methods such as wavelet transform, Fourier transform, and Variational Mode Decomposition (VMD) to analyze frequency signals. Data-driven methods primarily analyze simulated curves or measured power grid data. Some literature proposes a node local feature extraction method based on ensemble empirical mode decomposition and fast Fourier transform. Addressing the mode aliasing problem inherent in this method, some scholars have further introduced VMD into frequency signal analysis. VMD transforms frequency signal decomposition into a variational constrained optimization problem, forcing each IMF component to exhibit narrowband characteristics in the frequency domain, thus mathematically overcoming the limitations of EEMD, which relies on empirical envelopes to extract components. Using VMD to extract frequency features can yield high-dimensional features of power grid frequency characteristics in different regions across different frequencies, providing a theoretical foundation for subsequent research. However, the above studies all extract steady-state frequency features. In transient scenarios, the global consistent evolution trend with common features of each node can easily obscure the local relative components that characterize the differences between nodes, making it difficult to effectively identify the distribution characteristics between node frequencies.

[0005] In summary, existing research has made significant progress in feature extraction and analysis of the spatial distribution characteristics of frequency response, but the following shortcomings still exist: On the one hand, analytical modeling methods rely on precise parameters, and the large-scale integration of new energy sources leads to errors; on the other hand, existing data-driven methods mostly focus on extracting frequency features of different regions from steady-state frequency signals, but in transient processes, since the global component reflecting the consistent evolution trend of each node in the frequency signal is dominant, it often masks the local relative oscillation components that characterize the differences between nodes, and there is also the influence of noise, making it difficult to effectively extract the distribution characteristics between nodes.

[0006] Therefore, effectively decoupling the global components from the local relative oscillating components is the primary prerequisite for transient frequency feature extraction. Meanwhile, considering that local components are essentially the time-domain superposition of multiple non-stationary oscillation modes, existing methods struggle to effectively identify the different oscillation characteristics of local components at each node during transient processes. How to effectively identify the distribution characteristics of the frequency response of each node under transient conditions becomes crucial for accurately quantifying frequency spatial characteristics, and further research is needed on the effective extraction, identification, and quantification analysis of the differences in node frequency responses. Summary of the Invention

[0007] To address the technical problems of effectively decoupling global components from local relative oscillation components and the difficulty of existing methods in effectively identifying the different oscillation characteristics of local components of each node during transient processes, this invention proposes a frequency response feature decomposition method that considers spatial distribution differences, thereby achieving accurate decomposition of the frequency response difference characteristics of different nodes.

[0008] The inventive concept of this invention is as follows: This invention is based on an improved Pearson similarity and MVMD decomposition method for extracting and identifying node frequency difference features, achieving accurate decomposition of frequency response difference features of different nodes. First, the similarity between the frequency curves of each node is analyzed based on the improved Pearson correlation coefficient. Second, the node with the highest similarity to other nodes is selected as the reference node, and its frequency curve is used as the global reference component. Based on this, the local relative oscillation components of each node are calculated. Finally, MVMD decomposition is performed on it, which can effectively identify the different oscillation characteristics of the local components of each node during the transient process. While removing noise, the local components of each node are decomposed into several IMF components corresponding to different oscillation characteristics, providing a feature basis for subsequent frequency prediction considering the spatial distribution difference features of nodes. The effectiveness of the proposed method is verified using the IEEE 39-node example.

[0009] To achieve the aforementioned objectives, the present invention employs the following technical solution: a frequency response feature decomposition method considering spatial distribution differences, comprising the following steps:

[0010] S1. Construct a global component screening model based on improved Pearson similarity. By calculating the improved Pearson similarity index between the frequency curves of each node, select the node with the highest overall similarity with other nodes as the reference node, and use the frequency response of the reference node as the global reference component to obtain the local relative oscillation component of each node.

[0011] S2. Construct an identification model for local relative oscillation components based on multivariate variational mode decomposition, and introduce a unified oscillation center constraint.

[0012] S3. The variational optimization problem is solved iteratively using the alternating direction multiplier method, and the local relative oscillation components of each node are decomposed into several eigenmode function components with different oscillation centers.

[0013] S4. Verify the effectiveness of global component filtering based on improved Pearson similarity;

[0014] S5. Verify the effectiveness of identifying the different oscillation characteristics of local components at each node through multivariate variational mode decomposition.

[0015] Further, in step S1, a spatial similarity matrix R based on the Pearson correlation coefficient is constructed to measure the degree of similarity between the frequency curve waveforms of different nodes. Specifically, it is assumed that the frequency signals of node i and node j are respectively... and Let its mean be:

[0016] ;

[0017] In the formula, The frequency signal at node i is represented by t, where t is the discrete-time sampling point. This represents the frequency signal of node j; This represents the mean of the frequency signal at node i. The mean of the frequency signal at node j is represented by T; T represents the total number of sampling points or the length of the time series of the frequency signal.

[0018] The ratio of the product of the covariance and the standard deviation between nodes i and j is the Pearson similarity coefficient between nodes i and j. :

[0019] ;

[0020] In the formula, Let Pearson's similarity coefficient be the coefficient between node i and node j. express and The Pearson correlation coefficient between them; This represents the covariance between the signals of node i and node j; This represents the standard deviation of the frequency signal at node i; This represents the standard deviation of the frequency signal at node j; The value range of is [-1, 1], and different values ​​correspond to different properties. If If >0, the frequency signals of node i and node j are positively correlated and show the same trend; if If the frequency signals of node i and node j tend to 1, then the frequency signals of node i and node j are strongly positively correlated; if <0, the frequency signals of node i and node j are negatively correlated and have opposite trends; if If = 0, then the frequency signals of node i and node j have no linear correlation.

[0021] Therefore, the spatial similarity matrix R, which characterizes the similarity between all nodes, is:

[0022] ;

[0023] In the formula, represents a spatial similarity matrix, which is used to characterize the Pearson similarity of frequency curves between all nodes; represents an element in the matrix , which represents the Pearson correlation coefficient of frequency curves between node i and node j; N is the number of system nodes, since the spatial similarity matrix is symmetric, and the diagonal elements are always 1;

[0024] Extract the upper triangular elements with i<j from the spatial similarity matrix R, and calculate its average value :

[0025] ;

[0026] In the formula, N represents the total number of nodes in the power system, represents the Pearson similarity coefficient between node i and node j, which reflects the average similarity of the frequency curve of node i in waveform shape to all remaining nodes except node i; The larger is, the more consistent the waveform trend of node i is with all remaining nodes except node i. On the basis of the Pearson correlation coefficient, Euclidean distance is introduced as a measure of amplitude consistency to reflect the amplitude proximity of frequency curves between nodes. Euclidean distance quantitatively describes the overall deviation of the numerical values of two curves by calculating the square root of the sum of squared differences of numerical values at corresponding moments of two sequences. The smaller the Euclidean distance, the closer the numerical values of the two curves are, and the higher the amplitude consistency. The Euclidean distance between node i and node j is defined as:

[0027] ;

[0028] In the formula, represents the norm of the difference between the frequency curves of node i and node j, represents the Euclidean distance between node i and node j ; is a summation symbol, which means cumulative summation for t from 1 to T;

[0029] Thus, the Euclidean distance matrix D is constructed:

[0030] ;

[0031] In the formula, represents the Euclidean distance of the frequency curves between node i and node j. The diagonal element is 0, which means the distance between a node and itself is zero. The Euclidean distance matrix is a symmetric matrix, that is ;

[0032] To measure the overall similarity of node i with the other nodes in terms of amplitude, the average Euclidean distance of node i is defined. :

[0033] ;

[0034] In the formula, This represents the average Euclidean distance in amplitude between the frequency curve of node i and all other nodes except node i. The smaller the distance, the closer the frequency curve of node i is to the other nodes in amplitude, that is, the higher the amplitude consistency.

[0035] Based on the aforementioned spatial similarity matrix and Euclidean distance matrix, an improved Pearson similarity index is constructed that comprehensively considers both shape similarity and amplitude consistency. As an overall similarity index for node i:

[0036] ;

[0037] In the formula, The average Pearson similarity of node i represents the degree of similarity in waveform trends; the larger the value, the more consistent the shapes. , which is a weighting coefficient used to balance the contributions of the average Pearson similarity and the average Euclidean distance; This represents the average Euclidean distance of node i, used to measure the magnitude of amplitude deviation. The smaller the value, the closer the amplitudes are.

[0038] Selecting the improved Pearson similarity index The maximum node m is used as the reference node, and the frequency response of the node is... As a global reference component, the local relative oscillation components of each node are obtained by subtracting the frequency signal of the reference node m from the frequency signal of each node, as shown in the following equation:

[0039] (9);

[0040] In the formula, This represents the frequency response of the reference node m. This represents the original frequency signal of node i.

[0041] Further, in step S2, constructing an identification model for local relative oscillation components based on multivariate variational mode decomposition (MVMD) is a node frequency difference feature extraction method based on MVMD decomposition. This transforms the decomposition process of the local relative oscillation components of each node into variational constraint optimization in the frequency domain, with the optimization objective being to minimize the sum of the bandwidths of each component within its corresponding frequency band. Through iterative optimization, the local relative oscillation components of each node are decomposed into multiple narrowband IMF components with different oscillation centers. This includes the following steps:

[0042] S21. Assume the local relative oscillation component frequency signals of N nodes. It consists of K IMF components with finite bandwidth. Each IMF component corresponds to an oscillating component with a different rate of change in the node frequency response, and they are concentrated around a certain oscillation center. Nearby, and applying a unified oscillation center constraint to IMF components with similar oscillation characteristics across multiple nodes, the IMF components obtained from the frequency signal decomposition of each node are... As shown in the following formula:

[0043] (10);

[0044] In the formula, This is the time-domain representation of the k-th IMF component, where K represents the preset total number of IMF components, k represents the node index, and t represents the discrete-time sampling point. First, each IMF component... The signal is transformed into a complex signal using the Hilbert transform, and then converted into the corresponding analytic signal. As shown in the following formula:

[0045] (11);

[0046] In the formula, for The real part of the expression is obtained after the Hilbert transform. for The imaginary part of the signal, analytic signal The amplitude and phase information of each IMF component are separated.

[0047] Introducing the Oscillation Center Index Analyzing the signals of each IMF component The corresponding spectrum is along their common oscillation center. The oscillation components, originally distributed across different oscillation centers, are uniformly shifted to the vicinity of the zero frequency band and converted into baseband signals. The expression for the objective function of the multi-node system is as follows:

[0048] (12);

[0049] In the formula, Let represent the complex analytical form of the frequency signal corresponding to the nth node under the kth IMF component. This is a spectrum shifting operator used to shift the analytic signals of each IMF component at each node. The corresponding multi-node spectral matrix is ​​along its common oscillation center. The signal is shifted to analyze the spectral signals of IMF components with the same oscillation rate across multiple nodes. Here, k represents the number of IMF components, and n represents the number of nodes. To express differentiation, This represents the square of the L2 norm of a vector;

[0050] The constrained optimization problem of MVMD is expressed as:

[0051] (13);

[0052] In the formula, 'st' indicates the minimum value of equation (12), and 'st' represents the constraint condition.

[0053] Since the frequency signals of each node satisfy the reconstruction constraints, multiple sets of linear equality constraints corresponding to the total number of nodes are introduced. An augmented Lagrangian method is used to process these constraints, and the corresponding augmented Lagrangian function is expressed as follows:

[0054] (14);

[0055] In the formula, This represents the augmented Lagrange function. This represents the k-th IMF component. This represents the unified oscillation center corresponding to the k-th IMF component. Represents the Lagrange multipliers. This represents the analytic signal of the k-th IMF. The first term is the core objective function of the MVMD algorithm, which calculates the norm of the gradient function of the multi-node spectral matrix after spectral shifting. Its physical meaning is that the entire system has the same oscillation center. Under the unified constraints, minimize the spectral bandwidth of the IMF components. This is a penalty factor used to adjust bandwidth. The larger the value, the narrower the spectral bandwidth of the decomposed IMF components; the second term is a quadratic penalty term, which measures the residual between the sum of all extracted IMF components and the original local oscillation components.

[0056] Further, step S3 includes:

[0057] For the variational model of MVMD, the variables to be solved include each IMF component. , corresponding oscillation center and Lagrange multipliers Within the ADMM framework, by constructing an augmented Lagrangian function and employing an alternating minimization strategy, the original problem is decomposed into several iteratively solvable subproblems: In the (m+1)th iteration, it is assumed that the oscillation center of the previous iteration... With the Given constants, the IMF components at each node are minimized by minimizing the Lagrange function. The update of the k-th IMF component at this time can be expressed in the following optimized form:

[0058] (15);

[0059] In the formula, The Fourier transform form of the local relative oscillation component of node n. It represents the frequency domain sum of all IMF components at node n, except for the k-th IMF component; This represents half of the Lagrange multiplier corresponding to the k-th IMF component of node n; multiplier term To correct the operator's participation in the update of IMF components, the accumulated bias is fed back into the current IMF component, and the denominator term... This indicates that the components that cause the oscillations of each decomposed node to have similar speeds are concentrated at their respective oscillation centers. Within a narrow band, the IMF components of the frequency signals at each node are gradually corrected.

[0060] After updating based on the IMF components, the common oscillation center is further updated using the updated IMF components of each node. The update formula for the oscillation center is:

[0061] (16);

[0062] In the formula, This represents the unified oscillation center of the k-th IMF component in the (m+1)-th iteration. This represents the frequency domain representation of the k-th IMF component in the (m+1)-th iteration; This represents the total energy of the k-th IMF component of all nodes n; It represents the sum of the weighted spectral energies of the k-th IMF component of all nodes n.

[0063] Furthermore, in step S4, the solution includes the following steps:

[0064] S41: Set up the disturbance and collect data from each node after the disturbance.

[0065] An improved IEEE 39-node system model was constructed, and a load shedding disturbance was set at node 39. Transient frequency response data of each node were collected.

[0066] S42: Calculate the improved Pearson similarity index for each node.

[0067] Calculate the average Pearson correlation coefficient and average Euclidean distance of each node with all other nodes, select different weight coefficients, calculate the improved Pearson similarity index of each node, compare the results of the reference node under different weight coefficient values, and select the reference node.

[0068] S43: Using the selected reference node frequency response as the global reference component, calculate the local relative oscillation components of each node.

[0069] Furthermore, in step S5, the solution includes the following steps:

[0070] S51: Use the local relative oscillation components of each node as the input signal;

[0071] S52: Initialize each IMF component and its oscillation center, and solve iteratively using the alternating direction multiplier method until the convergence condition is met;

[0072] S53: Output the IMF components of different nodes.

[0073] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0074] 1. This invention can accurately identify the different oscillation characteristics of local components of each node, providing a characteristic basis for subsequent frequency stability analysis that takes into account the differences in the spatial distribution of nodes. It overcomes the problem in traditional variational mode decomposition methods where components with the same oscillation speed are assigned to different oscillation centers due to the lack of a unified oscillation center constraint.

[0075] 2. This invention constructs an improved Pearson similarity index that comprehensively considers waveform shape similarity and amplitude consistency by building a spatial similarity matrix and an Euclidean distance matrix. The node with the largest improved Pearson similarity index is selected as the reference node, and its frequency response is used as the global reference component. The local relative oscillation component is obtained by subtracting the global reference component from the frequency signal of each node. This effectively decouples the global component and the local relative oscillation component in the transient frequency response, overcoming the defect of existing methods that make it difficult to extract the local relative oscillation component because the global component dominates in the transient process.

[0076] 3. This invention introduces a unified oscillation center constraint to decompose the local relative oscillation components of each node into several eigenmode function components with different oscillation centers. This ensures that components with the same oscillation speed at each node are assigned to the same oscillation center, thereby effectively identifying the different oscillation characteristics of the local components of each node during the transient process. This overcomes the shortcomings of traditional variational mode decomposition methods, which assign components with the same oscillation speed at different nodes to different oscillation centers due to the independent application of oscillation center constraints to each node, and cannot compare the differences between nodes under the same oscillation mode. Attached Figure Description

[0077] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used together with the embodiments of the invention to explain the invention and do not constitute a limitation thereof.

[0078] Figure 1 This is a graph showing the frequency curves of each node in this invention.

[0079] Figure 2 is a heatmap of the Pearson similarity matrix in this invention.

[0080] Figure 3 is a heat map of the Euclidean distance matrix in this invention.

[0081] Figure 4 shows the curves of the local relative oscillation components of all nodes in this invention.

[0082] Figure 5 These are the time-domain and frequency-domain plots of each IMF component under the traditional VMD decomposition in this invention.

[0083] Figure 6 These are the time-domain and frequency-domain plots of the IMF components decomposed by MVMD in this invention. Detailed Implementation

[0084] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. Of course, the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0085] Example 1: See Figure 1 and Figure 6 The present invention provides a technical solution for frequency response feature decomposition considering spatial distribution differences, comprising the following steps:

[0086] S1. Construct a global component screening model based on improved Pearson similarity. By calculating the improved Pearson similarity index between the frequency curves of each node, select the node with the highest overall similarity with other nodes as the reference node, and use the frequency response of the reference node as the global reference component to obtain the local relative oscillation component of each node.

[0087] S2. Construct an identification model for local relative oscillation components based on multivariate variational mode decomposition, and introduce a unified oscillation center constraint.

[0088] S3. The variational optimization problem is solved iteratively using the alternating direction multiplier method, and the local relative oscillation components of each node are decomposed into several eigenmode function components with different oscillation centers.

[0089] S4. Verify the effectiveness of global component filtering based on improved Pearson similarity;

[0090] S5. Verify the effectiveness of identifying the different oscillation characteristics of local components at each node through multivariate variational mode decomposition.

[0091] Further, in step S1, a spatial similarity matrix R based on the Pearson correlation coefficient is constructed to measure the degree of similarity between the frequency curve waveforms of different nodes. Specifically, it is assumed that the frequency signals of node i and node j are respectively... and Let its mean be:

[0092] ;

[0093] In the formula, The frequency signal at node i is represented by t, where t is the discrete-time sampling point. This represents the frequency signal of node j; This represents the mean of the frequency signal at node i. The mean of the frequency signal at node j is represented by T; T represents the total number of sampling points or the length of the time series of the frequency signal.

[0094] The ratio of the product of the covariance and the standard deviation between nodes i and j is the Pearson similarity coefficient between nodes i and j. :

[0095] ;

[0096] In the formula, Let Pearson's similarity coefficient be the coefficient between node i and node j. express and The Pearson correlation coefficient between them; This represents the covariance between the signals of node i and node j; This represents the standard deviation of the frequency signal at node i; This represents the standard deviation of the frequency signal at node j; The value range of is [-1, 1], and different values ​​correspond to different properties. If If >0, the frequency signals of node i and node j are positively correlated and show the same trend; if If the frequency signals of node i and node j tend to 1, then the frequency signals of node i and node j are strongly positively correlated; if <0, the frequency signals of node i and node j are negatively correlated and have opposite trends; if If = 0, then the frequency signals of node i and node j have no linear correlation.

[0097] Therefore, the spatial similarity matrix R, which characterizes the similarity between all nodes, is:

[0098] ;

[0099] In the formula, represents the spatial similarity matrix, which is used to characterize the Pearson similarity of frequency curves between all nodes; represents the element in matrix , which is the Pearson correlation coefficient of frequency curves between node i and node j; N is the number of system nodes, since the spatial similarity matrix is symmetric, and its diagonal elements are always 1;

[0100] extract upper triangular elements satisfying i<j from the spatial similarity matrix R, and calculate its average value :

[0101] ;

[0102] In the formula, N represents the total number of nodes in the power system, represents the Pearson similarity coefficient between node i and node j, which reflects the average similarity degree between the frequency curve of node i and all remaining nodes except node i in terms of waveform shape; The larger the value is, the more consistent the waveform trend of node i is with all remaining nodes except node i. Based on the Pearson correlation coefficient, Euclidean distance is introduced as a measure of amplitude consistency to reflect the amplitude proximity degree of frequency curves between nodes. Euclidean distance quantitatively describes the overall deviation of the numerical values of two curves by calculating the square root of the sum of squared differences of numerical values at corresponding moments of two sequences. A smaller Euclidean distance indicates that the numerical values of the two curves are closer and the amplitude consistency is higher. The Euclidean distance between node i and node j is defined as :

[0103] ;

[0104] In the formula, represents the norm of the difference between frequency curves of node i and node j, represents the Euclidean distance between node i and node j ; is the summation symbol, which represents cumulative summation of t from 1 to T;

[0105] thus the Euclidean distance matrix D is constructed:

[0106] ;

[0107] In the formula, Let represent the Euclidean distance between the frequency curves of node i and node j. A diagonal element of 0 indicates that the distance between a node and itself is zero. The Euclidean distance matrix is ​​a symmetric matrix. ;

[0108] To measure the overall similarity of node i with the other nodes in terms of amplitude, the average Euclidean distance of node i is defined. :

[0109] ;

[0110] In the formula, This represents the average Euclidean distance in amplitude between the frequency curve of node i and all other nodes except node i. The smaller the distance, the closer the frequency curve of node i is to the other nodes in amplitude, that is, the higher the amplitude consistency.

[0111] Based on the aforementioned spatial similarity matrix and Euclidean distance matrix, an improved Pearson similarity index is constructed that comprehensively considers both shape similarity and amplitude consistency. As an overall similarity index for node i:

[0112] ;

[0113] In the formula, The average Pearson similarity of node i represents the degree of similarity in waveform trends; the larger the value, the more consistent the shapes. , which is a weighting coefficient used to balance the contributions of the average Pearson similarity and the average Euclidean distance; This represents the average Euclidean distance of node i, used to measure the magnitude of amplitude deviation. The smaller the value, the closer the amplitudes are.

[0114] Selecting the improved Pearson similarity index The maximum node m is used as the reference node, and the frequency response of the node is... As a global reference component, the local relative oscillation components of each node are obtained by subtracting the frequency signal of the reference node m from the frequency signal of each node, as shown in the following equation:

[0115] (9);

[0116] In the formula, This represents the frequency response of the reference node m. This represents the original frequency signal of node i.

[0117] Further, in step S2, constructing an identification model for local relative oscillation components based on multivariate variational mode decomposition (MVMD) is a node frequency difference feature extraction method based on MVMD decomposition. This transforms the decomposition process of the local relative oscillation components of each node into variational constraint optimization in the frequency domain, with the optimization objective being to minimize the sum of the bandwidths of each component within its corresponding frequency band. Through iterative optimization, the local relative oscillation components of each node are decomposed into multiple narrowband IMF components with different oscillation centers. This includes the following steps:

[0118] S21. Assume the local relative oscillation component frequency signals of N nodes. It consists of K IMF components with finite bandwidth. Each IMF component corresponds to an oscillating component with a different rate of change in the node frequency response, and they are concentrated around a certain oscillation center. Nearby, and applying a unified oscillation center constraint to IMF components with similar oscillation characteristics across multiple nodes, the IMF components obtained from the frequency signal decomposition of each node are... As shown in the following formula:

[0119] (10);

[0120] In the formula, This is the time-domain representation of the k-th IMF component, where K represents the preset total number of IMF components, k represents the node index, and t represents the discrete-time sampling point. First, each IMF component... The signal is transformed into a complex signal using the Hilbert transform, and then converted into the corresponding analytic signal. As shown in the following formula:

[0121] (11);

[0122] In the formula, for The real part of the expression is obtained after the Hilbert transform. for The imaginary part of the signal, analytic signal The amplitude and phase information of each IMF component are separated.

[0123] Introducing the Oscillation Center Index Analyzing the signals of each IMF component The corresponding spectrum is along their common oscillation center. The oscillation components, originally distributed across different oscillation centers, are uniformly shifted to the vicinity of the zero frequency band and converted into baseband signals. The expression for the objective function of the multi-node system is as follows:

[0124] (12);

[0125] In the formula, Let represent the complex analytical form of the frequency signal corresponding to the nth node under the kth IMF component. This is a spectrum shifting operator used to shift the analytic signals of each IMF component at each node. The corresponding multi-node spectral matrix is ​​along its common oscillation center. The signal is shifted to analyze the spectral signals of IMF components with the same oscillation rate across multiple nodes. Here, k represents the number of IMF components, and n represents the number of nodes. To express differentiation, This represents the square of the L2 norm of a vector;

[0126] The constrained optimization problem of MVMD is expressed as:

[0127] (13);

[0128] In the formula, 'st' indicates the minimum value of equation (12), and 'st' represents the constraint condition.

[0129] Since the frequency signals of each node satisfy the reconstruction constraints, multiple sets of linear equality constraints corresponding to the total number of nodes are introduced. An augmented Lagrangian method is used to process these constraints, and the corresponding augmented Lagrangian function is expressed as follows:

[0130] (14);

[0131] In the formula, This represents the augmented Lagrange function. This represents the k-th IMF component. This represents the unified oscillation center corresponding to the k-th IMF component. Represents the Lagrange multipliers. This represents the analytic signal of the k-th IMF. The first term is the core objective function of the MVMD algorithm, which calculates the norm of the gradient function of the multi-node spectral matrix after spectral shifting. Its physical meaning is that the entire system has the same oscillation center. Under the unified constraints, minimize the spectral bandwidth of the IMF components. This is a penalty factor used to adjust bandwidth. The larger the value, the narrower the spectral bandwidth of the decomposed IMF components; the second term is a quadratic penalty term, which measures the residual between the sum of all extracted IMF components and the original local oscillation components.

[0132] Further, step S3 includes:

[0133] For the variational model of MVMD, the variables to be solved include each IMF component. , corresponding oscillation center and Lagrange multipliers Within the ADMM framework, by constructing an augmented Lagrangian function and employing an alternating minimization strategy, the original problem is decomposed into several iteratively solvable subproblems: In the (m+1)th iteration, it is assumed that the oscillation center of the previous iteration... With the Given constants, the IMF components at each node are minimized by minimizing the Lagrange function. The update of the k-th IMF component at this time can be expressed in the following optimized form:

[0134] (15);

[0135] In the formula, The Fourier transform form of the local relative oscillation component of node n. It represents the frequency domain sum of all IMF components at node n, except for the k-th IMF component; This represents half of the Lagrange multiplier corresponding to the k-th IMF component of node n; multiplier term To correct the operator's participation in the update of IMF components, the accumulated bias is fed back into the current IMF component, and the denominator term... This indicates that the components that cause the oscillations of each decomposed node to have similar speeds are concentrated at their respective oscillation centers. Within a narrow band, the IMF components of the frequency signals at each node are gradually corrected.

[0136] After updating based on the IMF components, the common oscillation center is further updated using the updated IMF components of each node. The update formula for the oscillation center is:

[0137] (16);

[0138] In the formula, This represents the unified oscillation center of the k-th IMF component in the (m+1)-th iteration. This represents the frequency domain representation of the k-th IMF component in the (m+1)-th iteration; This represents the total energy of the k-th IMF component of all nodes n; It represents the sum of the weighted spectral energies of the k-th IMF component of all nodes n.

[0139] Furthermore, in step S4, the solution includes the following steps:

[0140] S41: Set up the disturbance and collect data from each node after the disturbance.

[0141] An improved IEEE 39-node system model was constructed, and a load shedding disturbance was set at node 39. Transient frequency response data of each node were collected.

[0142] S42: Calculate the improved Pearson similarity index for each node.

[0143] Calculate the average Pearson correlation coefficient and average Euclidean distance of each node with all other nodes, select different weight coefficients, calculate the improved Pearson similarity index of each node, compare the results of the reference node under different weight coefficient values, and select the reference node.

[0144] S43: Using the selected reference node frequency response as the global reference component, calculate the local relative oscillation components of each node.

[0145] Furthermore, in step S5, the solution includes the following steps:

[0146] S51: Use the local relative oscillation components of each node as the input signal;

[0147] S52: Initialize each IMF component and its oscillation center, and solve iteratively using the alternating direction multiplier method until the convergence condition is met;

[0148] S53: Output the IMF components of different nodes.

[0149] To verify the effectiveness of the node frequency difference feature extraction method based on improved Pearson similarity proposed in this invention, a case study was conducted based on the improved IEEE 39-node system model.

[0150] To verify the effectiveness of the proposed method, this embodiment uses the PSASP simulation platform to build an improved IEEE 39-node system containing wind turbines and photovoltaic units for simulation experiments. In the original IEEE 39 standard system, multiple 1.5MW doubly-fed induction generator (DFIG) wind turbines replace synchronous generators G06 and G08, and 1.5MW photovoltaic units replace synchronous generators G05 and G09 with the same capacity. All DFIG wind turbines are equipped with virtual inertia and droop control. Through these replacements, a high-proportion renewable energy power system with 6 synchronous generators and 4 renewable energy units is constructed, achieving a renewable energy output penetration rate of 38.97%, effectively simulating the operating characteristics of a novel power system with low inertia and weak damping.

[0151] Table 1 Parameter Table for Each Synchronous Generator Unit

[0152]

[0153] A simulation of the improved IEEE 39-node system model was performed. To simulate the system's operating state after an active power disturbance, this section sets the load shedding at node 39 to 880MW. Gaussian white noise was injected into each node. Since the initial values ​​of the random number generators corresponding to each node are different, the generated noise sequences are uncorrelated in the time domain, and the noise sequences of each node are independent of each other, thus simulating the independent measurement error environment of measuring points at different physical locations in a real wide-area measurement system. The doubly-fed wind turbines in each wind farm all use virtual inertia and droop control for frequency modulation. PMU devices are deployed at all nodes to record the frequency response data of each node at a sampling rate of 100Hz. The algorithms in this embodiment are all implemented based on Python programming.

[0154] To visually demonstrate the frequency response of each node after the disturbance, Figure 1 Frequency response curves for 39 nodes of the system after perturbation are presented. To quantify the spatial similarity characteristics of the frequency signals of each node, the original frequency signals of each node were first collected. For any two nodes, the linear correlation between them was calculated based on the Pearson correlation coefficient. Finally, the correlation coefficients of all node pairs were arranged in a 39×39 matrix and visualized using a heatmap, where red represents a correlation coefficient close to 1 and blue represents a correlation coefficient close to -1. Figure 2 The heatmap shown is a Pearson correlation coefficient matrix. From... Figure 2 As can be seen, the core area of ​​the heatmap is predominantly red, indicating a strong consistency between the local frequencies of most nodes within the system and the oscillation signals. This aligns with the fundamental physical law that power systems exhibit significant global synchronization characteristics. Simultaneously, it can be observed that the Pearson correlation coefficient in the area where new energy units are connected is lower than in the area dominated by traditional synchronous generators, mostly appearing yellow. This is because the equivalent inertia of the system decreases after new energy replaces synchronous generators, resulting in a greater rate of frequency change during disturbances. Therefore, the frequency response of new energy grid-connected nodes and their surrounding areas differs from that of traditional generator areas, manifested as a decrease in the Pearson correlation coefficient, reflecting the spatial distribution characteristics of the system's frequency response after new energy integration. However, a distinct blue-green patch appears in the area centered around nodes 39, 1, and 9, indicating that the correlation coefficients between these nodes and the remaining nodes are significantly lower. Node 39 is a load shedding node. Due to the power deficit and its proximity to node 37 of the new energy unit, it has the lowest Pearson similarity to the frequency response of the system reference node. Nodes 1, 9 and 39 are directly adjacent in the power grid topology. Due to the coupling effect of their power disturbances, their frequency synchronization decreases. The color difference in this local area also verifies the spatial coupling characteristic of the frequency responses of adjacent nodes in the topology.

[0155] While the Pearson correlation coefficient effectively characterizes the similarity of frequency waveforms at each node, it fails to reflect the closeness of frequency amplitudes at each node. This invention introduces a Euclidean distance matrix to measure the amplitude consistency of each node, the results of which are as follows: Figure 3 As shown in the diagram. Unlike the Pearson correlation coefficient, which indicates a higher similarity, Euclidean distance is a dissimilarity indicator. In the Euclidean distance matrix, the smaller the value, the higher the similarity between two nodes. It can be seen that nodes 39, 1, and 9 (disturbance sources / weak inertia / new energy nodes) have larger Euclidean distances than other nodes, appearing in blue. This indicates that not only do the frequency curves of these nodes differ significantly from the reference node, but their amplitudes also deviate significantly from the reference node. The Euclidean distance heatmap shows a significant low-value distribution in the central region. In the Euclidean distance metric, smaller and redder values ​​represent smaller amplitude differences between nodes and higher Euclidean distance similarity. The extremely low Euclidean distance between node pairs within the region indicates a high degree of consistency in their initial frequency change rate, maximum frequency deviation extreme, and quasi-steady-state frequency after disturbance. Furthermore, this high consistency reflects strong electrical coupling between nodes. This indicates that they are closely related in the system topology and have similar inertia support levels, resulting in a consistent global evolution trend in the frequencies of each node.

[0156] To reasonably balance waveform shape and amplitude consistency, this paper constructs an improved Pearson similarity index and conducts a sensitivity analysis on the weighting coefficients, the results of which are shown in Table 2. It can be observed that the weight of Euclidean distance gradually increases with the increase of the weighting coefficients. To ensure high correlation in waveform shape while maximizing amplitude consistency, the optimal reference node selected in this paper is node 4, which satisfies both high correlation and small amplitude deviation.

[0157] Table 2. Sensitivity analysis results with different weighting coefficients

[0158]

[0159] As shown in Table 2, if only the traditional Pearson correlation coefficient is used as the evaluation criterion, then node 18, which has the highest waveform shape correlation, will tend to be selected as the reference node. With the introduction of weighting coefficients, the evaluation system begins to consider the Euclidean distance between nodes, i.e., amplitude consistency. When the weighting coefficients are in the range [0.01, 0.40], the selected optimal reference node is consistently node 4. Therefore, although node 4 and node 18 show consistent waveform correlation (both reaching 0.9931), the average Euclidean distance of node 4 (0.1350) is smaller than that of node 18 (0.1384), indicating that node 4 is more representative in terms of amplitude similarity. Thus, the improved Pearson similarity index can effectively correct the shortcomings of the traditional Pearson index in amplitude deviation, selecting a reference node with more accurate physical meaning. Considering all these factors, this paper selects node 4 as the global reference node.

[0160] Using node 4 as the global reference node, the local relative oscillation components of each node are obtained by subtracting the global reference component from the frequency signal of each node, such as... Figure 3 , Figure 4 As shown, the local components exhibit different degrees of deviation between different nodes. These differences arise from the electrical distance between nodes, the location of the new energy unit connection, and the distance from the disturbance point, reflecting the spatial distribution differences in the frequency response of each node relative to the reference node 4 under disturbance. Nodes 39, 1, and 9 are near the disturbance source and deviate significantly from the frequency response of the reference node 4. Overall, the local components are mainly used to characterize the degree of deviation of the node frequency response from the system reference node.

[0161] 2) To verify the effectiveness of the node frequency difference feature identification method based on multivariate variational mode decomposition proposed in this invention, a numerical example was performed based on the improved IEEE 39-node system model.

[0162] To further verify the superiority of MVMD in handling multi-node frequency eigenvalue decomposition, this section compares the performance differences between the MVMD method and the traditional VMD method. First, the traditional VMD method is used for decomposition, yielding the time-domain and frequency-domain plots of each IMF component, as shown below. Figure 5 As shown, traditional VMD has significant limitations in multi-node frequency eigenvalue decomposition. The fluctuation periods of the IMF components in different oscillation modes differ significantly, such as in... Figure 5In the time-domain plot of the IMF1 component, it is evident that there are components with significantly different oscillation periods within the same IMF component. The oscillation frequency of the IMF1 component at some nodes is significantly lower or higher than that at other nodes, meaning that components with different oscillation speeds are forcibly grouped into the same IMF component. This is due to the problem that traditional VMD decomposes each node independently, leading to the forced classification of IMF components with different oscillation characteristics within each node into the same oscillation mode, thus making it impossible to compare the differences between nodes under the same oscillation mode.

[0163] To address the limitations of traditional VMD in multi-node decomposition, which fails to clearly separate components with varying oscillation rates at each node, this paper introduces MVMD decomposition. By incorporating a unified oscillation center constraint, joint decomposition of multi-node frequency signals can be achieved, as shown in the following figure. Figure 6 As shown. Comparison Figure 5 and Figure 6 The results show that MVMD decomposition has significant advantages. Figure 6 The oscillation periods of IMF1 to IMF5 decrease sequentially, while the oscillation center frequencies of each IMF component increase sequentially. IMF5 is a noise component, with its oscillation center mainly concentrated between 10-50 Hz and a peak around 25 Hz. As shown in the figure, a clear spectral stratification is evident from the low-frequency oscillations of IMF1 to the high-frequency oscillations of IMF5. In summary, the MVMD method can effectively identify the different oscillation characteristics of local components at each node during transient processes.

[0164] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A frequency response eigenvalue decomposition method considering spatial distribution differences, characterized in that, Includes the following steps S1. Construct a global component screening model based on improved Pearson similarity. By calculating the improved Pearson similarity index between the frequency curves of each node, select the node with the highest overall similarity with other nodes as the reference node, and use the frequency response of the reference node as the global reference component to obtain the local relative oscillation component of each node. S2. Construct an identification model for local relative oscillation components based on multivariate variational mode decomposition, and introduce a unified oscillation center constraint. S3. The variational optimization problem is solved iteratively using the alternating direction multiplier method, and the local relative oscillation components of each node are decomposed into several eigenmode function components with different oscillation centers. S4. Verify the effectiveness of global component filtering based on improved Pearson similarity; S5. Verify the effectiveness of identifying the different oscillation characteristics of local components at each node through multivariate variational mode decomposition.

2. The frequency response feature decomposition method considering spatial distribution differences according to claim 1, characterized in that, In step S1, a spatial similarity matrix R based on the Pearson correlation coefficient is constructed to measure the degree of similarity between the frequency curve waveforms of different nodes. Specifically, it is assumed that the frequency signals of node i and node j are respectively... and Let its mean be: ； In the formula, The frequency signal at node i is represented by t, where t is the discrete-time sampling point. This represents the frequency signal of node j; This represents the mean of the frequency signal at node i. This represents the mean value of the frequency signal at node j; T represents the total number of sampling points for the frequency signal or the length of the time series; The ratio of the product of the covariance and the standard deviation between nodes i and j is the Pearson similarity coefficient between nodes i and j. : ； In the formula, Let Pearson's similarity coefficient be the coefficient between node i and node j. express and The Pearson correlation coefficient between them; This represents the covariance between the signals of node i and node j; This represents the standard deviation of the frequency signal at node i; This represents the standard deviation of the frequency signal at node j; The value range of is [-1, 1], and different values ​​correspond to different properties. If >0, the frequency signals of node i and node j are positively correlated and show the same trend; if If the frequency signals of node i and node j tend to 1, then the frequency signals of node i and node j are strongly positively correlated; if <0, the frequency signals of node i and node j are negatively correlated and have opposite trends; if If =0, then the frequency signals of node i and node j have no linear correlation. Therefore, the spatial similarity matrix R, which characterizes the similarity between all nodes, is: ； In the formula, This represents the spatial similarity matrix, used to characterize the Pearson similarity of the frequency curves among all nodes; Representation matrix The elements in the table represent the Pearson correlation coefficients of the frequency curves between node i and node j; N is the number of nodes in the system. Extract upper triangular elements satisfying i<j from the spatial similarity matrix R, and calculate the average value thereof : ； In the formula, N represents the total number of nodes in the power system. The Pearson similarity coefficient represents the similarity between node i and node j. It reflects the average similarity in waveform shape between the frequency curve of node i and all other nodes; Define the Euclidean distance between node i and node j. for: ； In the formula, Let represent the norm of the difference in frequency curves between node i and node j. This represents the Euclidean distance between node i and node j. ; The summation symbol indicates that t is summed from 1 to T. Therefore, the Euclidean distance matrix D is constructed as follows: ； In the formula, Let represent the Euclidean distance between the frequency curves of node i and node j. A diagonal element of 0 indicates that the distance between a node and itself is zero. The Euclidean distance matrix is ​​a symmetric matrix. ; To measure the overall similarity of node i with the other nodes in terms of amplitude, the average Euclidean distance of node i is defined. : ； In the formula, This represents the average Euclidean distance in amplitude of the frequency curve of node i from all other nodes; Based on the aforementioned spatial similarity matrix and Euclidean distance matrix, an improved Pearson similarity index is constructed that comprehensively considers both shape similarity and amplitude consistency. As an overall similarity index for node i: ； In the formula, The average Pearson similarity of node i represents the degree of similarity in waveform trends; , which is a weighting coefficient used to balance the contributions of the average Pearson similarity and the average Euclidean distance; This represents the average Euclidean distance of node i, used to measure the magnitude of the magnitude deviation; Selecting the improved Pearson similarity index The maximum node m is used as the reference node, and the frequency response of the node is... As a global reference component, the local relative oscillation components of each node are obtained by subtracting the frequency signal of the reference node m from the frequency signal of each node, as shown in the following equation: (9)； In the formula, This represents the frequency response of the reference node m. This represents the original frequency signal of node i.

3. The frequency response feature decomposition method considering spatial distribution differences according to claim 1, characterized in that, In step S2, the identification model of local relative oscillation components based on multivariate variational mode decomposition is constructed. This is a node frequency difference feature extraction method based on multivariate variational mode decomposition (MVMD). The decomposition process of local relative oscillation components of each node is transformed into variational constraint optimization in the frequency domain. The optimization objective is to minimize the sum of the bandwidths of each component in the corresponding frequency band. Through iterative optimization, the local relative oscillation components of each node are decomposed into multiple narrowband IMF components with different oscillation centers. The steps include: S21. Assume the local relative oscillation component frequency signals of N nodes. It consists of K IMF components with finite bandwidth. Each IMF component corresponds to an oscillating component with a different rate of change in the node frequency response, and they are concentrated around a certain oscillation center. Nearby, and applying a unified oscillation center constraint to IMF components with similar oscillation characteristics across multiple nodes, the IMF components obtained from the frequency signal decomposition of each node are... As shown in the following formula: (10)； In the formula, This is the time-domain representation of the k-th IMF component, where K represents the preset total number of IMF components, k represents the node index, and t represents the discrete-time sampling point. First, each IMF component... The signal is transformed into a complex signal using the Hilbert transform, and then converted into the corresponding analytic signal. As shown in the following formula: (11)； In the formula, for The real part of the expression is obtained after the Hilbert transform. for The imaginary part of the signal, analytic signal The amplitude and phase information of each IMF component are separated. Introducing the Oscillation Center Index Analyzing the signals of each IMF component The corresponding spectrum is along their common oscillation center. The oscillation components, originally distributed across different oscillation centers, are uniformly shifted to the vicinity of the zero frequency band and converted into baseband signals. The expression for the objective function of the multi-node system is as follows: (12)； In the formula, Let represent the complex analytical form of the frequency signal corresponding to the nth node under the kth IMF component. This is a spectrum shifting operator used to shift the analytic signals of each IMF component at each node. The corresponding multi-node spectral matrix is ​​along its common oscillation center. The signal is shifted to enable analysis of the spectral signals of IMF components with the same oscillation rate across multiple nodes. Here, k represents the number of IMF components, and n represents the number of nodes. To express differentiation, This represents the square of the L2 norm of a vector; The constrained optimization problem of MVMD is expressed as: (13)； In the formula, This indicates that the minimum value of equation (12) is being sought, and st represents the constraint condition; Since the frequency signals of each node satisfy the reconstruction constraints, multiple sets of linear equality constraints corresponding to the total number of nodes are introduced. An augmented Lagrangian method is used to process these constraints, and the corresponding augmented Lagrangian function is expressed as follows: (14)； In the formula, This represents the augmented Lagrange function. This represents the k-th IMF component. This represents the unified oscillation center corresponding to the k-th IMF component. Represents the Lagrange multipliers. This represents the analytic signal of the k-th IMF. The first term is the core objective function of the MVMD algorithm, which calculates the norm of the gradient function of the multi-node spectral matrix after spectral shifting. Its physical meaning is that the entire system has the same oscillation center. Under the unified constraints, minimize the spectral bandwidth of the IMF components. This is a penalty factor used to adjust bandwidth. The larger the value, the narrower the spectral bandwidth of the decomposed IMF components; the second term is a quadratic penalty term, which measures the residual between the sum of all extracted IMF components and the original local oscillation components.

4. The frequency response feature decomposition method considering spatial distribution differences according to claim 1, characterized in that, Step S3 includes: For the variational model of MVMD, the variables to be solved include each IMF component. , corresponding oscillation center and Lagrange multipliers Within the ADMM framework, by constructing an augmented Lagrangian function and employing an alternating minimization strategy, the original problem is decomposed into several iteratively solvable subproblems: In the (m+1)th iteration, it is assumed that the oscillation center of the previous iteration... With the Given constants, the IMF components at each node are minimized by minimizing the Lagrange function. The update of the k-th IMF component at this time can be expressed in the following optimized form: (15)； In the formula, The Fourier transform form of the local relative oscillation component of node n. It represents the frequency domain sum of all IMF components at node n, except for the k-th IMF component; This represents half of the Lagrange multiplier corresponding to the k-th IMF component of node n; multiplier term To correct the operator's participation in the update of IMF components, the accumulated bias is fed back into the current IMF component, and the denominator term... This indicates that the components that cause the oscillations of each decomposed node to have similar speeds are concentrated at their respective oscillation centers. Within a narrow band, the IMF components of the frequency signals of each node are gradually corrected; After updating based on the IMF components, the common oscillation center is further updated using the updated IMF components of each node. The update formula for the oscillation center is: (16)； In the formula, This represents the unified oscillation center of the k-th IMF component in the (m+1)-th iteration. This represents the frequency domain representation of the k-th IMF component in the (m+1)-th iteration; This represents the total energy of the k-th IMF component of all nodes n; It represents the sum of the weighted spectral energies of the k-th IMF component of all nodes n.

5. The frequency response feature decomposition method considering spatial distribution differences according to claim 1, characterized in that, In step S4, the solution includes the following steps: S41: Set up the disturbance and collect data from each node after the disturbance. An improved IEEE 39-node system model was constructed, and a load shedding disturbance was set at node 39. Transient frequency response data of each node were collected. S42: Calculate the improved Pearson similarity index for each node. Calculate the average Pearson correlation coefficient and average Euclidean distance of each node with all other nodes, select different weight coefficients, calculate the improved Pearson similarity index of each node, compare the results of the reference node under different weight coefficient values, and select the reference node. S43: Using the selected reference node frequency response as the global reference component, calculate the local relative oscillation components of each node.

6. The frequency response feature decomposition method considering spatial distribution differences according to claim 1, characterized in that, In step S5, the solution includes the following steps: S51: Use the local relative oscillation components of each node as the input signal; S52: Initialize each IMF component and its oscillation center, and solve iteratively using the alternating direction multiplier method until the convergence condition is met; S53: Output the IMF components of different nodes.

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