Power distribution network high-resistance grounding fault detection method based on partial differential morphology

Abstract

The invention discloses a power distribution network high-resistance grounding fault detection method based on partial differential morphology, and the method comprises the steps: defining a partial differential morphology optimization model based on the idea of a partial differential equation, simulating a morphological evolution process of a signal through a nonlinear partial differential equation, and achieving the processing and analysis of the local features of the signal. The method gets rid of dependence on discrete structure elements, and converts mathematical morphology operation into a continuous process driven by a partial differential equation, so that signal processing is more flexible and has physical significance. On the basis, a partial differential morphological operator and a discretization implementation method thereof are defined, a partial differential morphological spectrum concept is introduced, and a theoretical tool is provided for signal multi-scale analysis. The method is applied to high-resistance grounding fault detection of the power distribution network, and the robustness and accuracy of a detection system in a low-sampling-frequency and strong-noise environment are remarkably improved through partial differential morphological spectrum analysis.

Description

Technical Field

[0001] This invention relates to the field of signal processing technology, and in particular to a method for detecting high-resistivity grounding faults in power distribution networks based on partial differential morphology. Background Technology

[0002] Traditional mathematical morphology, as a signal processing tool based on set theory, performs morphological operations on signals through structuring elements, demonstrating significant application value in fields such as image processing, pattern recognition, and power system fault diagnosis. However, this method faces a fundamental challenge: it relies on pre-set fixed structuring elements, making it difficult to adaptively capture the local continuous features of signals. Especially when processing complex and variable fault signals in power systems, it cannot fully identify subtle morphological changes in the signals, thus limiting the accuracy and robustness of feature extraction.

[0003] High-resistance grounding fault detection in distribution networks is a major technical challenge for ensuring the safe operation of power systems. When a fault occurs, the grounding impedance can reach several thousand ohms, and the fault current amplitude is close to the normal load level, making it difficult for traditional overcurrent protection devices to effectively identify. Existing detection methods based on transient signal analysis are easily limited by factors such as line topology, noise interference, and sampling frequency, resulting in a trade-off between sensitivity and reliability. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings and disadvantages of the prior art and propose a method for detecting high-resistivity grounding faults in distribution networks based on partial differential morphology. This method can effectively achieve adaptive processing and multi-scale analysis of local signal features, and complete the reliable detection of high-resistivity grounding faults in distribution networks.

[0005] To achieve the above objectives, the technical solution provided by this invention is: a method for detecting high-resistivity grounding faults in distribution networks based on partial differential morphology, comprising the following steps:

[0006] S1: At the first moment after a distribution network fault or disturbance occurs, collect the zero-sequence current data within the time window at the beginning of the feeder after that moment;

[0007] S2: Calculate the partial differential morphological spectrum of zero-sequence current data based on the defined partial differential morphological discretization operator and the mathematical expression of the partial differential morphological spectrum, and construct a dataset containing multiple partial differential morphological spectra. Specifically, based on the idea of ​​partial differential equations, a partial differential morphological optimization model is defined. This model simulates the morphological evolution process of the signal through nonlinear partial differential equations, enabling the processing and analysis of local signal features. Unlike traditional morphology, the partial differential morphological optimization model dynamically adjusts the intensity of morphological operations through a continuous-scale diffusion process, preserving important edge information while significantly improving the refinement of feature extraction. A partial differential morphological discretization operator is derived, aiming to effectively preserve the key morphological features of one-dimensional signals while smoothing them. Based on the concept of traditional morphological spectra, a partial differential morphological spectrum is defined. Unlike the coarse-grained description of signal shape and size distribution in traditional morphological spectra, the partial differential morphological spectrum characterizes the sensitivity of the scale factor to area changes during signal processing, quantitatively describing the signal's variation characteristics on a continuous scale and accurately capturing the details of signal changes with scale.

[0008] S3: Perform cluster analysis on the obtained partial differential morphology spectra, calculate the Wasserstein distance between each partial differential morphology spectrum in the dataset, and use the improved K-means clustering algorithm to classify the partial differential morphology spectra of the obtained zero-sequence current data into two categories: normal or disturbance signals and high-resistance ground fault signals. The improved K-means clustering algorithm combines Wasserstein distance with traditional K-means clustering. This algorithm enhances the ability to identify overlapping distribution areas by introducing a density weighting strategy. From the perspective of clustering stability, the introduction of density weighting can effectively reduce the oscillation amplitude of center updates at the numerical iteration level.

[0009] S4: For the new zero-sequence current data input at the next moment, repeat the above steps S2 and S3. Based on the Wasserstein distance between the partial differential morphology spectrum of each new zero-sequence current data and the partial differential morphology spectrum of each category centroid, classify the category of each new zero-sequence current data into the category closest to the category centroid. In real time, distinguish between normal or disturbance signals and high-resistance grounding fault signals, thereby detecting whether a high-resistance grounding fault has occurred in the distribution network.

[0010] Furthermore, in step S2, a partial differential morphological optimization model is used to iteratively optimize the zero-sequence current data;

[0011] Based on the ideas of partial differential morphology, a partial differential morphological optimization model is proposed. For a given original one-dimensional signal f(x), the optimized signal is denoted as u(x). The objective function V(u(x)) of the partial differential morphological optimization model is described as follows:

[0012]

[0013] In the formula, u(x) is the function after processing by the partial differential morphological optimization model, and x is the independent variable of the one-dimensional signal. Φ(u′(x)) is the first derivative of u(x), i.e., the gradient. Φ(u′(x)) is the regularization term. λ is the balance parameter and λ>0, which is used to adjust the relative weights of the regularization term and the fidelity term. ψ(u(x)-f(x)) is the fidelity term, which is used to maintain the similarity between the processed signal and the original signal.

[0014] The fidelity term is ψ(u(x)-f(x))=|u(x)-f(x)| 2 The regularization term is set as follows:

[0015]

[0016] In the formula, It is the second derivative of u(x), the function The sign function reflects the convexity or concavity of the signal at local points. It is convex. This indicates concavity, which allows for adaptive adjustment of the smoothing direction based on the local convexity and concavity of the signal. 's' is a scaling factor used to control the degree to which regularization suppresses the gradient, and 's > 0'. If the value of 's' is larger, then... The faster the decay, the weaker the regularization amplitude when the gradient increases, and the more obvious the edges are preserved; conversely, if the value of s is smaller, the gradient will not be suppressed quickly once it increases, and the partial differential morphological optimization model tends to smooth the convex and concave edges more strongly; therefore, the scale factor s corresponds to the size of the structuring element in traditional morphology. Based on this, by minimizing the objective function V(u(x)), this partial differential morphological optimization model effectively preserves the convex and concave structure and significant edges of the signal while smoothing noise and small fluctuations.

[0017] To minimize the objective function V(u(x)), the variational method is used to obtain u(x) that satisfies the optimal conditions, resulting in the continuity equation:

[0018]

[0019] To achieve numerical computation and solution, the above continuity equation is discretized, and the central difference approximation is adopted. The first and second derivatives of u(x) are as follows:

[0020]

[0021] In the formula, This represents the signal value at coordinate i in the nth iteration. This represents the signal value at coordinate i+1 in the nth iteration. Let Δx represent the signal value at coordinate i-1 in the nth iteration, and Δx be the spatial step size.

[0022] Substituting the first and second derivatives of u(x) into the above continuity equation, we obtain the mathematical expression for the partial differential morphological discretization operator as follows:

[0023]

[0024] In the formula, f represents the signal value at coordinate i in the (n+1)th iteration. i (x) represents the signal value at coordinate i in the original one-dimensional data, and Δs is the scale step size.

[0025] Furthermore, in step S2, a partial differential morphological spectrum is constructed using a partial differential morphological model based on physical driving characteristics.

[0026] Based on the concept of traditional morphological spectrum, a partial differential morphological spectrum, called PPS, is defined to characterize the sensitivity of the scale factor s to area changes during signal processing. Its mathematical expression is:

[0027]

[0028] In the formula, PPS(f(x),s) represents the sensitivity of the quantized signal f(x) to area changes when processed under different scale factors s. The larger the scale factor s, the larger the structuring element used, resulting in a smoother signal but also more loss of detail. s A(f(x)) represents the signal obtained by performing a series of morphological operations on the original one-dimensional signal f(x) using a structuring element with a scale factor of s. s (x) represents the processed signal f. s The area of ​​(x), u(x,s) is the signal u(x) processed by the partial differential morphological discretization operator at the scale factor s, Ω is the domain of the signal, which is independent of the scale factor s. Therefore, by exchanging the order of differentiation and integration, the partial differential morphological spectrum integral is defined as:

[0029]

[0030] The partial differential morphological spectrum integral energy can be decomposed into a structure evolution term. With fidelity constraint sum:

[0031]

[0032] When λ = 0, the partial differential morphological spectrum degenerates into pure morphological evolution:

[0033]

[0034] Partial differential morphology spectrometry (PDS) quantitatively describes the area distribution of a signal at different processing scale factors during diffusion, revealing the dynamic changes of the signal in multi-scale morphological operations. This helps in the analysis and understanding of the signal's structure and behavior. In other words, by using PDS, we can accurately capture the details of how the signal changes with scale factors, thus achieving a more in-depth and refined quantitative description in signal processing.

[0035] Furthermore, in step S3, an improved K-means clustering algorithm based on Wasserstein distance is used, as detailed below:

[0036] For two discrete probability distributions Q1=(q 1-1 ,q 1-2 ,...,q 1-v ...,q 1-m ) and Q2=(q 2-1 ,q 2-2 ,...,q 2-v ,...,q 2-m ), where q 1-v Let q represent the probability of the v-th event or state in Q1. 2-v Let v represent the probability of the v-th event or state in Q2. Q1 and Q2 have the same number of m possible outcomes, and the p-order Wasserstein distance between them is defined as:

[0037]

[0038] In the formula, W p (Q1,Q2) represents the p-th order Wasserstein distance between Q1 and Q2, where p is the order of the distance, a real number satisfying p≥1, γ~∏(Q1,Q2) represents all joint distributions of Q1 and Q2, with marginal distributions Q1 and Q2 respectively, Π(Q1,Q2) represents the set of all such joint distributions, inf denotes the infimum, i.e., the minimum value achievable among all possible joint distributions, E (X,Y)～γ [·] denotes the mathematical expectation of the joint distribution γ, (X,Y) are pairs of random variables drawn from γ, ||XY|| denotes the norm distance between variables X and Y, and ||X|| p +||Y|| p <∞ ensures the existence of the expectation, i.e., the distribution has a finite p-th moment; for each joint distribution γ, calculate the mathematical expectation E of all sampling distances ||XY||;

[0039] A Hadamard matrix is ​​a special type of orthogonal matrix where all elements are either +1 or -1, and the rows and columns are mutually orthogonal. The Hadamard matrix is ​​recursively defined as follows:

[0040]

[0041] The Hadamard matrix is ​​of size N×N, and its order N is an integer power of 2. N Let t represent a Hadamard matrix of order N, where t is a positive integer used for exponentiation. In the recursive definition, t represents the current exponent such that the matrix order is 2. t ;

[0042] By integrating morphological gradient operations with the Hadamard matrix, an improved gradient morphological spectrum based on the Hadamard matrix, called HGPS, is proposed. This improved gradient morphological spectrum utilizes the rectangular wave structure elements generated by the Hadamard matrix, which expands the diversity of feature extraction and improves the robustness of signal processing.

[0043] For the original one-dimensional signal in Representing the real number domain, the improved gradient morphological spectrum of the original one-dimensional signal f(x) is defined as follows:

[0044]

[0045] In the formula, For the selected Hadamard matrix, This represents the original one-dimensional signal f(x) in a given Hadamard matrix. The morphological gradient area spectrum value is a scalar value used to quantify the local abrupt changes of the original one-dimensional signal f(x) at a specific scale a and row index b. Denotes the scale parameter of the matrix, where Let represent all integers, and b be the index of the selected row number. Indicates the use of the Hadamard matrix The corresponding structuring element performs morphological gradient operation on the original one-dimensional signal f(x) to obtain the result function. A[·] is the area operator, which calculates the absolute area of ​​the morphological gradient result. The improved gradient morphological spectrum captures the morphological differences of the signal at multiple scales through the rectangular wave structuring elements corresponding to Hadamard matrices of different orders, thereby achieving a refined characterization of the local abrupt change features of the signal.

[0046] The improved K-means clustering algorithm enhances the ability to identify overlapping distribution regions by introducing a density-weighted strategy. In specific implementation, if the Wasserstein distance between a certain improved gradient morphology spectrum and the improved gradient morphology spectrum of the cluster center is smaller, and there are several samples with similar distribution to this improved gradient morphology spectrum nearby, then the weight is increased during the iterative optimization process; otherwise, it is decreased.

[0047] A set of improved gradient morphological spectra based on the Hadamard matrix for different signals HPGS j This represents the j-th improved gradient morphology spectrum, where each improved gradient morphology spectrum is a d-dimensional vector. Let represent the d-dimensional real number field, where d is determined by the feature extraction process;

[0048] The process of improving the K-means clustering algorithm is as follows:

[0049] S31: Randomly select K initial cluster centers and define the initial cluster center set. Where K is the number of clusters, that is, the number of clusters to be divided into for the data. Represents the initial center of the k-th cluster;

[0050] S32: Calculate the Wasserstein distance between each improved gradient morphology spectrum in set D and the improved gradient morphology spectrum of each cluster center, and assign each improved gradient morphology spectrum to the nearest cluster center;

[0051] S33: Calculate local density weights:

[0052]

[0053] In the formula, ω k,j These are local density weights used to calculate the weights between the improved gradient morphology spectrum of the j-th cluster center and the improved gradient morphology spectrum of the k-th cluster center. The square of the Wasserstein distance between the improved gradient morphology spectrum of the j-th cluster center and the improved gradient morphology spectrum of the k-th cluster center is represented by the following expression: Let exp(·) be the average of the squared Wasserstein distances between all improved gradient morphological spectra in cluster k and the improved gradient morphological spectra of the cluster center, and let exp(·) be the exponential function.

[0054] S34: Update cluster center:

[0055]

[0056] In the formula, at this time Let argmin be the updated k-th cluster center, and let argmin represent the parameters that minimize the objective function; here, it is used to find the optimal cluster center. HPGS represents the improved gradient morphology spectrum of the j′-th signal. j′ A probability distribution space with finite second moments over the d-dimensional real number field. This represents the summation of all improved gradient morphological spectra in the k-th cluster, where n is the summation of all improved gradient morphological spectra. k HPGS represents the number of improved gradient morphology spectra assigned to the k-th cluster. k,jThis represents the j-th improved gradient morphology spectrum assigned to the k-th cluster. This represents the square of the Wasserstein distance between the j′-th improved gradient morphology spectrum and the j′-th improved gradient morphology spectrum in the k-th cluster;

[0057] S35: Iterate through steps S32 to S34 until convergence, i.e., the cluster centers do not change significantly;

[0058] Through the above steps, the improved K-means clustering algorithm not only fully utilizes the advantages of Wasserstein distance in measuring distribution differences, but also improves the adaptability of the K-means clustering algorithm to complex distributions through a density-weighted strategy, thus exhibiting higher accuracy and stability in signal classification tasks.

[0059] Furthermore, in step S4, a high-resistance grounding fault in the distribution network is detected, as detailed below:

[0060] When a disturbance occurs in the measurement signal of the distribution network, but does not reach the protection action threshold, the disturbance signal is recorded and uploaded to the main station for fault identification. Specifically, a partial differential morphology optimization model and a partial differential morphology discretization operator are used to calculate the partial differential morphology spectrum of the disturbance signal, constructing a dataset containing multiple partial differential morphology spectra. Cluster analysis is performed on the obtained partial differential morphology spectra, calculating the Wasserstein distance between each partial differential morphology spectrum in the dataset. An improved K-means clustering algorithm is used to classify the partial differential morphology spectrum of the obtained disturbance signal into two categories: normal or disturbance signals and high-resistance ground fault signals. The partial differential morphology spectrum of the disturbance signal is assigned to the category closest to the centroid of the category, distinguishing between normal or disturbance signals and high-resistance ground fault signals in real time, and detecting whether a high-resistance ground fault has occurred in the distribution network.

[0061] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0062] 1. This invention defines a partial differential morphology optimization model and a partial differential morphology discretization operator, and introduces the concept of partial differential morphology spectrum, providing a theoretical tool for multi-scale signal analysis.

[0063] 2. The partial differential morphological discretization operator defined in this invention significantly reduces the interference of noise background while preserving the abrupt change positions of the original signal, and the extracted perturbation features are smoother and more consistent with the original features.

[0064] 3. The partial differential morphological spectrum constructed based on the physical driving characteristics of the partial differential morphological model provides an effective tool for further quantitatively describing the changes of the signal at continuous scales by quantitatively describing the area distribution of the signal at different processing scale factors during the diffusion process. It reveals the dynamic changes of the signal in multi-scale morphological operations and helps to analyze and understand the structure and behavior of the signal.

[0065] 4. The improved K-means clustering algorithm based on Wasserstein distance not only makes full use of the advantages of Wasserstein distance in measuring distribution differences, but also improves the adaptability of K-means clustering algorithm to complex distributions through density weighting strategy, thus showing higher accuracy and stability in signal classification tasks. Attached Figure Description

[0066] Figure 1 This is a schematic diagram of the overall process of the method of the present invention.

[0067] Figure 2 This is a flowchart illustrating the improved K-means clustering algorithm based on Wasserstein distance. Detailed Implementation

[0068] The present invention will be further described in detail below with reference to the embodiments and accompanying drawings, but the embodiments of the present invention are not limited thereto.

[0069] like Figure 1 As shown in the figure, this embodiment discloses a method for detecting high-resistivity grounding faults in distribution networks based on partial differential morphology, the specific details of which are as follows:

[0070] S1: At the first moment after a distribution network fault or disturbance occurs, collect the zero-sequence current data within the time window at the beginning of the feeder after that moment;

[0071] S2: Calculate the partial differential morphological spectrum of zero-sequence current data based on the defined partial differential morphological discretization operator and the mathematical expression of the partial differential morphological spectrum, and construct a dataset containing multiple partial differential morphological spectra. Based on the idea of ​​partial differential equations, a novel mathematical morphological optimization model is defined. This model simulates the morphological evolution of signals through nonlinear partial differential equations, enabling the processing and analysis of local signal features. Unlike traditional morphology, the partial differential morphological optimization model dynamically adjusts the intensity of morphological operations through a continuous-scale diffusion process, preserving important edge information while significantly improving the refinement of feature extraction. A partial differential morphological discretization operator is derived, aiming to effectively preserve key morphological features while smoothing one-dimensional signals. Based on the concept of traditional morphological spectra, a partial differential morphological spectrum (PPS) is defined. Unlike the coarse-grained description of signal shape and size distribution in traditional morphological spectra, the partial differential morphological spectrum characterizes the sensitivity of the scale factor to area changes during signal processing, quantitatively describing the signal's variation characteristics on a continuous scale and accurately capturing the details of signal changes with scale.

[0072] A partial differential morphological optimization model was used to iteratively optimize the zero-sequence current data;

[0073] For a given original one-dimensional signal f(x), the optimized signal is denoted as u(x). The objective function V(u(x)) of the partial differential morphological optimization model is described as follows:

[0074]

[0075] In the formula, u(x) is the function after processing by the partial differential morphological optimization model, and x is the independent variable of the one-dimensional signal. Φ(u′(x)) is the first derivative of u(x), i.e., the gradient. Φ(u′(x)) is the regularization term. λ is the balance parameter and λ>0, which is used to adjust the relative weights of the regularization term and the fidelity term. ψ(u(x)-f(x)) is the fidelity term, which is used to maintain the similarity between the processed signal and the original signal.

[0076] The fidelity term is ψ(u(x)-f(x))=|u(x)-f(x)| 2 The regularization term is set as follows:

[0077]

[0078] In the formula, It is the second derivative of u(x), the function The sign function reflects the convexity or concavity of the signal at local points. It is convex. This indicates concavity, which allows for adaptive adjustment of the smoothing direction based on the local convexity and concavity of the signal. 's' is a scaling factor used to control the degree to which regularization suppresses the gradient, and 's' > 0. If the value of 's' is larger, then... The faster the decay, the weaker the regularization amplitude when the gradient increases, and the more obvious the edges are preserved; conversely, if the value of s is smaller, the gradient will not be suppressed quickly once it increases, and the partial differential morphological optimization model tends to smooth the convex and concave edges more strongly; therefore, the scale factor s corresponds to the size of the structuring element in traditional morphology. Based on this, by minimizing the objective function V(u(x)0), this partial differential morphological optimization model effectively preserves the convex and concave structure and significant edges of the signal while smoothing noise and small fluctuations.

[0079] To minimize the objective function V(u(x)), the variational method is used to obtain u(x) that satisfies the optimal conditions, resulting in the continuity equation:

[0080]

[0081] To achieve numerical computation and solution, the above continuity equation is discretized, and the central difference approximation is adopted. The first and second derivatives of u(x) are as follows:

[0082]

[0083] In the formula, This represents the signal value at coordinate i in the nth iteration. This represents the signal value at coordinate i+1 in the nth iteration. Let Δx represent the signal value at coordinate i-1 in the nth iteration, and Δx be the spatial step size.

[0084] Substituting the first and second derivatives of u(x) into the above continuity equation, we obtain the mathematical expression for the partial differential morphological discretization operator as follows:

[0085]

[0086] In the formula, f represents the signal value at coordinate i in the (n+1)th iteration. i (x) represents the signal value of coordinate i in the original one-dimensional data, and Δs is the scale step size;

[0087] The results show that the partial differential morphology discretization operator significantly reduces the interference of noise background while preserving the signal abrupt change position, and the extracted perturbation features are smoother and more consistent with the original features.

[0088] A partial differential morphological spectrum was constructed using a partial differential morphological model based on physical driving characteristics;

[0089] Based on the concept of traditional morphological spectrum, a partial differential morphological spectrum, called the PDE-based pattern spectrum (PPS), is defined to characterize the sensitivity of the scale factor *s* to area changes during signal processing. Its mathematical expression is as follows:

[0090]

[0091] In the formula, PPS(f(x),s) represents the sensitivity of the quantized signal f(x) to area changes when processed under different scale factors s. The larger the scale factor s, the larger the structuring element used, resulting in a smoother signal but also more loss of detail. s A(f(x)) represents the signal obtained by performing a series of morphological operations on the original one-dimensional signal f(x) using a structuring element with a scale factor of s. s (x) represents the processed signal f. s The area of ​​(x), u(x,s) is the signal u(x) processed by the partial differential morphological discretization operator at the scale factor s, Ω is the domain of the signal, which is independent of the scale factor s. Therefore, by exchanging the order of differentiation and integration, the partial differential morphological spectrum integral is defined as:

[0092]

[0093] The partial differential morphological spectrum integral energy can be decomposed into a structure evolution term. With fidelity constraint sum:

[0094]

[0095] When λ = 0, the partial differential morphological spectrum degenerates into pure morphological evolution:

[0096]

[0097] Partial differential morphology spectrometry (PDS) quantitatively describes the area distribution of a signal at different processing scale factors during diffusion, revealing the dynamic changes of the signal in multi-scale morphological operations. This helps in the analysis and understanding of the signal's structure and behavior. In other words, by using PDS, we can accurately capture the details of how the signal changes with scale factors, thus achieving a more in-depth and refined quantitative description in signal processing.

[0098] S3: Perform cluster analysis on the obtained partial differential morphology spectra, calculate the Wasserstein distance between each partial differential morphology spectrum in the dataset, and use the improved K-means clustering algorithm to classify the partial differential morphology spectra of the obtained zero-sequence current data into two categories: normal or disturbance signals and high-resistance ground fault signals. The improved K-means clustering algorithm combines Wasserstein distance with traditional K-means clustering. This algorithm enhances the ability to identify overlapping distribution areas by introducing a density weighting strategy. From the perspective of clustering stability, the introduction of density weighting can effectively reduce the oscillation amplitude of center updates at the numerical iteration level.

[0099] like Figure 2 As shown, an improved K-means clustering algorithm based on Wasserstein distance was used, as detailed below:

[0100] For two discrete probability distributions Q1=(q 1-1 ,q 1-2 ,...,q 1-v ...,q 1-m ) and Q2=(q 2-1 ,q 2-2 ,...,q 2-v ,...,q 2-m ), where q 1-v Let q represent the probability of the v-th event or state in Q1. 2-v Let v represent the probability of the v-th event or state in Q2. Q1 and Q2 have the same number of m possible outcomes, and the p-order Wasserstein distance between them is defined as:

[0101]

[0102] In the formula, W p (Q1,Q2) represents the p-th order Wasserstein distance between Q1 and Q2, where p is the order of the distance, a real number satisfying p≥1, γ~Π(Q1,Q2) represents all joint distributions of Q1 and Q2, with marginal distributions Q1 and Q2 respectively, Π(Q1,Q2) represents the set of all such joint distributions, inf denotes the infimum, i.e., the minimum value achievable among all possible joint distributions, E (X,Y)～γ [·] denotes the mathematical expectation of the joint distribution γ, (X,Y) are pairs of random variables drawn from γ, ||XY|| denotes the norm distance between variables X and Y, and ||X|| p +||Y|| p <∞ ensures the existence of the expectation, i.e., the distribution has a finite p-th moment; for each joint distribution γ, calculate the mathematical expectation E of all sampling distances ||XY||;

[0103] A Hadamard matrix is ​​a special type of orthogonal matrix where all elements are either +1 or -1, and the rows and columns are mutually orthogonal. The Hadamard matrix is ​​recursively defined as follows:

[0104]

[0105] The Hadamard matrix is ​​of size N×N, and its order N is an integer power of 2. N Let t represent a Hadamard matrix of order N, where t is a positive integer used for exponentiation. In the recursive definition, t represents the current exponent such that the matrix order is 2. t ;

[0106] By integrating morphological gradient operations with the Hadamard matrix, an improved gradient morphological spectrum based on the Hadamard matrix, called HGPS, is proposed. This improved gradient morphological spectrum utilizes the rectangular wave structure elements generated by the Hadamard matrix, which expands the diversity of feature extraction and improves the robustness of signal processing.

[0107] For the original one-dimensional signal in Representing the real number domain, the improved gradient morphological spectrum of the original one-dimensional signal f(x) is defined as follows:

[0108]

[0109] In the formula, For the selected Hadamard matrix, This represents the original one-dimensional signal f(x) in a given Hadamard matrix. The morphological gradient area spectrum value is a scalar value used to quantify the local abrupt changes of the original one-dimensional signal f(x) at a specific scale a and row index b. Denotes the scale parameter of the matrix, where Let represent all integers, and b be the index of the selected row number. Indicates the use of the Hadamard matrix The corresponding structuring element performs morphological gradient operation on the original one-dimensional signal f(x) to obtain the result function. A[·] is the area operator, which calculates the absolute area of ​​the morphological gradient result. The improved gradient morphological spectrum captures the morphological differences of the signal at multiple scales through the rectangular wave structuring elements corresponding to Hadamard matrices of different orders, thereby achieving a refined characterization of the local abrupt change features of the signal.

[0110] The improved K-means clustering algorithm enhances the ability to identify overlapping distribution regions by introducing a density-weighted strategy. In practice, if the Wasserstein distance between an improved gradient morphology spectrum and the improved gradient morphology spectrum of the cluster center is smaller, and there are several samples with similar distributions to this improved gradient morphology spectrum nearby, then the weight is increased during the iterative optimization process; otherwise, it is decreased.

[0111] A set of improved gradient morphological spectra based on the Hadamard matrix for different signals HPGS j This represents the j-th improved gradient morphology spectrum, where each improved gradient morphology spectrum is a d-dimensional vector. Let represent the d-dimensional real number field, where d is determined by the feature extraction process;

[0112] The process of improving the K-means clustering algorithm is as follows:

[0113] S31: Randomly select K initial cluster centers and define the initial cluster center set. Where K is the number of clusters, that is, the number of clusters to be divided into for the data. Represents the initial center of the k-th cluster;

[0114] S32: Calculate the Wasserstein distance between each improved gradient morphology spectrum in set D and the improved gradient morphology spectrum of each cluster center, and assign each improved gradient morphology spectrum to the nearest cluster center;

[0115] S33: Calculate local density weights:

[0116]

[0117] In the formula, ω k,j These are local density weights used to calculate the weights between the improved gradient morphology spectrum of the j-th cluster center and the improved gradient morphology spectrum of the k-th cluster center. The square of the Wasserstein distance between the improved gradient morphology spectrum of the j-th cluster center and the improved gradient morphology spectrum of the k-th cluster center is represented by the following expression: Let exp(·) be the average of the squared Wasserstein distances between all improved gradient morphological spectra in cluster k and the improved gradient morphological spectra of the cluster center, and let exp(·) be the exponential function.

[0118] S34: Update cluster center:

[0119]

[0120] In the formula, at this time Let argmin be the updated k-th cluster center, and let argmin represent the parameters that minimize the objective function; here, it is used to find the optimal cluster center. HPGS represents the improved gradient morphology spectrum of the j′-th signal. j′ A probability distribution space with finite second moments over the d-dimensional real number field. This represents the summation of all improved gradient morphological spectra in the k-th cluster, where n is the summation of all improved gradient morphological spectra. k HPGS represents the number of improved gradient morphology spectra assigned to the k-th cluster. k,j This represents the j-th improved gradient morphology spectrum assigned to the k-th cluster. This represents the square of the Wasserstein distance between the j′-th improved gradient morphology spectrum and the j′-th improved gradient morphology spectrum in the k-th cluster;

[0121] S35: Iterate through steps S32 to S34 until convergence, i.e., the cluster centers do not change significantly;

[0122] Through the above steps, the improved K-means clustering algorithm not only fully utilizes the advantages of Wasserstein distance in measuring distribution differences, but also improves the adaptability of the K-means clustering algorithm to complex distributions through a density-weighted strategy, thus exhibiting higher accuracy and stability in signal classification tasks.

[0123] S4: For the newly input zero-sequence current data at the next moment, repeat steps S2 and S3 above. Based on the Wasserstein distance between the partial differential morphology spectrum of each newly input zero-sequence current data and the partial differential morphology spectrum of each category centroid, classify the data into the category closest to the category centroid. This allows for real-time differentiation between normal or disturbance signals and high-resistance grounding fault signals, detecting whether a high-resistance grounding fault has occurred in the distribution network. Specific details are as follows:

[0124] When a disturbance occurs in the measurement signal of the distribution network, but does not reach the protection action threshold, the disturbance signal is recorded and uploaded to the main station for fault identification. Specifically, a partial differential morphology optimization model and a partial differential morphology discretization operator are used to calculate the partial differential morphology spectrum of the disturbance signal, constructing a dataset containing multiple partial differential morphology spectra. Cluster analysis is performed on the obtained partial differential morphology spectra, calculating the Wasserstein distance between each partial differential morphology spectrum in the dataset. An improved K-means clustering algorithm is used to classify the partial differential morphology spectrum of the obtained disturbance signal into two categories: normal or disturbance signals and high-resistance ground fault signals. The partial differential morphology spectrum of the disturbance signal is assigned to the category closest to the centroid of the category, distinguishing between normal or disturbance signals and high-resistance ground fault signals in real time, and detecting whether a high-resistance ground fault has occurred in the distribution network.

[0125] To comprehensively evaluate the adaptability and effectiveness of the improved high-resistance grounding fault detection scheme under complex operating conditions, this study uses the classic Emanuel high-resistance grounding fault model for simulation research. This model has the advantages of simple structure, intuitive parameter setting and no need for complex differential calculations, and can realistically simulate the randomness, asymmetry and non-stationary arc characteristics present in high-resistance grounding faults (HIF).

[0126] Considering that 6-66kV distribution networks generally adopt a neutral point non-effective grounding operation mode, this study builds a typical 10kV resonant grounding distribution network simulation model based on the PSCAD / EMTDC platform. The main transformer has a rated voltage of 110kV / 10.5kV, uses arc suppression coil grounding, and the distribution network topology includes overhead lines and cable lines. A 1MW photovoltaic module is connected to the system, with a load power of 1MW, and the sampling frequency is set to 10kHz.

[0127] In the simulation environment, zero-sequence current sensors are installed at the beginning of each feeder to collect zero-sequence current data for two power frequency cycles after a fault or disturbance in real time, and to perform morphological spectrum calculation and feature extraction based on this data. Furthermore, this study constructs a complete signal acquisition and data processing workflow based on the RTDS real-time digital simulation platform.

[0128] When calculating the partial differential morphological spectrum, a spatial step size Δx = 0.01, an initial scale s = 0.1, a scale step size Δs = 0.1, and a fidelity term weight λ = 0.2 were used. The overall performance of the scheme is shown in Table 1.

[0129] Table 1. Detection performance of partial differential morphological scheme in radial distribution network

[0130]

[0131] In a noise-free environment, all sampling frequencies (10kHz, 5kHz, and 1kHz) achieved 100% across all four performance indicators, verifying the reliability of the method under ideal conditions. As noise intensity increases, the performance indicators gradually decrease but remain at a high level. For example, at 30dB noise, the overall performance of the method remained stable above 99.4%, demonstrating its adaptability to typical noise levels in the measurement environment. Furthermore, the performance differences between different sampling frequencies were small, indicating that the method is insensitive to changes in the sampling rate.

[0132] To verify the feasibility of the proposed solution in a real-world scenario, actual distribution network fault waveform data was used for testing. Taking a radially connected distribution network as an example, under a 10kHz sampling frequency and noise-free conditions, the calculated cluster center coordinates for the four types of events were: [120.02 75.32 65.11 61.79 61.84], [142.65 87.20 75.17 65.44 70.69], [340.19 245.67 241.02 206.39 181.63], [218.05 199.63 206.88 174.30 162.97]. Based on these cluster centers, the actual waveform data was identified, and the results are shown in Table 2. All cases had the smallest Wasserstein distance to high-resistance grounding faults, indicating that the solution has a good ability to distinguish high-resistance grounding faults. In the table, HIF represents a high-resistance ground fault, LIF represents a low-resistance ground fault, CS represents capacitor switching, and LS represents load switching.

[0133] Table 2. Identification results of actual cases based on partial differential morphology spectrum.

[0134]

[0135] Experimental Conclusions: This invention proposes a method for detecting high-resistivity grounding faults in distribution networks based on partial differential morphology. The constructed partial differential morphological optimization model achieves a dynamic balance between edge preservation and noise suppression, exhibiting stronger local adaptive capabilities compared to traditional morphological operators. The proposed partial differential morphological spectrum effectively captures the multi-resolution morphological structure information of fault signals through continuous-scale integration, demonstrating more stable feature extraction capabilities within the 10kHz to 1kHz sampling range, laying the foundation for accurate characterization of fault signals.

[0136] Experimental results demonstrate that the proposed scheme exhibits good adaptability in complex power distribution networks. Verification using actual fault recording data further confirms that the proposed scheme achieves a 100% success rate in detecting high-resistivity faults, while also effectively distinguishing between interference events such as capacitor switching and load switching. These results highlight the robustness and practicality of the partial differential morphological spectrum scheme in complex power grid environments. To further promote the engineering application of this method in smart new power distribution networks, future research will focus on the adaptive optimization mechanism of morphological parameters and explore FPGA-based hardware implementation to improve its real-time performance and engineering feasibility, making it worthy of widespread adoption.

[0137] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A partial differential morphology-based method for detecting high impedance ground faults in a power distribution network, characterized in that, Comprise the following steps: S1: at the first time after the occurrence of the power distribution network fault or disturbance, collect the zero sequence current data in the time window of the feeder start after this time; S2: calculate the partial differential morphological spectrum of the zero sequence current data according to the defined partial differential morphological discretization operator and the mathematical expression of the partial differential morphological spectrum, and construct a data set containing multiple partial differential morphological spectra; wherein, based on the idea of partial differential equation, a partial differential morphological optimization model is defined, the morphological evolution process of the signal is simulated through the nonlinear partial differential equation, the processing and analysis of the local characteristics of the signal are realized, and unlike the traditional morphological method, the partial differential morphological optimization model dynamically adjusts the intensity of the morphological operation through the diffusion process of continuous scale, retains important edge information, and significantly improves the fine degree of feature extraction, and the partial differential morphological discretization operator is derived, which aims to smooth the one-dimensional signal while effectively retaining its key morphological characteristics; based on the concept of traditional morphological spectrum, a partial differential morphological spectrum is defined, which is different from the coarse-grained description of signal shape size distribution in traditional morphological spectrum, and the partial differential morphological spectrum represents the sensitivity of scale factor to area change in signal processing process, which can quantitatively describe the change characteristics of signal in continuous scale and accurately capture the details of signal change with scale; S3: cluster analysis is performed on the obtained partial differential morphological spectrum, the Wasserstein distance between each partial differential morphological spectrum in the data set is calculated, and the improved Kmeans clustering algorithm is used to classify the partial differential morphological spectrum of the zero sequence current data into two categories: normal or disturbance signal and high resistance ground fault signal; wherein, the improved Kmeans clustering algorithm combines the Wasserstein distance with the traditional K-means clustering, and the algorithm enhances the recognition ability of the distribution overlap area by introducing the density weighting strategy, and from the perspective of clustering stability, the introduction of density weighting can effectively reduce the oscillation amplitude of center update at the numerical iteration level; S4: for the newly input zero sequence current data at the next time, repeat the above steps S2 and S3, and classify the new input zero sequence current data according to the Wasserstein distance between the partial differential morphological spectrum of each new input zero sequence current data and the partial differential morphological spectrum of each class center, and the partial differential morphological spectrum of each new input zero sequence current data is classified into the class closest to the class center, so as to distinguish normal or disturbance signal from high resistance ground fault signal in real time, thereby detecting whether high resistance ground fault occurs in the power distribution network.

2. The partial differential morphology based method for detection of high impedance ground faults in power distribution network as claimed in claim 1 wherein, In step S2, the partial differential morphological optimization model is used to iteratively optimize the zero sequence current data; Based on the idea of partial differential morphological optimization model, for a given original one-dimensional signal f(x), the optimized signal is denoted as u(x), and the objective function V(u(x)) of the partial differential morphological optimization model is described as: where u(x) is a function after the partial differential morphological optimization model processing, x is an independent variable of one-dimensional signal, is a first derivative of u(x) i.e. gradient, Φ(u'(x)) is a regularization term, λ is a balance parameter and λ > 0, used to adjust the relative weight of the regularization term and fidelity term, ψ(u(x)-f(x)) is a fidelity term, used to keep the similarity between the processed signal and the original signal; The fidelity term is ψ(u(x) - f(x)) = |u(x) - f(x)| 2 The regularization term is set as: In the formula, It is the second derivative of u(x), the function The sign function reflects the convexity or concavity of the signal at local points. It is convex. This indicates concavity, which allows for adaptive adjustment of the smoothing direction based on the local convexity and concavity of the signal. 's' is a scaling factor used to control the degree to which regularization suppresses the gradient, and 's > 0'. If the value of 's' is larger, then... The faster the decay, the weaker the regularization amplitude when the gradient increases, and the more obvious the edges are preserved; conversely, if the value of s is smaller, the gradient will not be suppressed quickly once it increases, and the partial differential morphological optimization model tends to smooth the convex and concave edges more strongly; therefore, the scale factor s corresponds to the size of the structuring element in traditional morphology. Based on this, by minimizing the objective function V(u(x)), this partial differential morphological optimization model effectively preserves the convex and concave structure and significant edges of the signal while smoothing noise and small fluctuations. In order to minimize the objective function V(u(x)), the variational method is used to obtain u(x) that satisfies the optimal condition, and the continuous equation is obtained: To realize the numerical calculation solution, the above continuous equation is discretized, and the central difference approximation is adopted, and the first order derivative and the second order derivative of u(x) are respectively: wherein denotes the signal value of coordinate i in the n-th iteration, denotes the signal value of coordinate i+1 in the n-th iteration, denotes the signal value of coordinate i-1 in the n-th iteration, Δx is the spatial step size; The first order derivative and the second order derivative of u(x) are substituted into the above continuous equation, and the mathematical expression of the partial differential morphological discretization operator is as follows: wherein denotes the signal value of coordinate i in the n+1 iteration, f i (x) denotes the signal value of coordinate i in the original one-dimensional data, Δs is the scale step.

3. The partial differential morphology based method for detection of high impedance ground faults in power distribution network as claimed in claim 2 wherein, In step S2, a partial differential morphological model based on physical driving characteristics is adopted to construct a partial differential morphological spectrum; Based on the concept of traditional morphological spectrum, a partial differential morphological spectrum, referred to as PPS, is defined to represent the sensitivity of the area change to the scale factor s in the signal processing process, and the mathematical expression is as follows: where PPS(f(x),s) represents the sensitivity of the area variation of the quantized signal f(x) when it is processed with different scale factors s, the larger the scale factor s, the larger the structuring element used, the smoother the processed signal, the more details lost, f s (x) represents the signal obtained after a series of morphological operations with structuring element of scale factor s on the original one-dimensional signal f(x), A(f s (x)) represents the area of the processed signal f s (x), u(x,s) is the signal u(x) processed by the partial differential morphological discretization operator with scale factor s, Ω is the signal definition domain, which is independent of the scale factor s, thus the order of differentiation and integration is exchanged to obtain the partial differential morphological spectrum integral definition: This partial differential morphological spectrum integral can be decomposed into a structural evolution term and a fidelity constraint term : When λ = 0, the partial differential morphological spectrum degenerates into pure morphological evolution: The partial differential morphological spectrum quantitatively describes the area distribution of the signal at different processing scale factors in the diffusion process, and can reveal the dynamic changes of the signal in multi-scale morphological operation, helping to analyze and understand the structure and behavior of the signal, that is, by using the partial differential morphological spectrum, the details of the signal change with the scale factor can be accurately captured, so that more in-depth and fine quantitative description can be realized in signal processing.

4. The partial differential morphology based method for detection of high impedance ground faults in power distribution network as claimed in claim 3 wherein, In step S3, an improved Kmeans clustering algorithm based on Wasserstein distance is adopted, and the specific case is as follows: For two discrete probability distributions Q1 = (q 1-1 ,q 1-2 ,...,q 1-v ,...,q 1-m ) and Q2 = (q 2-1 ,q 2-2 ,...,q 2-v ,...,q 2-m ), where q 1-v represents the probability of the vth event or state in Q1, q 2-v represents the probability of the vth event or state in Q2, Q1, Q2 have the same number m of possible outcomes, the pth order Wasserstein distance between them is defined as: where W p (Q1, Q2) denotes the p-order Wasserstein distance between Q1 and Q2, p is the order of the distance, is a real number and satisfies p > 1, γ ~ Π(Q1, Q2) is the set of all joint distributions of Q1 and Q2, whose marginal distributions are Q1 and Q2 respectively, Π(Q1, Q2) denotes the set of all such joint distributions, inf denotes the infimum, i.e. the minimum value that can be achieved among all possible joint distributions, E (X,Y)～γ [·] denotes the mathematical expectation with respect to the joint distribution γ, (X, Y) is a pair of random variables drawn from γ, ||X - Y|| denotes the norm distance between the variable X and the variable Y, ||X||p p +||Y||p p < ∞ ensures that the expectation exists, i.e. the distribution has finite p-order moments; for each joint distribution γ, the mathematical expectation E of all the sampled distances ||X - Y|| is calculated; The Hadamard matrix is a special orthogonal matrix, all elements of which are +1 or -1, and the rows and columns are orthogonal to each other, and the recursive definition of the Hadamard matrix is as follows: The Hadamard matrix is of size N x N, the matrix order N is an integer power of 2, H N represents a Hadamard matrix of order N, t represents a positive integer, is used for exponential expression, in the recursive definition, t represents the current index, so that the matrix order is 2 t ; By integrating morphological gradient operation and Hadamard matrix, an improved gradient morphological spectrum based on Hadamard matrix is proposed, referred to as HGPS, which uses the rectangular wave structure element generated by Hadamard matrix to expand the diversity of feature extraction and improve the robustness of signal processing; For the original one-dimensional signal where denotes the real field, the improved gradient morphological spectrum of the original one-dimensional signal f(x) is defined as follows: wherein is a selected Hadamard matrix, denotes the morphological gradient area spectrum value of the original one-dimensional signal f(x) under a given Hadamard matrix , which is a scalar value, is used to quantify the local mutation characteristics of the original one-dimensional signal f(x) under a specific scale a and row index b, denotes the scale parameter of the matrix, wherein represents all integers, and b is the index of the selected number of rows, denotes the use of a Hadamard matrix corresponding to the result function obtained after the morphological gradient operation on the original one-dimensional signal f(x) by the structural element, A[·] is an area operator, and the absolute area of the morphological gradient result is calculated. The improved gradient morphological spectrum captures the morphological differences of the signal at multiple scales through the rectangular wave structure elements corresponding to different order Hadamard matrices, and realizes the fine characterization of the local mutation characteristics of the signal; The improved Kmeans clustering algorithm enhances the recognition ability of the distribution overlap area by introducing the density weighting strategy, and in specific implementation, if the Wasserstein distance between a certain improved gradient morphological spectrum and the improved gradient morphological spectrum of the cluster center is smaller, and there are several samples similar to the distribution of this improved gradient morphological spectrum nearby, the weight will be increased in the iterative optimization process, and vice versa; A set of improved gradient morphological profiles based on Hadamard matrix for different signals where HPGS j represents the jth improved gradient morphological profile, each of which is a d-dimensional vector, denotes a d-dimensional real number field, d is determined by the feature extraction process; The process of the improved Kmeans clustering algorithm is as follows: S31: randomly select K initial cluster centers, define an initial cluster center set where K is the number of clusters, i.e., the number of clusters into which the data is to be divided, representing the initial center of the kth cluster; S32: Calculate the Wasserstein distance between each improved gradient morphological spectrum in set D and each improved gradient morphological spectrum of the cluster center, and assign each improved gradient morphological spectrum to the nearest cluster center; S33: Calculate the local density weight: where ω k,j is the local density weight used to calculate the weight between the jth improved gradient morphological spectrum and the improved gradient morphological spectrum of the kth cluster center, represents the square of the Wasserstein distance between the jth improved gradient morphological spectrum and the improved gradient morphological spectrum of the kth cluster center, is the average of the square of the Wasserstein distance between all improved gradient morphological spectra in the cluster k and the improved gradient morphological spectrum of the cluster center, and exp(·) is the exponential function. S34: Update the cluster center: where is the updated kth cluster center, arg min denotes taking the argument that minimizes the objective function, and here is used to find the optimal cluster centers, is the updated kth cluster center, arg min denotes taking the argument that minimizes the objective function, and here is used to find the optimal cluster centers, denotes the improved gradient shape pattern (HPGS) of the jth signal j′ is the space of probability distributions with finite second moment on the d-dimensional real number field, denotes the summation of all improved gradient shape patterns in the kth cluster, n k is the number of improved gradient shape patterns assigned to the kth cluster, HPGS k,j denotes the jth improved gradient shape pattern assigned to the kth cluster, denotes the square of the Wasserstein distance between the jth improved gradient shape pattern and the jth improved gradient shape pattern in the kth cluster; S35: Repeat steps S32 to S34 until convergence, that is, the cluster center does not change significantly; Through the above steps, the improved Kmeans clustering algorithm not only makes full use of the advantage of Wasserstein distance in measuring the difference between distributions, but also improves the adaptability of the Kmeans clustering algorithm to complex distributions through the density weighting strategy, thereby showing higher accuracy and stability in the signal classification task.

5. The partial differential morphology based method for detection of high impedance ground faults in power distribution network as claimed in claim 4 wherein, In step S4, the high resistance grounding fault of the power distribution network is detected, and the specific case is as follows: When the measurement signal of the power distribution network is disturbed, but does not reach the protection action threshold, the disturbance signal is recorded and uploaded to the master station for fault identification. Specifically, the partial differential morphological optimization model and the partial differential morphological discretization operator are used to calculate the partial differential morphological spectrum of the disturbance signal, a data set containing multiple partial differential morphological spectra is constructed, clustering analysis is performed on the obtained partial differential morphological spectrum, the Wasserstein distance between each partial differential morphological spectrum in the data set is calculated, and the improved Kmeans clustering algorithm is used to classify the partial differential morphological spectrum of the disturbance signal into two categories: normal or disturbance signal and high resistance grounding fault signal. The partial differential morphological spectrum of the disturbance signal is classified into the category closest to the category center, and the normal or disturbance signal and the high resistance grounding fault signal are distinguished in real time, and it is detected whether the high resistance grounding fault of the power distribution network occurs.

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