A power distribution network node power quality dynamic management demand quantification method

By constructing voltage and harmonic characteristic curves and combining them with voltage influence factors and harmonic propagation factors, the problem of insufficient dynamic assessment of power quality issues in distribution networks has been solved, enabling precise management of key nodes and time periods, and improving management efficiency and resource utilization.

CN120875675BActive Publication Date: 2026-07-21SICHUAN UNIV
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Patent Information

Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
SICHUAN UNIV
Filing Date
2025-07-28
Publication Date
2026-07-21

AI Technical Summary

Technical Problem

Existing technologies lack dynamic assessment methods for power quality issues in distribution networks, and cannot effectively identify critical periods and the coupling effects between nodes. This results in a lack of targeted and scientific allocation of governance resources, leading to low governance efficiency and resource waste.

Method used

By constructing characteristic curves of voltage exceeding limits and harmonic exceedance, extracting characteristic indicators of governance needs, introducing voltage influence factors and harmonic propagation factors, revising governance priority indicators, and identifying key nodes and time periods.

Benefits of technology

It enables dynamic identification and precise management of power quality problems in the distribution network, improves system stability and resource utilization efficiency, and enhances the accuracy and scientific nature of management.

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Abstract

The application discloses a power distribution network node power quality dynamic management demand quantification method, and belongs to the technical field of power distribution network power quality management. The method comprises the following steps: constructing voltage out-of-limit characteristic curves and harmonic over-limit characteristic curves of each node, and extracting node power quality management demand characteristic indexes; constructing a node power quality problem characteristic vector based on the node power quality management demand characteristic indexes; introducing an ideal worst management state vector, constructing a worst node reference characteristic, and constructing an initial management priority index based on the worst node reference characteristic and the node power quality problem characteristic vector; introducing a voltage influence factor and a harmonic propagation factor, and correcting the initial management priority index based on the voltage influence factor and the harmonic propagation factor; and identifying the most urgent key node in the power distribution network at different time sections based on the corrected initial management priority index. The method can effectively improve the accuracy of power distribution network power quality management.
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Description

Technical Field

[0001] This application relates to the field of power quality management technology for distribution networks, and in particular to a method for quantifying the dynamic management needs of power quality at distribution network nodes. Background Technology

[0002] In recent years, with the rapid integration of numerous distributed energy resources (DERs) into the distribution network, especially photovoltaic power generation systems, user-side energy storage devices, and electric vehicle charging facilities, the power grid's operation mode and power flow characteristics have undergone significant changes. The distribution system, originally dominated by centralized dispatch, is gradually evolving into a dynamic and complex system encompassing multiple power sources and various types of loads. Due to the intermittent and uncertain characteristics of distributed resources, power quality issues such as voltage fluctuations, sags, voltage limits, and harmonics in the distribution network are becoming increasingly prominent.

[0003] Currently, power quality management relies primarily on static assessments, lacking dynamic evaluation methods that consider temporal variations and differences in node characteristics. Specifically, existing research and engineering practices suffer from the following shortcomings: Lack of temporal feature extraction capabilities: It is difficult to characterize the fluctuations and suddenness of power quality problems across different time periods, making it impossible to accurately identify "critical periods"; Insufficient ability to identify node governance priorities: Most methods rely on 24-hour averages or weighted values, ignoring the time-dependent outbreak characteristics of power quality problems; Insufficient consideration of system coupling relationships: Power quality problems are propagating; a problem at one node may affect other nodes, but existing indicators are mostly based on node-specific data, lacking assessment of systemic contagion effects; Lack of targeted and scientific allocation of governance resources: Given limited power quality governance resources, precise control of "critical nodes and critical periods" is impossible, leading to low governance efficiency and resource waste.

[0004] Therefore, how to construct an analysis system that can effectively identify and quantify the power quality governance needs of nodes, and fully consider the temporal changes, the severity of problems and the coupling effects between nodes, has become a key technical problem that needs to be solved in the intelligent governance of distribution networks. Summary of the Invention

[0005] To address the aforementioned shortcomings in existing technologies, this application provides a method for quantifying the dynamic power quality management requirements of distribution network nodes, which solves the problem of power quality management that does not fully consider temporal changes, the severity of problems, and the coupling effects between nodes.

[0006] To achieve the aforementioned objectives, the technical solution adopted in this application is as follows: This application provides a method for quantifying the dynamic power quality management requirements of distribution network nodes, including: S1: Perform fundamental power flow calculation and harmonic power flow calculation on the distribution network under the multi-resource grid connection scenario, obtain the voltage and harmonic data of each node, and obtain the voltage over-limit characteristic curve and harmonic over-limit characteristic curve of each node. S2: Based on the voltage over-limit characteristic curve and harmonic exceedance characteristic curve of each node, extract the characteristic indicators of power quality management needs of the nodes. S3: Construct a feature vector of power quality problems for nodes based on the characteristic indicators of power quality governance needs at the nodes; S4: Introduce the ideal worst-case governance state vector, construct the worst-case node reference feature, and construct the initial governance priority index based on the worst-case node reference feature and the node's power quality problem feature vector; S5: Introduce voltage influence factor and harmonic propagation factor, and revise the initial governance priority index based on voltage influence factor and harmonic propagation factor; S6: Based on the revised initial governance priority index, identify the key nodes in the distribution network with the most urgent governance needs at different time segments.

[0007] Further, S2 includes: S201: Based on the voltage over-limit characteristic curves of each node, obtain the closed area of ​​the UT characteristic trajectory; S202: Based on the harmonic exceedance characteristic curves of each node, obtain the closed area of ​​the HT characteristic trajectory; S203: Based on the voltage over-limit characteristic curve and harmonic over-limit characteristic curve of each node, the characteristic trajectory symmetry is obtained.

[0008] Furthermore, the closed area of ​​the UT characteristic trajectory includes the area enclosed by the voltage over-limit characteristic curve and the horizontal axis from the moment before the voltage over-limit occurs to the end of the moment.

[0009] Furthermore, the closed area of ​​the HT characteristic trajectory includes the area enclosed by the harmonic exceedance characteristic curve and the horizontal axis from the moment before the occurrence of harmonic exceedance to the end of the moment.

[0010] Furthermore, the symmetry of the feature trajectory is characterized using the Hausdorff distance, and the representation expression is:

[0011] in, For the characteristic trajectory symmetry, and These respectively represent the voltage over-limit characteristic curve and the harmonic over-limit characteristic curve. The set of data points within the time interval, and These respectively represent the voltage over-limit characteristic curve and the harmonic over-limit characteristic curve. Data points within the time interval, This is the one-way Hausdorff distance.

[0012] Furthermore, in S3, based on the characteristic indicators of node power quality governance needs, a feature vector of power quality problems for the node is constructed, including: Based on the closed area of ​​the UT feature trajectory, the closed area of ​​the HT feature trajectory, and the symmetry of the feature trajectory, a power quality problem feature vector for each node within each time segment is constructed. The expression for the power quality problem feature vector of the node is as follows:

[0013] in, Time section internal nodes The feature vector of power quality problems The severity of node voltage exceeding limits. The severity of node harmonic exceedance, This relates to the coupling between voltage and harmonic problems.

[0014] Further, S4 includes: S401: Introduce the ideal worst-case governance state vector to construct the worst-case node reference feature at each time step:

[0015] S402: The distance between the power quality problem feature vector of a node and the reference feature of the worst node is calculated using the normalized Euclidean distance. The calculation formula is as follows:

[0016] in, For nodes At any moment The distance between the power quality problem characteristics and the worst-case node reference characteristics. For a moment Next node The The value of each indicator, For a moment Lowering the target Reference value, This represents three indicators; S403: Based on the distance between the power quality problem feature vectors of nodes and the reference features of the worst-performing node, an initial governance priority index is constructed. The expression for the initial governance priority index is as follows:

[0017] in, For nodes At any moment The initial governance priority indicators are as follows: It is a positive number.

[0018] Further, S5 includes: S501: Introduce a voltage impact factor to measure the impact of node-injected disturbances on the overall network voltage state. The expression for the voltage impact factor is:

[0019] in, For nodes Voltage influence factor, For nodes The amount of injected power disturbance. For nodes Voltage changes, This represents the total number of system nodes. S502: Introducing a harmonic propagation factor that characterizes the ability of a nodal-injected harmonic to propagate and amplify in the system. The expression for the harmonic propagation factor is:

[0020] in, For nodes The harmonic propagation factor, For nodes The Second harmonic current For the node Injection unit amplitude After the second harmonic current, the node The Subharmonic voltage It is the set of harmonic orders; S503: Based on the voltage influence factor and harmonic propagation factor, the initial governance priority index is modified. The modification formula is as follows:

[0021] in, For nodes At any moment The revised governance priority indicators below and This is a weighting adjustment coefficient for adjusting the voltage influence factor and harmonic propagation factor on the priority of node dynamic governance.

[0022] The beneficial effects of this application are: This application provides a method for quantifying the dynamic power quality management requirements of distribution network nodes. By constructing voltage over-limit characteristic curves and harmonic exceedance characteristic curves, it systematically depicts the power quality change process of each node in the distribution network over 24 hours. It can more sensitively identify voltage sags or harmonic bursts, has strong dynamic time-series feature extraction capabilities, and introduces voltage influence factors and harmonic propagation factors to quantify the impact of node disturbances on the entire network. At the same time, it optimizes the management ranking through a coupling correction mechanism, improves the overall system stability and resource utilization efficiency, and can effectively improve the accuracy, economy and scientific nature of power quality management in distribution networks. Attached Figure Description

[0023] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other embodiments can be obtained based on these drawings.

[0024] Figure 1 This is a flowchart illustrating a method for quantifying the dynamic power quality management requirements of a distribution network node, as provided in an embodiment of this application.

[0025] Figure 2 This is a schematic diagram of the UT and HT characteristic curves of a certain node provided in an embodiment of this application.

[0026] Figure 3 This is a schematic diagram of relevant parameters for a node power quality governance demand characteristic index provided in an embodiment of this application. Detailed Implementation

[0027] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art based on this application are within the scope of protection of this application.

[0028] This application provides a method for quantifying the dynamic power quality management requirements of distribution network nodes. This method can be found in [reference needed]. Figure 1 , Figure 1 The diagram shown is a flowchart illustrating a method for quantifying the dynamic power quality management requirements of a distribution network node according to an embodiment of this application, including: S1: Perform fundamental power flow calculation and harmonic power flow calculation on the distribution network under the multi-resource grid connection scenario, obtain the voltage and harmonic data of each node, and obtain the voltage over-limit characteristic curve and harmonic over-limit characteristic curve of each node.

[0029] In one embodiment of this application, fundamental wave power flow calculation and harmonic power flow calculation are performed on the distribution network under photovoltaic, energy storage, and charging pile grid connection to obtain voltage and harmonic data of each node at each moment, and then the voltage over-limit characteristic curve (UT characteristic curve) and harmonic exceedance characteristic curve (HT characteristic curve) of each node are obtained. Figure 2 As shown, by combining the UT characteristic curve and the HT characteristic curve, we can initially determine whether a node is a node to be addressed. The judgment rules are as follows: If the node is not a node to be addressed, its curve is relatively flat and close to the coordinate axis; if the node is a node to be addressed, its curve has obvious peaks, and the characteristic curve will show large peaks or multiple continuous or discrete peaks, indicating that the node has a continuous and unstable voltage exceeding the limit or harmonic exceeding the standard, and the node has a greater need for addressing.

[0030] S2: Based on the voltage over-limit characteristic curve and harmonic exceedance characteristic curve of each node, extract the characteristic indicators of power quality governance requirements of the nodes.

[0031] In one embodiment of this application, based on Figure 2 Three feature indicators can be extracted: the closed area y1 of the UT feature trajectory, the closed area y2 of the HT feature trajectory, and the symmetry of the feature trajectory y3. These indicators are used to quantify the power quality management needs of nodes at different times. The relevant calculation parameters are labeled in [the relevant section]. Figure 3 The calculation method for the indicators is as follows.

[0032] S201: Based on the voltage over-limit characteristic curves of each node, obtain the closed area of ​​the UT characteristic trajectory.

[0033] In one embodiment of this application, the closed area y1 of the UT characteristic trajectory represents the area enclosed by the UT characteristic curve and the x-axis at a certain moment. Its value is related to the magnitude and time of the voltage over-limit occurrence at the node, and the size of the area characterizes the severity of the voltage over-limit occurrence at that moment. The closed area of ​​the characteristic trajectory is calculated from the moment before the voltage over-limit occurrence to the moment before that occurrence, and its calculation formula is as follows:

[0034] in, and These represent the nodes in Time and The voltage exceeds the limit at any time.

[0035] S202: Based on the harmonic exceedance characteristic curves of each node, obtain the closed area of ​​the HT characteristic trajectory.

[0036] In one embodiment of this application, the closed area y2 of the HT characteristic trajectory represents the area enclosed by the HT characteristic curve and the x-axis at a certain moment. Its value is related to the total harmonic distortion rate (THD) of the voltage when the node experiences harmonic exceedance and the time. The size of the area characterizes the severity of the harmonic exceedance at that moment. The closed area of ​​the characteristic trajectory is calculated from the moment before the occurrence of harmonic exceedance to the end of that moment, and its calculation formula is as follows:

[0037] in, and These represent the nodes in Time and The total harmonic distortion (THD) of the voltage at which harmonics exceed the limit at any given time. Similarly, if If there is no harmonic exceedance at a given time point, but there is harmonic exceedance at the preceding or following time point, then... The closed area of ​​the characteristic trajectory at time HT is equal to the area of ​​a triangle.

[0038] S203: Based on the voltage over-limit characteristic curve and harmonic over-limit characteristic curve of each node, the characteristic trajectory symmetry is obtained.

[0039] In one embodiment of this application, the characteristic trajectory symmetry y3 characterizes the changing trend of another variable when a node experiences voltage over-limit or harmonic over-limit at a certain moment. The trajectory symmetry is characterized by the Hausdorff distance, as shown in the following equation:

[0040] in, and These respectively represent the voltage over-limit characteristic curve and the harmonic over-limit characteristic curve. The set of data points within the time interval, and These respectively represent the voltage over-limit characteristic curve and the harmonic over-limit characteristic curve. Data points within the time interval, this application in Within each time period, several points are selected on the two characteristic curves as data points for calculation. This is the one-way Hausdorff distance.

[0041] Due to the limited resources available for distributed flexible governance, improving the accuracy and targeting of power quality governance requires identifying the critical nodes in the distribution network with the most urgent governance needs at different time points. Traditional methods often obtain the daily governance priority by weighting and superimposing the 24-hour power quality indicators of nodes, which fails to reflect the time-specific and sudden characteristics of power quality problems at nodes and ignores the potential system coupling relationships between nodes.

[0042] To this end, this application further proposes a time-based governance priority quantification method based on multi-index distance criteria to improve the pertinence and system coordination of power quality governance decisions. (Based on nodes...) UT characteristic curve closed area HT characteristic curve closed area Symmetry index of trajectory Using the node as the core indicator, a feature vector is constructed for each node at each time step. The distance to the worst-case governance reference state is introduced as the basis for priority quantification. Finally, the priority is corrected through the coupling effect of the node, realizing the dynamic evaluation of the node's power quality governance priority at different time periods. The specific steps are as follows: S3: Construct a feature vector of power quality problems for nodes based on the characteristic indicators of power quality governance needs at the nodes.

[0043] In one embodiment of this application, based on three indices—the closed area of ​​the UT characteristic trajectory, the closed area of ​​the HT characteristic trajectory, and the symmetry of the characteristic trajectory—at each time segment... Internally, build nodes The characteristics of power quality problems are represented by a vector as follows:

[0044] in, Time section internal nodes The feature vector of power quality problems The severity of node voltage exceeding limits. The severity of node harmonic exceedance, This relates to the coupling between voltage and harmonic problems.

[0045] S4: Introduce the ideal worst-case governance state vector, construct the worst-case node reference feature, and construct the initial governance priority index based on the worst-case node reference feature and the node's power quality problem feature vector.

[0046] In one embodiment of this application, to quantify the time-segmented governance priorities of nodes, this application introduces an ideal worst-case governance state vector as a reference benchmark to construct the worst-case node reference features at each time step:

[0047] node At any moment The distance between the current power quality problem and the reference state is calculated using the normalized Euclidean distance:

[0048] in, For nodes At any moment The distance between the power quality problem characteristics and the worst-case node reference characteristics. For a moment Next node The The value of each indicator, For a moment Lowering the target Reference value, This represents three indicators; node At any moment The initial governance priority index is defined as:

[0049] in, For nodes At any moment The initial governance priority indicators are as follows: It is an extremely small positive number that is greater than zero, used to prevent the denominator from being zero. The larger the index, the more significant the node. At any moment The closer to the worst power quality condition, the higher the priority for remediation.

[0050] S5: Introduce voltage influence factor and harmonic propagation factor, and revise the initial governance priority index based on voltage influence factor and harmonic propagation factor.

[0051] In one embodiment of this application, complex electrical connections exist between nodes in a distribution network. Voltage disturbances and harmonic components can propagate through electrical channels in the system, causing power quality problems at one node to have a coupled impact on other nodes. Therefore, simply relying on the node's own power quality indicators to assess governance priorities may ignore the amplification effect or contagion characteristics of nodes at the system level, making it difficult to achieve the optimal resource scheduling strategy. To address this, this application introduces a node voltage impact factor and a harmonic propagation factor to correct the initial priority indicators. Specifically: S501: Introduce a voltage impact factor to measure the impact of node-injected disturbances on the overall network voltage state. The expression for the voltage impact factor is:

[0052] in, For nodes Voltage influence factor, For nodes The amount of injected power disturbance. For nodes Voltage changes, This represents the total number of system nodes. S502: Introducing a harmonic propagation factor that characterizes the ability of a nodal-injected harmonic to propagate and amplify in the system. The expression for the harmonic propagation factor is:

[0053] in, For nodes The harmonic propagation factor, For nodes The Second harmonic current For the node Injection unit amplitude After the second harmonic current, the node The Subharmonic voltage It is the set of harmonic orders; S503: Based on the voltage influence factor and harmonic propagation factor, the initial governance priority index is modified. The modification formula is as follows:

[0054] in, For nodes At any moment The revised governance priority indicators below and This is a weighting adjustment coefficient used to adjust the contribution of the voltage influence factor and harmonic propagation factor to the priority of node dynamic governance. In this embodiment, it is taken as... = =0.5.

[0055] S6: Based on the revised initial governance priority index, identify the key nodes in the distribution network with the most urgent governance needs at different time segments.

[0056] After coupling correction, key nodes that have amplification effects or are prone to triggering chain reactions in the system can be identified, further improving the system coordination of governance strategies.

[0057] This application provides a method for quantifying the dynamic power quality management requirements of distribution network nodes. By constructing voltage over-limit characteristic curves and harmonic exceedance characteristic curves, it systematically depicts the power quality change process of each node in the distribution network over 24 hours. It can more sensitively identify voltage sags or harmonic bursts, has strong dynamic time-series feature extraction capabilities, and introduces voltage influence factors and harmonic propagation factors to quantify the impact of node disturbances on the entire network. At the same time, it optimizes the management ranking through a coupling correction mechanism, improves the overall system stability and resource utilization efficiency, and can effectively improve the accuracy, economy and scientific nature of power quality management in distribution networks.

[0058] It should be noted that those skilled in the art will recognize that the embodiments described herein are for the purpose of helping readers understand the principles of this application, and should be understood as not limiting the scope of protection of this application to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this application without departing from the essence of this application, and these modifications and combinations are still within the scope of protection of this application.

Claims

1. A method for quantifying the dynamic power quality management requirements of distribution network nodes, characterized in that, include: S1: Perform fundamental power flow calculation and harmonic power flow calculation on the distribution network under the multi-resource grid connection scenario, obtain the voltage and harmonic data of each node, and obtain the voltage over-limit characteristic curve and harmonic over-limit characteristic curve of each node. S2: Based on the voltage over-limit characteristic curve and harmonic exceedance characteristic curve of each node, extract the characteristic indicators of power quality management needs of the nodes. S2 includes: S201: Based on the voltage over-limit characteristic curves of each node, obtain the closed area of ​​the UT characteristic trajectory; S202: Based on the harmonic exceedance characteristic curves of each node, obtain the closed area of ​​the HT characteristic trajectory; S203: Based on the voltage over-limit characteristic curve and harmonic over-limit characteristic curve of each node, the symmetry of the characteristic trajectory is obtained; S3: Construct a feature vector of power quality problems for nodes based on the characteristic indicators of power quality governance needs at the nodes; The S3 section constructs a power quality problem feature vector for nodes based on the node power quality governance demand characteristic indicators, including: Based on the closed area of ​​the UT feature trajectory, the closed area of ​​the HT feature trajectory, and the symmetry of the feature trajectory, a power quality problem feature vector for each node within each time segment is constructed. The expression for the power quality problem feature vector of the node is as follows: in, Time section internal nodes The feature vector of power quality problems The severity of node voltage exceeding limits. The severity of node harmonic exceedance, The coupling between voltage and harmonic problems; S4: Introduce the ideal worst-case governance state vector, construct the worst-case node reference feature, and construct the initial governance priority index based on the worst-case node reference feature and the node's power quality problem feature vector; S5: Introduce voltage influence factor and harmonic propagation factor, and revise the initial governance priority index based on voltage influence factor and harmonic propagation factor; The S5 includes: S501: Introduce a voltage impact factor to measure the impact of node-injected disturbances on the overall network voltage state. The expression for the voltage impact factor is: in, For nodes Voltage influence factor, For nodes The amount of injected power disturbance. For nodes Voltage changes, This represents the total number of system nodes. S502: Introducing a harmonic propagation factor that characterizes the ability of a nodal-injected harmonic to propagate and amplify in the system. The expression for the harmonic propagation factor is: in, For nodes The harmonic propagation factor, For nodes The Second harmonic current For the node Injection unit amplitude After the second harmonic current, the node The Subharmonic voltage It is the set of harmonic orders; S503: Based on the voltage influence factor and harmonic propagation factor, the initial governance priority index is modified. The modification formula is as follows: in, For nodes At any moment The revised governance priority indicators below For nodes At any moment The initial governance priority indicators are as follows: and The weighting adjustment coefficients for voltage influence factor and harmonic propagation factor on the priority of node dynamic governance; S6: Based on the revised initial governance priority index, identify the key nodes in the distribution network with the most urgent governance needs at different time segments.

2. The method for quantifying the dynamic power quality management requirements of distribution network nodes according to claim 1, characterized in that, The closed area of ​​the UT characteristic trajectory includes the area enclosed by the voltage over-limit characteristic curve and the horizontal axis from the moment before the voltage over-limit occurs to the end of the moment.

3. The method for quantifying the dynamic power quality management requirements of distribution network nodes according to claim 1, characterized in that, The closed area of ​​the HT characteristic trajectory includes the area enclosed by the harmonic exceedance characteristic curve and the horizontal axis from the moment before the occurrence of harmonic exceedance to the end of the moment.

4. The method for quantifying the dynamic power quality management requirements of distribution network nodes according to claim 1, characterized in that, The symmetry of the characteristic trajectory is characterized using the Hausdorff distance, and the expression for this characteristic is: in, For the characteristic trajectory symmetry, and These respectively represent the voltage over-limit characteristic curve and the harmonic over-limit characteristic curve. The set of data points within the time interval, and These respectively represent the voltage over-limit characteristic curve and the harmonic over-limit characteristic curve. Data points within the time interval, This is the one-way Hausdorff distance.

5. The method for quantifying the dynamic power quality management requirements of distribution network nodes according to claim 4, characterized in that, The S4 includes: S401: Introduce the ideal worst-case governance state vector to construct the worst-case node reference feature at each time step: S402: Calculate the distance between the power quality problem feature vector of a node and the reference feature of the worst-case node using normalized Euclidean distance. The calculation formula is as follows: in, For nodes At any moment The distance between the power quality problem characteristics and the worst-case node reference characteristics. For a moment Next node The The value of each indicator, For a moment Lowering the target Reference value, This represents three indicators; S403: Based on the distance between the power quality problem feature vectors of nodes and the reference features of the worst-performing node, an initial governance priority index is constructed. The expression for the initial governance priority index is as follows: in, For nodes At any moment The initial governance priority indicators are as follows: It is a positive number.