Cascading failure analysis method for urban infrastructure based on high-order network model

Abstract

The present application relates to the field of urban construction technology, and provides a method for analyzing cascading failures of urban infrastructure based on a high-order network model. In this method, a multi-layer network model of the urban infrastructure system is constructed, and a simple complex structure is used to characterize the interactions between multiple network nodes in the multi-layer network model to obtain a high-order network model of the urban infrastructure system; then, the node failure mechanism and the regional failure mechanism of the high-order network model are set, and the high-order network model is simulated for cascading failures under a preset cascading failure scenario; finally, based on the network comprehensive factor of the cascading failure of the defined high-order network model, according to the cascading failure simulation results, the high-risk nodes in the high-order network model are identified to improve the resilience of the urban infrastructure system corresponding to the high-risk nodes. In this way, high-risk nodes of urban infrastructure can be more effectively identified, and support and basis can be provided for improving the resilience and risk resistance of urban infrastructure systems.

Description

Technical Field

[0001] The present application relates to the field of urban construction technology, and in particular to a method for analyzing cascading failures of urban infrastructure based on a high-order network model. Background Art

[0002] Critical urban infrastructure, such as power grids, water supply systems, natural gas systems, and transportation networks, is an important component of maintaining social stability. These systems form a highly complex multi-layer network with extensive and deep interdependence. The failure of a single node or component may trigger a cascade effect through inter-layer coupling, leading to interruption and functional failure of the entire system. These effects propagate in a nonlinear and unpredictable manner, making risk assessment and management increasingly challenging. Summary of the Invention

[0003] The purpose of this application is to provide a method for analyzing cascading failures of urban infrastructure equipment based on a high-order network model to solve or alleviate the problems existing in the above-mentioned prior art.

[0004] In order to achieve the above objectives, this application provides the following technical solutions:

[0005] The present application provides a method for analyzing cascading failures of urban infrastructure equipment based on a high-order network model, comprising: step S101, constructing a multi-layer network model of an urban infrastructure system, and characterizing the interactions between multiple network nodes in the multi-layer network model using a simple complex structure to obtain a high-order network model of the urban infrastructure system; wherein the connections between different network nodes in the high-order network model represent functional dependencies between different urban infrastructure systems;

[0006] Step S102: setting a node failure mechanism and a regional failure mechanism of the high-order network model, and performing a cascading failure simulation on the high-order network model under a preset cascading failure scenario;

[0007] Step S103: Based on the defined network comprehensive factor of the cascading failure of the high-order network model and according to the cascading failure simulation result, identify the urban infrastructure system corresponding to the high-risk node in the high-order network model.

[0008] Preferably, in step S101, the key components of the urban infrastructure system are described as the network nodes, and the interactions between different network nodes are described based on the power supply relationship;

[0009] Classifying each of the network nodes into different network layers according to the functions of the urban infrastructure systems, and characterizing the functional dependencies between different urban infrastructure systems through connections between different network layers;

[0010] Geographic coordinates are assigned to each of the network nodes, and all the network nodes are projected onto a geographic base layer to construct the multi-layer network model.

[0011] Preferably, in step S102, in response to a failure of a single network node in the high-order network model due to an internal failure or a failure of other dependent network nodes in the network layer where the single network node is located, the node failure mechanism is triggered;

[0012] In response to multiple network nodes in a region failing synchronously due to external factors in the high-order network model, the regional failure mechanism is triggered.

[0013] Preferably, in step S102, in response to triggering the node failure mechanism and the failure of the network node corresponding to the substation, the cascading failure of the high-order network model under traditional first-order interaction and high-order network interaction is simulated, and the number of failed nodes in each time step is tracked;

[0014] In response to triggering the regional failure mechanism, multiple network nodes in the high-order network model fail simultaneously, and the dynamic changes of fault propagation and node recovery in each time step are simulated respectively, and the number of failed nodes in each time step and the distribution of failed nodes within the network layer of the high-order network model and between the network layers are recorded to determine the fault propagation range and fault propagation rate in the high-order network model.

[0015] Preferably, in step S102, in the node failure mechanism, according to the formula:

[0016]

[0017] Determine the network nodes when the high-order network model performs cascading failure simulation At the moment Cascading failure probability Where, For the network node Cascade failure rate under traditional first-order interactions, For the network node the rate of spread under high-order network interactions; For the network node The number of failed first-order neighbor nodes, For the network node The number of failed high-order neighbor nodes;

[0018] In the regional failure mechanism, according to the formula:

[0019]

[0020] Determine the network nodes when the high-order network model performs cascading failure simulation At the moment Regional failure probability Where, For the network node geographical coordinates; To simulate the affected area when external factors affect the synchronization of nodes in a specific area based on the regional failure mechanism, Affected areas outside the area.

[0021] Preferably, in step S102, according to the formula:

[0022]

[0023] Determine the fault propagation index when simulating cascading failures for the high-order network model ; Wherein, the fault propagation index is used to quantify the propagation range and recovery speed of cascading faults in the high-order network model;

[0024] Where, is the cumulative number of faulty network nodes when cascading failure simulation is performed on the high-order network model, The time required for network nodes to recover stability and no longer change when cascading failure simulation is performed on the high-order network model. is the maximum value of the time step under different simulation conditions when performing cascading failure simulation on the high-order network model.

[0025] Preferably, in step S103, according to the formula:

[0026]

[0027] Determine the network integration factor ;

[0028] Where, The network node in the high-order network model The total number of direct connections in all said network layers, is the total number of network nodes in the high-order network model; The network node in the high-order network model With the network node The shortest path length between The network node in the high-order network model With the network node The adjacency matrix of the connections between The adjacency matrix The maximum eigenvalue of

[0029] Preferably, in step S103, the network comprehensive factor of each network node in the high-order network model is calculated based on the cascading failure simulation result, and the network node corresponding to the largest network comprehensive factor is determined as a high-risk node in the high-order network model.

[0030] Preferably, the method further comprises: according to the formula:

[0031]

[0032] Determine the self-repair probability of the urban infrastructure system corresponding to the network node in the high-order network model Where, is the intrinsic recovery rate of the network node, The duration of a cascading failure in the high-order network model.

[0033] Preferably, the method further includes: verifying the rationality and effectiveness of the high-order network model through Jaccard similarity of different fault modes in the high-order network model.

[0034] Beneficial effects:

[0035] In the method for analyzing cascading failures of urban infrastructure equipment based on a high-order network model provided in an embodiment of the present application, a multi-layer network model of the urban infrastructure system is first constructed, and a simple complex structure is used to characterize the interactions between multiple network nodes in the multi-layer network model to obtain a high-order network model of the urban infrastructure system; then, the node failure mechanism and regional failure mechanism of the high-order network model are set, and a cascading failure simulation is performed on the high-order network model under a preset cascading failure scenario; finally, based on the network comprehensive factor of the cascading failure of the defined high-order network model and according to the cascading failure simulation results, high-risk nodes in the high-order network model are identified to improve the resilience of the urban infrastructure system corresponding to the high-risk nodes. In this way, through the propagation of cascading failures in the urban infrastructure system, high-risk nodes of the urban infrastructure can be more effectively identified, providing support and basis for cascading failure analysis of urban critical infrastructure, mitigating the impact of cascading failures, shortening network recovery time, and improving the resilience and risk resistance of the urban infrastructure system. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] The drawings and descriptions that constitute part of this application are intended to provide a further understanding of this application. The exemplary embodiments and descriptions of this application are intended to explain this application and do not constitute an improper limitation on this application. Among them:

[0037] Figure 1 A flowchart of a method for analyzing cascading failures of urban infrastructure equipment based on a high-order network model according to some embodiments of the present application is provided;

[0038] Figure 2 A logical diagram of a method for analyzing cascading failures of urban infrastructure equipment based on a high-order network model according to some embodiments of the present application;

[0039] Figure 3 A schematic diagram of a multi-layer network model of an urban infrastructure system provided according to some embodiments of the present application;

[0040] Figure 4 A schematic diagram of a node failure mechanism provided according to some embodiments of the present application;

[0041] Figure 5 A schematic diagram of a regional failure mechanism provided according to some embodiments of the present application;

[0042] Figure 6 A comparative schematic diagram of fault propagation of network nodes under high-order interactions according to some embodiments of the present application;

[0043] Figure 7 A schematic diagram of the number of failed nodes at different times according to some embodiments of the present application;

[0044] Figure 8 A schematic diagram of the robustness of different network node combinations under the self-healing mechanism provided according to some embodiments of the present application. DETAILED DESCRIPTION

[0045] The present application will be described in detail below with reference to the accompanying drawings and in conjunction with embodiments. Each example is provided by way of explanation of the present application and does not limit the present application. In fact, it will be clear to those skilled in the art that modifications and variations can be made in the present application without departing from the scope or spirit of the present application. For example, a feature shown or described as part of one embodiment can be used in another embodiment to produce yet another embodiment. All other embodiments obtained by those of ordinary skill in the art based on the embodiments in the present invention should fall within the scope of protection of the embodiments of the present invention.

[0046] In the risk effect analysis of urban infrastructure systems, traditional risk analysis methods are mostly based on pairwise interaction models, which only consider direct connections between nodes and cannot accurately describe the complex dynamic relationships between multiple nodes. However, high-order network models, by introducing multi-node interactions, can reflect the behavioral characteristics of complex systems and their multi-layer dependencies. They can reveal the propagation laws of faults at a higher level and provide an important tool for accurately predicting systemic risks.

[0047] Based on this, the embodiment of the present application proposes a method for analyzing cascading failures of urban infrastructure based on a high-order network model, which converts the urban infrastructure system into a high-order network model as the basic framework for cascading failure analysis; compares the behavior of the traditional first-order model with that of the high-order model, and more effectively identifies high-risk nodes by analyzing changes in cascading failure rate, propagation speed, and overall system impact, providing theoretical support for cascading failure analysis and self-healing mechanism optimization of critical urban infrastructure.

[0048] like Figures 1 to 8 As shown, the urban infrastructure cascading failure analysis method based on the high-order network model includes:

[0049] Step S101: construct a multi-layer network model of the urban infrastructure system, and use a simple complex structure to characterize the interaction between multiple network nodes in the multi-layer network model to obtain a high-order network model of the urban infrastructure system.

[0050] Among them, the connections between different network nodes in the high-order network model represent the functional dependencies between different urban infrastructure systems.

[0051] In this application, functional data and geographic information of critical urban infrastructure are collected, including key nodes and connections in power, transportation, communications, and water supply systems. Key components of urban infrastructure systems are described as network nodes, and the interactions between different network nodes are described based on power supply relationships. Based on the function of the urban infrastructure system, each network node is categorized into different network layers (e.g., power layer, transportation layer, water supply layer, etc.), and the functional dependencies between different urban infrastructure systems are characterized through connections between different network layers. In other words, critical urban infrastructure is represented as a multi-layer directed graph, where each node corresponds to a critical infrastructure system, such as a power generator, gas station, or water treatment plant. Edges connecting nodes represent functional dependencies between nodes, and inter-layer connections are used to describe the interactions between different urban infrastructure systems, primarily power supply relationships (e.g., power plants require gas supply, and water systems rely on electricity). Geographic coordinates are assigned to each network node, and all network nodes are projected onto the geographic base layer, ultimately constructing a multi-layer network model containing multiple network nodes and their connections.

[0052] Furthermore, a simple complex structure is used to characterize the interactions between multiple network nodes. High-dimensional geometric structures (such as triangles for interactions between three network nodes and tetrahedrons for connections between four network nodes) are used to analyze the complex dynamic characteristics of interconnected urban infrastructure systems. The impact of group-level interactions on the overall behavior of the network is analyzed. After configuring the internal connections of each network layer, cross-network layer edges are established to represent the interdependence between various urban infrastructure systems. For example, the power grid, as the core of urban infrastructure, transports energy from power generation facilities through the power grid to end users. In addition, other infrastructure systems, while maintaining their own internal connections, obtain energy from the power grid through the substation network to ensure the normal operation of their equipment.

[0053] Step S102: setting a node failure mechanism and a regional failure mechanism of the high-order network model, and performing a cascading failure simulation on the high-order network model under a preset cascading failure scenario.

[0054] In a multi-layer high-order network model, the status of a network node is jointly affected by its directly connected neighbors (i.e., first-order neighbors) within the network layer in which it is located, its higher-order neighbors connected through multi-network node interactions, and network nodes from other network layers through inter-network layer connections. On this basis, node-based failure mechanisms (node ​​failure mechanisms) and region-based failure mechanisms (regional failure mechanisms) are defined, and the urban infrastructure system is analyzed through a high-order network model.

[0055] When a single network node fails due to internal faults or the failure of other dependent network nodes in the network layer where the node resides, the node failure mechanism is triggered, triggering a cascading effect through the direct connections between network nodes. When multiple network nodes within a region fail simultaneously due to external factors, the regional failure mechanism is triggered. In other words, the regional failure mechanism affects multiple network nodes within a specific area simultaneously, and the triggering factor (such as a natural disaster or large-scale system failure) causes the simultaneous failure of multiple network nodes within the area.

[0056] In this application, a high-order network model is simulated under a pre-defined cascading failure scenario (high-order reinforcement and self-healing mechanism). In this model, high-order reinforcement is used to account for the more complex interdependencies of second-order or higher-order connections between network nodes. These connections involve the interaction of more than two network nodes. Regional and traditional network node state changes are instead represented as the cumulative impact of multiple network nodes on the state change of a single network node.

[0057] A network node failure directly causes failures in its directly connected neighboring network nodes. This direct failure propagation is defined as a first-order interaction. A network node failure, through a more complex path (possibly involving multiple intermediate network nodes), causes failures in other network nodes. This failure propagation is defined as a higher-order network interaction (i.e., higher-order reinforcement). In other words, a first-order interaction involves only the influence of two directly connected network nodes. A higher-order network interaction involves interactions between more than two network nodes, capturing more complex interdependencies within the network.

[0058] When simulating cascading failures through a high-order network model, when the node failure mechanism is triggered and the network node corresponding to the substation fails, the evolution of the cascading failure of the high-order network model under traditional first-order interactions (i.e., first-order neighbors) and high-order network interactions (i.e., high-order neighbors connected by multiple network nodes) is simulated, and the number of faulty nodes at each time step is tracked.

[0059] For multi-layer high-order network models, the cascading failure dynamics and node criticality are analyzed based on probability and centrality methods. The failure probability is determined by the first-order propagation probability and the higher-order propagation probability. The probability of a network node transitioning from a functional state to a faulty state depends on the faulty state of its neighboring nodes and is affected by direct connections and higher-order group interactions. Specifically, when a high-order network model triggers a node failure mechanism, according to the formula:

[0060]

[0061] Determine the network nodes when performing cascading failure simulation on high-order network models At the moment Cascading failure probability Where, For network nodes Cascade failure rate under traditional first-order interactions, For network nodes the rate of spread under high-order network interactions; For network nodes The number of failed first-order neighbor nodes, For network nodes The number of failed high-order neighbor nodes.

[0062] When simulating cascading failures through a high-order network model, when the regional failure mechanism is triggered, multiple network nodes in the high-order network model fail synchronously. The dynamic changes of fault propagation and node recovery at each time step are simulated separately, and the number of failed nodes (faulty network nodes) in each time step and the distribution of failed nodes within and between network layers in the high-order network model are recorded to determine the fault propagation range and fault propagation rate in the high-order network model.

[0063] The regional failure mechanism (regional failure mechanism) simulates the synchronous impact of external factors on nodes in a specific region. The failure probability of network nodes within the affected region is set to 1, and the failure probability of network nodes outside the affected region is set to 0. Specifically, when the high-order network model triggers the regional failure mechanism, according to the formula:

[0064]

[0065] Determine the network nodes for cascading failure simulation in high-order network models At the moment Regional failure probability Where, For network nodes The geographical coordinates of To simulate the affected area when external factors affect the synchronization of nodes in a specific area based on the regional failure mechanism, Affected areas outside the area.

[0066] In the cascading failure scenario of high-order reinforcement and self-healing mechanisms in the cascade, a high-order network model is used to simulate cascading failures of urban infrastructure systems. In this case, a self-healing mechanism is introduced (network nodes recover from a faulty state and remain stable during cascading failure simulations), and the transition of network nodes from failure (faulty state) to functional state is quantified by the recovery probability. Specifically, according to the formula:

[0067]

[0068] Determine the self-repair probability of urban infrastructure systems corresponding to network nodes in high-order network models Where, is the intrinsic recovery rate of the network nodes, The failure duration of a cascading failure in a high-order network model.

[0069] At the same time, in order to comprehensively evaluate the fault status, the cascading failures in the urban infrastructure system are evaluated by the stable time and the total number of failures during the cascading failure simulation. Among them, the stable time step (Stable TimeNormlized) represents the number of time steps required for the cascading failure to reach a stable state during the cascading failure simulation (the time required for the network nodes to recover and no longer change when the cascading failure simulation is performed on the high-order network model), at which time no further fault propagation will occur; the total number of failures (Total Failures Normalized) is the cumulative number of failed network nodes (faulty network nodes) during the fault propagation process; it reflects the overall degree of interruption of the entire system in the high-order network model. Here, for the entire high-order network model, the fault propagation index is used To quantify the propagation range and recovery speed of cascading failures, specifically, according to the formula:

[0070]

[0071] Determining the fault propagation exponent when simulating cascading failures in high-order network models Where, is the cumulative number of faulty network nodes when cascading failure simulation is performed on the high-order network model, The time required for network nodes to recover stability and no longer change when cascading failure simulation is performed on high-order network models. is the maximum value of the time step under different simulation conditions when simulating cascading failures for high-order network models.

[0072] Step S103 : Based on the defined network comprehensive factor of the cascading failure of the high-order network model and according to the cascading failure simulation results, identify the urban infrastructure systems corresponding to the high-risk nodes in the high-order network model.

[0073] In this application, degree centrality is used to measure the number of direct connections between a network node and other network nodes in a high-order network model, closeness centrality is used to measure whether a network node is located on the shortest path between other network nodes, and eigenvector centrality is used to measure the connection between a network node and other high-centrality network nodes. Furthermore, the network comprehensive factor is calculated based on degree centrality, closeness centrality, and eigenvector centrality. , used to identify key network nodes. Specifically, according to the formula:

[0074]

[0075] Determine the network integration factor Where, is the network node in the high-order network model The total number of direct connections in all network layers, is the total number of network nodes in the high-order network model; is the network node in the high-order network model With network nodes The shortest path length between is the network node in the high-order network model With network nodes The adjacency matrix of the connections between is the adjacency matrix The maximum eigenvalue of .

[0076] Then, based on the results of the cascading failure simulation, the network comprehensive factor of each network node in the high-order network model is calculated, and the network node corresponding to the largest network comprehensive factor is identified as the high-risk node in the high-order network model. The resilience of the urban infrastructure system corresponding to the high-risk node is improved. For example, by strengthening the capabilities of key components such as substations, water treatment plants and data centers, the risk of cross-system failure propagation can be effectively reduced.

[0077] In a specific application scenario, the node P3-1 (substation node) is selected for failure. By tracking the number of faulty nodes at each time step, the evolution of cascading failures under traditional first-order interactions and high-order network interactions is analyzed. Since high-order interactions in urban infrastructure systems are usually indirect and receive a buffering effect, the propagation rate of network nodes under high-order network interactions is calculated. The value is set to 0.5 to prevent high-order effects from excessively impacting cascading propagation. Introducing high-order interactions into the high-order network model significantly accelerates the speed and range of cascading failures. The dependencies between interconnected network nodes, such as substations, energy storage systems, and data centers, exacerbate system stress and further drive cascading effects.

[0078] For regional faults, we simulate an external disaster (such as an earthquake) that causes multiple network nodes in a region to fail simultaneously, and analyze their propagation. The fault propagation process is discretized into multiple short time steps. Each time step simulates the dynamic changes in fault propagation and node recovery. In each time step, we record the number of failed nodes and their distribution within and between network layers to analyze the scope and rate of fault propagation.

[0079] Regarding the self-healing mechanism and its role in fault recovery, the recovery probability and the first repair delay time are defined as the key parameters of the self-healing mechanism. The self-healing capabilities of network nodes are classified according to the defined key parameters, and the recovery probability of each network node is listed ( : first-order recovery probability, : high-order recovery probability) and repair delay, as shown in Table 1. Manual repair intervention was introduced after the 50th simulation time step. Prior to this, the network nodes relied solely on their black start capabilities (self-healing mechanism). The results show that with higher recovery probabilities, the urban infrastructure systems corresponding to the network nodes recovered faster, and the number of failed network nodes decreased rapidly, demonstrating that the self-healing mechanism can effectively mitigate cascading effects. As shown in Table 2, when the network node recovery probability (μ) is between 0.3 and 0.5, the self-healing mechanism exhibits significant fluctuations due to the complex dependencies between urban infrastructure systems. For example, after power is restored to the substation, the restoration of the transportation system may increase power demand, affecting power restoration.

[0080] Table 1 System propagation parameters and repair delay

[0081]

[0082] Table 2 Curve integral area under different μ values

[0083]

[0084] By analyzing the impact of self-healing network nodes on the propagation of cascading faults, a black start (self-healing mechanism) was implemented at the substation level, and robustness monitoring was performed over the first 50 time steps. The results showed that the substation's black start capability could maintain basic operations, but it was unable to recover its initial robustness. The robustness of network node combinations with different network comprehensive factors varied at each time point, indicating that different network node combinations have different strengths in the recovery process.

[0085] In this application, to further verify the rationality and effectiveness of a high-order network model of urban infrastructure systems, we evaluated the Jaccard similarity of different failure modes within the high-order network model. Specifically, we abstracted the cascading propagation data from multiple incidents into network nodes and connected edges, and compared the Jaccard similarities of different failure modes (as shown in Table 3). The similarity for node failures was approximately 88%, and for regional failures was approximately 80%. The higher similarity indicates that the model accurately simulates localized failures, while the slightly lower similarity for regional failures reflects the model's complexity. Here, a similarity of 80% indicates that the model can effectively reproduce large-scale regional cascading effects.

[0086] Table 3 Jaccard similarity of accident cases

[0087]

[0088] In this application, we effectively capture complex multi-node interactions based on high-order networks, thereby gaining a deeper understanding of the characteristics of cascading failures in urban infrastructure systems. We apply this to urban infrastructure systems by constructing a multi-layer network model of urban infrastructure and combining the node status influence relationship of the multi-layer network model with the node-based failure mechanism and the region-based failure mechanism. Through simulation models, we synthesize the cascading failure probability. , failure probability based on region , self-repair probability , Network Comprehensive Factor (NSF), Fault Propagation Index and other indicators are used to comprehensively evaluate the dynamic behavior of cascading failures in urban infrastructure systems. By analyzing and identifying high-risk nodes that have a significant impact on fault propagation, the overall resilience and risk resistance of the system can be improved.

[0089] In the description of the present invention, the terms "one embodiment," "some embodiments," "example," "specific example," or "some examples" mean that a specific feature, structure, material, or characteristic described in conjunction with that embodiment or example is included in at least one embodiment or example of the present invention. In this specification, the exemplary expressions of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.

[0090] The foregoing description is merely a preferred embodiment of the present application and is not intended to limit the present application. Various modifications and variations are readily apparent to those skilled in the art. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present application shall be included within the scope of protection of the present application.

Claims

1. A method for analyzing cascading failures of urban infrastructure equipment based on a high-order network model, characterized in that: include: Step S101: Construct a multi-layer network model of the urban infrastructure system, and use a simple complex structure to characterize the interactions between multiple network nodes in the multi-layer network model to obtain a high-order network model of the urban infrastructure system; wherein, according to the function of the urban infrastructure system, each network node is classified into a different network layer, and the functional dependency between different urban infrastructure systems is characterized by the connection between different network layers; Step S102: Setting the node failure mechanism and the regional failure mechanism of the high-order network model, and performing a cascading failure simulation on the high-order network model based on the high-order reinforcement and self-healing mechanism under a preset cascading failure scenario; wherein, in the node failure mechanism, according to the formula: ； Determine the network nodes when the high-order network model performs cascading failure simulation At the moment Cascading failure probability Where, For the network node Cascade failure rate under traditional first-order interactions, For the network node the rate of spread under higher-order network interactions; For the network node The number of failed first-order neighbor nodes, For the network node The number of failed high-order neighbor nodes; In the regional failure mechanism, according to the formula: ； Determine the network nodes when the high-order network model performs cascading failure simulation At the moment Regional failure probability Where, For the network node geographical coordinates; To simulate the affected area when external factors affect the synchronization of nodes in a specific area based on the regional failure mechanism, Affected areas Areas outside Step S103: According to the cascading failure simulation result, according to the formula: ； Determine the network comprehensive factor of each of the network nodes in the cascading failure of the high-order network model , to identify the urban infrastructure system corresponding to the high-risk nodes in the high-order network model; Where, The network node in the high-order network model The total number of direct connections in all network layers, is the total number of network nodes in the high-order network model; The network node in the high-order network model With the network node The shortest path length between The network node in the high-order network model With the network node The adjacency matrix of the connections between The adjacency matrix The maximum eigenvalue of .

2. The urban infrastructure cascading failure analysis method based on a high-order network model according to claim 1 is characterized in that: In step S101, Describing the key components of the urban infrastructure system as network nodes, and describing the interactions between different network nodes based on power supply relationships; Geographic coordinates are assigned to each of the network nodes, and all the network nodes are projected onto a geographic base layer to construct the multi-layer network model.

3. The urban infrastructure cascading failure analysis method based on a high-order network model according to claim 1 is characterized in that: In step S102, In response to a single network node in the high-order network model failing due to an internal failure or a failure of other dependent network nodes in the network layer where the single network node is located, triggering the node failure mechanism; In response to multiple network nodes in a region failing synchronously due to external factors in the high-order network model, the regional failure mechanism is triggered.

4. The urban infrastructure cascading failure analysis method based on a high-order network model according to claim 1 is characterized in that: In step S102, In response to the node failure mechanism being triggered and the network node corresponding to the substation failing, simulating the evolution of the cascading failure of the high-order network model under traditional first-order interactions and high-order network interactions, and tracking the number of failed nodes at each time step; In response to triggering the regional failure mechanism, multiple network nodes in the high-order network model fail simultaneously, and the dynamic changes of fault propagation and node recovery in each time step are simulated respectively, and the number of failed nodes in each time step and the distribution of failed nodes within the network layer of the high-order network model and between the network layers are recorded to determine the fault propagation range and fault propagation rate in the high-order network model.

5. The urban infrastructure cascading failure analysis method based on a high-order network model according to claim 1 is characterized in that: In step S102, according to the formula: ； Determine the fault propagation index when simulating cascading failures for the high-order network model ; Wherein, the fault propagation index is used to quantify the propagation range and recovery speed of cascading faults in the high-order network model; Where, is the cumulative number of faulty network nodes when cascading failure simulation is performed on the high-order network model, The time required for network nodes to recover stability and no longer change when cascading failure simulation is performed on the high-order network model. is the maximum value of the time step under different simulation conditions when performing cascading failure simulation on the high-order network model.

6. The urban infrastructure cascading failure analysis method based on a high-order network model according to claim 1 is characterized in that: In step S103, The network node corresponding to the largest network comprehensive factor is determined as a high-risk node in the high-order network model.

7. The urban infrastructure cascading failure analysis method based on a high-order network model according to any one of claims 1 to 6, characterized in that: Also includes: According to the formula: ； Determine the self-repair probability of the urban infrastructure system corresponding to the network node in the high-order network model ; Where, is the intrinsic recovery rate of the network node, The duration of a cascading failure in the high-order network model.

8. The urban infrastructure cascading failure analysis method based on a high-order network model according to any one of claims 1 to 6, characterized in that: Also includes: The rationality and effectiveness of the high-order network model are verified by using the Jaccard similarity of different failure modes in the high-order network model.

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